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Journal articles on the topic 'Spacetime algebra'

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Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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1

KAY, BERNARD S. "THE PRINCIPLE OF LOCALITY AND QUANTUM FIELD THEORY ON (NON GLOBALLY HYPERBOLIC) CURVED SPACETIMES." Reviews in Mathematical Physics 04, spec01 (December 1992): 167–95. http://dx.doi.org/10.1142/s0129055x92000194.

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In the context of a linear model (the covariant Klein Gordon equation) we review the mathematical and conceptual framework of quantum field theory on globally hyperbolic spacetimes, and address the question of what it might mean to quantize a field on a non globally hyperbolic spacetime. Our discussion centres on the notion of F-locality which we introduce and which asserts there is a net of local algebras such that every neighbourhood of every point contains a globally hyperbolic subneighbourhood of that point for which the field algebra coincides with the algebra one would obtain were one to regard the subneighbourhood as a spacetime in its own right and quantize — with some choice of time-orientation — according to the standard rules for quantum field theory on globally hyperbolic spacetimes. We show that F-locality is a property of the standard field algebra construction for globally hyperbolic spacetimes, and argue that it (or something similar) should be imposed as a condition on any field algebra construction for non globally hyperbolic spacetimes. We call a spacetime for which there exists a field algebra satisfying F-locality F-quantum compatible and argue that a spacetime which did not satisfy something similar to this condition could not arise as an approximate classical description of a state of quantum gravity and would hence be ruled out physically. We show that all F-quantum compatible spacetimes are time orientable. We also raise the issue of whether chronology violating spacetimes can be F-quantum compatible, giving a special model — a massless field theory on the “four dimensional spacelike cylinder” — which is F-quantum compatible, and a (two dimensional) model — a massless field theory on Misner space — which is not. We discuss the possible relevance of this latter result to Hawking’s recent Chronology Protection Conjecture.

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2

DA ROCHA, R., and J. VAZ. "CONFORMAL STRUCTURES AND TWISTORS IN THE PARAVECTOR MODEL OF SPACETIME." International Journal of Geometric Methods in Modern Physics 04, no. 04 (June 2007): 547–76. http://dx.doi.org/10.1142/s0219887807002193.

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Some properties of the Clifford algebras [Formula: see text] and [Formula: see text] are presented, and three isomorphisms between the Dirac–Clifford algebra [Formula: see text] and [Formula: see text] are exhibited, in order to construct conformal maps and twistors, using the paravector model of spacetime. The isomorphism between the twistor space inner product isometry group SU(2,2) and the group $pin+(2,4) is also investigated, in the light of a suitable isomorphism between [Formula: see text] and [Formula: see text]. After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint representation of $pin+(2,4), acting on paravectors. Twistors are then presented via the paravector model of Clifford algebras and related to conformal maps in the Clifford algebra over the Lorentzian ℝ4,1 spacetime. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac–Clifford algebra [Formula: see text] using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose, the Clifford algebra over ℝ4,1 is also used to describe conformal maps, instead of ℝ2,4. Our formalism sheds some new light on the use of the paravector model and generalizations.

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3

GRESNIGT, N. G., P. F. RENAUD, and P. H. BUTLER. "THE STABILIZED POINCARE–HEISENBERG ALGEBRA: A CLIFFORD ALGEBRA VIEWPOINT." International Journal of Modern Physics D 16, no. 09 (September 2007): 1519–29. http://dx.doi.org/10.1142/s0218271807010857.

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The stabilized Poincare–Heisenberg algebra (SPHA) is a Lie algebra of quantum relativistic kinematics generated by fifteen generators. It is obtained from imposing stability conditions after combining the Lie algebras of quantum mechanics and relativity. In this paper, we show how the sixteen-dimensional real Clifford algebras Cℓ(1,3) and Cℓ(3,1) can both be used to generate the SPHA. The Clifford algebra path to the SPHA avoids the traditional stability considerations. It is conceptually easier and more straightforward to work with a Clifford algebra. The Clifford algebra path suggests that the next evolutionary step toward a theory of physics at the interface of GR and QM might be to depart from working in spacetime and instead to work in spacetime–momentum.

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4

KOVAČEVIĆ, DOMAGOJ, and STJEPAN MELJANAC. "KAPPA-MINKOWSKI SPACETIME, KAPPA-POINCARÉ HOPF ALGEBRA AND REALIZATIONS." International Journal of Geometric Methods in Modern Physics 09, no. 06 (August 3, 2012): 1261009. http://dx.doi.org/10.1142/s0219887812610099.

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The κ-Minkowski spacetime and Lorentz algebra are unified in unique Lie algebra. Introducing commutative momenta, a family of κ-deformed Heisenberg algebras and κ-deformed Poincaré algebras are defined. They are determined by the matrix depending on momenta. Realizations and star product are defined and analyzed in general. The relation among the coproduct of momenta, realization and the star product is pointed out. Hopf algebra of the Poincaré algebra, related to the covariant realization, is presented in unified covariant form. Left–right dual realizations and dual algebra are introduced and considered. The generalized involution and the star inner product are defined and analyzed. Partial integration and deformed trace property are obtained in general. The translation invariance of the star product is pointed out.

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5

DASZKIEWICZ, MARCIN. "CANONICAL AND LIE-ALGEBRAIC TWIST DEFORMATIONS OF GALILEI ALGEBRA." Modern Physics Letters A 23, no. 07 (March 7, 2008): 505–17. http://dx.doi.org/10.1142/s0217732308026479.

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We describe various nonrelativistic contractions of two classes of twisted Poincaré algebra: canonical one (θμν-deformation) and the one leading to Lie-algebraic models of noncommutative spacetimes. The cases of contraction-independent and contraction-dependent twist parameters are considered. We obtain five models of noncommutative nonrelativistic spacetimes, in particular, two new Lie-algebraic nonrelativistic deformations of spacetime, respectively, with quantum time/classical space and with quantum space/classical time.

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6

Obaidullah, U., and Sameerah Jamal. "pp-wave potential functions: A complete study using Noether symmetries." International Journal of Geometric Methods in Modern Physics 18, no. 07 (March 18, 2021): 2150108. http://dx.doi.org/10.1142/s0219887821501085.

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In this paper, we examine the functional forms of the potentials [Formula: see text] that emerge from the Lagrangian of the pp-wave spacetime. To facilitate this investigation, Noether symmetries are employed as well as their linear combinations and subalgebras. We exploit the geometric fact that Noether point symmetries of geodesic Lagrangians are generated from the Homothetic algebra of spacetimes. Thus, we provide a complete analysis of the potentials of this spacetime, which are split into 14 isometry categories.

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7

Moia, Alessandro. "Noncommutative Spacetime Symmetries from Covariant Quantum Mechanics." Advances in High Energy Physics 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/4042314.

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In the last decades, noncommutative spacetimes and their deformed relativistic symmetries have usually been studied in the context of field theory, replacing the ordinary Minkowski background with an algebra of noncommutative coordinates. However, spacetime noncommutativity can also be introduced into single-particle covariant quantum mechanics, replacing the commuting operators representing the particle’s spacetime coordinates with noncommuting ones. In this paper, we provide a full characterization of a wide class of physically sensible single-particle noncommutative spacetime models and the associated deformed relativistic symmetries. In particular, we prove that they can all be obtained from the standard Minkowski model and the usual Poincaré transformations via a suitable change of variables. Contrary to previous studies, we find that spacetime noncommutativity does not affect the dispersion relation of a relativistic quantum particle, but only the transformation properties of its spacetime coordinates under translations and Lorentz transformations.

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8

MORETTI, VALTER. "ASPECTS OF NONCOMMUTATIVE LORENTZIAN GEOMETRY FOR GLOBALLY HYPERBOLIC SPACETIMES." Reviews in Mathematical Physics 15, no. 10 (December 2003): 1171–217. http://dx.doi.org/10.1142/s0129055x03001886.

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Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally-hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a C*-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of C*-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called loci, are realized as the elements of the inductive limit of the spaces of the algebraic states on the C*-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the role of a Lorentzian metric. Specializing back the formalism to the usual globally-hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events.

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9

SCIPIONI, R. "G-ALGEBRA AND CURVED SPACETIME." Modern Physics Letters A 10, no. 23 (July 30, 1995): 1705–9. http://dx.doi.org/10.1142/s0217732395001824.

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The recently proved nonexistence of generalized statistics in curved spacetime is reconsidered in the framework of g-algebra; we show in a rigorous way that for asymptotic states, a Fermi or Bose statistics in the “in” region always evolves a Bose or Fermi statistics in the “out” region; the new approach, however, permits one to infer that this fact might not be true when considering intermediate states.

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10

Roque, Waldir L., and Renato P. Dos Santos. "Computer algebra in spacetime embedding." Journal of Symbolic Computation 12, no. 3 (September 1991): 381–89. http://dx.doi.org/10.1016/s0747-7171(08)80156-3.

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11

Hestenes, David. "Spacetime physics with geometric algebra." American Journal of Physics 71, no. 7 (July 2003): 691–714. http://dx.doi.org/10.1119/1.1571836.

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12

Hestenes, David. "Curvature calculations with spacetime algebra." International Journal of Theoretical Physics 25, no. 6 (June 1986): 581–88. http://dx.doi.org/10.1007/bf00670472.

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13

Ballesteros, Angel, Giulia Gubitosi, and Flavio Mercati. "Interplay between Spacetime Curvature, Speed of Light and Quantum Deformations of Relativistic Symmetries." Symmetry 13, no. 11 (November 5, 2021): 2099. http://dx.doi.org/10.3390/sym13112099.

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Recent work showed that κ-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincaré algebra of special-relativistic symmetries, one can toggle the curvature parameter Λ, the Planck scale quantum deformation parameter κ and the speed of light parameter c to move to the well-studied κ-Poincaré algebra, the (quantum) (A)dS algebra, the (quantum) Galilei and Carroll algebras and their curved versions. In this review, we survey the properties and relations of these algebras of relativistic symmetries and their associated noncommutative spacetimes, emphasizing the nontrivial effects of interplay between curvature, quantum deformation and speed of light parameters.

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14

Fiorenza, Domenico, Hisham Sati, and Urs Schreiber. "Super-Lie n-algebra extensions, higher WZW models and super-p-branes with tensor multiplet fields." International Journal of Geometric Methods in Modern Physics 12, no. 02 (January 29, 2015): 1550018. http://dx.doi.org/10.1142/s0219887815500188.

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We formalize higher-dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type σ-model branes (open brane ending on background brane) are encoded precisely in (super-)L∞-extension theory and how the resulting "extended (super-)space-times" formalize spacetimes containing σ-model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super-p-brane spectrum of superstring/M-theory is realized this way, including the pure σ-model branes (the "old brane scan") but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional (11D) spacetime with an M2-brane condensate turns out to be the "M-theory super-Lie algebra". We also observe that in this formulation there is a simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11D sugra/M-theory spacetime, and of (prequantum) S-duality for type IIB string theory. Finally we give the non-perturbative description of all this by higher WZW-type σ-models on higher super-orbispaces with higher WZW terms in stacky differential cohomology.

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15

Perini, Claudio, and Gabriele Nunzio Tornetta. "A scale-covariant quantum spacetime." Reviews in Mathematical Physics 26, no. 04 (May 2014): 1450006. http://dx.doi.org/10.1142/s0129055x14500068.

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A noncommutative spacetime admitting dilation symmetry was briefly mentioned in the seminal work [8] of Doplicher, Fredenhagen and Roberts. In this paper, we explicitly construct the model in detail and carry out an indepth analysis. The C*-algebra that describes this quantum spacetime is determined, and it is shown that it admits an action by *-automorphisms of the dilation group, along with the expected Poincaré covariance. In order to study the main physical properties of this scale-covariant model, a free scalar neutral field is introduced as an investigation tool. Our key results are then the loss of locality and the irreducibility, or triviality, of special field algebras associated with regions of the ordinary Minkowski spacetime. It turns out, in the conclusions, that this analysis allows also to argue on viable ways of constructing a full conformally covariant model for quantum spacetime.

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16

Kritov, Alexander. "Gravitation with Cosmological Term, Expansion of the Universe as Uniform Acceleration in Clifford Coordinates." Symmetry 13, no. 3 (February 24, 2021): 366. http://dx.doi.org/10.3390/sym13030366.

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This paper presents a novel approach to the cosmological constant problem by the use of the Clifford algebras of space Cl3,0 and anti-space Cl0,3 with a particular focus on the paravector representation, emphasizing the fact that both algebras have a center represented just by two coordinates. Since the paravector representation allows assigning the scalar element of grade 0 to the time coordinate, we consider the relativity in such two-dimensional spacetime for a uniformly accelerated frame with the constant acceleration 3H0c. Using the Rindler coordinate transformations in two-dimensional spacetime and then applying it to Minkowski coordinates, we obtain the FLRW metric, which in the case of the Clifford algebra of space Cl3,0 corresponds to the anti-de Sitter (AdS) flat (k=0) case, the negative cosmological term and an oscillating model of the universe. The approach with anti-Euclidean Clifford algebra Cl0,3 leads to the de Sitter model with the positive cosmological term and the exact form of the scale factor used in modern cosmology.

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17

SHABBIR, GHULAM, and SUHAIL KHAN. "CLASSIFICATION OF CYLINDRICALLY SYMMETRIC STATIC SPACETIMES ACCORDING TO THEIR KILLING VECTOR FIELDS IN TELEPARALLEL THEORY OF GRAVITATION." Modern Physics Letters A 25, no. 07 (March 7, 2010): 525–33. http://dx.doi.org/10.1142/s0217732310032007.

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In this paper we classify cylindrically symmetric static spacetimes according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the teleparallel Killing vector fields are 3, 4, 6 or 10 which are the same in numbers as in general relativity. In case of 3, 4 or 6 the teleparallel Killing vector fields are multiple of the corresponding Killing vector fields in general relativity by some function of r. In the case of 10 Killing vector fields the spacetime becomes Minkowski spacetime and all the torsion components are zero. The Killing vector fields in this case are exactly the same as in general relativity. Here we also discuss the Lie algebra in each case. It is important to note that this classification also covers the plane symmetric static spacetimes.

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18

Obukhov, Valeriy V. "Algebra of the Symmetry Operators of the Klein–Gordon–Fock Equation for the Case When Groups of Motions G3 Act Transitively on Null Subsurfaces of Spacetime." Symmetry 14, no. 2 (February 9, 2022): 346. http://dx.doi.org/10.3390/sym14020346.

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The algebras of the symmetry operators for the Hamilton–Jacobi and Klein–Gordon–Fock equations are found for a charged test particle, moving in an external electromagnetic field in a spacetime manifold on the isotropic (null) hypersurface, of which a three-parameter groups of motions acts transitively. We have found all admissible electromagnetic fields for which such algebras exist. We have proved that an admissible field does not deform the algebra of symmetry operators for the free Hamilton–Jacobi and Klein–Gordon–Fock equations. The results complete the classification of admissible electromagnetic fields, in which the Hamilton–Jacobi and Klein–Gordon–Fock equations admit algebras of motion integrals that are isomorphic to the algebras of operators of the r-parametric groups of motions of spacetime manifolds if (r≤4).

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19

JANYŠKA, JOSEF, and MARCO MODUGNO. "HERMITIAN VECTOR FIELDS AND SPECIAL PHASE FUNCTIONS." International Journal of Geometric Methods in Modern Physics 03, no. 04 (June 2006): 719–54. http://dx.doi.org/10.1142/s0219887806001351.

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We start by analyzing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.

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20

Kerner, Richard. "Spacetime Symmetries andZ3-graded Quark Algebra." Journal of Physics: Conference Series 343 (February 8, 2012): 012056. http://dx.doi.org/10.1088/1742-6596/343/1/012056.

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21

Demir, Süleyman, and Murat Tanışlı. "Spacetime algebra for the reformulation of fluid field equations." International Journal of Geometric Methods in Modern Physics 14, no. 05 (April 13, 2017): 1750075. http://dx.doi.org/10.1142/s021988781750075x.

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In the light of the analogy between electromagnetism and fluid dynamics, the Maxwell-type equations of compressible fluids are reformulated on the basis of spacetime algebra. In this paper, it is proved that this algebra provides an efficient mathematical tool for describing fluid fields in a compact and elegant way. Moreover, the fluid wave equation in terms of potentials are derived in a form similar to electromagnetic and gravitational counterparts previously derived using spacetime algebra.

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22

CRANE, LOUIS. "RELATIONAL SPACETIME, MODEL CATEGORIES AND QUANTUM GRAVITY." International Journal of Modern Physics A 24, no. 15 (June 20, 2009): 2753–75. http://dx.doi.org/10.1142/s0217751x0904614x.

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We propose a mathematically concrete way of modelling the suggestion that in quantum gravity the spacetime manifold disappears. We replace the underlying point set topological space with several apparently different models, which are actually related by pairs of adjoint functors from rational homotopy theory. One is a discrete approximation to the causal null path space derived from the multiple images in the spacetime theory of gravitational lensing, described as an object in the model category of differential graded Lie algebras. Another of our models appears as a thickening of spacetime, which we interpret as a formulation of relational geometry. This model is produced from the finite dimensional differential graded algebra of differential forms which can be transmitted out of a finite region consistent with the Bekenstein bound by another functor, called geometric realisation. The thickening of spacetime, which we propose as a version of relational spacetime, has a surprizingly rich structure. Information which would make up a spin bundle over spacetime is contained in it, making it possible to include fermionic fields in a geometric state sum over it. Avenues toward constructing an actual quantum theory of gravity on our models are given a preliminary exploration.

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23

Bokhari, Ashfaque H., and A. H. Kara. "Gauge symmetries and isometries in higher dimensions." Modern Physics Letters A 35, no. 37 (October 1, 2020): 2050310. http://dx.doi.org/10.1142/s0217732320503101.

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We study the invariance properties of five-dimensional metrics and their corresponding geodesic equations of motion. In this context a number of five-dimensional models of the Einstein–Gauss–Bonnet (EGB) theory leading to black holes, wormholes and spacetime horns arising in a variety of situations are discussed in the context of variational symmetries of which each vector field, via Noether’s theorem (NT), provides a nontrivial conservation law. In particular, it is shown that algebraic structure of isometries and the variational conservation laws of the five-dimensional Einstein–Bonnet metric extend consistently from the well-known Minkowski, de-Sitter and Schwarzschild four-dimensional spacetimes to the considered five-dimensional ones. In the equivalent five-dimensional case, the maximal algebra of kvs is fifteen with eight additional Noether symmetries. Also, whereas the constant curvature five-dimensional case leads to fifteen kvs and one additional Noether symmetry and seven plus one in the minimal case, a number of metrics of the EGB theory in five dimensions give rise to algebras isomorphic a seven-dimensional algebra of kvs and a single additional Noether symmetry.

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24

CADONI, MARIANO. "STATISTICAL ENTROPY OF THE SCHWARZSCHILD BLACK HOLE." Modern Physics Letters A 21, no. 24 (August 10, 2006): 1879–87. http://dx.doi.org/10.1142/s0217732306021165.

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We derive the statistical entropy of the Schwarzschild black hole by considering the asymptotic symmetry algebra near the [Formula: see text] boundary of the spacetime at past null infinity. Using a two-dimensional description and the Weyl invariance of black hole thermodynamics this symmetry algebra can be mapped into the Virasoro algebra generating asymptotic symmetries of anti-de Sitter spacetime. Using Lagrangian methods we identify the stress–energy tensor of the boundary conformal field theory and calculate the central charge of the Virasoro algebra. The Bekenstein–Hawking result for the black hole entropy is regained using Cardy's formula. Our result strongly supports a nonlocal realization of the holographic principle.

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25

Mandal, Susobhan. "Existence of conserved quantities and their algebra in curved spacetime." International Journal of Modern Physics A 35, no. 26 (September 20, 2020): 2050162. http://dx.doi.org/10.1142/s0217751x20501626.

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In general relativity, finding out the geodesics of a given spacetime manifold is an important task because it determines which classical processes are dynamically forbidden. Conserved quantities play an important role in solving the geodesic equations of a general spacetime manifold. Furthermore, knowing all possible conserved quantities of a system gives information about the hidden symmetries of that system since conserved quantities are deeply connected with the symmetries of the system. These are very important in their own right. Conserved quantities are also useful to capture certain features of spacetime manifold for an asymptotic observer. In this article, we show the existence of these conserved charges and their algebra in a generic curved spacetime for a class of dynamical systems with the Hamiltonians quadratic and linear in momentum and spin.

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26

CHAKRABORTY, SOMDEB, and PARIJAT DEY. "WESS–ZUMINO–WITTEN MODEL FOR GALILEAN CONFORMAL ALGEBRA." Modern Physics Letters A 28, no. 38 (December 4, 2013): 1350176. http://dx.doi.org/10.1142/s0217732313501769.

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In this note, we construct a Wess–Zumino–Witten model based on the Galilean conformal algebra in two-spacetime dimensions, which is a nonrelativistic analogue of the relativistic conformal algebra. We obtain exact background corresponding to σ-models in six dimensions (the dimension of the group manifold) and a central charge c = 6. We carry out a Sugawara type construction to verify the conformal invariance of the model. Further, we discuss the feasibility of the background obtained as a physical spacetime metric.

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27

Tanaka, Sho. "Contracted Representation of Yang's Spacetime Algebra and Buniy-Hsu-Zee's Discrete Spacetime." Foundations of Physics Letters 19, no. 6 (November 2006): 567–78. http://dx.doi.org/10.1007/s10702-006-1010-9.

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28

Mendes, R. Vilela. "An extended Dirac equation in noncommutative spacetime." Modern Physics Letters A 31, no. 15 (May 17, 2016): 1650089. http://dx.doi.org/10.1142/s0217732316500899.

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Stabilizing, by deformation, the algebra of relativistic quantum mechanics a noncommutative spacetime geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation which has both a massless and a large mass solution. The nature of the solutions is discussed as well as the effects of coupling the two solutions.

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29

AGOSTINI, ALESSANDRA, GIOVANNI AMELINO-CAMELIA, MICHELE ARZANO, ANTONINO MARCIANÒ, and RUGGERO ALTAIR TACCHI. "GENERALIZING THE NOETHER THEOREM FOR HOPF-ALGEBRA SPACETIME SYMMETRIES." Modern Physics Letters A 22, no. 24 (August 10, 2007): 1779–86. http://dx.doi.org/10.1142/s0217732307024280.

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Over these past few years several quantum-gravity research groups have been exploring the possibility that in some Planck-scale nonclassical descriptions of spacetime one or another form of nonclassical spacetime symmetries might arise. One of the most studied scenarios is based on the use of Hopf algebras, but previous attempts were not successful in deriving constructively the properties of the conserved charges one would like to obtain from the Hopf structure, and this in turn did not allow a crisp physical characterization of the new concept of spacetime symmetry. Working within the example of κ-Minkowski noncommutative spacetime, known to be particularly troublesome from this perspective, we observe that these past failures in the search of the charges originated from not recognizing the crucial role that the noncommutative transformation parameters play in the symmetry analysis. We show that, if indeed one introduces appropriate noncommutative transformation parameters, all the steps of the Noether analysis can be easily performed, obtaining an explicit formula for the charges carried by fields that are solutions of the equation of motion.

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30

Baylis, William E. "Comment on `Dirac theory in spacetime algebra'." Journal of Physics A: Mathematical and General 35, no. 22 (May 24, 2002): 4791–96. http://dx.doi.org/10.1088/0305-4470/35/22/401.

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31

Agostini, Alessandra. "Hopf-Algebra Description of Noncommutative-Spacetime Symmetries." Czechoslovak Journal of Physics 53, no. 11 (November 2003): 955–61. http://dx.doi.org/10.1023/b:cjop.0000010518.07542.61.

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32

Doran, Chris, Anthony Lasenby, and Stephen Gull. "States and operators in the spacetime algebra." Foundations of Physics 23, no. 9 (September 1993): 1239–64. http://dx.doi.org/10.1007/bf01883678.

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33

MELJANAC, S., D. MELJANAC, A. SAMSAROV, and M. STOJIĆ. "κ-DEFORMED SNYDER SPACETIME." Modern Physics Letters A 25, no. 08 (March 14, 2010): 579–90. http://dx.doi.org/10.1142/s0217732310032652.

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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. Deformed Leibniz rule, the coproduct structure and star product are found. Special cases, particularly Snyder and κ-Minkowski in Maggiore-type realizations are discussed. Our construction leads to a new class of deformed special relativity theories.

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34

Mitsopoulos, Antonios, Michael Tsamparlis, and Andronikos Paliathanasis. "Constructing the CKVs of Bianchi III and V spacetimes." Modern Physics Letters A 34, no. 39 (December 19, 2019): 1950326. http://dx.doi.org/10.1142/s0217732319503267.

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We determine the conformal algebra of Bianchi III and Bianchi V spacetimes or, equivalently, we determine all Bianchi III and Bianchi V spacetimes which admit a proper conformal Killing vector (CKV). The algorithm that we use has been developed in [M. Tsamparlis et al.Class. Quantum. Grav. 15, 2909 (1998)] and concerns the computation of the CKVs of decomposable spacetimes. The main point of this method is that a decomposable space admits a CKV if the reduced space admits a gradient homothetic vector, the latter being possible only if the reduced space is flat or a space of constant curvature. We apply this method in a stepwise manner starting from the two-dimensional spacetime which admits an infinite number of CKVs and we construct step by step the Bianchi III and V spacetimes by assuming that CKVs survive as we increase the dimension of the space. We find that there is only one Bianchi III and one Bianchi V spacetime which admit at maximum one proper CKV. In each case, we determine the CKV and the corresponding conformal factor. As a first application in these two spacetimes, we study the kinematics of the comoving observers and the dynamics of the corresponding cosmological fluid. As a second application, we determine in these spacetimes generators of the Lie symmetries of the wave equation.

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35

HOLLANDS, STEFAN. "RENORMALIZED QUANTUM YANG–MILLS FIELDS IN CURVED SPACETIME." Reviews in Mathematical Physics 20, no. 09 (October 2008): 1033–172. http://dx.doi.org/10.1142/s0129055x08003420.

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We present a proof that the quantum Yang–Mills theory can be consistently defined as a renormalized, perturbative quantum field theory on an arbitrary globally hyperbolic curved, Lorentzian spacetime. To this end, we construct the non-commutative algebra of observables, in the sense of formal power series, as well as a space of corresponding quantum states. The algebra contains all gauge invariant, renormalized, interacting quantum field operators (polynomials in the field strength and its derivatives), and all their relations such as commutation relations or operator product expansion. It can be viewed as a deformation quantization of the Poisson algebra of classical Yang–Mills theory equipped with the Peierls bracket. The algebra is constructed as the cohomology of an auxiliary algebra describing a gauge fixed theory with ghosts and anti-fields. A key technical difficulty is to establish a suitable hierarchy of Ward identities at the renormalized level that ensures conservation of the interacting BRST-current, and that the interacting BRST-charge is nilpotent. The algebra of physical interacting field observables is obtained as the cohomology of this charge. As a consequence of our constructions, we can prove that the operator product expansion closes on the space of gauge invariant operators. Similarly, the renormalization group flow is proved not to leave the space of gauge invariant operators. The key technical tool behind these arguments is a new universal Ward identity that is formulated at the algebraic level, and that is proven to be consistent with a local and covariant renormalization prescription. We also develop a new technique to accomplish this renormalization process, and in particular give a new expression for some of the renormalization constants in terms of cycles.

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36

BALACHANDRAN, A. P., and M. MARTONE. "SPACETIME FROM SYMMETRY: THE MOYAL PLANE FROM THE POINCARÉ–HOPF ALGEBRA." Modern Physics Letters A 24, no. 23 (July 30, 2009): 1811–21. http://dx.doi.org/10.1142/s0217732309031144.

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We show how to get a noncommutative product for functions on spacetime starting from the deformation of the coproduct of the Poincaré group using the Drinfel'd twist. Thus it is easy to see that the commutative algebra of functions on spacetime (ℝ4) can be identified as the set of functions on the Poincaré group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincaré group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and involves cohomological features. As is known, spacetime algebra fixes the coproduct on the diffeomorphism group of the manifold. We now see that the influence is reciprocal: they are strongly tied.

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37

Ma, Wen-Xiu. "Integrable Nonlocal PT-Symmetric Modified Korteweg-de Vries Equations Associated with so(3, \({\mathbb{R}}\))." Symmetry 13, no. 11 (November 19, 2021): 2205. http://dx.doi.org/10.3390/sym13112205.

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We construct integrable PT-symmetric nonlocal reductions for an integrable hierarchy associated with the special orthogonal Lie algebra so(3,R). The resulting typical nonlocal integrable equations are integrable PT-symmetric nonlocal complex reverse-spacetime and real reverse-spacetime modified Korteweg-de Vries equations associated with so(3,R).

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38

CHRYSSOMALAKOS, C., and E. OKON. "GENERALIZED QUANTUM RELATIVISTIC KINEMATICS: A STABILITY POINT OF VIEW." International Journal of Modern Physics D 13, no. 10 (December 2004): 2003–34. http://dx.doi.org/10.1142/s0218271804006632.

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We apply Lie algebra deformation theory to the problem of identifying the stable form of the quantum relativistic kinematical algebra. As a warm up, given Galileo's conception of spacetime as input, some modest computer code we wrote zeroes in on the Poincaré-plus-Heisenberg algebra in about a minute. Further ahead, along the same path, lies a three-dimensional deformation space, with an instability double cone through its origin. We give physical as well as geometrical arguments supporting our view that moment, rather than position operators, should enter as generators in the Lie algebra. With this identification, the deformation parameters give rise to invariant length and mass scales. Moreover, standard quantum relativistic kinematics of massive, spinless particles corresponds to non-commuting moment operators, a purely quantum effect that bears no relation to spacetime non-commutativity, in sharp contrast to earlier interpretations.

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39

DAPPIAGGI, CLAUDIO, VALTER MORETTI, and NICOLA PINAMONTI. "RIGOROUS STEPS TOWARDS HOLOGRAPHY IN ASYMPTOTICALLY FLAT SPACETIMES." Reviews in Mathematical Physics 18, no. 04 (May 2006): 349–415. http://dx.doi.org/10.1142/s0129055x0600270x.

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Scalar QFT on the boundary ℑ+at future null infinity of a general asymptotically flat 4D spacetime is constructed using the algebraic approach based on Weyl algebra associated to a BMS-invariant symplectic form. The constructed theory turns out to be invariant under a suitable strongly-continuous unitary representation of the BMS group with manifest meaning when the fields are interpreted as suitable extensions to ℑ+of massless minimally coupled fields propagating in the bulk. The group theoretical analysis of the found unitary BMS representation proves that such a field on ℑ+coincides with the natural wave function constructed out of the unitary BMS irreducible representation induced from the little group Δ, the semidirect product between SO(2) and the two-dimensional translations group. This wave function is massless with respect to the notion of mass for BMS representation theory. The presented result proposes a natural criterion to solve the long-standing problem of the topology of BMS group. Indeed the found natural correspondence of quantum field theories holds only if the BMS group is equipped with the nuclear topology rejecting instead the Hilbert one. Eventually, some theorems towards a holographic description on ℑ+of QFT in the bulk are established at level of C*-algebras of fields for asymptotically flat at null infinity spacetimes. It is proved that preservation of a certain symplectic form implies the existence of an injective *-homomorphism from the Weyl algebra of fields of the bulk into that associated with the boundary ℑ+. Those results are, in particular, applied to 4D Minkowski spacetime where a nice interplay between Poincaré invariance in the bulk and BMS invariance on the boundary at null infinity is established at the level of QFT. It arises that, in this case, the *-homomorphism admits unitary implementation and Minkowski vacuum is mapped into the BMS invariant vacuum on ℑ+.

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40

Medeiros, Paul de, and S. Prem Kumar. "Spacetime Virasoro algebra from strings on zero radiusAdS3." Journal of High Energy Physics 2003, no. 12 (December 18, 2003): 043. http://dx.doi.org/10.1088/1126-6708/2003/12/043.

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41

Ribeiro, M. A., and C. R. Paiva. "Relativistic optics in moving media with spacetime algebra." European Physical Journal Applied Physics 49, no. 3 (February 3, 2010): 33003. http://dx.doi.org/10.1051/epjap/2009146.

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42

Dressel, Justin, Konstantin Y. Bliokh, and Franco Nori. "Spacetime algebra as a powerful tool for electromagnetism." Physics Reports 589 (August 2015): 1–71. http://dx.doi.org/10.1016/j.physrep.2015.06.001.

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43

Hussain, Tahir, Khudija Shaheen, and Faiza Saleem. "Homothetic matter collineations of static plane symmetric spacetimes." International Journal of Geometric Methods in Modern Physics 16, no. 12 (October 23, 2019): 1950182. http://dx.doi.org/10.1142/s0219887819501822.

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In this paper, we present a complete classification of static plane symmetric spacetimes via their homothetic symmetries of the energy–momentum tensor, known as homothetic matter collineations (HMCs). The HMC equations for these spacetimes are derived and then solved by considering the degeneracy and non-degeneracy of the energy–momentum tensor. In the former case, we have obtained 6, 11 and infinite number of HMCs, while in the latter case, the solution of HMC equations yields 6-, 7-, 8-, 10- and 11-dimensional algebra of HMCs. The obtained HMCs generate some differential constraints involving the components of the energy–momentum tensor. Some examples of static plane symmetric spacetime metrics satisfying these constraints are provided and the physical interpretations of these metrics are discussed.

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44

Kibaroğlu, Salih, and Oktay Cebecioğlu. "Gauge theory of the Maxwell and semi-simple extended (anti) de Sitter algebra." International Journal of Modern Physics D 30, no. 10 (June 15, 2021): 2150075. http://dx.doi.org/10.1142/s0218271821500759.

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In this paper, a semi-simple and Maxwell extension of the (anti) de Sitter algebra is constructed. Then, a gauge-invariant model has been presented by gauging the Maxwell semi-simple extension of the (anti) de Sitter algebra. We firstly construct a Stelle–West-like model action for five-dimensional spacetime in which the effects of spontaneous symmetry breaking have been taken into account. In doing so, we get an extended version of Einstein’s field equations. Next, we decompose the five-dimensional extended Lie algebra and establish a MacDowell–Mansouri-like action that contains the Einstein–Hilbert term, the cosmological term as well as new terms coming from Maxwell extension in four-dimensional spacetime where the torsion-free condition is assumed. Finally, we have shown that both models are equivalent for an appropriately chosen gauge condition.

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45

Loret, Niccoló, Stjepan Meljanac, Flavio Mercati, and Danijel Pikutić. "Vectorlike deformations of relativistic quantum phase-space and relativistic kinematics." International Journal of Modern Physics D 26, no. 11 (September 19, 2017): 1750123. http://dx.doi.org/10.1142/s0218271817501231.

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We study a family of noncommutative spacetimes constructed by one four-vector. The large set of coordinate commutation relations described in this way includes many cases that are widely studied in the literature. The Hopf-algebra symmetries of these noncommutative spacetimes, as well as the structures of star product and twist are introduced and considered at first order in the deformation, described by four parameters. We also study the deformations to relativistic kinematics implied by this framework, and calculate the most general expression for the momentum dependence of the Lorentz transformations on momenta, which is an effect that is required by consistency. At the end of the paper we analyse the phenomenological consequences of this large family of vectorlike deformations on particles propagation in spacetime. This leads to a set of characteristic phenomenological effects.

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46

Joyce, William P. "Gauge freedom of Dirac theory in complexified spacetime algebra." Journal of Physics A: Mathematical and General 35, no. 22 (May 24, 2002): 4737–47. http://dx.doi.org/10.1088/0305-4470/35/22/306.

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47

Joyce, William P. "Reply to Comment on `Dirac theory in spacetime algebra'." Journal of Physics A: Mathematical and General 35, no. 22 (May 24, 2002): 4797–98. http://dx.doi.org/10.1088/0305-4470/35/22/402.

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48

Kovačević, D., and S. Meljanac. "Kappa-Minkowski spacetime, kappa-Poincaré Hopf algebra and realizations." Journal of Physics A: Mathematical and Theoretical 45, no. 13 (March 20, 2012): 135208. http://dx.doi.org/10.1088/1751-8113/45/13/135208.

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49

Gull, Stephen, Anthony Lasenby, and Chris Doran. "Electron paths, tunnelling, and diffraction in the spacetime algebra." Foundations of Physics 23, no. 10 (October 1993): 1329–56. http://dx.doi.org/10.1007/bf01883782.

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50

PIERCE, DAVID M. "AN EXTENDED N=1 FERMIONIC SUPERCURRENT AND ITS ASSOCIATED N=2 SUPERCONFORMAL ALGEBRA." Modern Physics Letters A 10, no. 38 (December 14, 1995): 2967–77. http://dx.doi.org/10.1142/s0217732395003100.

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An extended free fermionic construction of the internal N=1 worldsheet supercurrent for four-dimensional superstring theory is given. We show how it can describe theories with massless fermions, and we discuss the corresponding N=2 superconformal algebra. As an intermediate step, we show that an internal N=2 global superconformal invariance occurs in any superstring theory with massless fermions at tree level. To demonstrate this fact, we give the N=2 supercurrents for a model with N=1 spacetime supersymmetry and a model without spacetime supersymmetry.

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