We show that the Euclidean Snyder non-commutative space implies infinitely many different physical predictions. The distinct frameworks are specified by generalized uncertainty relations underlying deformed Heisenberg algebras. Considering the one-dimensional case in the minisuperspace arena, the bouncing Universe dynamics of loop quantum cosmology can be recovered.
We construct a scalar field theory on the Snyder non-commutative space-time. The symmetry underlying the Snyder geometry is deformed at the co-algebraic level only, while its Poincaré algebra is undeformed. The Lorentz sector is undeformed at both algebraic and co-algebraic level, but the co-product for momenta (defining the star-product) is non-co-associative. The Snyder-deformed Poincaré group is described by a non-co-associative Hopf algebra. The definition of the interacting theory in terms of a non-associative star-product is thus questionable. We avoid the non-associativity by the use of…
A Snyder model generated by the noncommutative coordinates and Lorentz generators close a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. It leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formu…
In this paper we study the Kepler problem in the non commutative Snyder scenario. We characterize the deformations in the Poisson bracket algebra under a mimic procedure from quantum standard formulations and taking into account a general recipe to build the noncommutative phase space coordinates (in the sense of Poisson brackets). We obtain an expression to the deformed potential, and then the consequences in the precession of the orbit of Mercury are calculated. This result allows us to find an estimated value for the non commutative deformation parameter introduced.
We have developed a unified scheme for studying Non-Commutative algebras based on Generalized Uncertainty Principle (GUP) and Snyder form in a relativistically covariant point particle Lagrangian (or symplectic) framework. Even though the GUP based algebra and Snyder algebra are very distinct, the more involved latter algebra emerges from an approximation of the Lagrangian model of the former algebra. Deformed Poincare generators for the systems that keep space-time symmetries of the relativistic particle models have been studied thoroughly. From a purely constrained dynamical analysis perspec…
In this paper, we construct for the first time the non-commutative fluid with the deformed Poincare invariance. To this end, the realization formalism of the noncommutative spaces is employed and the results are particularized to the Snyder space. The non-commutative fluid generalizes the fluid model in the action functional formulation to the noncommutative space. The fluid equations of motion and the conserved energy-momentum tensor are obtained.
We study a noncanonical Hilbert space representation of the polymer quantum mechanics. It is shown that Heisenberg algebra get some modifications in the constructed setup from which a generalized uncertainty principle will naturally come out. Although the extracted physical results are the same as those obtained from the standard canonical representation, the noncanonical representation may be notable in view of its possible connection with the generalized uncertainty theories suggested by string theory. In this regard, by considering an Snyder-deformed Heisenberg algebra we show that since th…
A system of relativistic Snyder particles with mutual two-body interaction that lives in a Non-Commutative Snyder geometry is studied. The underlying novel symplectic structure is a coupled and extended version of (single particle) Snyder algebra. In a recent work by Casalbuoni and Gomis, Phys.Rev. D90, 026001 (2014), a system of interacting conventional particles (in commutative spacetime) was studied with special emphasis on it's Conformal Invariance. Proceeding along the same lines we have shown that our interacting Snyder particle model is also conformally invariant. Moreover, the conforma…
We construct two types of scalar field theory on Snyder space-time. The first one is based on the natural momenta addition inherent to the coset momentum space. This construction uncovers a non-associative deformation of the Poincaré symmetries. The second one considers Snyder space-time as a subspace of a larger non-commutative space. We discuss different possibilities to restrict the extra-dimensional scalar field theory to a theory living only on Sndyer space-time and present the consequences of these restrictions on the Poincaré symmetries. We show moreover how the non-associative approach…
We show that the UV-regime at the Lifshitz point $z=3$ is equivalent to work with a momenta manifold whose topology is the same as that of an $AdS_2\times S^2$ space. According to Snyder's theory, curved momentum space is related to non-commutative quantized spacetime. In this sense, our analysis suggests an equivalence between Horava-Lifshitz and Snyder's theory.
We describe, in an algebraic way, the $κ$-deformed extended Snyder models, that depend on three parameters $β, κ$ and $λ$, which in a suitable algebra basis are described by the de Sitter algebras ${o}(1,N)$. The commutation relations of the algebra contain a parameter $λ$, which is used for the calculations of perturbative expansions. For such $κ$-deformed extended Snyder models we consider the Heisenberg double with dual generalized momenta sector, and provide the respective generalized quantum phase space depending on three parameters mentioned above. Further, we study for these models an a…
Different deformations of the Poincare symmetries have been identified for various non-commutative spaces (e.g. $κ$-Minkowski, $sl(2,R)$, Moyal). We present here the deformation of the Poincare symmetries related to Snyder space-time. The notions of smooth "K-loop", a non-associative generalization of Abelian Lie groups, and its infinitesimal counterpart given by the Lie triple system are the key objects in the construction.
In this article we define and quantize a truncated form of the nonassociative and noncommutative Snyder phi^4 field theory using the functional method in momentum space. More precisely, the action is approximated by expanding up to the linear order in the Snyder deformation parameter beta, producing an effective model on commutative spacetime for the computation of the two-, four- and six-point functions. The two- and four-point functions at one loop have the same structure as at the tree level, with UV divergences faster than in the commutative theory. The same behavior appears in the six-poi…
We discuss a generalisation of the Snyder model that includes all the possible deformations of the Heisenberg algebra compatible with Lorentz invariance, in terms of realisations of the noncommutative geometry. The corresponding deformed addition of momenta, the twist and the $R$-matrix are calculated to first order in the deformation parameters for all models. In the particular case of the Snyder realisation, the exact formula for the twist is obtained.
We present Lie-algebraic deformations of Minkowski space with undeformed Poincare algebra. These deformations interpolate between Snyder and kappa-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. Deformed Leibniz rule, the coproduct structure and star product are found. Special cases, particularly Snyder and kappa-Minkowski in Maggiore-type realizations are discussed. Our construction leads to a new class of deformed special relativity theories.
We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate peturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT.
The relativistic Lorentz-covariant quantum space-times obtained by Snyder can be described by the coset generators of (anti) de-Sitter algebras. Similarly, the Lorentz-covariant quantum phase spaces introduced by Yang, which contain additionally quantum curved fourmomenta and quantum-deformed relativistic Heisenberg algebra, can be defined by suitably chosen coset generators of conformal algebras. We extend such algebraic construction to the respective superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY Snyder model) and indicate also how to obtain the quantum relativistic…
Long time ago, C.N. Yang proposed a model of noncommutative spacetime that generalized the Snyder model to a curved background. In this paper we review his proposal and the generalizations that have been suggested during the years. In particular, we discuss the most general algebras that contain as subalgebras both de Sitter and Snyder algebras, preserving Lorentz invariance, and are generated by a two-parameter deformation of the canonical Heisenberg algebra. We also define their realizations on quantum phase space, giving explicit examples, both exact and in terms of a perturbative expansion…
We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $κ$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $κ$-deformed Snyder to Snyder space is constructed. Deformed Leibniz rule, the coproduct structure and star product are found. Special cases, particularly Snyder and $κ$-Minkowski in Maggiore-type realizations are discussed.
We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $κ$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are however isomorphic at the level of Hopf algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $κ$-deformed S…
We study an interacting $λ\,ϕ^4_{\star}$ scalar field defined on Snyder-de Sitter space. Due to the noncommutativity as well as the curvature of this space, the renormalization of the two-point function differs from the commutative case. In particular, we show that the theory in the limit of small curvature and noncommutativity is described by a model similar to the Grosse-Wulkenhaar one. Moreover, very much akin to what happens in the Grosse-Wulkenhaar model, our computation demonstrates that there exists a fixed point in the renormalization group flow of the harmonic and mass terms.
We discuss the construction of a scalar field theory with momentum space given by a coset. By introducing a generalized Fourier transform, we show how the dual scalar field theory actually lives in Snyder's space-time. As a side-product we identify a star product realization of Snyder's non-commutative space, but also the deformation of the Poincare symmetries necessary to have these symmetries realized in Snyder's space-time. A key feature of the construction is that the star product is non-associative.
Families of vector-like deformed relativistic quantum phase spaces and corresponding realizations are analyzed. Method for general construction of star product is presented. Corresponding twist, expressed in terms of phase space coordinates, in Hopf algebroid sense is presented. General linear realizations are considered and corresponding twists, in terms of momenta and Poincaré-Weyl generators or $\mathfrak{gl}(n)$ generators, are constructed and R-matrix is discussed. Classification of linear realizations leading to vector-like deformed phase spaces is given. There are 3 types of spaces: $i)…
We construct and analyze a phase diagram of a self-interacting matrix field coupled to curvature of the non-commutative truncated Heisenberg space. The model reduces to the renormalizable Grosse-Wulkenhaar model in an infinite matrix size limit and exhibits a purely non-commutative non-uniformly ordered phase. Particular attention is given to scaling of model's parameters. We additionally provide the infinite matrix size limit for the disordered to ordered phase transition line.
We show that a possible violation of the Robertson-Schrödinger uncertainty principle may signal the existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in phase-space (even if it violates the Robertson-Schrödinger uncertainty principle) is always a quantum state of an appropriate non-commutative extension of quantum mechanics. Conversely, all canonical non-commutative extensions of quantum mechanics display states that violate the Robertson-Schrödinger uncertainty principle.
After a critical overview of the Generalized Uncertainty Principle (GUP) applied to compact objects, we propose a texture of Heisenberg uncertainty principle in curved spacetimes (CHUP). CHUP allows to write down physically motivated STUR (spacetime uncertainty relations) in a generic background for a non commutative spacetime in terms of tetrad variables. In order to study possible quantum effects for compact astrophysical objects as white dwarf, neutron stars and black holes, an expression for quantum fluctuations is outlined. As a result, contrary to GUP-based claims, we found no evidence f…
The noncommutative spacetimes associated to the $κ$-Poincaré relativistic symmetries and their "non-relativistic" (Galilei) and "ultra-relativistic" (Carroll) limits are indistinguishable, since their coordinates satisfy the same algebra. In this work, we show that the three quantum kinematical models can be differentiated when looking at the associated spaces of time-like worldlines. Specifically, we construct the noncommutative spaces of time-like geodesics with $κ$-Galilei and $κ$-Carroll symmetries as contractions of the corresponding $κ$-Poincaré space and we show that these three spaces …
Appearance of adiabatic geometric phase shift in the context of noncommutative quantum mechanics is studied using an exactly solvable model of 2D simple harmonic oscilator in Moyal plane, where momentum non-commutativity are also considered along with spatial noncommutativity. After finding a suitable Bopp shift, that bridges the noncommutative phase space operators with their effective commutative counterparts, having their dependence on the non-commutative parameters, we study the adiabatic evolution in the Heisenberg picture. An explicit expression for the geometric phase shift under adiaba…
Our familiar Newton's laws allow determination of both position and velocity of any object precisely. Early nineteenth century saw the birth of quantum mechanics where all measurements must obey Heisenberg's uncertainty principle. Basically, we cannot simultaneously measure with precision, both position and momentum of particles in the microscopic atomic world. A natural extension will be to assume that space becomes fuzzy as we approach the study of early universe. That is, all the components of position cannot be simultaneously measured with precision. Such a space is called non-commutative …
We study all the symmetries of the free Schrödinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schrödinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in …
$N$ "exotic" [alias non-commutative] particles with masses $m_a$, charges $e_a$ and non-commutative parameters $θ_a$, moving in a uniform magnetic field $B$, separate into center-of-mass and internal motions if Kohn's condition $e_a/m_a=\const$ is supplemented with $e_aθ_a=\const.$ Then the center-of-mass behaves as a single exotic particle carrying the total mass and charge of the system, $M$ and $e$, and a suitably defined non-commutative parameter $Θ$. For vanishing electric field off the critical case $eΘB\neq1$, the particles perform the usual cyclotronic motion with modified but equal fr…
Unified graded differential algebra, generated by $κ$-Minkowski noncommutative (NC) coordinates, Lorentz generators and anticommuting one-forms, is constructed. It is compatible with $κ$-Poincaré-Hopf algebra. For time- and space-like deformations, the super-Jacobi identities are not satisfied. By introducing additional generator, interpreted as exterior derivative, we find a new unique algebra that satisfies all super-Jacobi identities. It is universal and valid for all type of deformations (time-, space-, and light-like). For time-like deformations this algebra coincides with the one in \cit…
We show the emergence of Berry phase in a forced harmonic oscillator system placed in the quantum space-time of Moyal type, where the time 't' is also an operator. An effective commutative description of the system gives a time dependent generalised harmonic oscillator system with perturbation linear in position and momentum. The system is then diagonalised to get a generalised harmonic oscillator and then its adiabatic evolution over time-period $\mathcal{T}$ is studied in Heisenberg picture to compute the expression of geometric phase-shift.
