Structure of correlated worldline theories of quantum gravity

A. O. Barvinsky, D. Carney, and P. C. E. Stamp

Phys. Rev. D 98, 084052 – Published 29 October 2018

Abstract

We consider the general form of correlated worldline (CWL) theories of quantum gravity. We show that one can have two different kinds of CWL theory, in which the generating functional is written as either a sum or a product over multiple copies of the coupled matter and gravitational fields. In both versions, the paths in a functional formulation are correlated via gravity itself, causing a breakdown of the superposition principle; however, the product form survives consistency tests not satisfied by the summed form. To better understand the structure of these two theories, we show how to perform diagrammatic expansions in the gravitational coupling for each version of CWL theory, using particle propagation and scalar fields as examples. We explicitly calculate contributions to two-point and four-point functions, again for each version of the theory, up to second order in the gravitational coupling.

Received 21 June 2018

DOI:https://doi.org/10.1103/PhysRevD.98.084052

Physics Subject Headings (PhySH)

Alternative gravity theories General relativity Open quantum systems & decoherence Quantum gravity Quantum-to-classical transition

Gravitation, Cosmology & AstrophysicsQuantum Information, Science & TechnologyGeneral PhysicsNonlinear Dynamics

Authors & Affiliations

A. O. Barvinsky¹, D. Carney^2,3,4, and P. C. E. Stamp^2,5,6

¹Theory Department, Lebedev Physics Institute, Leninsky Prospect 53, Moscow 119991, Russia
²Pacific Institute of Theoretical Physics, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia V6T 1Z1, Canada
³Joint Center for Quantum Information and Computer Science, 3100 Atlantic Building, University of Maryland, College Park, Maryland 20742, USA
⁴Joint Quantum Institute, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899, USA
⁵Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia V6T 1Z1, Canada
⁶School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand

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Vol. 98, Iss. 8 — 15 October 2018

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Images

Figure 1
The contour involved in the ring diagrams for the generating functional in this paper. In (a), we show in bold (black) the contour for a matter particle or field in proper time, extending from $t = - \infty$ up to $t = \infty$ and back again; it is then closed at $t = - \infty$ around the “temperature cylinder” of circumference $2 π / k T$ . In (b), we show how such a propagator is represented in this paper.
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Figure 2
Feynman-Keldysh diagrams contributing to the generating functional $Q [J]$ for summed CWL theory, when we set the external current $J (x) = 0$ ; compare Eqs. (10) and (11). In (a), we see a term contributing to $Q_{1} [J]$ ; in (b), we see a term contributing to $Q_{2} [J]$ ; and in (c), we see a term contributing to $Q_{3} [J]$ . The scalar field propagator is shown in black, and the graviton propagator is in (hatched) red.
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Figure 3
Typical diagrams contributing to a correlator in summed CWL theory [cf. Eq. (19)]. Here, we show graphs for $G_{2}^{+ -} (x, x^{'})$ , a two-point correlator in which one current insertion is made on the forward ( $+$ ) path and one is made on the backward ( $-$ ) path. Contributions come from all of the ${\tilde{Q}}_{n} [J]$ in $Q [J]$ . In (a), we see a contribution from $Q_{1} [J]$ ; in (b), we see a contribution from $Q_{2} [J]$ . The scalar field propagator is shown in black, the graviton propagator is shown in (hatched) red, and current insertions are shown in blue dotted lines.
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Figure 4
Contributions to the particle propagator $K (x, x^{'})$ for the summed version of CWL theory. In (a), we show the contribution $Δ K_{1} (x, x^{'})$ in Eq. (31); in (b), we show the contribution $Δ K_{2}^{A} (x, x^{'})$ of Eq. (39); and in (c), we show the contribution $Δ K_{2}^{B} (x, x^{'})$ of Eq. (42). The particle paths are shown in black, and the graviton paths are shown in hatched red.
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Figure 5
Leading-order graviton terms contributing to the two-point correlator $G_{a, a^{'}}^{(n)} (x, x^{'})$ for product CWL theory—these corrections are calculated in the text. The four second-order (in graviton insertions) graphs are (a) a disconnected graph, (b) a tadpole graph, (c) a one-loop self-energy graph, and (d) a seagull graph. For $a \neq a^{'}$ , all the diagrams vanish. For given $a = a^{'}$ , there are $n$ possible choices for the loop copy number $i$ in case (b) but only one choice in case (c). The scalar field propagator is shown in black, and the graviton propagator is shown in hatched red.
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Figure 6
Lowest-order terms contributing to the four-point correlator $G_{a_{1}, a_{2}, a_{3}, a_{4}}^{(n)} (x_{1}, x_{2}, x_{3}, x_{4})$ for product CWL theory—these scattering corrections are calculated in the text. The three second-order (in graviton insertions) graphs are usually called direct, exchange, and crossed graphs. The scalar field propagator is shown in black, and the graviton propagator is shown in hatched red.
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