We discuss the obstruction to the construction of a multiparticle field theory on a $kappa$-Minkowski noncommutative spacetime: the existence of multilocal functions which respect the deformed symmetries of the problem. This construction is only poss
ible for a light-like version of the commutation relations, if one requires invariance of the tensor product algebra under the coaction of the $kappa$-Poincare group. This necessitates a braided tensor product. We study the representations of this product, and prove that $kappa$-Poincare-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 $kappa$-Poincare-invariant quantum field theory, and identify some open problems.
We present a procedure for quantizing complex projective spaces $mathbb{CP}^{p,q}$, $qge 1$, as well as construct relevant star products on these spaces. The quantization is made unique with the demand that it preserves the full isometry algebra of t
he metric. Although the isometry algebra, namely $su(p+1,q)$, is preserved by the quantization, the Killing vectors generating these isometries pick up quantum corrections. The quantization procedure is an extension of one applied recently to Euclidean $AdS_2$, where it was found that all quantum corrections to the Killing vectors vanish in the asymptotic limit, in addition to the result that the star product trivializes to pointwise product in the limit. In other words, the space is asymptotically anti-de Sitter making it a possible candidate for the $AdS/CFT$ correspondence principle. In this article, we find indications that the results for quantized Euclidean $AdS_2$ can be extended to quantized $mathbb{CP}^{p,q}$, i.e., noncommutativity is restricted to a limited neighborhood of some origin, and these quantum spaces approach $mathbb{CP}^{p,q}$ in the asymptotic limit.
Noncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the sc
ale - and ultimately the concept of a point - makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal-Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.
Using the methods of ordinary quantum mechanics we study $kappa$-Minkowski space as a quantum space described by noncommuting self-adjoint operators, following and enlarging arXiv:1811.08409. We see how the role of Fourier transforms is played in thi
s case by Mellin transforms. We briefly discuss the role of transformations and observers.
We review the approach to the standard model of particle interactions based on spectral noncommutative geometry. The paper is (nearly) self-contained and presents both the mathematical and phenomenological aspects. In particular the bosonic spectral
action and the fermionic action are discussed in detail, and how they lead to phenomenology. We also discuss the Euclidean vs. Lorentz issues and how to go beyond the standard model in this framework.
I will discuss some aspects of the concept of point in quantum gravity, using mainly the tool of noncommutative geometry. I will argue that at Plancks distances the very concept of point may lose its meaning. I will then show how, using the spectral
action and a high momenta expansion, the connections between points, as probed by boson propagators, vanish. This discussion follows closely [1] (Kurkov-Lizzi-Vassilevich Phys. Lett. B 731 (2014) 311, [arXiv:1312.2235 [hep-th]].
We consider a noncommutative field theory with space-time $star$-commutators based on an angular noncommutativity, namely a solvable Lie algebra: the Euclidean in two dimension. The $star$-product can be derived from a twist operator and it is shown
to be invariant under twisted Poincare transformations. In momentum space the noncommutativity manifests itself as a noncommutative $star$-deformed sum for the momenta, which allows for an equivalent definition of the $star$-product in terms of twisted convolution of plane waves. As an application, we analyze the $lambda phi^4$ field theory at one-loop and discuss its UV/IR behaviour. We also analyze the kinematics of particle decay for two different situations: the first one corresponds to a splitting of space-time where only space is deformed, whereas the second one entails a non-trivial $star$-multiplication for the time variable, while one of the three spatial coordinates stays commutative.
We review the noncommutative approach to the standard model. We start with the introduction if the mathematical concepts necessary for the definition of noncommutative spaces, and manifold in particular. This defines the framework of spectral geometr
y. This is applied to the standard model of particle interaction, discussing the fermionic and bosonic spectral action. The issues relating to the calculation of the mass of the Higgs are discussed, as well as the role of neutrinos and Wick rotations. Finally, we present the possibility of solving the problem of the Higgs mass by considering a pregeometric grand symmetry.
We study the dimensional aspect of the geometry of quantum spaces. Introducing a physically motivated notion of the scaling dimension, we study in detail the model based on a fuzzy torus. We show that for a natural choice of a deformed Laplace operat
or, this model demonstrates quite non-trivial behaviour: the scaling dimension flows from 2 in IR to 1 in UV. Unlike another model with the similar property, the so-called Horava-Lifshitz model, our construction does not have any preferred direction. The dimension flow is rather achieved by a rearrangement of the degrees of freedom. In this respect the number of dimensions is deceptive. Some physical consequences are discussed.
Motivated by the Dirac idea that fundamental constant are dynamical variables and by conjectures on quantum structure of spacetime at small distances, we consider the possibility that Planck constant $hbar$ is a time depending quantity, undergoing ra
ndom gaussian fluctuations around its measured constant mean value, with variance $sigma^2$ and a typical correlation timescale $Delta t$. We consider the case of propagation of a free particle and a one--dimensional harmonic oscillator coherent state, and show that the time evolution in both cases is different from the standard behaviour. Finally, we discuss how interferometric experiments or exploiting coherent electromagnetic fields in a cavity may put effective bounds on the value of $tau= sigma^2 Delta t$.
