Orbit method for quantum corner symmetries

Irreducible representations of the staggered fermion symmetry group

Program for assigning molecular orbitals to irreducible representations of symmetry groups

We give a general bosonic construction of oscillator-like unitary irreducible representations (UIR) of non-compact groups whose coset spaces with respect to their maximal compact subgroups are Hermitian symmetric. With the exception of E7(7), they include all the non-compact invariance groups of extended supergravity theories in four dimensions. These representations have the remarkable property that each UIR is uniquely determined by an irreducible representation of the maximal compact subgroup. We study the connection between our construction, the Hermitian symmetric spaces and the Tits-Koecher construction of the Lie algebras of corresponding groups. We then give the bosonic construction of the Lie algebra ofE 7(7) in SU(8), SO(8) and U(7) bases and study its properties. Application of our method toE 7(7) leads to reducible unitary representations.

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+.

A symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg commutation relations that are fundamental to quantum mechanics must be valid in all of these physical states. This paper shows that the maximal quantum symmetry group, whose projective representations preserve the Heisenberg commutation relations in this manner, is the inhomogeneous symplectic group. The projective representations are equivalent to the unitary representations of the central extension of the inhomogeneous symplectic group. This centrally extended group is the semidirect product of the cover of the symplectic group and the Weyl-Heisenberg group. Its unitary irreducible representations are computed explicitly using the Mackey representation theorems for semidirect product groups.

Weyl–Pedersen calculus for some semidirect products of nilpotent Lie groups

We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup P 0 by a one-parameter group ℝ*+ = {r: r > 0} of automorphisms of P 0. This construction is determined by a faithful unitary representation of P 0 (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as r → 0. We apply this construction to the current groups of maximal parabolic subgroups in the groups of motions of the n-dimensional real and complex Lobachevsky spaces. The obtained representations of the current groups of parabolic subgroups uniquely extend to the groups of currents with values in the groups O(n, 1) and U(n, 1). This gives a new description of the representations, constructed in the 1970s and realized in the Fock space, of the current groups of the latter groups. The key role in our construction is played by the so-called special representation of the parabolic subgroup P and a remarkable σ-finite measure (Lebesgue measure) ℒ on the space of distributions.

A general theory of a unified construction of the oscillator-like unitary irreducible representations (UIR) of non-compact groups and supergroups is presented. Particle state as well as coherent state bases for these UIRs are given and the case of SU(m,p/n+q) is treated in detail. Applications of this theory to the construction of unitary representations of non-compact groups and supergroups of extended supergravity theories, with particular emphasis on E7(7) and OSp(8/4,IR) are also discussed.

We use the method of group contractions to relate wavelet analysis and Gabor analysis. Wavelet analysis is associated with unitary irreducible representations of the affine group while the Gabor analysis is associated with unitary irreducible representations of the Heisenberg group. We obtain unitary irreducible representations of the Heisenberg group as contractions of representations of the extended affine group. Furthermore, we use these contractions to relate the two analyses, namely, we contract coherent states, resolutions of the identity, and tight frames. In order to obtain the standard Gabor frame, we construct a family of time localized wavelet frames that contract to that Gabor frame. Starting from a standard wavelet frame, we construct a family of frequency localized wavelet frames that contract to a nonstandard Gabor frame. In particular, we deform Gabor frames to wavelet frames.

Unitary cocycle representations of the Galilean line group: Quantum mechanical principle of equivalence

The concept of a unitary representation(UR) of a super Lie group is formulated via super Harish-Chandra pairs. For super semidirectproducts the classical Wigner-Mackey theory of little groups works perfectly in the supersymmetric setting, and leads to a description of all of their unitary irreducible representations (UIR). The Clifford structure of the representations and the concept of super multiplets all make sense in the general context of super semidirect products, which includes all cases studied by the physicists, and leads to many of their major predictions: multiplet structure (both for minimal and extended supersymmetry), and the famous susy partners.

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.

We introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero. The theory is based on Kirillov's method of orbits for nilpotent groups over local fields. When the semisimple group is a classical group, we prove that the new theory is essentially equivalent to Howe's theory of N-rank (see [Ho4], [L2], [Sc]). Therefore our results provide a systematic generalization of the notion of a small representation (in the sense of Howe) to exceptional groups. However, unlike previous works that used ad hoc methods to study different types of classical groups (and some exceptional ones; see [We], [LS]), our definition is simultaneously applicable to both classical and exceptional groups. The most important result of this article is a general “purity” result for unitary representations which demonstrates how similar partial results in these authors' works should be formulated and proved for an arbitrary semisimple group in the language of Kirillov's theory. The purity result is a crucial step toward studying small representations of exceptional groups. New results concerning small unitary representations of exceptional groups will be published in a forthcoming paper [S]

Unitary continuous projective representations of symmetry groups operating in Hilbert spaces of one-particle wave functions are studied from the point of view of physical equivalence, which does not always agree with projective equivalence. Global and local equivalence between projective representations is defined and the local equivalence classes within a given global equivalence class are determined from group-theoretical properties, especially the exponents of the symmetry group. Examples are given of cosmological symmetry groups describing one-dimensional models of universes, where a free particle respectively a particle in a uniform external field are described by projectively equivalent representations of the symmetry group, which are globally but not locally equivalent.

The use of a method based on the Clebsch-Gordan reduction in terms of typical variables is described for the solution of three important problems of crystal physics and of the structural phase transition theory: 1. decomposition of any tensor into bases of irreducible representations of symmetry group, 2. construction of thermodynamic potential of given symmetry in various variables, 3. construction of gradient invariants which are used in the theory of incommensurate structures. The method is elucidated on an example of crystallographic point groupD4 and on the family of its isomorphs. On the basis of a table of Clebsch-Gordan products for the groupD4 there are given: 1. tables of tensorial bases up to fourth rank tensors for the groupsD4,C4v, andD2d, 2. the so-called “extended integrity basis” for the groupD4 in components of polarization and of strain tensor, 3. the so-called “typical extended integrity basis” for the isomorphic familyD4 On the basis of which any extended integrity basis for groups of the familyD4 and in any set of variables can be deduced, 4. the so-called Lifshits-type invariants and quadratic gradient invariants for all magnetic groups of the Laue classD4.