Schedule. 9:30 to 11:30 (BST) Wednesdays via MS Teams.
This course aims to introduce the basic setting of quantum mechanics from a conceptual point of view, starting with an algebraic axiomatization of the process of measuring quantities in a physical experiment (introduced essentially by I. Segal), and proceeding to motivate the need for non-commutativity from experimental observations. Quantum mechanics emerges then as the dynamics of a non-commutative probability space with Heisenberg's commutation relations as describing a “maximally non-commutative” situation. If time permits we will cover the construction of quantum dynamics via Euclidean path integrals.
A rough outline of the topics is as follows:
QM as a theory of measurement. The Stern-Gerlach experiment and quantum phenomena. Mathematical model of a physical system. Operational definition and properties of observables and states. Duality, algebraic structure. C*-algebras. Classical mechanics from commutative observation al Example of the description of a classical mechanical system.
Elements of the theory of C* algebras. Definition and first properties, spectrum of an element, spectral radius, multiplicative functionals, Abelian Banach algebras, maximal ideals and multiplicative functionals, Gelfand transform, Gelfand-Naimark theorem on the structure of commutative C* algebras, functional calculus. Properties of positive elements, positive linear functionals. Gelfand–Naimark–Segal representations, representations of non-commutative C* algebras, the non-commutative Gelfand–Naimark theorem, irreducible representations and pure states, pure state of abelian algebras are multiplicative, probabilistic interpretation, folium of a representation, Fell's theorem and physical interpretation
Non-commutativity and quantum phenomena. necessity of non-commutativity to describe quantum phenomenology, Heisenberg's indetermination principle and Planck's constant, Schrödinger-Robertson inequality, canonical commutation relations (CCR), impossibility to realize the CCR with bounded operators, probabilistic phenomena in non-commutative algebras, construction of discrete systems of complementary observables. Canonical commutation relations as limits of discrete ones.
The Weyl algebra. Baker-Campbell-Hausdorff formula and relation with Heisenberg's commutation relations. Observables as *-homomorphism and unitary representations of R on Hilbert space. Heisenberg group and von Neumann theorem on the uniqueness of irreducible regular representations of Weyl algebra in finite dimensions, Schrödinger representation, irreducibility. Determination of the vacuum in the Schrödinger representation. Gaussian representation of Weyl relations. Reducible representations.
Dynamics of a quantum system. Unitary representation of weakly continuous automorphism on an invariant state. Link between unitary groups and self-adjoint semigroups. Energy operator. Semigroups and *-homomorphism, Bernstein theorem on totally monotone functions. Ground states and Wightman functions.
Euclidean approach. (if time permits) Axioms for Wightman functions. Axioms for Schwinger functions and the reconstruction theorem. Reflection positivity. Reflection positivity for Markov processes and construction or reflection positive processes via SDEs.
Strocchi, F. An Introduction to the Mathematical Structure of Quantum Mechanics: A Short Course for Mathematicians. 2 edition. New Jersey: World Scientific Publishing Company, 2008.
I. E. Segal. Postulates for General Quantum Mechanics. The Annals of Mathematics, 48(4):930, oct 1947.
Segal, E. E., and George Whitelaw Mackey. Mathematical Problems of Relativistic Physics. Providence, RI: American Mathematical Society, 1963.
Rudolf Haag. Local Quantum Physics: Fields, Particles, Algebras. Texts and Monographs in Physics. Springer, 2nd rev. and enl. ed edition, 1996.
Meyer, Paul A. Quantum Probability for Probabilists. Springer, 1995.
The material I will present has already been used to deliver the course “Functional Integration and Quantum Mechanics” in Bonn in SS2020 [link] were you can find scripts and resources useful to peek on the specific contents of this year's course.
Note 1 (version 1, 23/4/24) [pdf]
Week 1 | (24.4.24) The Stern–Gerlach experiment. Axiomatization of the measurement process: observable, states and their duality. The C* algebraic setting, basic definition, C* condition, spectrum of an element. [script]
Week 2 | (1.5.24) The C* algebraic setting (continued) : spectral theory of commutative Banach algebras, Gelfand transform, positive elements, states. [script]
(leftovers from Week 2) Gelfand–Naimark–Segal representation and the Gelfand–Naimark theorem, pure states and irreducible representations, folium of a representation, Fell's theorem, the quantum world.
Week 3 | (8.5.24) Heisenberg indetermination principle, Schrödinger-Robertson inequality, canonical commutation relations (CCR), probabilistic phenomena in non-commutative algebras, construction of discrete systems of complementary observables. Canonical commutation relations as limits of discrete ones. Weyl algebra, Baker-Campbell-Hausdorff formula and relation with Heisenberg's commutation relations.
Week 4 | (15.5.24) Weyl algebra, Heisenberg group and von Neumann theorem on the uniqueness of irreducible regular representations of Weyl algebra in finite dimensions, Schrödinger representation, irreducibility. Determination of the vacuum in the Schrödinger representation. Gaussian representation of Weyl relations. Reducible representations.
Week 5 | (22.5.24) Dynamics of a quantum system: unitary representation of weakly continuous automorphism on an invariant state
Week 6 | (29.5.24) Semigroups and *-homomorphism, Bernstein theorem on totally monotone functions. Link between unitary groups and self-adjoint semigroups. Energy operator. Positive energy condition. Link between unitary groups and self-adjoint semigroups continued. Ground states and Wightman functions.
Week 7 | (5.6.24) Axioms for Schwinger functions and the reconstruction theorem. Reflection positivity for Gaussian processes.
Week 8 | (19.6.24) [Warning: no lecture on June 12th!] Nelson's positivity and stochastic processes. Reflection positivity for Markov processes and construction or reflection positive processes via SDEs.