Quantum mechanics and quantum field theory from algebraic and geometric viewpoints.
Quantum mechanics can be obtained as a classical theory with a restricted set of observables.
Generalizations of QM can be obtained similarly. These theories (deterministic physical theories) can be described as geometric theories ( theories where the starting point is the set of states).
Decoherence can be proved in general geometric theories placed in a random environment. One can derive probabilities from decoherence.
Theories with translation symmetry=quantum field theories
without fields.
In theories with translation symmetry, one can define particles and quasiparticles.
In the algebraic approach to quantum theory, we can use asymptotic commutativity or the cluster property to define the inclusive scattering matrix and prove its existence.
The inclusive scattering matrix can be expressed in terms of generalized Green functions (Keldysh Green functions) on shell.
The notion of an inclusive scattering matrix can be introduced in geometric theories.
Calculations in the geometric approach are as easy as in the conventional approach (and sometimes easier).
L-functionals. Applications to the infrared problem in QED.
Generalizations of QM can be obtained similarly. These theories (deterministic physical theories) can be described as geometric theories ( theories where the starting point is the set of states).
Decoherence can be proved in general geometric theories placed in a random environment. One can derive probabilities from decoherence.
Theories with translation symmetry=quantum field theories
without fields.
In theories with translation symmetry, one can define particles and quasiparticles.
In the algebraic approach to quantum theory, we can use asymptotic commutativity or the cluster property to define the inclusive scattering matrix and prove its existence.
The inclusive scattering matrix can be expressed in terms of generalized Green functions (Keldysh Green functions) on shell.
The notion of an inclusive scattering matrix can be introduced in geometric theories.
Calculations in the geometric approach are as easy as in the conventional approach (and sometimes easier).
L-functionals. Applications to the infrared problem in QED.
Lecturer
Date
19th March ~ 28th May, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday | 10:40 - 12:15 | Shuangqing-B534 | ZOOM 11 | 435 529 7909 | BIMSA |
Prerequisite
I will freely use the notions of Poisson brackets, Lie algebra of a topological group, derivation of algebra, convex set, convex envelope of a set. I will remind definitions of these notions; a review of them can be found in \Appendices A1 and A2 to my book [1].
Syllabus
The course will be divided into several independent ( or almost independent) parts.
1. General questions
This part contains the formulation of some basic definitions and results with special attention to the review of deterministic theories and their relation to classical theories with a restricted set of observables (following Chapter 4 of my book [1] and paper [2]) and to the notion of inclusive scattering matrix.
This part is not a prerequisite for understanding the main part of the course.
2. Quantum mechanics and quantum field theory from algebraic and geometric viewpoints.
This part follows Chapters 1 and 2 of book [1] and paper [3].
The main topics: algebraic and geometric approaches to quantum theory, GNS construction, Weyl and Clifford algebras, superspace, solitons as classical analogs of particles, quantum particles and quasiparticles, scattering matrix and inclusive scattering matrix, Green functions and generalized Green functions,
LSZ formula for the scattering matrix and its analog for the inclusive scattering matrix. Appearance of the inclusive scattering matrix in the formalism of L-functionals.
3. Quantization
This part is devoted to the rigorous quantization of symplectic vector spaces and Lagrangian subspaces of these spaces. To quantize, we should fix Darboux coordinates; the result does not depend on this choice (up to a constant factor). In the formalism of L-functionals, it is not necessary to fix Darboux coordinates. These ideas can be used to quantize quadratic action functionals, in particular, the Dirichlet functional, and to develop a simplified version of operator formalism in CFT.
Paper in preparation.
4. Quantum electrodynamics in the formalism of L-functionals
The conventional scattering matrix vanishes in quantum electrodynamics because every process is accompanied by the emission of soft photons. However, the inclusive scattering matrix
does exist; it is infrared finite.
Paper in preparation (with I. Frolov)
5. Introduction to homological algebra and derived geometry. Applications to physics
Differential graded algebras. Homology. Euler characteristic and Lefschetz Trace formula. Quasi-isomorphism. BRST quantization. Projective modules and projective resolution. Koszul-Tate resolution. Cohomology of Lie algebras.
Supermanifolds. Q-manifolds and QP-manifolds. Derived geometry.
BV formalism.
Book in preparation
6. Local field theories.
Sewing of two domains. Axiomatics of conformal theory. BV formalism and BFV formalism. Cattaneo-Mnev-Reshetikhin theory.
Book in preparation.
7. New approach to string theory and superstring theory [4],[5]
A classical theory with first-class constraints can be quantized in terms of the BRST formalism. One can describe a broad class of physical quantities for quantum theory and show that multiloop amplitudes of string theory are among these quantities.
1. General questions
This part contains the formulation of some basic definitions and results with special attention to the review of deterministic theories and their relation to classical theories with a restricted set of observables (following Chapter 4 of my book [1] and paper [2]) and to the notion of inclusive scattering matrix.
This part is not a prerequisite for understanding the main part of the course.
2. Quantum mechanics and quantum field theory from algebraic and geometric viewpoints.
This part follows Chapters 1 and 2 of book [1] and paper [3].
The main topics: algebraic and geometric approaches to quantum theory, GNS construction, Weyl and Clifford algebras, superspace, solitons as classical analogs of particles, quantum particles and quasiparticles, scattering matrix and inclusive scattering matrix, Green functions and generalized Green functions,
LSZ formula for the scattering matrix and its analog for the inclusive scattering matrix. Appearance of the inclusive scattering matrix in the formalism of L-functionals.
3. Quantization
This part is devoted to the rigorous quantization of symplectic vector spaces and Lagrangian subspaces of these spaces. To quantize, we should fix Darboux coordinates; the result does not depend on this choice (up to a constant factor). In the formalism of L-functionals, it is not necessary to fix Darboux coordinates. These ideas can be used to quantize quadratic action functionals, in particular, the Dirichlet functional, and to develop a simplified version of operator formalism in CFT.
Paper in preparation.
4. Quantum electrodynamics in the formalism of L-functionals
The conventional scattering matrix vanishes in quantum electrodynamics because every process is accompanied by the emission of soft photons. However, the inclusive scattering matrix
does exist; it is infrared finite.
Paper in preparation (with I. Frolov)
5. Introduction to homological algebra and derived geometry. Applications to physics
Differential graded algebras. Homology. Euler characteristic and Lefschetz Trace formula. Quasi-isomorphism. BRST quantization. Projective modules and projective resolution. Koszul-Tate resolution. Cohomology of Lie algebras.
Supermanifolds. Q-manifolds and QP-manifolds. Derived geometry.
BV formalism.
Book in preparation
6. Local field theories.
Sewing of two domains. Axiomatics of conformal theory. BV formalism and BFV formalism. Cattaneo-Mnev-Reshetikhin theory.
Book in preparation.
7. New approach to string theory and superstring theory [4],[5]
A classical theory with first-class constraints can be quantized in terms of the BRST formalism. One can describe a broad class of physical quantities for quantum theory and show that multiloop amplitudes of string theory are among these quantities.
Reference
An essential part of the course follows my short book "Quantum mechanics and quantum field theory from algebraic and geometric viewpoints" ( Springer, 2024)
1. Quantum mechanics and quantum field theory from algebraic and geometric viewpoints.
This part follows Chapters 1 and 2 of book [1]
The main topics: algebraic and geometric approaches to quantum theory, GNS construction, Weyl and Clifford algebras, superspace, solitons as classical analogs of particles, quantum particles and quasiparticles, scattering matrix and inclusive scattering matrix, Green functions and generalized Green functions,
LSZ formula for the scattering matrix and its analog for the inclusive scattering matrix. Appearance of the inclusive scattering matrix in the formalism of L-functionals.
2. General questions
This part contains the formulation of some basic definitions and results with special attention to the review of deterministic theories and their relation to classical theories with a restricted set of observables (following Chapter 4 of my book [1] and paper [2]) and to the notion of inclusive scattering matrix.
1. Quantum mechanics and quantum field theory from algebraic and geometric viewpoints.
This part follows Chapters 1 and 2 of book [1]
The main topics: algebraic and geometric approaches to quantum theory, GNS construction, Weyl and Clifford algebras, superspace, solitons as classical analogs of particles, quantum particles and quasiparticles, scattering matrix and inclusive scattering matrix, Green functions and generalized Green functions,
LSZ formula for the scattering matrix and its analog for the inclusive scattering matrix. Appearance of the inclusive scattering matrix in the formalism of L-functionals.
2. General questions
This part contains the formulation of some basic definitions and results with special attention to the review of deterministic theories and their relation to classical theories with a restricted set of observables (following Chapter 4 of my book [1] and paper [2]) and to the notion of inclusive scattering matrix.
Video Public
Yes
Notes Public
Yes
Language
English