Nonrelativistic Contraction Research Articles

Non-relativistic gravity theory based on an enlargement of the extended Bargmann algebra

In this work we study a non-relativistic three dimensional Chern-Simons gravity theory based on an enlargement of the Extended Bargmann algebra. A finite nonrelativistic Chern-Simons gravity action is obtained through the non-relativistic contraction of a particular U(1) enlargement of the so-called AdS-Lorentz algebra. We show that the non-relativistic gravity theory introduced here reproduces the Maxwellian Exotic Bargmann gravity theory when a flat limit ℓ → ∞ is applied. We also present an alternative procedure to obtain the non-relativistic versions of the AdS-Lorentz and Maxwell algebras through the semigroup expansion method.

Non-relativistic fields from arbitrary contracting backgrounds

We discuss a non-relativistic contraction of massive and massless field theories minimally coupled to gravity. Using the non-relativistic limiting procedure introduced in our previous work, we (re-)derive non-relativistic field theories of massive and massless spins 0 to 3/2 coupled to torsionless Newton–Cartan backgrounds. We elucidate the relativistic origin of the Newton–Cartan central charge gauge field and explain its relation to particle number conservation.

Generalized twist deformations of Poincaré and Galilei quantum groups

The three quantum groups dual to the generalized twist deformed Poincaré Hopf algebras are provided with use of FRT procedure. Their Galilean counterparts are obtained by nonrelativistic contraction scheme.

From Lorentzian to Galilean (2+1) gravity: Drinfelʼd doubles, quantization and noncommutative spacetimes

It is shown that the canonical classical r-matrix arising from the Drinfelʼd double (DD) structure underlying the two-fold centrally extended (2+1) Galilean and Newton–Hooke (NH) Lie algebras (with either zero or non-zero cosmological constant Λ, respectively) originates as a well-defined non-relativistic contraction of a specific class of canonical r-matrices associated with the DD structure of the (2+1) (anti)-de Sitter Lie algebra. The full quantum group structure associated with such (2+1) Galilean and NH DD is presented, and the corresponding noncommutative spacetimes are shown to contain a commuting ‘absolute time’ coordinate together with two noncommutative space coordinates , whose commutator is a function of the cosmological constant Λ and of the (central) ‘quantum time’ coordinate . Thus, the Chern–Simons approach to Galilean (2+1) gravity can be consistently understood as the appropriate non-relativistic limit of the Lorentzian theory, and their associated quantum group symmetries (which do not fall into the family of so-called kappa-deformations) can also be derived from the (anti)-de Sitter quantum doubles through a well-defined quantum group contraction procedure.

(In)finite extensions of algebras from their İnönü–Wigner contractions

The method to obtain massive non-relativistic states from the Poincaré algebra is twofold. First, following İnönü and Wigner, the Poincaré algebra has to be contracted to the Galilean one. Second, the Galilean algebra has to be extended to include the central mass operator. We show that the central extension might be properly encoded in the non-relativistic contraction. In fact, any İnönü–Wigner contraction of one algebra to another corresponds to an infinite tower of Abelian extensions of the latter. The proposed method is straightforward and holds for both central and non-central extensions. Apart from the Bargmann (non-zero mass) extension of the Galilean algebra, our list of examples includes the Weyl algebra obtained from an extension of the contracted SO(3) algebra, the Carrollian (ultrarelativistic) contraction of the Poincaré algebra, the exotic Newton–Hooke algebra and some others.This paper is dedicated to the memory of Laurent Houart (1967–2011).

Galilean conformal mechanics from nonlinear realizations

We apply the nonlinear realizations method for constructing new Galilean conformal mechanics models. Our starting point is the Galilean conformal algebra which is a nonrelativistic contraction of its relativistic counterpart. We calculate Maurer-Cartan one-forms, examine various choices of the relevant coset spaces, and consider the geometric inverse Higgs-type constraints which reduce the number of the independent coset parameters and, in some cases, provide dynamical equations. New Galilean conformally invariant actions are derived in arbitrary space-time dimension $D=d+1$ (no central charges), as well as in the special dimension $D=2+1$ with one exotic central charge. We obtain new classical mechanics models which extend the standard ($D=0+1$) conformal mechanics in the presence of $d$ nonvanishing space dimensions.

Formula omitted] extended Galilei superalgebras with central charges

We perform a nonrelativistic contraction of N-extended Poincaré superalgebra with internal symmetry U(N) and general set of N(N−1) real central charges. We show that for even N=2k and particular choice of the dependence of Zij on light velocity c one gets the N-extended Galilei superalgebra with unchanged number of central charges and compact internal symmetry algebra U(k;H)=USp(2k). The Hamiltonian positivity condition is modified only by one central charge. If we put all the central charges equal to zero one gets the 2k-extended Galilei superalgebra as the subalgebra of recently introduced extended Galilei conformal superalgebra (de Azcárraga, Lukierski (2009) [1] and Sakaguchi [2]).

Generalized twist deformations of Poincaré and Galilei Hopf algebras

The three new deformed Poincaré Hopf algebras are constructed with the use of twist procedure. The corresponding relativistic space-times providing the sum of canonical and Lie-algebraic type of noncommutativity are proposed. Finally, the nonrelativistic contraction limits to the corresponding Galilei Hopf algebras are performed.

Nonrelativistic Contraction of a Canonically Deformed Super-Poincaré Hopf Algebra

We consider the nonrelativistic contraction of a canonically deformed \({\mathcal{N}} = 1\) super-Poincare Hopf algebra. In such a way we get the new quantum supergroup – a canonically twisted super-Galilei Hopf algebra equipped with the Clifford-like spinorial sector of the corresponding super-space. Besides, we also propose the new Lie-algebraic twist deformation of a super-Galilei algebra leading to classical time and quantum space as well as to the Clifford-like fermionic part of the twisted \({\mathcal{N}} = 1\) superspace.

CANONICAL AND LIE-ALGEBRAIC TWIST DEFORMATIONS OF κ-POINCARÉ AND CONTRACTIONS TO κ-GALILEI ALGEBRAS

We propose canonical and Lie-algebraic twist deformations of κ-deformed Poincaré Hopf algebra which leads to the generalized κ-Minkowski space–time relations. The corresponding deformed κ-Poincaré quantum groups are also calculated. Finally, we perform the nonrelativistic contraction limit to the corresponding twisted Galilean algebras and dual Galilean quantum groups.

CANONICAL, LIE-ALGEBRAIC AND QUADRATIC TWIST DEFORMATIONS OF GALILEI GROUP

New Galilei quantum groups dual to the Hopf algebras proposed in Ref. 1 are obtained by the nonrelativistic contraction procedures. The corresponding Lie-algebraic and quadratic quantum spacetimes are identified with the translation sectors of considered algebras.

Quantum relativistic oscillator. II. Nonrelativistic limit and phenomenological justification.

It is shown that the quantum relativistic oscillator with noncommuting internal momentum and position has as its nonrelativistic contraction limit the usual three-dimensional oscillator. The nonrelativistic limits for representations with integer and half-integer spin are compared with the quark-model picture. The spectrum predicted by the quantum relativistic oscillator with rotations is compared with experimental hadron spectra.

Non-relativistic de sitter space-time and the harmonic oscillator

A non-relativistic contraction of the de Sitter group O(3,2) is shown to act as a dynamical group for the three-dimensional harmonic oscillator in a way which preserves many of the features of the relationship between the Galileo group and the free particle.