On Infinite Tensor Products of Projective Unitary Representations

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.

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.

The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics.We put aside the relativistic symmetry and show that the maximal quantum symmetry that leaves the Heisenberg commutation relations invariant is the projective representations of the conformally scaled inhomogeneous symplectic group. The Weyl-Heisenberg group and noncommutative structure arises directly because the quantum phase requires projective representations.We then consider the relativistic implications of the quantum phase that lead to the Born line element and the projective representations of an inhomogeneous unitary group that defines a noninertial quantum theory. (Understanding noninertial quantum mechanics is a prelude to understanding quantum gravity.) The remarkable properties of this symmetry and its limits are studied.

The Mackey little group procedure is used to classify the projective irreducible unitary representations of the group SU(2)ΛR4. Whereas an arbitrary ordinary representation of this group plays the role of both spectral and dynamical group for the three-dimensional hydrogen atom, it is found that one of its projective representations acts as spectral group for the two-dimensional isotropic oscillator. The projective representations of the Lie algebra of SU(2)ΛR4 are treated as ordinary representations of the central extension of this Lie algebra, and shift operator techniques used to obtain the matrix elements of its representatives. The connection with the isotropic oscillator is shown to be related to the realizability of the extended Lie algebra by two-boson creation and annihilation operators.

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 estimation of unknown unitary operation when the set of possible unitary operations is given by a projective unitary representation of a compact group. We show that neither indefinite causal order strategy nor adaptive strategy improves the performance of this estimation when error function satisfies group covariance. That is, the optimal parallel strategy gives the optimal performance even under indefinite causal order strategy and adaptive strategy. To study this problem, we newly introduce the concept of generalized positive operator valued measure (GPOVM), and its convariance condition. Using these concepts, we show the above statement.

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.

Previous discussions of the bases and projection operators for projective representations are analyzed and alternatives are proposed. Detailed proofs are provided for a result which is often assumed or proved within unacceptable limitations, namely that the form of the projection operators which is standard for vector representations is also valid for unitary projective representations. These proofs provide necessary conditions for this result, and they are constructed in terms of the definition given for the bases of projective representations. The calculation of Clebsch–Gordan coefficients for projective representations by means of the projection operators is discussed. Whereas in the method of Dirl the work is carried out entirely in terms of the matrix representations, and the symmetrization of the bases has to be considered in a second step, all the work of this paper is conducted starting from the symmetrization of the bases of the projective representations, so that those two steps are carried out simultaneously.

1. Let Γn be the representation group or spin group (4, 9) of Sn. Then the irreducible representations of Γn are of two distinct types. These are (a) ordinary representations, which are the irreducible representations of the symmetric group and (b) spin or projective representations. Corresponding to every partition (λ) = (λ1, λ2, …, λm) of n with λ1 > λ2 > … > λm > 0 there is an irreducible spin representation 〈λ〉 of Γn.

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.

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.

We answer the questions, "What does Chern-Simons theory assign to a point?" and "What kind of mathematical object does Chern-Simons theory assign to a point?" Our answer to the first question is representations of the based loop group. More precisely, we identify a certain class of projective unitary representations of the based loop group [Formula: see text] We define the fusion product of such representations, and we prove that, modulo certain conjectures, the Drinfel'd center of that representation category of [Formula: see text] is equivalent to the category of positive energy representations of the free loop group [Formula: see text]† The abovementioned conjectures are known to hold when the gauge group is abelian or of type [Formula: see text] Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: They are tensor categories that are equivalent to their bicommutant inside [Formula: see text], the category of bimodules over a hyperfinite [Formula: see text] factor. We prove that, modulo certain conjectures, the category of representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type [Formula: see text].

q-Analogues of regularisation theorems for linear and projective representations of the symmetric group

We show that for an odd prime r>3 and an integer g>1, in the projective representation given by the SO(3) Witten-Reshitikhin-Turaev theory at an rth root of unity, the image of the mapping class group of a surface of genus g is dense.

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.