Geometry Of Momentum Space Research Articles

Relative-locality distant observers and the phenomenology of momentum-space geometry

We study the translational invariance of the relative-locality framework proposed in Amelino-Camelia et al (2011 Phys. Rev. D 84 084010), which had been previously established only for the case of a single interaction. We provide an explicit example of boundary conditions at endpoints of worldlines, which indeed ensures the desired translational invariance for processes involving several interactions, even when some of the interactions are causally connected (particle exchange). We illustrate the properties of the associated relativistic description of distant observers within the example of a κ-Poincaré-inspired momentum-space geometry, with de Sitter metric and parallel transport governed by a non-metric and torsionful connection. We find that in such a theory, simultaneously emitted massless particles do not reach simultaneously a distant detector, as expected in light of the findings of Freidel and Smolin (2011 arXiv:1103.5626) on the implications of non-metric connections. We also show that the theory admits a free-particle limit, where the relative-locality results of Amelino-Camelia et al (2011 Phys. Lett. B 700 150) are reproduced. We establish that the torsion of the κ-Poincaré connection introduces a small (but observably large) dependence of the time of detection, for simultaneously emitted particles, on some properties of the interactions producing the particles at the source.

RELATIVE LOCALITY: A DEEPENING OF THE RELATIVITY PRINCIPLE

We describe a recently introduced principle of relative locality which we propose governs a regime of quantum gravitational phenomena accessible to experimental investigation. This regime comprises phenomena in which ℏ and GN may be neglected, while their ratio, the Planck mass [Formula: see text], is important. We propose that Mp governs the scale at which momentum space may have a curved geometry. We find that there are striking consequences for the concept of locality. The description of events in spacetime now depends on the energy used to probe it. But there remains an invariant description of physics in phase space. There is furthermore a reasonable expectation that the geometry of momentum space can be measured experimentally using astrophysical observations.

Relative locality and the soccer ball problem

We consider the behavior of macroscopic bodies within the framework of relative locality [G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, and L. Smolin, arXiv:1101.0931]. This is a recent proposal for Planck scale modifications of the relativistic dynamics of particles which are described as arising from deformations in the geometry of momentum space. We consider and resolve a common objection against such proposals, which is that, even if the corrections are small for elementary particles in current experiments, they are huge when applied to composite systems such as soccer balls, planets, and stars, with energies ${E}_{\mathrm{macro}}$ much larger than ${M}_{P}$. We show that this soccer ball problem does not arise within the framework of relative locality because the nonlinear effects for the dynamics of a composite system with $N$ elementary particles appear at most of order ${E}_{\mathrm{macro}}/N\ifmmode\cdot\else\textperiodcentered\fi{}{M}_{P}$.

Principle of relative locality

We propose a deepening of the relativity principle according to which the invariant arena for non-quantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in spacetimes are constructed by observers, but different observers, separated from each other by translations, construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them. This framework, in which absolute locality is replaced by relative locality, results from deforming momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of momentum space geometry, such as its curvature, torsion and non-metricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle all measurable by appropriate experiments. We also discuss a natural set of physical hypotheses which singles out the cases of momentum space with a metric compatible connection and constant curvature.

Relative locality: a deepening of the relativity principle

We describe a recently introduced principle of relative locality which we propose governs a regime of quantum gravitational phenomena accessible to experimental investigation. This regime comprises phenomena in which $${\hbar}$$ and G N may be neglected, while their ratio, the Planck mass $${M_p =\sqrt{\hbar / G_N}}$$ , is important. We propose that M p governs the scale at which momentum space may have a curved geometry. We find that there are striking consequences for the concept of locality. The description of events in spacetime now depends on the energy used to probe it. But there remains an invariant description of physics in phase space. There is furthermore a reasonable expectation that the geometry of momentum space can be measured experimentally using astrophysical observations.

Momentum space metric, nonlocal operator, and topological insulators

Momentum space of a gapped quantum system is a metric space: it admits a notion of distance reflecting properties of its quantum ground state. By using this quantum metric, we investigate geometric properties of momentum space. In particular, we introduce a non-local operator which represents distance square in real space and show that this corresponds to the Laplacian in curved momentum space, and also derive its path integral representation in momentum space. The quantum metric itself measures the second cumulant of the position operator in real space, much like the Berry gauge potential measures the first cumulant or the electric polarization in real space. By using the non-local operator and the metric, we study some aspects of topological phases such as topological invariants, the cumulants and topological phase transitions. The effect of interactions to the momentum space geometry is also discussed.

Amplitudes in theN=4supersymmetric Yang-Mills theory from quantum geometry of momentum space

We discuss multiloop MHV amplitudes in the N=4 SYM theory in terms of effective gravity in the momentum space with IR regulator branes as degrees of freedom. Kinematical invariants of external particles yield the moduli spaces of complex or Kahler structures which are the playgrounds for the Kodaira-Spencer(KS) or Kahler type gravity. We suggest fermionic representation of the loop MHV amplitudes in the N=4 SYM theory assuming the identification of the IR regulator branes with KS fermions in the B model and Lagrangian branes in A model. The two-easy mass box diagram is related to the correlator of fermionic currents on the spectral curve in B model or hyperbolic volume in the A model and it plays the role of a building block in the whole picture. The BDS-like anzatz has the interpretation as the semiclassical limit of a fermionic correlator. It is argued that fermionic representation implies a kind of integrability on the moduli spaces. We conjecture the interpretation of the reggeon degrees of freedom in terms of the open strings stretched between the IR regulator branes.

The $so(d + 2, 2)$ minimal representation and ambient tractors: the conformal geometry of momentum space

Tractor Calculus is a powerful tool for analyzing Weyl invariance; although fundamentally linked to the Cartan connection, it may also be arrived at geometrically by viewing a conformal manifold as the space of null rays in a Lorentzian ambient space. For dimension d conformally flat manifolds we show that the (d+2)-dimensional Fefferman--Graham ambient space corresponds to the momentum space of a massless scalar field. Hence on the one hand the null cone parameterizes physical momentum excitations, while on the other hand, null rays correspond to points in the underlying conformal manifold. This allows us to identify a fundamental set of tractor operators with the generators of conformal symmetries of a scalar field theory in a momentum representation. Moreover, these constitute the minimal representation of the non-compact conformal Lie symmetry algebra of the scalar field with Kostant--Kirillov dimension d+1. Relaxing the conformally flat requirement, we find that while the conformal Lie algebra of tractor operators is deformed by curvature corrections, higher relations in the enveloping algebra corresponding to the minimal representation persist. We also discuss potential applications of these results to physics and conformal geometry.

The Snyder space-time quantization, q-deformations, and ultraviolet divergences

The Snyder idea [Phys. Rev. 71 (1947) 38; 72 (1947) 68] to introduce the quantized space-time coordinates or, respectively, the curved momentum space is the oldest example of using the non-commutative geometry in physics. The goal of this letter is to draw attention to the connection between the geometry of momentum space and the character of singularities of the principal field-theoretical quantities. The well-known fact is that the structure of these singularites defines the character of ultraviolet divergences. In the Snyder theory, the problem of multiplication of singular field-theoretic functions is modified in a major way as compared with the usual Quantum Field Theory (QFT), and as a consequence, no ultraviolet divergences occur. It is shown that the modification of singularities is connected with the finite-difference (deformed) realizations of the Poincaré algebra.

Momentum space simulation of proton spectra in heavy ion reactions.

A new and transparent model for the theoretical simulation of proton singles spectra from heavy ion collisions is introduced. Working with few assumptions about momentum space geometry in heavy ion collisions, this model is able to explain the double differential singles cross sections. Predictions for proton-proton coincidence data are also obtained in the framework of this model and compared with data.