Quantum mechanics and quantum field theory. Algebraic and geometric approaches

TL;DR

This paper explores algebraic and geometric frameworks for quantum mechanics and quantum field theory, deriving standard and generalized formulas for quantum probabilities and scattering processes, and highlighting the classical nature of quantum theories.

Contribution

It introduces a geometric approach to quantum theory that extends beyond traditional quantum mechanics, including new formulations of scattering matrices and their relation to Green functions.

Findings

Standard quantum probabilities derived from decoherence.

Inclusive scattering matrix expressed via generalized Green functions.

Quantum theories can be viewed as classical theories with limited observables.

Abstract

This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory including some recent results. It is based on the algebraic approach where the starting point is a star-algebra and on the geometric approach where the starting point is a convex set of states. Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in the geometric approach. Particles are defined as elementary excitations of the ground state (and quasiparticles as elementary excitations of any translation invariant state). The conventional scattering matrix does not work for quasiparticles (and even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the notion of inclusive scattering matrix, closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) and the inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics. The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of the symbol of an operator; these arguments can be applied also in the geometric approach. The notion of inclusive scattering matrix makes sense in the geometric approach (but it seems that one cannot give a definition of the conventional scattering matrix in this situation). The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices can measure only a part of observables.