Free Fermi and Bose Fields in TQFT and GBF

Robert Oeckl

Symmetry, integrability and geometry: methods and applications, Tome 9 (2013)

Résumé

We present a rigorous and functorial quantization scheme for linear fermionic and bosonic field theory targeting the topological quantum field theory (TQFT) that is part of the general boundary formulation (GBF). Motivated by geometric quantization, we generalize a previous axiomatic characterization of classical linear bosonic field theory to include the fermionic case. We proceed to describe the quantization scheme, combining a Fock space quantization for state spaces with the Feynman path integral for amplitudes. We show rigorously that the resulting quantum theory satisfies the axioms of the TQFT, in a version generalized to include fermionic theories. In the bosonic case we show the equivalence to a previously developed holomorphic quantization scheme. Remarkably, it turns out that consistency in the fermionic case requires state spaces to be Krein spaces rather than Hilbert spaces. This is also supported by arguments from geometric quantization and by the explicit example of the Dirac field theory. Contrary to intuition from the standard formulation of quantum theory, we show that this is compatible with a consistent probability interpretation in the GBF. Another surprise in the fermionic case is the emergence of an algebraic notion of time, already in the classical theory, but inherited by the quantum theory. As in earlier work we need to impose an integrability condition in the bosonic case for all TQFT axioms to hold, due to the gluing anomaly. In contrast, we are able to renormalize this gluing anomaly in the fermionic case.

Détail

Keywords: general boundary formulation; topological quantum field theory; fermions; free field theory; functorial quantization; foundations of quantum theory; quantum field theory.

Bibliographie

[1] Atiyah M., “Topological quantum field theories”, Inst. Hautes Études Sci. Publ. Math., 1988, no. 68, 175–186 | MR | Zbl

[2] Baez J. C., Segal I. E., Zhou Z. F., Introduction to algebraic and constructive quantum field theory, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1992 | MR | Zbl

[3] Bleuler K., “Eine neue Methode zur Behandlung der longitudinalen und skalaren Photonen”, Helvetica Phys. Acta, 23 (1950), 567–586 | MR | Zbl

[4] Colosi D., General boundary quantum field theory in de Sitter spacetime, arXiv: 1010.1209

[5] Gupta S. N., “Theory of longitudinal photons in quantum electrodynamics”, Proc. Phys. Soc. Sect. A, 63 (1950), 681–691 | MR | Zbl

[6] Oeckl R., “Affine holomorphic quantization”, J. Geom. Phys., 62 (2012), 1373–1396, arXiv: 1104.5527 | DOI | MR | Zbl

[7] Oeckl R., “General boundary quantum field theory: foundations and probability interpretation”, Adv. Theor. Math. Phys., 12 (2008), 319–352, arXiv: hep-th/0509122 | MR | Zbl

[8] Oeckl R., “General boundary quantum field theory: timelike hypersurfaces in the Klein–Gordon theory”, Phys. Rev. D, 73 (2006), 065017, 13 pp., arXiv: hep-th/0509123 | DOI | MR

[9] Oeckl R., “Holomorphic quantization of linear field theory in the general boundary formulation”, SIGMA, 8 (2012), 050, 31 pp., arXiv: 1009.5615 | DOI | MR

[10] Oeckl R., “Observables in the general boundary formulation”, Quantum Field Theory and Gravity (Regensburg, 2010), Birkhäuser, Basel, 2012, 137–156, arXiv: 1101.0367 | DOI | MR

[11] Oeckl R., “Probabilities in the general boundary formulation”, J. Phys. Conf. Ser., 67 (2007), 012049, 6 pp., arXiv: hep-th/0612076 | DOI

[12] Oeckl R., “Schrödinger–Feynman quantization and composition of observables in general boundary quantum field theory”, arXiv: 1201.1877

[13] Oeckl R., “States on timelike hypersurfaces in quantum field theory”, Phys. Lett. B, 622 (2005), 172–177, arXiv: hep-th/0505267 | DOI | MR | Zbl

[14] Oeckl R., “Two-dimensional quantum Yang–Mills theory with corners”, J. Phys. A: Math. Theor., 41 (2008), 135401, 20 pp., arXiv: hep-th/0608218 | DOI | MR | Zbl

[15] Peskin M. E., Schroeder D. V., An introduction to quantum field theory, Addison-Wesley Publishing Company, Reading, MA, 1995 | MR

[16] Reeh H., Schlieder S., “Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten Felden”, Nuovo Cimento, 22 (1961), 1051–1068 | DOI | MR

[17] Segal G., “The definition of conformal field theory”, Differential Geometrical Methods in Theoretical Physics, Como, 1987, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 250, Kluwer Acad. Publ., Dordrecht, 1988, 165–171 | MR

[18] Segal G., “The definition of conformal field theory”, Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., 308, Cambridge University Press, Cambridge, 2004, 421–577 | MR | Zbl

[19] Woodhouse N., Geometric quantization, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1980 | MR