Showing papers on "Unitary state published in 1969"

Singular Integrals and the Principal Series, IV

Abstract: The intertwining operators that have been constructed for all the series of unitary representations appearing in the Plancherel formula of a connected real semisimple Lie group of matrices are given a new normalization and then applied in two ways. The first is to obtain dimension formulas for the commuting algebras of the unitary representations in question. The second is to establish the existence of complementary series. These bear the same relationship to the unitary representations under study that the complementary series found earlier bear to the principal series.

Unitary closed loops in reggeized feynman theory.

Abstract: We use the factorized tree graphs of the generalized Veneziano model to construct, through the tree theorem, all graphs with one closed loop. While unitary in the sense of perturbation theory, the loops contain a new "particle counting" divergence related to the very large number of daughters in the trees.

Combinatorial Structure of State Vectors in Un. I. Hook Patterns for Maximal and Semimaximal States in Un

Abstract: It is shown that, in the boson‐operator realization, the state vectors of the unitary groups Un—in the canonical chain Un⊃Un−1⊃⋯⊃U1—can be obtained ab initio by a combinatorial probabilistic method. From the Weyl branching law, a general state vector in Un is uniquely specified in the canonical chain; the algebraic determination of such a general state vector is in principle known (Cartan‐Main theorem) from the state vector of highest weight; the explicit procedure is a generalization of the SU(2) lowering‐operator technique. The present combinatorial method gives the normalization of these state vectors in terms of a new generalization of the combinatorial entity, the Nakayama hook, which generalization arises ab initio from a probabilistic argument in a natural way in the lowering procedure. It is the advantage of our general hook concept that it recasts those known algebraic results into a most economical algorithm which clarifies the structure of the boson‐operator realization of the Un representations.

Canonical Transformations and Spectra of Quantum Operators

Abstract: The claim that unitary transformations in quantum mechanics correspond to the canonical transformations of classical mechanics is not correct. The spectra of operators produced by unitary transformation of Cartesian coordinate and momentum operators (q, k) are necessarily continuous over the entire real domain of their eigenvalues. Operators with spectra which are not everywhere continuous are generated from (q, k) by one‐sided unitary transformations U for which U†U = 1 but for which UU† commutes with either q or k (but not both). If UU† commutes with k, the new coordinates and momenta (r, s) satisfy commutation relations [sm, rn] = 2πi1δm,n, [sm, sn] = 0, but [rm, rn] ≠ 0; (r, s) are canonical only for one‐dimensional systems. The properties of one‐sided unitary transformations are described; they are characterized by φ(K), the eigenvalue of UU†. The one‐dimensional case for which the one‐sided unitary transformation is canonical is discussed in detail. A prescription is given for obtaining the operator...

Products of hermitian matrices and symmetries

Abstract: It is the main purpose of this note to prove that every complex matrix with real determinant is the product of four hermitian matrices; Theorem 2 is an actually stronger result. Every real square matrix is the product of two real hermitian matrices [1]; this is a special case of our Theorem 1 which is of interest in itself, if it is indeed new. Theorem 3 was motivated by a theorem of Halmos and Kakutani [3 ] who proved that every unitary operator on an infinitedimensional Hilbert space is the product of four symmetries (i.e., operators that are hermitian and unitary). We also show that the number of factors in these results cannot be reduced in general.

Representation theory of Sp/4/ and SO/5/

Abstract: The basis states for all the irreducible unitary representations of Sp(4) are constructed by means of a calculus of boson operators. The Gel'fand states are explicitly expanded in terms of their constituent Weyl patterns. In terms of these states the matrix elements of finite rotations in five dimensions are obtained.

Remarks on Poincaré generators, unitary transformations and helicity representations

Abstract: Coester and Chakrabarti have proposed independently unitary transformations diagonalizing the helicity operator and connecting the [m, s] and [0,s] I.U.R. ofP+↑ through the limiting processm→0. The unitary transformation giving theexplicit equivalence of these contributions is calculated. When used for defining a helicity basis for [m, s] I.U.R., the Chakrabarti transformation is found to be convenient to obtain the associated transformation laws under infinitesimal Lorentz transformations. The corresponding transformation laws form=0 follow directly in agreement with Fronsdal's results.

Finite Transformations and Basis States of SU(n)

Abstract: A convenient parameterization for finite transformations of SU(n) is developed which explicitly exhibits the special unitary subgroups. This is also used to parameterize the defining space. Higher‐dimensional representations are discussed. The question of which representations carry the trivial representation of SU(n − 1) is considered, as well as the parameterization of these states. Application is made to the transformations and basis states of SU(3).

On perturbations of the spectrum of unitary operators

Abstract: We study the stability of the essential spectrum of unitary operators under nuclear and compact perturbations.

Classical Fields on Spacelike Mass Shells

Abstract: We define classical fields, corresponding to unitary representations of the inhomogeneous Lorentz group with M 2 < 0, which belong to the discrete series. These fields satisfy Bargmann‐Wigner equations which are given in explicit matrix form.