The Statistical Mechanics of the N-particle system is derived from Quantum Random Field Theory. An asymptotically measure preserving, flow operator has been obtained from the QFT evolution operator. A basic assumption is that the field Lagrangian is a generalized, infinitely divisible random field, and the interaction Hamiltonian varies in time in a step-wise manner at the transition lattice space-time points. Macroscopic boundary conditions are imposed on the wave functions. The density matrix, the partition function, the Bose-Einstein, the Fermi-Dirac distributions and the thermodynamic temperature follow. The equilibrium or non-equilibrium temperature have been obtained as functionals of the interaction Hamiltonian. The chemical potential has also been derived from the theory.
The Statistical Mechanics of the N-particle system is derived from Quantum Random Field Theory. An asymptotically measure preserving, flow operator has been obtained from the QFT evolution operator. A basic assumption is that the field Lagrangian is a generalized, infinitely divisible random field, and the interaction Hamiltonian varies in time in a step-wise manner at the transition lattice space-time points. Macroscopic boundary conditions are imposed on the wave functions. The density matrix, the partition function, the Bose-Einstein, the Fermi-Dirac distributions and the thermodynamic temperature follow. The equilibrium or non-equilibrium temperature have been obtained as functionals of the interaction Hamiltonian. The chemical potential has also been derived from the theory.
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.
A Lorentz invariant and Hermitian statistical field theory is presented. Its derivation is based on the random field theory, the evolution operator and the path integration. It yields the density matrix, the Gibbs states and the temperatures for equilibrium, nonequilibrium and variable particles numbers states. The chemical potential follows from a spontaneous energy renormalization of the Hamiltonian in the evolution operator. The generalization to any field other than scalar fields is straightforward.
The quantum field theory in one space + one time dimension described by the Lagrangian $\mathfrak{L} = (\frac{i}{2}){\ensuremath{\varphi}}^{*}{\stackrel{}{\ensuremath{\partial}}}_{0}\ensuremath{\varphi}\ensuremath{-}({\ensuremath{\partial}}_{1}{\ensuremath{\varphi}}^{*})({\ensuremath{\partial}}_{1}\ensuremath{\varphi})\ensuremath{-}c{|\ensuremath{\varphi}|}^{4}$ is studied for systems with finite temperature and particle density. Using momentum-space techniques previously developed, a graphical procedure is obtained for calculating inner products of many-particle scattering state wave functions. The unitarity of the wave operator $U(0,\ensuremath{-}\ensuremath{\infty})$ is demonstrated as a pattern of graphical cancellations. An operator formulation of statistical mechanics is derived in which partition functions are given in terms of matrix elements having the form of diagonal (forward) inner products. The importance of noncommutativity of the forward limit and the $i\ensuremath{\epsilon}\ensuremath{\rightarrow}0$ limit is noted and traced to the presence of forward singular graphs in the inner product. Combining this observation with wave-operator unitarity, we obtain a graphical recipe for calculating N-body partition functions. The thermodynamics first described by Yang and Yang is obtained by summation of the fugacity series for the grand partition function.
Thermodynamics is used to describe the bulk properties of matter in or near equilibrium. Many scientists, notably Boyle, Carnot, Clausius, Gay-Lussac, Gibbs, Joule, Kelvin, and Rumford, contributed to the development of the field over three centuries. Quantities such as mass, pressure, energy, and so on are readily defined and measured. Classical statistical mechanics attempts to understand thermodynamics by the application of classical mechanics to the microscopic particles making up the system. Great progress in this field was made by physicists like Boltzmann and Maxwell. Temperature, entropy, particle number, and chemical potential are thus understandable in terms of the microscopic nature of matter. Classical mechanics is inadequate in many circumstances however, and ultimately must be replaced by quantum mechanics. In fact, the ultraviolet catastrophe encountered by the application of classical mechanics and electromagnetism to blackbody radiation was one of the problems that led to the development of quantum theory. The development of quantum statistical mechanics was achieved by a number of twentieth century physicists, most notably Planck, Einstein, Fermi, and Bose. The purpose of this chapter is to give a mini-review of the basic concepts of quantum statistical mechanics as applied to noninteracting systems of particles. This will set the stage for the functional integral representation of the partition function, which is a cornerstone of modern relativistic quantum field theory and the quantum statistical mechanics of interacting particles and fields. Ensembles One normally encounters three types of ensemble in equilibrium statistical mechanics. The microcanonical ensemble is used to describe an isolated system that has a fixed energy E , a fixed particle number N , and a fixed volume V .
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 four part series of lectures on the connection of statistical mechanics and quantum field theory. The general principles relating statistical mechanics and the path integral formulation of quantum field theory are presented in the first lecture. These principles are then illustrated in lecture 2 by a presentation of the theory of the Ising model for H=0, where both the homogeneous and randomly inhomogeneous models are treated and the scaling theory and the relation with Fredholm determinants and Painlev{é} equations is presented. In lecture 3 we consider the Ising model with H=0, where the relation with gauge theory is used to discuss the phenomenon of confinement. We conclude in the last lecture with a discussion of quantum spin diffusion in one dimensional chains and a presentation of the chiral Potts model which illustrates the physical effects that can occur when the Euclidean and Minkowski regions are not connected by an analytic continuation. (To be published as part of the Proceedings of the Sixth Annual Theoretical Physics Summer School of the Australian National University which was held in Canberra during Jan. 1994.)