The semigroup of statistical operators, and the unitary group of time translation operators generated by the same Hamiltonian of a nonrelativistic fermion field, are naturally imbedded in a holomorphic half-plane semigroup. The statistical expectation values of products of time-dependent operators are then boundary values of holomorphic functions which allow a Bochner integral representation with a Cauchy kernel. The general time-temperature-dependent Green's functions permit a concise spectral representation. It is suggested that a thermodynamic perturbation theory should treat the Heisenberg and Bloch equations simultaneously in terms of a perturbation theory for holomorphic half-plane semigroups generated by semibounded self-adjoint Hamiltonians.
The semigroup of statistical operators, and the unitary group of time translation operators generated by the same Hamiltonian of a nonrelativistic fermion field, are naturally imbedded in a holomorphic half-plane semigroup. The statistical expectation values of products of time-dependent operators are then boundary values of holomorphic functions which allow a Bochner integral representation with a Cauchy kernel. The general time-temperature-dependent Green's functions permit a concise spectral representation. It is suggested that a thermodynamic perturbation theory should treat the Heisenberg and Bloch equations simultaneously in terms of a perturbation theory for holomorphic half-plane semigroups generated by semibounded self-adjoint Hamiltonians.
This chapter introduces the holomorphic representation of quantum mechanics, because it allows a study of the properties of boson systems both from the point of view of evolution and of quantum statistical physics (in the so-called second quantization formalism). Path integrals then take the form of integrals over trajectories in phase space in a complex parametrization. A parallel formalism, based on integration over anti-commuting type or Grassmann variables, makes it possible to handle fermions in a way quite analogous to bosons. The corresponding path integral representation of the statistical operator is then derived. The holomorphic formalism is specially well adapted to a study of the harmonic oscillator and, more generally, of perturbed harmonic oscillators. As an illustration, the formalism is applied to the Bose-Einstein condensation. This chapter also considers complex integrals and Wick's theorem, kernel of operators, general Hamiltonians, partition function, and the quantum Bose gas.
(D.O.E.) under contract #DE-AC02-76ER03069, and by the Fonds pour la Formation de Chercheurs et l’Aide à la Recherche. A free non-relativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Non-local boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schrödinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian. 1 I.
This article presents functional integral representations for the heat semigroups with the infinitesimal generators given by self-adjoint Hamiltonians describing an interaction of a non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation.Special attention is paid to definition of the "time-ordered Hilbert space-valued stochastic integrals associated with a family of isometries from a Hilbert space t,o another one" and semigroup techniques.Some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schr\"odinger operators with classical vector potentials.
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.
A system of bosons interacting by 2-body forces is treated in momentum space in the Heisenberg representation, assuming a statistically homogeneous and stationary state. The principle of weak statistical dependence of different momentum modes, previously applied to turbulence dynamics, is used to obtain an exact expansion expressing quadrilinear (2-body) time-displaced correlation functions of the second-quantized field variables in terms of bilinear time-displaced correlation functions and average impulse-response functions. The last-named functions are defined in terms of the exact Green's operator, for infinitesimal sources, of the nonlinear Heisenberg equations of motion. A set of coupled integral equations are then obtained which determine both the bilinear correlation functions and the response functions. By retaining in these equations only the lowest terms in the cited expansion, a theory is obtained which includes particle "self-energy" interactions with the medium of all orders. A modification of the theory is briefly outlined which includes, at finite temperatures, all the processes described at absolute zero by the Brueckner approximation and, in addition, 3-body effects, iterated to all orders, which are omitted in that approximation.
In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.