Journal of Mathematical Physics

Proof and Refinements of an Inequality of Feynman

Kurt Symanzik

Published Jul 1, 1965• 74 citations

Abstract

A lower bound given by Feynman for the quantum mechanical free energy of an oscillator is proved, refined, and generalized. The method, application of a classical inequality to path integrals, also gives upper bounds for one-temperature Green's functions.

DOI: 10.1063/1.1704383

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Journal of Mathematical Physics

Proof and Refinements of an Inequality of Feynman

Kurt Symanzik

Published Jul 1, 1965• 74 citations

Publisher

Abstract

DOI: 10.1063/1.1704383

Related papers

Extension to the case of a magnetic field of Feynman’s path-integral upper bound on the ground-state energy: Application to the Fröhlich polaron

J. T. Devreese, F. Brosens

The Feynman inequality ${\mathit{E}}_{\mathit{G}}$ \ensuremath{\le} ${\mathit{E}}_{\mathrm{trial}}$ +${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S- ${\mathit{S}}_{\mathrm{trial}}$ 〉/\ensuremath{\beta} for path integrals provides a powerful upper bound on the ground-state energy ${\mathit{E}}_{\mathit{G}}$ of a large variety of systems. ${\mathit{E}}_{\mathrm{trial}}$ is the ground-state energy of some trial system with action ${\mathit{S}}_{\mathrm{trial}}$ for imaginary values of the time variable, and S is the action (also expressed in imaginary time variables) of the system under study. \ensuremath{\beta}=1/ ${\mathit{k}}_{\mathit{B}}$ T, where ${\mathit{k}}_{\mathit{B}}$ is the Boltzmann constant and T the temperature. However, the Feynman inequality is not a priori justified for a system in a magnetic field, because imaginary terms subsist in the action also after transforming to imaginary time variables. Replacing or extending this inequality when magnetic fields are present has therefore been a long-standing problem. In the present paper we solve this problem. We first derive an inequality, providing an upper bound for the ground-state energy, that is valid even in the case of a nonzero magnetic field, ${\mathrm{E}}_{\mathrm{G}}$ \ensuremath{\le} ${\mathrm{E}}_{\mathrm{trial}}$ +〈\ensuremath{\infty}\ensuremath{\Vert}scrT{ ${\mathit{markU}}_{\mathrm{trial}}$ (\ensuremath{\infty},-\ensuremath{\infty})[V(0)- ${\mathit{V}}_{\mathrm{trial}}$ (0)]}-\ensuremath{\infty}〉,for a system with Hamiltonian ${\mathrm{H}}_{0}$ +V. T time-ordering operator, and ${\mathit{U}}_{\mathrm{trial}}$ is the time evolution operator of a trial system with Hamiltonian ${\mathit{H}}_{0}$ + ${\mathit{V}}_{\mathrm{trial}}$ in the interaction representation, with the interactions V(t) and ${\mathit{V}}_{\mathrm{trial}}$ (t) switched on adiabatically. Because of the time ordering, retardation effects are also properly taken into account. The contribution of the magnetic field is included in the unperturbed Hamiltonian ${\mathit{H}}_{0}$ . If the time-dependent integrands occurring in the matrix element in the right-hand side of our generalized inequality satisfy certain analyticity conditions in the complex-time plane, this inequality reduces to the Feynman inequality for path integrals. If these analyticity conditions are not satisfied, our generalized inequality may introduce supplementary terms ${\mathit{E}}^{\mathrm{DB}}$ in the right-hand side of the Feynman upper bound, ${\mathrm{E}}_{\mathrm{G}}$ \ensuremath{\le} ${\mathrm{E}}_{\mathrm{t}\mathbf{i}\mathrm{al}}$ +${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S- ${\mathit{S}}_{\mathrm{trial}}$ 〉/\ensuremath{\beta} + ${\mathit{E}}^{\mathrm{DB}}$ ,because different branch lines or singularities have to be taken into account in the transformation to imaginary time variables. As an important illustration, our generalized inequality is applied to the problem of the Fr\"ohlich polaron in a magnetic field. From the generalization of the Feynman an inequality derived in the present paper, we determine the conditions to be imposed on the variational parameters in the trial action, such that the Feynman upper bound in its original form remains valid for a polaron in a magnetic field. Some limiting cases are studied analytically to illustrate the relevance of our additional constraints on the variational parameters of the trial system. In the free-particle limit and for a particular value of one of the variational parameters, we explicitly derive the contributions from the branch lines in the complex-time plane which arise if these additional constraints are not satisfied.The Feynman inequality ${\mathit{E}}_{\mathit{G}}$\ensuremath{\le}${\mathit{E}}_{\mathrm{trial}}$+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-${\mathit{S}}_{\mathrm{trial}}$〉/\ensuremath{\beta} for path integrals provides a powerful upper bound on the ground-state energy ${\mathit{E}}_{\mathit{G}}$ of a large variety of systems. ${\mathit{E}}_{\mathrm{trial}}$ is the ground-state energy of some trial system with action ${\mathit{S}}_{\mathrm{trial}}$ for imaginary values of the time variable, and S is the action (also expressed in imaginary time variables) of the system under study. \ensuremath{\beta}=1/${\mathit{k}}_{\mathit{B}}$T, where ${\mathit{k}}_{\mathit{B}}$ is the Boltzmann constant and T the temperature. However, the Feynman inequality is not a priori justified for a system in a magnetic field, because imaginary terms subsist in the action also after transforming to imaginary time variables. Replacing or extending this inequality when magnetic fields are present has therefore been a long-standing problem. In the present paper we solve this problem. We first derive an inequality, providing an upper bound for the ground-state energy, that is valid even in the case of a nonzero magnetic field, ${\mathrm{E}}_{\mathrm{G}}$\ensuremath{\le}${\mathrm{E}}_{\mathrm{trial}}$+〈\ensuremath{\infty}\ensuremath{\Vert}scrT{${\mathit{markU}}_{\mathrm{trial}}$ (\ensuremath{\infty},-\ensuremath{\infty})[V(0)-${\mathit{V}}_{\mathrm{trial}}$(0)]}-\ensuremath{\infty}〉,for a system with Hamiltonian ${\mathrm{H}}_{0}$+V. T time-ordering operator, and ${\mathit{U}}_{\mathrm{trial}}$ is the time evolution operator of a trial system with Hamiltonian ${\mathit{H}}_{0}$+${\mathit{V}}_{\mathrm{trial}}$ in the interaction representation, with the interactions V(t) and ${\mathit{V}}_{\mathrm{trial}}$(t) switched on adiabatically. Because of the time ordering, retardation effects are also properly taken into account. The contribution of the magnetic field is included in the unperturbed Hamiltonian ${\mathit{H}}_{0}$. If the time-dependent integrands occurring in the matrix element in the right-hand side of our generalized inequality satisfy certain analyticity conditions in the complex-time plane, this inequality reduces to the Feynman inequality for path integrals. If these analyticity conditions are not satisfied, our generalized inequality may introduce supplementary terms ${\mathit{E}}^{\mathrm{DB}}$ in the right-hand side of the Feynman upper bound, ${\mathrm{E}}_{\mathrm{G}}$\ensuremath{\le}${\mathrm{E}}_{\mathrm{t}\mathbf{i}\mathrm{al}}$+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-${\mathit{S}}_{\mathrm{trial}}$〉/\ensuremath{\beta} +${\mathit{E}}^{\mathrm{DB}}$,because different branch lines or singularities have to be taken into account in the transformation to imaginary time variables. As an important illustration, our generalized inequality is applied to the problem of the Fr\"ohlich polaron in a magnetic field. From the generalization of the Feynman an inequality derived in the present paper, we determine the conditions to be imposed on the variational parameters in the trial action, such that the Feynman upper bound in its original form remains valid for a polaron in a magnetic field. Some limiting cases are studied analytically to illustrate the relevance of our additional constraints on the variational parameters of the trial system. In the free-particle limit and for a particular value of one of the variational parameters, we explicitly derive the contributions from the branch lines in the complex-time plane which arise if these additional constraints are not satisfied.

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Peter H. Kleban

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Variational upper bound on the scattering length

Donald Gelman

Previous work on the application to scattering theory of the Feynman path-integral formulation of nonrelativistic quantum mechanics is extended. By using the regularization $\mathrm{Tr}({e}^{\ensuremath{-}\ensuremath{\beta}H}\ensuremath{-}{e}^{\ensuremath{-}\ensuremath{\beta}{H}^{0}})$ of the divergent partition function $\mathrm{Tr}{e}^{\ensuremath{-}\ensuremath{\beta}H}$, where ${H}^{0}$ is the free-particle Hamiltonian, $H={H}^{0}+V$ , and $V$ is a short-range potential of the form $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ which cannot support a bound state, we obtain a variational upper bound on the scattering length. This bound is expressed in terms of path integrals containing a trial functional $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$, and reduces to the natural analog of the Rayleigh-Ritz result when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ is of the standard form $\ensuremath{-}\ensuremath{\int}\frac{[{E}_{K}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}]dt}{\ensuremath{\hbar}}$, with ${E}_{K}$ the kinetic energy and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ a trial potential. Since $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ need not be of this form, the bound contains new features.Previous work on the application to scattering theory of the Feynman path-integral formulation of nonrelativistic quantum mechanics is extended. By using the regularization $\mathrm{Tr}({e}^{\ensuremath{-}\ensuremath{\beta}H}\ensuremath{-}{e}^{\ensuremath{-}\ensuremath{\beta}{H}^{0}})$ of the divergent partition function $\mathrm{Tr}{e}^{\ensuremath{-}\ensuremath{\beta}H}$, where ${H}^{0}$ is the free-particle Hamiltonian, $H={H}^{0}+V$, and $V$ is a short-range potential of the form $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ which cannot support a bound state, we obtain a variational upper bound on the scattering length. This bound is expressed in terms of path integrals containing a trial functional $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$, and reduces to the natural analog of the Rayleigh-Ritz result when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ is of the standard form $\ensuremath{-}\ensuremath{\int}\frac{[{E}_{K}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}]dt}{\ensuremath{\hbar}}$, with ${E}_{K}$ the kinetic energy and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ a trial potential. Since $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ need not be of this form, the bound contains new features.

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Path Integral of Schrödinger’s Equation

Hocine Boukabcha, Salah Eddine Aid, Amina Ghobrini

The path integral is a powerful tool for studying quantum mechanics because it has the merit of establishing the connection between classical mechanics and quantum mechanics. This formalism quickly gained prominence in various fields of theoretical physics, including its generalization to quantum field theory, quantum mechanics, and statistical physics. Using the Feynman propagator, we can calculate the partition function, the free energy, wave functions, and the energy spectrum of the considered physical system. Moreover, the Feynman formalism finds broad applications in geophysics and in the field of financial sciences.The path integral is a powerful tool for studying quantum mechanics because it has the merit of establishing the connection between classical mechanics and quantum mechanics. This formalism quickly gained prominence in various fields of theoretical physics, including its generalization to quantum field theory, quantum mechanics, and statistical physics. Using the Feynman propagator, we can calculate the partition function, the free energy, wave functions, and the energy spectrum of the considered physical system. Moreover, the Feynman formalism finds broad applications in geophysics and in the field of financial sciences.

IntechOpen eBooksSep 29, 20233 citations

Derivation of a Lower Bound on the Free Energy

Robert H. T. Yeh

Journal of Mathematical PhysicsMay 1, 19704 citations

Three useful bounds in quantum mechanics -- easily obtained by Wiener integration

Hajo Leschke, Rainer Ruder

Jan 1, 20080 citations

The Feynman inequality

{\mathit{E}}_{\mathit{G}}

\ensuremath{\le}

{\mathit{E}}_{\mathrm{trial}}

+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-

{\mathit{S}}_{\mathrm{trial}}

〉/\ensuremath{\beta} for path integrals provides a powerful upper bound on the ground-state energy

{\mathit{E}}_{\mathit{G}}

of a large variety of systems.

{\mathit{E}}_{\mathrm{trial}}

is the ground-state energy of some trial system with action

{\mathit{S}}_{\mathrm{trial}}

for imaginary values of the time variable, and S is the action (also expressed in imaginary time variables) of the system under study. \ensuremath{\beta}=1/

{\mathit{k}}_{\mathit{B}}

T, where

{\mathit{k}}_{\mathit{B}}

is the Boltzmann constant and T the temperature. However, the Feynman inequality is not a priori justified for a system in a magnetic field, because imaginary terms subsist in the action also after transforming to imaginary time variables. Replacing or extending this inequality when magnetic fields are present has therefore been a long-standing problem. In the present paper we solve this problem. We first derive an inequality, providing an upper bound for the ground-state energy, that is valid even in the case of a nonzero magnetic field,

{\mathrm{E}}_{\mathrm{G}}

\ensuremath{\le}

{\mathrm{E}}_{\mathrm{trial}}

+〈\ensuremath{\infty}\ensuremath{\Vert}scrT{

{\mathit{markU}}_{\mathrm{trial}}

(\ensuremath{\infty},-\ensuremath{\infty})[V(0)-

{\mathit{V}}_{\mathrm{trial}}

(0)]}-\ensuremath{\infty}〉,for a system with Hamiltonian

{\mathrm{H}}_{0}

+V. T time-ordering operator, and

{\mathit{U}}_{\mathrm{trial}}

is the time evolution operator of a trial system with Hamiltonian

{\mathit{H}}_{0}

{\mathit{V}}_{\mathrm{trial}}

in the interaction representation, with the interactions V(t) and

{\mathit{V}}_{\mathrm{trial}}

(t) switched on adiabatically. Because of the time ordering, retardation effects are also properly taken into account. The contribution of the magnetic field is included in the unperturbed Hamiltonian

{\mathit{H}}_{0}

. If the time-dependent integrands occurring in the matrix element in the right-hand side of our generalized inequality satisfy certain analyticity conditions in the complex-time plane, this inequality reduces to the Feynman inequality for path integrals. If these analyticity conditions are not satisfied, our generalized inequality may introduce supplementary terms

{\mathit{E}}^{\mathrm{DB}}

in the right-hand side of the Feynman upper bound,

{\mathrm{E}}_{\mathrm{G}}

\ensuremath{\le}

{\mathrm{E}}_{\mathrm{t}\mathbf{i}\mathrm{al}}

+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-

{\mathit{S}}_{\mathrm{trial}}

〉/\ensuremath{\beta} +

{\mathit{E}}^{\mathrm{DB}}

,because different branch lines or singularities have to be taken into account in the transformation to imaginary time variables. As an important illustration, our generalized inequality is applied to the problem of the Fr\"ohlich polaron in a magnetic field. From the generalization of the Feynman an inequality derived in the present paper, we determine the conditions to be imposed on the variational parameters in the trial action, such that the Feynman upper bound in its original form remains valid for a polaron in a magnetic field. Some limiting cases are studied analytically to illustrate the relevance of our additional constraints on the variational parameters of the trial system. In the free-particle limit and for a particular value of one of the variational parameters, we explicitly derive the contributions from the branch lines in the complex-time plane which arise if these additional constraints are not satisfied.

{H}^{0}

is the free-particle Hamiltonian,

H={H}^{0}+V

, and

V

is a short-range potential of the form $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ which cannot support a bound state, we obtain a variational upper bound on the scattering length. This bound is expressed in terms of path integrals containing a trial functional $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$, and reduces to the natural analog of the Rayleigh-Ritz result when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ is of the standard form $\ensuremath{-}\ensuremath{\int}\frac{[{E}_{K}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}]dt}{\ensuremath{\hbar}}$, with

{E}_{K}

the kinetic energy and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ a trial potential. Since $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}$ need not be of this form, the bound contains new features.