V. V. Varlamov
$P$-, $T$-, $C$-transformations of the Dirac field in the de Sitter space are studied in the framework of an automorphism set of Clifford algebras. Finite group structure of the discrete transformations is elucidated. It is shown that $CPT$ groups of the Dirac field in Minkowski and de Sitter spaces are isomorphic.
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A group structure of the discrete transformations (parity, time reversal and charge conjugation) for spinor field in de Sitter space are studied in terms of extraspecial finite groups. Two $CPT$ groups are introduced, the first group from an analysis of the de Sitter-Dirac wave equation for spinor field, and the second group from a purely algebraic approach based on the automorphism set of Clifford algebras. It is shown that both groups are isomorphic to each other.
$CPT$ groups for spinor fields in de Sitter and anti-de Sitter spaces are defined in the framework of automorphism groups of Clifford algebras. It is shown that de Sitter spaces with mutually opposite signatures correspond to Clifford algebras with different algebraic structure that induces an essential difference of $CPT$ groups associated with these spaces. $CPT$ groups for charged particles are considered with respect to phase factors on the various spinor spaces related with real subalgebras of the simple Clifford algebra over the complex field (Dirac algebra). It is shown that $CPT$ group…
We construct the inequivalent irreducible representations (IIR's) of the CPT groups of the Dirac field operator \hatψ and the electromagnetic quantum potential \hat{A}_μ. The results are valid both for free and interacting (QED) fields. Also, and for the sake of completeness, we construct the IIR's of the CPT group of the Dirac equation.
$CPT$ groups of higher spin fields are defined in the framework of automorphism groups of Clifford algebras associated with the complex representations of the proper orthochronous Lorentz group. Higher spin fields are understood as the fields on the Poincaré group which describe orientable (extended) objects. A general method for construction of $CPT$ groups of the fields of any spin is given. $CPT$ groups of the fields of spin-1/2, spin-1 and spin-3/2 are considered in detail. $CPT$ groups of the fields of tensor type are discussed. It is shown that tensor fields correspond to particles of th…
We find out that both the matrix and the operator CPT groups of the spin-3/2 field (with or without mass) are respectively isomorphic to $D_4\rtimes\mathbb{Z}_2$ and $Q\times\mathbb{Z}_2$. These groups are exactly the same groups as for the Dirac field, though there is no a priori reason why they should coincide.
In quantum information context, the groups generated by Pauli spin matrices, and Dirac gamma matrices, are known as the single qubit Pauli group P, and two-qubit Pauli group P2, respectively. It has been found [M. Socolovsky, Int. J. Theor. Phys. 43, 1941 (2004)] that the CPT group of the Dirac equation is isomorphic to P. One introduces a two-qubit entangling orthogonal matrix S basically related to the CPT symmetry. With the aid of the two-qubit swap gate, the S matrix allows the generation of the three-qubit real Clifford group and, with the aid of the Toffoli gate, the Weyl group W(E8) is …
An algebraic description of basic discrete symmetries (space inversion P, time reversal T, charge conjugation C and their combinations PT, CP, CT, CPT) is studied. Discrete subgroups {1,P,T,PT} of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. The fundamental automorphisms form a finite group of order 4. The charge conjugation is represented by a pseudoautomorphism of the Clifford algebra. Such a description allows one to extend the automorphism group. It is shown that an extend…
Universal coverings of the orthogonal groups and their extensions are studied in terms of Clifford-Lipschitz groups. An algebraic description of basic discrete symmetries (space inversion $P$, time reversal $T$, charge conjugation $C$ and their combinations $PT$, $CP$, $CT$, $CPT$) is given. Discrete subgroups $\{1, P, T, PT\}$ of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. The fundamental automorphisms form a finite group of order 4. The charge conjugation is represented by …
Using the standard representation of the Dirac equation we show that, up to signs, there exist only TWO SETS of consistent solutions for the matrices of charge conjugation (C), parity (P), and time reversal (T). In both cases, P^2=-1, and then two succesive applications of the parity transformation to spin 1/2 fields NECESSARILY amounts to a 2πrotation. Each of these sets generates a non abelian group of sixteen elements, G_1 and G_2, which are non isomorphic subgroups of the Dirac algebra, which, being a Clifford algebra, gives a geometric nature to the generators, in particular to C. It turn…
Matrix elements and spherical functions of irreducible representations of the de Sitter group are studied on the various homogeneous spaces of this group.
de Sitter Relativity Group
本文基于作者近期合著专著的框架，简述了 $1+4$ 维 de Sitter (dS$_4$) 群及其 Lie 代数与（余）伴随轨道，指出后者可作为 dS$_4$ 框架下的经典初等系统。
We study the explicit form of Poincaré and discrete transformations of flavor states in a two-flavor scalar model, which represents the simplest example of the field mixing. Because of the particular form of the flavor vacuum condensate, we find that the aforementioned symmetries are spontaneously broken. The ensuing vacuum stability group is identified with the Euclidean group E(3). With the help of Fabri-Picasso theorem, we show that flavor vacua with different time labels and in different Lorentz frames are unitarily inequivalent to each other and they constitute a manifold of zero-flavor-c…
This work is intended to investigate the geometry of anti-de Sitter spacetime (AdS), from the point of view of the Laplacian Comparison Theorem (LCT), and to give another description of the hyperbolical embedding standard formalism of the de Sitter and anti-de Sitter spacetimes in a pseudo-Euclidean spacetime. It is shown how to reproduce some geometrical properties of AdS, from the LCT in AdS, choosing suitable functions that satisfy basic properties of Riemannian geometry. We also introduce and discuss the well-known embedding of a 4-sphere and a 4-hyperboloid in a 5-dimensional pseudo-Eucli…
Matrix elements and spherical functions of irreducible representations of the de Sitter group are studied on the various homogeneous spaces of this group. It is shown that a universal covering of the de Sitter group gives rise to quaternion Euler angles. An explicit form of Casimir and Laplace-Beltrami operators on the homogeneous spaces is given. Different expressions of the matrix elements and spherical functions are given in terms of multiple hypergeometric functions both for finite-dimensional and unitary representations of the principal series of the de Sitter group.
We show that the induced representations of the de Sitter isometry group proposed many years ago by Nachtmann are equivalent to those derived from our general theory of external symmetry. These methods complete each other leading to a coherent theory of covariant fields with spin on the de Sitter spacetime. Some technical details of these representations are presented here for the first time.
We study the convergence of inner products of free fields over the homogeneous spaces of the de Sitter group and show that the convergence of inner products in the of $N$-particle states is defined by the asymptotic behavior of hypergeometric functions. We calculate the inner product for two-particle states on the four-dimensional hyperboloid in detail.
We show that the CPT groups of QED emerge naturally from the PT and P (or T) subgroups of the Lorentz group. We also find relationships between these discrete groups and continuous groups, like the connected Lorentz and Poincaré groups and their universal coverings.
Curved momentum spaces associated to the $κ$-deformation of the (3+1) de Sitter and Anti-de Sitter algebras are constructed as orbits of suitable actions of the dual Poisson-Lie group associated to the $κ$-deformation with non-vanishing cosmological constant. The $κ$-de Sitter and $κ$-Anti-de Sitter curved momentum spaces are separately analysed, and they turn out to be, respectively, half of the (6+1)-dimensional de Sitter space and half of a space with $SO(4,4)$ invariance. Such spaces are made of the momenta associated to spacetime translations and the "hyperbolic" momenta associated to boo…
We show that the CPT group of the Dirac field emerges naturally from the PT and P (or T) subgroups of the Lorentz group.
Quantum field theories on de Sitter spacetime with global U(1) gauge symmetry are deformed using the joint action of the internal symmetry group and a one-parameter group of boosts. The resulting theory turns out to be wedge-local and non-isomorphic to the initial one for a class of theories, including the free charged Dirac field. The properties of deformed models coming from inclusions of CAR-algebras are studied in detail.
We present a covariant quantization of the free "massive" spin-3/2 fields in four-dimensional de Sitter space-time based on analyticity in the complexified pseudo-Riemannian manifold. The field equation is obtained as an eigenvalue equation of the Casimir operator of the de Sitter group. The solutions are calculated in terms of coordinate-independent de Sitter plane-waves in tube domains and the null curvature limit is discussed. We give the group theoretical content of the field equation. The Wightman two-point function $S^{i \bar j}_{αα'}(x,x')$ is calculated. We introduce the spinor-vector …
This paper is a continuation and elaboration of our work quant-ph/0206057 (Nucl. Phys. B, 1968, 7, 79) where some approach to the variable-mass problem were proposed. Here we have found a concret realization of irreducible representations of the inhomogeneous group P(1,n) - the group of translations and rotations in (1+n)-dimensional Minkowski space in two classes (when P_0^2-P_k^2>0 and P_0^2-P_k^2<0). All the P(1,n)-invariant equations of the Schrodinger-Foldy type are written down. Some questions of a physical interpretation of the quantum, mechanical scheme based on the inhomogeneous de Si…
The de Sitter group and its representations: a window on the notion of de Sitterian elementary systems
本文回顾了de Sitter时空中基本系统的构造，基于dS群的酉不可约表示，探讨了从经典到量子的过渡及热力学解释，并构建了与Minkowski表述平行的一致量子场论。
We consider the Lie group $\mathbb{R}^D_κ$ generated by the Lie algebra of $κ$-Minkowski space. Imposing the invariance of the metric under the pull-back of diffeomorphisms induced by right translations in the group, we show that a unique right invariant metric is associated with $\mathbb{R}^D_κ$. This metric coincides with the metric of de Sitter space-time. We analyze the structure of unitary representations of the group $\mathbb{R}^D_κ$ relevant for the realization of the non-commutative $κ$-Minkowski space by embedding into $(2D-1)$-dimensional Heisenberg algebra. Using a suitable set of g…
We classified in arXiv:2201.04152 certain 10d supergravity solutions with a 4d de Sitter, Minkowski or anti-de Sitter spacetime. We then found new solutions in previously unexplored classes. In this paper we study their properties, compare them to swampland conjectures, and make new observations. Using new numerical tools, we first identify all Lie algebras underlying the 6d group manifolds, allowing us to discuss their compactness. We then investigate scale separation, and prove related no-go theorems. Last but not least, we automatize and analyze the stability of all solutions. This leads us…
An analogue of the Newton-Wigner position operator is defined for a massive neutral scalar field in de Sitter space. The one-particle subspace of the theory, consisting of positive-energy solutions of the Klein-Gordon equation selected by the Hadamard condition, is identified with an irreducible representation of de Sitter group. Postulates of localizability analogous to those written by Wightman for fields in Minkowski space are formulated on it, and a unique solution is shown to exist. A simple expression for the time-evolution of the operator is presented.
This paper is a continuation and elaboration of our brief notice quant-ph/0206057 (Nucl. Phys. B, 1968, 7, 79) where some approach to the variable-mass problem was proposed. Here we have found a definite realization of irreducible representations of the inhomogeneous group P(1,n), the group of translations and rotations in (1+n)-dimensional Minkowski space, in two classes (when P_0^2-P_k^2>0 and P_0^2-P_k^2<0). All P(1,n)-invariant equations of the Schrodinger-Foldy type are written down. Some equations of physical interpretation of the quantal scheme based on the inhomogeneous de Sitter group…
Experimental evidences and theoretical motivations lead to consider the curved space-time relativity based on the de Sitter group $SO_0(1,4)$ or $Sp(2,2)$ as an appealing substitute to the flat space-time Poincare relativity. Quantum elementary systems are then associated to unitary irreducible representations of that simple Lie group. At the lowest limit of the discrete series lies a remarkable family of scalar representations involving Krein structures and related undecomposable representation cohomology which deserves to be thoroughly studied in view of quantization of the corresponding car…
We present a simple group representation analysis of massive, and particularly ``partially massless'', fields of arbitrary spin in de Sitter spaces of any dimension. The method uses bulk to boundary propagators to relate these fields to Euclidean conformal ones at one dimension lower. These results are then used to revisit an old question: can a consistent de Sitter supergravity be constructed, at least within its intrinsic horizon?
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