This chapter focuses on Quantum Mechanics and Quantum Field Theory in a euclidean formulation. This means that, in general, it discusses the matrix elements of the quantum statistical operator eβH (the density matrix at thermal equilibrium), where H is the hamiltonian and β is the inverse temperature. The chapter begins by first deriving the path integral representation of matrix elements of the quantum statistical operator for hamiltonians of the simple form p 2/2m + V (q). Comparing classical statistical physics in one space dimension and quantum statistical physics of the particle, it introduces statistical correlation functions and discusses their quantum interpretation. It then explicitly calculates the path integral corresponding to a harmonic oscillator in a time-dependent external force. This result can be used to reduce the evaluation of path integrals in the case of analytic potentials to perturbation theory. The chapter shows on a first example that path integrals are especially well suited to the study of the classical limit, by relating a quantum and classical partition function. The appendix explains some general properties of the two-point function, and use the semi-classical approximation of the partition function to derive Bohr–Sommerfeld's quantization condition.
This chapter focuses on Quantum Mechanics and Quantum Field Theory in a euclidean formulation. This means that, in general, it discusses the matrix elements of the quantum statistical operator eβH (the density matrix at thermal equilibrium), where H is the hamiltonian and β is the inverse temperature. The chapter begins by first deriving the path integral representation of matrix elements of the quantum statistical operator for hamiltonians of the simple form p 2/2m + V (q). Comparing classical statistical physics in one space dimension and quantum statistical physics of the particle, it introduces statistical correlation functions and discusses their quantum interpretation. It then explicitly calculates the path integral corresponding to a harmonic oscillator in a time-dependent external force. This result can be used to reduce the evaluation of path integrals in the case of analytic potentials to perturbation theory. The chapter shows on a first example that path integrals are especially well suited to the study of the classical limit, by relating a quantum and classical partition function. The appendix explains some general properties of the two-point function, and use the semi-classical approximation of the partition function to derive Bohr–Sommerfeld's quantization condition.
Abstract Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.
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.
We establish a rather deep relation between statistical mechanics and quantum mechanics at an imaginary time. For the quantum field theory, this implies a relation between finite temperature quantum field theory in Minkowski space and quantum field theory in Euclidean space with one compact Euclidean time direction. The quantum mechanical language being a proper formulation of statistical physics as a molecular theory is basically the study of quantum equilibrium as well as non-equilibrium systems. A quantum system found in a thermodynamic equilibrium state with an external classical system (thermostat or thermal reservoir) may be characterized by fixed values of thermodynamic parameters of the system, say, for example, temperature T , volume V and number of particles N . In this way the total energy of the thermostat is assumed to be much greater than the energy of the system. The behavior of the quantum system may be assumed to be independent of the nature of the thermostat. The most convenient object of the quantum theory to describe such non-isolated systems is the density matrix since the partition function does not determine any local thermodynamic quantities. It is for this reason that we introduced quantum mechanical and quantum statistical density matrices which are necessary to do the investigation via path integration. This enables us to make a transition to the quasi-classical approximation of quantum statistical mechanics, which, in turn, enables us to evaluate the first quantum correction to the classical Boltzmann distribution function.
Chapter 2 constructed a path integral representation of the matrix elements of the statistical operator e-βH. This chapter extends the construction to hamiltonians which are general functions of phase space variables. This results in integrals over trajectories or paths in phase space. When the hamiltonian is quadratic in the momentum variables, the integral over momenta is gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is somewhat ambiguous, reflecting the problem of order between quantum operators. Hamiltonians which are general quadratic functions of momentum variables provide other important examples, and these are analyzed thoroughly. Such hamiltonians arise in the quantization of the motion on Riemannian manifolds. The analysis is illustrated by the quantization of the free motion on the sphere SN-1.
Abstract This chapter presents a short review of path integrals in quantum mechanics. This is followed by a detailed discussion of the evaluation of the semi-classical propagator, involving quadratic expansions of the action and the calculation of determinants. The concept of functionals is introduced for multi-dimensional cases, using a system linearly coupled to many oscillators as an example. The use of path integrals in statistical mechanics is described and quantum corrections to the classical limit are obtained. The connection with Green functions and level density is presented, with an excursion to periodic orbit theory and to the Gutzwiller formula for the oscillating part of the single particle level density. Many-body systems with separable interactions are treated by the Hubbard-Stratonovich transformation. For quantum corrections to the partition function, the static- and perturbed static path approximations are developed. The breakdown for unstable modes is discussed and a possible extension to damped motion is sketched.
The aim of this chapter is to introduce the concept of the Feynman path integral. As well as developing the general construction scheme, particular emphasis is placed on establishing the interconnections between the quantum mechanical path integral, classical Hamiltonian mechanics, and classical statistical mechanics. The practice of path integration is discussed in the context of several pedagogical applications. As well as the canonical examples of a quantum particle in a single and a double potential well, we discuss the generalization of the path integral scheme to tunneling of extended objects (quantum fields), dissipative and thermally assisted quantum tunneling, and the quantum mechanical spin. In this chapter we will temporarily leave the arena of many-body physics and second quantization and, at least superficially, return to single-particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of functional field integral methods for many-body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher-dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various techniques and concepts of field theory, namely stationary phase analyses of functional integrals, the Euclidean formulation of field theory, instanton techniques, and the role of topological concepts in field theory, will be motivated and introduced in this chapter.