Time-Conjugation in a Unified Quantum Theory for Hermitian and Non-Hermitian Electronic Systems under Time-Reversal Symmetry

We propose a reformulation of the mathematical formalism of many-electron quantum theory that rests entirely on the physical properties of the electronic system under investigation, rather than conventional mathematical assumption of Hermitian operators in Hilbert space. The formalism is based on a modified dot-product that replaces the familiar complex-conjugation in Hilbert space Symmetry 2021, 13, x. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/symmetry Article Time-Conjugation in a Unified Quantum Theory for Hermitian and Non-Hermitian Electronic Systems under Time-Reversal Symmetry


Introduction
The title of Schrödinger's 1926 paper [1] ("Quantization as Eigenvalue Problem") aptly expresses the essence of the conceptual leap from classical to quantum mechanics. Schrödinger's time-independent eigenvalue equation restricts the energies Ei of an N-particle system ℋ to numerical values for which a corresponding state function ψi satisfies Equation (1). As every student learns, the classical Hamiltonian (total energy) of the system is first made operator-valued by replacing each particle momentum pλ of the kinetic energy ( , energy of motion) with the differential operator -iℏ∇λ, then adding the classical-like potential energy ( , energy of position) to obtain the quantal ℋ = + whose eigenproperties (1) provide the quantal description.
In the present context, we focus on the quantum aspects of electronic systems, where the sharp distinctions from classical conceptions were first recognized [2]. More specifically, we adopt the conventional quantum chemical framework of non-relativistic N-electron theory in the Born-Oppenheimer approximation [3,4] where nuclei are treated as

Introduction
The title of Schrödinger's 1926 paper [1] ("Quantization as Eigenvalue Problem") aptly expresses the essence of the conceptual leap from classical to quantum mechanics. Schrödinger's time-independent eigenvalue equation restricts the energies E i of an N-particle system H to numerical values for which a corresponding state function ψ i satisfies Equation (1). As every student learns, the classical Hamiltonian (total energy) of the system is first made operator-valued by replacing each particle momentum p λ of the kinetic energy (K, energy of motion) with the differential operator -ih∇ λ , then adding the classical-like potential energy (V, energy of position) to obtain the quantal H = K + V whose eigenproperties (1) provide the quantal description.
In the present context, we focus on the quantum aspects of electronic systems, where the sharp distinctions from classical conceptions were first recognized [2]. More specifically, we adopt the conventional quantum chemical framework of non-relativistic N-electron theory in the Born-Oppenheimer approximation [3,4] where nuclei are treated as classical particles of fixed mass and charge that contribute (along with possible external electric and magnetic fields) to total potential energy V. Of course, a parallel treatment might start with vibrational degrees of freedom (phonons) as the system of interest for quantization.

Introduction
The title o aptly expresses Schrödinger's ti restricts the ene sponding state Hamiltonian (to particle momen operator -iℏ∇λ, obtain the quan In the pres the sharp distin cally, we adopt tron theory in  , and (ii) the operators for H and other physical properties of the system are Hermitian. Both properties are based on a chosen form of dot-product, which provides the scalar measure of proximity that is characteristic of the space. In this description, allowed operators (including H) must have domain and range properly matched to the space described by the chosen dot-product. The goal of the present work is to examine how alternative choices of dot product can achieve a more unified description of bound and dissipative (quasi-bound) states of general atomic and molecular systems. As we show below, the modified dot product is based on intrinsic symmetries of the physical system H (as specified by the associated complete set of commuting observables), and thereby is required to be consistent with physical details of the chosen system rather than of fixed mathematical type.

Coordinates and Dot Products for Stationary and Dissipative States of Many-Electron Systems
For Hilbert space, the well-known dot-product d

Time-Conjugation in a Unified Quantum Theory and Non-Hermitian Electronic Systems under Ti Symmetry F. Weinhold
Theoretical Chemistry Institute and Department of Chemistry, University of WI 53706, USA; weinhold@chem.wisc.edu Abstract: We propose a reformulation of the mathematical formal theory that rests entirely on the physical properties of the electronic ther than conventional mathematical assumption of Hermitian ope malism is based on a modified dot-product that replaces the familiar space ℌ (fixed for all physical systems) by time-conjugation in -spa spin, magnetic field, or other explicit t-dependence of the system H ing different spatial structure for different systems. The usual Hermitian ators is thereby generalized to a self-t-adjoint ("t-reversible") charac generalized theorems of virial and hypervirial type. The -space values of measurable properties and the Born-probabilistic interpr underlie the present quantum theory of measurement, while also poral" behavior of internal decay (tunneling-type) phenomena from

Introduction
The title of Schrödinger's 1926 paper [1] ("Quantizatio aptly expresses the essence of the conceptual leap from clas Schrödinger's time-independent eigenvalue equation restricts the energies Ei of an N-particle system ℋ to numeri sponding state function ψi satisfies Equation (1). As every Hamiltonian (total energy) of the system is first made operat particle momentum pλ of the kinetic energy ( , energy of m operator -iℏ∇λ, then adding the classical-like potential energ obtain the quantal ℋ = + whose eigenproperties (1) prov In the present context, we focus on the quantum aspects the sharp distinctions from classical conceptions were first r cally, we adopt the conventional quantum chemical framewo tron theory in the Born-Oppenheimer approximation [3,4]

Time-Conjugation in a Unified Quantum Theory for Hermitian and Non-Hermitian Electronic Systems under Time-Reversal Symmetry
F. Weinhold

Introduction
The title of Schrödinger's 1926 paper [1] ("Quantization as Eigenvalue Problem") aptly expresses the essence of the conceptual leap from classical to quantum mechanics. Schrödinger's time-independent eigenvalue equation restricts the energies Ei of an N-particle system ℋ to numerical values for which a corresponding state function ψi satisfies Equation (1). As every student learns, the classical Hamiltonian (total energy) of the system is first made operator-valued by replacing each particle momentum pλ of the kinetic energy ( , energy of motion) with the differential operator -iℏ∇λ, then adding the classical-like potential energy ( , energy of position) to obtain the quantal ℋ = + whose eigenproperties (1) provide the quantal description.
In the present context, we focus on the quantum aspects of electronic systems, where the sharp distinctions from classical conceptions were first recognized [2]. More specifically, we adopt the conventional quantum chemical framework of non-relativistic N-electron theory in the Born-Oppenheimer approximation [3,4] where nuclei are treated as where the integration is over all allowed values of real-space cartesian variables. The key characteristic of Hilbert space is that the dot-product of any function with itself is required to be real, positive, and finite,

Abstract:
We propose a reformulation of the mathematical formalism of many-electron q theory that rests entirely on the physical properties of the electronic system under investiga ther than conventional mathematical assumption of Hermitian operators in Hilbert space. malism is based on a modified dot-product that replaces the familiar complex-conjugation in space ℌ (fixed for all physical systems) by time-conjugation in -space (as generated by the spin, magnetic field, or other explicit t-dependence of the system Hamiltonian ℋ of interes ing different spatial structure for different systems. The usual Hermitian requirement for physic ators is thereby generalized to a self-t-adjoint ("t-reversible") character, leading to correspo generalized theorems of virial and hypervirial type. The -space reformulation preserves values of measurable properties and the Born-probabilistic interpretations of state functi underlie the present quantum theory of measurement, while also properly distinguishin poral" behavior of internal decay (tunneling-type) phenomena from that of applied fields w

Introduction
The title of Schrödinger's 1926 paper [1] ("Quantization as Eigenvalue Pro aptly expresses the essence of the conceptual leap from classical to quantum mec Schrödinger's time-independent eigenvalue equation restricts the energies Ei of an N-particle system ℋ to numerical values for which a sponding state function ψi satisfies Equation (1). As every student learns, the c Hamiltonian (total energy) of the system is first made operator-valued by replacin

Abstract:
We propose a reformulation of the math theory that rests entirely on the physical properties ther than conventional mathematical assumption o malism is based on a modified dot-product that repla space ℌ (fixed for all physical systems) by time-conj spin, magnetic field, or other explicit t-dependence ing different spatial structure for different systems. The ators is thereby generalized to a self-t-adjoint ("t-re generalized theorems of virial and hypervirial typ values of measurable properties and the Born-pro underlie the present quantum theory of measurem poral" behavior of internal decay (tunneling-type) p

Introduction
The title of Schrödinger's 1926 paper [1] aptly expresses the essence of the conceptual Schrödinger's time-independent eigenvalue eq

Abstract:
We propose a reformulation of the mathematical formalism of many-electron quantum theory that rests entirely on the physical properties of the electronic system under investigation, rather than conventional mathematical assumption of Hermitian operators in Hilbert space. The formalism is based on a modified dot-product that replaces the familiar complex-conjugation in Hilbert space ℌ (fixed for all physical systems) by time-conjugation in -space (as generated by the specific spin, magnetic field, or other explicit t-dependence of the system Hamiltonian ℋ of interest), yielding different spatial structure for different systems. The usual Hermitian requirement for physical operators is thereby generalized to a self-t-adjoint ("t-reversible") character, leading to correspondingly generalized theorems of virial and hypervirial type. The -space reformulation preserves the real values of measurable properties and the Born-probabilistic interpretations of state functions that underlie the present quantum theory of measurement, while also properly distinguishing "temporal" behavior of internal decay (tunneling-type) phenomena from that of applied fields with par-

Introduction
The title of Schrödinger's 1926 paper [1] ("Quantization as Eigenvalue Problem") aptly expresses the essence of the conceptual leap from classical to quantum mechanics. Schrödinger's time-independent eigenvalue equation

Abstract:
We propose a reformulation of the mathematical formalism of many-elect theory that rests entirely on the physical properties of the electronic system under inv ther than conventional mathematical assumption of Hermitian operators in Hilbert s malism is based on a modified dot-product that replaces the familiar complex-conjugat space ℌ (fixed for all physical systems) by time-conjugation in -space (as generated b spin, magnetic field, or other explicit t-dependence of the system Hamiltonian ℋ of in ing different spatial structure for different systems. The usual Hermitian requirement for p ators is thereby generalized to a self-t-adjoint ("t-reversible") character, leading to cor generalized theorems of virial and hypervirial type. The -space reformulation pres values of measurable properties and the Born-probabilistic interpretations of state f underlie the present quantum theory of measurement, while also properly distingu poral" behavior of internal decay (tunneling-type) phenomena from that of applied fie

Introduction
The title of Schrödinger's 1926 paper [1] ("Quantization as Eigenvalu aptly expresses the essence of the conceptual leap from classical to quantum Schrödinger's time-independent eigenvalue equation

Abstract:
We propose a reformulation of the mathe theory that rests entirely on the physical properties o ther than conventional mathematical assumption of malism is based on a modified dot-product that replac space ℌ (fixed for all physical systems) by time-conju spin, magnetic field, or other explicit t-dependence o ing different spatial structure for different systems. The u ators is thereby generalized to a self-t-adjoint ("t-rev generalized theorems of virial and hypervirial type values of measurable properties and the Born-prob underlie the present quantum theory of measurem poral" behavior of internal decay (tunneling-type) p

Time-Conjugation in a Unified
Quantum Theory for Hermitian and -based conceptions are considered to apply only to stationary bound-state solutions of the t-independent regime. Description of more general scattering or resonance-type phenomena instead requires the non-eigenvalue form of Schrödinger's t-dependent wave equation, viz., As shown by Dirac [9], the usual non-relativistic Hamiltonian must then be replaced with kinetic energy terms that depend linearly on momenta to avoid conflicts with special relativity in the high-energy limit. In either case, Equation (5) becomes the starting point for describing possible explicit or implicit time-dependence in H. When all such t-dependence is absent, solution of (5) reduces in well-known fashion to (1) by merely taking Ψ and ψ to be related by a phase factor (Ψ = e −iht ψ) that cancels out of any d

Abstract:
We propose a reformulation of the mathe theory that rests entirely on the physical properties o ther than conventional mathematical assumption of malism is based on a modified dot-product that repla space ℌ (fixed for all physical systems) by time-conj spin, magnetic field, or other explicit t-dependence ing different spatial structure for different systems. The u ators is thereby generalized to a self-t-adjoint ("t-rev generalized theorems of virial and hypervirial typ values of measurable properties and the Born-prob underlie the present quantum theory of measurem poral" behavior of internal decay (tunneling-type) p ametric t-dependence on an external clock. The t-pr "c-product" that was previously found useful in com ing resonances.

Introduction
The title of Schrödinger's 1926 paper [1] aptly expresses the essence of the conceptual l Schrödinger's time-independent eigenvalue eq restricts the energies Ei of an N-particle system sponding state function ψi satisfies Equation Hamiltonian (total energy) of the system is fir particle momentum pλ of the kinetic energy ( operator -iℏ∇λ, then adding the classical-like p obtain the quantal ℋ = + whose eigenprop In the present context, we focus on the qu the sharp distinctions from classical conceptio cally, we adopt the conventional quantum chem tron theory in the Born-Oppenheimer approx

ℋψi =
evaluation. The mathematical framework expressed by Equations (1)-(5) has enjoyed widespread success. In particular, the structural form of the eigenvalue-type Equation (1) for tindependent systems and the first-order differential form of Equation (5) for t-dependent systems appears secure [9]. However, in certain respects the d Symmetry 2021, 13, x. https://doi.org/10.3390/xxxxx

Time-Conjugation in a Unified Quantum and Non-Hermitian Electronic Systems un Symmetry F. Weinhold
Theoretical Chemistry Institute and Department of Chemis WI 53706, USA; weinhold@chem.wisc.edu Abstract: We propose a reformulation of the mathe theory that rests entirely on the physical properties o ther than conventional mathematical assumption of malism is based on a modified dot-product that replac space ℌ (fixed for all physical systems) by time-conju spin, magnetic field, or other explicit t-dependence o ing different spatial structure for different systems. The u ators is thereby generalized to a self-t-adjoint ("t-rev generalized theorems of virial and hypervirial type values of measurable properties and the Born-prob underlie the present quantum theory of measurem poral" behavior of internal decay (tunneling-type) ph ametric t-dependence on an external clock. The t-pro "c-product" that was previously found useful in com ing resonances.

Introduction
The title of Schrödinger's 1926 paper [1] aptly expresses the essence of the conceptual le Schrödinger's time-independent eigenvalue eq restricts the energies Ei of an N-particle system sponding state function ψi satisfies Equation ( Hamiltonian (total energy) of the system is firs particle momentum pλ of the kinetic energy ( operator -iℏ∇λ, then adding the classical-like p obtain the quantal ℋ = + whose eigenprop In the present context, we focus on the qua the sharp distinctions from classical conception cally, we adopt the conventional quantum chem tron theory in the Born-Oppenheimer approx   (2)-(4) give an imperfect fit to the full range of quantum phenomena of the physical world, as we now wish to discuss.
The traditional Hilbert-space view (2) of quantum theory was most dramatically called into question by the remarkable work of Balslev and Combes [10,11] in the early 1970s. For a broad range of Coulombic and other analytic potentials of physical interest, their work showed that a complex-rotated eigenvalue equation gives exact eigenvalues both for the low-energy bound states (where W i = E i , cf. Equation (1)) as well as for the physically important quasi-bound resonance states ψ r that lie in the highenergy continuum region above the ionization threshold. In the latter case, W r becomes complex-valued, yielding both the position E r and width Γ r of the spectral feature as well as the corresponding lifetime τ r of the quasi-bound species, spectral position E r and width Γ r are observable properties of a resonance spectral feature, and it was therefore quite surprising that a time-type observable τ r could be extracted from what appears nominally to be the time-independent (eigenvalue-type) form (6) of Schrödinger's equation. Moreover, the bonanza of additional spectral information in Equations (6)-(8) is achieved by abandoning the Hilbert space formulation (2) of dot-product and the Hermitian requirement (4) on H θ . Specifically, H θ is constructed from H by complex-rotation of each particle coordinate r λ by angle θ into the complex plane, r λ → r λ e iθ (9) thereby introducing non-Hermitian character in H θ and other violations of traditional mathematical assumptions. It is noteworthy that the resonance-type (finite lifetime) phenomena encompassed by Balslev-Combes theory are not restricted to the high-energy spectroscopic domain of field-free atoms and molecules. Similar resonance characteristics extend to all states of atoms and molecules in the presence of an electric field, where the idealized field-free energy levels E i become Stark-shifted and broadened (Γ i > 0) by the ionizing effect of even infinitesimal field strengths. Thus, the Balslev-Combes (non-Hermitian) formulation should be considered as the more fundamental mathematical conception underlying the quantum mechanics of atomic and molecular phenomena.
Superficially, the Hermitian condition (4) may seem indispensable to the quantum theory of measurement in two respects: (i) assuring real values of the expectation values d Article

Time-Conjugation in a Unified Quantum Theory for Hermitian and Non-Hermitian Electronic Systems under Time-Reversal Symmetry F. Weinhold
Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Madison, Madison, WI 53706, USA; weinhold@chem.wisc.edu Abstract: We propose a reformulation of the mathematical formalism of many-electron quantum theory that rests entirely on the physical properties of the electronic system under investigation, rather than conventional mathematical assumption of Hermitian operators in Hilbert space. The formalism is based on a modified dot-product that replaces the familiar complex-conjugation in Hilbert space ℌ (fixed for all physical systems) by time-conjugation in -space (as generated by the specific spin, magnetic field, or other explicit t-dependence of the system Hamiltonian ℋ of interest), yielding different spatial structure for different systems. The usual Hermitian requirement for physical operators is thereby generalized to a self-t-adjoint ("t-reversible") character, leading to correspondingly generalized theorems of virial and hypervirial type. The -space reformulation preserves the real values of measurable properties and the Born-probabilistic interpretations of state functions that underlie the present quantum theory of measurement, while also properly distinguishing "temporal" behavior of internal decay (tunneling-type) phenomena from that of applied fields with parametric t-dependence on an external clock. The t-product represents a further generalization of the "c-product" that was previously found useful in complex coordinate-rotation studies of autoionizing resonances.

Time-Conjugation in a Unified Q and Non-Hermitian Electronic Sy Symmetry F. Weinhold
Theoretical Chemistry Institute and Depa WI 53706, USA; weinhold@chem.wisc.ed Abstract: We propose a reformulati theory that rests entirely on the phys ther than conventional mathematica malism is based on a modified dot-pr space ℌ (fixed for all physical system spin, magnetic field, or other explicit ing different spatial structure for differe ators is thereby generalized to a selfgeneralized theorems of virial and values of measurable properties and underlie the present quantum theor poral" behavior of internal decay (tu ametric t-dependence on an externa "c-product" that was previously fou ing resonances.

Time-Conjugation in a Unified
Quantum Theory for Hermitian and (ψ i ,ψ j ) = δ ij ). Although mathematical Hermiticity (4) is sufficient to satisfy these two requirements, it admits many operators that have no conceivable relationship to measurable properties of a realistic physical system. However, as we show below, a consistent basis for the quantum theory of measurement can also be achieved in a properly generalized conception of dot-Symmetry 2021, 13, 808 4 of 11 product that departs from Hilbert's envisioned mathematical framework, but seamlessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) and the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident connection to the "c-product" formulation [12], as introduced in the late 1970s to derive generalized virial and hypervirial theorems for the Balslev-Combes domain that are fully analogous to those in the traditional Hermitian domain. In this complex ( it admits many operators that have no conceivable relationship to measurable properties of a realistic physical system. However, as we show below, a consistent basis for the quantum theory of measurement can also be achieved in a properly generalized conception of dot-product that departs from Hilbert's envisioned mathematical framework, but seamlessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) and the idealized resonance-free domain (1)-(4) of traditional bound-state applications. The revised dot-product conception to be introduced below has an evident connection to the "c-product" formulation [12], as introduced in the late 1970s to derive generalized virial and hypervirial theorems for the Balslev-Combes domain that are fully analogous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-product dℭ is defined by merely omitting complex conjugation of the first function, viz.
As a result, there is no longer assurance that dℭ(ψ,ψ) is real or positive. The Hermitian condition (4) is similarly replaced by as the condition for "self-c-adjoint" (or "c-conjugate") character of ℋ or other operators.
The dℭ-based formulation is contrary to traditional Hilbert-space and Hermitan operator conceptions, but allows the Balslev-Combes theory of dilation-analytic operators to be developed in remarkably parallel fashion for what has become a vast array of non-Hermitian quantum mechanical applications [13]. The present work aims at still further departures from traditional mathematical conceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum mechanics. In effect, we abandon all associations with purely mathematical constructions of a dot-product, seeking instead a physical-based construction that depends on the system of interest, and more specifically on its explicit t-dependence. In this manner, the system itself dictates the t-dependent type of "conjugacy" between operators (or "proximity" between functions) that is characteristic of the associated -space. The generalized formalism automatically reverts to the proper type of operator-conjugacy and function-proximity measure appropriate to the t-dependence of system ℋ, assuring consistent and unified description of all known system types.
The change in focus from states/functions to systems/operators in the -space can be achieved by adopting an operator-based method to uniquely specify the state of the system, as we now describe. For a non-degenerate state, it is well-known that measurement of the eigenvalue Ei is sufficient to identify the state function ψi uniquely, whereas in the case of degeneracies, additional measurements of a complete set of commuting observables ("symmetries" of ℋ [14]) serve to resolve the degeneracies and complete the unique specification of state. Operators of this ℋ-specific complete commuting set thus become the central focus of measurements on system ℋ, constituting only a small subset of the operators (many completely unphysical) meeting the mathematical Hermitian requirement of Equation (4).

Time-Conjugation in Systems with Time, Spin, or Magnetic Field Dependence
Consider a general Hamiltonian ℋ in the coordinate representation, it admits many operators that have no conceivable relationship to measurable propert of a realistic physical system. However, as we show below, a consistent basis for the qu tum theory of measurement can also be achieved in a properly generalized conception dot-product that departs from Hilbert's envisioned mathematical framework, but sea lessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) a the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident conn tion to the "c-product" formulation [12], as introduced in the late 1970s to derive gener ized virial and hypervirial theorems for the Balslev-Combes domain that are fully ana gous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-pr uct dℭ is defined by merely omitting complex conjugation of the first function, viz.
dℭ(ψ,φ) ≡ ∫ψ (r1,r2,...,rN) φ(r1,r2,...,rN) dr1dr2 ... drN, all ψ, φ ∈ ℭ ( As a result, there is no longer assurance that dℭ(ψ,ψ) is real or positive. The Hermit condition (4) is similarly replaced by dℭ(ψ,ℋφ) = dℭ(ℋψ, φ), all ψ, φ ∈ ℭ ( as the condition for "self-c-adjoint" (or "c-conjugate") character of ℋ or other operato The dℭ-based formulation is contrary to traditional Hilbert-space and Hermitan opera conceptions, but allows the Balslev-Combes theory of dilation-analytic operators to developed in remarkably parallel fashion for what has become a vast array of non-H mitian quantum mechanical applications [13]. The present work aims at still further departures from traditional mathematical c ceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum m chanics. In effect, we abandon all associations with purely mathematical constructions a dot-product, seeking instead a physical-based construction that depends on the syst of interest, and more specifically on its explicit t-dependence. In this manner, the syst itself dictates the t-dependent type of "conjugacy" between operators (or "proximity" tween functions) that is characteristic of the associated -space. The generalized form ism automatically reverts to the proper type of operator-conjugacy and function-prox ity measure appropriate to the t-dependence of system ℋ, assuring consistent and unif description of all known system types.
The change in focus from states/functions to systems/operators in the -space can achieved by adopting an operator-based method to uniquely specify the state of the syste as we now describe. For a non-degenerate state, it is well-known that measurement of eigenvalue Ei is sufficient to identify the state function ψi uniquely, whereas in the case degeneracies, additional measurements of a complete set of commuting observab ("symmetries" of ℋ [14]) serve to resolve the degeneracies and complete the unique sp ification of state. Operators of this ℋ-specific complete commuting set thus become central focus of measurements on system ℋ, constituting only a small subset of the op ators (many completely unphysical) meeting the mathematical Hermitian requiremen Equation (4).

Time-Conjugation in Systems with Time, Spin, or Magnetic Field Dependence
Consider a general Hamiltonian ℋ in the coordinate representation, is defined by merely omitting complex conjugation of the first function, viz. it admits many operators that have no conceivable relationship to measurable properties of a realistic physical system. However, as we show below, a consistent basis for the quantum theory of measurement can also be achieved in a properly generalized conception of dot-product that departs from Hilbert's envisioned mathematical framework, but seamlessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) and the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident connec-tion to the "c-product" formulation [12], as introduced in the late 1970s to derive generalized virial and hypervirial theorems for the Balslev-Combes domain that are fully analogous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-product dℭ is defined by merely omitting complex conjugation of the first function, viz. r1,r2,...,rN) φ(r1,r2,...,rN) dr1dr2 ... drN, all ψ, φ ∈ ℭ As a result, there is no longer assurance that dℭ(ψ,ψ) is real or positive. The Hermitian condition (4) is similarly replaced by as the condition for "self-c-adjoint" (or "c-conjugate") character of ℋ or other operators.
The dℭ-based formulation is contrary to traditional Hilbert-space and Hermitan operator conceptions, but allows the Balslev-Combes theory of dilation-analytic operators to be developed in remarkably parallel fashion for what has become a vast array of non-Hermitian quantum mechanical applications [13]. The present work aims at still further departures from traditional mathematical con-ceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum mechanics. In effect, we abandon all associations with purely mathematical constructions of a dot-product, seeking instead a physical-based construction that depends on the system of interest, and more specifically on its explicit t-dependence. In this manner, the system itself dictates the t-dependent type of "conjugacy" between operators (or "proximity" between functions) that is characteristic of the associated -space. The generalized formalism automatically reverts to the proper type of operator-conjugacy and function-proximity measure appropriate to the t-dependence of system ℋ, assuring consistent and unified description of all known system types.
The change in focus from states/functions to systems/operators in the -space can be achieved by adopting an operator-based method to uniquely specify the state of the system, as we now describe. For a non-degenerate state, it is well-known that measurement of the eigenvalue Ei is sufficient to identify the state function ψi uniquely, whereas in the case of degeneracies, additional measurements of a complete set of commuting observables ("symmetries" of ℋ [14]) serve to resolve the degeneracies and complete the unique specification of state. Operators of this ℋ-specific complete commuting set thus become the central focus of measurements on system ℋ, constituting only a small subset of the operators (many completely unphysical) meeting the mathematical Hermitian requirement of Equation (4).

Time-Conjugation in Systems with Time, Spin, or Magnetic Field Dependence
Consider a general Hamiltonian ℋ in the coordinate representation, (ψ,ϕ) ≡ ψ(r 1 ,r 2 , . . . ,r N ) ϕ(r 1 ,r 2 , . . . ,r N ) dr 1 dr 2 . . . dr N , all ψ, ϕ ∈ Symmetry 2021, 13, x FOR PEER REVIEW 4 it admits many operators that have no conceivable relationship to measurable proper of a realistic physical system. However, as we show below, a consistent basis for the qu tum theory of measurement can also be achieved in a properly generalized conception dot-product that departs from Hilbert's envisioned mathematical framework, but sea lessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident conn tion to the "c-product" formulation [12], as introduced in the late 1970s to derive gene ized virial and hypervirial theorems for the Balslev-Combes domain that are fully ana gous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-pr uct dℭ is defined by merely omitting complex conjugation of the first function, viz.
dℭ(ψ,φ) ≡ ∫ψ (r1,r2,...,rN) φ(r1,r2,...,rN) dr1dr2 ... drN, all ψ, φ ∈ ℭ ( As a result, there is no longer assurance that dℭ(ψ,ψ) is real or positive. The Hermi condition (4) is similarly replaced by dℭ(ψ,ℋφ) = dℭ(ℋψ, φ), all ψ, φ ∈ ℭ as the condition for "self-c-adjoint" (or "c-conjugate") character of ℋ or other operat The dℭ-based formulation is contrary to traditional Hilbert-space and Hermitan opera conceptions, but allows the Balslev-Combes theory of dilation-analytic operators to developed in remarkably parallel fashion for what has become a vast array of non-H mitian quantum mechanical applications [13]. The present work aims at still further departures from traditional mathematical c ceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum m chanics. In effect, we abandon all associations with purely mathematical construction a dot-product, seeking instead a physical-based construction that depends on the syst of interest, and more specifically on its explicit t-dependence. In this manner, the syst itself dictates the t-dependent type of "conjugacy" between operators (or "proximity" tween functions) that is characteristic of the associated -space. The generalized form ism automatically reverts to the proper type of operator-conjugacy and function-prox ity measure appropriate to the t-dependence of system ℋ, assuring consistent and unif description of all known system types.
The change in focus from states/functions to systems/operators in the -space can achieved by adopting an operator-based method to uniquely specify the state of the syst as we now describe. For a non-degenerate state, it is well-known that measurement of eigenvalue Ei is sufficient to identify the state function ψi uniquely, whereas in the cas degeneracies, additional measurements of a complete set of commuting observab ("symmetries" of ℋ [14]) serve to resolve the degeneracies and complete the unique sp ification of state. Operators of this ℋ-specific complete commuting set thus become central focus of measurements on system ℋ, constituting only a small subset of the op ators (many completely unphysical) meeting the mathematical Hermitian requiremen Equation (4).

Time-Conjugation in Systems with Time, Spin, or Magnetic Field Dependence
Consider a general Hamiltonian ℋ in the coordinate representation, As a result, there is no longer assurance that d it admits many operators that have no conceivable relationship to measurable properties of a realistic physical system. However, as we show below, a consistent basis for the quantum theory of measurement can also be achieved in a properly generalized conception of dot-product that departs from Hilbert's envisioned mathematical framework, but seamlessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) and the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident connec-tion to the "c-product" formulation [12], as introduced in the late 1970s to derive generalized virial and hypervirial theorems for the Balslev-Combes domain that are fully analogous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-product dℭ is defined by merely omitting complex conjugation of the first function, viz.
The present work aims at still further departures from traditional mathematical con-ceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum mechanics. In effect, we abandon all associations with purely mathematical constructions of a dot-product, seeking instead a physical-based construction that depends on the system of interest, and more specifically on its explicit t-dependence. In this manner, the system itself dictates the t-dependent type of "conjugacy" between operators (or "proximity" between functions) that is characteristic of the associated -space. The generalized formalism automatically reverts to the proper type of operator-conjugacy and function-proximity measure appropriate to the t-dependence of system ℋ, assuring consistent and unified description of all known system types.
The change in focus from states/functions to systems/operators in the -space can be achieved by adopting an operator-based method to uniquely specify the state of the system, as we now describe. For a non-degenerate state, it is well-known that measurement of the eigenvalue Ei is sufficient to identify the state function ψi uniquely, whereas in the case of degeneracies, additional measurements of a complete set of commuting observables ("symmetries" of ℋ [14]) serve to resolve the degeneracies and complete the unique specification of state. Operators of this ℋ-specific complete commuting set thus become the central focus of measurements on system ℋ, constituting only a small subset of the operators (many completely unphysical) meeting the mathematical Hermitian requirement of Equation (4).

Time-Conjugation in Systems with Time, Spin, or Magnetic Field Dependence
Consider a general Hamiltonian ℋ in the coordinate representation, it admits many operators that have no conceivable relationship to measurable properties of a realistic physical system. However, as we show below, a consistent basis for the quantum theory of measurement can also be achieved in a properly generalized conception of dot-product that departs from Hilbert's envisioned mathematical framework, but seamlessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) and the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident connec-tion to the "c-product" formulation [12], as introduced in the late 1970s to derive generalized virial and hypervirial theorems for the Balslev-Combes domain that are fully analogous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-product dℭ is defined by merely omitting complex conjugation of the first function, viz. r1,r2,...,rN) φ(r1,r2,...,rN) dr1dr2 ... drN, all ψ, φ ∈ ℭ (10) As a result, there is no longer assurance that dℭ(ψ,ψ) is real or positive. The Hermitian condition (4) is similarly replaced by dℭ(ψ,ℋφ) = dℭ(ℋψ, φ), all ψ, φ ∈ ℭ (11) as the condition for "self-c-adjoint" (or "c-conjugate") character of ℋ or other operators. The dℭ-based formulation is contrary to traditional Hilbert-space and Hermitan operator conceptions, but allows the Balslev-Combes theory of dilation-analytic operators to be developed in remarkably parallel fashion for what has become a vast array of non-Hermitian quantum mechanical applications [13].
The present work aims at still further departures from traditional mathematical con-ceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum mechanics. In effect, we abandon all associations with purely mathematical constructions of a dot-product, seeking instead a physical-based construction that depends on the system of interest, and more specifically on its explicit t-dependence. In this manner, the system itself dictates the t-dependent type of "conjugacy" between operators (or "proximity" between functions) that is characteristic of the associated -space. The generalized formalism automatically reverts to the proper type of operator-conjugacy and function-proximity measure appropriate to the t-dependence of system ℋ, assuring consistent and unified description of all known system types.
The change in focus from states/functions to systems/operators in the -space can be achieved by adopting an operator-based method to uniquely specify the state of the system, as we now describe. For a non-degenerate state, it is well-known that measurement of the eigenvalue Ei is sufficient to identify the state function ψi uniquely, whereas in the case of degeneracies, additional measurements of a complete set of commuting observables ("symmetries" of ℋ [14]) serve to resolve the degeneracies and complete the unique specification of state. Operators of this ℋ-specific complete commuting set thus become the central focus of measurements on system ℋ, constituting only a small subset of the operators (many completely unphysical) meeting the mathematical Hermitian requirement of Equation (4).

Time-Conjugation in Systems with Time, Spin, or Magnetic Field Dependence
Consider a general Hamiltonian ℋ in the coordinate representation, it admits many operators that have no conceivable relationship to measurable properties of a realistic physical system. However, as we show below, a consistent basis for the quantum theory of measurement can also be achieved in a properly generalized conception of dot-product that departs from Hilbert's envisioned mathematical framework, but seamlessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) and the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident connec-tion to the "c-product" formulation [12], as introduced in the late 1970s to derive generalized virial and hypervirial theorems for the Balslev-Combes domain that are fully analogous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-product dℭ is defined by merely omitting complex conjugation of the first function, viz. r1,r2,...,rN) φ(r1,r2,...,rN) dr1dr2 ... drN, all ψ, φ ∈ ℭ (10) As a result, there is no longer assurance that dℭ(ψ,ψ) is real or positive. The Hermitian condition (4) is similarly replaced by dℭ(ψ,ℋφ) = dℭ(ℋψ, φ), all ψ, φ ∈ ℭ (11) as the condition for "self-c-adjoint" (or "c-conjugate") character of ℋ or other operators. The dℭ-based formulation is contrary to traditional Hilbert-space and Hermitan operator conceptions, but allows the Balslev-Combes theory of dilation-analytic operators to be developed in remarkably parallel fashion for what has become a vast array of non-Hermitian quantum mechanical applications [13].
The present work aims at still further departures from traditional mathematical con-ceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum mechanics. In effect, we abandon all associations with purely mathematical constructions of a dot-product, seeking instead a physical-based construction that depends on the system of interest, and more specifically on its explicit t-dependence. In this manner, the system itself dictates the t-dependent type of "conjugacy" between operators (or "proximity" between functions) that is characteristic of the associated -space. The generalized formalism automatically reverts to the proper type of operator-conjugacy and function-proximity measure appropriate to the t-dependence of system ℋ, assuring consistent and unified description of all known system types.
The change in focus from states/functions to systems/operators in the -space can be achieved by adopting an operator-based method to uniquely specify the state of the system, as we now describe. For a non-degenerate state, it is well-known that measurement of the eigenvalue Ei is sufficient to identify the state function ψi uniquely, whereas in the case of degeneracies, additional measurements of a complete set of commuting observables ("symmetries" of ℋ [14]) serve to resolve the degeneracies and complete the unique specification of state. Operators of this ℋ-specific complete commuting set thus become the central focus of measurements on system ℋ, constituting only a small subset of the operators (many completely unphysical) meeting the mathematical Hermitian requirement of Equation (4).

Time-Conjugation in Systems with Time, Spin, or Magnetic Field Dependence
Consider a general Hamiltonian ℋ in the coordinate representation, it admits many operators that have no conceivable relationship to measurable properties of a realistic physical system. However, as we show below, a consistent basis for the quantum theory of measurement can also be achieved in a properly generalized conception of dot-product that departs from Hilbert's envisioned mathematical framework, but seamlessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) and the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident connec-tion to the "c-product" formulation [12], as introduced in the late 1970s to derive generalized virial and hypervirial theorems for the Balslev-Combes domain that are fully analogous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-product dℭ is defined by merely omitting complex conjugation of the first function, viz.
The present work aims at still further departures from traditional mathematical con-ceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum mechanics. In effect, we abandon all associations with purely mathematical constructions of a dot-product, seeking instead a physical-based construction that depends on the system of interest, and more specifically on its explicit t-dependence. In this manner, the system itself dictates the t-dependent type of "conjugacy" between operators (or "proximity" between functions) that is characteristic of the associated -space. The generalized formalism automatically reverts to the proper type of operator-conjugacy and function-proximity measure appropriate to the t-dependence of system ℋ, assuring consistent and unified description of all known system types.
The change in focus from states/functions to systems/operators in the -space can be achieved by adopting an operator-based method to uniquely specify the state of the system, as we now describe. For a non-degenerate state, it is well-known that measurement of the eigenvalue Ei is sufficient to identify the state function ψi uniquely, whereas in the case of degeneracies, additional measurements of a complete set of commuting observables ("symmetries" of ℋ [14]) serve to resolve the degeneracies and complete the unique specification of state. Operators of this ℋ-specific complete commuting set thus become the central focus of measurements on system ℋ, constituting only a small subset of the operators (many completely unphysical) meeting the mathematical Hermitian requirement of Equation (4).

Time-Conjugation in Systems with Time, Spin, or Magnetic Field Dependence
Consider a general Hamiltonian ℋ in the coordinate representation, it admits many operators that have no conceivable relationship to measurable properties of a realistic physical system. However, as we show below, a consistent basis for the quantum theory of measurement can also be achieved in a properly generalized conception of dot-product that departs from Hilbert's envisioned mathematical framework, but seamlessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) and the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident connec-tion to the "c-product" formulation [12], as introduced in the late 1970s to derive generalized virial and hypervirial theorems for the Balslev-Combes domain that are fully analogous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-product dℭ is defined by merely omitting complex conjugation of the first function, viz.
The present work aims at still further departures from traditional mathematical con-ceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum mechanics. In effect, we abandon all associations with purely mathematical constructions of a dot-product, seeking instead a physical-based construction that depends on the system of interest, and more specifically on its explicit t-dependence. In this manner, the system itself dictates the t-dependent type of "conjugacy" between operators (or "proximity" between functions) that is characteristic of the associated -space. The generalized formalism automatically reverts to the proper type of operator-conjugacy and function-proximity measure appropriate to the t-dependence of system ℋ, assuring consistent and unified description of all known system types.
The change in focus from states/functions to systems/operators in the -space can be achieved by adopting an operator-based method to uniquely specify the state of the system, as we now describe. For a non-degenerate state, it is well-known that measurement of the eigenvalue Ei is sufficient to identify the state function ψi uniquely, whereas in the case of degeneracies, additional measurements of a complete set of commuting observables ("symmetries" of ℋ [14]) serve to resolve the degeneracies and complete the unique specification of state. Operators of this ℋ-specific complete commuting set thus become the central focus of measurements on system ℋ, constituting only a small subset of the operators (many completely unphysical) meeting the mathematical Hermitian requirement of Equation (4).

Time-Conjugation in Systems with Time, Spin, or Magnetic Field Dependence
Consider a general Hamiltonian ℋ in the coordinate representation, -based formulation is contrary to traditional Hilbert-space and Hermitan operator conceptions, but allows the Balslev-Combes theory of dilation-analytic operators to be developed in remarkably parallel fashion for what has become a vast array of non-Hermitian quantum mechanical applications [13].
The present work aims at still further departures from traditional mathematical conceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum mechanics. In effect, we abandon all associations with purely mathematical constructions of a dot-product, seeking instead a physical-based construction that depends on the system of interest, and more specifically on its explicit t-dependence. In this manner, the system itself dictates the t-dependent type of "conjugacy" between operators (or "proximity" between functions) that is characteristic of the associated T-space. The generalized formalism automatically reverts to the proper type of operator-conjugacy and function-proximity measure appropriate to the t-dependence of system H, assuring consistent and unified description of all known system types.
The change in focus from states/functions to systems/operators in the T-space can be achieved by adopting an operator-based method to uniquely specify the state of the system, as we now describe. For a non-degenerate state, it is well-known that measurement of the eigenvalue E i is sufficient to identify the state function ψ i uniquely, whereas in the case of degeneracies, additional measurements of a complete set of commuting observables ("symmetries" of H [14]) serve to resolve the degeneracies and complete the unique specification of state. Operators of this H-specific complete commuting set thus become the central focus of measurements on system H, constituting only a small subset of the operators (many completely unphysical) meeting the mathematical Hermitian requirement of Equation (4).
"Spatial" aspects of quantum mechanical structure are most clearly exhibited in Dirac's elegant bra ("〈")-ket ("〉") notation, without which coherent discussion of quantum mechanical fundamentals cannot proceed. For present notational needs, we employ braces to distinguish conventional

Time-Conjugation in a Unified Quantum Theory for He and Non-Hermitian Electronic Systems under Time-Rev Symmetry F. Weinhold
Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Mad WI 53706, USA; weinhold@chem.wisc.edu Abstract: We propose a reformulation of the mathematical formalism of many-el theory that rests entirely on the physical properties of the electronic system under i ther than conventional mathematical assumption of Hermitian operators in Hilber malism is based on a modified dot-product that replaces the familiar complex-conju space ℌ (fixed for all physical systems) by time-conjugation in -space (as generate spin, magnetic field, or other explicit t-dependence of the system Hamiltonian ℋ o ing different spatial structure for different systems. The usual Hermitian requirement fo ators is thereby generalized to a self-t-adjoint ("t-reversible") character, leading to generalized theorems of virial and hypervirial type. The -space reformulation p values of measurable properties and the Born-probabilistic interpretations of sta underlie the present quantum theory of measurement, while also properly disti poral" behavior of internal decay (tunneling-type) phenomena from that of applied ametric t-dependence on an external clock. The t-product represents a further gene "c-product" that was previously found useful in complex coordinate-rotation stud ing resonances. -space dot-products ("〈ψ|ϕ〉") from T-space counterparts ("{ψ|ϕ}") in equations to follow. If further distinction between

Time-Conjugation in a Unified Quan and Non-Hermitian Electronic System Symmetry F. Weinhold
Theoretical Chemistry Institute and Department WI 53706, USA; weinhold@chem.wisc.edu Abstract: We propose a reformulation of t theory that rests entirely on the physical pro ther than conventional mathematical assum malism is based on a modified dot-product t space ℌ (fixed for all physical systems) by t spin, magnetic field, or other explicit t-dep ing different spatial structure for different syste ators is thereby generalized to a self-t-adjoi generalized theorems of virial and hyperv values of measurable properties and the B underlie the present quantum theory of m poral" behavior of internal decay (tunnelin ametric t-dependence on an external clock. "c-product" that was previously found use ing resonances.  it admits many operators that have no conceivable relationship to measurable propertie of a realistic physical system. However, as we show below, a consistent basis for the quan tum theory of measurement can also be achieved in a properly generalized conception o dot-product that departs from Hilbert's envisioned mathematical framework, but seam lessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) an the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident connec tion to the "c-product" formulation [12], as introduced in the late 1970s to derive genera ized virial and hypervirial theorems for the Balslev-Combes domain that are fully analo gous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-prod uct dℭ is defined by merely omitting complex conjugation of the first function, viz.
The present work aims at still further departures from traditional mathematical con -space (10), and T-space variants is required, alternative

F. Weinhold
Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin WI 53706, USA; weinhold@chem.wisc.edu Abstract: We propose a reformulation of the mathematical formalism of ma theory that rests entirely on the physical properties of the electronic system u ther than conventional mathematical assumption of Hermitian operators in H malism is based on a modified dot-product that replaces the familiar complexspace ℌ (fixed for all physical systems) by time-conjugation in -space (as ge spin, magnetic field, or other explicit t-dependence of the system Hamiltonia ing different spatial structure for different systems. The usual Hermitian requirem ators is thereby generalized to a self-t-adjoint ("t-reversible") character, leadi generalized theorems of virial and hypervirial type. The -space reformula values of measurable properties and the Born-probabilistic interpretations underlie the present quantum theory of measurement, while also properly poral" behavior of internal decay (tunneling-type) phenomena from that of a ametric t-dependence on an external clock. The t-product represents a furthe "c-product" that was previously found useful in complex coordinate-rotation ing resonances. it admits many operators that have no conceivable relationship to measurable properties of a realistic physical system. However, as we show below, a consistent basis for the quantum theory of measurement can also be achieved in a properly generalized conception of dot-product that departs from Hilbert's envisioned mathematical framework, but seamlessly incorporates both the Balslev-Combes domain of resonance phenomena (6)-(9) and the idealized resonance-free domain (1)-(4) of traditional bound-state applications.
The revised dot-product conception to be introduced below has an evident connec-tion to the "c-product" formulation [12], as introduced in the late 1970s to derive generalized virial and hypervirial theorems for the Balslev-Combes domain that are fully analogous to those in the traditional Hermitian domain. In this complex (ℭ) domain, the c-product dℭ is defined by merely omitting complex conjugation of the first function, viz.
The present work aims at still further departures from traditional mathematical con-ceptions of Hermitian operators in Hilbert space as the intrinsic "home" of quantum mechanics. In effect, we abandon all associations with purely mathematical constructions of 〈ϕ|ψ〉, T 〈ϕ|ψ〉 symbolism can be adopted.

Time-Conjugation in Systems with Time, Spin, or Magnetic Field Dependence
Consider a general Hamiltonian H in the coordinate representation, H = H(r, t, B, S) (12) with arguments that exhibit the explicit dependence (if any) on time t, magnetic field B, and quantum-mechanical spin S [15], as well as spatial position r (now symbolizing the collective coordinates of the N-particle system). The r-dependence which is to be rotated into the complex plane can be distinguished by defining a formal time-conjugated (overbar) Hamiltonian H, H ≡ H(r, −t, −B, −S*) (13) A similar procedure defines the time-conjugated operator A ≡ A(r, −t, −B, −S*) of a general dynamical variable A(r, t, B, S). All such operators necessarily have the properties = A = A, AB = A B and so forth.
With a similar convention, it is also possible to define the t-conjugated counterpart (ψ) of any physical wave function ψ that could arise in the ordinary time-forward regime. To do so, let ψ be uniquely specified in the usual manner by the complete commuting set of observables A, B, . . . for which it is the common eigenfunction. The t-conjugate ψ can then be defined as the corresponding common eigenfunction of the t-conjugated operators A, B, . . . [where Equation (27) below specifies the "corresponding" eigenvalue that completes the definition]. In particular, if ψ satisfies the t-dependent Schrödinger equation (as suggested by the "equivalence" of Ψ, ψ mentioned above) (H − ih ∂/∂t)ψ = 0 (14) then ψ must satisfy the time-conjugated equation Although additional steps are required to find the specific ψ corresponding to a given "forward" wave function ψ, general time-reversal symmetry suggests that functions having the desired properties must exist, and must satisfy = ψ = ψ, ψϕ = ψ ϕ, Aψ = A ψ, and so forth. Note particularly that the wave function ψ(r,t) need not be the same as ψ(r,−t).
It is now convenient to introduce the t-adjoint, denoted A ‡ , of an operator A by the definition {A ‡ ψ|ϕ} = {ψ|Aϕ}, all ψ, ϕ ∈ T Equation (18) shows that the t-adjoint A ‡ is related to the ordinary Hermitian adjoint A † by the equation The general conservation of probability, Equation (19), is thus equivalent to the requirement that the Hamiltonian be self-t-adjoint, We may also call such operators "reversible" (or "t-reversible") to suggest their intimate relationship to time-reversal symmetry.
The reversibility property (A = A ‡ ) characterizes observables that are even functions of time in the classical sense, while skew-reversibility (A = −A ‡ ) characterizes those that are odd in time. We assume more generally that each A commutes with its own t-adjoint, [A, A ‡ ] = 0, and hence shares with it a common set of eigenfunctions {u k }. Let the corresponding eigenvalues of A, A ‡ be denoted by a k and a k ‡ , respectively, Au k = a k ·u k (24) A ‡ u k = a k ‡ u k (25) and let the t-conjugate eigenvalue equation of A similarly be written as Au k = a k u k (26) Consistency demands that the t-conjugate eigenvalues satisfy a k = a k ‡ (27) since Equations (16) and (21) require that, for every "non-exceptional" u k with non-zero t-product with itself ({u k |u k } = 0), Equation (27) completes the definition of t-conjugate wave functions ψ. Table 1 summarizes the specific forms of common operators A for the variously defined "conjugacy" types (A, A*, A † , A ‡ ). For example, if r (of length r) denotes the "ruler" (scale) for the collective position coordinate, the total electronic energy H(r) = T (r) + V(r) of an isolated atom or molecule includes kinetic energy T (homogeneous of degree −2; "scaling as r −2 ") plus Coulombic potential energy V (scaling as r −1 ) so that (cf. rows 8, 9 of Table 1) H θ = T θ + V θ = T (re iθ ) + V(re iθ ) = e −2iθ T (r) + e −iθ V(r) (29) and correspondingly H θ * = H θ † = e 2iθ T (r) + e iθ V(r)  If the potential energy includes an additional contribution n K(r) that scales as r n (e.g., harmonic oscillator, n = 2), the corresponding complex-rotated n V θ is n V θ = n V(re iθ ) = e inθ n V(r) (31) with n V θ * = n V θ † = e −inθ n V(r), and so forth. The reversibility condition (23) characterizes the Hamiltonian operators of known physical systems. One can verify from Table 1 that the t-adjoints of the fundamental dynamical variables r, p, S satisfy r ‡ = r, p ‡ = −p, S ‡ = −S, and thus coincide with the corresponding time-reversed operators ( [4], pp. 667-669). Observables A that are real functions of such variables will similarly satisfy A ‡ = A rev , which suffices ( [4], p. 645) to insure that the property H ‡ = H is consistent with known time reversal symmetry of the Hamiltonian operator. Note however that A ‡ need not coincide with A rev for more general operators; for example, c ‡ = c for a general complex number (regarded as a multiplicative operator), whereas c rev = c* ( [4], p. 640). One could imagine many potentials that are formally Hermitian, but would violate fundamental time-reversal symmetry, such as r × B, r·S, (r·p + p·r)/2, and so forth (cf. Table 1). While the Hermitian condition H † = H allows such unphysical possibilities, the reversibility condition (23) excludes these and similar terms (to any odd power) from the Hamiltonian. The class of reversible Hamiltonians is in this sense more restrictive and "physical" than that of Hermitian Hamiltonians.
In another sense, however, the reversible Hamiltonians are more general than Hermitian Hamiltonians, in that they properly include the complex-rotated kinetic (T ) and potential (V) components of the Hamiltonian H(re iθ ) = e -2iθ T + V(re iθ ) needed to treat resonance phenomena [10]. This follows from the fact, Equation (22), that a complex number is intrinsically reversible, (e iθ ) ‡ = e iθ , and from the observation that V θ (r) = V(re iθ ) is reversible whenever V(r) itself is. Thus, the formalism based on t-products and t-reversible Hamiltonians readily incorporates the dilatation analyticity of Balslev-Combes theory, whereas that based on conventional scalar products and Hermitian operators does not.
One can readily verify that the t-product reduces to the ordinary scalar product whenever the relevant Hamiltonian is Hermitian. Thus, whenever H = H † = H ‡ , one finds