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arXiv:2203.07795 · Progress of Theoretical and Experimental Physics

Reality from maximizing overlap in the periodic complex action theory

Keiichi Nagao, Holger Bech Nielsen

Published Jul 23, 2022• 2 citations
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Abstract

Abstract We study the periodic complex action theory (CAT) by imposing a periodic condition in the future-included CAT where the time integration is performed from the past to the future, and extend a normalized matrix element of an operator $\hat{\mathcal {O}}$, which is called the weak value in the real action theory, to another expression $\langle \hat{\mathcal {O}} \rangle _{\mathrm{periodic}~\mathrm{time}}$. We present two theorems stating that $\langle \hat{\mathcal {O}} \rangle _{\mathrm{periodic}~\mathrm{time}}$ becomes real for $\hat{\mathcal {O}}$ being Hermitian with regard to a modified inner product that makes a given non-normal Hamiltonian $\hat{H}$ normal. The first theorem holds for a given period tp in a case where the number of eigenstates having the maximal imaginary part B of the eigenvalues of $\hat{H}$ is just one, while the second one stands for tp selected such that the absolute value of the transition amplitude is maximized in a case where B ≤ 0 and |B| is much smaller than the distances between any two real parts of the eigenvalues of $\hat{H}$. The latter proven via a number-theoretical argument suggests that, if our universe is periodic, then even the period could be an adjustment parameter to be determined in the Feynman path integral. This is a variant type of the maximization principle that we previously proposed.

DOI
10.1093/ptep/ptac102

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Keiichi Nagao, HOLGER BECH NIELSEN

We study a complex action theory (CAT) whose path runs over not only past but also future. We show that, if we regard a matrix element defined in terms of the future state at time TB and the past state at time TA as an expectation value in the CAT, then we are allowed to have the Heisenberg equation, Ehrenfest's theorem, and the conserved probability current density. In addition, we show that the expectation value at the present time t of a future-included theory for large TB − t and large t − TA corresponds to that of a future-not-included theory with a proper inner product for large t − TA. Hence, the CAT with future explicitly present in the formalism and influencing in principle the past is not excluded phenomenologically, because the effects are argued to be very small in the present era. Furthermore, we explicitly derive the Hamiltonian for the future state via a path integral, and confirm that it is given by the Hermitian conjugate of the Hamiltonian for the past state.

arXiv:1205.3706•Progress of Theoretical and Experimental Physics•Feb 1, 2013•5 citations
Abstract We study the periodic complex action theory (CAT) by imposing a periodic condition in the future-included CAT where the time integration is performed from the past to the future, and extend a normalized matrix element of an operator O^\hat{\mathcal {O}}O^, which is called the weak value in the real action theory, to another expression ⟨O^⟩periodic time\langle \hat{\mathcal {O}} \rangle _{\mathrm{periodic}~\mathrm{time}}⟨O^⟩periodic time​. We present two theorems stating that ⟨O^⟩periodic time\langle \hat{\mathcal {O}} \rangle _{\mathrm{periodic}~\mathrm{time}}⟨O^⟩periodic time​ becomes real for O^\hat{\mathcal {O}}O^ being Hermitian with regard to a modified inner product that makes a given non-normal Hamiltonian H^\hat{H}H^ normal. The first theorem holds for a given period tp in a case where the number of eigenstates having the maximal imaginary part B of the eigenvalues of H^\hat{H}H^ is just one, while the second one stands for tp selected such that the absolute value of the transition amplitude is maximized in a case where B ≤ 0 and |B| is much smaller than the distances between any two real parts of the eigenvalues of H^\hat{H}H^. The latter proven via a number-theoretical argument suggests that, if our universe is periodic, then even the period could be an adjustment parameter to be determined in the Feynman path integral. This is a variant type of the maximization principle that we previously proposed.
In the complex action theory whose path runs over not only past but also future, we study a normalized matrix element of an operator |O^\hat {\mathcal O}O^| defined in terms of the future state at the latest time |TBT_BTB​| and the past state at the earliest time |TAT_ATA​| with a proper inner product that makes normal a given Hamiltonian that is non-normal at first. We present a theorem that states that, provided that the operator |O^\hat {\mathcal O}O^| is |QQQ|-Hermitian, i.e., Hermitian with regard to the proper inner product, the normalized matrix element becomes real and time-develops under a |QQQ|-Hermitian Hamiltonian for the past and future states selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. Furthermore, we give a possible procedure to formulate the |QQQ|-Hermitian Hamiltonian in terms of |QQQ|-Hermitian coordinate and momentum operators, and construct a conserved probability current density.
In the future-included complex and real action theories whose paths run over not only the past but also the future, we briefly review the theorem on the normalized matrix element of an operator O^\hat{\cal O}O^, which is defined in terms of the future and past states with a proper inner product IQI_QIQ​ that makes a given Hamiltonian normal. The theorem states that, provided that the operator O^\hat{\cal O}O^ is QQQ-Hermitian, i.e. Hermitian with regard to the proper inner product IQI_QIQ​, the normalized matrix element becomes real and time-develops under a QQQ-Hermitian Hamiltonian for the past and future states selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. Discussing what the theorem implicates, we speculate that the future-included complex action theory would be the most elegant quantum theory.
In the future-included real action theory whose path runs over not only past but also future, we demonstrate a theorem, which states that the normalized matrix element of a Hermitian operator |O^\hat{\mathcal O}O^| defined in terms of the future state at the final time |TBT_BTB​| and the fixed past state at the initial time |TAT_ATA​| becomes real for the future state selected such that the absolute value of the transition amplitude from the past state to the future state is maximized. This is a special version of our previously proposed theorem for the future-included complex action theory. We find that though the maximization principle leads to the reality of the normalized matrix element in the future-included real action theory, it does not specify the future and past states so much as in the case of the future-included complex action theory. In addition, we argue that the normalized matrix element seems to be more natural than the usual expectation value. Thus we speculate that the functional integral formalism of quantum theory could be most elegant in the future-included complex action theory.
Abstract In a special representation of complex action theory that we call “future-included,” we study a harmonic oscillator model defined with a non-normal Hamiltonian H^\hat{H}H^, in which a mass mmm and an angular frequency ω\omegaω are taken to be complex numbers. In order for the model to be sensible some restrictions on mmm and ω\omegaω are required. We draw a phase diagram in the plane of the arguments of mmm and ω\omegaω, according to which the model is classified into several types. In addition, we formulate two pairs of annihilation and creation operators, two series of eigenstates of the Hamiltonians H^\hat{H}H^ and H^†\hat{H}^\daggerH^†, and coherent states. They are normalized in a modified inner product IQI_QIQ​, with respect to which the Hamiltonian H^\hat{H}H^ becomes normal. Furthermore, applying to the model the maximization principle that we previously proposed, we obtain an effective theory described by a Hamiltonian that is QQQ-Hermitian, i.e. Hermitian with respect to the modified inner product IQI_QIQ​. The generic solution to the model is found to be the “ground” state. Finally we discuss what the solution implies.
In quantum theory its action is usually taken to be real, but we can consider another theory whose action is complex. In addition, in the Feynman path integral, the time integration is usually performed over the period between the initial time |TAT_ATA​| and some specific time, say, the present time |ttt|⁠. Besides such a future-not-included theory, we can consider the future-included theory, in which not only the past state |∣A(TA)⟩| A(T_A) \rangle∣A(TA​)⟩| at the initial time |TAT_ATA​| but also the future state |∣B(TB)⟩| B(T_B) \rangle∣B(TB​)⟩| at the final time |TBT_BTB​| is given at first, and the time integration is performed over the whole period from the past to the future. Thus quantum theory can be classified into four types, according to whether its action is real or not, and whether the future is included or not. We argue that, if a theory is described with a complex action, then such a theory is suggested to be the future-included theory, rather than the future-not-included theory. Otherwise persons living at different times would see different histories of the universe.
We study a complex action theory (CAT) whose path runs over not only past but also future. We show that, if we regard a matrix element defined in terms of the future state at time TB and the past state at time TA as an expectation value in the CAT, then we are allowed to have the Heisenberg equation, Ehrenfest's theorem, and the conserved probability current density. In addition, we show that the expectation value at the present time t of a future-included theory for large TB − t and large t − TA corresponds to that of a future-not-included theory with a proper inner product for large t − TA. Hence, the CAT with future explicitly present in the formalism and influencing in principle the past is not excluded phenomenologically, because the effects are argued to be very small in the present era. Furthermore, we explicitly derive the Hamiltonian for the future state via a path integral, and confirm that it is given by the Hermitian conjugate of the Hamiltonian for the past state.