Communications in Mathematical Physics

, Volume 360, Issue 1, pp 439–479 | Cite as

A Cohomological Perspective on Algebraic Quantum Field Theory

  • Eli Hawkins
Open Access


Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model.

This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited.

To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory.


  1. 1.
    Bär, C.: Green-hyperbolic operators on globally hyperbolic spacetimes. Commun. Math. Phys. 333(3), 1585–1615 (2015). arXiv:1310.0738 [math-ph]
  2. 2.
    Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization. Ann. Phys. 111, 61–151 (1977)CrossRefMATHADSGoogle Scholar
  3. 3.
    Benini, M., Schenkel, A.: Quantum field theories on categories fibered in groupoids. Commun. Math. Phys. 356(1), 19–64 (2017). arXiv:1610.06071 [math-ph]
  4. 4.
    Brunetti R., Fredenhagen K., Imani P., Rejzner K.: The locality axiom in quantum field theory and tensor products of C *-algebras. Rev. Math. Phys. 26(6), 1450010 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle—a new paradigm for local quantum field theory. Commun. Math. Phys. 237(1–2), 31–68 (2003)MathSciNetCrossRefMATHADSGoogle Scholar
  6. 6.
    Buchholz D., D’Antoni C., Fredenhagen K.: The universal structure of local algebras. Commun. Math. Phys. 111(1), 123–135 (1987)MathSciNetCrossRefMATHADSGoogle Scholar
  7. 7.
    Buchholz, D., Lechner, G., Summers, S.: Warped convolutions, Rieffel deformations and the construction of quantum field theories. Commun. Math. Phys. 304(1), 95–123 (2011). arXiv:1005.2656 [math-ph]
  8. 8.
    Dütsch M., Fredenhagen K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219(1), 5–30 (2001)MathSciNetCrossRefMATHADSGoogle Scholar
  9. 9.
    Dütsch M., Fredenhagen K.: Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity. Rev. Math. Phys. 16(10), 1291–1348 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fewster, C.J.: Endomorphisms and automorphisms of locally covariant quantum field theories. Rev. Math. Phys. 25(5), 1350008 (2013)Google Scholar
  11. 11.
    Fewster, C.J.: An analogue of the Coleman–Mandula theorem for quantum field theory in curved apacetimes. Commun. Math. Phys. (2017). arXiv:1609.02705 [math-ph]
  12. 12.
    Fredenhagen, K.: The structure of local algebras of observables. In: Operator Algebras and Mathematical Physics (Iowa City, Iowa, 1985). Contemp. Math., vol. 62, pp. 153–165. Amer. Math. Soc., Providence, RI (1987)Google Scholar
  13. 13.
    Fredenhagen, K.: The structure of local algebras of observables in relativistic quantum field theory. In: Topics in Quantum Field Theory and Spectral Theory (Reinhardsbrunn, 1985), Rep. MATH, 86-1, pp. 55–67. Akad. Wiss. DDR, Berlin (1986)Google Scholar
  14. 14.
    Gerstenhaber M.: The cohomology structure of an associative ring. Ann. Math. (2) 78, 267–288 (1963)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gerstenhaber M.: On the deformation of rings and algebras. Ann. Math. (2) 79, 59–103 (1964)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gerstenhaber M., Schack Samuel D.: The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set. Trans. Am. Math. Soc. 310(1), 135–165 (1988)MathSciNetMATHGoogle Scholar
  17. 17.
    Gerstenhaber, M., Schack, S.D.: Algebraic cohomology and deformation theory. In: Deformation Theory of Algebras and Structures and Applications (Il Ciocco, 1986). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 247, pp. 11–264. Kluwer Acad. Publ., Dordrecht (1988)Google Scholar
  18. 18.
    Guido D., Longo R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181(1), 11–35 (1996)MathSciNetCrossRefMATHADSGoogle Scholar
  19. 19.
    Haag, R.: Local quantum physics. Texts and Monographs in Physics. Springer, Berlin (1996)Google Scholar
  20. 20.
    Hawking, S.W., Ellis, G.F.R.: The large scale structure of space–time. Cambridge Monographs on Mathematical Physics, vol. 1. Cambridge University Press, London (1973)Google Scholar
  21. 21.
    Hawkins, E., Rejzner, K.: The Star Product in Interacting Quantum Field Theory. arXiv:1612.09157 [math-ph]
  22. 22.
    Hochschild G.: On the cohomology groups of an associative algebra. Ann. Math. (2) 46, 58–67 (1945)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003). arXiv:q-alg/9709040
  24. 24.
    Leinster, T.: Basic Bicategories. arXiv:math/9810017 [math.CT]
  25. 25.
    Leinster, T.: Higher Operads, Higher Categories. In: London Mathematical Society Lecture Note Series, vol. 298. Cambridge University Press, Cambridge (2004). arXiv:math/0305049 [math.CT]
  26. 26.
    Mitchell B.: Rings with several objects. Adv. Math. 8, 1–161 (1972)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Peierls R.E.: The commutation laws of relativistic field theory. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 214(1117), 143–157 (1952)MathSciNetCrossRefMATHADSGoogle Scholar
  28. 28.
    Müller, O., Sánchez, M.: Lorentzian manifolds isometrically embeddable in \({\mathbb{L}^N}\). Trans. Am. Math. Soc. 363(10), 5367–5379 (2011). arXiv:0812.4439 [math.DG]
  29. 29.
    Rieffel, M.A.: Deformation quantization for actions of \({\mathbb{R}^d}\). Mem. Am. Math. Soc. 106(506), x+93 (1993)Google Scholar
  30. 30.
    Rejzner K.: Perturbative Algebraic Quantum Field Theory—An Introduction for Mathematicians. Springer, New York (2016)CrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsThe University of YorkYorkUK

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