Algebraic quantum field theory

Algebraic quantum field theory (AQFT) is an application to local quantum physics of

, and mappings between those.

Haag–Kastler axioms

Let ${\displaystyle {\mathcal {O}}}$ be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a net ${\displaystyle \{{\mathcal {A}}(O)\}_{O\in {\mathcal {O}}}}$ of von Neumann algebras ${\displaystyle {\mathcal {A}}(O)}$ on a common Hilbert space ${\displaystyle {\mathcal {H}}}$ satisfying the following axioms:^[1]

• Isotony: ${\displaystyle O_{1}\subset O_{2}}$ implies ${\displaystyle {\mathcal {A}}(O_{1})\subset {\mathcal {A}}(O_{2})}$.
• Causality: If ${\displaystyle O_{1}}$ is space-like separated from ${\displaystyle O_{2}}$, then ${\displaystyle [{\mathcal {A}}(O_{1}),{\mathcal {A}}(O_{2})]=0}$.
• Poincaré covariance: A strongly continuous unitary representation ${\displaystyle U({\mathcal {P}})}$ of the Poincaré group ${\displaystyle {\mathcal {P}}}$ on ${\displaystyle {\mathcal {H}}}$ exists such that ${\displaystyle {\mathcal {A}}(gO)=U(g){\mathcal {A}}(O)U(g)^{*},\,\,g\in {\mathcal {P}}.}$
• Spectrum condition: The joint spectrum ${\displaystyle \mathrm {Sp} (P)}$ of the energy-momentum operator ${\displaystyle P}$ (i.e. the generator of space-time translations) is contained in the closed forward lightcone.
• Existence of a vacuum vector: A cyclic and Poincaré-invariant vector ${\displaystyle \Omega \in {\mathcal {H}}}$ exists.

The net algebras ${\displaystyle {\mathcal {A}}(O)}$ are called local algebras and the C* algebra ${\displaystyle {\mathcal {A}}:={\overline {\bigcup _{O\in {\mathcal {O}}}{\mathcal {A}}(O)}}}$ is called the quasilocal algebra.

Category-theoretic formulation

Let Mink be the

${\displaystyle {\mathcal {A}}}$ from Mink to uC*alg, the category of in uC*alg (isotony).

The

norm topology

of ${\displaystyle {\mathcal {A}}(M)}$ (

Poincaré covariance

).

Minkowski space has a causal structure. If an open set V lies in the causal complement of an open set U, then the image of the maps

${\displaystyle {\mathcal {A}}(i_{U,U\cup V})}$

and

${\displaystyle {\mathcal {A}}(i_{V,U\cup V})}$

(spacelike commutativity). If ${\displaystyle {\bar {U}}}$ is the causal completion of an open set U, then ${\displaystyle {\mathcal {A}}(i_{U,{\bar {U}}})}$ is an isomorphism (primitive causality).

A state with respect to a C*-algebra is a positive linear functional over it with unit norm. If we have a state over ${\displaystyle {\mathcal {A}}(M)}$, we can take the "partial trace" to get states associated with ${\displaystyle {\mathcal {A}}(U)}$ for each open set via the

structure.

According to the

of ${\displaystyle {\mathcal {A}}(M).}$ . This is the vacuum sector.

QFT in curved spacetime

More recently, the approach has been further implemented to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is in particular suitable to generalize the renormalization procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in presence of a black hole have been obtained.^{[citation needed]}

References

Further reading

• MR 0165864
• MR 1405610
• Brunetti, Romeo; Fredenhagen, Klaus; Verch, Rainer (2003). "The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Field Theory".
  S2CID 13950246
  .
• Brunetti, Romeo; Dütsch, Michael; Fredenhagen, Klaus (2009). "Perturbative Algebraic Quantum Field Theory and the Renormalization Groups".
  S2CID 15493763
  .
• ISBN 978-3-642-02780-2
  .
• Brunetti, Romeo; Dappiaggi, Claudio;
  ISBN 978-3-319-21353-8
  .
• ISBN 978-3-319-25901-7
  .
• Hack, Thomas-Paul (2016). Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes. SpringerBriefs in Mathematical Physics. Vol. 6. Springer.
  S2CID 119657309
  .
• Dütsch, Michael (2019). From Classical Field Theory to Perturbative Quantum Field Theory. Progress in Mathematical Physics. Vol. 74. Birkhäuser.
  S2CID 126907045
  .
• Yau, Donald (2019). Homotopical Quantum Field Theory. World Scientific.
  S2CID 119168109
  .
• Dedushenko, Mykola (2023). "Snowmass white paper: The quest to define QFT". International Journal of Modern Physics A. 38 (4n05).
  S2CID 247450696
  .

External links

• Local Quantum Physics Crossroads 2.0 – A network of scientists working on local quantum physics
• Papers – A database of preprints on algebraic QFT
• Algebraic Quantum Field Theory – AQFT resources at the University of Hamburg