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arXiv:1512.00356 · Communications in Mathematical Physics

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

Gonzalo A. Bley, Lawrence E. Thomas

Published Jan 28, 2017• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/s00220-017-2834-9

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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.
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.
The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman–Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock “fixes” the defects of the Feynman integral. Possible applications where the potential in the Schrödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper.
A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.
Given the classical dynamics of a non-relativistic particle in terms of a Hamiltonian or an action, it is relatively straightforward to obtain the non-relativistic quantum mechanics (NRQM) of the system. These standard procedures, based on either the Hamiltonian or the path integral, however, do not work in the case of a relativistic particle. As a result we do not have a single-particle description of relativistic quantum mechanics (RQM). Instead, the correct approach requires a transmutation of dynamical variables from the position and momentum of a single particle to a field and its canonical momentum. Particles, along with antiparticles , reappear in a very nontrivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe a relativistic and non-relativistic free particle implies that obtaining the NRQM limit of QFT is conceptually nontrivial. I examine this limit in several approaches (like, for e.g., Hamiltonian dynamics, Lagrangian and Hamiltonian path integrals, field theoretic description etc.) and identify the precise issues which arise when one attempts to obtain the NRQM from QFT in each of these approaches. The dichotomy of NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot , in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. This fact has several implications, especially for the operational notion of space coordinates in quantum theory. A close examination of these issues, which arise when quantum theory is combined with special relativity, could offer insights in the context of attempts to combine quantum theory with general relativity.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.