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Deformed Relativistic Symmetry Principles

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Modified and Quantum Gravity

Part of the book series: Lecture Notes in Physics ((LNP,volume 1017))

Abstract

We review the main features of models where relativistic symmetries are deformed at the Planck scale. We cover the motivations, links to other quantum gravity approaches, describe in some detail the most studied theoretical frameworks, including Hopf algebras, relative locality, and other scenarios with deformed momentum space geometry, discuss possible phenomenological consequences, and point out current open questions.

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Notes

  1. 1.

    This formula is to be understood as indicating the lowest-order correction to the standard special-relativistic expression in powers of the particle’s energy over the Planck energy, where the order is given by the positive integer n and \(\eta \) is a dimensionless parameter indicating the strength of the effect at the Planck scale. In general, formulas considering all-order corrections go beyond this simple power-law expression, see e.g. Sect. 2.3.3.

  2. 2.

    While in special relativity c is the maximum allowed speed, in DSR it is to be understood as the speed of low-energy massless particles. And the Planck energy is a relativistic invariant, but is not necessarily the maximum allowed energy. It might be the case in some specific models, but it is not true in general.

  3. 3.

    From now on we set \(c=1\).

  4. 4.

    One might also consider deformations of the other relativistic symmetries, but here we will only focus on boosts for simplicity.

  5. 5.

    In the following we will sometimes use a simplified notation omitting the explicit indication of the four-vector index \(\mu \).

  6. 6.

    By definition, elements of the dual of a Hilbert space \(\mathcal {H}\) are continuous linear maps from \(\mathcal {H}\) to \(\mathbb {C}\). Given the inner product \(\langle \mathbf {k}'|\mathbf {k}\rangle \) on \(\mathcal {H}\), it is evident that bra \(\langle \mathbf {k}| \) is an element of the dual space.

  7. 7.

    As we will explain in the following section, in some approaches to DSR based purely on the geometry of momentum space one assumes to be in a “semiclassical” regime of quantum gravity, such that the Planck constant \(\hbar \) and the Newton constant G vanish, but their ratio is fixed and finite. In this regime, one can build an energy scale \(E_P\) but not a length scale \(L_P\to 0\). In the context of Hopf algebra and non-commutative geometry, this is not the regime that is considered, since one needs a constant with dimensions of length to govern space-time noncommutativity as in (2.34).

  8. 8.

    We adopt a semiclassical approximation, so that symmetry generators act on the momentum space coordinates via Poisson brackets. The properties of the generators of the Hopf algebra are inherited by the Poisson brackets with the convention that, if \([G, f(P_{\mu })]= i h(P_{\mu })\), then \(\{G,f(p_{\mu })\} = h(p_{\mu })\), for any generator G of the Hopf algebra. The functions f, h, take as argument the translation generators \(P_{\mu }\) in the first case, and the momentum space coordinates \(p_{\mu }\) in the second one. This approximation is justified in the “semiclassical” limit we mentioned in the previous footnote and further described in Sect. 2.4.

  9. 9.

    A similar feature as the one we are discussing here for boosts exists for rotation transformations, see [44].

  10. 10.

    A more general expression applies when considering finite transformations [44]; however, here we only discuss the first order in \(\xi \).

  11. 11.

    Here we are using Poisson brackets instead of commutators because we are taking the semiclassical limit which turns a Hops algebra into a Poisson-Lie algebra.

  12. 12.

    This is a completely analogous construction to the one of general relativity where momentum space is the cotangent space of the space-time manifold at a point in spacetime.

  13. 13.

    See [89] for alternative, but physically equivalent, prescriptions.

  14. 14.

    This result was recently rederived using a line element in phase space for a multi-particle system in [91].

  15. 15.

    Note that in the previous section the squared distance was identified with the squared of the distance in momentum space, but any function of the Casimir will be also a Casimir.

  16. 16.

    Note that the metric \(g_{\mu \nu }\) is the inverse of \(g^{\mu \nu }\).

  17. 17.

    We have reabsorbed the coefficient \(c_1\) in the scale \(\kappa \).

  18. 18.

    This is the de Sitter metric written in the comoving coordinate system used in Refs. [44, 92].

  19. 19.

    Of course one might reach different conclusions concerning time shift when using different bases of the \(\kappa \)-Poincaré algebra. For example, using the the classical basis of \(\kappa \)-Poincaré, there could be an absence of time shifts for massless particles with different energies [119]. Within the relative locality framework this can be understood in terms of the non-invariance of physical predictions under momentum space diffeomorphisms [120]. Moreover, depending on the effective scheme used for studying this effect, different time delay formulas are obtained, and may not lead to a time delay [119, 121,122,123].

  20. 20.

    Remember that we set the low-energy particle velocity \(c=1\).

  21. 21.

    The result of [64] was obtaining starting from a deformation of the relativistic transformations in de Sitter spacetime, which is maximally symmetric, and then deducing the time shift in FRW via a slicing procedure, first devised in [68].

  22. 22.

    As we mentioned in the previous subsection, even propagation effects might allow to distinguish between the two frameworks, since there could be a different dependence on the redshift of the source.

  23. 23.

    We write all formulas up to the first order in \(\frac {\eta }{E_P}\). For ultra-relativistic electrons and positrons one can consider \(m_e \frac {\eta }{E_P}\simeq 0\).

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Authors and Affiliations

  1. Dipartimento di Fisica “Ettore Pancini”, Università di Napoli Federico II, Naples, Italy

    Michele Arzano & Giulia Gubitosi

  2. INFN, Sezione di Napoli, Naples, Italy

    Michele Arzano & Giulia Gubitosi

  3. Departamento de Matemáticas y Computación, Universidad de Burgos, Burgos, Spain

    José Javier Relancio

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  1. Michele Arzano
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  2. Giulia Gubitosi
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  3. José Javier Relancio
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Correspondence to Giulia Gubitosi .

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  1. ZARM, University of Bremen, Bremen, Germany

    Christian Pfeifer

  2. ZARM, University of Bremen, Bremen, Germany

    Claus Lämmerzahl

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Arzano, M., Gubitosi, G., Relancio, J.J. (2023). Deformed Relativistic Symmetry Principles. In: Pfeifer, C., Lämmerzahl, C. (eds) Modified and Quantum Gravity. Lecture Notes in Physics, vol 1017. Springer, Cham. https://doi.org/10.1007/978-3-031-31520-6_2

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