Abstract
The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras, since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We show then that, fixing any principal U(1)-bundle, there exists a suitable category of subbundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.
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References
Ashtekar A., Sen A.: On the role of space-time topology in quantum phenomena: superselection of charge and emergence of nontrivial vacua. J. Math. Phys. 21, 526 (1980)
Atiyah M.F.: Complex analytic connections in fibre bundles. Trans. Am. Math. Soc. 85, 181–207 (1957)
Baum H.: Eichfeldtheorie: Eine Einführung in die Differentialgeometrie auf Faserbündeln. Springer, Berlin (2009)
Benini, M., Dappiaggi, C., Schenkel, A.: Quantum field theory on affine bundles. Annales Henri Poincare 15, 171 (2014). arXiv:1210.3457 [math-ph]
Benini, M., Dappiaggi, C., Schenkel, A.: Quantized Abelian principal connections on Lorentzian manifolds. Commun. Math. Phys. 330, 123 (2014). arXiv:1303.2515 [math-ph]
Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle: a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31 (2003) [math-ph/0112041]
Bär, C.: Green-hyperbolic operators on globally hyperbolic spacetimes. arXiv:1310.0738 [math-ph]
Bär, C., Ginoux, N.: CCR- versus CAR-quantization on curved spacetimes. In: Finster F., Müller O., Nardmann M., Tolksdorf J., Zeidler E. (eds.) Quantum field theory and gravity, pp. 183–206 (2012)
Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorenzian manifolds and quantization. Zürich: European Mathematical Society (2007). arXiv:0806.1036 [math.DG]
Binz E., Honegger R., Rieckers A.: Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space. J. Math. Phys. 45, 2885 (2004)
Bratteli O., Robinson D.W.: Operator algebras and quantum statistical mechanics. Volume 2 of Equilibrium states. Models in quantum statistical mechanics. Springer, Berlin (1996)
Dimock J.: Quantized electromagnetic field on a manifold. Rev. Math. Phys. 4, 223 (1992)
Dappiaggi, C., Lang, B.: Quantization of Maxwell’s equations on curved backgrounds and general local covariance. Lett. Math. Phys.101, 265 (2012). arXiv:1104.1374 [gr-qc]
Dappiaggi, C., Siemssen, D.: Hadamard states for the vector potential on asymptotically flat spacetimes. Rev. Math. Phys. 25, 1350002 (2013). arXiv:1106.5575 [gr-qc]
Edwards R.E.: Functional analysis. Theory and applications. Holt, Rinehart and Winston, New York (1965)
Fewster, C.J., Hunt, D.S.: Quantization of linearized gravity in cosmological vacuum spacetimes. Rev. Math. Phys. 25, 1330003 (2013). arXiv:1203.0261 [math-ph]
Finster, F., Strohmaier, A.: Gupta–Bleuler quantization of the Maxwell field in globally hyperbolic space-times. arXiv:1307.1632 [math-ph]
Haag R., Kastler D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)
Hack, T.-P., Schenkel, A.: Linear bosonic and fermionic quantum gauge theories on curved spacetimes. Gen. Rel. Gravit. 45, 877 (2013). arXiv:1205.3484 [math-ph]
Kobayashi S., Nomizu K.: Foundations of differential geometry Vol. I. Wiley Classics Library, A Wiley-Interscience Publication, Wiley, New York (1996)
Leyland, P., Roberts, J., Testard, D.: Duality for quantum free fields. CPT-78/P-1016
Manuceau J., Sirugue M., Testard D., Verbeure A.: The smallest C*-algebra for canonical commutations relations. Commun. Math. Phys. 32, 231 (1973)
Pfenning, M.J.: Quantization of the Maxwell field in curved spacetimes of arbitrary dimension. Class. Quant. Gravity 26, 135017 (2009). arXiv:0902.4887 [math-ph]
Sanders, K.: A note on spacelike and timelike compactness. Class. Quant. Gravity 30, 115014 (2013). arXiv:1211.2469 [math-ph]
Sanders, K., Dappiaggi, C., Hack, T.-P.: Electromagnetism, local covariance, the Aharonov–Bohm effect and Gauss’ law. Commun. Math. Phys. 328, 625 (2014). arXiv:1211.6420 [math-ph]
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Communicated by Y. Kawahigashi
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Benini, M., Dappiaggi, C., Hack, TP. et al. A C*-Algebra for Quantized Principal U(1)-Connections on Globally Hyperbolic Lorentzian Manifolds. Commun. Math. Phys. 332, 477–504 (2014). https://doi.org/10.1007/s00220-014-2100-3
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DOI: https://doi.org/10.1007/s00220-014-2100-3