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Formulation of Complex Action Theory

Keiichi Nagao, Holger Bech Nielsen

Published Dec 1, 2011• 13 citations
OpenAlex

Abstract

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.

Related papers

Formulation of Complex Action Theory II -- Momentum and Hamiltonian --

Keiichi Nagao, HOLGER BECH NIELSEN

In the complex action theory (CAT) we explicitly examine how the momentum and Hamiltonian are defined from the Feynman path integral (FPI) point of view based on the complex coordinate formalism of our foregoing paper. After reviewing the formalism briefly, we describe in FPI with a Lagrangian the time development of a $\xi$-parametrized wave function, which is a solution to an eigenvalue problem of a momentum operator. Solving this eigenvalue problem, we derive the momentum, Hamiltonian, and Schrodinger equation. Oppositely, starting from the Hamiltonian we derive the Lagrangian in FPI, and we are led to the momentum relation again via the saddle point for $p$. This study confirms that the momentum and Hamiltonian in the CAT have the same forms as those in the real action theory. We also show the third derivation of the momentum relation via the saddle point for $q$.

arXiv:1105.1294•May 6, 2011•0 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

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

K. Nagao, H. Bech Nielsen

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.

arXiv:1502.00385•Progress of Theoretical and Experimental Physics•May 11, 2015•6 citations

Formulation of Complex Action Theory

K. Nagao, H. B. Nielsen

We formulate a complex action theory which includes operators of coordinate and momentum $\hat{q}$ and $\hat{p}$ being replaced with non-hermitian operators $\hat{q}_{new}$ and $\hat{p}_{new}$, and their eigenstates ${}_m

arXiv:1104.3381•Progress of Theoretical Physics•Dec 1, 2011•14 citations

Formalism of a harmonic oscillator in the future-included complex action theory

Keiichi Nagao, HOLGER BECH NIELSEN

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 $\hat{H}$, in which a mass $m$ and an angular frequency $\omega$ are taken to be complex numbers. In order for the model to be sensible some restrictions on $m$ and $\omega$ are required. We draw a phase diagram in the plane of the arguments of $m$ and $\omega$, 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 $\hat{H}$ and $\hat{H}^\dagger$, and coherent states. They are normalized in a modified inner product $I_Q$, with respect to which the Hamiltonian $\hat{H}$ 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 $Q$-Hermitian, i.e. Hermitian with respect to the modified inner product $I_Q$. The generic solution to the model is found to be the “ground” state. Finally we discuss what the solution implies.

arXiv:1902.01424•Progress of Theoretical and Experimental Physics•Apr 26, 2019•2 citations

Lagrangian and phase-space formulations

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.

Cambridge University Press eBooks•Mar 11, 2004•0 citations
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.
In the complex action theory (CAT) we explicitly examine how the momentum and Hamiltonian are defined from the Feynman path integral (FPI) point of view based on the complex coordinate formalism of our foregoing paper. After reviewing the formalism briefly, we describe in FPI with a Lagrangian the time development of a ξ\xiξ-parametrized wave function, which is a solution to an eigenvalue problem of a momentum operator. Solving this eigenvalue problem, we derive the momentum, Hamiltonian, and Schrodinger equation. Oppositely, starting from the Hamiltonian we derive the Lagrangian in FPI, and we are led to the momentum relation again via the saddle point for ppp. This study confirms that the momentum and Hamiltonian in the CAT have the same forms as those in the real action theory. We also show the third derivation of the momentum relation via the saddle point for qqq.
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.
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.
We formulate a complex action theory which includes operators of coordinate and momentum q^\hat{q}q^​ and p^\hat{p}p^​ being replaced with non-hermitian operators q^new\hat{q}_{new}q^​new​ and p^new\hat{p}_{new}p^​new​, and their eigenstates ${}_m
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.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.