Abstract
This chapter further develops the position in which spacetime is primary which we set up in Chaps. 6, 10, 11, 12 and 26. We now consider this at the quantum level using path integral formulations for gauge theory and for general relativity. The general-relativistic case involves various further subtleties; most facets of the Problem of Time have counterparts here; the inner product problem, however, has been traded for a measure problem. By facet interference, this is furthermore a formidable diffeomorphism-invariant measure problem. This book has so far addressed the Frozen Formalism Problem facet of the Problem of Time by identifying it as stemming from Background Independence’s Temporal Relationalism aspect, which we implemented by reformulating the principles of dynamics and canonical quantum theory. We now extend this by additionally forging a Temporal Relationalism implementing version of the path integral formulation. We finally outline approaches to physics which make concurrent use of both canonical and path-integral theory.
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Notes
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- 2.
This Measure Problem is, additionally, a technical trade-off [193], in the sense of having it instead of the Canonical Approach’s Operator Ordering Problem.
- 3.
Or almost-discrete, e.g. involving an ancillary continuum sample space in the Causal Sets Approach or eventually taking a continuum limit in the Causal Dynamical Triangulation Approach.
References
Anderson, E.: The problem of time and quantum cosmology in the relational particle mechanics arena. arXiv:1111.1472
Barrett, J.W., Crane, L.: Relativistic spin networks and quantum gravity. J. Math. Phys. 39, 3296 (1998)
Bojowald, M., Hoehn, P.A., Tsobanjan, A.: An effective approach to the problem of time. Class. Quantum Gravity 28, 035006 (2011). arXiv:1009.5953
Bojowald, M., Hoehn, P.A., Tsobanjan, A.: Effective approach to the problem of time: general features and examples. Phys. Rev. D 83, 125023 (2011). arXiv:1011.3040
Carlip, S.: Quantum Gravity in \(2 + 1\) Dimensions. Cambridge University Press, Cambridge (1998)
DeWitt, B.S.: Quantum theory of gravity. II. The manifestly covariant theory. Phys. Rev. 160, 1195 (1967)
DeWitt, B.S.: Quantum theory of gravity. III. Applications of the covariant theory. Phys. Rev. 160, 1239 (1967)
Dittrich, B., Hoehn, P.A.: Constraint analysis for variational discrete systems. J. Math. Phys. 54, 093505 (2013). arXiv:1303.4294
Fadde’ev, L.D.: The Feynman integral for singular Lagrangians. Theor. Math. Phys. 1, 1 (1969)
Fredenhagen, K., Rejzner, K.: Batalin-Vilkovisky formalism in the functional approach to classical field theory. Commun. Math. Phys. 314, 93 (2012)
Gibbons, G.W., Hawking, S.W., Perry, M.: Nucl. Phys. B 138, 141 (1978)
Halliwell, J.J., Louko, J.: Steepest descent contours in the path integral approach to quantum cosmology. 3. A general method with applications to minisuperspace models. Phys. Rev. D 42, 3997 (1990)
Halvorson, H., Mueger, M.: Algebraic quantum field theory. math-ph/0602036
Hartle, J.B., Schleich, K.: The conformal rotation in linearized gravity. In: Batalin, I., Isham, C.J., Vilkovilsky, G.A. (eds.) Quantum Field Theory and Quantum Statistics, vol. 2. Hilger, Bristol (1987)
Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1992)
Hoehn, P.A., Kubalova, E., Tsobanjan, A.: Effective relational dynamics of a nonintegrable cosmological model. Phys. Rev. D 86, 065014 (2012). arXiv:1111.5193
Isham, C.J.: Quantum gravity—an overview. In: Isham, C.J., Penrose, R., Sciama, D.W. (eds.) Quantum Gravity 2: A Second Oxford Symposium. Clarendon, Oxford (1981)
Isham, C.J.: Aspects of Quantum Gravity. Lectures Given at Conference: C85–07-28.1 (Scottish Summer School 1985:0001), available on KEK archive
Kiefer, C.: Quantum Gravity. Clarendon, Oxford (2004)
Loll, R.: Discrete approaches to quantum gravity in four dimensions. Living Rev. Relativ. 1, 13 (1998). gr-qc/9805049
Muga, G., Sala Mayato, R., Egusquiza, I. (eds.): Time in Quantum Mechanics, vol. 1. Springer, Berlin (2008)
Muga, G., Sala Mayato, R., Egusquiza, I. (eds.): Time in Quantum Mechanics, vol. 2. Springer, Berlin (2010)
Oriti, D.: Disappearance and emergence of space and time in quantum gravity. arXiv:1302.2849
Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Perseus Books, Reading (1995)
Sundermeyer, K.: Constrained Dynamics. Springer, Berlin (1982)
Weinberg, S.: The Quantum Theory of Fields. Vol. II. Modern Applications. Cambridge University Press, Cambridge (1995)
Williams, R.M., Tuckey, P.A.: Regge calculus: a bibliography and brief review. Class. Quantum Gravity 9, 1409 (1992)
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Anderson, E. (2017). Spacetime Primary Approaches: Path Integrals. In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_52
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