Relativistic implications of the quantum phase

Noninertial transformations on time–position–momentum–energy space {t, q, p, e} with invariant Born–Green metric and the symplectic metric −de ∧ dt + dp ∧ dq are studied. This group of transformations contains the Lorentz group as the inertial special case and, in the limit of small forces and velocities, reduces to the expected Hamilton transformations leaving invariant the symplectic metric and the nonrelativistic line element ds2 = −dt2. The transformations bound relative velocities by c and relative forces by b. Spacetime is no longer an invariant subspace but is relative to noninertial observer frames. In the limit of b → ∞, spacetime is invariant. Born was lead to the metric by a concept of reciprocity between position and momentum degrees of freedom and for this reason we call this reciprocal relativity. For large b, such effects will almost certainly only manifest in a quantum regime. Wigner showed that special relativistic quantum mechanics follows from the projective representations of the inhomogeneous Lorentz group. Projective representations of a Lie group are equivalent to the unitary representations of its central extension. The same method of projective representations for the inhomogeneous group is used to define the quantum theory in the noninertial case. The central extension of the inhomogeneous group is the cover of the quaplectic group . is the Weyl–Heisenberg group. The group, and the associated Heisenberg commutation relations central to quantum mechanics, results directly from requiring projective representations. A set of second-order wave equations result from the representations of the Casimir operators.

We consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recovering for the phase-retrieval problem. For a given multiplier $$\mu $$ of a finite abelian group G, we show that the representation dimension of any irreducible $$\mu $$ -projective representation of G is exactly the rank of the symmetric multiplier matrix associated with $$\mu $$ . With the help of this result we are able to prove that every irreducible $$\mu $$ -projective representation of a finite abelian group G admits a frame vector with the maximal span property, and obtain a complete characterization for all such frame vectors. Consequently the complement of the set of all the maximal span frame vectors for any projective unitary representation of any finite abelian group is Zariski-closed. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements.

This is a review paper on the generalization of Euclidean as well as pseudo-Euclidean groups of interest in quantum mechanics. The Weyl–Heisenberg groups, Hn, together with the Euclidean, En, and pseudo-Euclidean Ep,q, groups are two families of groups with a particular interest due to their applications in quantum physics. In the present manuscript, we show that, together, they give rise to a more general family of groups, Kp,q, that contain Hp,q and Ep,q as subgroups. It is noteworthy that properties such as self-similarity and invariance with respect to the orientation of the axes are properly included in the structure of Kp,q. We construct generalized Hermite functions on multidimensional spaces, which serve as orthogonal bases of Hilbert spaces supporting unitary irreducible representations of groups of the type Kp,q. By extending these Hilbert spaces, we obtain representations of Kp,q on rigged Hilbert spaces (Gelfand triplets). We study the transformation laws of these generalized Hermite functions under Fourier transform.

We initiate a study of infinite tensor products of projective unitary representations of a discrete group G. Special attention is given to regular representations twisted by 2-cocycles and to projective representations associated with CCR-representations of bilinear maps.Detailed computations are presented in the case where G is a finitely generated free abelian group.We also discuss an extension problem about product type actions of G, where the projective representation theory of G plays a central role.

With the Bondi‐Metzner‐Sachs (BMS) group in general relativity as the main motivation and example, a theorem is proved which may be described as follows. Let G be a complex semisimple, connected and simply connected Lie group with compact real form K , and let A be a metrizable, complete and locally convex real topological vector space on which there is a continuous G action. Consider the semidirect product topological group Gx s A (which is, in general, infinite‐dimensional) constructed naturally out of G and A . If the set of equivalence classes of irreducible representations of K in A satisfies certain hypotheses, then the second cohomology group of G x s A in the sense of continuous group cohomology is trivial. When G = SL (2, C ) and A is an appropriate function space of real‐valued functions of the 2‐sphere endowed with a specific G action (e.g. A may consist of C k , k ⩾ 3, real‐valued functions defined on the 2‐sphere), the semidirect product group is the universal cover of the BMS group. The theorem implies the existence of lifting of the projective unitary representations of the BMS group to the linear unitary representations of its universal cover. In the quantum context when we consider massless quantum fields at null infinity of a non‐stationary, asymptotically Minkowskian space‐time, in place of the projective unitary representations of the BMS group, there is no loss of generality in considering the linear unitary representations of its universal cover instead.

It is shown that every continuous unitary projective representation of a Polish group can be lifted to a Borel multiplier representation (i.e., to a representation “up to a Borel factor”) and that this, in turn, can be derived from a continuous (ordinary) representation of a Polish group obtained from a central topological extension of the group considered by the multiplicative group of all complex numbers of absolute value 1. One determines the factors of all Borel multiplier representations of the Bondi-Metzner-Sacks group when the subgroup of “supertranslations” is the additive group of a separable real Hilbert space.

In this chapter, we establish some basic facts on Clifford theory for induced projective representations. We follow quite closely the exposition for the case of ordinary representations in Chap. 2 and we give the first applications of the theory. Another fundamental application is in Sect. 10.2 on irreducible projective representations of finite Abelian groups.

Thermal fluctuations in time-dependent quantum processes are treated by a constant-temperature generalization of Wigner's formulation of quantum mechanics in phase space. To this end, quantum Nosè–Hoover dynamics is defined by generalizing the Moyal bracket. Computational applications of the formalism, together with further theoretical developments, are discussed.

We extend the discussion of projective group representations in quaternionic Hilbert space that was given in our recent book. The associativity condition for quaternionic projective representations is formulated in terms of unitary operators and then analyzed in terms of their generator structure. The multi-centrality and centrality assumptions are also analyzed in generator terms, and implications of this analysis are discussed.

An eigenfunction method is applied to reduce the regular projective representations (Reps) of finite groups to obtain their irreducible projective Reps. Anti-unitary groups are treated specially, where the decoupled factor systems and modified Schur’s lemma are introduced. We discuss the applications of irreducible Reps in many-body physics. It is shown that in symmetry protected topological phases, geometric defects or symmetry defects may carry projective Rep of the symmetry group; while in symmetry enriched topological phases, intrinsic excitations (such as spinons or visons) may carry projective Rep of the symmetry group. We also discuss the applications of projective Reps in problems related to spectrum degeneracy, such as in search of models without sign problem in quantum Monte Carlo simulations.

Kostant’s quantization procedure is applied for constructing irreducible projective representations of a solvable Lie group from symplectic homogeneous spaces on which the group acts. When specialized to a certain class of such groups, including the exponential ones, the technique exposed in the present paper provides a complete parametrization of all irreducible projective representations.

In this paper, we introduce the concepts of rough projective representations of rough groups and discuss their relations with rough 2-cocycle. In particular, we deduce that any rough projective representation associated with a rough multiplier is equivalent to a rough representation if and only if a rough multiplier is a rough coboundary.

On Schur multiplier and projective representations of Heisenberg groups

After a detailed investigation of some fundamental general orthogonal symmetry properties of incommensurate condensed systems, the line groups are applied for describing the structure of such modulated crystals. It is demonstrated, that use of projective representations of these groups may give a more refined description of such structures and significantly extends applicability of the diffuse scattering formalism.

This paper continues the study of the nature and interdependence of the axioms of relativistic field theory; attention is focused on the notion of relativistic invariance. The central result of the present paper is the derivation of necessary and sufficient conditions for a representation of the covariant free field to admit a unitary representation of the inhomogeneous Lorentz group associated with the field operator. It is shown that only the standard Fock-Cook representation has this property. The relevance of the requirement that the Lorentz group is represented by a unitary family associated with the field operator is exhibited by an analysis of the covariant representations of Shale and Segal. These representations involve extremal states which are not pure, and group representations by intertwining operators.