Abstract This paper is the second part of a series that develops the mathematical framework necessary for studying field theories on “T-Minkowski” noncommutative spacetimes. These spacetimes constitute a class of noncommutative geometries, introduced in Part I, that are each invariant under distinct quantum-group deformations of the Poincaré group. All these noncommutative geometries possess certain physically desirable characteristics, which allow me to develop all the tools of differential geometry and functional analysis that are necessary in order to build consistent and T-Poincaré-invariant noncommutative classical field theories.

Spain

Flavio Mercati

T-Minkowski Noncommutative Spacetimes II: Classical Field Theory Gimbal lock On the consequences of twisted Poincaré symmetry upon QFT on Moyal noncommutative spaces

Abstract This paper introduces and investigates a class of noncommutative spacetimes that I will call “T-Minkowski,” whose quantum Poincaré group of isometries exhibits unique and physically motivated characteristics. Notably, the coordinates on the Lorentz subgroup remain commutative, while the deformation is confined to the translations (hence the T in the name), which act like an integrable set of vector fields on the Lorentz group. This is similar to Majid’s bicrossproduct construction, although my approach allows the description of spacetimes with commutators that include a constant matrix as well as terms that are linear in the coordinates (the resulting structure is that of a centrally extended Lie algebra). Moreover, I require that one can define a covariant braided tensor product representation of the quantum Poincaré group, describing the algebra of N-points. This also implies that a 4D bicovariant differential calculus exists on the noncommutative spacetime. The resulting models can all be described in terms of a numerical triangular R-matrix through RTT relations (as well as RXX, RXY, and RXdX relations for the homogeneous spacetime, the braiding, and the differential calculus). The R-matrices I find are in one-to-one correspondence with the triangular r-matrices on the Poincaré group without quadratic terms in the Lorentz generators. These have been classified, up to automorphisms, by Zakrzewski, and amount to 16 inequivalent models. This paper is the first of a series, focusing on the identification of all the quantum Poincaré groups that are allowed by my assumptions, as well as the associated quantum homogeneous spacetimes, differential calculi, and braiding constructions.

T-Minkowski Noncommutative Spacetimes I: Poincaré Groups, Differential Calculi, and Braiding On the consequences of twisted Poincaré symmetry upon QFT on Moyal noncommutative spaces

In this study, we construct a <math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>-dimensional, relativistic, free, complex scalar quantum field theory on the noncommutative spacetime known as lightlike <math><mi>κ</mi></math>-Minkowski. The associated <math><mrow><mi>κ</mi></mrow></math>-Poincaré quantum group of isometries is triangular, and its quantum R matrix enables the definition of a braided algebra of <math><mrow><mi>N</mi></mrow></math> points that retains <math><mrow><mi>κ</mi></mrow></math>-Poincaré invariance. Leveraging our recent findings, we can now represent the generators of the deformed oscillator algebra as nonlinear redefinitions of undeformed oscillators, which are nonlocal in momentum space. The deformations manifest at the multiparticle level, as the one-particle states are identical to the undeformed ones. We successfully introduce a covariant and involutive deformed flip operator using the R matrix. The corresponding deformed (anti)symmetrization operators are covariant and idempotent, allowing for a well-posed definition of multiparticle states, a result long sought in quantum field theory on <math><mrow><mi>κ</mi></mrow></math>-Minkowski. We find that <math><mrow><mi>P</mi></mrow></math> and <math><mrow><mi>T</mi></mrow></math> are not symmetries of the theory, although <math><mrow><mi>P</mi><mi>T</mi></mrow></math> (and hence <math><mrow><mi>C</mi><mi>P</mi><mi>T</mi></mrow></math>) is. We conclude by noticing that identical particles appear distinguishable in the new theory, and discuss the fate of the Pauli exclusion principle in this setting.

Italy

Giuseppe Fabiano

Multiparticle states in braided lightlike <math><mrow><mi>κ</mi></mrow></math>-Minkowski noncommutative QFT

We discuss the construction of a free scalar quantum field theory on <math><mi>κ</mi></math>-Minkowski noncommutative spacetime. We do so in terms of <math><mi>κ</mi></math>-Poincaré-invariant <math><mi>N</mi></math>-point functions, i.e., multilocal functions which respect the deformed symmetries of the spacetime. As shown in a previous paper by some of us, this is only possible for a lightlike version of the commutation relations, which allow the construction of a covariant algebra of <math><mi>N</mi></math> points that generalizes the <math><mi>κ</mi></math>-Minkowski commutation relations. We solve the main shortcoming of our previous approach, which prevented the development of a fully covariant quantum field theory: the emergence of a non-Lorentz-invariant boundary of momentum space. To solve this issue, we propose to “extend” momentum space by introducing a class of new Fourier modes and we prove that this approach leads to a consistent definition of the Pauli-Jordan function, which turns out to be undeformed with respect to the commutative case. We finally address the quantization of our scalar field and obtain a deformed, <math><mi>κ</mi></math>-Poincaré-invariant, version of the bosonic oscillator algebra.

Maria Grazia Di Luca

New class of plane waves for <math><mi>κ</mi></math>-noncommutative quantum field theory

A novel approach to study the properties of models with quantum-deformed relativistic symmetries relies on a noncommutative space of worldlines rather than the usual noncommutative spacetime. In this setting, spacetime can be reconstructed as the set of events, that are identified as the crossing of different worldlines. We lay down the basis for this construction for the κ-Poincaré model, analyzing the fuzzy properties of κ-deformed time-like worldlines and the resulting fuzziness of the reconstructed events.

Angel Ballesteros

Giulia Gubitosi

Ivan Gutierrez-Sagredo

Fuzzy worldlines with κ-Poincaré symmetries

We discuss the obstruction to the construction of a multiparticle field theory on a <math><mi>κ</mi></math>-Minkowski noncommutative spacetime: the existence of multilocal functions which respect the deformed symmetries of the problem. This construction is only possible for a lightlike version of the commutation relations, if one requires invariance of the tensor product algebra under the coaction of the <math><mi>κ</mi></math>-Poincaré group. This necessitates a braided tensor product. We study the representations of this product, and prove that <math><mi>κ</mi></math>-Poincaré-invariant <math><mi>N</mi></math>-point functions belong to an Abelian subalgebra, and are therefore commutative. We use this construction to define the 2-point Whightman and Pauli–Jordan functions, which turn out to be identical to the undeformed ones. We finally outline how to construct a free scalar <math><mi>κ</mi></math>-Poincaré-invariant quantum field theory, and identify some open problems.

Fedele Lizzi

<math><mrow><mi>κ</mi></mrow></math>-Poincaré comodules, braided tensor products, and noncommutative quantum field theory

A useful concept in the development of physical models on the κ-Minkowski noncommutative spacetime is that of a curved momentum space. This structure is not unique: several inequivalent momentum space geometries have been identified. Some are associated to a different assumption regarding the signature of spacetime (i.e. Lorentzian vs. Euclidean), but there are inequivalent momentum spaces that can be associated to the same signature and even the same group of symmetries. Moreover, in the literature there are two approaches to the definition of these momentum spaces, one based on the right- (or left-)invariant metrics on the Lie group generated by the κ-Minkowski algebra. The other is based on the construction of 5-dimensional matrix representation of the κ-Minkowski coordinate algebra. Neither approach leads to a unique construction. Here, we find the relation between these two approaches and introduce a unified approach, capable of describing all momentum spaces, and identify the corresponding quantum group of spacetime symmetries. We reproduce known results and get a few new ones. In particular, we describe the three momentum spaces associated to the κ-Poincaré group, which are half of a de Sitter, anti-de Sitter or Minkowski space, and we identify what distinguishes them. Moreover, we find a new momentum space with the geometry of a light cone, associated to a κ-deformation of the Carroll group.

Mattia Manfredonia

The Momentum Spaces of -Minkowski noncommutative spacetime

We study the limits to the localizability of events and reference frames in the <math><mi>κ</mi></math>-Minkowski quantum spacetime. Our main tool will be a representation of the <math><mi>κ</mi></math>-Minkowski commutation relations between coordinates, and the operator and measurement theory borrowed from ordinary quantum mechanics. Spacetime coordinates are described by operators on a Hilbert space, and a complete set of commuting observables cannot contain the radial coordinate and time at the same time. The transformation between the complete sets turns out to be the Mellin transform, which allows us to discuss the localizability properties of states both in space and in time. We then discuss the transformation rules between inertial observers, which are described by the quantum <math><mi>κ</mi></math>-Poincaré group. These too are subject to limitations in the localizability of states, which impose further restrictions on the ability of an observer to localize events defined in a different observer’s reference frame.

France

Timothé Poulain

Localization and reference frames in <math><mi>κ</mi></math>-Minkowski spacetime

Noncommutative spacetimes are a proposed effective description of the low-energy regime of Quantum Gravity. Defining the microcausality relations of a scalar quantum field theory on the κ-Minkowski noncommutative spacetime allows us to define for the first time a notion of light-cone in a quantum spacetime. This allows us to reach two conclusions. First, the majority of the literature on κ-Minkowski suggests that this spacetime allows superluminal propagation of particles. The structure of the light-cone we introduced allows to rule this out, thereby excluding the possibility of constraining the relevant models with observations of in-vacuo dispersion of Gamma Ray Burst photons. Second, we are able to reject a claim made in Neves et al. (2010) [33], according to which the light-cone of the κ-Minkowski spacetime has a ‘blurry’ region of Planck-length thickness, independently of the distance of the two events considered. Such an effect would be hopeless to measure. Our analysis reveals that the thickness of the region where the notion of timelike and spacelike separations blurs grows like the square root of the distance. This magnifies the effect, e.g. in the case of cosmological distances, by 30 orders of magnitude.

Matteo Sergola

Light cone in a quantum spacetime

We study a complex free scalar field theory on a noncommutative background spacetime called <math><mi>κ</mi></math>-Minkowski. We ensure the Lorentz invariance of the theory by assuming an “elliptic de Sitter” topology for momentum space. To define covariant quantization rules, we introduce a noncommutative Pauli-Jordan function, which is invariant under <math><mi>κ</mi></math>-deformed Poincaré transformations, as well as diffeomorphisms of momentum space. We derive the algebra of creation and annihilation operators implied by our Pauli-Jordan function. Finally, we use our construction to study the structure of the light cone in <math><mrow><mi>κ</mi></mrow></math>-Minkowski spacetime, and to derive its physical consequences.