Nonunitary Representations Research Articles

Multiplicity-free, unitary and nonunitary irreducible representations of SL(3,R)

Multiplicity-free, irreducible representations of the group ∼(SL(3,R)) are obtained from SU(2) subgroup representations by a constructive method. It is observed that there exist two series of unitary representations with k contents {k0,k0+1,k0+2,...}k0≥3, {k0,k0+2,k0+4,...} k0=0, 1, 1/2 and finite-dimensional representations with k content {k0+1,k0+3,...,k0+2n+1} k0=1, 1/2 {2,4,6,...,2n}, n=1,2,...,k0=0.

Representations of the Poincaré group associated with unstable particles

The problem of a relativistically covariant description of unstable particles is reexamined. We follow the approach which associates a unitary reducible representation of the Poincar\'e group with a larger isolated system, and compare it with the one ascribing a nonunitary irreducible representation to the unstable particle alone. It is shown that the problem originates in the choice of the subspace ${\mathcal{H}}_{u}$ of the state Hilbert space which could be related to the unstable particle. Translational invariance of ${\mathcal{H}}_{u}$ is proved to be incompatible with unitarity of the boosts. Further, we propose a concrete choice of ${\mathcal{H}}_{u}$ and argue that in most cases of the actual experimental arrangements this subspace is effectively one dimensional. A correct slow-down for decay of the moving particles is obtained.

Microscopic foundation for the particle-vibrator model

The tools for deriving microscopically the particle-vibrator model are presented in the form of a new fermion-boson expansion in which valence particles or holes added to a closed shell are treated essentially as fermion-degrees of freedom, while particle-hole excitations of the closed shell are treated as bosons. The Pauli principle is completely fulfilled without redundancy. Both the nonunitary (Dyson) and unitary representations are given.

Intertwining operators for semisimple groups, II

The purpose of the present paper is to expand the use of intertwining operators for semisimple Lie groups. In an earlier form (see [20]), these operators were meromorphic continuations of integral operators and exhibited equivalences among nonunitary principal series representations, those induced from a minimal parabolic subgroup. We demonstrated that there was an intimate connection between these operators and the irreducibility of the principal series, on the one hand, and the unitarity of the analytically continued representations (the complementary series), on the other hand. In the intervening years we have generalized the setting for intertwining operators significantly, and we have learned of new applications. For the most part, the expansion in the setting is that minimal parabolic subgroups have now been replaced by arbitrary parabolic subgroups M A N , and the representations that are studied are induced from a representation of M A N that is irreducible unitary on M, is one-dimensional on A, and is trivial on N. The expanded theory allows us to determine the degree of reducibility of all series of representations appearing in the Plancherel formula of the group, and to study complementary series attached to them. Such intertwining operators have also now found major applications in classifying representations (see Langlands [29] and Knapp-Zuckerman [-25]) and appear to have significance for some problems in number theory. Our objective in this paper is threefold: to develop the analytic properties of intertwining operators in what seems to be the appropriate degree of generality, to give a dimension formula for the commuting algebras of the unitary representations induced when the representation of M is in the discrete series and the character of A is unitary (and to give some further insights into these representations), and to illustrate a technique for dealing with complementary series. To make it possible to be more specific, we introduce some notation. Let G be a reductive group whose identity component has compact center; the precise axioms for G are given in w 1. Fix a Cartan involution 0 for G, and let P be a parabolic subgroup of G. Then P c~ OP decomposes into the product of commuting subgroups M and A, where A is a vector group and M satisfies the

Matrix elements for infinitesimal operators of the groups U(p+q) and U(p,q) in a U(p) ×U(q) basis. I

In this article explicit expressions are obtained for the action of the infinitesimal operators of the principal nonunitary series representations of the groups U(p,q) in a U(p) ×U(q) basis. It is moreover shown how the finite dimensional irreducible representations of the group U(p,q) and the group U(p+q) with respect to a U(p) ×U(q) basis are obtained from the principal nonunitary series representations of the group U(p,q).

Representation matrix elements and Clebsch–Gordan coefficients of the semisimple Lie groups

We give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.

Linear representations of any dimensional Lorentz group and computation formulas for their matrix elements

The representation matrix elements of SO(n,1) are discussed in a space spanned by the representation matrix elements of the maximal compact subgroup SO(n). A multiplier of the representation corresponding to the boost of SO(n,1) is completely determined by requiring the commutation relations of SO(n,1) for the differential operators of the multiplier representation and of the parameter group of SO(n). It is shown that the bases of the space, the representation matrix elements of SO(n), are classified by the group chains of the first and the second parameter groups of SO(n), whose differential operators commute with each other, and the characteristic numbers of SO(n,1) are the same as those of the first parameter group of SO(n−1) and a complex number appearing in the multiplier. By using the scalar product defined in the space, the matrix elements for the differential operators and the computation formulas for the representation corresponding to the boost of SO(n,1) are given for all unitary representations of SO(n,1) and useful formulas containing the d matrix elements of SO(n) are obtained. By making use of these results, even for the nonunitary representation of SO(n,1) the matrix elements for the differential operators and the computation formula for the representation corresponding to the boost are obtained by defining the matrix elements with respect to the bases of the space. It is also pointed out that the unitary representations (the complementary series) corresponding to some value of the parameter, which appear in the classification using only the matrix elements of the generators, should not be included in our classification table because of divergence of the normalization integral. The continuation to SO(n+1) and the contraction to ISO(n) from the principal series are discussed.

Nonunitary representations of MH(2) and the groupax+b and Laplace transformation

Nonunitary representations of MH(2) and the groupax+b and Laplace transformation

Space-time operators and the incompatibility of quantum mechanics with both finite-dimensional spinor fields and Lagrangian dynamics in the context of special relativity

It is shown that the present procedure used to quantize relativistic systems is inconsistent. Three mutually supporting arguments are given to sustain this conclusion. First, it is noted that the use of wave functions which transform according to any representation of 0(1, 3) whether finite or infinite dimensional is inappropriate because such a description allows for too many degrees of freedom. The phenomenon of Thomas precession indicates that internal structure such as spin and multipole moments must be described by mass shell (rest system) three-tensors rather than by unconstrained four-tensors. Second, even if representations of 0(1, 3) are employed, the momentum space construction for position-time operators, which is quite general and is applicable in any Euclidean or pseudo-Euclidean space, requires that the infinite-dimensional UIRs of 0(1, 3) be used rather than the finite-dimensional, nonunitary spinor representations. Third, various anomalous features of the customary kinematic formalism can be readily understood provided that this formalism is viewed as anad hoc blend of two other formalisms which, while self-consistent, are incompatible except for the trivial case of free one-particle states. These criticisms focus attention on a number of specific weaknesses of the kinematic foundations of relativistic quantum mechanics and relativistic quantum field theory. These weaknesses are sufficiently serious to require a radical revision of the current theory even at the kinematic level.

A note on nonunitary principal series representations

It is proven in a rather elementary way that any nonunitary principal series representation of a semisimple Lie group G is of finite length, having as a trivial consequence that the set of infinitesimal equivalence classes of quasisimple irreducible representations of G with a given infinitesimal character is finite.

Topological Irreducibility of Nonunitary Representations of Group Extensions

A functional-analytic approach to the study of the topological irreducibility of certain nonunitary induced representations is set forth. The methods contrast, and in some sense, encompass those first initiated by E. Thieleker in [4], and are amenable to complete irreducibility questions as well. Several sufficient conditions for topological irreducibility are established. A sufficient condition for reducibility is also presented—the latter serving to explain an interesting counterexample due to J. M. G. Fell.

Classification of the irreducible representations of semisimple Lie groups

We obtain a classification of the irreducible (nonunitary) representations of a connected semisimple Lie group G, in terms of their restriction to a maximal compact subgroup K of G. (A classification in terms of analytic properties of the representations has been given by R. P. Langlands [(1973), mimeographed notes, Institute for Advanced Study, Princeton, NJ] for linear groups.) We first define a norm on the representations of K: if mu in K, mu is a nonnegative real number. Then if pi in G, mu is called a lowest K-type of pi if mu is minimal among the K-types occurring in pi. We announce a parameterization of the set of representations containing mu as a lowest K-type by the orbits of a finite group acting in a complex vector space (the dual of the vector part of a certain Cartan subgroup of G), and the result that mu necessarily occurs with multiplicity one in such representations.

Mackey's Theorem for Nonunitary Representations

Mackey's Theorem for Nonunitary Representations

Mackey’s theorem for nonunitary representations

We present an analog to the Intertwining Number Theorem of Mackey in a setting which arises naturally in the study of the representation theory of p-adic linear groups.

Topological irreducibility of nonunitary representations of group extensions

A functional-analytic approach to the study of the topological irreducibility of certain nonunitary induced representations is set forth. The methods contrast, and in some sense, encompass those first initiated by E. Thieleker in [4], and are amenable to complete irreducibility questions as well. Several sufficient conditions for topological irreducibility are established. A sufficient condition for reducibility is also presented—the latter serving to explain an interesting counterexample due to J. M. G. Fell.

Interacting quantum fields and the chronometric principle

A form of interaction in quantum field theory is described that is physically intrinsic rather than superimposed via a postulated nonlinearity on a hypothetical free field. It derives from the extension to general symmetries of the distinction basic for the chronometric cosmology between the physical (driving) and the observed energies, together with general precepts of quantum field theory applicable to nonunitary representations. The resulting interacting field is covariant, causal, involves real particle production, and is devoid of nontrivial ultraviolet divergences. Possible physical applications are discussed.

Representations of the universal covering group of SU(1,1) and their bilinear and trilinear invariant forms

The unitary irreducible and many nonunitary representations of the universal covering group of SU(1,1) are given in a realization on certain spaces of functions. We discuss intertwining operators for these representations and their connection with the discrete series. The tensor product decomposition is performed by means of an integral transformation. A completeness relation for these integral kernels is derived.

Indecomposable representations of Poincaré group and quantization of weak gravitational and electromagnetic fields

In this paper a quantization procedure of the electromagnetic and the weak gravitational fields, in an indefinite (but non-negative) metric space of states, is presented. We show that the covariance axiom can only be formulated with non-unitary representations (of the Poincaré group) whose restrictions to the one-particle states are indecomposable. In the gravitational case the corresponding indecomposable representation describes six states of helicity for gravitons. The quantized field appears as six interacting massless fields of helicities 2, −2,1, −1,0 and 0. These six values of helicity correspond to the six modes of polarization in the most general metric theory of gravity. For the electromagnetic field one obtains an indecomposable representation of the Poincaré group with helicities 1, −1 and 0 corresponding to transverse and longitudinal photons.

Time graph of the unstable particle and the nonunitary representations of the Poincaré group

The second-quantization scheme for neutral scalar fields describing an unstable particle is formally developed within the framework of the nonunitary representations of the Poincar\'e group, ${{\overline{\mathcal{P}}}_{+}}^{\ensuremath{\uparrow}}$. The fields satisfy the postulates of conventional field theories with a modified spectral condition. The exponential decay law for the time graph of the unstable particle is derived from a study of the asymptotic behavior of the causal Green's function.

The 2:1 anisotropic oscillator, separation of variables and symmetry group in Bargmann space

We present a detailed analysis of the separation of variables for the time-dependent Schrödinger equation for the anisotropic oscillator with a 2:1 frequency ratio. This reduces essentially to the time-independent one, where the known separability in Cartesian and parabolic coordinates applies. The eigenvalue problem in parabolic coordinates is a multiparameter one which is solved in a simple manner by transforming the system to Bargmann’s Hilbert space. There, the degeneracy space appears as a subspace of homogeneous polynomials which admit unique representations of a solvable symmetry algebra s3 in terms of first order operators. These representations, as well as their conjugate representations, are then integrated to indecomposable finite-dimensional nonunitary representations of the corresponding group S3. It is then shown that the two separable coordinate systems correspond to precisely the two orbits of the factor algebra s3/u (1) [u (1) generated by the Hamiltonian] under the adjoint action of the group. We derive some special function identitites for the new polynomials which occur in parabolic coordinates. The action of S3 induces a nonlinear canocical transformation in phase space which leaves the Hamiltonian invariant. We discuss the differences with previous works which present su (2) as the algebra responsible for the degeneracy of the two-dimensional anisotropic oscillator.