A Structural Approach to Relativistic Symmetry: Dual Relativity and the Lorentz–Heisenberg Algebra

Abstract

This paper studies a realization-theoretic problem inside the standard Lorentz-covariant Fourier-dual framework on $L^2\left({\mathbb{R}}^{3,1}\right)$: whether position-space and momentum-space geometric translations can be placed on equal structural footing without leaving the ordinary X- and K-polarized realizations. Working on the common Schwartz core $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, we first isolate a Fourier-compatibility obstruction: Fourier transform exchanges geometric translations with character actions, while the Poincaré algebra contains at most one Lorentz-covariant abelian translation ideal. The main result is that, within the resulting Fourier-compatible realization class, the minimal operator-generated Lie algebra is the Lorentz–Heisenberg algebra. We then determine the full center of its universal enveloping algebra, derive the normalized Lorentz-bivector invariants, orbit data, and connected stabilizers in nondegenerate sectors, and show that the orbit variable is a normalized Lorentz bivector rather than a momentum vector. Finally, for fixed spectral elements in the dual translation sectors, we derive the associated scalar, Dirac, and vector equations in position and momentum space and show that, in the regular polarized realizations, the represented Heisenberg sector induces dual local abelian phase groups, compatible covariant derivatives, curvatures, and primary Dirac–Maxwell systems.

1. Introduction

1.1. State of the Art and Standard Mathematical Background

Relativistic quantum symmetry is classically organized by the Poincaré group and its unitary representations, beginning with the work of Wigner and Bargmann and developed further in the modern representation-theoretic literature [1,2,3,4,5,6,7]. In parallel, harmonic analysis, Fourier duality, geometric quantization, and the orbit method describe how translation groups, projective representations, and coadjoint-orbit data connect Lie theory with physical realizations [8,9,10,11,12,13,14].

A recurring theme across these pieces of literature is the attempt to place position and momentum on more nearly symmetric footing. Historically, this theme goes back to Born’s reciprocity idea [15] and to the early Lie-algebraic proposals of Snyder and Yang [16,17]; for the broader classification viewpoint on relativistic kinematics, see also [18]. More recent approaches include canonical/quaplectic and related phase-space symmetry frameworks [19,20,21,22,23], together with stabilized or deformed Poincaré–Heisenberg and Yang-type models [24,25,26,27,28,29,30,31]. String-theoretic doubled-coordinate formulations such as double field theory [32,33], as well as quantum-spacetime or noncommutative-spacetime approaches [34,35], are conceptually nearby in motivation but structurally distinct from the present setting, since neither a doubled spacetime coordinate set nor a deformation of the spacetime coordinate algebra is assumed here.

1.2. The Specific Problem Addressed in This Paper

The present paper addresses a narrower realization-theoretic problem. We do not deform the Poincaré algebra, introduce additional invariant scales, or assume a Heisenberg normal subgroup from the outset. Instead, we remain within the standard Lorentz-covariant X- and K-polarized realizations on $L^2\left({\mathbb{R}}^{3,1}\right)$, with all basic operator identities first formulated on the common dense invariant Schwartz core $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, and ask the following:

Within the fixed Lorentz-covariant Fourier-dual realization framework, what minimal operator-generated Lie algebra is forced if one requires two Lorentz-covariant geometric translation sectors to stand on equal structural footing?

The key preliminary fact is standard but decisive: Fourier transform exchanges geometric translations with character actions, not with geometric translations [8,9]. The obstruction is therefore internal to the standard X- and K-polarized realizations themselves. In the ordinary Poincaré framework, spacetime translations form the unique Lorentz-covariant abelian translation ideal. Consequently, Fourier duality alone does not place the two geometric translation sectors on equal kinematical footing.

1.3. Guiding Principle and Main Result

We formulate the Dual Relativity Principle: the geometric origin structures of position space and momentum space should not be intrinsically privileged relative to one another unless a structural obstruction forces such an asymmetry. The point of this paper is not to classify all abstract Lorentz-equivariant doubled translation extensions, nor to carry out a realization-independent Mackey classification of an integrated Lorentz–Heisenberg group. The main theorem is sharper and internal to the fixed realization class:

Within the Fourier-compatible operator realization class studied here, the minimal operator-generated Lie algebra forced by the Dual Relativity Principle is the Lorentz–Heisenberg algebra.

The derivation proceeds in three logically distinct stages. First, we isolate the Fourier-compatibility obstruction in the standard polarized realizations. Second, we show that the Poincaré algebra cannot realize the required doubled geometric translation structure because it contains at most one Lorentz-covariant abelian translation ideal. Third, we prove that the unique Lorentz-equivariant scalar central extension of the doubled translation module is realized, in this fixed class, as the mixed Heisenberg bracket, and hence that the resulting operator-generated Lie algebra is the Lorentz–Heisenberg algebra.

1.4. What Is Standard and What Is New

To keep the standard material and the original contribution clearly separated, the manuscript is divided into a preliminary part and a contribution part.

Section 2 and Section 3 are preliminary. They collect the standard Fourier-analytic, operator-theoretic, and Lorentz-covariant background needed later, and they isolate the Fourier-compatibility obstruction in a self-contained form.

The original contributions begin in Section 4. More precisely, the main new results of the manuscript are the following.

(i)

We formulate the Fourier-compatible dual realization class singled out by the Dual Relativity Principle and prove that the Poincaré algebra cannot realize the required doubled geometric translation structure.

(ii)

We prove that, within this fixed realization class, the operator-generated Lie algebra is the Lorentz–Heisenberg algebra, and that the mixed Heisenberg bracket is rigid up to normalization of the central generator.

(iii)

We determine the full center of the universal enveloping algebra, introduce the normalized commuting Lorentz bivectors in nondegenerate irreducible sectors, and derive the corresponding normalized Casimir data.

(iv)

We classify the nondegenerate orbit sectors and their connected stabilizers. In particular, the orbit variable is shown to be a normalized Lorentz bivector rather than a momentum vector, and the classification produces a mixed regular sector with no analogue in the standard Wigner vector-orbit picture.

(v)

For fixed spectral elements in the dual translation sectors, we derive the corresponding scalar, Dirac, and vector equations in momentum space and position space.

(vi)

In the regular polarized realizations, we show that the represented Heisenberg sector induces dual local abelian phase groups, compatible covariant derivatives, antisymmetric curvatures, and primary Dirac–Maxwell systems.

(vii)

Within that same regular-realization package, the canonical abelian phase covariantization also admits an independent object-level phase-coupling refinement: fundamentally neutral objects are obtained by vanishing abelian phase coupling without changing the fixed irreducible nondegenerate Lorentz–Heisenberg sector.

1.5. Position of the Result in the Literature

This paper is therefore sharply delimited relative to existing work. It differs from the standard Wigner–Bargmann–Poincaré framework because the basic problem is not the classification of one-particle momentum-vector sectors, but the compatibility of two Lorentz-covariant geometric translation sectors inside a fixed Fourier-dual realization class. It differs from canonical/quaplectic, Yang-type, and stabilized/deformed Poincaré–Heisenberg programs because the Lorentz–Heisenberg algebra is not assumed from the outset: it is derived once the Dual Relativity Principle and the Fourier-compatible realization constraints are imposed. It also differs from realization-independent cohomological extension theory because the main theorem is not “all possible abstract extensions”, but “the algebra forced once the standard Fourier-dual realization class is fixed”. It also differs from relative-locality and modern Born-reciprocal doubled-geometry programs [36,37], where momentum-space or doubled phase-space geometry is postulated a priori, whereas here, the relevant K-space gauge and curvature structures are derived inside the undeformed Fourier-compatible realization class.

1.6. Structure of the Manuscript

Section 2 and Section 3 constitute the preliminary part. They fix the common Fourier/operator-theoretic framework and establish the Fourier-compatibility obstruction. Section 4, Section 5, Section 6, Section 7 and Section 8 contain the original results.

More precisely, Section 4 derives the Lorentz–Heisenberg algebra from the Dual Relativity Principle inside the Fourier-compatible realization class. Section 5 determines the full center of the universal enveloping algebra and the normalized Casimir data. Section 6 classifies the nondegenerate orbit sectors and connected stabilizers and compares the resulting bivector-orbit picture with the Wigner framework. Section 7 develops the fixed-spectral scalar, Dirac, and vector dynamics in momentum and position space. Section 8 derives the dual local abelian gauge structures in the regular polarized realizations and the associated primary Dirac–Maxwell systems.

2. Preliminaries: Standard Fourier-Covariant Framework in X and K

This section is purely preliminary. Its role is only to fix notation, conventions, and the standard operator identities used later; none of the statements below are claimed as new. For standard background on Schwartz-space Fourier analysis, self-adjointness and spectral calculus for the regular polarized realizations, and Lie-theoretic background on $\mathfrak{so}(3,1)$, see [6,7,9,38,39].

Unless explicitly stated otherwise, Lie algebras are over $\mathbb{R}$, universal enveloping algebras are over $\mathbb{C}$, and representation spaces are complex. Complexification is denoted by ${(·)}_{\mathbb{C}}$. All operator identities are first understood on the common dense invariant core

$$\mathcal{S}\left({\mathbb{R}}^{3,1}\right),$$

and closures or joint spectral calculi are invoked only through the standard realization facts summarized in Proposition 3.

Remark 1 (Real Lie algebras and physics normalization). 

The abstract Lie algebras $\mathfrak{so}(3,1)$, ${\mathfrak{h}}_9$, and $\mathfrak{g}$ are real Lie algebras, whereas the realized operators are written in the standard physics normalization and therefore carry factors of i in their commutators. Equivalently, one may pass to the real basis

$${\mathsf{M}}_{\mu \nu }:=-i{M}_{\mu \nu },{\mathsf{T}}_{\mu }:=-i{T}_{\mu },{\mathsf{Q}}_{\mu }:=-i{Q}_{\mu },$$

without changing the underlying real Lie-algebraic content.

We work on

$${\mathcal{H}}_X=L^2({\mathbb{R}}^{3,1},d^{4}x),{\mathcal{H}}_K=L^2({\mathbb{R}}^{3,1},d^{4}k),$$

with Lebesgue measure. The Schwartz space

$$\mathcal{S}\left({\mathbb{R}}^{3,1}\right)\subset {\mathcal{H}}_X,{\mathcal{H}}_K$$

is stable under Fourier transform, polynomial multiplication, and partial differentiation. Indices are raised and lowered with

$${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,1,1),x_{\mu }:={\eta }_{\mu \nu }{x}^{\nu },k_{\mu }:={\eta }_{\mu \nu }{k}^{\nu },$$

and we fix ${ϵ}_{0123}=+1$. We also write

$${\partial }^{x^{\mu }}:={\eta }^{\mu \nu }{\partial }_{x^{\nu }},{\partial }^{k^{\mu }}:={\eta }^{\mu \nu }{\partial }_{k^{\nu }}.$$

Definition 1 (Fourier convention). 

For $f\in \mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, define

$$\left(\mathcal{F}f\right)\left(k\right)=\widehat{f}\left(k\right):={\displaystyle \frac{1}{{\left(2\pi \right)}^2}}{\int }_{{\mathbb{R}}^{3,1}}{e}^{-ik·x}f\left(x\right)d^{4}x,k·x:=k_{\mu }{x}^{\mu },$$

with inverse

$$\left({\mathcal{F}}^{-1}\widehat{f}\right)\left(x\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^2}}{\int }_{{\mathbb{R}}^{3,1}}{e}^{+ik·x}\widehat{f}\left(k\right)d^{4}k.$$

Proposition 1 (Fourier duality of differentiation and multiplication). 

For every $f\in \mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, writing $\widehat{f}:=\mathcal{F}f$, one has

$$\begin{array}{cc}\hfill \mathcal{F}\left[{\partial }_{x^{\mu }}f\right]\left(k\right)& =i{k}_{\mu }\widehat{f}\left(k\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \mathcal{F}\left[x_{\mu }f\right]\left(k\right)& =i{\partial }_{k^{\mu }}\widehat{f}\left(k\right),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\mathcal{F}}^{-1}\left[k_{\mu }\widehat{f}\right]\left(x\right)& =-i{\partial }_{x^{\mu }}f\left(x\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\mathcal{F}}^{-1}\left[{\partial }_{k^{\mu }}\widehat{f}\right]\left(x\right)& =-i{x}_{\mu }f\left(x\right).\hfill \end{array}$$

(4)

Equivalently, on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$,

$$-i{\partial }_{x^{\mu }}={\mathcal{F}}^{-1}{k}_{\mu }\mathcal{F},x_{\mu }={\mathcal{F}}^{-1}\left(i{\partial }_{k^{\mu }}\right)\mathcal{F},$$

(5)

and hence

$$k_{\mu }=\mathcal{F}\left(-i{\partial }_{x^{\mu }}\right){\mathcal{F}}^{-1},i{\partial }_{k^{\mu }}=\mathcal{F}{x}_{\mu }{\mathcal{F}}^{-1}.$$

(6)

Proof. 

These are standard consequences of integration by parts and differentiation under the integral sign on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$; see [9] (Chapter 1). □

Definition 2 (Canonical operators and Lorentz generators). 

On $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, define

$$\begin{array}{cccc}\hfill {\widehat{x}}_{\mu }^{\left(X\right)}f& :=x_{\mu }f,\hfill & \hfill {\widehat{k}}_{\mu }^{\left(X\right)}f& :=-i{\partial }_{x^{\mu }}f,\hfill \end{array}$$

(7)

$$\begin{array}{cccc}\hfill {\widehat{x}}_{\mu }^{\left(K\right)}\widehat{f}& :=i{\partial }_{k^{\mu }}\widehat{f},\hfill & \hfill {\widehat{k}}_{\mu }^{\left(K\right)}\widehat{f}& :=k_{\mu }\widehat{f}.\hfill \end{array}$$

(8)

The Lorentz Lie algebra $\mathfrak{so}(3,1)$ is generated by antisymmetric elements

$$M_{\mu \nu }=-M_{\nu \mu },\mu ,\nu =0,\dots ,3,$$

with

$$[M_{\mu \nu },M_{\rho \sigma }]=i\left({\eta }_{\nu \rho }{M}_{\mu \sigma }-{\eta }_{\mu \rho }{M}_{\nu \sigma }-{\eta }_{\nu \sigma }{M}_{\mu \rho }+{\eta }_{\mu \sigma }{M}_{\nu \rho }\right),$$

(9)

and

$$\mathfrak{so}{(3,1)}_{\mathbb{C}}\cong {\mathfrak{sl}}_2\left(\mathbb{C}\right)\oplus {\mathfrak{sl}}_2\left(\mathbb{C}\right).$$

Its standard differential realizations are

$$\begin{array}{cc}\hfill M_{\mu \nu }^{\left(X\right)}& =i\left(x_{\mu }{\partial }_{x^{\nu }}-x_{\nu }{\partial }_{x^{\mu }}\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill M_{\mu \nu }^{\left(K\right)}& =i\left(k_{\mu }{\partial }_{k^{\nu }}-k_{\nu }{\partial }_{k^{\mu }}\right).\hfill \end{array}$$

(11)

Proposition 2 (Canonical commutators, Fourier transport, and Lorentz covariance). 

On $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$,

$$\begin{array}{cccccc}\hfill [{\widehat{x}}_{\mu }^{\left(X\right)},{\widehat{x}}_{\nu }^{\left(X\right)}]& =0,\hfill & \hfill [{\widehat{k}}_{\mu }^{\left(X\right)},{\widehat{k}}_{\nu }^{\left(X\right)}]& =0,\hfill & \hfill [{\widehat{x}}_{\mu }^{\left(X\right)},{\widehat{k}}_{\nu }^{\left(X\right)}]& =i{\eta }_{\mu \nu }\mathbb{I},\hfill \end{array}$$

(12)

$$\begin{array}{cccccc}\hfill [{\widehat{x}}_{\mu }^{\left(K\right)},{\widehat{x}}_{\nu }^{\left(K\right)}]& =0,\hfill & \hfill [{\widehat{k}}_{\mu }^{\left(K\right)},{\widehat{k}}_{\nu }^{\left(K\right)}]& =0,\hfill & \hfill [{\widehat{x}}_{\mu }^{\left(K\right)},{\widehat{k}}_{\nu }^{\left(K\right)}]& =i{\eta }_{\mu \nu }\mathbb{I},\hfill \end{array}$$

(13)

and

$${\widehat{x}}_{\mu }^{\left(K\right)}=\mathcal{F}{\widehat{x}}_{\mu }^{\left(X\right)}{\mathcal{F}}^{-1},{\widehat{k}}_{\mu }^{\left(K\right)}=\mathcal{F}{\widehat{k}}_{\mu }^{\left(X\right)}{\mathcal{F}}^{-1},M_{\mu \nu }^{\left(K\right)}=\mathcal{F}{M}_{\mu \nu }^{\left(X\right)}{\mathcal{F}}^{-1}.$$

(14)

Moreover, the canonical operators transform as Lorentz vectors:

$$\begin{array}{cc}\hfill [M_{\mu \nu }^{\left(X\right)},{\widehat{x}}_{\rho }^{\left(X\right)}]& =i\left({\eta }_{\nu \rho }{\widehat{x}}_{\mu }^{\left(X\right)}-{\eta }_{\mu \rho }{\widehat{x}}_{\nu }^{\left(X\right)}\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill [M_{\mu \nu }^{\left(X\right)},{\widehat{k}}_{\rho }^{\left(X\right)}]& =i\left({\eta }_{\nu \rho }{\widehat{k}}_{\mu }^{\left(X\right)}-{\eta }_{\mu \rho }{\widehat{k}}_{\nu }^{\left(X\right)}\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill [M_{\mu \nu }^{\left(K\right)},{\widehat{x}}_{\rho }^{\left(K\right)}]& =i\left({\eta }_{\nu \rho }{\widehat{x}}_{\mu }^{\left(K\right)}-{\eta }_{\mu \rho }{\widehat{x}}_{\nu }^{\left(K\right)}\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill [M_{\mu \nu }^{\left(K\right)},{\widehat{k}}_{\rho }^{\left(K\right)}]& =i\left({\eta }_{\nu \rho }{\widehat{k}}_{\mu }^{\left(K\right)}-{\eta }_{\mu \rho }{\widehat{k}}_{\nu }^{\left(K\right)}\right).\hfill \end{array}$$

(18)

In particular, the X- and K-polarized realizations are unitarily equivalent.

Proof. 

The commutator identities are immediate from the definitions, and the Fourier transport identities follow from Proposition 1. The Lorentz-covariance relations are the standard differential-representation formulas for the vector representation; see, for example, ref. [7] (Chapter 1). □

Proposition 3 (Standard operator-theoretic facts for the regular polarized realizations). 

On

$${\mathcal{H}}_X=L^2({\mathbb{R}}^{3,1},d^{4}x),{\mathcal{H}}_K=L^2({\mathbb{R}}^{3,1},d^{4}k),$$

the multiplication operators $x_{\mu }$ and $k_{\mu }$, initially defined on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, are self-adjoint and form strongly commuting families. The differential operators

$$-i{\partial }_{x^{\mu }}\mathit{on}{\mathcal{H}}_X,i{\partial }_{k^{\mu }}\mathit{on}{\mathcal{H}}_K,$$

with initial domain $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$ are essentially self-adjoint; their closures form strongly commuting self-adjoint families and satisfy

$$\overline{-i{\partial }_{x^{\mu }}}={\mathcal{F}}^{-1}\overline{k_{\mu }}\mathcal{F},\overline{i{\partial }_{k^{\mu }}}=\mathcal{F}\overline{x_{\mu }}{\mathcal{F}}^{-1}.$$

Consequently, the joint functional calculi and the exponentials used later are the canonical ones attached to these self-adjoint commuting families.

Proof. 

These are standard facts for the Schrödinger-type realizations on Schwartz space and their Fourier transforms; see [38,39]. □

3. Preliminaries: Translation Representations and the Fourier-Compatibility Obstruction

This section still belongs to the preliminary part of this paper. Using the notation and Fourier convention fixed in Section 2, it records the standard geometric and character translation actions in the X- and K-polarized realizations together with their Fourier transport properties; the individual formulas are standard consequences of Fourier analysis and the translation representation theory of ${\mathbb{R}}^{3,1}$; see [8,9,38]. The point is not that the abelian group ${\mathbb{R}}^{3,1}$ admits inequivalent abstract unitary representations, but that, within the standard X- and K-polarized realizations, Fourier transform exchanges geometric translations with character representations rather than with geometric translations. This section makes that structural obstruction precise and thereby supplies the final preliminary input for Section 4, where it is converted into a minimality theorem for the operator-generated symmetry algebra.

We proceed in three steps. First, we record the natural geometric translation actions in X-space and K-space. Second, we identify their Fourier images. Third, we show that the Fourier image of a geometric translation action is a character action, so that the two geometric translation sectors are not intertwined by Fourier duality.

3.1. Translation Actions in the Two Polarizations

On ${\mathcal{H}}_X=L^2({\mathbb{R}}^{3,1},d^{4}x)$, define the geometric translation representation

$$\left(T_a^{\left(X\right)}f\right)\left(x\right):=f(x-a),a\in {\mathbb{R}}^{3,1}.$$

(19)

On the common invariant core $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, its infinitesimal operators are

$$T_{\mu }^{\left(X\right)}=-i{\partial }_{x^{\mu }}.$$

(20)

On ${\mathcal{H}}_K=L^2({\mathbb{R}}^{3,1},d^{4}k)$, define the geometric translation representation

$$\left(T_b^{\left(K\right)}\widehat{f}\right)\left(k\right):=\widehat{f}(k-b),b\in {\mathbb{R}}^{3,1},$$

(21)

whose infinitesimal operators on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$ are

$$T_{\mu }^{\left(K\right)}=+i{\partial }_{k^{\mu }}.$$

(22)

Independently of these geometric actions, the same group ${\mathbb{R}}^{3,1}$ also acts by characters:

$$\begin{array}{cc}\hfill \left(Q_a^{\left(K\right)}\widehat{f}\right)\left(k\right)& :=e^{-ik·a}\widehat{f}\left(k\right),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \left(Q_b^{\left(X\right)}f\right)\left(x\right)& :=e^{ix·b}f\left(x\right).\hfill \end{array}$$

(24)

On $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, the corresponding infinitesimal operators are the multiplication operators

$$Q_{\mu }^{\left(K\right)}=k_{\mu },Q_{\mu }^{\left(X\right)}=x_{\mu }.$$

(25)

Thus, each polarization carries two natural ${\mathbb{R}}^{3,1}$-actions: one geometric and one by characters. The question is which of these are related by Fourier transform.

3.2. Fourier Transport of the Translation Actions

The answer is the following.

Proposition 4 (Fourier transport and non-intertwining of geometric translations). 

For all $a,b\in {\mathbb{R}}^{3,1}$, one has

$$\begin{array}{cc}\hfill \mathcal{F}{T}_a^{\left(X\right)}{\mathcal{F}}^{-1}& =Q_a^{\left(K\right)},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \mathcal{F}{Q}_b^{\left(X\right)}{\mathcal{F}}^{-1}& =T_b^{\left(K\right)}.\hfill \end{array}$$

(27)

Consequently, Fourier transform exchanges geometric translations with character representations, not with geometric translations. In particular, for every $a\ne 0$,

$$\mathcal{F}{T}_a^{\left(X\right)}{\mathcal{F}}^{-1}\ne T_a^{\left(K\right)}.$$

(28)

Equivalently, within the standard Fourier-dual realizations, Fourier transform does not identify spacetime shifts with momentum-space shifts.

This is the Fourier-compatibility obstruction used in Section 4: the obstruction lies not in the abstract representation theory of the abelian group ${\mathbb{R}}^{3,1}$ but in the fact that Fourier duality does not intertwine the two geometric translation sectors.

Proof. 

Let $f\in \mathcal{S}\left({\mathbb{R}}^{3,1}\right)$. Using the Fourier convention of Definition 1,

$$\begin{array}{cc}\hfill \left(\mathcal{F}{T}_a^{\left(X\right)}f\right)\left(k\right)& ={\displaystyle \frac{1}{{\left(2\pi \right)}^2}}{\int }_{{\mathbb{R}}^{3,1}}{e}^{-ik·x}f(x-a)d^{4}x\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left(2\pi \right)}^2}}{\int }_{{\mathbb{R}}^{3,1}}{e}^{-ik·(y+a)}f\left(y\right)d^{4}y(y:=x-a)\hfill \\ \hfill & =e^{-ik·a}\widehat{f}\left(k\right)\hfill \\ \hfill & =\left(Q_a^{\left(K\right)}\widehat{f}\right)\left(k\right).\hfill \end{array}$$

Hence

$$\mathcal{F}{T}_a^{\left(X\right)}{\mathcal{F}}^{-1}=Q_a^{\left(K\right)},$$

which proves (26).

Likewise,

$$\begin{array}{cc}\hfill \left(\mathcal{F}{Q}_b^{\left(X\right)}f\right)\left(k\right)& ={\displaystyle \frac{1}{{\left(2\pi \right)}^2}}{\int }_{{\mathbb{R}}^{3,1}}{e}^{-ik·x}{e}^{ix·b}f\left(x\right)d^{4}x\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left(2\pi \right)}^2}}{\int }_{{\mathbb{R}}^{3,1}}{e}^{-i(k-b)·x}f\left(x\right)d^{4}x\hfill \\ \hfill & =\widehat{f}(k-b)\hfill \\ \hfill & =\left(T_b^{\left(K\right)}\widehat{f}\right)\left(k\right),\hfill \end{array}$$

so

$$\mathcal{F}{Q}_b^{\left(X\right)}{\mathcal{F}}^{-1}=T_b^{\left(K\right)},$$

which proves (27).

To prove (28), fix $a\ne 0$. Choose $\widehat{f}\in \mathcal{S}\left({\mathbb{R}}^{3,1}\right)$ such that $\widehat{f}\left(0\right)\ne 0$ and $\widehat{f}(-a)=0$; for example, take a nonzero Schwartz function supported in a sufficiently small neighborhood of 0 not containing $-a$. Then

$$\left(Q_a^{\left(K\right)}\widehat{f}\right)\left(0\right)=e^{-i0·a}\widehat{f}\left(0\right)=\widehat{f}\left(0\right)\ne 0,$$

whereas

$$\left(T_a^{\left(K\right)}\widehat{f}\right)\left(0\right)=\widehat{f}(-a)=0.$$

Hence $Q_a^{\left(K\right)}\ne T_a^{\left(K\right)}$, and therefore

$$\mathcal{F}{T}_a^{\left(X\right)}{\mathcal{F}}^{-1}=Q_a^{\left(K\right)}\ne T_a^{\left(K\right)}.$$

This proves the claimed non-intertwining. □

The conclusion is the structural input needed for the next section.

4. Dual Relativity and the Lorentz–Heisenberg Algebra

This section begins the original part of the manuscript. The preliminary sections established the Fourier-compatibility obstruction: in the standard X- and K-polarized realizations, Fourier transform exchanges geometric translations with character actions, while the Poincaré algebra contains at most one Lorentz-covariant abelian translation ideal. The Dual Relativity Principle therefore cannot be implemented inside the ordinary Poincaré algebra and forces an enlarged kinematical algebra.

The argument proceeds in four steps. First, we isolate the abstract doubled translation data suggested by the principle. Second, we fix the Fourier-compatible realization class to which all later minimality statements refer. Third, we prove the affine obstruction inside $\mathfrak{iso}(3,1)$. Fourth, we show that the unique Lorentz-equivariant scalar central extension of the doubled translation space is realized as the mixed Heisenberg bracket. Hence, within this fixed realization class, the operator-generated Lie algebra is the Lorentz–Heisenberg algebra. Throughout, we keep separate the three logically distinct levels: the guiding principle, the realization class, and the Lie algebra forced within that class.

Principle 1 (Dual Relativity Principle). 

A relativistic kinematical symmetry algebra should not intrinsically privilege the geometric origin structure of position space over that of momentum space, or conversely. Consistently with Fourier duality between the X- and K-realizations, and given that their Lorentz sectors already encode frame orientation on equal footing, the symmetry algebra should likewise place the corresponding geometric translation sectors, which determine the respective affine origins, on equal structural footing, unless excluded by a structural obstruction.

4.1. Abstract Doubled Translation Data and Fourier-Compatible Realizations

We first isolate the abstract doubled translation data and then fix the realization class in which the Dual Relativity Principle will be implemented. This keeps the module-theoretic input separate from the concrete operator realization.

Let $V\cong {\mathbb{R}}^{3,1}$ denote the standard real vector representation of $\mathfrak{so}(3,1)$.

Definition 3 (Translation module). 

A translation module is a copy of V endowed with the standard $\mathfrak{so}(3,1)$-action.

Definition 4 (Dual translation pair). 

A dual translation pair consists of two translation modules $V_X$ and $V_K$, together with a nonzero Lorentz-invariant bilinear pairing

$$\eta :V_X\otimes V_K\to \mathbb{R}.$$

Remark 2 (Realization of the dual translation pair). 

In the standard X- and K-polarized realizations considered below, the abstract modules are represented by

$$V_X=\left\{\begin{array}{cc}\mathrm{span}\left\{T_{\mu }^{\left(X\right)}\right\}\hfill & \mathit{in}X,\hfill \\ \mathrm{span}\left\{Q_{\mu }^{\left(K\right)}\right\}\hfill & \mathit{in}K,\hfill \end{array}\right.V_K=\left\{\begin{array}{cc}\mathrm{span}\left\{Q_{\mu }^{\left(X\right)}\right\}\hfill & \mathit{in}X,\hfill \\ \mathrm{span}\left\{T_{\mu }^{\left(K\right)}\right\}\hfill & \mathit{in}K.\hfill \end{array}\right.$$

All later minimality statements are relative to the following realization class.

Definition 5 (Fourier-compatible dual realization). 

Relative to the fixed Hilbert spaces ${\mathcal{H}}_X,{\mathcal{H}}_K$ and Fourier transform $\mathcal{F}$, a Fourier-compatible dual realization of a dual translation pair consists of strongly continuous unitary representations

$$U_T,U_Q:{\mathbb{R}}^{3,1}\to \mathcal{U}\left({\mathcal{H}}_X\right),{\widehat{U}}_T,{\widehat{U}}_Q:{\mathbb{R}}^{3,1}\to \mathcal{U}\left({\mathcal{H}}_K\right),$$

together with strongly continuous unitary Lorentz actions

$$U_{\Lambda }^{\left(X\right)}:{\mathrm{SO}}^{+}(3,1)\to \mathcal{U}\left({\mathcal{H}}_X\right),{\widehat{U}}_{\Lambda }^{\left(K\right)}:{\mathrm{SO}}^{+}(3,1)\to \mathcal{U}\left({\mathcal{H}}_K\right),$$

such that, for all $a,b\in {\mathbb{R}}^{3,1}$ and $\Lambda \in {\mathrm{SO}}^{+}(3,1)$, the following hold:

(i) 

On ${\mathcal{H}}_X$, the two sectors are realized by

$$\begin{array}{cc}\hfill \left(U_T\left(a\right)f\right)\left(x\right)& =f(x-a),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \left(U_Q\left(b\right)f\right)\left(x\right)& =e^{i{b}^{\mu }{x}_{\mu }}f\left(x\right).\hfill \end{array}$$

(30)

(ii) 

On ${\mathcal{H}}_K$, the Fourier-dual realizations are

$$\begin{array}{cc}\hfill \left({\widehat{U}}_T\left(a\right)\widehat{f}\right)\left(k\right)& =e^{-i{k}^{\mu }{a}_{\mu }}\widehat{f}\left(k\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \left({\widehat{U}}_Q\left(b\right)\widehat{f}\right)\left(k\right)& =\widehat{f}(k-b).\hfill \end{array}$$

(32)

(iii) 

The X- and K-space realizations are related by Fourier duality:

$$\begin{array}{cc}\hfill {\widehat{U}}_T\left(a\right)& =\mathcal{F}{U}_T\left(a\right){\mathcal{F}}^{-1},\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\widehat{U}}_Q\left(b\right)& =\mathcal{F}{U}_Q\left(b\right){\mathcal{F}}^{-1},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\widehat{U}}_{\Lambda }^{\left(K\right)}& =\mathcal{F}{U}_{\Lambda }^{\left(X\right)}{\mathcal{F}}^{-1}.\hfill \end{array}$$

(35)

(iv) 

Each translation sector is Lorentz-covariant:

$$\begin{array}{cc}\hfill U_{\Lambda }^{\left(X\right)}{U}_T\left(a\right){\left(U_{\Lambda }^{\left(X\right)}\right)}^{-1}& =U_T(\Lambda a),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill U_{\Lambda }^{\left(X\right)}{U}_Q\left(b\right){\left(U_{\Lambda }^{\left(X\right)}\right)}^{-1}& =U_Q(\Lambda b),\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\widehat{U}}_{\Lambda }^{\left(K\right)}{\widehat{U}}_T\left(a\right){\left({\widehat{U}}_{\Lambda }^{\left(K\right)}\right)}^{-1}& ={\widehat{U}}_T(\Lambda a),\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\widehat{U}}_{\Lambda }^{\left(K\right)}{\widehat{U}}_Q\left(b\right){\left({\widehat{U}}_{\Lambda }^{\left(K\right)}\right)}^{-1}& ={\widehat{U}}_Q(\Lambda b).\hfill \end{array}$$

(39)

For later use, define the packaged operators

$$\begin{array}{cc}\hfill U_X(a,b)& :=U_Q\left(b\right)U_T\left(a\right),\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\widehat{U}}_K(a,b)& :={\widehat{U}}_Q\left(b\right){\widehat{U}}_T\left(a\right).\hfill \end{array}$$

(41)

Explicitly,

$$\begin{array}{cc}\hfill \left(U_X(a,b)f\right)\left(x\right)& =e^{i{b}^{\mu }{x}_{\mu }}f(x-a),\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill \left({\widehat{U}}_K(a,b)\widehat{f}\right)\left(k\right)& =e^{-i{a}^{\mu }{k}_{\mu }}\widehat{f}(k-b).\hfill \end{array}$$

(43)

Whenever infinitesimal generators are invoked, they are understood on the common dense invariant core

$$\mathcal{S}\left({\mathbb{R}}^{3,1}\right)\subset L^2\left({\mathbb{R}}^{3,1}\right).$$

4.2. Affine Obstruction Inside the Poincaré Algebra

Having fixed the realization framework, we next show that the usual Poincaré algebra cannot realize the doubled geometric translation structure required by the principle. The point is that the Poincaré algebra has room for only one Lorentz-covariant abelian translation ideal.

Let

$$\mathfrak{iso}(3,1)=\mathfrak{so}(3,1)⋉V$$

denote the Poincaré algebra.

Proposition 5 (Uniqueness of the Lorentz vector ideal). 

Let $I\subset \mathfrak{iso}(3,1)$ be an abelian ideal invariant under the adjoint action of $\mathfrak{so}(3,1)$ and isomorphic, as an $\mathfrak{so}(3,1)$-module, to the vector representation V. Then $I=V$.

Requiring an ideal, rather than merely a subalgebra, encodes the translation sector as a normal subgroup of the corresponding kinematical group, i.e., the affine structure underlying spacetime translations.

Proof. 

As an $\mathfrak{so}(3,1)$-module,

$$\mathfrak{iso}(3,1)=\mathfrak{so}(3,1)\oplus V.$$

Let

$${\pi }_{\mathrm{ad}}:\mathfrak{iso}(3,1)\to \mathfrak{so}(3,1)$$

be the canonical $\mathfrak{so}(3,1)$-equivariant projection. Since $I\cong V$ is irreducible, the restriction ${\pi }_{\mathrm{ad}}{|}_I$ is either zero or injective.

We first show that injectivity is impossible. If ${\pi }_{\mathrm{ad}}{|}_I$ were injective, then I would define a nonzero $\mathfrak{so}(3,1)$-equivariant embedding

$$V\hookrightarrow \mathfrak{so}(3,1).$$

After complexification,

$$V_{\mathbb{C}}\cong \left(\frac{1}{2},\frac{1}{2}\right),\mathfrak{so}{(3,1)}_{\mathbb{C}}\cong (1,0)\oplus (0,1),$$

and these $\mathfrak{so}{(3,1)}_{\mathbb{C}}$-modules are non-isomorphic; see, for example, ([6], Chapters 5–6) and ([7], Chapter V). Hence

$${Hom}_{\mathfrak{so}{(3,1)}_{\mathbb{C}}}\left(V_{\mathbb{C}},\mathfrak{so}{(3,1)}_{\mathbb{C}}\right)=0.$$

By complexification of Hom,

$${Hom}_{\mathfrak{so}(3,1)}{(V,\mathfrak{so}(3,1))}_{\mathbb{C}}\cong {Hom}_{\mathfrak{so}{(3,1)}_{\mathbb{C}}}\left(V_{\mathbb{C}},\mathfrak{so}{(3,1)}_{\mathbb{C}}\right),$$

so already

$${Hom}_{\mathfrak{so}(3,1)}(V,\mathfrak{so}(3,1))=0,$$

contradicting injectivity.

Therefore ${\pi }_{\mathrm{ad}}{|}_I=0$, so $I\subseteq V$. Since V is irreducible as an $\mathfrak{so}(3,1)$-module and $I\ne 0$, it follows that $I=V$. □

Corollary 1. 

The Poincaré algebra admits at most one Lorentz-covariant abelian translation ideal.

Thus the Dual Relativity Principle cannot be realized inside $\mathfrak{iso}(3,1)$; any realization with two Lorentz-covariant geometric translation sectors must enlarge the Poincaré algebra.

4.3. Lorentz-Invariant Scalar Central Extensions of the Doubled Translation Space

The preceding obstruction identifies the problem. We now determine the minimal new bracket that can couple the two translation sectors while remaining Lorentz equivariant.

Let

$$\mathfrak{t}:=V_X\oplus V_K$$

be the doubled translation space, regarded for the moment as an abelian Lie algebra. We write

$$H_{\mathrm{CE}}^{•}(\mathfrak{t},\mathbb{R})$$

for its Chevalley–Eilenberg cohomology with coefficients in the trivial $\mathfrak{t}$-module $\mathbb{R}$. Because $\mathfrak{t}$ is abelian, the Chevalley–Eilenberg differential vanishes, and hence

$$H_{\mathrm{CE}}^2(\mathfrak{t},\mathbb{R})\cong {\Lambda }^2\left({\mathfrak{t}}^{*}\right).$$

This is the standard Chevalley–Eilenberg cohomology of Lie algebras, and the interpretation of $H_{\mathrm{CE}}^2$ as the classifying space of scalar central extensions is standard; see [40,41].

The next lemma identifies the Lorentz-invariant tensor needed to extract the scalar central extensions.

Lemma 1 (Invariant tensor structure). 

For the Lorentz vector representation V,

$${(V\otimes V)}^{{\mathrm{SO}}^{+}(3,1)}=\mathrm{span}\left\{\eta \right\},{\left({\Lambda }^{2}V\right)}^{{\mathrm{SO}}^{+}(3,1)}=\left\{0\right\}.$$

Proof. 

Using the complexified Lorentz-representation decomposition

$$V_{\mathbb{C}}\cong \left(\frac{1}{2},\frac{1}{2}\right),$$

one has

$${\Lambda }^2{V}_{\mathbb{C}}\cong (1,0)\oplus (0,1),{Sym}^2{V}_{\mathbb{C}}\cong (1,1)\oplus (0,0);$$

see, for example, refs. [6] (Chapters 5–6) and [7] (Chapter V). Hence ${\Lambda }^{2}V$ is the adjoint module and contains no invariant vectors, while ${Sym}^{2}V$ contains a unique invariant line generated by the Minkowski metric. Therefore

$${(V\otimes V)}^{{\mathrm{SO}}^{+}(3,1)}=\mathrm{span}\left\{\eta \right\},{\left({\Lambda }^{2}V\right)}^{{\mathrm{SO}}^{+}(3,1)}=\left\{0\right\}.$$

□

Proposition 6 (Invariant cohomology for scalar central extensions). 

$$H_{\mathrm{CE}}^2{(\mathfrak{t},\mathbb{R})}^{{\mathrm{SO}}^{+}(3,1)}\cong \mathbb{R}.$$

Equivalently, Lorentz-equivariant central extensions of $\mathfrak{t}$ by a one-dimensional trivial module are unique up to scale.

Proof. 

Using

$${\Lambda }^2\left({\mathfrak{t}}^{*}\right)\cong {\Lambda }^2\left(V_X^{*}\right)\oplus (V_X^{*}\otimes V_K^{*})\oplus {\Lambda }^2\left(V_K^{*}\right),$$

the first and third summands contain no invariant elements by Lemma 1. The mixed summand has a one-dimensional invariant subspace generated by the Minkowski pairing $\eta$. Hence the invariant part of $H_{\mathrm{CE}}^2(\mathfrak{t},\mathbb{R})$ is one-dimensional. □

Corollary 2 (Canonical central cocycle). 

Up to normalization, the unique Lorentz-invariant scalar-valued 2-cocycle on $\mathfrak{t}$ is

$$\omega \left((t,q),(t^{\prime },q^{\prime })\right)=\eta (t,q^{\prime })-\eta (t^{\prime },q).$$

(44)

Equivalently,

$$\omega (T_{\mu },Q_{\nu })={\eta }_{\mu \nu },\omega (T_{\mu },T_{\nu })=0,\omega (Q_{\mu },Q_{\nu })=0.$$

This cocycle is the abstract algebraic precursor of the Heisenberg bracket. The next step is to show that, in the Fourier-compatible operator realizations fixed above, this bracket is not merely allowed: it is forced.

4.4. Operator Realization and Emergence of the Lorentz–Heisenberg Algebra

We now pass from the abstract extension data to the concrete Fourier-compatible operator realizations. Proposition 7 identifies the mixed Heisenberg bracket in the X-polarized realization. The Lorentz-covariance relations used below are the standard differential-representation formulas already recorded in Proposition 2; the corresponding K-polarized statements follow either by the same computation or by Fourier conjugation.

Proposition 7 (Weyl relation and infinitesimal generators). 

Let $U_T\left(a\right)$ and $U_Q\left(b\right)$ be as in Definition 5. Then

$$U_T\left(a\right)U_Q\left(b\right)=e^{-i{a}^{\mu }{b}_{\mu }}{U}_Q\left(b\right)U_T\left(a\right),a,b\in {\mathbb{R}}^{3,1}.$$

(45)

On $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, the infinitesimal generators in the X-space realization are

$$T_{\mu }^{\left(X\right)}=-i{\partial }_{x^{\mu }},Q_{\mu }^{\left(X\right)}=x_{\mu },$$

(46)

and they satisfy

$$[T_{\mu }^{\left(X\right)},T_{\nu }^{\left(X\right)}]=0,[Q_{\mu }^{\left(X\right)},Q_{\nu }^{\left(X\right)}]=0,[T_{\mu }^{\left(X\right)},Q_{\nu }^{\left(X\right)}]=-i{\eta }_{\mu \nu }\mathbb{I}.$$

(47)

Proof. 

We first compute the group-level relation, because it already displays the phase factor that will differentiate to the Heisenberg bracket. For $f\in \mathcal{S}\left({\mathbb{R}}^{3,1}\right)$,

$$\begin{array}{cc}\hfill \left(U_T\left(a\right)U_Q\left(b\right)f\right)\left(x\right)& =U_T\left(a\right)\left(e^{i{b}^{\mu }{x}_{\mu }}f\left(x\right)\right)\hfill \\ \hfill & =e^{i{b}^{\mu }(x_{\mu }-a_{\mu })}f(x-a)\hfill \\ \hfill & =e^{-i{a}^{\mu }{b}_{\mu }}\left(U_Q\left(b\right)U_T\left(a\right)f\right)\left(x\right),\hfill \end{array}$$

which proves (45).

We next pass to infinitesimal generators. By Definition 5,

$$T_{\mu }^{\left(X\right)}=-i{\partial }_{x^{\mu }},Q_{\mu }^{\left(X\right)}=x_{\mu }.$$

Since partial derivatives commute, one has

$$[T_{\mu }^{\left(X\right)},T_{\nu }^{\left(X\right)}]=0.$$

Since multiplication operators commute, one has

$$[Q_{\mu }^{\left(X\right)},Q_{\nu }^{\left(X\right)}]=0.$$

Finally, for $f\in \mathcal{S}\left({\mathbb{R}}^{3,1}\right)$,

$$[T_{\mu }^{\left(X\right)},Q_{\nu }^{\left(X\right)}]f=[-i{\partial }_{x^{\mu }},x_{\nu }]f=-i\left({\partial }_{x^{\mu }}{x}_{\nu }\right)f=-i{\eta }_{\mu \nu }f,$$

which proves the mixed commutator in (47). □

Remark 3 (Abstract central element and operator identity). 

Throughout this section, $\mathbf{1}$ denotes the abstract central generator of the Lie algebra or of its universal enveloping algebra, whereas $\mathbb{I}$ denotes the identity operator on a realization space. These two symbols are kept distinct.

We can now package the previous computations into the Lie algebra forced by the Fourier-compatible realization class.

Definition 6 (Lorentz–Heisenberg algebra). 

The Lorentz–Heisenberg algebra is the Lie algebra

$$\mathfrak{g}=\mathfrak{so}(3,1)⋉{\mathfrak{h}}_9,{\mathfrak{h}}_9=\mathrm{span}\{T_{\mu },Q_{\mu },\mathbf{1}\},$$

with generators $M_{\mu \nu }=-M_{\nu \mu }$, $T_{\mu }$, $Q_{\mu }$, and central element $\mathbf{1}$, satisfying

$$\begin{array}{cc}\hfill [M_{\mu \nu },M_{\rho \sigma }]& =i({\eta }_{\nu \rho }{M}_{\mu \sigma }-{\eta }_{\mu \rho }{M}_{\nu \sigma }-{\eta }_{\nu \sigma }{M}_{\mu \rho }+{\eta }_{\mu \sigma }{M}_{\nu \rho }),\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill [M_{\mu \nu },T_{\rho }]& =i({\eta }_{\nu \rho }{T}_{\mu }-{\eta }_{\mu \rho }{T}_{\nu }),\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill [M_{\mu \nu },Q_{\rho }]& =i({\eta }_{\nu \rho }{Q}_{\mu }-{\eta }_{\mu \rho }{Q}_{\nu }),\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill [T_{\mu },T_{\nu }]& =0,[Q_{\mu },Q_{\nu }]=0,\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill [T_{\mu },Q_{\nu }]& =-i{\eta }_{\mu \nu }\mathbf{1},[\mathbf{1},\mathfrak{g}]=0.\hfill \end{array}$$

(52)

Remark 4 (Jacobi identity on the common invariant core). 

The relations (48)–(52) are not imposed independently of the realized operator system: they are precisely the commutator relations satisfied by the corresponding operators on the common invariant core $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$. Since the commutator of linear operators on a common invariant domain automatically satisfies the Jacobi identity,

$$[A,[B,C\left]\right]+[B,[C,A\left]\right]+[C,[A,B\left]\right]=0,$$

the abstract bracket determined by (48)–(52) defines a Lie algebra.

Theorem 1 (Operator-generated realization theorem in the Fourier-compatible class). 

Let ${\mathfrak{g}}_{\mathrm{gen}}$ be the Lie algebra of operators on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$ generated by

$$\{M_{\mu \nu },T_{\mu },Q_{\mu },\mathbb{I}\}$$

arising from a Fourier-compatible dual realization in the sense of Definition 5. Then

$${\mathfrak{g}}_{\mathrm{gen}}\cong \mathfrak{so}(3,1)⋉{\mathfrak{h}}_9.$$

Hence, within the Fourier-compatible realization class of Definition 5, the operator-generated Lie algebra is the Lorentz–Heisenberg algebra.

Proof. 

The proof has two parts. We first identify the relations satisfied by the realized generators, and then show that these relations already exhaust the generated Lie algebra.

By Proposition 7,

$$[T_{\mu },T_{\nu }]=0,[Q_{\mu },Q_{\nu }]=0,[T_{\mu },Q_{\nu }]=-i{\eta }_{\mu \nu }\mathbb{I}.$$

By Proposition 2, together with the identifications

$$T_{\rho }=-i{\partial }_{x^{\rho }},Q_{\rho }=x_{\rho },$$

the standard differential Lorentz generators satisfy

$$[M_{\mu \nu },T_{\rho }]=i({\eta }_{\nu \rho }{T}_{\mu }-{\eta }_{\mu \rho }{T}_{\nu }),[M_{\mu \nu },Q_{\rho }]=i({\eta }_{\nu \rho }{Q}_{\mu }-{\eta }_{\mu \rho }{Q}_{\nu }).$$

Together with the standard Lorentz relations for $M_{\mu \nu }$, these are exactly the defining relations (48)–(52), after representing the abstract central generator $\mathbf{1}$ by $\mathbb{I}$.

Therefore there is a surjective Lie algebra homomorphism

$$\varphi :\mathfrak{so}(3,1)⋉{\mathfrak{h}}_9⟶{\mathfrak{g}}_{\mathrm{gen}}$$

sending abstract generators to the corresponding realized operators.

It remains to prove injectivity. Let

$$L:=\sum _{\mu <\nu }{\alpha }^{\mu \nu }{M}_{\mu \nu }+{\displaystyle \sum _{\mu =0}^3}{a}^{\mu }{T}_{\mu }+{\displaystyle \sum _{\mu =0}^3}{b}^{\mu }{Q}_{\mu }+c\mathbb{I}$$

act as the zero operator on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$. In the X-polarized realization,

$$M_{\mu \nu }=i(x_{\mu }{\partial }_{x^{\nu }}-x_{\nu }{\partial }_{x^{\mu }}),T_{\mu }=-i{\partial }_{x^{\mu }},Q_{\mu }=x_{\mu }.$$

Hence L is a first-order differential operator whose principal symbol is

$${\sigma }_1\left(L\right)(x,\xi )=\sum _{\mu <\nu }{\alpha }^{\mu \nu }i(x_{\mu }{\xi }_{\nu }-x_{\nu }{\xi }_{\mu })-{\displaystyle \sum _{\mu =0}^3}i{a}^{\mu }{\xi }_{\mu }.$$

Because $L=0$, one has ${\sigma }_1\left(L\right)\equiv 0$. Setting $x=0$ gives

$${\displaystyle \sum _{\mu =0}^3}{a}^{\mu }{\xi }_{\mu }=0\forall \xi ,$$

hence $a^{\mu }=0$ for all $\mu$. Therefore

$$\sum _{\mu <\nu }{\alpha }^{\mu \nu }(x_{\mu }{\xi }_{\nu }-x_{\nu }{\xi }_{\mu })=0\forall x,\xi ,$$

so all ${\alpha }^{\mu \nu }=0$. Thus L reduces to the multiplication operator

$$L={\displaystyle \sum _{\mu =0}^3}{b}^{\mu }{x}_{\mu }+c\mathbb{I}.$$

Since $L=0$ on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, the affine polynomial

$${\displaystyle \sum _{\mu =0}^3}{b}^{\mu }{x}_{\mu }+c$$

vanishes identically, and therefore $b^{\mu }=0$ for all $\mu$ and $c=0$. Hence

$${\left\{M_{\mu \nu }\right\}}_{\mu <\nu },{\left\{T_{\mu }\right\}}_{\mu =0}^3,{\left\{Q_{\mu }\right\}}_{\mu =0}^3,\mathbb{I}$$

are linearly independent, so

$$dim{\mathfrak{g}}_{\mathrm{gen}}=15.$$

On the abstract side,

$$dim\left(\mathfrak{so}(3,1)⋉{\mathfrak{h}}_9\right)=6+9=15.$$

Therefore $\varphi$ is a surjective homomorphism between Lie algebras of the same finite dimension and is consequently an isomorphism. □

Remark 5 (Relation to nearby enlarged kinematics constructions). 

The Lorentz–Heisenberg algebra derived here should not be identified with either the canonical/quaplectic constructions, where a Weyl–Heisenberg normal subgroup is assumed from the outset [19,20,21], or with stabilized/deformed Poincaré–Heisenberg and Yang-type algebras obtained by deformation or stability principles [24,25,26,27]. The point proved here is more specific: within the Fourier-compatible realization class, the mixed Heisenberg bracket is forced by the coexistence of two Lorentz-covariant geometric translation sectors.

Theorem 1 identifies the operator-generated Lie algebra in the fixed realization class. The next theorem records the corresponding rigidity statement: in any faithful Fourier-compatible realization, the mixed bracket is necessarily one-dimensional, central, and proportional to the Minkowski pairing. Thus, up to normalization of the central generator, the resulting Lie algebra is again the Lorentz–Heisenberg algebra.

Theorem 2 (Rigidity of the mixed bracket in the Fourier-compatible class). 

Let $\mathfrak{g}$ be a Lie algebra admitting a faithful realization by operators on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$ satisfying the Fourier-compatible realization conditions above. Then the mixed bracket between the two translation modules is necessarily one-dimensional and central:

$$[V_X,V_K]=\mathbb{R}\mathbf{1}.$$

More precisely, after fixing a generator $\mathbf{1}$ of this central line, the mixed bracket is uniquely determined up to normalization, and may be written in the form

$$[T_{\mu },Q_{\nu }]=-i{\eta }_{\mu \nu }\mathbf{1}.$$

In particular, after fixing the generator $\mathbf{1}$ of the one-dimensional central line $[V_X,V_K]$, the Lie subalgebra generated by

$$\mathfrak{so}(3,1)\oplus V_X\oplus V_K$$

is isomorphic to the Lorentz–Heisenberg algebra.

Proof. 

By Proposition 7, every Fourier-compatible realization satisfies

$$[T_{\mu },Q_{\nu }]=-i{\eta }_{\mu \nu }\mathbb{I}$$

on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$. Since the realization is faithful, the operator $\mathbb{I}$ corresponds to a nonzero central element $\mathbf{1}\in \mathfrak{g}$. Hence all mixed commutators lie in the central line $\mathbb{R}\mathbf{1}$.

Because the Minkowski form ${\eta }_{\mu \nu }$ is nondegenerate, the span of the mixed commutators is exactly that line:

$$[V_X,V_K]=\mathbb{R}\mathbf{1}.$$

Choosing $\mathbf{1}$ as a generator of this one-dimensional central subspace fixes the bracket up to overall normalization, yielding

$$[T_{\mu },Q_{\nu }]=-i{\eta }_{\mu \nu }\mathbf{1}.$$

The final statement is now immediate: after choosing $\mathbf{1}$ as a generator of the one-dimensional central space $[V_X,V_K]$, the Lorentz action and the mixed bracket are exactly the defining relations (48)–(52); hence the subalgebra generated by $\mathfrak{so}(3,1)\oplus V_X\oplus V_K$ is isomorphic to the Lorentz–Heisenberg algebra. □

5. Casimir Structure and Central Extension

The purpose of this section is to identify the intrinsic central invariants of the Lorentz–Heisenberg algebra and then to convert those abstract invariants into the scalar data carried by irreducible representations.

The logic of the section is as follows. We first construct a Lorentz bivector $J_{\mu \nu }\in U\left(\mathfrak{g}\right)$ that commutes with the full Heisenberg translation sector. This produces the natural quadratic central elements $C_1$ and $C_2$. We then determine the full center of $U\left(\mathfrak{g}\right)$ in two stages: first by identifying the center of $U\left({\mathfrak{h}}_9\right)$, and then by determining the centralizer of $U\left({\mathfrak{h}}_9\right)$ inside $U\left(\mathfrak{g}\right)$. Finally, we pass from the abstract center to irreducible representations, where the central elements act by scalars. Those scalars are the invariants used in the orbit analysis of Section 6.

5.1. Translation-Commuting Lorentz Bivectors

We begin with the structural object that drives the entire section. The goal is to separate from the Lorentz generators a bivector-valued combination that still transforms covariantly under the Lorentz action but commutes with all Heisenberg translation generators. Once such an element is available, Lorentz-invariant quadratic contractions of it will automatically be candidates for central elements.

Define

$$J_{\mu \nu }:=\mathbf{1}{M}_{\mu \nu }+Q_{\mu }{T}_{\nu }-Q_{\nu }{T}_{\mu }\in U\left(\mathfrak{g}\right).$$

(53)

Thus $J_{\mu \nu }=-J_{\nu \mu }$. The next proposition shows that the family $\left\{J_{\mu \nu }\right\}$ has exactly the two properties needed later: it commutes with the translation sector and transforms as a Lorentz bivector.

Proposition 8 (Translation-commuting Lorentz bivectors). 

The elements $J_{\mu \nu }$ satisfy

$$\begin{array}{cc}\hfill [J_{\mu \nu },T_{\rho }]& =0,\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill [J_{\mu \nu },Q_{\rho }]& =0,\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill [M_{\alpha \beta },J_{\mu \nu }]& =i\left({\eta }_{\beta \mu }{J}_{\alpha \nu }-{\eta }_{\alpha \mu }{J}_{\beta \nu }-{\eta }_{\beta \nu }{J}_{\alpha \mu }+{\eta }_{\alpha \nu }{J}_{\beta \mu }\right),\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill [J_{\mu \nu },J_{\rho \sigma }]& =i\left({\eta }_{\nu \rho }{J}_{\mu \sigma }-{\eta }_{\mu \rho }{J}_{\nu \sigma }-{\eta }_{\nu \sigma }{J}_{\mu \rho }+{\eta }_{\mu \sigma }{J}_{\nu \rho }\right)\mathbf{1}.\hfill \end{array}$$

(57)

Proof. 

The proof is organized so that each step supplies one ingredient needed for the next.

• Step 1: isolate the Heisenberg-sector correction term. Set

  $$A_{\mu \nu }:=Q_{\mu }{T}_{\nu }-Q_{\nu }{T}_{\mu }.$$

  This is the non-Lorentz part of $J_{\mu \nu }$. Using (52), the centrality of $\mathbf{1}$, and the Leibniz rule, we compute its commutators with the translation generators:

  $$\begin{array}{cc}\hfill [A_{\mu \nu },T_{\rho }]& =[Q_{\mu },T_{\rho }]T_{\nu }-[Q_{\nu },T_{\rho }]T_{\mu }\hfill \\ \hfill & =i{\eta }_{\mu \rho }\mathbf{1}{T}_{\nu }-i{\eta }_{\nu \rho }\mathbf{1}{T}_{\mu }\hfill \\ \hfill & =-i({\eta }_{\nu \rho }{T}_{\mu }-{\eta }_{\mu \rho }{T}_{\nu })\mathbf{1},\hfill \\ \hfill [A_{\mu \nu },Q_{\rho }]& =Q_{\mu }[T_{\nu },Q_{\rho }]-Q_{\nu }[T_{\mu },Q_{\rho }]\hfill \\ \hfill & =-i{\eta }_{\nu \rho }{Q}_{\mu }\mathbf{1}+i{\eta }_{\mu \rho }{Q}_{\nu }\mathbf{1}\hfill \\ \hfill & =-i({\eta }_{\nu \rho }{Q}_{\mu }-{\eta }_{\mu \rho }{Q}_{\nu })\mathbf{1}.\hfill \end{array}$$
• Step 2: prove that $J_{\mu \nu }$ commutes with the translation sector. Since $J_{\mu \nu }=\mathbf{1}{M}_{\mu \nu }+A_{\mu \nu }$, the Lorentz-covariance relations (49) and (50), together with the identities from Step 1, give

  $$[J_{\mu \nu },T_{\rho }]=\mathbf{1}[M_{\mu \nu },T_{\rho }]+[A_{\mu \nu },T_{\rho }]=0,$$

  and similarly

  $$[J_{\mu \nu },Q_{\rho }]=\mathbf{1}[M_{\mu \nu },Q_{\rho }]+[A_{\mu \nu },Q_{\rho }]=0.$$

  This proves (54) and (55).
• Step 3: prove Lorentz covariance of $J_{\mu \nu }$. Applying the Leibniz rule once more and using (49) and (50), one finds

  $$[M_{\alpha \beta },A_{\mu \nu }]=i\left({\eta }_{\beta \mu }{A}_{\alpha \nu }-{\eta }_{\alpha \mu }{A}_{\beta \nu }-{\eta }_{\beta \nu }{A}_{\alpha \mu }+{\eta }_{\alpha \nu }{A}_{\beta \mu }\right).$$

  Combining this with $J_{\mu \nu }=\mathbf{1}{M}_{\mu \nu }+A_{\mu \nu }$ and the Lorentz commutator relation (48) gives (56).
• Step 4: compute the commutator algebra of the $J_{\mu \nu }$. By (54) and (55), each $J_{\mu \nu }$ commutes with the generators of ${\mathfrak{h}}_9$, hence with every element of $U\left({\mathfrak{h}}_9\right)$, in particular with every $A_{\rho \sigma }$. Therefore

  $$[J_{\mu \nu },J_{\rho \sigma }]=[J_{\mu \nu },\mathbf{1}{M}_{\rho \sigma }+A_{\rho \sigma }]=\mathbf{1}[J_{\mu \nu },M_{\rho \sigma }].$$

  Using (56) and antisymmetry of $J_{\mu \nu }$, this is exactly (57). □

The proposition isolates the key object of the section: $J_{\mu \nu }$ behaves as a Lorentz bivector but is invisible to the Heisenberg translation sector. The next corollary extracts from this bivector the quadratic central elements that will eventually generate the full center.

Corollary 3 (Quadratic central elements). 

The elements

$$\begin{array}{cc}\hfill C_0& :=\mathbf{1},\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill C_1& :=\frac{1}{2}{J}_{\mu \nu }{J}^{\mu \nu },\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill C_2& :=\frac{1}{4}{ϵ}_{\mu \nu \rho \sigma }{J}^{\mu \nu }{J}^{\rho \sigma }\hfill \end{array}$$

(60)

belong to $Z\left(U\right(\mathfrak{g}\left)\right)$.

Proof. 

By (54) and (55), the elements $C_0,C_1,C_2$ commute with $U\left({\mathfrak{h}}_9\right)$. By (56), the family $J_{\mu \nu }$ transforms as a Lorentz bivector under the adjoint action of $\mathfrak{so}(3,1)$. Since ${\eta }_{\mu \nu }$ and ${ϵ}_{\mu \nu \rho \sigma }$ are Lorentz-invariant tensors, the contractions defining $C_1$ and $C_2$ commute with all $M_{\alpha \beta }$. Hence $C_0,C_1,C_2$ commute with every generator of $\mathfrak{g}$, and therefore lie in $Z\left(U\right(\mathfrak{g}\left)\right)$. □

5.2. The Center of $U\left({\mathfrak{h}}_9\right)$ and the Centralizer of the Heisenberg Sector

We now move from explicit central elements to the structure of the full center. The strategy is to first understand the Heisenberg part on its own, and then determine which elements of $U\left(\mathfrak{g}\right)$ commute with that Heisenberg sector. The first step is therefore to compute $Z\left(U\left({\mathfrak{h}}_9\right)\right)$.

Proposition 9 (Center of the Heisenberg enveloping algebra). 

$$Z\left(U\left({\mathfrak{h}}_9\right)\right)=\mathbb{C}\left[\mathbf{1}\right].$$

Proof. 

The proof reduces the question from the enveloping algebra to its associated graded Poisson algebra.

Equip $U\left({\mathfrak{h}}_9\right)$ with its standard PBW filtration

$$F_{n}U\left({\mathfrak{h}}_9\right):=\mathrm{span}\{X_1\cdots X_r:r\le n,X_j\in {\mathfrak{h}}_9\},F_{-1}U\left({\mathfrak{h}}_9\right):=0.$$

Then

$$grU\left({\mathfrak{h}}_9\right)\cong S\left({\mathfrak{h}}_9\right)\cong \mathbb{C}[t_0,\dots ,t_3,q_0,\dots ,q_3,z],$$

where

$$t_{\mu }:=\sigma \left(T_{\mu }\right),q_{\mu }:=\sigma \left(Q_{\mu }\right),z:=\sigma \left(\mathbf{1}\right).$$

The commutator on $U\left({\mathfrak{h}}_9\right)$ induces on $grU\left({\mathfrak{h}}_9\right)$ the Poisson bracket

$$\{t_{\mu },q_{\nu }\}=-i{\eta }_{\mu \nu }z,\{t_{\mu },t_{\nu }\}=0,\{q_{\mu },q_{\nu }\}=0,\{z,·\}=0.$$

(61)

Let $X\in Z\left(U\left({\mathfrak{h}}_9\right)\right)$, and suppose $X\ne 0$. If n is its filtration degree, then

$$X\in F_{n}U\left({\mathfrak{h}}_9\right)\setminus F_{n-1}U\left({\mathfrak{h}}_9\right).$$

Because X commutes with every $T_{\mu }$ and $Q_{\mu }$, its principal symbol $\sigma \left(X\right)$ Poisson commutes with every $t_{\mu }$ and $q_{\mu }$. Using (61) and the derivation property of the Poisson bracket, one obtains

$$\begin{array}{cc}\hfill 0=\{\sigma \left(X\right),t_{\mu }\}& =iz{\displaystyle \sum _{\nu =0}^3}{\eta }_{\mu \nu }{\displaystyle \frac{\partial \sigma \left(X\right)}{\partial q_{\nu }}},\hfill \\ \hfill 0=\{\sigma \left(X\right),q_{\mu }\}& =-iz{\displaystyle \sum _{\nu =0}^3}{\eta }_{\nu \mu }{\displaystyle \frac{\partial \sigma \left(X\right)}{\partial t_{\nu }}}.\hfill \end{array}$$

Now $\mathbb{C}[t_{\mu },q_{\mu },z]$ is an integral domain, $z\ne 0$ in this ring, and $\left({\eta }_{\mu \nu }\right)$ is nondegenerate. Therefore

$${\displaystyle \frac{\partial \sigma \left(X\right)}{\partial q_{\nu }}}=0,{\displaystyle \frac{\partial \sigma \left(X\right)}{\partial t_{\nu }}}=0(\nu =0,\dots ,3),$$

so $\sigma \left(X\right)\in \mathbb{C}\left[z\right]$. Because $\sigma \left(X\right)$ is homogeneous, there exists $c\in \mathbb{C}$ such that

$$\sigma \left(X\right)=c{z}^n.$$

Set $p\left(\lambda \right):=c{\lambda }^n\in \mathbb{C}\left[\lambda \right]$. Then $p\left(\mathbf{1}\right)\in \mathbb{C}\left[\mathbf{1}\right]\subset Z\left(U\left({\mathfrak{h}}_9\right)\right)$ and

$$\sigma \left(p\left(\mathbf{1}\right)\right)=c{z}^n=\sigma \left(X\right).$$

Hence $X-p\left(\mathbf{1}\right)$ is again central but has a strictly smaller filtration degree. Induction on the filtration degree shows that $X\in \mathbb{C}\left[\mathbf{1}\right]$. The reverse inclusion is immediate because $\mathbf{1}$ is central. □

Remark 6 (Nondegenerate sector convention). 

All PBW, centralizer, and center computations below are understood in the fixed nondegenerate quotient

$$U_{{\hslash }_{\mathrm{eff}}}\left(\mathfrak{g}\right):=U\left({\mathfrak{g}}_{\mathbb{C}}\right)/\langle \mathbf{1}-{\hslash }_{\mathrm{eff}}{1}_U\rangle ,{\hslash }_{\mathrm{eff}}\ne 0,$$

so $\mathbf{1}$ is invertible there, with ${\mathbf{1}}^{-1}={\hslash }_{\mathrm{eff}}^{-1}{1}_U$.

The preceding proposition shows that the Heisenberg sector contributes no central elements beyond powers of the distinguished central generator $\mathbf{1}$. We now use this fact to identify the entire centralizer of $U\left({\mathfrak{h}}_9\right)$ inside $U\left(\mathfrak{g}\right)$.

Define

$${\mathcal{A}}_J:=\mathrm{the}\mathrm{subalgebra}\mathrm{of}U\left(\mathfrak{g}\right)\mathrm{generated}\mathrm{by}\mathbf{1}\mathrm{and}{J}_{\mu \nu }.$$

Proposition 10 (PBW basis adapted to $J_{\mu \nu }$ and centralizer of the Heisenberg sector). 

For an ordered basis ${\left\{L_a\right\}}_{a=1}^6$ of $\mathfrak{so}(3,1)$, let ${\left\{J_a\right\}}_{a=1}^6$ be the corresponding ordered family obtained from (53). Then

(i) 

For multi-indices

$$\alpha =({\alpha }_0,\dots ,{\alpha }_3),\beta =({\beta }_0,\dots ,{\beta }_3),\gamma =({\gamma }_1,\dots ,{\gamma }_6),$$

the ordered monomials

$$T^{\alpha }{Q}^{\beta }{\mathbf{1}}^m{J}^{\gamma },\alpha ,\beta \in {\mathbb{N}}_0^4,\gamma \in {\mathbb{N}}_0^6,m\in {\mathbb{N}}_0,$$

(62)

form a basis of $U\left(\mathfrak{g}\right)$.

(ii) 

One has

$${\mathrm{Cent}}_{U\left(\mathfrak{g}\right)}\left(U\left({\mathfrak{h}}_9\right)\right)={\mathcal{A}}_J.$$

Proof. 

The two parts are linked: part (i) provides the coordinate system in $U\left(\mathfrak{g}\right)$ needed to prove part (ii).

• Step 1: construct a PBW basis adapted to $J_{\mu \nu }$. Write

  $$L^{\gamma }:=L_1^{{\gamma }_1}\dots L_6^{{\gamma }_6}.$$

  By the PBW theorem [7,42], the ordered monomials

  $$T^{\alpha }{Q}^{\beta }{\mathbf{1}}^m{L}^{\gamma }$$

  form a basis of $U\left(\mathfrak{g}\right)$.

For each a, one has

$$J_a=\mathbf{1}{L}_a+u_a,u_a\in U\left({\mathfrak{h}}_9\right),$$

because $J_a$ is obtained from (53) by replacing the abstract Lorentz generator $M_{\mu \nu }$ with $L_a$. Hence

$$J^{\gamma }={\mathbf{1}}^{\left|\gamma \right|}{L}^{\gamma }+(\mathrm{a}\mathrm{linear}\mathrm{combination}\mathrm{of}\mathrm{terms}\mathrm{with}\mathrm{strictly}\mathrm{smaller}\mathrm{Lorentz}\mathrm{degree}),$$

where $\left|\gamma \right|:={\gamma }_1+\cdots +{\gamma }_6$. Therefore

$$T^{\alpha }{Q}^{\beta }{\mathbf{1}}^m{J}^{\gamma }=T^{\alpha }{Q}^{\beta }{\mathbf{1}}^{m+\left|\gamma \right|}{L}^{\gamma }+(\mathrm{terms}\mathrm{of}\mathrm{smaller}\mathrm{Lorentz}\mathrm{degree}).$$

Thus the change of ordered generators from $\left\{L_a\right\}$ to $\left\{J_a\right\}$ is triangular with diagonal entry 1, so the monomials (62) form a basis of $U\left(\mathfrak{g}\right)$.

• Step 2: compute the centralizer of $U\left({\mathfrak{h}}_9\right)$. By (54) and (55), every $J_{\mu \nu }$ commutes with $U\left({\mathfrak{h}}_9\right)$, and $\mathbf{1}$ is central. Therefore

  $${\mathcal{A}}_J\subseteq {\mathrm{Cent}}_{U\left(\mathfrak{g}\right)}\left(U\left({\mathfrak{h}}_9\right)\right).$$

Conversely, let

$$X\in {\mathrm{Cent}}_{U\left(\mathfrak{g}\right)}\left(U\left({\mathfrak{h}}_9\right)\right).$$

By part (i), X has a unique expansion

$$X=\sum _{\gamma }{a}_{\gamma }{J}^{\gamma },a_{\gamma }\in U\left({\mathfrak{h}}_9\right),$$

with only finitely many nonzero coefficients $a_{\gamma }$. For any $Y\in U\left({\mathfrak{h}}_9\right)$,

$$0=[Y,X]=\sum _{\gamma }[Y,a_{\gamma }]J^{\gamma },$$

because each $J^{\gamma }$ commutes with $U\left({\mathfrak{h}}_9\right)$. By the uniqueness of the basis expansion (62), one has

$$[Y,a_{\gamma }]=0\mathrm{for}\mathrm{every}\gamma \mathrm{and}\mathrm{every}Y\in U\left({\mathfrak{h}}_9\right).$$

Thus every coefficient $a_{\gamma }$ lies in $Z\left(U\left({\mathfrak{h}}_9\right)\right)$, so Proposition 9 gives

$$a_{\gamma }\in \mathbb{C}\left[\mathbf{1}\right].$$

Hence $X\in {\mathcal{A}}_J$, proving the reverse inclusion. □

At this stage, the problem of determining the full center of $U\left(\mathfrak{g}\right)$ has been reduced to the smaller algebra ${\mathcal{A}}_J$. The remaining step is to compute the center of ${\mathcal{A}}_J$ itself.

5.3. The Full Center Theorem

The final algebraic ingredient is to identify ${\mathcal{A}}_J$ with a Rees algebra built from $U\left(\mathfrak{so}\right(3,1\left)\right)$. This converts the center computation into a filtered version of the familiar center of the Lorentz enveloping algebra.

Theorem 3 (Full center theorem). 

With $C_0,C_1,C_2$ defined by (58)–(60), one has

$$Z\left(U\left(\mathfrak{g}\right)\right)=\mathbb{C}[C_0,C_1,C_2]=\mathbb{C}[\mathbf{1},C_1,C_2].$$

(63)

In particular, $C_0,C_1,C_2$ are algebraically independent and generate the full center of $U\left(\mathfrak{g}\right)$.

Proof. 

The proof has three steps: reduction to ${\mathcal{A}}_J$, identification of ${\mathcal{A}}_J$ with a Rees algebra, and computation of the center of that Rees algebra.

• Step 1: reduce from $U\left(\mathfrak{g}\right)$ to ${\mathcal{A}}_J$. Let $X\in Z\left(U\right(\mathfrak{g}\left)\right)$. Since X commutes with $U\left({\mathfrak{h}}_9\right)$, Proposition 10 implies

  $$X\in {\mathcal{A}}_J.$$

  Therefore

  $$Z\left(U\left(\mathfrak{g}\right)\right)=Z\left({\mathcal{A}}_J\right).$$
• Step 2: realize ${\mathcal{A}}_J$ as a Rees algebra. Let $\mathfrak{s}$ be an abstract copy of $\mathfrak{so}(3,1)$ with ordered basis ${\left\{L_a\right\}}_{a=1}^6$. Equip $U\left(\mathfrak{s}\right)$ with its standard PBW filtration $F_{•}U\left(\mathfrak{s}\right)$, and define

  $$\mathcal{R}\left(\mathfrak{s}\right):=\underset{n\ge 0}{⨁}{F}_{n}U\left(\mathfrak{s}\right)t^n\subset U\left(\mathfrak{s}\right)\left[t\right],$$

  where t is a central indeterminate.

The assignment

$$t⟼\mathbf{1},t{L}_{\mu \nu }⟼J_{\mu \nu }$$

extends to a surjective algebra homomorphism

$$\Phi :\mathcal{R}\left(\mathfrak{s}\right)\to {\mathcal{A}}_J.$$

To prove injectivity, note that the elements

$$t^{m+\left|\gamma \right|}{L}^{\gamma },m\in {\mathbb{N}}_0,\gamma \in {\mathbb{N}}_0^6,$$

form a basis of $\mathcal{R}\left(\mathfrak{s}\right)$, whereas Proposition 10 shows that the elements

$${\mathbf{1}}^m{J}^{\gamma }$$

form a basis of ${\mathcal{A}}_J$. Since

$$\Phi \left(t^{m+\left|\gamma \right|}{L}^{\gamma }\right)={\mathbf{1}}^m{J}^{\gamma },$$

the map $\Phi$ is bijective. Hence

$${\mathcal{A}}_J\cong \mathcal{R}\left(\mathfrak{s}\right).$$

• Step 3: compute the center of the Rees algebra. Because t is central, localization at the powers of t gives

  $$\mathcal{R}\left(\mathfrak{s}\right)\left[t^{-1}\right]\cong U\left(\mathfrak{s}\right)[t,t^{-1}].$$

  Therefore

  $$Z\left(\mathcal{R}\left(\mathfrak{s}\right)\left[t^{-1}\right]\right)=Z\left(U\left(\mathfrak{s}\right)\right)[t,t^{-1}].$$

For $\mathfrak{s}\cong \mathfrak{so}(3,1)$, the standard Lorentz Casimirs

$${\Omega }_1:=\frac{1}{2}{L}_{\mu \nu }{L}^{\mu \nu },{\Omega }_2:=\frac{1}{4}{ϵ}_{\mu \nu \rho \sigma }{L}^{\mu \nu }{L}^{\rho \sigma }$$

generate the center of $U\left(\mathfrak{s}\right)$; see, for example, [7,42]. Hence

$$Z\left(U\left(\mathfrak{s}\right)\right)=\mathbb{C}[{\Omega }_1,{\Omega }_2].$$

Hence

$$Z\left(\mathcal{R}\left(\mathfrak{s}\right)\left[t^{-1}\right]\right)=\mathbb{C}[t,t^{-1},{\Omega }_1,{\Omega }_2].$$

Let $X\in Z\left(\mathcal{R}\right(\mathfrak{s}\left)\right)$. Viewing X inside the localized algebra, write

$$X=\sum _{b,c\ge 0}{f}_{bc}\left(t\right){\Omega }_1^b{\Omega }_2^c,f_{bc}\left(t\right)\in \mathbb{C}[t,t^{-1}],$$

with only finitely many nonzero coefficients. Expand

$$f_{bc}\left(t\right)=\sum _{a\in \mathbb{Z}}{\lambda }_{a,b,c}{t}^a.$$

Since $\mathbb{C}[t,t^{-1},{\Omega }_1,{\Omega }_2]$ is a polynomial algebra in independent variables, the monomials $t^a{\Omega }_1^b{\Omega }_2^c$ are linearly independent. Therefore, because $X\in \mathcal{R}\left(\mathfrak{s}\right)\subset U\left(\mathfrak{s}\right)[t,t^{-1}]$, each monomial with ${\lambda }_{a,b,c}\ne 0$ must itself lie in the Rees algebra.

Now ${\Omega }_1$ and ${\Omega }_2$ each have PBW degree 2, so ${\Omega }_1^b{\Omega }_2^c$ has PBW degree $2b+2c$. Hence

$$t^a{\Omega }_1^b{\Omega }_2^c\in \mathcal{R}\left(\mathfrak{s}\right)⟺a\ge 2b+2c.$$

Therefore

$$f_{bc}\left(t\right)\in t^{2b+2c}\mathbb{C}\left[t\right]\mathrm{for}\mathrm{every}(b,c),$$

and so

$$Z\left(\mathcal{R}\left(\mathfrak{s}\right)\right)=\mathbb{C}[t,t^2{\Omega }_1,t^2{\Omega }_2].$$

Transporting this description through $\Phi$ gives

$$\Phi \left(t\right)=\mathbf{1}=C_0,\Phi \left(t^2{\Omega }_1\right)=\frac{1}{2}{J}_{\mu \nu }{J}^{\mu \nu }=C_1,\Phi \left(t^2{\Omega }_2\right)=\frac{1}{4}{ϵ}_{\mu \nu \rho \sigma }{J}^{\mu \nu }{J}^{\rho \sigma }=C_2.$$

Thus

$$Z\left(U\left(\mathfrak{g}\right)\right)=\mathbb{C}[\mathbf{1},C_1,C_2].$$

It remains to prove algebraic independence. Suppose

$$P(C_0,C_1,C_2)=0\mathrm{for}\mathrm{some}P\in \mathbb{C}[u,v,w].$$

Applying ${\Phi }^{-1}$ yields

$$P(t,t^2{\Omega }_1,t^2{\Omega }_2)=0\mathrm{in}\mathcal{R}\left(\mathfrak{s}\right).$$

After localizing at t, this becomes an identity in $\mathbb{C}[t,t^{-1},{\Omega }_1,{\Omega }_2]$. Distinct monomials $u^a{v}^b{w}^c$ map to distinct monomials $t^{a+2b+2c}{\Omega }_1^b{\Omega }_2^c$, which are linearly independent. Hence $P=0$. Therefore $C_0,C_1,C_2$ are algebraically independent. □

5.4. Irreducible Sectors and Normalized Casimir Scalars

The algebraic part of the section is now complete: we know the full center of $U\left(\mathfrak{g}\right)$. We next translate this abstract result into representation-theoretic data. In an irreducible representation, every central element acts by a scalar, and those scalars are exactly the invariants needed in the next section.

Proposition 11 (Central character and Casimir scalars). 

Let $\rho :U\left(\mathfrak{g}\right)\to End\left(\mathcal{H}\right)$ be an irreducible representation. Then

$$\rho \left(\mathbf{1}\right)={\hslash }_{\mathrm{eff}}\mathbb{I}$$

(64)

for a unique scalar ${\hslash }_{\mathrm{eff}}\in \mathbb{C}$. If ρ is unitary, then ${\hslash }_{\mathrm{eff}}\in \mathbb{R}$. Moreover, there exist unique scalars ${\lambda }_1,{\lambda }_2\in \mathbb{C}$ such that

$$\rho \left(C_1\right)={\lambda }_1\mathbb{I},\rho \left(C_2\right)={\lambda }_2\mathbb{I}.$$

(65)

If ${\hslash }_{\mathrm{eff}}\ne 0$, then there exist unique scalars $c_1,c_2\in \mathbb{C}$ such that

$$\rho \left(C_0\right)={\hslash }_{\mathrm{eff}}\mathbb{I},\rho \left(C_1\right)={\hslash }_{\mathrm{eff}}^2{c}_1\mathbb{I},\rho \left(C_2\right)={\hslash }_{\mathrm{eff}}^2{c}_2\mathbb{I}.$$

(66)

Proof. 

Since $\mathbf{1}$ is central, Schur’s lemma [6,43] gives (64). In the unitary case, ${\hslash }_{\mathrm{eff}}\in \mathbb{R}$ because

$$[\rho \left(T_{\mu }\right),\rho \left(Q_{\nu }\right)]=-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I}$$

is skew-adjoint.

By Theorem 3, the elements $C_1$ and $C_2$ are central. Hence Schur’s lemma again implies that $\rho \left(C_1\right)$ and $\rho \left(C_2\right)$ are scalar operators, proving (65). If ${\hslash }_{\mathrm{eff}}\ne 0$, writing those scalars as ${\hslash }_{\mathrm{eff}}^2{c}_1$ and ${\hslash }_{\mathrm{eff}}^2{c}_2$ defines $c_1$ and $c_2$ uniquely, which yields (66). □

Remark 7 (Normalized commuting Lorentz generators). 

Suppose now that ρ is irreducible and nondegenerate, i.e., ${\hslash }_{\mathrm{eff}}\ne 0$. Then the normalized operators

$${\tilde{J}}_{\mu \nu }:={\hslash }_{\mathrm{eff}}^{-1}\rho \left(J_{\mu \nu }\right)$$

(67)

satisfy

$$[{\tilde{J}}_{\mu \nu },\rho \left(T_{\rho }\right)]=0,[{\tilde{J}}_{\mu \nu },\rho \left(Q_{\rho }\right)]=0,$$

and

$$[{\tilde{J}}_{\mu \nu },{\tilde{J}}_{\rho \sigma }]=i\left({\eta }_{\nu \rho }{\tilde{J}}_{\mu \sigma }-{\eta }_{\mu \rho }{\tilde{J}}_{\nu \sigma }-{\eta }_{\nu \sigma }{\tilde{J}}_{\mu \rho }+{\eta }_{\mu \sigma }{\tilde{J}}_{\nu \rho }\right).$$

Thus $\left\{{\tilde{J}}_{\mu \nu }\right\}$ furnish a genuine Lorentz copy in the representation. In the remainder of the manuscript, we retain the notation ${\tilde{J}}_{\mu \nu }$ for these normalized operators and do not relabel them as $J_{\mu \nu }$. The scalar orbit invariants are the numbers $c_1,c_2$, whereas $C_1,C_2$ remain the abstract central elements of $U\left(\mathfrak{g}\right)$.

The interpretation is now immediate. In every irreducible representation, the central generator $\mathbf{1}$ acts by the scalar ${\hslash }_{\mathrm{eff}}$. Thus the effective Planck scale arises here as a structural consequence of the Lorentz–Heisenberg algebra rather than as an external deformation parameter. The degenerate sector ${\hslash }_{\mathrm{eff}}=0$ is therefore distinguished already at the level of the central character: in that sector, the mixed Heisenberg commutator acts trivially, so the nondegenerate quantum scale carried by ${\hslash }_{\mathrm{eff}}\ne 0$ disappears. In this precise sense, the sector ${\hslash }_{\mathrm{eff}}=0$ may be regarded as the classical-scale analogue of the theory. Correspondingly, it does not admit the normalization (67), it does not carry the normalized Lorentz bivector data ${\tilde{J}}_{\mu \nu }$, and it must be treated separately from the orbit-theoretic analysis of Section 6, which is intrinsically restricted to the nondegenerate case ${\hslash }_{\mathrm{eff}}\ne 0$ (see

Table 1. Central invariants of the Lorentz–Heisenberg algebra in a nondegenerate irreducible sector ${\hslash }_{\mathrm{eff}}\ne 0$.

Central Element	Definition in $\mathit{U}\left(\mathfrak{g}\right)$	Action in a Nondegenerate Irreducible Sector
$C_0$	$\mathbf{1}$	$\rho \left(C_0\right)={\hslash }_{\mathrm{eff}}\mathbb{I}$
$C_1$	$\frac{1}{2}{J}_{\mu \nu }{J}^{\mu \nu }$	$\rho \left(C_1\right)={\hslash }_{\mathrm{eff}}^2{c}_1\mathbb{I}$
$C_2$	$\frac{1}{4}{ϵ}_{\mu \nu \rho \sigma }{J}^{\mu \nu }{J}^{\rho \sigma }$	$\rho \left(C_2\right)={\hslash }_{\mathrm{eff}}^2{c}_2\mathbb{I}$

).

6. Orbit Stratification by the Normalized Casimir Scalars

The purpose of this section is to convert the normalized Casimir data obtained in Section 5 into an explicit orbit classification for the normalized commuting Lorentz bivector. The argument proceeds in four steps.

First, we fix the nondegenerate irreducible sector and identify the normalized orbit variable $\tilde{J}$. Second, we rewrite its two Lorentz invariants in terms of the boost–rotation decomposition and the associated complex vector $z\in {\mathbb{C}}^3$. Third, we classify the ${\mathrm{SO}}^{+}(3,1)$ orbits and compute the connected stabilizers of the canonical representatives. Finally, we compare the resulting bivector-orbit picture with the standard vector-orbit classification of Wigner.

Throughout this section, fix an irreducible representation with

$$\rho \left(C_0\right)={\hslash }_{\mathrm{eff}}\mathbb{I},{\hslash }_{\mathrm{eff}}\ne 0,$$

and let ${\tilde{J}}_{\mu \nu }$ denote the normalized operators defined in (67). Thus, on the given irreducible sector,

$$\frac{1}{2}{\tilde{J}}_{\mu \nu }{\tilde{J}}^{\mu \nu }=c_1\mathbb{I},\frac{1}{4}{ϵ}_{\mu \nu \rho \sigma }{\tilde{J}}^{\mu \nu }{\tilde{J}}^{\rho \sigma }=c_2\mathbb{I},$$

(68)

equivalently,

$$\rho \left(C_1\right)={\hslash }_{\mathrm{eff}}^2{c}_1\mathbb{I},\rho \left(C_2\right)={\hslash }_{\mathrm{eff}}^2{c}_2\mathbb{I}.$$

If ${\hslash }_{\mathrm{eff}}=0$, the normalization of Remark 7 is unavailable, so the bivector-orbit classification below is not defined.

We identify the normalized Lorentz copy with its dual by means of the invariant bilinear form

$$\langle X,Y\rangle :=\frac{1}{2}{X}_{\mu \nu }{Y}^{\mu \nu }.$$

Hence adjoint and coadjoint orbit classifications coincide. When convenient, we write

$$\tilde{J}\in {\Lambda }^2\left({\mathbb{R}}^{3,1}\right)$$

for the bivector whose components in the standard coordinate basis are ${\tilde{J}}_{\mu \nu }$.

6.1. Boost–Rotation Decomposition and the Complex Orbit Variable

We first rewrite the normalized bivector in coordinates adapted to the Lorentz action. Let

$$B_i:={\tilde{J}}_{0i},R_1:={\tilde{J}}_{23},R_2:={\tilde{J}}_{31},R_3:={\tilde{J}}_{12},$$

so that every normalized bivector may be written uniquely as

$$\tilde{J}=\mathbf{b}·B+\mathbf{r}·R,\mathbf{b},\mathbf{r}\in {\mathbb{R}}^3.$$

(69)

A direct computation gives

$$c_1=\mathbf{r}·\mathbf{r}-\mathbf{b}·\mathbf{b},c_2=-2\mathbf{b}·\mathbf{r}.$$

(70)

The next step is to package the pair $(c_1,c_2)$ into a single complex quadratic invariant. Define

$$z:=\mathbf{r}-i\mathbf{b}\in {\mathbb{C}}^3.$$

(71)

In the fixed nondegenerate irreducible sector under consideration, the normalized generators ${\tilde{J}}_{\mu \nu }$ furnish a real copy of $\mathfrak{so}(3,1)$. After complexification, set

$$X_i:=\frac{1}{2}(R_i+i{B}_i),Y_i:=\frac{1}{2}(R_i-i{B}_i).$$

Then

$$[X_i,X_j]=i{ϵ}_{ijk}{X}_k,[Y_i,Y_j]=i{ϵ}_{ijk}{Y}_k,[X_i,Y_j]=0.$$

Lemma 2 (Self-dual identification and induced $\mathrm{SO}(3,\mathbb{C})$ action). 

Let

$${\Lambda }_{+}^2:={\mathrm{span}}_{\mathbb{C}}\{X_1,X_2,X_3\}\subset {\Lambda }^2{\left({\mathbb{R}}^{3,1}\right)}_{\mathbb{C}}.$$

Then ${\Lambda }_{+}^2$ is invariant under the complexified adjoint action of ${\mathrm{SO}}^{+}(3,1)$, and the coefficient map

$${\Lambda }_{+}^2\to {\mathbb{C}}^3,z_1{X}_1+z_2{X}_2+z_3{X}_3\mapsto (z_1,z_2,z_3),$$

identifies that action with the standard complex orthogonal action of $\mathrm{SO}(3,\mathbb{C})$. In particular, if

$$\tilde{J}=z·X+\overline{z}·Y,$$

then under the proper orthochronous Lorentz action, the self-dual coefficient vector transforms by

$$z⟼gz\mathit{for}\mathit{some}g\in \mathrm{SO}(3,\mathbb{C}),$$

and therefore $z·z$ is Lorentz-invariant.

Proof. 

The relations $[X_i,X_j]=i{ϵ}_{ijk}{X}_k$ show that ${\Lambda }_{+}^2$ is a 3-dimensional complex Lie algebra isomorphic to $\mathfrak{so}(3,\mathbb{C})$. In the basis $(X_1,X_2,X_3)$, the restriction of the complexified invariant bilinear form is the standard complex quadratic form on ${\mathbb{C}}^3$. Hence the induced action on coefficients is the standard orthogonal action of $\mathrm{SO}(3,\mathbb{C})$. The final statement follows immediately. □

Thus the complex spans of $\left\{X_i\right\}$ and $\left\{Y_i\right\}$ give the two chiral summands of

$$\mathfrak{so}{(3,1)}_{\mathbb{C}}\cong {\mathfrak{sl}}_2\left(\mathbb{C}\right)\oplus {\mathfrak{sl}}_2\left(\mathbb{C}\right).$$

Since

$$R_i=X_i+Y_i,B_i=-i(X_i-Y_i),$$

the bivector decomposition (69) becomes

$$\tilde{J}=(\mathbf{r}-i\mathbf{b})·X+(\mathbf{r}+i\mathbf{b})·Y=z·X+\overline{z}·Y.$$

By Lemma 2, z is the coefficient vector of the self-dual part of $\tilde{J}$, and under the proper orthochronous Lorentz action, it transforms through the standard complex orthogonal action of $\mathrm{SO}(3,\mathbb{C})$. In particular,

$$z·z=(\mathbf{r}-i\mathbf{b})·(\mathbf{r}-i\mathbf{b})=c_1+i{c}_2$$

(72)

is Lorentz-invariant.

Before classifying all orbits, we isolate the simple locus, since it is exactly the locus on which the second invariant vanishes.

Proposition 12 (Simple bivectors). 

A normalized bivector $\tilde{J}\in {\Lambda }^2\left({\mathbb{R}}^{3,1}\right)$ is simple; i.e., there exist vectors $n,v\in {\mathbb{R}}^{3,1}$ such that

$$\tilde{J}=n\wedge v,$$

if and only if $c_2=0$.

Proof. 

The purpose of the proof is to translate decomposability of a bivector into the second Casimir invariant.

A bivector in four dimensions is simple if and only if it is decomposable, equivalently if and only if

$$\tilde{J}\wedge \tilde{J}=0.$$

Let $(e_0,e_1,e_2,e_3)$ be the standard oriented basis of ${\mathbb{R}}^{3,1}$, with ${ϵ}_{0123}=+1$. Writing

$$\tilde{J}=\frac{1}{2}{\tilde{J}}_{\mu \nu }{e}^{\mu }\wedge e^{\nu },$$

one has the exact identity in ${\Lambda }^4\left({\mathbb{R}}^{3,1}\right)$,

$$\tilde{J}\wedge \tilde{J}={\displaystyle \frac{1}{4}}{ϵ}_{\mu \nu \rho \sigma }{\tilde{J}}^{\mu \nu }{\tilde{J}}^{\rho \sigma }{e}^0\wedge e^1\wedge e^2\wedge e^3.$$

By (68), the coefficient is precisely $c_2$. Hence

$$\tilde{J}\wedge \tilde{J}=c_2{e}^0\wedge e^1\wedge e^2\wedge e^3.$$

Since $e^0\wedge e^1\wedge e^2\wedge e^3\ne 0$, it follows that

$$\tilde{J}\wedge \tilde{J}=0⟺c_2=0.$$

This is exactly the desired criterion. □

6.2. Orbit Classification

We now classify the Lorentz orbits themselves. Equation (72) shows that the pair $(c_1,c_2)$ is the natural invariant data. The only subtlety is the singular locus $c_1=c_2=0$, where two distinct orbits occur.

Theorem 4 (Orbit classification in the nondegenerate sector). 

Fix a nondegenerate irreducible sector ${\hslash }_{\mathrm{eff}}\ne 0$, and let $\tilde{J}$ denote the corresponding normalized commuting Lorentz bivector. Then the ${\mathrm{SO}}^{+}(3,1)$ orbit of $\tilde{J}$ is classified by the pair $(c_1,c_2)$, except on the common zero locus $c_1=c_2=0$, where one must distinguish the zero orbit from the unique nonzero null orbit.

More precisely, exactly one of the following occurs.

(i) 

Zero orbit.

$$\tilde{J}=0,c_1=0,c_2=0.$$

(ii) 

Regular orbit.

$$(c_1,c_2)\ne (0,0).$$

Then $\tilde{J}$ is Lorentz-conjugate to a unique orbit of the form

$${\tilde{J}}^{\left(0\right)}(\theta ,\psi )=\theta {\tilde{J}}_{01}+\psi {\tilde{J}}_{23},(\theta ,\psi )\ne (0,0),$$

(73)

with

$$c_1=-{\theta }^2+{\psi }^2,c_2=-2\theta \psi .$$

(74)

Equivalently, if

$$r:=\sqrt{c_1^2+c_2^2},$$

one may choose $\theta ,\psi \in \mathbb{R}$ satisfying

$${\theta }^2={\displaystyle \frac{r-c_1}{2}},{\psi }^2={\displaystyle \frac{r+c_1}{2}},\theta \psi =-{\displaystyle \frac{c_2}{2}}.$$

(75)

The ambiguity $(\theta ,\psi )\mapsto (-\theta ,-\psi )$ does not produce a distinct orbit.

(iii) 

Nonzero null orbit.

$$\tilde{J}\ne 0,c_1=0,c_2=0.$$

Then $\tilde{J}$ is Lorentz-conjugate to

$$N:={\tilde{J}}_{01}-{\tilde{J}}_{13}=B_1+R_2.$$

(76)

This orbit is null in the sense of (70); under the real Lie algebra identification

$$\mathfrak{so}(3,1)\cong {\mathfrak{sl}}_2{\left(\mathbb{C}\right)}_{\mathbb{R}},$$

it is the unique nonzero nilpotent orbit.

Proof. 

The proof has three steps: first the regular case, then the singular locus $c_1=c_2=0$, and finally the nilpotent interpretation of the nonzero null orbit.

• Step 1: classification of the regular orbits. Set

$$\alpha :=c_1+i{c}_2=z·z.$$

Assume first that $\alpha \ne 0$. Then $z\ne 0$. Choose a square root ${\alpha }^{1/2}$ of $\alpha$, and define

$$u:={\alpha }^{-1/2}z.$$

Then

$$u·u={\alpha }^{-1}(z·z)=1.$$

Thus u is a unit vector for the nondegenerate complex quadratic form on ${\mathbb{C}}^3$. By the complex Gram–Schmidt process, u extends to an orthonormal basis of ${\mathbb{C}}^3$. The matrix with these basis vectors as columns lies in $\mathrm{SO}(3,\mathbb{C})$ and sends z to

$$({\alpha }^{1/2},0,0).$$

Since the Lorentz action on $\tilde{J}$ induces the standard $\mathrm{SO}(3,\mathbb{C})$ action on z, this gives a Lorentz-conjugate representative with

$$z=({\alpha }^{1/2},0,0).$$

Write

$${\alpha }^{1/2}=\psi -i\theta ,\theta ,\psi \in \mathbb{R}.$$

Then, by (71),

$$\mathbf{r}=(\psi ,0,0),\mathbf{b}=(\theta ,0,0),$$

so the corresponding bivector is

$${\tilde{J}}^{\left(0\right)}(\theta ,\psi )=\theta {\tilde{J}}_{01}+\psi {\tilde{J}}_{23}.$$

Equation (74) follows immediately from (72). Conversely, (75) is obtained by solving (74) for ${\theta }^2$, ${\psi }^2$, and $\theta \psi$.

It remains to identify the residual ambiguity. Replacing ${\alpha }^{1/2}$ by $-{\alpha }^{1/2}$ replaces $(\theta ,\psi )$ by $(-\theta ,-\psi )$. This does not produce a distinct orbit because the rotation $exp\left(\pi R_2\right)\in {\mathrm{SO}}^{+}(3,1)$ sends

$${\tilde{J}}_{01}\mapsto -{\tilde{J}}_{01},{\tilde{J}}_{23}\mapsto -{\tilde{J}}_{23}.$$

Hence $(c_1,c_2)\ne (0,0)$ determines a unique regular orbit.

• Step 2: classification on the singular locus $c_1=c_2=0$. Assume now that $\alpha =0$. If $z=0$, then

$$\mathbf{r}=0,\mathbf{b}=0,$$

so $\tilde{J}=0$, giving the zero orbit. Suppose instead that $z\ne 0$. Then z lies on the nonzero null cone of the quadratic form

$$q\left(z\right):=z·z.$$

By Witt’s theorem for the nondegenerate complex quadratic space $({\mathbb{C}}^3,q)$, the orthogonal group $\mathrm{O}(3,\mathbb{C})$ acts transitively on nonzero isotropic vectors; see [44]. Composing, if necessary, with the determinant-1 element

$$\mathrm{diag}(-1,-1,1)\in \mathrm{SO}(3,\mathbb{C}),$$

one sees that $\mathrm{SO}(3,\mathbb{C})$ acts transitively on the nonzero null cone as well. Hence every nonzero null vector is $\mathrm{SO}(3,\mathbb{C})$-conjugate to

$$(-i,1,0).$$

By (71), this corresponds to

$$\mathbf{b}=(1,0,0),\mathbf{r}=(0,1,0),$$

hence to the representative

$$N=B_1+R_2={\tilde{J}}_{01}-{\tilde{J}}_{13}.$$

Thus there is exactly one nonzero orbit in the common zero locus

$$c_1=c_2=0.$$

• Step 3: identification of the nonzero null orbit as nilpotent. Under the real Lie-algebra isomorphism

$$\mathfrak{so}(3,1)\cong {\mathfrak{sl}}_2{\left(\mathbb{C}\right)}_{\mathbb{R}},$$

the representative N corresponds to a nonzero nilpotent element of ${\mathfrak{sl}}_2\left(\mathbb{C}\right)$. Every nonzero nilpotent element of ${\mathfrak{sl}}_2\left(\mathbb{C}\right)$ is conjugate to

$$\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),$$

because a nonzero nilpotent matrix has rank 1, so one may choose v with $Av\ne 0$, and then the basis $(Av,v)$ puts A into Jordan form. Therefore ${\mathfrak{sl}}_2\left(\mathbb{C}\right)$ has exactly one nonzero nilpotent adjoint orbit, and the nonzero null orbit above is precisely that orbit. □

6.3. Connected Stabilizers

The next step is to determine the connected stabilizers of the canonical orbit representatives. This identifies the analogue of the little-group data for the bivector classification.

Proposition 13 (Connected stabilizers of the canonical orbit representatives). 

Let $H_{\tilde{J}}\subset {\mathrm{SO}}^{+}(3,1)$ denote the stabilizer of the normalized bivector $\tilde{J}$ under the adjoint action, and let $H_{\tilde{J}}^{\circ }$ denote its identity component. Then the connected stabilizers in the three orbit sectors are as follows.

(i) 

Regular sector. If

$$\tilde{J}={\tilde{J}}^{\left(0\right)}(\theta ,\psi )=\theta {\tilde{J}}_{01}+\psi {\tilde{J}}_{23},(\theta ,\psi )\ne (0,0),$$

then

$$Lie\left(H_{\tilde{J}}\right)=\mathrm{span}\{{\tilde{J}}_{01},{\tilde{J}}_{23}\},H_{\tilde{J}}^{\circ }\cong {\mathrm{SO}}^{+}(1,1)\times \mathrm{SO}\left(2\right).$$

(ii) 

Nonzero null sector. If

$$\tilde{J}=N:={\tilde{J}}_{01}-{\tilde{J}}_{13},$$

then, writing $H_N:=H_{\tilde{J}}$,

$$Lie\left(H_N\right)=\mathrm{span}\{N,\tilde{N}\},\tilde{N}:={\tilde{J}}_{23}-{\tilde{J}}_{02},$$

with

$$[N,\tilde{N}]=0,H_N^{\circ }\cong {\mathbb{R}}^2.$$

(iii) 

Zero sector. If $\tilde{J}=0$, then

$$H_{\tilde{J}}={\mathrm{SO}}^{+}(3,1),H_{\tilde{J}}^{\circ }={\mathrm{SO}}^{+}(3,1).$$

Proof. 

We first compute the stabilizer Lie algebra and then integrate it to the identity component of the stabilizer subgroup.

Let $e_1,e_2,e_3$ denote the standard basis of ${\mathbb{R}}^3$. For $\alpha ,\beta ,u,v\in {\mathbb{R}}^3$, write

$$\alpha ·R:={\displaystyle \sum _{i=1}^3}{\alpha }_i{R}_i,\beta ·B:={\displaystyle \sum _{i=1}^3}{\beta }_i{B}_i,$$

and similarly for $u·B$ and $v·R$. Set

$$X=\alpha ·R+\beta ·B,Y=v·R+u·B.$$

Using the Lorentz commutation relations, one computes

$$[X,Y]=i\left((\alpha \times v-\beta \times u)·R+(\alpha \times u+\beta \times v)·B\right).$$

(77)

Step 1: regular representative. Take

$$Y=\psi R_1+\theta B_1.$$

Then $[X,Y]=0$ is equivalent, by (77), to

$$(\psi \alpha -\theta \beta )\times e_1=0,(\theta \alpha +\psi \beta )\times e_1=0.$$

Since $(\theta ,\psi )\ne (0,0)$, the matrix

$$\left(\begin{array}{cc}\psi & -\theta \\ \theta & \psi \end{array}\right)$$

is invertible. Therefore

$$\alpha \times e_1=0,\beta \times e_1=0,$$

so $\alpha$ and $\beta$ are both parallel to $e_1$. Hence

$$Lie\left(H_{\tilde{J}}\right)=\mathrm{span}\{R_1,B_1\}=\mathrm{span}\{{\tilde{J}}_{23},{\tilde{J}}_{01}\}.$$

To identify the connected stabilizer, note that $R_1$ and $B_1$ commute, and the corresponding one-parameter subgroups

$$exp\left(\mathbb{R}{B}_1\right)\subset {\mathrm{SO}}^{+}(3,1),exp\left(\mathbb{R}{R}_1\right)\subset {\mathrm{SO}}^{+}(3,1)$$

act on the orthogonal planes $(x^0,x^1)$ and $(x^2,x^3)$, respectively. Their product

$$K_{\tilde{J}}:=exp\left(\mathbb{R}{B}_1\right)exp\left(\mathbb{R}{R}_1\right)$$

is therefore a connected Lie subgroup isomorphic to

$${\mathrm{SO}}^{+}(1,1)\times \mathrm{SO}\left(2\right),$$

with

$$Lie\left(K_{\tilde{J}}\right)=Lie\left(H_{\tilde{J}}\right).$$

Since $K_{\tilde{J}}\subseteq H_{\tilde{J}}$, one has

$$K_{\tilde{J}}\subseteq H_{\tilde{J}}^{\circ }.$$

Conversely, $K_{\tilde{J}}$ and $H_{\tilde{J}}^{\circ }$ are connected Lie subgroups of the same matrix Lie group with the same Lie algebra. Hence they coincide. Therefore

$$H_{\tilde{J}}^{\circ }=K_{\tilde{J}}\cong {\mathrm{SO}}^{+}(1,1)\times \mathrm{SO}\left(2\right).$$

Step 2: null representative. Take

$$N=B_1+R_2.$$

Then $[X,N]=0$ is equivalent to

$$\alpha \times e_2-\beta \times e_1=0,\alpha \times e_1+\beta \times e_2=0.$$

Writing these equations componentwise gives

$${\alpha }_3=0,{\beta }_3=0,{\beta }_2=-{\alpha }_1,{\beta }_1={\alpha }_2.$$

Hence

$$X={\alpha }_1(R_1-B_2)+{\alpha }_2(R_2+B_1)={\alpha }_1({\tilde{J}}_{23}-{\tilde{J}}_{02})+{\alpha }_2({\tilde{J}}_{01}-{\tilde{J}}_{13}),$$

which proves

$$Lie\left(H_N\right)=\mathrm{span}\{N,\tilde{N}\},\tilde{N}:={\tilde{J}}_{23}-{\tilde{J}}_{02}.$$

A direct computation from the Lorentz commutation relations gives

$$[N,\tilde{N}]=0.$$

Therefore

$$K_N:=\{exp\left(sN\right)exp\left(t\tilde{N}\right):s,t\in \mathbb{R}\}$$

is a connected abelian Lie subgroup of ${\mathrm{SO}}^{+}(3,1)$ with Lie algebra $\mathrm{span}\{N,\tilde{N}\}$.

To identify $K_N$, consider

$$\Phi :{\mathbb{R}}^2\to K_N,\Phi (s,t)=exp\left(sN\right)exp\left(t\tilde{N}\right).$$

Because N and $\tilde{N}$ commute,

$$\Phi (s,t)=exp(sN+t\tilde{N}).$$

If $\Phi (s,t)=I$, then $exp(sN+t\tilde{N})=I$. But $sN+t\tilde{N}$ is nilpotent in the Lorentz algebra representation, and a nilpotent matrix has an exponential equal to the identity only if the matrix itself vanishes. Hence

$$sN+t\tilde{N}=0.$$

Since N and $\tilde{N}$ are linearly independent, it follows that $s=t=0$. Thus $\Phi$ is injective, and therefore

$$K_N\cong {\mathbb{R}}^2.$$

Again $K_N\subseteq H_N^{\circ }$ and

$$Lie\left(K_N\right)=Lie\left(H_N^{\circ }\right),$$

so the connected subgroup correspondence yields

$$H_N^{\circ }=K_N\cong {\mathbb{R}}^2.$$

Step 3: zero representative. If $\tilde{J}=0$, then every element of ${\mathrm{SO}}^{+}(3,1)$ stabilizes $\tilde{J}$. Hence

$$H_{\tilde{J}}={\mathrm{SO}}^{+}(3,1).$$

Since ${\mathrm{SO}}^{+}(3,1)$ is connected,

$$H_{\tilde{J}}^{\circ }={\mathrm{SO}}^{+}(3,1).$$

□

For a summary of these results, see

Table 2. Orbit strata determined by the Casimir data in the nondegenerate sector ${\hslash }_{\mathrm{eff}}\ne 0$, together with the excluded degenerate central sector for reference.

Sector	Conditions	Representative	${\mathit{H}}_{\tilde{\mathit{J}}}^{\circ }$
Excluded degeneratecentral sector	${\hslash }_{\mathrm{eff}}=0$	no normalizedbivector orbit	—
Zero orbit	${\hslash }_{\mathrm{eff}}\ne 0,c_1=c_2=0,\tilde{J}=0$	0	${\mathrm{SO}}^{+}(3,1)$
Null/nilpotent orbit	${\hslash }_{\mathrm{eff}}\ne 0,c_1=c_2=0,\tilde{J}\ne 0$	${\tilde{J}}_{01}-{\tilde{J}}_{13}$	${\mathbb{R}}^2$
Simple boost-type orbit	${\hslash }_{\mathrm{eff}}\ne 0,c_2=0,c_1<0$	$\sqrt{-c_1}{\tilde{J}}_{01}$	${\mathrm{SO}}^{+}(1,1)\times \mathrm{SO}\left(2\right)$
Simple rotation-type orbit	${\hslash }_{\mathrm{eff}}\ne 0,c_2=0,c_1>0$	$\sqrt{c_1}{\tilde{J}}_{23}$	${\mathrm{SO}}^{+}(1,1)\times \mathrm{SO}\left(2\right)$
Mixed regular orbit	${\hslash }_{\mathrm{eff}}\ne 0,c_2\ne 0$	$\theta {\tilde{J}}_{01}+\psi {\tilde{J}}_{23}$	${\mathrm{SO}}^{+}(1,1)\times \mathrm{SO}\left(2\right)$

.

6.4. Comparison with Wigner’s Orbit Classification

We conclude by situating the preceding bivector-orbit classification relative to the standard Wigner picture. The comparison is useful for orientation, but the two frameworks classify different geometric objects.

In Wigner’s analysis, one studies irreducible unitary representations of the Poincaré group, interpreted physically as one-particle sectors. The Lorentz-covariant orbit variable is a momentum vector

$$p_{\mu }\in V^{*}\cong {\mathbb{R}}^{3,1},$$

arising from the unique abelian translation ideal of the Poincaré algebra, and the classification is organized by Lorentz orbits of $p_{\mu }$ together with irreducible representations of the corresponding little groups [1,2,3].

By contrast, in the present manuscript, the Lorentz–Heisenberg algebra is realized on field spaces, and the orbit-theoretic analysis concerns field-sector realizations rather than a classification of particle states. After fixing a nondegenerate central character ${\hslash }_{\mathrm{eff}}\ne 0$, the minimal Lorentz-covariant object commuting with both translation sectors is not a vector but the normalized bivector ${\tilde{J}}_{\mu \nu }$. The resulting orbit data are therefore

$$({\hslash }_{\mathrm{eff}},c_1,c_2),$$

together with the connected stabilizer of the corresponding bivector orbit.

On the simple locus $c_2=0$, the three nonzero orbit types provide the closest analogue of the Wigner trichotomy:

$$c_2=0,c_1<0(\mathrm{boost}-\mathrm{type}),c_1=c_2=0,\tilde{J}\ne 0\left(\mathrm{null}\right),c_2=0,c_1>0(\mathrm{rotation}-\mathrm{type}).$$

These correspond, at the level of orbit stratification only, to the timelike, null, and spacelike momentum sectors in the vector-valued Wigner classification. The analogy stops there: the orbit variable here is a Lorentz bivector acting in field-sector realizations, not a momentum vector labeling particle representations.

There is also a structural difference in the stabilizers. In Wigner’s vector classification, the timelike and spacelike sectors have different little connected groups. In the present bivector classification, by contrast, the boost-type sector $c_2=0,c_1<0$, the rotation-type sector $c_2=0,c_1>0$, and the mixed regular sector $c_2\ne 0$ all have the same connected stabilizer form,

$$H_{\tilde{J}}^{\circ }\cong {\mathrm{SO}}^{+}(1,1)\times \mathrm{SO}\left(2\right),$$

whereas the nonzero null orbit has connected stabilizer

$$H_{\tilde{J}}^{\circ }\cong {\mathbb{R}}^2.$$

The mixed regular sector is the genuinely new feature of the bivector classification. It is characterized by $c_2\ne 0,$ or equivalently by non-simple normalized bivectors. This sector has no vector-valued analogue in Wigner’s classification. The reason is that a Lorentz vector orbit is governed by a single quadratic invariant $p_{\mu }{p}^{\mu }$, whereas a Lorentz bivector admits two independent Lorentz invariants, namely $c_1$ and $c_2$. Accordingly, the condition $c_2\ne 0$ singles out an intrinsic regular orbit sector recording an essential mixed boost–rotation component that cannot be removed by a Lorentz transformation.

The present manuscript determines the orbit data and connected stabilizers arising in the Fourier-compatible field-space realizations studied here. It does not attempt a realization-independent Mackey-type classification of all irreducible unitary representations of an integrated Lorentz–Heisenberg group. In this respect, it is adjacent to, but distinct from, the canonical quaplectic literature [19,20,21], where semidirect products with normal Heisenberg subgroups are analyzed by Mackey theory and Casimir equations. Here, after fixing the central character, the relevant orbit variable is a normalized Lorentz bivector rather than a momentum vector.

7. Fixed Spectral Elements and Dynamics in Momentum and Position Space

The purpose of this section is to show that, once a fixed spectral element is chosen in one of the two translation sectors, the realized Lorentz–Heisenberg algebra determines Lorentz-covariant scalar, spinor, and vector dynamics in the dual polarization. No new Lie-algebraic input is introduced: the section uses only the common operator-theoretic framework of Section 2 and the Fourier-compatible Lorentz–Heisenberg realization of Section 4.

The argument has two steps. First, we isolate the quadratic operator identities that provide the distinguished Lorentz scalars in the regular K- and X-polarized realizations. Second, we derive the K-polarized scalar, Dirac, and vector equations for a fixed quadratic value $x^{*2}$. The corresponding X-polarized equations for a fixed quadratic value $k^{*2}$ are then obtained by the same Fourier-dual mechanism; to avoid repetition, we state these equations and their action functionals explicitly but do not repeat the parallel derivations.

7.1. Common Operator Input

By Definition 2, the canonical operators in the two polarizations are

$${\widehat{x}}_{\mu }^{\left(K\right)}=i{\partial }_{k^{\mu }},{\widehat{k}}_{\mu }^{\left(K\right)}=k_{\mu },{\widehat{x}}_{\mu }^{\left(X\right)}=x_{\mu },{\widehat{k}}_{\mu }^{\left(X\right)}=-i{\partial }_{x^{\mu }}.$$

Hence, one has the quadratic operator identities

$${\left({\widehat{x}}^{\left(K\right)}\right)}^{\mu }{\widehat{x}}_{\mu }^{\left(K\right)}=-{\partial }_{k^{\mu }}{\partial }^{k^{\mu }},{\left({\widehat{k}}^{\left(X\right)}\right)}^{\mu }{\widehat{k}}_{\mu }^{\left(X\right)}=-{\partial }_{x^{\mu }}{\partial }^{x^{\mu }}.$$

(78)

Fix now

$$x^{*}\in {\left(V_X^{\left(K\right)}\right)}^{*},k^{*}\in {\left(V_K^{\left(X\right)}\right)}^{*},$$

and write

$$x^{*2}:=x^{*\mu }{x}_{\mu }^{*},k^{*2}:=k^{*\mu }{k}_{\mu }^{*}.$$

The equations below depend only on these Lorentz-invariant quadratic values.

7.2. K-Polarized Dynamics for Fixed $x^{*}$

We first work in the K-polarized realization. The structural point is that the operator

$${\left({\widehat{x}}^{\left(K\right)}\right)}^{\mu }{\widehat{x}}_{\mu }^{\left(K\right)}$$

already supplies the distinguished Lorentz scalar from which the momentum-space scalar, spinor, and vector equations are built.

7.2.1. Scalar Sector

Let $\widehat{f}\left(k\right)$ be a complex scalar field. The scalar equation associated with the fixed spectral element $x^{*}$ is

$$\left({\left({\widehat{x}}^{\left(K\right)}\right)}^{\mu }{\widehat{x}}_{\mu }^{\left(K\right)}-x^{*2}\right)\widehat{f}\left(k\right)=0,$$

(79)

that is,

$$\left(-{\partial }_{k^{\mu }}{\partial }^{k^{\mu }}-x^{*2}\right)\widehat{f}\left(k\right)=0.$$

(80)

Its action functional is

$$S_{\mathrm{sc}}^{\left(K\right)}[\widehat{f};x^{*}]:={\int }_{{\mathbb{R}}^{3,1}}\left({\partial }_{k^{\mu }}{\widehat{f}}^{*}{\partial }^{k^{\mu }}\widehat{f}-x^{*2}{\widehat{f}}^{*}\widehat{f}\right)d^{4}k.$$

(81)

Proposition 14 (Scalar Euler–Lagrange equation). 

For compactly supported smooth variations of $\widehat{f}$ and ${\widehat{f}}^{*}$, the Euler–Lagrange equation of (81) is (80).

Proof. 

Treat $\widehat{f}$ and ${\widehat{f}}^{*}$ as independent variables. Writing ${\mathcal{L}}_{\mathrm{sc}}^{\left(K\right)}$ for the integrand of (81), one has

$${\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{sc}}^{\left(K\right)}}{\partial {\widehat{f}}^{*}}}=-x^{*2}\widehat{f},{\displaystyle \frac{\partial {\mathcal{L}}_{\mathrm{sc}}^{\left(K\right)}}{\partial \left({\partial }_{k^{\mu }}{\widehat{f}}^{*}\right)}}={\partial }^{k^{\mu }}\widehat{f}.$$

The Euler–Lagrange equation therefore gives

$$-x^{*2}\widehat{f}-{\partial }_{k^{\mu }}{\partial }^{k^{\mu }}\widehat{f}=0,$$

which is (80). □

7.2.2. Dirac Factorization

Let $\widehat{f}\left(k\right)$ be a Dirac spinor field, with

$$\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }\mathbb{I}.$$

Choose

$${\sigma }_{x^{*}}\in \mathbb{C},{\sigma }_{x^{*}}^2=x^{*2},$$

and define

$${\mathcal{D}}_{{\sigma }_{x^{*}}}^{\left(K\right)}:=i{\gamma }^{\mu }{\partial }_{k^{\mu }}-{\sigma }_{x^{*}}.$$

(82)

Proposition 15 (Dirac factorization). 

One has

$$\left(i{\gamma }^{\mu }{\partial }_{k^{\mu }}+{\sigma }_{x^{*}}\right)\left(i{\gamma }^{\nu }{\partial }_{k^{\nu }}-{\sigma }_{x^{*}}\right)=-{\partial }_{k^{\mu }}{\partial }^{k^{\mu }}-x^{*2}.$$

(83)

Thus

$${\mathcal{D}}_{{\sigma }_{x^{*}}}^{\left(K\right)}\widehat{f}\left(k\right)=0$$

(84)

factorizes (80).

Proof. 

Since ${\sigma }_{x^{*}}$ is scalar,

$$(i{\gamma }^{\mu }{\partial }_{k^{\mu }}+{\sigma }_{x^{*}})(i{\gamma }^{\nu }{\partial }_{k^{\nu }}-{\sigma }_{x^{*}})={\left(i{\gamma }^{\mu }{\partial }_{k^{\mu }}\right)}^2-{\sigma }_{x^{*}}^2.$$

Moreover,

$${\left(i{\gamma }^{\mu }{\partial }_{k^{\mu }}\right)}^2=-{\displaystyle \frac{1}{2}}\{{\gamma }^{\mu },{\gamma }^{\nu }\}{\partial }_{k^{\mu }}{\partial }_{k^{\nu }}=-{\partial }_{k^{\mu }}{\partial }^{k^{\mu }}.$$

Using ${\sigma }_{x^{*}}^2=x^{*2}$ proves the claim. □

The corresponding Dirac action is

$$S_{\mathrm{D}}^{\left(K\right)}[\widehat{f};{\sigma }_{x^{*}}]:={\int }_{{\mathbb{R}}^{3,1}}{\widehat{f}}^{•}\left(i{\gamma }^{\mu }{\partial }_{k^{\mu }}-{\sigma }_{x^{*}}\right)\widehat{f}{d}^{4}k,{\widehat{f}}^{•}:={\widehat{f}}^{†}{\gamma }^0.$$

(85)

7.2.3. Vector Sector

Let ${\widehat{A}}_{\mu }\left(k\right)$ be a vector field on momentum space, and define

$${\widehat{F}}_{\mu \nu }:={\partial }_{k^{\mu }}{\widehat{A}}_{\nu }-{\partial }_{k^{\nu }}{\widehat{A}}_{\mu }.$$

(86)

Using only the realized differential operators and the scalar $x^{*2}$, consider the Lorentz-invariant action

$$S_{\mathrm{vec}}^{\left(K\right)}[\widehat{A};x^{*}]:={\int }_{{\mathbb{R}}^{3,1}}\left(-{\displaystyle \frac{1}{4}}{\widehat{F}}^{\mu \nu }{\widehat{F}}_{\mu \nu }+{\displaystyle \frac{1}{2}}{x}^{*2}{\widehat{A}}^{\nu }{\widehat{A}}_{\nu }\right)d^{4}k.$$

(87)

For $x^{*2}\ne 0$, this is the Proca-type action determined by the fixed spectral element $x^{*}$; for $x^{*2}=0$, it reduces to the Maxwell action.

Proposition 16 (Vector Euler–Lagrange equation). 

For compactly supported smooth variations in ${\widehat{A}}_{\mu }$, the Euler–Lagrange equation of (87) is

$${\partial }_{k^{\mu }}{\widehat{F}}^{\mu \nu }+x^{*2}{\widehat{A}}^{\nu }=0.$$

(88)

Taking the divergence yields

$$x^{*2}{\partial }_{k^{\nu }}{\widehat{A}}^{\nu }=0.$$

(89)

Hence on every non-null sector $x^{*2}\ne 0$,

$${\partial }_{k^{\nu }}{\widehat{A}}^{\nu }=0.$$

Proof. 

Varying (87) gives

$$\delta {\widehat{F}}_{\mu \nu }={\partial }_{k^{\mu }}\delta {\widehat{A}}_{\nu }-{\partial }_{k^{\nu }}\delta {\widehat{A}}_{\mu }.$$

Using antisymmetry of ${\widehat{F}}_{\mu \nu }$,

$$\delta S_{\mathrm{vec}}^{\left(K\right)}={\int }_{{\mathbb{R}}^{3,1}}\left(-{\widehat{F}}^{\mu \nu }{\partial }_{k^{\mu }}\delta {\widehat{A}}_{\nu }+x^{*2}{\widehat{A}}^{\nu }\delta {\widehat{A}}_{\nu }\right)d^{4}k.$$

After integration by parts and discarding the boundary term,

$$\delta S_{\mathrm{vec}}^{\left(K\right)}={\int }_{{\mathbb{R}}^{3,1}}\left({\partial }_{k^{\mu }}{\widehat{F}}^{\mu \nu }+x^{*2}{\widehat{A}}^{\nu }\right)\delta {\widehat{A}}_{\nu }{d}^{4}k.$$

Since the variations are arbitrary, (88) follows. Applying ${\partial }_{k^{\nu }}$ and using

$${\partial }_{k^{\nu }}{\partial }_{k^{\mu }}{\widehat{F}}^{\mu \nu }=0$$

gives (89). □

Corollary 4 (Null sector). 

On the null locus $x^{*2}=0$, the action (87) reduces to

$$S_{\mathrm{M}}^{\left(K\right)}\left[\widehat{A}\right]=-{\displaystyle \frac{1}{4}}{\int }_{{\mathbb{R}}^{3,1}}{\widehat{F}}^{\mu \nu }{\widehat{F}}_{\mu \nu }{d}^{4}k,$$

(90)

and the Euler–Lagrange equation reduces to

$${\partial }_{k^{\mu }}{\widehat{F}}^{\mu \nu }=0.$$

(91)

7.3. X-Polarized Dynamics for Fixed $k^{*}$

We now pass to the Fourier-dual X-polarized realization. No new structural input is needed: the same constructions apply after the exchange

$$({\widehat{x}}^{\left(K\right)},k,x^{*})⟷({\widehat{k}}^{\left(X\right)},x,k^{*}).$$

To avoid duplicating the preceding derivations, we state the resulting X-space equations and action functionals directly.

Proposition 17 (Fourier-dual X-space equations and actions). 

Let $k^{*}\in {\left(V_K^{\left(X\right)}\right)}^{*}$ be fixed.

(i) 

Scalar sector.  For a complex scalar field $f\left(x\right)$,

$$\left({\left({\widehat{k}}^{\left(X\right)}\right)}^{\mu }{\widehat{k}}_{\mu }^{\left(X\right)}-k^{*2}\right)f\left(x\right)=0,$$

(92)

that is,

$$\left(-{\partial }_{x^{\mu }}{\partial }^{x^{\mu }}-k^{*2}\right)f\left(x\right)=0.$$

(93)

Its action is

$$S_{\mathrm{sc}}^{\left(X\right)}[f;k^{*}]:={\int }_{{\mathbb{R}}^{3,1}}\left({\partial }_{x^{\mu }}{f}^{*}{\partial }^{x^{\mu }}f-k^{*2}{f}^{*}f\right)d^{4}x.$$

(94)

(ii) 

Spinor sector. For a Dirac spinor field $f\left(x\right)$, choose

$${\sigma }_{k^{*}}\in \mathbb{C},{\sigma }_{k^{*}}^2=k^{*2}.$$

Define

$${\mathcal{D}}_{{\sigma }_{k^{*}}}^{\left(X\right)}:=i{\gamma }^{\mu }{\partial }_{x^{\mu }}-{\sigma }_{k^{*}}.$$

(95)

Then

$$\left(i{\gamma }^{\mu }{\partial }_{x^{\mu }}+{\sigma }_{k^{*}}\right)\left(i{\gamma }^{\nu }{\partial }_{x^{\nu }}-{\sigma }_{k^{*}}\right)=-{\partial }_{x^{\mu }}{\partial }^{x^{\mu }}-k^{*2},$$

(96)

and hence

$${\mathcal{D}}_{{\sigma }_{k^{*}}}^{\left(X\right)}f\left(x\right)=0.$$

(97)

Its action is

$$S_{\mathrm{D}}^{\left(X\right)}[f;{\sigma }_{k^{*}}]:={\int }_{{\mathbb{R}}^{3,1}}{f}^{•}\left(i{\gamma }^{\mu }{\partial }_{x^{\mu }}-{\sigma }_{k^{*}}\right)f{d}^{4}x,f^{•}:=f^{†}{\gamma }^0.$$

(98)

(iii) 

Vector sector. For a vector field $A_{\mu }\left(x\right)$, define

$$F_{\mu \nu }:={\partial }_{x^{\mu }}{A}_{\nu }-{\partial }_{x^{\nu }}{A}_{\mu }.$$

(99)

Then

$${\partial }_{x^{\mu }}{F}^{\mu \nu }+k^{*2}{A}^{\nu }=0.$$

(100)

If $k^{*2}\ne 0$, this is the Proca equation with the operator-determined Lorentz scalar $k^{*2}$; if $k^{*2}=0$, it reduces to the Maxwell equation

$${\partial }_{x^{\mu }}{F}^{\mu \nu }=0.$$

(101)

Its action is

$$S_{\mathrm{vec}}^{\left(X\right)}[A;k^{*}]:={\int }_{{\mathbb{R}}^{3,1}}\left(-{\displaystyle \frac{1}{4}}{F}^{\mu \nu }{F}_{\mu \nu }+{\displaystyle \frac{1}{2}}{k}^{*2}{A}^{\nu }{A}_{\nu }\right)d^{4}x.$$

(102)

Thus the X-polarized equations are the standard Klein–Gordon, Dirac, Proca, and Maxwell equations, with the Lorentz scalar $k^{*2}$ supplied by the fixed spectral element $k^{*}$ rather than inserted as an external mass parameter [3].

Proof. 

This is the Fourier-dual counterpart of the K-polarized construction. The scalar equation follows from (78); the Dirac factorization is obtained from Proposition 15 by the replacement

$$(x^{*},{\partial }_{k^{\mu }})⟷(k^{*},{\partial }_{x^{\mu }});$$

and the vector equation and action are obtained from Proposition 16 and (87) by the same exchange of variables. □

The conclusion is therefore structural: once a fixed spectral element is chosen in one translation sector, the realized Lorentz–Heisenberg algebra determines the corresponding scalar, spinor, and vector dynamics in the dual polarization. In the X-polarized realization, these are precisely the standard relativistic field equations, with the distinguished Lorentz scalar supplied by the operator structure itself.

8. Local $\mathit{U}\left(\mathbf{1}\right)$ Gauge Structure and Primary Dirac–Maxwell Systems

This section shows that, in the regular X- and K-polarized realizations of the represented Lorentz–Heisenberg algebra, the local abelian gauge sector and the associated primary Dirac–Maxwell systems are structural consequences of the represented Heisenberg sector itself. No independent gauge-theoretic axiom is introduced.

The decisive point is the distinction between the affine Weyl seed and the full local phase algebra. The mixed Heisenberg commutator first yields, through the Baker–Campbell–Hausdorff formula, a finite-dimensional affine Weyl sector. Its phase functions are affine in x or k, hence their derivatives are constant. Conjugation by those affine phases therefore produces only constant shifts of the differential generators, so the corresponding curvature must vanish. Nontrivial curvature appears only after passing to the full commutative phase algebra canonically generated, in the regular realization, by the strongly commuting coordinate family through joint functional calculus. For general phase functions in $C_{\mathrm{pol}}^{\infty }\left({\mathbb{R}}^{3,1}\right)$, the first derivatives need not be constant, so the induced shifts of the differential generators become x- or k-dependent multiplication operators; their antisymmetrized derivatives are exactly the curvature tensors.

Throughout, fix a nondegenerate irreducible sector

$$\rho \left(\mathbf{1}\right)={\hslash }_{\mathrm{eff}}\mathbb{I},{\hslash }_{\mathrm{eff}}\ne 0,$$

(103)

and choose nonzero real parameters $e,\widehat{e}$ such that

$${\hslash }_{\mathrm{eff}}=e\widehat{e}.$$

(104)

Write ${\rho }_X$ and ${\rho }_K$ for the corresponding regular X- and K-polarized realizations on ${\mathcal{H}}_X$ and ${\mathcal{H}}_K$. To avoid overloading the earlier notation $Q_{\mu }^{\left(X\right)},Q_{\mu }^{\left(K\right)},T_{\mu }^{\left(X\right)},T_{\mu }^{\left(K\right)}$ from Definition 5, we denote the represented generators in the present fixed sector by

$$\begin{array}{cccc}\hfill {\mathcal{Q}}_{\mu }^{\left(X\right)}& :={\rho }_X\left(Q_{\mu }\right)=e{x}_{\mu },\hfill & \hfill {\mathcal{T}}_{\mu }^{\left(X\right)}& :={\rho }_X\left(T_{\mu }\right)=-i\widehat{e}{\partial }_{x^{\mu }},\hfill \end{array}$$

(105)

$$\begin{array}{cccc}\hfill {\mathcal{Q}}_{\mu }^{\left(K\right)}& :={\rho }_K\left(Q_{\mu }\right)=\widehat{e}{k}_{\mu },\hfill & \hfill {\mathcal{T}}_{\mu }^{\left(K\right)}& :={\rho }_K\left(T_{\mu }\right)=ie{\partial }_{k^{\mu }}.\hfill \end{array}$$

(106)

These satisfy the Fourier-duality relations

$${\mathcal{Q}}_{\mu }^{\left(K\right)}=\mathcal{F}{\mathcal{T}}_{\mu }^{\left(X\right)}{\mathcal{F}}^{-1},{\mathcal{T}}_{\mu }^{\left(K\right)}=\mathcal{F}{\mathcal{Q}}_{\mu }^{\left(X\right)}{\mathcal{F}}^{-1},$$

(107)

together with the represented Heisenberg commutators

$$\begin{array}{cc}\hfill [{\mathcal{T}}_{\mu }^{\left(X\right)},{\mathcal{Q}}_{\nu }^{\left(X\right)}]& =-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I},\hfill \end{array}$$

(108)

$$\begin{array}{cc}\hfill [{\mathcal{Q}}_{\mu }^{\left(K\right)},{\mathcal{T}}_{\nu }^{\left(K\right)}]& =-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I}.\hfill \end{array}$$

(109)

Only the product ${\hslash }_{\mathrm{eff}}$ is intrinsic to the representation; the separate factors e and $\widehat{e}$ are auxiliary scalings adapted to the two polarized realizations.

To avoid duplicating Fourier-dual calculations, we state the formulas in both polarizations but carry out the derivations only in the K-polarized realization. The corresponding X-polarized statements follow by the same method after the replacements

$$k\leftrightarrow x,e\leftrightarrow \widehat{e},{\mathcal{T}}_{\mu }^{\left(K\right)}=ie{\partial }_{k^{\mu }}\leftrightarrow {\mathcal{T}}_{\mu }^{\left(X\right)}=-i\widehat{e}{\partial }_{x^{\mu }},{\mathcal{Q}}_{\mu }^{\left(K\right)}=\widehat{e}{k}_{\mu }\leftrightarrow {\mathcal{Q}}_{\mu }^{\left(X\right)}=e{x}_{\mu },$$

together with the Fourier-duality relations already established.

Let

$$C_{\mathrm{pol}}^{\infty }\left({\mathbb{R}}^{3,1}\right)$$

denote the space of real-valued smooth functions whose derivatives of all orders have at most polynomial growth. This class is used only as an analytic regularity class adapted to the regular polarized realizations. It is not an additional physical axiom. Its role is to provide a canonical phase algebra large enough to contain the affine BCH phase functions and, at the same time, to admit phase functions with nonconstant first derivatives, which is exactly what makes nontrivial curvature possible.

8.1. Affine Weyl Seed and Canonical Local Phase Algebra

We first isolate the two ingredients needed later. The represented Heisenberg commutator produces the affine Weyl sector through BCH, and the regular realization then enlarges that sector canonically to the full local phase algebra by joint functional calculus.

Definition 7 (Translation and affine phase exponentials). 

For $a,b\in {\mathbb{R}}^{3,1}$, define

$$\begin{array}{cccc}\hfill W_a^{\left(X\right)}& :=exp\left(-i{a}^{\mu }{\widehat{e}}^{-1}{\mathcal{T}}_{\mu }^{\left(X\right)}\right),\hfill & \hfill W_b^{\left(X\right)}& :=exp\left(i{b}^{\mu }{e}^{-1}{\mathcal{Q}}_{\mu }^{\left(X\right)}\right),\hfill \end{array}$$

(110)

$$\begin{array}{cccc}\hfill {\widehat{W}}_b^{\left(K\right)}& :=exp\left(i{b}^{\mu }{e}^{-1}{\mathcal{T}}_{\mu }^{\left(K\right)}\right),\hfill & \hfill {\widehat{W}}_a^{\left(K\right)}& :=exp\left(-i{a}^{\mu }{\widehat{e}}^{-1}{\mathcal{Q}}_{\mu }^{\left(K\right)}\right).\hfill \end{array}$$

(111)

Proposition 18 (Weyl relations and BCH normal forms). 

On $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, the ordered exponentials satisfy

$$\begin{array}{cc}\hfill W_a^{\left(X\right)}{W}_b^{\left(X\right)}& =e^{-i{a}^{\mu }{b}_{\mu }}{W}_b^{\left(X\right)}{W}_a^{\left(X\right)},\hfill \end{array}$$

(112)

$$\begin{array}{cc}\hfill {\widehat{W}}_b^{\left(K\right)}{\widehat{W}}_a^{\left(K\right)}& =e^{i{a}^{\mu }{b}_{\mu }}{\widehat{W}}_a^{\left(K\right)}{\widehat{W}}_b^{\left(K\right)}.\hfill \end{array}$$

(113)

Because the relevant commutators are central, the BCH formula truncates and yields

$$\begin{array}{cc}\hfill W_a^{\left(X\right)}{W}_b^{\left(X\right)}& =exp\left(i{e}^{-1}{b}^{\mu }{\mathcal{Q}}_{\mu }^{\left(X\right)}-i{\widehat{e}}^{-1}{a}^{\mu }{\mathcal{T}}_{\mu }^{\left(X\right)}-{\displaystyle \frac{i}{2}}{a}^{\mu }{b}_{\mu }\mathbb{I}\right),\hfill \end{array}$$

(114)

$$\begin{array}{cc}\hfill {\widehat{W}}_b^{\left(K\right)}{\widehat{W}}_a^{\left(K\right)}& =exp\left(i{e}^{-1}{b}^{\mu }{\mathcal{T}}_{\mu }^{\left(K\right)}-i{\widehat{e}}^{-1}{a}^{\mu }{\mathcal{Q}}_{\mu }^{\left(K\right)}+{\displaystyle \frac{i}{2}}{a}^{\mu }{b}_{\mu }\mathbb{I}\right).\hfill \end{array}$$

(115)

Equivalently,

$$\begin{array}{cc}\hfill exp\left(i{e}^{-1}{b}^{\mu }{\mathcal{Q}}_{\mu }^{\left(X\right)}-i{\widehat{e}}^{-1}{a}^{\mu }{\mathcal{T}}_{\mu }^{\left(X\right)}\right)& =e^{\frac{i}{2}{a}^{\mu }{b}_{\mu }}{W}_a^{\left(X\right)}{W}_b^{\left(X\right)}=e^{-\frac{i}{2}{a}^{\mu }{b}_{\mu }}{W}_b^{\left(X\right)}{W}_a^{\left(X\right)},\hfill \end{array}$$

(116)

$$\begin{array}{cc}\hfill exp\left(i{e}^{-1}{b}^{\mu }{\mathcal{T}}_{\mu }^{\left(K\right)}-i{\widehat{e}}^{-1}{a}^{\mu }{\mathcal{Q}}_{\mu }^{\left(K\right)}\right)& =e^{-\frac{i}{2}{a}^{\mu }{b}_{\mu }}{\widehat{W}}_b^{\left(K\right)}{\widehat{W}}_a^{\left(K\right)}=e^{\frac{i}{2}{a}^{\mu }{b}_{\mu }}{\widehat{W}}_a^{\left(K\right)}{\widehat{W}}_b^{\left(K\right)}.\hfill \end{array}$$

(117)

Moreover, on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$,

$$\begin{array}{cc}\hfill \left(W_a^{\left(X\right)}{W}_b^{\left(X\right)}f\right)\left(x\right)& =e^{i{b}^{\mu }(x_{\mu }-a_{\mu })}f(x-a),\hfill \end{array}$$

(118)

$$\begin{array}{cc}\hfill \left({\widehat{W}}_b^{\left(K\right)}{\widehat{W}}_a^{\left(K\right)}\widehat{f}\right)\left(k\right)& =e^{-i{a}^{\mu }(k_{\mu }-b_{\mu })}\widehat{f}(k-b).\hfill \end{array}$$

(119)

Accordingly,

$$\begin{array}{cc}\hfill \left(exp\left(i{e}^{-1}{b}^{\mu }{\mathcal{Q}}_{\mu }^{\left(X\right)}-i{\widehat{e}}^{-1}{a}^{\mu }{\mathcal{T}}_{\mu }^{\left(X\right)}\right)f\right)\left(x\right)& =e^{i{ϵ}_{a,b}^{\mathrm{aff}}\left(x\right)}f(x-a),\hfill \end{array}$$

(120)

$$\begin{array}{cc}\hfill \left(exp\left(i{e}^{-1}{b}^{\mu }{\mathcal{T}}_{\mu }^{\left(K\right)}-i{\widehat{e}}^{-1}{a}^{\mu }{\mathcal{Q}}_{\mu }^{\left(K\right)}\right)\widehat{f}\right)\left(k\right)& =e^{i{\widehat{ϵ}}_{a,b}^{\mathrm{aff}}\left(k\right)}\widehat{f}(k-b),\hfill \end{array}$$

(121)

where

$$\begin{array}{cc}\hfill {ϵ}_{a,b}^{\mathrm{aff}}\left(x\right)& :=b^{\mu }\left(x_{\mu }-\frac{1}{2}{a}_{\mu }\right),\hfill \end{array}$$

(122)

$$\begin{array}{cc}\hfill {\widehat{ϵ}}_{a,b}^{\mathrm{aff}}\left(k\right)& :=-a^{\mu }\left(k_{\mu }-\frac{1}{2}{b}_{\mu }\right).\hfill \end{array}$$

(123)

Thus the affine Weyl sector already yields phase-modulated translation operators, but only with affine phase functions.

Proof. 

We prove the K-space statements; the X-space formulas follow by the polarization replacements fixed above.

Set

$$A:=i{e}^{-1}{b}^{\mu }{\mathcal{T}}_{\mu }^{\left(K\right)},B:=-i{\widehat{e}}^{-1}{a}^{\mu }{\mathcal{Q}}_{\mu }^{\left(K\right)}.$$

Using (109),

$$[A,B]=i{a}^{\mu }{b}_{\mu }\mathbb{I}.$$

Since $[A,B]$ is central, BCH truncates

$$e^A{e}^B=e^{A+B+\frac{1}{2}[A,B]},e^B{e}^A=e^{A+B-\frac{1}{2}[A,B]},$$

which gives (115) and (117); Equation (113) is the equivalent central-commutator form

$$e^A{e}^B=e^B{e}^A{e}^{[A,B]}.$$

By (106),

$$A=-b^{\mu }{\partial }_{k^{\mu }},B=-i{a}^{\mu }{k}_{\mu },$$

hence

$$\left({\widehat{W}}_b^{\left(K\right)}\widehat{f}\right)\left(k\right)=\widehat{f}(k-b),\left({\widehat{W}}_a^{\left(K\right)}\widehat{f}\right)\left(k\right)=e^{-i{a}^{\mu }{k}_{\mu }}\widehat{f}\left(k\right).$$

Therefore

$$\begin{array}{cc}\hfill \left({\widehat{W}}_b^{\left(K\right)}{\widehat{W}}_a^{\left(K\right)}\widehat{f}\right)\left(k\right)& ={\widehat{W}}_b^{\left(K\right)}\left(e^{-i{a}^{\mu }{k}_{\mu }}\widehat{f}\left(k\right)\right)\hfill \\ \hfill & =e^{-i{a}^{\mu }(k_{\mu }-b_{\mu })}\widehat{f}(k-b),\hfill \end{array}$$

which proves (119). Using (117),

$$\begin{array}{cc}\hfill \left(exp(A+B)\widehat{f}\right)\left(k\right)& =e^{-\frac{i}{2}{a}^{\mu }{b}_{\mu }}\left({\widehat{W}}_b^{\left(K\right)}{\widehat{W}}_a^{\left(K\right)}\widehat{f}\right)\left(k\right)\hfill \\ \hfill & =e^{-i{a}^{\mu }(k_{\mu }-\frac{1}{2}{b}_{\mu })}\widehat{f}(k-b),\hfill \end{array}$$

which is (121) with (123). □

Proposition 18 identifies the affine Weyl seed forced by the represented Heisenberg commutator. We now pass to the decisive enlargement: the full local phase algebra generated canonically from the commuting coordinate family of the regular realization.

Proposition 19 (Canonical local phase algebra from the commuting coordinate family). 

In the regular polarized realizations, the families

$$e^{-1}{\mathcal{Q}}_{\mu }^{\left(X\right)}=x_{\mu },{\widehat{e}}^{-1}{\mathcal{Q}}_{\mu }^{\left(K\right)}=k_{\mu }$$

are strongly commuting self-adjoint operators. Hence, by their joint functional calculus, every real-valued function

$$ϵ,\widehat{ϵ}\in C_{\mathrm{pol}}^{\infty }\left({\mathbb{R}}^{3,1}\right)$$

canonically determines unitary operators

$$\begin{array}{cc}\hfill {\mathcal{U}}_{ϵ}^{\left(X\right)}& :=exp\left(iϵ\left(e^{-1}{\mathcal{Q}}^{\left(X\right)}\right)\right),\hfill \end{array}$$

(124)

$$\begin{array}{cc}\hfill {\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}& :=exp\left(i\widehat{ϵ}\left({\widehat{e}}^{-1}{\mathcal{Q}}^{\left(K\right)}\right)\right),\hfill \end{array}$$

(125)

acting on $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$ by

$$\begin{array}{cc}\hfill \left({\mathcal{U}}_{ϵ}^{\left(X\right)}f\right)\left(x\right)& =e^{iϵ\left(x\right)}f\left(x\right),\hfill \end{array}$$

(126)

$$\begin{array}{cc}\hfill \left({\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}\widehat{f}\right)\left(k\right)& =e^{i\widehat{ϵ}\left(k\right)}\widehat{f}\left(k\right).\hfill \end{array}$$

(127)

If

$$\left({\tau }_aϵ\right)\left(x\right):=ϵ(x-a),\left({\widehat{\tau }}_b\widehat{ϵ}\right)\left(k\right):=\widehat{ϵ}(k-b),$$

then

$$\begin{array}{cc}\hfill W_a^{\left(X\right)}{\mathcal{U}}_{ϵ}^{\left(X\right)}{\left(W_a^{\left(X\right)}\right)}^{-1}& ={\mathcal{U}}_{{\tau }_aϵ}^{\left(X\right)},\hfill \end{array}$$

(128)

$$\begin{array}{cc}\hfill {\widehat{W}}_b^{\left(K\right)}{\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}{\left({\widehat{W}}_b^{\left(K\right)}\right)}^{-1}& ={\widehat{\mathcal{U}}}_{{\widehat{\tau }}_b\widehat{ϵ}}^{\left(K\right)}.\hfill \end{array}$$

(129)

Therefore the regular polarized realizations canonically carry the full local phase groups

$${\mathcal{U}}_X\cong C_{\mathrm{pol}}^{\infty }({\mathbb{R}}^{3,1},U\left(1\right)),{\mathcal{U}}_K\cong C_{\mathrm{pol}}^{\infty }({\mathbb{R}}^{3,1},U\left(1\right)).$$

Moreover, the affine BCH phase functions ${ϵ}_{a,b}^{\mathrm{aff}}$ and ${\widehat{ϵ}}_{a,b}^{\mathrm{aff}}$ form a distinguished finite-dimensional subgroup of these canonical commutative phase algebras.

Proof. 

We prove the K-space statement. Since

$${\widehat{e}}^{-1}{\mathcal{Q}}_{\mu }^{\left(K\right)}=k_{\mu },$$

the family ${\left\{{\widehat{e}}^{-1}{\mathcal{Q}}_{\mu }^{\left(K\right)}\right\}}_{\mu =0}^3$ is the standard strongly commuting self-adjoint multiplication family on $L^2({\mathbb{R}}^{3,1},d^{4}k)$. By the joint spectral theorem, each real-valued

$$\widehat{ϵ}\in C_{\mathrm{pol}}^{\infty }\left({\mathbb{R}}^{3,1}\right)$$

determines canonically the self-adjoint multiplication operator $\widehat{ϵ}\left(k\right)$, and its exponential is multiplication by $e^{i\widehat{ϵ}\left(k\right)}$. This proves (127).

Next, ${\widehat{W}}_b^{\left(K\right)}$ acts by translation:

$$\left({\widehat{W}}_b^{\left(K\right)}\widehat{f}\right)\left(k\right)=\widehat{f}(k-b).$$

Hence

$$\begin{array}{cc}\hfill \left({\widehat{W}}_b^{\left(K\right)}{\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}{\left({\widehat{W}}_b^{\left(K\right)}\right)}^{-1}\widehat{f}\right)\left(k\right)& ={\widehat{W}}_b^{\left(K\right)}\left(e^{i\widehat{ϵ}\left(k\right)}\widehat{f}(k+b)\right)\hfill \\ \hfill & =e^{i\widehat{ϵ}(k-b)}\widehat{f}\left(k\right)\hfill \\ \hfill & =\left({\widehat{\mathcal{U}}}_{{\widehat{\tau }}_b\widehat{ϵ}}^{\left(K\right)}\widehat{f}\right)\left(k\right),\hfill \end{array}$$

which proves (129). The X-space formulas follow by the same argument, since $e^{-1}{\mathcal{Q}}_{\mu }^{\left(X\right)}=x_{\mu }$ is the corresponding commuting self-adjoint multiplication family on $L^2({\mathbb{R}}^{3,1},d^{4}x)$. □

The structural enlargement is now clear. The affine Weyl sector supplies only the affine phase functions ${ϵ}_{a,b}^{\mathrm{aff}}$ and ${\widehat{ϵ}}_{a,b}^{\mathrm{aff}}$, whose derivatives are constant; it therefore yields only constant shifts in the differential generators. By contrast, the full phase algebra generated by $C_{\mathrm{pol}}^{\infty }\left({\mathbb{R}}^{3,1}\right)$ allows nonconstant derivatives, and hence variable first-order coefficients after conjugation. This is the point at which nontrivial curvature becomes possible.

8.2. From the Full Local Phase Algebra to Covariant Derivatives

We now differentiate the phase action. For affine Weyl phases, the derivatives $\partial {ϵ}^{\mathrm{aff}}$ and $\partial {\widehat{ϵ}}^{\mathrm{aff}}$ are constant, so the induced shifts of ${\mathcal{T}}_{\mu }^{\left(X\right)}$ and ${\mathcal{T}}_{\mu }^{\left(K\right)}$ are constant and the corresponding curvature vanishes. For general $ϵ,\widehat{ϵ}\in C_{\mathrm{pol}}^{\infty }\left({\mathbb{R}}^{3,1}\right)$, however, the derivatives $\partial ϵ$ and $\partial \widehat{ϵ}$ are variable polynomial-growth functions. Conjugation by the associated phase multipliers therefore produces variable first-order coefficients, and after covariantization, their antisymmetrized derivatives yield the curvature tensors $F_{\mu \nu }$ and ${\widehat{F}}_{\mu \nu }$.

Lemma 3 (Conjugation formulas). 

For $ϵ,\widehat{ϵ}\in C_{\mathrm{pol}}^{\infty }\left({\mathbb{R}}^{3,1}\right)$,

$$\begin{array}{cc}\hfill {\mathcal{U}}_{ϵ}^{\left(X\right)}{\partial }_{x^{\mu }}{\left({\mathcal{U}}_{ϵ}^{\left(X\right)}\right)}^{-1}& ={\partial }_{x^{\mu }}-i{\partial }_{x^{\mu }}ϵ,\hfill \end{array}$$

(130)

$$\begin{array}{cc}\hfill {\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}{\partial }_{k^{\mu }}{\left({\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}\right)}^{-1}& ={\partial }_{k^{\mu }}-i{\partial }_{k^{\mu }}\widehat{ϵ}.\hfill \end{array}$$

(131)

Consequently,

$$\begin{array}{cc}\hfill {\mathcal{U}}_{ϵ}^{\left(X\right)}{\mathcal{T}}_{\mu }^{\left(X\right)}{\left({\mathcal{U}}_{ϵ}^{\left(X\right)}\right)}^{-1}& ={\mathcal{T}}_{\mu }^{\left(X\right)}-\widehat{e}{\partial }_{x^{\mu }}ϵ\mathbb{I},\hfill \end{array}$$

(132)

$$\begin{array}{cc}\hfill {\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}{\mathcal{T}}_{\mu }^{\left(K\right)}{\left({\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}\right)}^{-1}& ={\mathcal{T}}_{\mu }^{\left(K\right)}+e{\partial }_{k^{\mu }}\widehat{ϵ}\mathbb{I}.\hfill \end{array}$$

(133)

Proof. 

We prove the K-space identities. Using (127),

$$\begin{array}{cc}\hfill {\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}{\partial }_{k^{\mu }}{\left({\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}\right)}^{-1}\widehat{f}& =e^{i\widehat{ϵ}\left(k\right)}{\partial }_{k^{\mu }}\left(e^{-i\widehat{ϵ}\left(k\right)}\widehat{f}\left(k\right)\right)\hfill \\ \hfill & =\left({\partial }_{k^{\mu }}-i{\partial }_{k^{\mu }}\widehat{ϵ}\right)\widehat{f}\left(k\right),\hfill \end{array}$$

which proves (131). Multiplying by $ie$ gives (133). The X-space formulas follow by the same computation, using (126) and ${\mathcal{T}}_{\mu }^{\left(X\right)}=-i\widehat{e}{\partial }_{x^{\mu }}$. □

Define the covariant first-order operators

$$\begin{array}{cccc}\hfill D_{\mu }^{\left(X\right)}& :={\partial }_{x^{\mu }}+i{\displaystyle \frac{e}{\widehat{e}}}{A}_{\mu }\left(x\right),\hfill & \hfill {\mathcal{T}}_{D,\mu }^{\left(X\right)}& :=-i\widehat{e}{D}_{\mu }^{\left(X\right)}={\mathcal{T}}_{\mu }^{\left(X\right)}+e{A}_{\mu }\left(x\right)\mathbb{I},\hfill \end{array}$$

(134)

$$\begin{array}{cccc}\hfill {\widehat{D}}_{\mu }^{\left(K\right)}& :={\partial }_{k^{\mu }}+i{\displaystyle \frac{\widehat{e}}{e}}{\widehat{A}}_{\mu }\left(k\right),\hfill & \hfill {\mathcal{T}}_{D,\mu }^{\left(K\right)}& :=ie{\widehat{D}}_{\mu }^{\left(K\right)}={\mathcal{T}}_{\mu }^{\left(K\right)}-\widehat{e}{\widehat{A}}_{\mu }\left(k\right)\mathbb{I},\hfill \end{array}$$

(135)

where $A_{\mu },{\widehat{A}}_{\mu }\in C_{\mathrm{pol}}^{\infty }\left({\mathbb{R}}^{3,1}\right)$ are real-valued.

Proposition 20 (Structural gauge transformation laws). 

Under conjugation by the local phase groups,

$$\begin{array}{cc}\hfill {\mathcal{U}}_{ϵ}^{\left(X\right)}{D}_{\mu }^{\left(X\right)}{\left({\mathcal{U}}_{ϵ}^{\left(X\right)}\right)}^{-1}& ={\partial }_{x^{\mu }}+i{\displaystyle \frac{e}{\widehat{e}}}\left(A_{\mu }-{\displaystyle \frac{\widehat{e}}{e}}{\partial }_{x^{\mu }}ϵ\right),\hfill \end{array}$$

(136)

$$\begin{array}{cc}\hfill {\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}{\widehat{D}}_{\mu }^{\left(K\right)}{\left({\widehat{\mathcal{U}}}_{\widehat{ϵ}}^{\left(K\right)}\right)}^{-1}& ={\partial }_{k^{\mu }}+i{\displaystyle \frac{\widehat{e}}{e}}\left({\widehat{A}}_{\mu }-{\displaystyle \frac{e}{\widehat{e}}}{\partial }_{k^{\mu }}\widehat{ϵ}\right),\hfill \end{array}$$

(137)

or equivalently,

$$\begin{array}{cc}\hfill A_{\mu }^{\prime }\left(x\right)& =A_{\mu }\left(x\right)-{\displaystyle \frac{\widehat{e}}{e}}{\partial }_{x^{\mu }}ϵ\left(x\right),\hfill \end{array}$$

(138)

$$\begin{array}{cc}\hfill {\widehat{A}}_{\mu }^{\prime }\left(k\right)& ={\widehat{A}}_{\mu }\left(k\right)-{\displaystyle \frac{e}{\widehat{e}}}{\partial }_{k^{\mu }}\widehat{ϵ}\left(k\right).\hfill \end{array}$$

(139)

Proof. 

Using (130),

$$\begin{array}{cc}\hfill {\mathcal{U}}_{ϵ}^{\left(X\right)}{D}_{\mu }^{\left(X\right)}{\left({\mathcal{U}}_{ϵ}^{\left(X\right)}\right)}^{-1}& ={\mathcal{U}}_{ϵ}^{\left(X\right)}\left({\partial }_{x^{\mu }}+i{\displaystyle \frac{e}{\widehat{e}}}{A}_{\mu }\right){\left({\mathcal{U}}_{ϵ}^{\left(X\right)}\right)}^{-1}\hfill \\ \hfill & ={\partial }_{x^{\mu }}-i{\partial }_{x^{\mu }}ϵ+i{\displaystyle \frac{e}{\widehat{e}}}{A}_{\mu }\hfill \\ \hfill & ={\partial }_{x^{\mu }}+i{\displaystyle \frac{e}{\widehat{e}}}\left(A_{\mu }-{\displaystyle \frac{\widehat{e}}{e}}{\partial }_{x^{\mu }}ϵ\right),\hfill \end{array}$$

which is (136). The K-space formula follows identically from (131); the equivalent transformation laws (138) and (139) are just a rewriting of these identities. □

Remark 8 (Standard local $U\left(1\right)$ form). 

If one reparametrizes the phase variables by

$${\alpha }_X:={\displaystyle \frac{\widehat{e}}{e}}ϵ,{\alpha }_K:={\displaystyle \frac{e}{\widehat{e}}}\widehat{ϵ},$$

then (126), (127), (138), and (139) become

$$\begin{array}{cccc}\hfill f\left(x\right)& \mapsto e^{i(e/\widehat{e}){\alpha }_X\left(x\right)}f\left(x\right),\hfill & \hfill A_{\mu }& \mapsto A_{\mu }-{\partial }_{x^{\mu }}{\alpha }_X,\hfill \\ \hfill \widehat{f}\left(k\right)& \mapsto e^{i(\widehat{e}/e){\alpha }_K\left(k\right)}\widehat{f}\left(k\right),\hfill & \hfill {\widehat{A}}_{\mu }& \mapsto {\widehat{A}}_{\mu }-{\partial }_{k^{\mu }}{\alpha }_K.\hfill \end{array}$$

Thus both polarizations carry the standard local abelian $U\left(1\right)$ transformation law, with effective couplings $e/\widehat{e}$ and $\widehat{e}/e$.

8.3. Curvature as the Central Coefficient of the Covariantized Translation Commutator

The covariantization just constructed preserves the represented Heisenberg normalization. The curvature sector is then exactly the obstruction to commutativity of the covariantized differential generators.

Proposition 21 (Compatibility with ${\hslash }_{\mathrm{eff}}=e\widehat{e}$). 

The covariantized generators preserve the represented Heisenberg normalization:

$$\begin{array}{cc}\hfill [{\mathcal{T}}_{D,\mu }^{\left(X\right)},{\mathcal{Q}}_{\nu }^{\left(X\right)}]& =-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I},\hfill \end{array}$$

(140)

$$\begin{array}{cc}\hfill [{\mathcal{Q}}_{\mu }^{\left(K\right)},{\mathcal{T}}_{D,\nu }^{\left(K\right)}]& =-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I}.\hfill \end{array}$$

(141)

Proof. 

Since $A_{\mu }\left(x\right)$ commutes with ${\mathcal{Q}}_{\nu }^{\left(X\right)}=e{x}_{\nu }$,

$$[{\mathcal{T}}_{D,\mu }^{\left(X\right)},{\mathcal{Q}}_{\nu }^{\left(X\right)}]=[{\mathcal{T}}_{\mu }^{\left(X\right)},{\mathcal{Q}}_{\nu }^{\left(X\right)}]=-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I}.$$

Likewise, ${\widehat{A}}_{\mu }\left(k\right)$ commutes with ${\mathcal{Q}}_{\nu }^{\left(K\right)}=\widehat{e}{k}_{\nu }$, so

$$[{\mathcal{Q}}_{\mu }^{\left(K\right)},{\mathcal{T}}_{D,\nu }^{\left(K\right)}]=[{\mathcal{Q}}_{\mu }^{\left(K\right)},{\mathcal{T}}_{\nu }^{\left(K\right)}]=-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I}.$$

□

Definition 8 (Curvatures). 

Define

$$\begin{array}{cc}\hfill F_{\mu \nu }\left(x\right)& :={\partial }_{x^{\mu }}{A}_{\nu }-{\partial }_{x^{\nu }}{A}_{\mu },\hfill \end{array}$$

(142)

$$\begin{array}{cc}\hfill {\widehat{F}}_{\mu \nu }\left(k\right)& :={\partial }_{k^{\mu }}{\widehat{A}}_{\nu }-{\partial }_{k^{\nu }}{\widehat{A}}_{\mu }.\hfill \end{array}$$

(143)

Proposition 22 (Curvature commutators and gauge invariance). 

One has

$$\begin{array}{cc}\hfill [D_{\mu }^{\left(X\right)},D_{\nu }^{\left(X\right)}]& =i{\displaystyle \frac{e}{\widehat{e}}}{F}_{\mu \nu }\left(x\right)\mathbb{I},\hfill \end{array}$$

(144)

$$\begin{array}{cc}\hfill [{\widehat{D}}_{\mu }^{\left(K\right)},{\widehat{D}}_{\nu }^{\left(K\right)}]& =i{\displaystyle \frac{\widehat{e}}{e}}{\widehat{F}}_{\mu \nu }\left(k\right)\mathbb{I},\hfill \end{array}$$

(145)

and therefore

$$\begin{array}{cc}\hfill [{\mathcal{T}}_{D,\mu }^{\left(X\right)},{\mathcal{T}}_{D,\nu }^{\left(X\right)}]& =-i{\hslash }_{\mathrm{eff}}{F}_{\mu \nu }\left(x\right)\mathbb{I},\hfill \end{array}$$

(146)

$$\begin{array}{cc}\hfill [{\mathcal{T}}_{D,\mu }^{\left(K\right)},{\mathcal{T}}_{D,\nu }^{\left(K\right)}]& =-i{\hslash }_{\mathrm{eff}}{\widehat{F}}_{\mu \nu }\left(k\right)\mathbb{I}.\hfill \end{array}$$

(147)

Moreover, $F_{\mu \nu }$ and ${\widehat{F}}_{\mu \nu }$ are invariant under (138) and (139).

Proof. 

Since multiplication operators commute,

$$\begin{array}{cc}\hfill [D_{\mu }^{\left(X\right)},D_{\nu }^{\left(X\right)}]& =\left[{\partial }_{x^{\mu }}+i{\displaystyle \frac{e}{\widehat{e}}}{A}_{\mu },{\partial }_{x^{\nu }}+i{\displaystyle \frac{e}{\widehat{e}}}{A}_{\nu }\right]\hfill \\ \hfill & =i{\displaystyle \frac{e}{\widehat{e}}}\left({\partial }_{x^{\mu }}{A}_{\nu }-{\partial }_{x^{\nu }}{A}_{\mu }\right)\mathbb{I}=i{\displaystyle \frac{e}{\widehat{e}}}{F}_{\mu \nu }\left(x\right)\mathbb{I},\hfill \end{array}$$

which is (144). The K-space identity is identical in form:

$$[{\widehat{D}}_{\mu }^{\left(K\right)},{\widehat{D}}_{\nu }^{\left(K\right)}]=i{\displaystyle \frac{\widehat{e}}{e}}{\widehat{F}}_{\mu \nu }\left(k\right)\mathbb{I}.$$

Using ${\hslash }_{\mathrm{eff}}=e\widehat{e}$, one obtains

$$[{\mathcal{T}}_{D,\mu }^{\left(X\right)},{\mathcal{T}}_{D,\nu }^{\left(X\right)}]={(-i\widehat{e})}^2[D_{\mu }^{\left(X\right)},D_{\nu }^{\left(X\right)}]=-i{\hslash }_{\mathrm{eff}}{F}_{\mu \nu }\left(x\right)\mathbb{I},$$

and similarly

$$[{\mathcal{T}}_{D,\mu }^{\left(K\right)},{\mathcal{T}}_{D,\nu }^{\left(K\right)}]={\left(ie\right)}^2[{\widehat{D}}_{\mu }^{\left(K\right)},{\widehat{D}}_{\nu }^{\left(K\right)}]=-i{\hslash }_{\mathrm{eff}}{\widehat{F}}_{\mu \nu }\left(k\right)\mathbb{I}.$$

Finally, invariance of $F_{\mu \nu }$ and ${\widehat{F}}_{\mu \nu }$ follows immediately from (138)–(139), because antisymmetrized derivatives annihilate exact one-form shifts. □

Remark 9 (Momentum-space gauge precedent). 

As a mathematical precedent for gauge-covariant differentiation directly in momentum variables, compare the Berry-connection formalism reviewed in [45]. The present K-space construction is structurally different, since it is derived representation-theoretically from the Lorentz–Heisenberg realization rather than from adiabatic band geometry.

8.4. Primary Dirac–Maxwell Systems

We now apply the preceding structural results to the null-fiber dynamics of Section 7. Once the covariant derivatives have been fixed by the represented Heisenberg sector, no further structural input is needed. As before, the full derivation is given only in K-space, while the X-space system is recorded as the Fourier-dual counterpart.

Assume the standard Clifford relations

$$\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }\mathbb{I}.$$

(148)

8.4.1. Primary K-Space Dirac–Maxwell System

Let ${\widehat{f}}_x\left(k\right)$ be a Dirac spinor on a fixed x-spectral fiber, and write

$${\widehat{f}}_x^{•}:={\widehat{f}}_x^{†}{\gamma }^0,{\widehat{j}}^{\mu }\left(k\right):={\widehat{f}}_x^{•}\left(k\right){\gamma }^{\mu }{\widehat{f}}_x\left(k\right).$$

Consider the action

$${\mathcal{S}}_x^{\left(K\right)}[{\widehat{f}}_x,\widehat{A}]:={\int }_{{\mathbb{R}}^{3,1}}{d}^{4}k\left[{\widehat{f}}_x^{•}\left(i{\gamma }^{\mu }{\widehat{D}}_{\mu }^{\left(K\right)}-{\sigma }_{x^{*}}\right){\widehat{f}}_x-{\displaystyle \frac{1}{4}}{\widehat{F}}^{\mu \nu }{\widehat{F}}_{\mu \nu }\right].$$

(149)

Theorem 5 (Primary K-space Dirac–Maxwell system). 

On the null spectral fiber $x^{\mu }{x}_{\mu }=0$, the action (149) is invariant under

$${\widehat{f}}_x\left(k\right)\mapsto e^{i\widehat{ϵ}\left(k\right)}{\widehat{f}}_x\left(k\right),{\widehat{A}}_{\mu }\left(k\right)\mapsto {\widehat{A}}_{\mu }\left(k\right)-{\displaystyle \frac{e}{\widehat{e}}}{\partial }_{k^{\mu }}\widehat{ϵ}\left(k\right).$$

(150)

Its Euler–Lagrange equations are

$$\begin{array}{cc}\hfill \left(i{\gamma }^{\mu }{\widehat{D}}_{\mu }^{\left(K\right)}-{\sigma }_{x^{*}}\right){\widehat{f}}_x& =0,\hfill \end{array}$$

(151)

$$\begin{array}{cc}\hfill {\partial }_{k^{\nu }}{\widehat{F}}^{\nu \mu }& ={\displaystyle \frac{\widehat{e}}{e}}{\widehat{j}}^{\mu }.\hfill \end{array}$$

(152)

If

$${\widehat{f}}_x^{•}{\overleftarrow{\widehat{D}}}_{\mu }^{\left(K\right)}:={\partial }_{k^{\mu }}{\widehat{f}}_x^{•}-i{\displaystyle \frac{\widehat{e}}{e}}{\widehat{A}}_{\mu }{\widehat{f}}_x^{•},$$

then the adjoint Dirac equation is

$${\widehat{f}}_x^{•}\left(i{\overleftarrow{\widehat{D}}}_{\mu }^{\left(K\right)}{\gamma }^{\mu }+{\sigma }_{x^{*}}\right)=0,$$

(153)

and every solution satisfies

$${\partial }_{k^{\mu }}{\widehat{j}}^{\mu }=0.$$

(154)

Proof. 

Gauge invariance follows from (137). Since $x_{\mu }$ is fixed on the chosen spectral fiber, it commutes with the local phase multiplier, and ${\widehat{F}}_{\mu \nu }$ is gauge-invariant.

Variation with respect to ${\widehat{f}}_x^{•}$ gives (151). Variation with respect to ${\widehat{f}}_x$, after integration by parts, gives (153). Variation with respect to ${\widehat{A}}_{\mu }$ yields (152), because

$${\displaystyle \frac{\partial }{\partial {\widehat{A}}_{\mu }}}\left({\widehat{f}}_x^{•}i{\gamma }^{\nu }{\widehat{D}}_{\nu }^{\left(K\right)}{\widehat{f}}_x\right)=-{\displaystyle \frac{\widehat{e}}{e}}{\widehat{f}}_x^{•}{\gamma }^{\mu }{\widehat{f}}_x=-{\displaystyle \frac{\widehat{e}}{e}}{\widehat{j}}^{\mu }.$$

Finally,

$${\partial }_{k^{\mu }}{\widehat{j}}^{\mu }=\left({\widehat{f}}_x^{•}{\overleftarrow{\widehat{D}}}_{\mu }^{\left(K\right)}\right){\gamma }^{\mu }{\widehat{f}}_x+{\widehat{f}}_x^{•}{\gamma }^{\mu }{\widehat{D}}_{\mu }^{\left(K\right)}{\widehat{f}}_x,$$

and the two terms cancel by (151) and (153). □

8.4.2. Primary X-Space Dirac–Maxwell System

The X-polarized system is obtained by the same construction with x- and k-roles interchanged.

Let $f_k\left(x\right)$ be a Dirac spinor on a fixed k-spectral fiber, and write

$$f_k^{•}:=f_k^{†}{\gamma }^0,j^{\mu }\left(x\right):=f_k^{•}\left(x\right){\gamma }^{\mu }{f}_k\left(x\right).$$

Consider the action

$${\mathcal{S}}_x^{\left(K\right)}[{\widehat{f}}_x,\widehat{A}]:={\int }_{{\mathbb{R}}^{3,1}}{d}^{4}k\left[{\widehat{f}}_x^{•}\left(i{\gamma }^{\mu }{\widehat{D}}_{\mu }^{\left(K\right)}-{\sigma }_{x^{*}}\right){\widehat{f}}_x-{\displaystyle \frac{1}{4}}{\widehat{F}}^{\mu \nu }{\widehat{F}}_{\mu \nu }\right].$$

(155)

Theorem 6 (Primary X-space Dirac–Maxwell system). 

On the null spectral fiber $k^{\mu }{k}_{\mu }=0$, the action (155) is invariant under

$$f_k\left(x\right)\mapsto e^{iϵ\left(x\right)}{f}_k\left(x\right),A_{\mu }\left(x\right)\mapsto A_{\mu }\left(x\right)-{\displaystyle \frac{\widehat{e}}{e}}{\partial }_{x^{\mu }}ϵ\left(x\right).$$

(156)

Its Euler–Lagrange equations are

$$\begin{array}{cc}\hfill \left(i{\gamma }^{\mu }{D}_{\mu }^{\left(X\right)}-{\sigma }_{k^{*}}\right)f_k& =0,\hfill \end{array}$$

(157)

$$\begin{array}{cc}\hfill {\partial }_{x^{\nu }}{F}^{\nu \mu }& ={\displaystyle \frac{e}{\widehat{e}}}{j}^{\mu }.\hfill \end{array}$$

(158)

If

$$f_k^{•}{\overleftarrow{D}}_{\mu }^{\left(X\right)}:={\partial }_{x^{\mu }}{f}_k^{•}-i{\displaystyle \frac{e}{\widehat{e}}}{A}_{\mu }{f}_k^{•},$$

then the adjoint Dirac equation is

$$f_k^{•}\left(i{\overleftarrow{D}}_{\mu }^{\left(X\right)}{\gamma }^{\mu }+{\sigma }_{k^{*}}\right)=0,$$

(159)

and every solution satisfies

$${\partial }_{x^{\mu }}{j}^{\mu }=0.$$

(160)

Proof. 

This is the X-polarized counterpart of Theorem 5. Starting from (136), one repeats the same variational argument after the replacements

$$k\leftrightarrow x,{\widehat{A}}_{\mu }\leftrightarrow A_{\mu },{\displaystyle \frac{\widehat{e}}{e}}\leftrightarrow {\displaystyle \frac{e}{\widehat{e}}},{\widehat{f}}_x\leftrightarrow f_k.$$

This yields (157)–(160). □

8.5. Structural Conclusion

We can now summarize the logic of the section. The affine Weyl seed is forced by the represented Heisenberg commutator through BCH. The full local phase algebra is then forced, in the regular realization, by the joint functional calculus of the commuting coordinate family. Differentiating the induced phase action yields the gauge shift of the differential generators, from which the covariant derivatives, curvatures, and null-fiber Dirac–Maxwell systems follow.

Theorem 7 (Structural origin of the local $U\left(1\right)$ systems). 

Within the standard regular Fourier-dual realizations of the Lorentz–Heisenberg algebra, the affine Weyl phases, the full local $U\left(1\right)$ phase groups in both polarizations, the compatible connection transformation laws, the covariant first-order operators, the antisymmetric curvatures, and the resulting primary Dirac–Maxwell systems are all structural consequences of the represented Heisenberg sector. In particular, within the chosen regular polarized realizations, they do not require an additional independent gauge-symmetry postulate beyond the represented Lorentz–Heisenberg structure and the realization data used to construct the local phase algebra.

Proof. 

The affine Weyl seed is identified by Proposition 18. The full local phase algebra is canonically generated from the commuting coordinate family by Proposition 19. The infinitesimal phase action on the differential generators is given by Lemma 3, which fixes the gauge shift laws of Proposition 20. The represented Heisenberg normalization is preserved by Proposition 21, and the antisymmetric curvatures arise as the central coefficients of the covariantized translation commutators by Proposition 22. Substituting the resulting covariant derivatives into the null-fiber Dirac and Maxwell systems of Section 7 yields the K-space system of Theorem 5 and, by the same argument, the X-space system of Theorem 6. Thus the full local abelian gauge structure is induced by the represented Heisenberg sector itself. □

The logical hierarchy of the section is therefore

$$\begin{array}{cc}\hfill \mathrm{mixed}\mathrm{central}\mathrm{extension}& ⟹\mathrm{affine}\mathrm{Weyl}\mathrm{relations}\hfill \\ \hfill & ⟹\mathrm{full}\mathrm{local}\mathrm{phase}\mathrm{algebra}\hfill \\ \hfill & ⟹\mathrm{covariant}\mathrm{generators}\mathrm{and}\mathrm{antisymmetric}\mathrm{curvature}\hfill \\ \hfill & ⟹\mathrm{primary}\mathrm{Dirac}–\mathrm{Maxwell}\mathrm{systems}.\hfill \end{array}$$

This is the precise sense in which the local abelian gauge sector arises representation-theoretically from the Lorentz–Heisenberg framework.

8.6. Independent Phase-Coupling Refinement and Fundamentally Neutral Objects

The local abelian $U\left(1\right)$ sector derived above is structural and is fixed by the represented Heisenberg sector together with the realization data

$$\rho \left(\mathbf{1}\right)={\hslash }_{\mathrm{eff}}\mathbb{I},{\hslash }_{\mathrm{eff}}=e\widehat{e}\ne 0.$$

These data determine the realized Heisenberg generators and should not be identified with the abelian phase-coupling strength of every object carried by the same fixed nondegenerate sector. We therefore allow independent object-level phase couplings

$$q_X,q_K\in \mathbb{R}$$

without varying $e,\widehat{e}$ or ${\hslash }_{\mathrm{eff}}$.

Definition 9 (Independent phase-coupling covariantization). 

Define

$$\begin{array}{cccc}\hfill D_{\mu }^{(X,q_X)}& :={\partial }_{x^{\mu }}+i{q}_X{A}_{\mu }\left(x\right),\hfill & \hfill {\mathcal{T}}_{D,\mu }^{(X,q_X)}& :=-i\widehat{e}{D}_{\mu }^{(X,q_X)}={\mathcal{T}}_{\mu }^{\left(X\right)}+\widehat{e}{q}_X{A}_{\mu }\left(x\right)\mathbb{I},\hfill \end{array}$$

(161)

$$\begin{array}{cccc}\hfill {\widehat{D}}_{\mu }^{(K,q_K)}& :={\partial }_{k^{\mu }}+i{q}_K{\widehat{A}}_{\mu }\left(k\right),\hfill & \hfill {\mathcal{T}}_{D,\mu }^{(K,q_K)}& :=ie{\widehat{D}}_{\mu }^{(K,q_K)}={\mathcal{T}}_{\mu }^{\left(K\right)}-e{q}_K{\widehat{A}}_{\mu }\left(k\right)\mathbb{I}.\hfill \end{array}$$

(162)

No relation is imposed between $q_X,q_K$ and the realization data $e,\widehat{e}$.

Remark 10 (Canonical covariantization as a distinguished case). 

The canonical covariantization of (134)–(135) is recovered by the special choice

$$q_X={\displaystyle \frac{e}{\widehat{e}}},q_K={\displaystyle \frac{\widehat{e}}{e}},$$

for which

$$D_{\mu }^{(X,q_X)}=D_{\mu }^{\left(X\right)},{\widehat{D}}_{\mu }^{(K,q_K)}={\widehat{D}}_{\mu }^{\left(K\right)},$$

and

$${\mathcal{T}}_{D,\mu }^{(X,q_X)}={\mathcal{T}}_{D,\mu }^{\left(X\right)},{\mathcal{T}}_{D,\mu }^{(K,q_K)}={\mathcal{T}}_{D,\mu }^{\left(K\right)}.$$

Thus the structurally derived abelian phase sector is the distinguished canonical case inside a larger object-level coupling family.

Proposition 23 (Independent phase couplings preserve the represented Heisenberg sector). 

For every $q_X,q_K\in \mathbb{R}$,

$$\begin{array}{cc}\hfill [{\mathcal{T}}_{D,\mu }^{(X,q_X)},{\mathcal{Q}}_{\nu }^{\left(X\right)}]& =-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I},\hfill \end{array}$$

(163)

$$\begin{array}{cc}\hfill [{\mathcal{Q}}_{\mu }^{\left(K\right)},{\mathcal{T}}_{D,\nu }^{(K,q_K)}]& =-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I}.\hfill \end{array}$$

(164)

Moreover,

$$\begin{array}{cc}\hfill [D_{\mu }^{(X,q_X)},D_{\nu }^{(X,q_X)}]& =i{q}_X{F}_{\mu \nu }\left(x\right)\mathbb{I},\hfill \end{array}$$

(165)

$$\begin{array}{cc}\hfill [{\widehat{D}}_{\mu }^{(K,q_K)},{\widehat{D}}_{\nu }^{(K,q_K)}]& =i{q}_K{\widehat{F}}_{\mu \nu }\left(k\right)\mathbb{I},\hfill \end{array}$$

(166)

and therefore

$$\begin{array}{cc}\hfill [{\mathcal{T}}_{D,\mu }^{(X,q_X)},{\mathcal{T}}_{D,\nu }^{(X,q_X)}]& =-i{\widehat{e}}^2{q}_X{F}_{\mu \nu }\left(x\right)\mathbb{I},\hfill \end{array}$$

(167)

$$\begin{array}{cc}\hfill [{\mathcal{T}}_{D,\mu }^{(K,q_K)},{\mathcal{T}}_{D,\nu }^{(K,q_K)}]& =-i{e}^2{q}_K{\widehat{F}}_{\mu \nu }\left(k\right)\mathbb{I}.\hfill \end{array}$$

(168)

Proof. 

Since $A_{\mu }\left(x\right)$ commutes with ${\mathcal{Q}}_{\nu }^{\left(X\right)}=e{x}_{\nu }$,

$$[{\mathcal{T}}_{D,\mu }^{(X,q_X)},{\mathcal{Q}}_{\nu }^{\left(X\right)}]=[{\mathcal{T}}_{\mu }^{\left(X\right)},{\mathcal{Q}}_{\nu }^{\left(X\right)}]=-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I}.$$

Likewise, ${\widehat{A}}_{\mu }\left(k\right)$ commutes with ${\mathcal{Q}}_{\nu }^{\left(K\right)}=\widehat{e}{k}_{\nu }$, so

$$[{\mathcal{Q}}_{\mu }^{\left(K\right)},{\mathcal{T}}_{D,\nu }^{(K,q_K)}]=[{\mathcal{Q}}_{\mu }^{\left(K\right)},{\mathcal{T}}_{\nu }^{\left(K\right)}]=-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I}.$$

Because multiplication operators commute,

$$\begin{array}{cc}\hfill [D_{\mu }^{(X,q_X)},D_{\nu }^{(X,q_X)}]& =\left[{\partial }_{x^{\mu }}+i{q}_X{A}_{\mu },{\partial }_{x^{\nu }}+i{q}_X{A}_{\nu }\right]\hfill \\ \hfill & =i{q}_X\left({\partial }_{x^{\mu }}{A}_{\nu }-{\partial }_{x^{\nu }}{A}_{\mu }\right)\mathbb{I}=i{q}_X{F}_{\mu \nu }\left(x\right)\mathbb{I},\hfill \end{array}$$

and similarly

$$[{\widehat{D}}_{\mu }^{(K,q_K)},{\widehat{D}}_{\nu }^{(K,q_K)}]=i{q}_K{\widehat{F}}_{\mu \nu }\left(k\right)\mathbb{I}.$$

Multiplying by ${(-i\widehat{e})}^2=-{\widehat{e}}^2$ and ${\left(ie\right)}^2=-e^2$ gives (167) and (168). □

Definition 10 (Fundamentally neutral object). 

A fundamentally neutral object is one for which the abelian phase-coupling parameters vanish:

$$q_X=0,q_K=0.$$

Equivalently,

$$D_{\mu }^{(X,0)}={\partial }_{x^{\mu }},{\widehat{D}}_{\mu }^{(K,0)}={\partial }_{k^{\mu }},$$

and

$${\mathcal{T}}_{D,\mu }^{(X,0)}={\mathcal{T}}_{\mu }^{\left(X\right)},{\mathcal{T}}_{D,\mu }^{(K,0)}={\mathcal{T}}_{\mu }^{\left(K\right)}.$$

Proposition 24 (Neutral objects preserve the fixed nondegenerate Heisenberg sector). 

A fundamentally neutral object remains in the same fixed irreducible nondegenerate Lorentz–Heisenberg sector

$$\rho \left(\mathbf{1}\right)={\hslash }_{\mathrm{eff}}\mathbb{I},{\hslash }_{\mathrm{eff}}\ne 0,$$

but carries no abelian phase coupling. More precisely,

$$\begin{array}{cccc}\hfill [{\mathcal{T}}_{D,\mu }^{(X,0)},{\mathcal{Q}}_{\nu }^{\left(X\right)}]& =-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I},\hfill & \hfill [{\mathcal{T}}_{D,\mu }^{(X,0)},{\mathcal{T}}_{D,\nu }^{(X,0)}]& =0,\hfill \end{array}$$

(169)

$$\begin{array}{cccc}\hfill [{\mathcal{Q}}_{\mu }^{\left(K\right)},{\mathcal{T}}_{D,\nu }^{(K,0)}]& =-i{\eta }_{\mu \nu }{\hslash }_{\mathrm{eff}}\mathbb{I},\hfill & \hfill [{\mathcal{T}}_{D,\mu }^{(K,0)},{\mathcal{T}}_{D,\nu }^{(K,0)}]& =0.\hfill \end{array}$$

(170)

Proof. 

The mixed commutators are the special case $q_X=q_K=0$ of (163) and (164). The translation–translation commutators are the special case $q_X=q_K=0$ of (167) and (168). □

Remark 11. 

Neutrality is therefore implemented by vanishing abelian phase coupling, not by varying the realization parameters $e,\widehat{e}$ and not by altering the fixed central character ${\hslash }_{\mathrm{eff}}=e\widehat{e}$. The represented Heisenberg sector remains unchanged.

9. Conclusions

This paper solved a specific realization-theoretic problem in relativistic symmetry: whether the geometric translation sectors of position space and momentum space can be placed on equal structural footing within the standard Lorentz-covariant Fourier-dual framework. Working entirely in the X- and K-polarized realizations on $L^2\left({\mathbb{R}}^{3,1}\right)$, with all basic operator identities first formulated on the common invariant Schwartz core $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, this manuscript identified the decisive obstruction: Fourier transform exchanges geometric translations with character actions rather than intertwining the two geometric translation sectors [8,9]. Together with the uniqueness of the Lorentz-covariant abelian translation ideal inside the Poincaré algebra, this is the precise reason why the standard Poincaré framework does not already realize the required symmetry.

The manuscript was organized accordingly. Section 2 and Section 3 supplied the preliminary Fourier-analytic, operator-theoretic, and Lorentz-covariant background and isolated the obstruction in a self-contained form. The original contribution then began in Section 4, where the Dual Relativity Principle was imposed inside the fixed Fourier-compatible realization class and the minimal operator-generated Lie algebra was proved to be the Lorentz–Heisenberg algebra.

This is the precise sense in which the paper is axiomatically minimal. It does not begin by independently postulating a Heisenberg sector, a doubled phase-space group, or a local gauge symmetry. Rather, it begins with a single structural principle together with a fixed realization framework and determines what algebraic and representation-theoretic structure is forced inside that framework. In particular, the mixed Heisenberg bracket arises not as an arbitrary deformation but as the minimal Lorentz-equivariant scalar central extension compatible with the doubled geometric translation requirement.

Once this minimal realization is fixed, its consequences follow systematically. This paper determines the full center

$$Z\left(U\left(\mathfrak{g}\right)\right)=\mathbb{C}[C_0,C_1,C_2],$$

extracts the central character ${\hslash }_{\mathrm{eff}}$ and the normalized Casimir scalars $c_1,c_2$, introduces the normalized commuting Lorentz bivectors ${\tilde{J}}_{\mu \nu }$, and shows that the resulting orbit theory is governed by normalized Lorentz-bivector data rather than by momentum-vector data. In particular, the orbit classification yields a mixed regular sector $c_2\ne 0$ with no analogue in the standard Wigner vector-orbit picture. In this respect, the construction lies closer in spirit to orbit-theoretic and geometric-quantization viewpoints than to the particle-sector classification of Wigner and Bargmann, while remaining distinct from both.

The same principle-to-realization-to-consequence logic governs the dynamical and gauge-theoretic parts of the manuscript. For fixed spectral elements in the dual translation sectors, the realized operators determine Lorentz-covariant scalar, Dirac, and vector equations in momentum and position space. In the regular polarized realizations, the represented Heisenberg sector further induces dual local abelian phase groups, compatible covariant derivatives, antisymmetric curvatures, and the primary Dirac–Maxwell systems. Within that same regular-realization package, the canonical abelian covariantization also admits an independent object-level phase-coupling refinement: fundamentally neutral objects are obtained by vanishing abelian phase coupling without changing the fixed irreducible nondegenerate Lorentz–Heisenberg sector. Thus the local $U\left(1\right)$ structures studied here are not separately postulated ingredients; they are structural consequences of the realized Lorentz–Heisenberg framework.

The resulting position of this paper in the literature is therefore narrow and explicit. It is not a realization-independent classification of all Lorentz-equivariant doubled translation extensions, not a Mackey classification of an integrated Lorentz–Heisenberg group, and not a deformation of relativistic symmetry in the usual stabilized or noncommutative sense. Its contribution is instead to clearly separate three levels that are often conflated: the guiding symmetry principle, the admissible Fourier-compatible realization class, and the algebraic, orbit-theoretic, dynamical, and gauge-theoretic structures forced once that class is fixed.

Several natural problems remain open. The physical interpretation of the mixed regular sector deserves separate investigation. A realization-independent classification of abstract Lorentz-equivariant doubled translation extensions and of the corresponding integrated groups and unitary representations remains to be developed. It is also natural to ask whether the same structural mechanism persists in curved, deformed, or more general phase-space settings. These questions lie beyond the scope of the present article. What the paper establishes is the narrower but foundational statement required here: within the standard Fourier-compatible realization class, the Lorentz–Heisenberg algebra and its associated orbit, dynamical, and local abelian gauge structures are forced by the Dual Relativity Principle itself.