Abstract
The reformulation of classical theory done in Chap. 3 served as a preparation for constructing QFT models. The framework that we are going to use is deformation quantization combined with causal perturbation theory. To quantize a given theory described by the action S we first need to split S into a free part \(S_0\) (at most quadratic in field configurations) and the interaction term \(S_I\). Then, we quantize the theory defined by \(S_0\), using deformation quantization based on a Moyal-type formula, and in the final step we will re-introduce the interaction using causal perturbation theory. This last step will be discussed in Chap. 6, while the present chapter deals with deformation quantization.
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References
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. 111(1), 61–110 (1978)
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. II. Physical applications. Ann. Phys. 111(1), 111–151 (1978)
Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz quantization of Kähler manifolds an gl(N), \({N} \rightarrow \infty \) limits. Commun. Math. Phys. 165(2), 281–296 (1994)
Brunetti, R., Dütsch, M., Fredenhagen, K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13(5), 1541–1599 (2009)
Chilian, B., Fredenhagen, K.: The time slice axiom in perturbative quantum field theory on globally hyperbolic spacetimes. Commun. Math. Phys. 287(2), 513–522 (2008). arXiv:0802.1642v1 [math-ph]
DÃijtsch, M., Fredenhagen, K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219(1), 5–30 (2001)
Dütsch, M., Fredenhagen, K.: Perturbative algebraic field theory, and deformation quantization. Math. Phys. Math. Phys. Quantum Oper. Algebr. Asp. 30, 1–10 (2001)
Dereziński, J.: Gérard, C: Mathematics of Quantization and Quantum Fields. Cambridge University Press, Cambridge (2013)
Dito, J.: Star-product approach to quantum field theory: the free scalar field. Lett. Math. Phys. 20(2), 125–134 (1990)
Groenewold, H.J.: On the principles of elementary quantum mechanics. Physica 12(7), 405–460 (1946)
Hawkins, E.: An obstruction to quantization of the sphere. Commun. Math. Phys. 283(3), 675–699 (2008)
Kontsevich, M.: Deformation quantization of Poisson manifolds. I. Lett. Math. Phys. 66, 157–216 (2003)
Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529–553 (1996)
Rieffel, M.A.: Quantization and \(C^*\)-algebras. Contemp. Math. 167, 67–67 (1994)
Rieffel, M.A.: Questions on quantization. Contemp. Math. 228, 315–326 (1998)
Van Hove, L.: Sur certaines représentations unitaires d’un groupe infini de transformations, acad. roy. Belg. Cl. Sci. Mém. Collect 80(2), 29 (1951)
Waldmann, S.: Poisson-geometrie und deformationsquantisierung: Eine einführung. Springer, Berlin (2007)
Zahn, J.: The renormalized locally covariant Dirac field. Rev. Math. Phys. 26, 1330012 (2014)
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Rejzner, K. (2016). Deformation Quantization. In: Perturbative Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-25901-7_5
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DOI: https://doi.org/10.1007/978-3-319-25901-7_5
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