# Coulomb Solutions from Improper Pseudo-Unitary Free Gauge Field Operator Translations

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

_{l}, q

_{m}under mild natural requirements as generators of unitary transformations on a Hilbert space up to unitary equivalence, provided F is finite. Already for the case F = 1 it is straightforward to show that an algebra fulfilling the commutation relations Equation (1) cannot be represented by operators defined on a finite-dimensional Hilbert space $\mathscr{H}$

_{f}, since (ħ = 1)

_{1},

_{3}and the proper orthochronous Lorentz group ${\mathcal{L}}_{+}^{\u2020}$

^{2}= P

_{μ}P

^{μ}of a relativistic quantum field theory describing interacting fields. One should note here that the particles in the present sense like, e.g., a neutron or an atom, can be viewed as composite objects, and the notion elementary system might be more appropriate. Then, objects like quark and gluons can be viewed as elementary particles, although they do not appear in the physical spectrum of the Standard Model. The job of the corresponding elementary fields as carriers of charges is rather to implement the principle of causality and to allow for a kind of coordinatization of an underlying physical theory and to finally extract the algebra of observables. The type and number of the elementary fields appearing in a theory is rather unrelated to the physical spectrum of empirically observable particles, i.e., elementary systems.

^{2}is absent for states with an electric charge as a direct consequence of Gauss’ law, and one finds that the Lorentz symmetry is not implementable in a sector of states with nonvanishing electric charge, an issue that also will be an aspect of the forthcoming discussion. Such problems are related to the fact that the Poincaré symmetry is an overidealization related to global considerations of infinite flat space-time, whereas physical measurements have a local character. The expression infraparticle has been coined for charged particles like the electron accompanied by a dressing field of massless particles [13].

^{2}> 0 and a (half-)integer spin parameter s. In the massless case, the unitary irreducible representations of ${\overline{\mathcal{P}}}_{+}^{\uparrow}$, which have played an important rôle in quantum field theory so far, are those that describe particles with a given non-negative (half-)integer helicity.

_{Ξ}

_{,α}of ${\overline{\mathcal{P}}}_{+}^{\uparrow}$ [14], which are related to the so-called string-localized quantum fields [15]. These representations can be labeled by two parameters: 0 < Ξ < ∞ and $\alpha \in \{0,\frac{1}{2}\}$. The representations describe massless objects with a spin operator along the momentum having the unbounded spectrum {0, ±1, ±2,…} for α = 0 and $\{\pm \frac{1}{2},\pm \frac{3}{2},\dots \}$ for $\alpha =\frac{1}{2}$. There are still ongoing investigations in order to find out whether string-localized quantum fields will have any direct application in future quantum field theories [16]. Since the infinite spin representations can be distinguished by the continuous parameter Ξ, they are also called continuous spin representations, a naming that sometimes leads to some confusion about the helicity spectrum, which is quantized but infinite.

## 2. The Electromagnetic Field

^{0123}= 1 = −ϵ

_{0123}

^{μν}invariant since

^{μ}. Since for a pure gauge ${A}_{pg}^{\mu}={\partial}^{\mu}\mathrm{\chi}$

^{0}= Φ.

_{unph}, which in the case of the so-called Feynman gauge is chosen according to

^{μ}describing a non-interacting massless spin-1 field from a more general point of view become

^{μ}can be gauged away by a suitable scalar χ that solves

_{0}fulfilling □χ

_{0}(x) = 0. The formal strategy described above works well even after quantization for QED. However, when gauge fields couple to themselves, special care is needed.

^{ν}

_{μ}A

^{μ}= 0 leads to

^{3}-direction is given by

^{0}) is a normalization factor, whereas the corresponding left-handed plane wave is given by

^{3}-direction (k

^{3}< 0), one has

_{1}, Σ

_{2}, and Σ

_{3}defined by the totally antisymmetric tensor in three dimensions ${\epsilon}_{lmn}=\frac{1}{2}(l-m)(m-n)(n-l)$

_{j}= ∂/∂x

^{j}, j = 1, 2, 3)

^{μ}by, where 1

_{3}denotes the 3 × 3 identity matrix, and Γ

_{j}= Σ

_{j}= −Γ

^{j}for j = 1, 2, 3, Equation (36) finally reads

^{+}(1, 3) by the isomorphic complex orthogonal group $SO(3,\u2102)$, preserving the conditions imposed by Equation (38).

## 3. Lorentz-Covariant Quantization of the Free Gauge Field

^{μ}(x) are acting on a Fock–Hilbert space $\mathrm{\mathcal{F}}$ with positive-definite norm, and since the free field

## 4. Charge and Gauge Transformations as Field Translations

^{0}(x) on the Fock vacuum |0⟩ since

^{0}(x) → A′

^{0}(x) preserves the skew-adjointness of A

^{0}.

^{μ}(x) = ∂

^{μ}χ(x) with a smooth scalar χ rapidly decreasing in space-like directions and fulfilling the wave equation $\square \mathrm{\chi}(x)=0,\tilde{Q}$ becomes a BRST-generator ${\tilde{Q}}_{g}$ of free field gauge transformations [24]. Introducing emission and absorption operators for unphysical photons, which are combinations of time-like and longitudinal states, according to

_{phys}⊂$\mathrm{\mathcal{F}}$ contains no free unphysical photons

## 5. Static Fields

^{μ}(x) by additional classical fields q

^{μ}(x), which are solutions of the wave equation. This minor defect if one wants to describe static fields can be remedied by adding a time-dependence to the classical q

^{μ}-fields, which become ${\tilde{q}}^{\mu}({x}_{0},\overrightarrow{k})={q}^{\mu}(\overrightarrow{k}){e}^{i{k}^{0}{x}^{0}}$. With the sometimes more suggestive notation $t={x}^{0},\omega ={k}^{0}=|\overrightarrow{k}|$ and the definitions

## 6. Particle Numbers

_{μ}are Schwartz test functions, i.e., when the ${a}_{\mu}{(\overrightarrow{k})}^{\u2020}$ are smeared with smooth functions of rapid decrease. Coulomb fields do not belong to this class of functions.

_{0}by e

^{0}= ct, c = 1)

## 7. Conclusions

^{3}-theory, where the interaction Hamiltonian density is given by the normally ordered third order monomial of a free uncharged (massless) scalar field and a coupling constant λ

_{n}can be viewed as well-defined, already regularized expressions. Second, infrared divergences are also present in Equation (130). This is not astonishing, since the T

_{n}’s are operator-valued distributions, and therefore must be smeared out by test functions in $S({\mathbb{R}}^{4n})$. One may therefore introduce a test function $g(x)\in S({\mathbb{R}}^{4})$ that plays the role of an “adiabatic switching” and provides a cutoff in the long-range part of the interaction, which can be considered as a natural infrared regulator [31,32]. Then, according to Epstein and Glaser, the infrared regularized S-matrix is given by

## Acknowledgments

## Appendix

#### Position and Momentum Operators

^{n}

^{−1}= 0, and finally one is successively lead to a contradiction

_{β}acting on Lebesgue square integrable wave functions $\mathrm{\Psi}\in {\mathcal{L}}^{2}(\mathbb{R})$ according to

^{−iαq}is also unitary and therefore

## Conflicts of Interest

**PACS classifications:**11.10.-z Field theory; 11.10.Jj Asymptotic problems and properties; 11.15.-q Gauge field theories; 11.30.-j Symmetry and conservation laws.

**MSC classifications:**81S05, 81T05, 81T10, 81T13, 81T70.

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Aste, A.
Coulomb Solutions from Improper Pseudo-Unitary Free Gauge Field Operator Translations. *Symmetry* **2014**, *6*, 1037-1057.
https://doi.org/10.3390/sym6041037

**AMA Style**

Aste A.
Coulomb Solutions from Improper Pseudo-Unitary Free Gauge Field Operator Translations. *Symmetry*. 2014; 6(4):1037-1057.
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Aste, Andreas.
2014. "Coulomb Solutions from Improper Pseudo-Unitary Free Gauge Field Operator Translations" *Symmetry* 6, no. 4: 1037-1057.
https://doi.org/10.3390/sym6041037