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.)
We review the formulation of quantum mechanics for identical spinning particles with wavefunctions that are singlevalued when permuted configurations are identified. The identification requires the spins to be smoothly permuted along with position variables, so spin is represented in a position-dependent ‘transported basis’, rather than the usual fixed basis. The simplest transported basis, constructed in terms of spins represented as pairs of commuting harmonic oscillators, gives the correct connection between spin and statistics. More complicated constructions can give the wrong exchange sign. The theory is generalized to incorporate additional properties such as isospin, color and strangeness. Some remarks about the relation between this approach and those based on relatitivity and/or field theory are given.
The traditional standard theory of quantum mechanics is unable to solve the spin-statistics problem, i.e. to justify the utterly important \qo{Pauli Exclusion Principle} but by the adoption of the complex standard relativistic quantum field theory. In a recent paper (Ref. [1]) we presented a proof of the spin-statistics problem in the nonrelativistic approximation on the basis of the \qo{Conformal Quantum Geometrodynamics}. In the present paper, by the same theory the proof of the Spin-Statistics Theorem is extended to the relativistic domain in the general scenario of curved spacetime. The relativistic approach allows to formulate a manifestly step-by-step Weyl gauge invariant theory and to emphasize some fundamental aspects of group theory in the demonstration. No relativistic quantum field operators are used and the particle exchange properties are drawn from the conservation of the intrinsic helicity of elementary particles. It is therefore this property, not considered in the Standard Quantum Mechanics, which determines the correct spin-statistics connection observed in Nature [1]. The present proof of the Spin-Statistics Theorem is simpler than the one presented in Ref. [1], because it is based on symmetry group considerations only, without having recourse to frames attached to the particles.