Holomorphic quantization of linear field theory in the general boundary formulation

Robert Oeckl

Symmetry, integrability and geometry: methods and applications, Tome 8 (2012)

Résumé

We present a rigorous quantization scheme that yields a quantum field theory in general boundary form starting from a linear field theory. Following a geometric quantization approach in the Kähler case, state spaces arise as spaces of holomorphic functions on linear spaces of classical solutions in neighborhoods of hypersurfaces. Amplitudes arise as integrals of such functions over spaces of classical solutions in regions of spacetime. We prove the validity of the TQFT-type axioms of the general boundary formulation under reasonable assumptions. We also develop the notions of vacuum and coherent states in this framework. As a first application we quantize evanescent waves in Klein–Gordon theory.

Détail

Keywords: geometric quantization, topological quantum field theory, coherent states, foundations of quantum theory, quantum field theory.

Bibliographie

[1] Atiyah M., “Topological quantum field theories”, Inst. Hautes Études Sci. Publ. Math., 1988, 175–186 | DOI | MR | Zbl

[2] Bargmann V., “On a Hilbert space of analytic functions and an associated integral transform. Part I”, Comm. Pure Appl. Math., 14 (1961), 187–214 | DOI | MR | Zbl

[3] Bargmann V., Remarks on a Hilbert space of analytic functions, Proc. Nat. Acad. Sci. USA, 48 (1962), 199–204 | DOI | MR | Zbl

[4] Bochner S., Harmonic analysis and the theory of probability, University of California Press, Berkeley and Los Angeles, 1955 | MR

[5] Colosi D., On the structure of the vacuum state in general boundary quantum field theory, arXiv: 0903.2476

[6] Colosi D., $S$-matrix in de Sitter spacetime from general boundary quantum field theory, arXiv: 0910.2756

[7] Colosi D., Oeckl R., “On unitary evolution in quantum field theory in curved spacetime”, Open Nuclear Part. Phys. J., 4 (2011), 13–20 ; arXiv: 0912.0556 | DOI

[8] Colosi D., Oeckl R., “$S$-matrix at spatial infinity”, Phys. Lett. B, 665 (2008), 310–313 ; arXiv: 0710.5203 | DOI | MR

[9] Colosi D., Oeckl R., “Spatially asymptotic $S$-matrix from general boundary formulation”, Phys. Rev. D, 78 (2008), 025020, 22 pp. ; arXiv: 0802.2274 | DOI

[10] Colosi D., Oeckl R., “States and amplitudes for finite regions in a two-dimensional Euclidean quantum field theory”, J. Geom. Phys., 59 (2009), 764–780 ; arXiv: 0811.4166 | DOI | MR | Zbl

[11] Conrady F., Doplicher L., Oeckl R., Rovelli C., Testa M., “Minkowski vacuum in background independent quantum gravity”, Phys. Rev. D, 69 (2004), 064019, 7 pp. ; arXiv: gr-qc/0307118 | DOI | MR

[12] Conrady F., Rovelli C., “Generalized Schrödinger equation in Euclidean field theory”, Internat. J. Modern Phys. A, 19 (2004), 4037–4068 ; arXiv: hep-th/0310246 | DOI | MR | Zbl

[13] Doplicher L., “Generalized Tomonaga–Schwinger equation from the Hadamard formula”, Phys. Rev. D, 70 (2004), 064037, 7 pp. ; arXiv: gr-qc/0405006 | DOI | MR

[14] Haag R., Kastler D., “An algebraic approach to quantum field theory”, J. Math. Phys., 5 (1964), 848–861 | DOI | MR | Zbl

[15] Lang S., Real and functional analysis, Graduate Texts in Mathematics, 142, 3rd ed., Springer-Verlag, New York, 1993 | DOI | MR | Zbl

[16] Oeckl R., “A “general boundary” formulation for quantum mechanics and quantum gravity”, Phys. Lett. B, 575 (2003), 318–324 ; arXiv: hep-th/0306025 | DOI | MR | Zbl

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

[18] Oeckl R., Against commutators, Talk at Perimeter Institute (Waterloo, Canada, January 20, 2009) available at http://pirsa.org/09010002/

[19] 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

[20] 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

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

[22] Oeckl R., “Schrödinger's cat and the clock: lessons for quantum gravity”, Classical Quantum Gravity, 20 (2003), 5371–5380 ; arXiv: gr-qc/0306007 | DOI | MR | Zbl

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

[24] Oeckl R., “The general boundary approach to quantum gravity”, Proceedings of the First International Conference on Physics (Tehran, 2004), Amirkabir University, Tehran, 2004, 257–265; arXiv: gr-qc/0312081

[25] 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

[26] Osterwalder K., Schrader R., “Axioms for Euclidean Green's functions”, Comm. Math. Phys., 31 (1973), 83–112 | DOI | MR | Zbl

[27] Osterwalder K., Schrader R., “Axioms for Euclidean Green's functions. II”, Comm. Math. Phys., 42 (1975), 281–305 | DOI | MR | Zbl

[28] 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

[29] Streater R.F., Wightman A.S., PCT, spin and statistics, and all that, W.A. Benjamin, Inc., New York – Amsterdam, 1964 | MR | Zbl

[30] Sudakov V.N., “Linear sets with quasi-invariant measure”, Dokl. Akad. Nauk SSSR, 127 (1959), 524–525 | MR | Zbl

[31] Turaev V.G., Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, 18, Walter de Gruyter Co., Berlin, 1994 | MR | Zbl

[32] Wald R.M., Quantum field theory in curved spacetime and black hole thermodynamics, Chicago Lectures in Physics, University of Chicago Press, Chicago, IL, 1994 | MR

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