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

Estimates on Functional Integrals of Non-Relativistic Quantum Field Theory, with Applications to the Nelson and Polaron Models

Gonzalo Alfredo Bley Delgado

Published Apr 26, 2016• 2 citations
Publisher

Abstract

In this dissertation we present a computational tool that allows one to provide lower bounds for the ground state energy of several quantum-mechanical Hamiltonians.Given a Hamiltonian H, its ground state energy can be expressed as a Feynman-Kac formula inf spec H = -lim T →∞ 1 T log E e A T , where A T is negative the time integral of the effective potential of the Schrödinger operator H, evaluated at a Brownian path on the time interval [0, T ].It is shown how a Clark-Ocone expansion for the action A T , namely an expansion as its deterministic expectation plus a random oscillatory part, represented as a stochastic integral, can lead to an upper bound for the exponential moment of A T , E e A T , of the form e BT +o(T ) , for some constant B. This leads, in particular, to a lower bound for H given by -B.The method is illustrated in two canonical models in non-relativistic quantum field theory: the Nelson and polaron models.

DOI
10.18130/v3qp13

Related papers

Estimates on Functional Integrals of Quantum Mechanics and Non-relativistic Quantum Field Theory

Gonzalo A. Bley, Lawrence E. Thomas

We provide a unified method for obtaining upper bounds for certain functional integrals appearing in quantum mechanics and non-relativistic quantum field theory, functionals of the form $${E[\rm {exp}(\mathcal{A}_T)]}$$ , the (effective) action $${\mathcal{A}_T}$$ being a function of particle trajectories up to time T. The estimates in turn yield rigorous lower bounds for ground state energies, via the Feynman–Kac formula. The upper bounds are obtained by writing the action for these functional integrals in terms of stochastic integrals. The method is illustrated in familiar quantum mechanical settings: for the hydrogen atom, for a Schrödinger operator with $${1/|x|^2}$$ potential with small coupling, and, with a modest adaptation of the method, for the harmonic oscillator. We then present our principal applications of the method, in the settings of non-relativistic quantum field theories for particles moving in a quantized Bose field, including the optical polaron and Nelson models.

arXiv:1512.00356•Communications in Mathematical Physics•Jan 28, 2017•9 citations

Direct path-integral treatment of the polaron problem

Arkady L. Kholodenko, Karl F. Freed

Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that $n\mathrm{th}$-order terms in the polaron coupling constant are obtained from $n\mathrm{th}$-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a $2n\mathrm{th}$-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.

Physical review. B, Condensed matter•Apr 15, 1983•23 citations

Ground-State Energy and Effective Mass of the Polaron

Élliott H. Lieb, Kazuo Yamazaki

The polaron Hamiltonian would be easily soluble were it not for the quartic term appearing therein. It is proposed to substitute for the quartic term a quadratic term having roughly the same properties, and in such a way that the ground-state energy of the new Hamiltonian is rigorously a lower bound for the true energy. With a very small amount of work one can obtain a lower bound as a continuous function of $\ensuremath{\alpha}$ for all values of $\ensuremath{\alpha}$. The result agrees fairly well with the results obtained by other methods. Using the equivalent Hamiltonian one can also obtain an analytic expression for the effective mass, although one cannot say it is a bound for the true effective mass. Futhermore, once one has obtained a lower bound for the energy as a continuous function of the parameters of the Hamiltonian, one can rigorously derive upper and lower bounds for the ground-state expectation values of various operators. For example, it can be shown that for large $\ensuremath{\alpha}$ and large k, $〈{{a}_{\mathrm{k}}}^{*}{a}_{\mathrm{k}}〉\ensuremath{\sim}{\mathrm{k}}^{\ensuremath{-}6}$ and not ${\mathbf{k}}^{\ensuremath{-}2}$ exp(-${\mathbf{k}}^{\ensuremath{-}2}$) as in Pekar's solution. Because of its simplicity, it is possible that this method may have application to other ground-state problems.

Physical Review•Aug 1, 1958•79 citations

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.

Physical review. B, Condensed matter•Mar 15, 1992•23 citations

Polaron ground-state energy inddimensions

Gurjav Ganbold, Гарий Владимирович Ефимов

The polaron path integral introduced by Feynman is generalized to the case of an electron moving in d space dimensions. A scheme of systematic calculations of the obtained path integral is developed to estimate the ground-state energy of the polaron in the same way for different values of the electron-phonon coupling constant \ensuremath{\alpha}. The leading-order term in this approach yields an upper bound to the polaron self-energy and improves Feynman's variational estimates for d=2 and d=3. A scaling relation between polaron self-energies for different space dimensions is obtained for this term. The next correction to this estimation is calculated by numerical integration for d=2 and d=3.

Physical review. B, Condensed matter•Aug 1, 1994•11 citations

Path-Integral Solutions of the Fröhlich Hamiltonian

Ashok Chatterjee, Soma Mukhopadhyay

This chapter presents a brief introduction to the path-integral method before applying it to the polaron problem. The coupling theory is based on the path-integral formulation of quantum mechanics developed by Feynman in 1955. The path-integral formalism proves to be a useful tool for many field-theoretic problems. Feynman's path-integral method as applied by Feynman to determine the polaron effective mass has an intuitive rather than formal basis. The comparison revealed unequivocally the superiority of the Feynman model over the entire range of the coupling constant. Schultz also simulated the Feynman approximation by replacing the whole lattice with a second, fictitious particle harmonically bound to the electron by a spring of constant k. Thus, the Feynman's strong-coupling energy is better than the Landau-Pekar energy with the Gaussian trial function. The numerical results reported by Marshall and Mills show that the second-order correction to the Feynman result for the ground state energy is less than 2% for the entire coupling rang.

CRC Press eBooks•Oct 9, 2018•0 citations
In this dissertation we present a computational tool that allows one to provide lower bounds for the ground state energy of several quantum-mechanical Hamiltonians.Given a Hamiltonian H, its ground state energy can be expressed as a Feynman-Kac formula inf spec H = -lim T →∞ 1 T log E e A T , where A T is negative the time integral of the effective potential of the Schrödinger operator H, evaluated at a Brownian path on the time interval [0, T ].It is shown how a Clark-Ocone expansion for the action A T , namely an expansion as its deterministic expectation plus a random oscillatory part, represented as a stochastic integral, can lead to an upper bound for the exponential moment of A T , E e A T , of the form e BT +o(T ) , for some constant B. This leads, in particular, to a lower bound for H given by -B.The method is illustrated in two canonical models in non-relativistic quantum field theory: the Nelson and polaron models.
We provide a unified method for obtaining upper bounds for certain functional integrals appearing in quantum mechanics and non-relativistic quantum field theory, functionals of the form E[exp(AT)]{E[\rm {exp}(\mathcal{A}_T)]}E[exp(AT​)] , the (effective) action AT{\mathcal{A}_T}AT​ being a function of particle trajectories up to time T. The estimates in turn yield rigorous lower bounds for ground state energies, via the Feynman–Kac formula. The upper bounds are obtained by writing the action for these functional integrals in terms of stochastic integrals. The method is illustrated in familiar quantum mechanical settings: for the hydrogen atom, for a Schrödinger operator with 1/∣x∣2{1/|x|^2}1/∣x∣2 potential with small coupling, and, with a modest adaptation of the method, for the harmonic oscillator. We then present our principal applications of the method, in the settings of non-relativistic quantum field theories for particles moving in a quantized Bose field, including the optical polaron and Nelson models.
Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that nthn\mathrm{th}nth-order terms in the polaron coupling constant are obtained from nthn\mathrm{th}nth-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a 2nth2n\mathrm{th}2nth-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.
The polaron Hamiltonian would be easily soluble were it not for the quartic term appearing therein. It is proposed to substitute for the quartic term a quadratic term having roughly the same properties, and in such a way that the ground-state energy of the new Hamiltonian is rigorously a lower bound for the true energy. With a very small amount of work one can obtain a lower bound as a continuous function of $\ensuremath{\alpha}$ for all values of $\ensuremath{\alpha}$. The result agrees fairly well with the results obtained by other methods. Using the equivalent Hamiltonian one can also obtain an analytic expression for the effective mass, although one cannot say it is a bound for the true effective mass. Futhermore, once one has obtained a lower bound for the energy as a continuous function of the parameters of the Hamiltonian, one can rigorously derive upper and lower bounds for the ground-state expectation values of various operators. For example, it can be shown that for large $\ensuremath{\alpha}$ and large k, $〈{{a}_{\mathrm{k}}}^{*}{a}_{\mathrm{k}}〉\ensuremath{\sim}{\mathrm{k}}^{\ensuremath{-}6}$ and not ${\mathbf{k}}^{\ensuremath{-}2}$ exp(-${\mathbf{k}}^{\ensuremath{-}2}$) as in Pekar's solution. Because of its simplicity, it is possible that this method may have application to other ground-state problems.
The Feynman inequality EG{\mathit{E}}_{\mathit{G}}EG​\ensuremath{\le}Etrial{\mathit{E}}_{\mathrm{trial}}Etrial​+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-Strial{\mathit{S}}_{\mathrm{trial}}Strial​〉/\ensuremath{\beta} for path integrals provides a powerful upper bound on the ground-state energy EG{\mathit{E}}_{\mathit{G}}EG​ of a large variety of systems. Etrial{\mathit{E}}_{\mathrm{trial}}Etrial​ is the ground-state energy of some trial system with action Strial{\mathit{S}}_{\mathrm{trial}}Strial​ 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/kB{\mathit{k}}_{\mathit{B}}kB​T, where kB{\mathit{k}}_{\mathit{B}}kB​ 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, EG{\mathrm{E}}_{\mathrm{G}}EG​\ensuremath{\le}Etrial{\mathrm{E}}_{\mathrm{trial}}Etrial​+〈\ensuremath{\infty}\ensuremath{\Vert}scrT{markUtrial{\mathit{markU}}_{\mathrm{trial}}markUtrial​ (\ensuremath{\infty},-\ensuremath{\infty})[V(0)-Vtrial{\mathit{V}}_{\mathrm{trial}}Vtrial​(0)]}-\ensuremath{\infty}〉,for a system with Hamiltonian H0{\mathrm{H}}_{0}H0​+V. T time-ordering operator, and Utrial{\mathit{U}}_{\mathrm{trial}}Utrial​ is the time evolution operator of a trial system with Hamiltonian H0{\mathit{H}}_{0}H0​+Vtrial{\mathit{V}}_{\mathrm{trial}}Vtrial​ in the interaction representation, with the interactions V(t) and Vtrial{\mathit{V}}_{\mathrm{trial}}Vtrial​(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 H0{\mathit{H}}_{0}H0​. 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 EDB{\mathit{E}}^{\mathrm{DB}}EDB in the right-hand side of the Feynman upper bound, EG{\mathrm{E}}_{\mathrm{G}}EG​\ensuremath{\le}Etial{\mathrm{E}}_{\mathrm{t}\mathbf{i}\mathrm{al}}Etial​+${\mathrm{lim}}_{\mathrm{\ensuremath{\beta}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈S-Strial{\mathit{S}}_{\mathrm{trial}}Strial​〉/\ensuremath{\beta} +EDB{\mathit{E}}^{\mathrm{DB}}EDB,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 polaron path integral introduced by Feynman is generalized to the case of an electron moving in d space dimensions. A scheme of systematic calculations of the obtained path integral is developed to estimate the ground-state energy of the polaron in the same way for different values of the electron-phonon coupling constant \ensuremath{\alpha}. The leading-order term in this approach yields an upper bound to the polaron self-energy and improves Feynman's variational estimates for d=2 and d=3. A scaling relation between polaron self-energies for different space dimensions is obtained for this term. The next correction to this estimation is calculated by numerical integration for d=2 and d=3.
This chapter presents a brief introduction to the path-integral method before applying it to the polaron problem. The coupling theory is based on the path-integral formulation of quantum mechanics developed by Feynman in 1955. The path-integral formalism proves to be a useful tool for many field-theoretic problems. Feynman's path-integral method as applied by Feynman to determine the polaron effective mass has an intuitive rather than formal basis. The comparison revealed unequivocally the superiority of the Feynman model over the entire range of the coupling constant. Schultz also simulated the Feynman approximation by replacing the whole lattice with a second, fictitious particle harmonically bound to the electron by a spring of constant k. Thus, the Feynman's strong-coupling energy is better than the Landau-Pekar energy with the Gaussian trial function. The numerical results reported by Marshall and Mills show that the second-order correction to the Feynman result for the ground state energy is less than 2% for the entire coupling rang.