Causality and momentum conservation from relative locality

Giovanni Amelino-Camelia, Stefano Bianco, Francesco Brighenti, and Riccardo Junior Buonocore

Phys. Rev. D 91, 084045 – Published 21 April 2015

Abstract

Theories involving curved momentum space, which recently became a topic of interest in the quantum-gravity literature, can, in general, violate many apparently robust aspects of our current description of the laws of physics, including relativistic invariance, locality, causality, and global momentum conservation. Here, we explore some aspects of the pathologies arising in generic theories involving curved momentum space for what concerns causality and momentum conservation. However, we also report results suggesting that when momentum space is maximally symmetric, and the theory is formulated relativistically, most notably including translational invariance with the associated relativity of spacetime locality, momentum is globally conserved and there is no violation of causality.

Received 5 June 2014

DOI:https://doi.org/10.1103/PhysRevD.91.084045

Authors & Affiliations

Giovanni Amelino-Camelia^1,2, Stefano Bianco^1,2, Francesco Brighenti^3,4, and Riccardo Junior Buonocore^1,5

¹Dipartimento di Fisica, Università “La Sapienza” P.le A. Moro 2, 00185 Roma, Italy
²Sez. Roma1 INFN, P.le A. Moro 2, 00185 Roma, Italy
³Dipartimento di Fisica e Astronomia dell’Università di Bologna, Via Irnerio 46, 40126 Bologna, Italy
⁴Sez. Bologna INFN, Via Irnerio 46, 40126 Bologna, Italy
⁵Department of Mathematics, King’s College London, The Strand, London WC2R 2LS, United Kingdom

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Vol. 91, Iss. 8 — 15 April 2015

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Images

Figure 1
We here show schematically a three-valent event marked by a $K^{(0)}$ that symbolizes a boundary term conventionally located at value $s_{0}$ of the affine parameter $s$ . The boundary term enforces (deformed) momentum conservation at the event.
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Figure 2
We give here a schematic description of a process composed of two causally connected events. The event at Alice could be the absorption of a photon by an atom and the event at Bob could be another absorption of a photon by the same atom. The implications of relative locality are visualized by describing Alice’s perspective on the process in the left panel and the perspective of Bob (distant from Alice and in relative rest with respect to Alice) in the right panel. According to Alice’s description, the first absorption event (which occurs in Alice’s origin of the reference frame) is local, but Alice’s inferences about the second absorption event (which occurs at Bob, far away from Alice) would characterize it as nonlocal. Bob has a relativistically specular viewpoint: Bob’s description of the second absorption event (which occurs in Bob’s origin of the reference frame) is local but Bob’s inferences about the first absorption event (which occurs at Alice, far away from Bob) would characterize it as nonlocal. This is how a pair of causally connected distant local events gets described in the presence of relative locality.
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Figure 3
We show schematically here two causal links: a pair of causally connected events is shown in red and another pair of causally connected events is shown in blue, but there is no causal connection [in spite of the coincidence of the events $K^{(0)}$ and $K^{(1)}$ ] between events where blue lines cross and events where red lines cross. We analyze this situation with the simplifying assumption that some of the particles involved (those described by dashed lines) have energies small enough that the Planck-scale effects of interest here can be safely neglected.
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Figure 4
The two causally connected pairs of events considered in this section can lead to a striking picture of distant inferences (because of relative locality) when $p_{1}^{'} ≃ p_{1}^{''}$ . In that case, the particle with coordinates ${z^{'}}^{μ}$ , which actually interacts at $K^{(2)}$ , in the coordinatization by distant observer Alice (top panel) appears to come out of the interaction $K^{(3)}$ . In turn, as we show in the bottom panel of the figure, Bob’s inferences about the events $K^{(0)}$ and $K^{(1)}$ (which are distant from Bob) are affected by peculiar relative-locality features.
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Figure 5
We show schematically here a pair of events causally connected by the exchange of two particles arranged in such a way that one would have a causal loop. Such causal loops are allowed, if one assumes a curvature of momentum space without enforcing translational invariance and the associated relativity of spacetime locality. Notice here that the time flow of the figure goes essentially from bottom to top: the lines $q$ , $y$ and $k$ , $z$ are incoming, while $q^{'}$ , $y^{'}$ and $k^{'}$ , $z^{'}$ are outgoing. For the other figures of this manuscript, we found it to be more convenient (and to obtain a more pleasant visualization) to show the time flow going from left to right, but for this figure we gave priority to having the same visualization adopted for the corresponding figure of Ref. [26] (Fig. 2 of that manuscript).
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Figure 6
We show here schematically two causally connected events that form a Möbius diagram. The laws of conservation at the two vertices are set up in such a way that the particle going out from the first vertex has its momentum appearing on the right-hand side of the composition law and its momentum also appears on the left-hand side of the composition of the momenta at the second vertex.
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