Abstract
Dynamical locality is a condition on a locally covariant physical theory, asserting that kinematic and dynamical notions of local physics agree. This condition was introduced in arXiv:1106.4785, where it was shown to be closely related to the question of what it means for a theory to describe the same physics on different spacetimes. In this paper, we consider in detail the example of the free minimally coupled Klein–Gordon field, both as a classical and quantum theory (using both the Weyl algebra and a smeared field approach). It is shown that the massive theory obeys dynamical locality, both classically and in quantum field theory, in all spacetime dimensions n ≥ 2 and allowing for spacetimes with finitely many connected components. In contrast, the massless theory is shown to violate dynamical locality in any spacetime dimension, in both classical and quantum theory, owing to a rigid gauge symmetry. Taking this into account (equivalently, working with the massless current) dynamical locality is restored in all dimensions n ≥ 2 on connected spacetimes, and in all dimensions n ≥ 3 if disconnected spacetimes are permitted. The results on the quantized theories are obtained using general results giving conditions under which dynamically local classical symplectic theories have dynamically local quantizations.
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Abraham R., Marsden J.E., Ratiu T.: Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences, vol. 75, 2nd edn. Springer, New York (1988)
Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and concrete categories: the joy of cats. Reprints in Theory and Applications of Categories, pp. 1–507. Wiley, New York (2006) (reprint of the 1990 original)
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 (1992)
Bär C., Ginoux N., Pfäffle F.: Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society (EMS), Zürich (2007)
Beem J.K., Ehrlich P.E., Easley K.L.: Global Lorentzian Geometry. Monographs and Textbooks in Pure and Applied Mathematics, vol. 202, 2nd edn. Marcel Dekker, New York (1996)
Binz E., Honegger R., Rieckers A.: Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space. J. Math. Phys. 45, 2885–2907 (2004)
Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000)
Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle: a new paradigm for local quantum physics. Commun. Math. Phys. 237, 31–68 (2003)
Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–785 (2002)
Brunetti R., Ruzzi G.: Superselection sectors and general covariance. I. Commun. Math. Phys. 270, 69–108 (2007)
Brunetti R., Ruzzi G.: Quantum charges and spacetime topology: the emergence of new superselection sectors. Commun. Math. Phys. 287, 523–563 (2009)
Buchholz D., Verch R.: Scaling algebras and renormalization group in algebraic quantum field theory. II. Instructive examples. Rev. Math. Phys. 10, 775–800 (1998)
Choquet-Bruhat Y.: General Relativity and the Einstein Equations. Oxford Mathematical Monographs. Oxford University Press, Oxford (2009)
Dappiaggi C.: Remarks on the Reeh-Schlieder property for higher spin free fields on curved spacetimes. Rev. Math. Phys. 23, 1035–1062 (2011)
Dappiaggi C., Fredenhagen K., Pinamonti N.: Stable cosmological models driven by a free quantum scalar field. Phys. Rev. D 77, 104015 (2008)
Dappiaggi, C., Lang, B.: Quantization of Maxwell’s equations on curved backgrounds and general local covariance (2011). arXiv:1104.1374
Degner A., Verch R.: Cosmological particle creation in states of low energy. J. Math. Phys. 51, 022302 (2010)
Dikranjan D., Tholen W.: Categorical Structure of Closure Operators. Mathematics and Its Applications, vol. 346. Kluwer, Dordrecht (1995)
Dimock J.: Algebras of local observables on a manifold. Commun. Math. Phys. 77, 219–228 (1980)
Dimock J.: Quantized electromagnetic field on a manifold. Rev. Math. Phys. 4, 223–233 (1992)
Ferguson, M.: Dynamical locality of the nonminimally coupled scalar field and enlarged algebra of Wick polynomials (2012). arXiv:1203.2151
Fewster, C.J.: Endomorphisms and automorphisms of locally covariant quantum field theories. ArXiv:1201.3295
Fewster C.J.: Quantum energy inequalities and local covariance. II. Categorical formulation. Gen. Relativ. Gravit. 39, 1855–1890 (2007)
Fewster, C.J.: On the notion of ‘the same physics in all spacetimes’. In: Finster F., Müller O., Nardmann M., Tolksdorf J., Zeidler E. (eds.) Quantum Field Theory and Gravity. Conceptual and Mathematical Advances in the Search for a Unified Framework. Birkhäuser, Basel (2012). ArXiv:1105.6202
Fewster C.J., Pfenning M.J.: A quantum weak energy inequality for spin-one fields in curved spacetime. J. Math. Phys. 44, 4480–4513 (2003)
Fewster C.J., Pfenning M.J.: Quantum energy inequalities and local covariance. I: globally hyperbolic spacetimes. J. Math. Phys. 47, 082303 (2006)
Fewster, C.J., Verch, R.: Dynamical locality and covariance: what makes a physical theory the same in all spacetimes? Annales H. Poincaré (2012) (to appear). ArXiv:1106.4785
Greub W.H.: Linear Algebra, 3rd edn. Die Grundlehren der Mathematischen Wissenschaften, Band 97. Springer, New York (1967)
Greub W.H.: Multilinear Algebra. Die Grundlehren der Mathematischen Wissenschaften, Band 136. Springer, New York (1967)
Haag R.: Local Quantum Physics: Fields, Particles, Algebras. Springer, Berlin (1992)
Hollands S.: Renormalized quantum Yang-Mills fields in curved spacetime. Rev. Math. Phys. 20(9), 1033–1172 (2008)
Hollands S., Wald R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289–326 (2001)
Hollands S., Wald R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345 (2002)
Landau L.J.: A note on extended locality. Commun. Math. Phys. 13, 246–253 (1969)
Mac Lane S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1998)
Manuceau J., Verbeure A.: Quasi-free states of the C.C.R.-algebra and Bogoliubov transformations. Commun. Math. Phys. 9, 293–302 (1968)
O’Neill B.: Semi-Riemannian Geometry. Academic Press, New York (1983)
Pfenning M.J.: Quantization of the Maxwell field in curved spacetimes of arbitrary dimension. Class. Quantum Gravity 26, 135017 (2009)
Reed, M., Simon, B.: Methods of modern mathematical physics. I, 2nd edn. Functional analysis. Academic Press (Harcourt Brace Jovanovich Publishers), New York (1980)
Sanders K.: On the Reeh-Schlieder property in curved spacetime. Commun. Math. Phys. 288, 271–285 (2009)
Schoch A.: On the simplicity of Haag fields. Int. J. Theor. Phys. 1, 107–113 (1968)
Streater R.F.: Spontaneous breakdown of symmetry in axiomatic theory. Proc. R. Soc. Ser. A 287, 510–518 (1965)
Verch R.: A spin-statistics theorem for quantum fields on curved spacetime manifolds in a generally covariant framework. Commun. Math. Phys. 223, 261–288 (2001)
Verch, R.: Local covariance, renormalization ambiguity, and local thermal equilibrium in cosmology. In: Finster F., Müller, O., Nardmann, M., Tolksdorf J., Zeidler, E. (eds.) Quantum Field Theory and Gravity. Conceptual and Mathematical Advances in the Search for a Unified Framework. Birkhäuser, Basel (2012). ArXiv:1105.6249
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Communicated by Klaus Fredenhagen.
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Fewster, C.J., Verch, R. Dynamical Locality of the Free Scalar Field. Ann. Henri Poincaré 13, 1675–1709 (2012). https://doi.org/10.1007/s00023-012-0166-z
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DOI: https://doi.org/10.1007/s00023-012-0166-z