• Functional Analysis

Overview diagrams

• topological vector spaces

• locally convex topological vector spaces

Basic concepts

• topological vector space

  • locally convex topological vector space

  • Banach Spaces

    • reflexive

  • Smith Spaces

  • Hilbert Spaces, Fréchet Spaces, Sobolev spaces, Lebesgue Spaces

  • Bornological Vector Spaces

  • Barrelled Vector Spaces

• linear operator

  • bounded, unbounded, self-adjoint, compact, Fredholm

  • spectrum of an operator

• operator algebras

  • functional calculus

Theorems

• Stone-Weierstrass theorem

• spectral theory

  • spectral theorem

  • Gelfand duality

• functional calculus

• Riesz representation theorem

• measure theory

Topics in Functional Analysis

• Bases

• Algebraic Theories in Functional Analysis

• An Elementary Treatment of Hilbert Spaces

• When are two Banach spaces isomorphic?

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

• classical, pre-quantum, quantum, perturbative quantum

• relativistic, Euclidean, thermal

Lagrangian field theory

• field (physics)

  • field bundle

  • field history

  • space of field histories

• Lagrangian density

  • Euler-Lagrange form, presymplectic current

  • Euler-Lagrange equations of motion

• locally variational field theory

• covariant phase space

  • Peierls-Poisson bracket

    • advanced and retarded propagator,

    • causal propagator

quantization

• geometric quantization of symplectic groupoids

• algebraic deformation quantization, star algebra

quantum mechanical system, quantum probability

• subsystem

• observables

  • field observables

  • local observables

  • polynomial observables

  • microcausal observables

  • operator algebra, C*-algebra, von Neumann algebra

• local net of observables

  • causal locality

  • Haag-Kastler axioms

  • Wightman axioms

  • field net

  • conformal net

• state on a star-algebra, expectation value

  • pure state

    wave function

    collapse of the wave function/conditional expectation value

  • mixed state, density matrix

  • space of quantum states

  • vacuum state

    • quasi-free state,

    • Hadamard state

    • Wightman propagator

• picture of quantum mechanics

free field quantization

• star algebra, Moyal deformation quantization

• Wick algebra

  • canonical commutation relations, Weyl relations

  • normal ordered product

• Fock space

gauge theories

• gauge symmetry

• BRST complex, BV-BRST formalism

• local BV-BRST complex

• BV-operator

• quantum master equation

• master Ward identity

• gauge anomaly

interacting field quantization

• causal perturbation theory, perturbative AQFT

• interaction

• S-matrix, scattering amplitude

  • causal additivity

  • time-ordered product, Feynman propagator

  • Feynman diagram, Feynman perturbation series

  • effective action

  • vacuum stability

• interacting field algebra

  • Bogoliubov's formula

  • quantum Møller operator

• adiabatic limit

  • infrared divergence

  • interacting vacuum

renormalization

• ("re-")normalization scheme

  • extension of distributions

  • ("re"-)normalization condition

  • quantum anomaly

• renormalization group

  • interaction vertex redefinition

  • Stückelberg-Petermann renormalization group

  • renormalization group flow/running coupling constants

• effective quantum field theory

  • UV cutoff

  • counterterms

  • relative effective action

  • Wilsonian RG, Polchinski flow equation

Theorems

States and observables

• order-theoretic structure in quantum mechanics

  • Alfsen-Shultz theorem

  • Harding-Döring-Hamhalter theorem

• Kochen-Specker theorem

• Bell's theorem

• Fell's theorem

• Gleason's theorem

• Wigner theorem

• Bub-Clifton theorem

• Kadison-Singer problem

Operator algebra

• Wick's theorem

• GNS construction

  • cyclic vector, separating vector

• modular theory

• Fell's theorem

• Stone-von Neumann theorem

• Haag's theorem

Local QFT

• Reeh-Schlieder theorem

• Bisognano-Wichmann theorem

• PCT theorem

• spin-statistics theorem

• DHR superselection theory

• Osterwalder-Schrader theorem (Wick rotation)

Perturbative QFT

• Schwinger-Dyson equation

• main theorem of perturbative renormalization