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A Short Introduction to the Quantum Formalism[S] A short introduction to the quantum formalism[s] François David Institut de Physique Théorique CNRS, URA 2306, F-91191 Gif-sur-Yvette, France CEA, IPhT, F-91191 Gif-sur-Yvette, France [email protected] These notes are an elaboration on: (i) a short course that I gave at the IPhT-Saclay in May- June 2012; (ii) a previous letter [Dav11] on reversibility in quantum mechanics. They present an introductory, but hopefully coherent, view of the main formalizations of quantum mechanics, of their interrelations and of their common physical underpinnings: causality, reversibility and locality/separability. The approaches covered are mainly: (ii) the canonical formalism; (ii) the algebraic formalism; (iii) the quantum logic formulation. Other subjects: quantum information approaches, quantum correlations, contextuality and non-locality issues, quantum measurements, interpretations and alternate theories, quantum gravity, are only very briefly and superficially discussed. Most of the material is not new, but is presented in an original, homogeneous and hopefully not technical or abstract way. I try to define simply all the mathematical concepts used and to justify them physically. These notes should be accessible to young physicists (graduate level) with a good knowledge of the standard formalism of quantum mechanics, and some interest for theoretical physics (and mathematics). These notes do not cover the historical and philosophical aspects of quantum physics. arXiv:1211.5627v1 [math-ph] 24 Nov 2012 Preprint IPhT t12/042 ii CONTENTS Contents 1 Introduction 1-1 1.1 Motivation . 1-1 1.2 Organization . 1-2 1.3 What this course is not! . 1-3 1.4 Acknowledgements . 1-3 2 Reminders 2-1 2.1 Classical mechanics . 2-1 2.1.1 Lagrangian formulation . 2-1 2.1.2 Hamiltonian formulation . 2-2 2.1.2.a - Phase space and Hamiltonian: . 2-2 2.1.2.b - Hamilton-Jacobi equation . 2-3 2.1.2.c - Symplectic manifolds . 2-4 2.1.2.d - Observables, Poisson brackets . 2-5 2.1.2.e - Dynamics, Hamiltonian flows: . 2-5 2.1.2.f - The Liouville measure . 2-6 2.1.2.g - Example: the classical spin . 2-6 2.1.2.h - Statistical states, distribution functions, the Liouville equation . 2-7 2.1.2.i - Canonical transformations . 2-8 2.1.2.j - Along the Hamiltonian flow . 2-9 2.1.3 The commutative algebra of observables . 2-10 2.1.4 "Axiomatics" . 2-11 2.2 Probabilities . 2-11 2.2.1 The frequentist point of view . 2-11 2.2.2 The Bayesian point of view . 2-12 2.2.3 Conditional probabilities . 2-12 2.3 Quantum mechanics: “canonical” formulation . 2-13 2.3.1 Principles . 2-13 2.3.1.a - Pure states: . 2-13 2.3.1.b - Observables: . 2-13 2.3.1.c - Measurements, Born principle: . 2-13 2.3.1.d - Unitary dynamics . 2-14 2.3.1.e - Multipartite systems: . 2-14 2.3.1.f - Correspondence principe, canonical quantization . 2-15 2.3.2 Representations of quantum mechanics . 2-15 2.3.2.a - The Schrödinger picture . 2-15 2.3.2.b - The Heisenberg picture . 2-16 2.3.3 Quantum statistics and the density matrix . 2-17 2.3.3.a - The density matrix . 2-17 2.3.3.b - Interpretations . 2-18 François David, 2012 Lecture notes – November 27, 2012 CONTENTS iii 2.3.3.c - The von Neumann entropy . 2-19 2.3.3.d - Application: Entanglement entropy . 2-19 2.3.3.e - Gibbs states . 2-20 2.3.3.f - Imaginary time formalism . 2-20 2.4 Path and functional integrals formulations . 2-21 2.4.1 Path integrals . 2-21 2.4.2 Field theories, functional integrals . 2-23 2.5 Quantum mechanics and reversibility . 2-24 2.5.1 Is quantum mechanics reversible or irreversible? . 2-24 2.5.2 Reversibility of quantum probabilities . 2-24 3 Algebraic quantum formalism 3-1 3.1 Introduction . 3-1 3.2 The algebra of observables . 3-1 3.2.1 The mathematical principles . 3-1 3.2.1.a - Observables . 3-2 3.2.1.b - The ∗-conjugation . 3-2 3.2.1.c - States . 3-2 3.2.2 Physical discussion . 3-2 3.2.2.a - Observables and causality . 3-2 3.2.2.b - The ∗-conjugation and reversibility . 3-3 3.2.2.c - States, mesurements and probabilities . 3-3 3.2.3 Physical observables and pure states . 3-4 3.2.3.a - Physical (symmetric) observables: . 3-4 3.2.3.b - Pure states: . 3-4 3.2.3.c - Bounded observables . 3-4 3.3 The C∗-algebra of observables . 3-4 3.3.1 The norm on observables, A is a Banach algebra . 3-4 3.3.2 The observables form a real C∗-algebra . 3-5 3.3.3 Spectrum of observables and results of measurements . 3-5 3.3.4 Complex C∗-algebras . 3-6 3.4 The GNS construction, operators and Hilbert spaces . 3-7 3.4.1 Finite dimensional algebra of observables . 3-7 3.4.2 Infinite dimensional real algebra of observables . 3-8 3.4.3 The complex case, the GNS construction . 3-9 3.5 Why complex algebras? . 3-10 3.5.1 Dynamics: . 3-11 3.5.2 Locality and separability: . 3-11 3.5.3 Quaternionic Hilbert spaces: . 3-12 3.6 Superselection sectors . 3-12 3.6.1 Definition . 3-12 3.6.2 A simple example: the particle on a circle . 3-12 3.6.3 General discussion . 3-13 3.7 von Neumann algebras . 3-14 3.7.1 Definitions . 3-14 3.7.2 Classification of factors . 3-15 3.7.3 The Tomita-Takesaki theory . 3-15 3.8 Locality and algebraic quantum field theory . 3-16 3.8.1 Algebraic quantum field theory in a dash . 3-16 IPhT 2012 Introduction to quantum formalism[s] iv CONTENTS 3.8.2 Axiomatic QFT . 3-19 3.8.2.a - Wightman axioms . 3-19 3.8.2.b - CPT and spin-statistics theorems . 3-19 3.9 Discussion . 3-20 4 The quantum logic formalism 4-1 4.1 Introduction: measurements as logic . 4-1 4.2 A presentation of the principles . 4-4 4.2.1 Projective measurements as propositions . 4-4 4.2.2 Causality, POSET’s and the lattice of propositions . 4-4 4.2.2.a - Causal order relation: . 4-5 4.2.2.b - AND (meet ^): . 4-5 4.2.2.c - Logical OR (join _): . 4-6 4.2.2.d - Trivial 1 and vacuous Æ propositions: . 4-6 4.2.3 Reversibility and orthocomplementation . 4-6 4.2.3.a - Negations a0 and 0a .............................. 4-6 4.2.3.b - Causal reversibility and negation . 4-7 4.2.3.c - Orthogonality . 4-8 4.2.4 Subsystems of propositions and orthomodularity . 4-9 4.2.4.a - What must replace distributivity? . 4-9 4.2.4.b - Sublattices and weak-modularity . 4-9 4.2.4.c - Orthomodular lattices . ..
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