Quantum Theory, Lecture 1

This course will begin with a short summary of some aspects of the history of Quantum Mechanics, which will include Einstein’s photon hypothesis, his analysis of monatomic quantum gases (including Bose-Einstein condensation for ideal Bose gases), and a modern interpretation of Heisenberg’s discovery of Matrix Mechanics. A brief review of the “deformation point of view” will be given, emphasizing the fact that the atomistic nature of matter can be understood as arising from “quantization”, i.e., from a “deformation” of continuum theories of matter. Subsequently, some of the key features of Quantum Physics distinguishing it from Classical Physics - Entanglement, Kochen-Specker Theorem, violation of Bell Inequalities, etc. - and some of the puzzling features of Quantum Mechanics will be recalled. A short presentation of the theory of indirect (weak) measurements and observations, as pioneered by Kraus, and of the phenomenon of “purification” will follow next. This will prepare the ground for a discussion of a novel general approach to Quantum Mechanics that claims to solve the so-called “measurement problem” and eliminates an undue role of “observers” in the formulation of Quantum Mechanics. It will then be time to consider some concrete applications of Quantum Theory. Presumably, examples of irreversible behavior exhibited by open systems in a quantum-mechanical description - including a derivation of the (first and the second) fundamental laws of thermodynamics, a brief review of the derivation of Brownian motion from unitary quantum dynamics and possibly of some further dynamical phenomena - will be discussed at the beginning of this section of the course. Afterwards, the foundations of Equilibrium Quantum Statistical Mechanics, including the KMS condition and its derivation by Haag, Hugenholtz and Winnink, will be reviewed. This formalism will then be applied to studying some phase transitions in Quantum Statistical Mechanics, (using the method of “infrared bounds”). The course will end more or less where it started: Aspects of the theory of interacting Bose gases, including the discussion of various limiting regimes useful to understand, for example, Bose-Einstein condensation, will be discussed in some detail.