Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1502.00385 · Progress of Theoretical and Experimental Physics

Reality and hermiticity from maximizing overlap in the future-included complex action theory

K. Nagao, H. Bech Nielsen

Published May 11, 2015• 6 citations
arXivHTMLPDF

Abstract

In the complex action theory whose path runs over not only past but also future, we study a normalized matrix element of an operator |$\hat {\mathcal O}$| defined in terms of the future state at the latest time |$T_B$| and the past state at the earliest time |$T_A$| 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 |$\hat {\mathcal O}$| is |$Q$|-Hermitian, i.e., Hermitian with regard to the proper inner product, the normalized matrix element becomes real and time-develops under a |$Q$|-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 |$Q$|-Hermitian Hamiltonian in terms of |$Q$|-Hermitian coordinate and momentum operators, and construct a conserved probability current density.

DOI
10.1093/ptep/ptv057

Related papers

Reality from maximizing overlap in the future-included theories

Keiichi Nagao, HOLGER BECH NIELSEN

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 $\hat{\cal O}$, which is defined in terms of the future and past states with a proper inner product $I_Q$ that makes a given Hamiltonian normal. The theorem states that, provided that the operator $\hat{\cal O}$ is $Q$-Hermitian, i.e. Hermitian with regard to the proper inner product $I_Q$, the normalized matrix element becomes real and time-develops under a $Q$-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.

arXiv:1710.02071•Oct 5, 2017•2 citations

Reality from maximizing overlap in the future-included real action theory

Keiichi Nagao, Holger Bech Nielsen

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 |$\hat{\mathcal O}$| defined in terms of the future state at the final time |$T_B$| and the fixed past state at the initial time |$T_A$| 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.

arXiv:1705.01585•Progress of Theoretical and Experimental Physics•Jun 30, 2017•1 citation

Reality from maximizing overlap in the periodic complex action theory

Keiichi Nagao, Holger Bech Nielsen

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.

arXiv:2203.07795•Progress of Theoretical and Experimental Physics•Jul 23, 2022•2 citations

Theory including future not excluded: Formulation of complex action theory II

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

Automatic hermiticity for mixed states

Keiichi Nagao, HOLGER BECH NIELSEN

Abstract We previously proposed a mechanism to effectively obtain, after a long time development, a Hamiltonian being Hermitian with regard to a modified inner product IQ that makes a given non-normal Hamiltonian normal by using an appropriately chosen Hermitian operator Q. We studied it for pure states. In this letter we show that a similar mechanism also works for mixed states by introducing density matrices to describe them and investigating their properties explicitly both in the future-not-included and future-included theories. In particular, in the latter, where not only a past state at the initial time TA but also a future state at the final time TB is given, we study a couple of candidates for it, and introduce a “skew density matrix” composed of both ensembles of the future and past states such that the trace of the product of it and an operator ${\cal O}$ matches a normalized matrix element of ${\cal O}$. We argue that the skew density matrix defined with IQ at the present time t for large TB − t and large t − TA approximately corresponds to another density matrix composed of only an ensemble of past states and defined with another inner product $I_{Q_J}$ for large t − TA.

arXiv:2209.11619•Progress of Theoretical and Experimental Physics•Feb 16, 2023•1 citation

Formulation of Complex Action Theory

Keiichi Nagao, Holger Bech Nielsen

We formulate a complex action theory which includes operators of coordinate and momentum q and p being replaced with non-hermitian operators qnew and pnew, and their eigenstates m new q| and m new p| with complex eigenvalues q and p. Introducing a philosophy of keeping the analyticity in path integration variables, we define a modified set of complex conjugate, real and imaginary parts, hermitian conjugates and bras, and explicitly construct qnew, pnew, m new q| and m new p| by formally squeezing coherent states.We also pose a theorem on the relation between functions on the phase space and the corresponding operators.Only in our formalism can we describe a complex action theory or a real action theory with complex saddle points in the tunneling effect etc. in terms of bras and kets in the functional integral.Furthermore, in a system with a non-hermitian diagonalizable bounded Hamiltonian, we show that the mechanism to obtain a hermitian Hamiltonian after a long time development proposed in our letter 11) works also in the complex coordinate formalism.If the hermitian Hamiltonian is given in a local form, a conserved probability current density can be constructed with two kinds of wave functions.

•Dec 1, 2011•13 citations
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 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.
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.
Abstract We previously proposed a mechanism to effectively obtain, after a long time development, a Hamiltonian being Hermitian with regard to a modified inner product IQ that makes a given non-normal Hamiltonian normal by using an appropriately chosen Hermitian operator Q. We studied it for pure states. In this letter we show that a similar mechanism also works for mixed states by introducing density matrices to describe them and investigating their properties explicitly both in the future-not-included and future-included theories. In particular, in the latter, where not only a past state at the initial time TA but also a future state at the final time TB is given, we study a couple of candidates for it, and introduce a “skew density matrix” composed of both ensembles of the future and past states such that the trace of the product of it and an operator O{\cal O}O matches a normalized matrix element of O{\cal O}O. We argue that the skew density matrix defined with IQ at the present time t for large TB − t and large t − TA approximately corresponds to another density matrix composed of only an ensemble of past states and defined with another inner product IQJI_{Q_J}IQJ​​ for large t − TA.
We formulate a complex action theory which includes operators of coordinate and momentum q and p being replaced with non-hermitian operators qnew and pnew, and their eigenstates m new q| and m new p| with complex eigenvalues q and p. Introducing a philosophy of keeping the analyticity in path integration variables, we define a modified set of complex conjugate, real and imaginary parts, hermitian conjugates and bras, and explicitly construct qnew, pnew, m new q| and m new p| by formally squeezing coherent states.We also pose a theorem on the relation between functions on the phase space and the corresponding operators.Only in our formalism can we describe a complex action theory or a real action theory with complex saddle points in the tunneling effect etc. in terms of bras and kets in the functional integral.Furthermore, in a system with a non-hermitian diagonalizable bounded Hamiltonian, we show that the mechanism to obtain a hermitian Hamiltonian after a long time development proposed in our letter 11) works also in the complex coordinate formalism.If the hermitian Hamiltonian is given in a local form, a conserved probability current density can be constructed with two kinds of wave functions.