We propose a stochastic interpretation of spacetime noncommutativity starting from the path integral formulation of quantum mechanical commutation relations. We discuss how the (non)commutativity of spacetime is inherently related to the continuity or discontinuity of paths in the path integral formulation. Utilizing Wiener processes, we demonstrate that continuous paths lead to commutative spacetime, whereas discontinuous paths correspond to noncommutative spacetime structures. As an example we introduce discontinuous paths from which the <math><mi>κ</mi></math>-Minkowski spacetime commutators can be obtained. Moreover we focus on modifications of the Leibniz rule for differentials acting on discontinuous trajectories. We show how these can be related to the deformed action of translation generators focusing, as a working example, on the <math><mi>κ</mi></math>-Poincaré algebra. Our findings suggest that spacetime noncommutativity can be understood as a result of fundamental discreteness in temporal and/or spatial evolution.

Italy

Michele Arzano

Germany

Folkert Kuipers

Stochastic origin of spacetime noncommutativity

Infrared effects in the scattering of particles in gravity and electrodynamics entail an exchange of relativistic angular momentum between pairs of particles and the gauge field. Due to this exchange particles can carry an asymptotically non-vanishing “pairwise” boost-like angular momentum proportional to the product of their couplings to the field. At the quantum level this asymptotic angular momentum suggests the existence of a new quantum number carried by multi-particle states. We argue that such quantum number is related to a modification of the action of the generators of Lorentz transformations on multi-particle states. We derive such a modification using a group-theoretic argument based on the little group of the conformal primary basis for asymptotic states. The corresponding representation is an extension of the ordinary multi-particle Fock representation of the Poincaré group. The new multi-particle states belonging to such representation no longer factorize into tensor products of one-particle states. Viewed from a gravitational point of view, our results provide evidence for a universal breakdown of the description of multi-particle sates in terms of tensor products of one-particle states due to infrared back-reaction.

Sweden

Francesco Alessio

A new pairwise boost quantum number from celestial states

We consider sets of states in conformal quantum mechanics associated to generators of time evolution whose orbits cover different regions of the time domain. States labelled by a continuous global time variable define the two-point correlation functions of the theory seen as a one-dimensional conformal field theory. Such states exhibit the structure of a thermofield double built on bipartite eigenstates of generators of non-global time evolution. In terms of the correspondence between radial conformal symmetries in Minkowski space-time and time evolution in conformal quantum mechanics proposed in [1, 2] these generators coincide with conformal Killing vectors tangent to worldlines of Milne and diamond observers at constant radius. The temperature of the thermofield double states in conformal quantum mechanics reproduces the temperatures perceived by such diamond and Milne observers. We calculate the entanglement entropy associated to the thermofield double states and obtain a UV divergent logarithmic behaviour analogous to known results in two-dimensional conformal field theory in which the entangling boundary is point-like.

Denmark

Alessandra D’Alise

Domenico Frattulillo

Entanglement entropy in conformal quantum mechanics

We argue that the notion of identical particles is no longer well defined in quantum systems governed by noncommutative deformations of spacetime symmetries. Such models are characterized by four-momentum space given by a non-Abelian Lie group. Our analysis is based on the observation that, for states containing more than one particle, only the total momentum of the system is a well-defined quantum number. Such total momentum is obtained from the non-Abelian composition of the particles individual momenta which are no longer uniquely defined. The main upshot of our analysis is that all previous attempts to construct Fock spaces for these models rested on wrong assumptions and indeed have been unsuccessful. We also show how the natural braiding of momentum quantum numbers which characterizes the exchange of factors in the tensor product of states is covariant under relativistic transformations, thus solving a long-standing problem in the field.

Poland

Jerzy Kowalski-Glikman

Quantum particles in noncommutative spacetime: An identity crisis

It is well known in the literature that the momentum space associated to the κ-Poincaré algebra is described by the Lie group <math><mi>A</mi><mi>N</mi><mo>(</mo><mn>3</mn><mo>)</mo></math>. In this letter we show that the full κ-Poincaré Hopf algebra structure can be obtained from rather straightforward group-theoretic manipulations starting from the Iwasawa decomposition of the <math><mrow><mi>SO</mi></mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo></math> group.

A group theoretic description of the -Poincaré Hopf algebra

The generators of radial conformal symmetries in Minkowski space-time can be mapped to the generators of time evolution in conformal quantum mechanics. Within this correspondence we show that in conformal quantum mechanics the state associated to the inertial vacuum in Minkowski space-time has the structure of a thermofield double. Such state is built from a bipartite “vacuum state”, the ground state of the generators of hyperbolic time evolution, which cover only part of the time domain. When time evolution is restricted to a finite time domain one obtains the temperature perceived by static diamond observers in the Minkowski vacuum. When time evolution is determined by dilations, covering only half of the time line, the temperature of the thermofield double corresponds to the non-vanishing temperature perceived by Milne observers whose proper time evolution is confined to the future cone (Milne universe) of Minkowski space-time. The two pictures are related by a conformal transformation on the real line. Our result provides a purely group theoretical derivation of the Milne and diamond temperatures and shows that the fundamental ingredient for vacuum thermal effects is the presence of a horizon rather than acceleration.

Vacuum thermal effects in flat space-time from conformal quantum mechanics

We present a construction of <math><mi>κ</mi></math>-deformed complex scalar field theory with the objective of shedding light on the way discrete symmetries and <math><mi>C</mi><mi>P</mi><mi>T</mi></math> invariance are affected by the deformation. Our starting point is the observation that, in order to have an appropriate action of Lorentz symmetries on antiparticle states, these should be described by four-momenta living on the complement of the portion of the de Sitter group manifold to which <math><mi>κ</mi></math>-deformed particle four-momenta belong. Once the equations of motions are properly worked out from the deformed action, we obtain that the particle and antiparticle are characterized by different mass-shell constraints, leading to a subtle form of departure from <math><mi>C</mi><mi>P</mi><mi>T</mi></math> invariance. The remaining part of our work is dedicated to a detailed description of the action of deformed Poincaré and discrete symmetries on the complex field.

Andrea Bevilacqua

Giacomo Rosati

Josua Unger

<math><mi>κ</mi></math>-deformed complex fields and discrete symmetries

It is shown that a general radial conformal Killing vector in Minkowski space-time can be associated to a generator of time evolution in conformal quantum mechanics. Among these conformal Killing vectors there is a class which maps causal diamonds in Minkowski space-time into themselves. The flow of such Killing vectors describes worldlines of accelerated observers with a finite lifetime within a causal diamond. Time evolution of static diamond observers is equivalent to time evolution in conformal quantum mechanics governed by a hyperbolic Hamiltonian and covering only a segment of the time axis. This indicates that the Unruh temperature perceived by static diamond observers in the vacuum state of inertial observers in Minkowski space-time can be obtained from the behaviour of the two-point functions of conformal quantum mechanics. The results presented suggest a group theoretical description of the recently proposed associated to null surfaces defined by light fronts in Minkowski space-time.

Conformal quantum mechanics of causal diamonds

We introduce a non-commutative deformation of the algebra of bipolar spherical harmonics supporting the action of the full Lorentz algebra. Our construction is close in spirit to the one of the non-commutative spherical harmonics associated to the fuzzy sphere and, as such, it leads to a maximal value of the angular momentum. We derive the action of Lorentz boost generators on such non-commutative spherical harmonics and show that it is compatible with the existence of a maximal angular momentum.

A fuzzy bipolar celestial sphere

We show that deformed relativistic kinematics, expected to emerge in a flat-spacetime limit of quantum gravity, predicts different lifetimes for particles and their antiparticles. This phenomenon is a consequence of Planck-scale modifications of the action of discrete symmetries. In particular we focus on deformations of the action of CPT derived from the κ-Poincaré algebra, the most studied example of Planck-scale deformation of relativistic symmetries. Looking at lifetimes of muons and anti-muons we are able to derive an experimental bound on the deformation parameter of <math><mi>κ</mi><mo>≳</mo><mn>4</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>14</mn></mrow></msup></math> GeV from measurements at the LHC. Such bound has the potential to reach the value of <math><mi>κ</mi><mo>≳</mo><mn>2</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></msup></math> GeV using measurements at the planned Future Circular Collider (FCC).