Showing papers on "Unitary state published in 1972"

Representations of the Bondi-Metzner-Sachs Group. I. Determination of the Representations

Abstract: Following a historical introduction, it is suggested that irreducible unitary representations of the Bondi-Metzner-Sachs group may be used to classify elementary particles in a quantum theory which takes ‘asymptotically flat5 gravitational fields into account. The unitary representations of the group induced from irreducible unitary representations of the connected little groups are all determined. It is shown that the connected little groups are all compact, so that the ‘spins’ of the corresponding particles are necessarily discrete, and the wave functions have a finite number of components. Furthermore, the spins are of precisely the observed type. This is in striking contrast to the situation for the Poincare group, for which the spins may be discrete or continuous. (The continuous spin wave functions are infinite-component.) It is concluded that the B.M.S. group may provide an explanation for the observed discreteness of the spins of elementary particles.

Some remarks on the ``null plane development'' of a relativistic quantum field theory

Abstract: We give conditions for the existence of field operators on so-called null planes and discuss some consequences of the necessary restriction of the test function space, concerning Haag's theorem and the possibility of unitary mappings intertwining between free fields of different masses. In the last section we discuss conditions under which a unitary representation of the dilatations in the null plane gives rise to a unitary representation of the dilatations in Minkowski space.

Cultural Regionalism and the Unitary State Idea in Belgium

Abstract: IKE many other contemporary nation-states, Belgium has suffered a decline in national cohesion as the result of a steady growth in the regional consciousness of its citizens. Although the major culture areas of Belgium-Flanders and Wallonia-were unimportant in its early evolution, the political situation is quite different today; for ethnic awareness between the two communal-linguistic regions has reached a point that threatens the viability of the state.1 Moreover, Brussels-Capital, which comprises the nineteen officially bilingual communes of the capital district, has emerged as a third region, with interests that are not compatible with those of either the Flemish or the Walloons (Fig. i). Nonetheless, Belgium remains a unitary state, even though the unitary system, with its high degree of centralized authority, is increasingly less adequate in meeting demands for a decentralization of political power. The challenge to centralized political control has reached worldwide dimensions in recent years. Yet about go percent of the world's sovereign states have a unitary form of government in which the principle of home rule is difficult to accommodate. Belgium, with its long experience in dealing with distinct cultural regionalism, is a crucible in which the government's ability to adjust to an increased level of disaffection with the unitary state is being severely tested. What happens in Belgium will be carefully observed by other states that are attempting to resolve similar problems. This paper identifies and evaluates the forces that have exacerbated regional differences in Belgium and examines the government's attempts to deal with requests for regional autonomy within the framework of a highly centralized bureaucracy. But the question of federalism cannot be ignored. Although the unitary state has existed in Belgium since its founding in 1830, it is by no means inviolate as a modus operandi. In fact, the point has often been made that a de facto fed-

Quantum Field Theories with Shadow States. I. Soluble Models

Abstract: To construct a finite local relativistic quantum field theory we may introduce an indefinite-metric vector space, but then to avoid conflict with unitarity we must consider only a selected subset of states to be physical. The remaining states participate in the dynamics but are not among the complete set of physical states as far as probability interpretation is concerned. These states are called shadow states. The $S$ matrix should be unitary when restricted to the physical states. In this paper we formulate and solve several simple models of field theories with shadow states and demonstrate the manner in which shadow states influence the dynamics and the structure of the scattering amplitude. The choice of a standing-wave boundary condition for the shadow states is shown to be completely consistent with the physical description of the scattering process in terms of wave packets. These methods are adapted to the study of low-energy pion-nucleon scattering in the following paper.

State Vectors with vanishing norm and the probability interpretation

Abstract: After an analysis of why theories with dipole states do not give a unitary physicalS-matrix automatically, a reasonable unitarization procedure is proposed working even for these pathological cases.

Theorem on unitary equivalence of fock representations

Abstract: We prove that two Fock states WJ and WK (not necessarily gauge invariant) on the CAR-algebra are unitarily equivalent if and only if J 2014 K I is a Hilbert-Schmidt operator. We calculate explicitly the norm diff erence ( ~ w J fl. Let (H, s) be a separable Euclidean space and J and K complex structures on (H, s), i. e. Consider the operators and let P = U P ), Q = V Q I be their polar decompositions, Q ), P and U commute with J and K ; consequently the dimension of Ker P is even or infinite; Q is a normal operator, therefore V can be chosen such that V+ =V, V2 -= 1. The same notations as in [1] are used : c1 = c1 (H, s) is the CAR-algebra and 03C9J is any pure quasi-free state on c1; J satisfies : J+ = J, J2 =1. THEOREM 1. Let the operator P be diagonalizable [i. e. «§;);gN orthonormal basis of H such that P d~ = ~~1, R (reals)], then lhere exists a family of subspaces of H invariant under J and K such (i) H = C Hn ; ANN. INST. POINCARE, A-XVI-2 7 88 J. MANUCEAU AND A. VERBEURE (ii) dim Ho and dim H i is even or infinite, dim Hn = 4 for n ~ 2; (iii) P ==~. i.n pn, where Pn H = Hn ; "’;"0 = 2, ~, ~ 1 = 2 and 2 ~tR 2 n for n ~ 2. Proof. Let F = Ker Q ; F and F ~ (orthogonal complement of F for s) are invariant for J and K. (a) Suppose Fl = § 0 ) ; then JK == ~ ~ is unitary and Hermitian, there exists a decomposition F = Ho + Hi such that P = Po + Pi, where Po and Pi are the orthogonal projection operators on Ho respectively Hi, which are invariant under J and K and therefore dim Ho and dim Hi is even or infinite. (b) Suppose F = 0 ~, let Hx be subspaces of H such that PHx = ~x H~. Because [P, J]== [P, K]= 0, the subspaces H:x are invariant for J and K. Remark that P2 + Q+ Q = 4, Q+ Q = Q 12; therefore Q has the same proper subspaces Hx as P ). Let Q Hx = p« Hx, then À~ + [J-~ = 4 for all a. Take any ~t, E H). and consider the subspaces generated by the real orthogonal set V ~)., J ~), J V u~, }. It is clear that H~ is a real subspace invariant under J and K of dimension four. In general H = F + F1 the results of (a) and (b) prove the theorem. Q. E. D. LEMMA. Let xj and 7TK be the Fock representations associated with J respectively K. If 03C0J and 1!K are unitarily equivalent then [J, K]+ has 2 as the only accumulation point of its spectrum. Proof. Let be any infinite orthonormal set of Hand

Spin and unitary spin in the dual resonance model.

Abstract: The recent developments in constructing dual amplitudes with spins and unitary symmetry are briefly reviewed.

A Second Foundation for Quantum Theory

Abstract: As shown in a previous paper 2, the combination of complementarity logic and probability calculus is sufficient to establish the essential features of the general formalism of quantum theory, known as statistical transformation theory. The advantages of this way of founding quantum theory as compared to other procedures lie in the following: (1) The complex-valuedness of the state vectors (unitary metric) is seen to be a direct consequence of complementarity logic.3 (2) The concept of physical quantity is reduced to the more general concept of physical property. Hence it can be proved that the operators representing physical quantities must be hypermaximal. (3) The existence of a Hamiltonian is not required.

Remarkable Connection between the Multi‐Veneziano Integrand and the Character of Special Unitary Groups

Abstract: The remarkable connection that exists between the multi‐Veneziano integrand at a fixed point of the external momenta and the character of special unitary groups is exhibited explicitly. Some remarks are made on such a connection for arbitrary external momenta.

Exact-unitary model calculations of three-particle scattering

Abstract: Publisher Summary The exact-unitary models of Cahill and Kowalski enable the construction of approximate amplitudes, which satisfy the constraints of three-particle unitarity, even above the break-up threshold. The Cahill and Kowalski models have the attractive feature of reducing to a hierarchy of on-shell one-dimensional integral equations and quadratures, even in the case of local two-particle interactions. The model was found to be quite successful in reproducing elastic differential cross-sections. This chapter describes the gross features of the break-up spectra. It presents some typical results for elastic scattering and break-up. The chapter discusses S-wave nucleon–nucleon interactions and the two nucleon scattering information from the Yamaguchi triplet- and singlet-spin separable interactions. The exact-unitarity is a sufficiently powerful constraint upon the elastic channel for the forward peak and all other essential features to be constructed. The unitary model break-up spectra indicate that exact unitarity alone is not sufficient to reproduce the quantitative features of the exact spectra. In particular, it seems that although the Cahill unitary model has an explicit final state interaction term, the non-unitary off-shell 3+3 intermediate states play an important role in the extreme final state interaction region.

On the adequacy of the unitary pole expansion

Abstract: Publisher Summary The separability of model nuclear interactions simplifies the three-body problem greatly. Therefore, the separable approximations of local potentials, like the unitary pole expansion (UPE), have been used frequently. In order to judge the quality of such approximations, one usually compares the full T-matrices for the original potential and its approximation. This chapter presents the comparison between the original potential V(r) and the potential V 1 (r), which generates the same on-shell T-matrix and bound state energy as V. It explains the corresponding inverse scattering problem to find the local potential V 1 (r), which is unique. The comparison between V and V 1 is easy because of their dependence on just one variable, whereas the corresponding T-matrices depend on three variables. Unitary pole approximation (UPA) and UPE may be not such good approximations as some three-body calculations seem to indicate. In particular, one should expect deviations at high energies and in many-body problems, the inner part of the inter-action plays a more important role.