CHAPTER III. The Quantum Statistics

Michel Le Bellac, G. Barton

Abstract This chapter links Parts I and II, which were devoted to statistical mechanics, and Part III, which deals with quantum field theory. In Part III we shall revert to the tools developed for studying critical phenomena, namely functional integrals (often called path integrals in this new connection); Feynman diagrams; renormalization; and the renormalization group. A few adaptations are all that these tools need in their new context, and we shall use some elementary examples to illustrate the links between problems in statistical and in quantum mechanics.

Keith S. Stowe

Jeffrey Alan Barrett

Abstract This chapter begins by discussing the foundations of Hugh Everett III's relative state theory, which he regards as a new, more general, and complete theory than the standard von Neumann–Dirac theory. It then talks about why Albert Einstein rejected Max Born's formulation of quantum mechanics where he described his new statistical interpretation of the wave function. It also provides a discussion on von Neumann's formulation of quantum mechanics and the standard theory, as well as a summary of the said theory. The chapter goes on to detail several arguments presented by Einstein regarding von Neumann's interpretation of quantum mechanics. It then presents von Neumann's psychophysical parallelism and Wigner's friend experiment.

William Band, Gordon Frank Newell

Michael Grady

The thesis consists of nine chapters and two appendices. In chapter II I discuss the connection given by the transfer matrix formulation between a classical statistical mechanical system and an associated quantum Hamiltonian, using as an example the familiar two- 4 dimensional Ising model. Next, in chapter III, I show how this applies to Baxter's 8-vertex model and the XYZ Hamiltonian, and how an infinite set of conserved charges for the XYZ model arises from properties of the transfer matrix. In Chapter IV I discuss the meaning of self-duality as it applies to quantum Hamiltonians, and develop a general form for a self-dual Hamiltonian. In Chapter V I show how one can find the explicit form for conserved charges in the XY model by a heuristic technique which takes advantage of the models' self-duality. In chapter VI it is shown how the solution to the XYZ model by the quantum inverse scattering method leads to the infinite set of conserved charges associated with it. The thesis culminates in the statement and proof of the theorem mentioned above, which stipulates a condition under which self-dual theories possess an infinite set of conserved charges (Chapter VII), after which I briefly discuss applications of the theorem (VIII) and draw some conclusions (IX).

Don Koks