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 SIGMA, 2012, Volume 8, 050, 31 pages (Mi sigma727)

Holomorphic quantization of linear field theory in the general boundary formulation

Robert Oeckl

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, C.P. 58190, Morelia, Michoacán, Mexico

Abstract: 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.

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

DOI: https://doi.org/10.3842/SIGMA.2012.050

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ArXiv: 1009.5615
MSC: 57R56; 81S10; 81T05; 81T20
Received: April 27, 2012; in final form August 3, 2012; Published online August 9, 2012
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Citation: Robert Oeckl, “Holomorphic quantization of linear field theory in the general boundary formulation”, SIGMA, 8 (2012), 050, 31 pp.

Citation in format AMSBIB
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\by Robert Oeckl
\paper Holomorphic quantization of linear field theory in the general boundary formulation
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\yr 2012
\vol 8
\papernumber 050
\totalpages 31
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\crossref{https://doi.org/10.3842/SIGMA.2012.050}
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This publication is cited in the following articles:
1. Daniele Colosi, Dennis Rätzel, “The Unruh Effect in General Boundary Quantum Field Theory”, SIGMA, 9 (2013), 019, 22 pp.
2. Robert Oeckl, “Free Fermi and Bose Fields in TQFT and GBF”, SIGMA, 9 (2013), 028, 46 pp.
3. Perez A., “The Spin-Foam Approach to Quantum Gravity”, Living Rev. Relativ., 16 (2013), 3
4. Banisch R. Hellmann F. Raetzel D., “The Unruh-Dewitt Detector and the Vacuum in the General Boundary Formalism”, Class. Quantum Gravity, 30:23 (2013), 235026
5. Colosi D. Raetzel D., “Quantum Field Theory on Timelike Hypersurfaces in Rindler Space”, Phys. Rev. D, 87:12 (2013), 125001
6. Dohse M., “Classical Klein-Gordon Solutions, Symplectic Structures, and Isometry Actions on Ads Spacetimes”, J. Geom. Phys., 70 (2013), 130–156
7. Arjang M., Zapata J.A., “Multisymplectic Effective General Boundary Field Theory”, Class. Quantum Gravity, 31:9 (2014)
8. Hoehn Ph.A., “Quantization of Systems With Temporally Varying Discretization. i. Evolving Hilbert Spaces”, J. Math. Phys., 55:8 (2014), 083508
9. Oeckl R., “Schrodinger-Feynman Quantization and Composition of Observables in General Boundary Quantum Field Theory”, Adv. Theor. Math. Phys., 19:2 (2015), 451–506
10. Homero G. Díaz-Maríin, “General Boundary Formulation for $n$-Dimensional Classical Abelian Theory with Corners”, SIGMA, 11 (2015), 048, 35 pp.
11. Colosi D., “in-Out Propagator in de Sitter Space From General Boundary Quantum Field Theory”, Phys. Lett. B, 748 (2015), 70–73
12. Dohse M., Oeckl R., “Complex Structures For An S-Matrix of Klein-Gordon Theory on Ads Spacetimes”, Class. Quantum Gravity, 32:10 (2015), 105007
13. Colosi D., “An Introduction To the General Boundary Formulation of Quantum Field Theory”, 7th international workshop dice2014 spacetime - matter - quantum mechanics, Journal of Physics Conference Series, 626, ed. Diosi L. Elze H. Fronzoni L. Halliwell J. Kiefer C. Prati E. Vitiello G., IOP Publishing Ltd, 2015, 012031
14. D. Colosi, M. Dohse, “The $S$-matrix in Schrödinger representation for curved spacetimes in general boundary quantum field theory”, J. Geom. Phys., 114 (2017), 65–84
15. H. G. Diaz-Marin, “Dirichlet to Neumann operator for abelian Yang–Mills gauge fields”, Int. J. Geom. Methods Mod. Phys., 14:11 (2017), 1750153
16. D. Colosi, M. Dohse, “Complex structures and quantum representations for scalar QFT in curved spacetimes”, Int. J. Theor. Phys., 56:11 (2017), 3359–3386
17. Homero G. Díaz-Marín, Robert Oeckl, “Quantum Abelian Yang–Mills Theory on Riemannian Manifolds with Boundary”, SIGMA, 14 (2018), 105, 31 pp.
18. Oeckl R., “Coherent States in Fermionic Fock-Krein Spaces and Their Amplitudes”, Coherent States and Their Applications: a Contemporary Panorama, Springer Proceedings in Physics, 205, ed. Antoine J. Bagarello F. Gazeau J., Springer-Verlag Berlin, 2018, 243–263
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