zbMATH — the first resource for mathematics

The \(S\)-matrix in Schrödinger representation for curved spacetimes in general boundary quantum field theory. (English) Zbl 1360.83022
Summary: We use the General Boundary Formulation (GBF) of Quantum Field Theory to compute the \(S\)-matrix for a general interacting scalar field in a wide class of curved spacetimes. As a by-product we obtain the general expression of the Feynman propagator for the scalar field, defined in the following three types of spacetime regions. First, there are the familiar interval regions (e.g. a time interval times all of space). Second, we consider the rod hypercylinder regions (all of time times a solid ball in space). Third, the tube hypercylinders (all of time times a solid shell in space) are related to interval regions, and result from removing a smaller rod from a concentric larger one. Using the Schrödinger representation for the quantum states combined with Feynman’s path integral quantization, we obtain the \(S\)-matrix as the asymptotic limit of the GBF amplitude associated with finite interval, and rod regions. For interval regions, whose boundary consists of two Cauchy surfaces, the asymptotic GBF-amplitude becomes the standard \(S\)-matrix. Our work generalizes previous results (obtained in Minkowski, Rindler, de Sitter, and Anti de Sitter spacetimes) to a wide class of curved spacetimes.

83C47 Methods of quantum field theory in general relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
81T20 Quantum field theory on curved space or space-time backgrounds
81S40 Path integrals in quantum mechanics
Full Text: DOI arXiv
[1] Oeckl, R., A “general boundary” formulation for quantum mechanics and quantum gravity, Phys. Lett. B, 575, 318-324, (2003) · Zbl 1094.81551
[2] Oeckl, R., General boundary quantum field theory: foundations and probability interpretation, Adv. Theor. Math. Phys., 12, 319-352, (2008) · Zbl 1210.81039
[3] Oeckl, R., General boundary quantum field theory: timelike hypersurfaces in Klein-Gordon theory, Phys. Rev. D, 73, (2006)
[4] Oeckl, R., Probabilites in the general boundary formulation, J. Phys.: Conf. Ser., 67, (2007)
[5] Colosi, D.; Oeckl, R., S-matrix at spatial infinity, Phys. Lett. B, 665, 310-313, (2008) · Zbl 1328.81218
[6] Colosi, D.; Oeckl, R., Spatially asymptotic S-matrix from general boundary formulation, Phys. Rev. D, 78, (2008)
[7] Colosi, D.; Oeckl, R., States and amplitudes for finite regions in a two-dimensional Euclidean quantum field theory, J. Geom. Phys., 59, 764-780, (2009) · Zbl 1164.81012
[8] Colosi, D.; Rätzel, D., Quantum field theory on timelike hypersurfaces in Rindler space, Phys. Rev. D, 87, (2013)
[9] D. Colosi, S-matrix in de Sitter spacetime from general boundary quantum field theory, 2009, arXiv:0910.2756.
[10] D. Colosi, General boundary quantum field theory in de Sitter spacetime, 2010, arXiv:1010.1209.
[11] Bernal, A.; Sánchez, M., Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Comm. Math. Phys., 257-1, 43, (2005) · Zbl 1081.53059
[12] Colosi, D.; Oeckl, R., On unitary evolution in quantum field theory in curved spacetime, Open Nucl. Part. Phys. J., 4, 13-20, (2011)
[13] Hatfield, B., Quantum field theory of point particles and strings, (1991), Perseus Books
[14] Woodhouse, N., Geometric quantization, (1991), Oxford University Press · Zbl 0466.58022
[15] Dohse, M., Classical Klein-Gordon solutions, symplectic structures and isometry actions on AdS spacetimes, J. Geom. Phys., 70, 130-156, (2013) · Zbl 1283.83005
[16] Corichi, A.; Cortez, J.; Quevedo, H., Schrödinger and Fock representation for a field theory on curved spacetime, Ann. Physics, 313, 446-478, (2004) · Zbl 1074.81054
[17] Jackiw, R., Analysis on infinite-dimensional manifolds - Schrödinger representation for quantized fields (in his book: diverse topics in theoretical and mathematical physics), (1995), World Scientific
[18] Oeckl, R., Holomorphic quantization of linear field theory in the general boundary formulation, SIGMA, 8, 50-81, (2012)
[19] D. Colosi, On the structure of the vacuum state in general boundary quantum field theory, 2009, arXiv:0903.2476.
[20] Oeckl, R., Free Fermi and Bose fields in TQFT and GBF, SIGMA, 9, 28-74, (2013) · Zbl 1283.81114
[21] Oeckl, R., Two-dimensional quantum Yang-Mills theory with corners, J. Phys. A, 41, (2008) · Zbl 1138.81036
[22] Birrell, N.; Davies, P., Quantum fields in curved space, (1982), Cambridge University Press · Zbl 0476.53017
[23] Casimir, H., On the attraction between two perfectly conducting plates, Proc. K. Ned. Akad. Wet., 51, 793, (1948) · Zbl 0031.19005
[24] Bordag, M.; Klimchitskaya, G.; Mohideen, U.; Mostepanenko, V., Advances in the Casimir effect, (2009), Oxford University Press
[25] Milton, K., The Casimir effect: physical manifestation of zero-point energy, (2002), World Scientific
[26] (Dalvit, D.; Milonni, P.; Roberts, D.; da Rosa, F., Casimir Physics, Lecture Notes in Physics, vol. 834, (2011), Springer) · Zbl 1220.81006
[27] Maldacena, J., The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys., 2, 231, (1998) · Zbl 0914.53047
[28] Balasubramanian, V.; Giddings, S.; Lawrence, A., What do CFTs tell us about anti-de Sitter spacetimes?, J. High Energy Phys., 03, 001, (1999) · Zbl 0965.81098
[29] Giddings, S., The boundary S-matrix and the AdS to CFT dictionary, Phys. Rev. Lett., 83, 2707-2710, (1999) · Zbl 0958.81151
[30] ’t Hooft, G., The scattering matrix approach for the quantum black hole: an overview, Internat. J. Modern Phys. A, 11, 4623, (1996) · Zbl 1044.81683
[31] Oeckl, R., The Schrödinger representation and its relation to the holomorphic representation in linear and affine field theory, J. Math. Phys., 53, (2012) · Zbl 1276.81115
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.