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.
We discuss the obstruction to the construction of a multiparticle field theory on a κ-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 κ-Poincaré group. This necessitates a braided tensor product. We study the representations of this product, and prove that κ-Poincaré-invariant N-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 κ-Poincaré-invariant quantum field theory, and identify some open problems.
Relative locality in κ-Poincaré Gubitosi, Giulia; Mercati, Flavio
Classical and quantum gravity,
07/2013, Letnik:
30, Številka:
14
Journal Article
Recenzirano
Odprti dostop
We show that the κ-Poincaré Hopf algebra can be interpreted in the framework of curved momentum space leading to relative locality. We study the geometric properties of the momentum space described ...by κ-Poincaré and derive the consequences for particle propagation and energy-momentum conservation laws in interaction vertices, obtaining for the first time a coherent and fully workable model of the deformed relativistic kinematics implied by κ-Poincaré. We describe the action of boost transformations on multi-particle systems, showing that the covariance of the composed momenta requires a dependence of the rapidity parameter on the particle momenta themselves. Finally, we show that this particular form of the boost transformations keeps the validity of the relativity principle, demonstrating the invariance of the equations of motion under boost transformations.
We study a complex free scalar field theory on a noncommutative background spacetime called κ-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 κ-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 κ-Minkowski spacetime, and to derive its physical consequences.
It is widely believed that special initial conditions must be imposed on any time-symmetric law if its solutions are to exhibit behavior of any kind that defines an "arrow of time." We show that this ...is not so. The simplest nontrivial time-symmetric law that can be used to model a dynamically closed universe is the Newtonian N-body problem with vanishing total energy and angular momentum. Because of special properties of this system (likely to be shared by any law of the Universe), its typical solutions all divide at a uniquely defined point into two halves. In each, a well-defined measure of shape complexity fluctuates but grows irreversibly between rising bounds from that point. Structures that store dynamical information are created as the complexity grows and act as "records." Each solution can be viewed as having a single past and two distinct futures emerging from it. Any internal observer must be in one half of the solution and will only be aware of the records of one branch and deduce a unique past and future direction from inspection of the available records.
Spherically symmetric, asymptotically flat solutions of Shape Dynamics were previously studied assuming standard falloff conditions for the metric and the momenta. These ensure that the spacetime is ...asymptotically Minkowski, and that the falloff conditions are Poincaré-invariant. These requirements however are not legitimate in Shape Dynamics, which does not make assumptions on the structure or regularity of spacetime. Analyzing the same problem in full generality, I find that the system is underdetermined, as there is one function of time that is not fixed by any condition and appears to have physical relevance. This quantity can be fixed only by studying more realistic models coupled with matter, and it turns out to be related to the dilatational momentum of the matter surrounding the region under study.
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