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Physical review. D/Physical review. D.

Time advance and probability conservation in P T -symmetric quantum mechanics

Philip D. Mannheim

Published Jul 25, 2025• 2 citations
Publisher

Abstract

When excited states decay with energy ${E}_{0}\ensuremath{-}i\mathrm{\ensuremath{\Gamma}}$ the time evolution operator $U(t)={e}^{\ensuremath{-}iHt}$ does not obey ${U}^{\ifmmode\dagger\else\textdagger\fi{}}(t)U(t)=I$. Nonetheless, probability conservation is not lost if one includes both decay and excitation with energy ${E}_{0}+i\mathrm{\ensuremath{\Gamma}}$, though it takes a different form. Specifically, if the eigenspectrum of a Hamiltonian is complete, then due to $CPT$ symmetry, a symmetry that holds for all physical systems, there must exist an operator $V$ that effects $VH{V}^{\ensuremath{-}1}={H}^{\ifmmode\dagger\else\textdagger\fi{}}$, so that ${V}^{\ensuremath{-}1}{U}^{\ifmmode\dagger\else\textdagger\fi{}}(t)VU(t)=I$. In consequence, because of probability conservation the time delay associated with decay must be accompanied by an equal and opposite time advance for excitation. Thus when a photon excites an atom the spontaneous emission of a photon from the excited state must occur without any decay time delay at all. An effect of this form together with an associated negative time delay has recently been reported by Sinclair et al. [PRX Quantum 3, 010314 (2022)] and Angulo et al. [arXiv:2409.03680].

DOI
10.1103/cgs8-d78m

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Time Advance and Probability Conservation in PT-Symmetric Quantum Mechanics

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When excited states decay the time evolution operator $U(t)=e^{-iHt}$ does not obey $U^{\dagger}(t)U(t)=I$. Nonetheless, probability conservation is not lost if one includes both excitation and decay, though it takes a different form. Specifically, if the eigenspectrum of a Hamiltonian is complete, then due to $CPT$ symmetry, a symmetry that holds for all physical systems, there must exist an operator $V$ that effects $VHV^{-1}=H^{\dagger}$, so that $V^{-1}U^{\dagger}(t)VU(t)=I$. In consequence, the time delay associated with decay must be accompanied by an equal and opposite time advance for excitation. Thus when a photon excites an atom the spontaneous emission of a photon from the excited state must occur without any decay time delay at all. An effect of this form together with an associated negative time delay appear to have recently been reported by Sinclair et. al., PRX Quantum \textbf{3}, 010314 (2022) and Angulo et. al., arXiv:2409.03680 [quant-ph].

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While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution. In this paper, we provide conditions that are both necessary and sufficient. We show that PT symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution. For any PT-symmetric Hamiltonian H, there always exists an operator V that relates H to its Hermitian adjoint according to VHV(-1)=H(†). When the energy spectrum of H is complete, Hilbert space norms <ψ(1)|V|ψ(2)> constructed with this V are always preserved in time. With the energy eigenvalues of a real secular equation being either real or appearing in complex conjugate pairs, we thus establish the unitarity of time evolution in both cases. We also establish the unitarity of time evolution for Hamiltonians whose energy spectra are not complete. We show that when the energy eigenvalues of a Hamiltonian are real and complete, the operator V is a positive Hermitian operator, which has an associated square root operator that can be used to bring the Hamiltonian to a Hermitian form. We show that systems with PT-symmetric Hamiltonians obey causality. We note that Hermitian theories are ordinarily associated with a path integral quantization prescription in which the path integral measure is real, while in contrast, non-Hermitian but PT-symmetric theories are ordinarily associated with path integrals in which the measure needs to be complex, but in which the Euclidean time continuation of the path integral is nonetheless real. Just as the second-order Klein-Gordon theory is stabilized against transitions to negative frequencies because its Hamiltonian is positive-definite, through PT symmetry, the fourth-order derivative Pais-Uhlenbeck theory can equally be stabilized.

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We attract attention to an interesting family of quantum systems where the generator $H_{(gen)}$ of time-evolution of wave functions is not equal to the Hamiltonian $H$. We describe the origin of the difference $H_{(gen)}-H$ and interpret it as a carrier of a compressed information about the other relevant observables.

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Note Dec. 8, 2022: B in A+B=C and p1A-p1B=p2C is negative for n1<n2 where n is the index of refraction. Both equations i.e. probability conservation and average momentum conservation are involved in this result i.e. B/A = (c2-c1)/(c1+c2) (1). In Parts I and II we considered the problem of a beam of light A moving in the positive x direction striking a medium at x=0 and reflecting B and refracting C. This is a probabilistic problem because a single photon has a probability to reflect and another to refract. We also noted that conservation of energy holds i.e. I(A)/c1 = I(B)/c1 + I(C)/c2 where I(A),I(B) and I(C) >0 are intensities . The goal is to find I(B) and I(C) given I(A), but there is only one equation and two unknowns. In Part I we argued that an energy conservation equation incorporates two equations representing square root type probabilities. In Part II we argued that a physical motivation for such a situation may be that in a steady state case, one has dynamic directional probabilities not just static ones. The static/classical probability is associated with the conservation of energy and groups by time i.e an incident beam exists during one time range and reflected and refracted beams during another. One, however, may create a steady state system so that all three beams exist at the same time, although the reflected and refracted beams at any instant are the result of incident photons which hit the medium at an earlier time. A steady state system should not rely on time, but still contains probabilities. There is thus no reason to group by time, but rather one groups by space with the incident and reflected beams in one region and the refracted in another. Introducing space grouped dynamic probabilities A,B,C one may write (using conservation of probability and average momentum): A+B = C ((1a)) and p1A -p1B = p2C ((1b)) where p1=n1p and p2=n2p and c1=c/n 1 and c2=c/n2. The second equation has the appearance of a conservation of energy equation if one places the term -p1B on the RHS. The problem is that A+B=C implies B must be negative if n1<n2. Thus dynamic probabilities may be negative (which is unusual) and the second equation cannot be conservation of energy although it is of the correct form. In Part II we noted one may multiply the two equations together and by using a difference of squares preserve the form of an energy conservation equation which now has AA, BB and CC terms which are necessarily positive and may be identified with I(A),I(B) and I(C). In this note we wish to consider the consequences of B being negative. This we suggest implies A and B interfere so that A+B yields a number smaller than A i.e. C for n1<n2. As a result the negative value of B allows for conservation of probability (or photons). At first this may seem a mathematical convenience, but the two equations ((1a)) and ((1b)) lead to physical consequences, namely the probabilities BB and CC in terms of AA. We thus speculate that there might be a physical mechanism associated with the minus value of B, the reflected ray. This physical property (which should exist for individual photons) ensures conservation of probability. If one considers a glass wedge (or thin film) in air one has reflection from the glass (n1<n2) bringing in a minus sign in B (for conservation with the incident beam). The refracted beam becomes the new incident beam within the glass and reflects at an n1>n2 interface so B’ is not negative. B and B’ may now interfere (depending on the thickness of the wedge) creating an interference pattern which must conserve probability overall. In general one would expect B and B’ to be independent, but each carries conservation of probability information (in a physical form) and this information is different causing interference to occur.

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P T symmetry in relativistic quantum mechanics

Carl M. Bender, Philip D. Mannheim

In nonrelativistic quantum mechanics and in relativistic quantum field theory, the time coordinate $t$ is a parameter and thus the time-reversal operator $\mathcal{T}$ does not actually reverse the sign of $t$. In contrast, in the five-dimensional approach to relativistic quantum mechanics introduced by Feynman, time $t$ is a quantum-mechanical operator. In this paper it is shown how one can use this five-dimensional approach to extend $\mathcal{T}$ and $\mathcal{P}\mathcal{T}$ symmetry from nonrelativistic to relativistic quantum mechanics and implement time-reversal as an operation that effects $\mathcal{T}\text{ }t\mathcal{T}=\ensuremath{-}t$ just as $\mathcal{P}$ effects $\mathcal{P}x\mathcal{P}=\ensuremath{-}x$, with $\mathcal{P}\mathcal{T}$ thus effecting $\mathcal{P}\mathcal{T}\text{ }{x}_{\ensuremath{\mu}}\mathcal{P}\mathcal{T}=\ensuremath{-}{x}_{\ensuremath{\mu}}$. Some illustrative relativistic quantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but $\mathcal{P}\mathcal{T}$ symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real.

arXiv:1107.0501•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 29, 2011•16 citations
When excited states decay with energy E0\ensuremath−i\ensuremathΓ{E}_{0}\ensuremath{-}i\mathrm{\ensuremath{\Gamma}}E0​\ensuremath−i\ensuremathΓ the time evolution operator U(t)=e\ensuremath−iHtU(t)={e}^{\ensuremath{-}iHt}U(t)=e\ensuremath−iHt does not obey U\ifmmode†\else\textdagger\fi(t)U(t)=I{U}^{\ifmmode\dagger\else\textdagger\fi{}}(t)U(t)=IU\ifmmode†\else\textdagger\fi(t)U(t)=I. Nonetheless, probability conservation is not lost if one includes both decay and excitation with energy E0+i\ensuremathΓ{E}_{0}+i\mathrm{\ensuremath{\Gamma}}E0​+i\ensuremathΓ, though it takes a different form. Specifically, if the eigenspectrum of a Hamiltonian is complete, then due to CPTCPTCPT symmetry, a symmetry that holds for all physical systems, there must exist an operator VVV that effects VHV\ensuremath−1=H\ifmmode†\else\textdagger\fiVH{V}^{\ensuremath{-}1}={H}^{\ifmmode\dagger\else\textdagger\fi{}}VHV\ensuremath−1=H\ifmmode†\else\textdagger\fi, so that V\ensuremath−1U\ifmmode†\else\textdagger\fi(t)VU(t)=I{V}^{\ensuremath{-}1}{U}^{\ifmmode\dagger\else\textdagger\fi{}}(t)VU(t)=IV\ensuremath−1U\ifmmode†\else\textdagger\fi(t)VU(t)=I. In consequence, because of probability conservation the time delay associated with decay must be accompanied by an equal and opposite time advance for excitation. Thus when a photon excites an atom the spontaneous emission of a photon from the excited state must occur without any decay time delay at all. An effect of this form together with an associated negative time delay has recently been reported by Sinclair et al. [PRX Quantum 3, 010314 (2022)] and Angulo et al. [arXiv:2409.03680].
When excited states decay the time evolution operator U(t)=e−iHtU(t)=e^{-iHt}U(t)=e−iHt does not obey U†(t)U(t)=IU^{\dagger}(t)U(t)=IU†(t)U(t)=I. Nonetheless, probability conservation is not lost if one includes both excitation and decay, though it takes a different form. Specifically, if the eigenspectrum of a Hamiltonian is complete, then due to CPTCPTCPT symmetry, a symmetry that holds for all physical systems, there must exist an operator VVV that effects VHV−1=H†VHV^{-1}=H^{\dagger}VHV−1=H†, so that V−1U†(t)VU(t)=IV^{-1}U^{\dagger}(t)VU(t)=IV−1U†(t)VU(t)=I. In consequence, the time delay associated with decay must be accompanied by an equal and opposite time advance for excitation. Thus when a photon excites an atom the spontaneous emission of a photon from the excited state must occur without any decay time delay at all. An effect of this form together with an associated negative time delay appear to have recently been reported by Sinclair et. al., PRX Quantum \textbf{3}, 010314 (2022) and Angulo et. al., arXiv:2409.03680 [quant-ph].
While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution. In this paper, we provide conditions that are both necessary and sufficient. We show that PT symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution. For any PT-symmetric Hamiltonian H, there always exists an operator V that relates H to its Hermitian adjoint according to VHV(-1)=H(†). When the energy spectrum of H is complete, Hilbert space norms <ψ(1)|V|ψ(2)> constructed with this V are always preserved in time. With the energy eigenvalues of a real secular equation being either real or appearing in complex conjugate pairs, we thus establish the unitarity of time evolution in both cases. We also establish the unitarity of time evolution for Hamiltonians whose energy spectra are not complete. We show that when the energy eigenvalues of a Hamiltonian are real and complete, the operator V is a positive Hermitian operator, which has an associated square root operator that can be used to bring the Hamiltonian to a Hermitian form. We show that systems with PT-symmetric Hamiltonians obey causality. We note that Hermitian theories are ordinarily associated with a path integral quantization prescription in which the path integral measure is real, while in contrast, non-Hermitian but PT-symmetric theories are ordinarily associated with path integrals in which the measure needs to be complex, but in which the Euclidean time continuation of the path integral is nonetheless real. Just as the second-order Klein-Gordon theory is stabilized against transitions to negative frequencies because its Hamiltonian is positive-definite, through PT symmetry, the fourth-order derivative Pais-Uhlenbeck theory can equally be stabilized.
Abstract Physicists believe that there is a corresponding conservation law for each principle of symmetry. In standard textbooks on quantum mechanics, however, it is asserted that the time-reversal symmetry does not lead to any conservation law. There seems to be a missing piece of the jigsaw puzzle of the connection between the principles of symmetry and the conservation laws. In this paper, we resolve this difficulty by considering the time-reversal invariance of the transition probability a basic physical requirement of the time-reversal symmetry. We show that this requirement, together with the time-reversal covariance of the Schrödinger equation, guarantees the hermiticity of the Hamiltonian which ensures the probability conservation. Thus we complete the missing piece by demonstrating that the time-reversal symmetry leads to the probability conservation. The present paper can help undergraduate students gain deeper insight into the connection between the principles of symmetry and the conservation laws in quantum mechanics.
We attract attention to an interesting family of quantum systems where the generator H(gen)H_{(gen)}H(gen)​ of time-evolution of wave functions is not equal to the Hamiltonian HHH. We describe the origin of the difference H(gen)−HH_{(gen)}-HH(gen)​−H and interpret it as a carrier of a compressed information about the other relevant observables.
Note Dec. 8, 2022: B in A+B=C and p1A-p1B=p2C is negative for n1<n2 where n is the index of refraction. Both equations i.e. probability conservation and average momentum conservation are involved in this result i.e. B/A = (c2-c1)/(c1+c2) (1). In Parts I and II we considered the problem of a beam of light A moving in the positive x direction striking a medium at x=0 and reflecting B and refracting C. This is a probabilistic problem because a single photon has a probability to reflect and another to refract. We also noted that conservation of energy holds i.e. I(A)/c1 = I(B)/c1 + I(C)/c2 where I(A),I(B) and I(C) >0 are intensities . The goal is to find I(B) and I(C) given I(A), but there is only one equation and two unknowns. In Part I we argued that an energy conservation equation incorporates two equations representing square root type probabilities. In Part II we argued that a physical motivation for such a situation may be that in a steady state case, one has dynamic directional probabilities not just static ones. The static/classical probability is associated with the conservation of energy and groups by time i.e an incident beam exists during one time range and reflected and refracted beams during another. One, however, may create a steady state system so that all three beams exist at the same time, although the reflected and refracted beams at any instant are the result of incident photons which hit the medium at an earlier time. A steady state system should not rely on time, but still contains probabilities. There is thus no reason to group by time, but rather one groups by space with the incident and reflected beams in one region and the refracted in another. Introducing space grouped dynamic probabilities A,B,C one may write (using conservation of probability and average momentum): A+B = C ((1a)) and p1A -p1B = p2C ((1b)) where p1=n1p and p2=n2p and c1=c/n 1 and c2=c/n2. The second equation has the appearance of a conservation of energy equation if one places the term -p1B on the RHS. The problem is that A+B=C implies B must be negative if n1<n2. Thus dynamic probabilities may be negative (which is unusual) and the second equation cannot be conservation of energy although it is of the correct form. In Part II we noted one may multiply the two equations together and by using a difference of squares preserve the form of an energy conservation equation which now has AA, BB and CC terms which are necessarily positive and may be identified with I(A),I(B) and I(C). In this note we wish to consider the consequences of B being negative. This we suggest implies A and B interfere so that A+B yields a number smaller than A i.e. C for n1<n2. As a result the negative value of B allows for conservation of probability (or photons). At first this may seem a mathematical convenience, but the two equations ((1a)) and ((1b)) lead to physical consequences, namely the probabilities BB and CC in terms of AA. We thus speculate that there might be a physical mechanism associated with the minus value of B, the reflected ray. This physical property (which should exist for individual photons) ensures conservation of probability. If one considers a glass wedge (or thin film) in air one has reflection from the glass (n1<n2) bringing in a minus sign in B (for conservation with the incident beam). The refracted beam becomes the new incident beam within the glass and reflects at an n1>n2 interface so B’ is not negative. B and B’ may now interfere (depending on the thickness of the wedge) creating an interference pattern which must conserve probability overall. In general one would expect B and B’ to be independent, but each carries conservation of probability information (in a physical form) and this information is different causing interference to occur.
In nonrelativistic quantum mechanics and in relativistic quantum field theory, the time coordinate ttt is a parameter and thus the time-reversal operator T\mathcal{T}T does not actually reverse the sign of ttt. In contrast, in the five-dimensional approach to relativistic quantum mechanics introduced by Feynman, time ttt is a quantum-mechanical operator. In this paper it is shown how one can use this five-dimensional approach to extend T\mathcal{T}T and PT\mathcal{P}\mathcal{T}PT symmetry from nonrelativistic to relativistic quantum mechanics and implement time-reversal as an operation that effects $\mathcal{T}\text{ }t\mathcal{T}=\ensuremath{-}t$ just as P\mathcal{P}P effects $\mathcal{P}x\mathcal{P}=\ensuremath{-}x$, with PT\mathcal{P}\mathcal{T}PT thus effecting $\mathcal{P}\mathcal{T}\text{ }{x}_{\ensuremath{\mu}}\mathcal{P}\mathcal{T}=\ensuremath{-}{x}_{\ensuremath{\mu}}$. Some illustrative relativistic quantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but PT\mathcal{P}\mathcal{T}PT symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real.