Linear kinematical groups

We prove a theorem which states that in an (n+1)-dimensional space-time (n3) the only linear kinematical groups which are compatible with the isotropy of space are the Lorentz and Galilei groups. The special casesn=1 andn=2 are also briefly discussed.On leave of absence from Instituto di Fisica dell' Universitá, Milano, Italy. A. v. Humboldt fellow.     

Exponential representations of the canonical commutation relations

A class of representations of the canonical commutation relations is investigated. These representations, which are called exponential representations, are given by explicit formulas. Exponential representations are thus comparable to tensor product representations in that one may compute useful criteria concerning various properties. In particular, they are all locally Fock, and non-trivial exponential representations are globally disjoint from the Fock representation. Also, a sufficient condition is obtained for two exponential representations not to be disjoint. An example is furnished by Glimm's model for the :⁴: interaction for boson fields in three space-time dimensions.     

Absence of symmetry breakdown and uniqueness of the vacuum for multicomponent field theories

Correlation inequalities are used to show that the two component (²)² model (with HD, D, HP, P boundary conditions) has a unique vacuum if the field does not develop a non-zero expectation value. It follows by a generalized Coleman theorem that in two space-time dimensions the vacuum is unique for all values of the coupling constant. In three space-time dimensions the vacuum is unique below the critical coupling constant.For then-componentP(||²)₂+₁ model, absence of continuous symmetry breaking, as goes to zero, is proven for all states which are translation invariant, satisfy the spectral condition, and are weak* limit points of finite volume states satisfyingN _loc and higher order estimates.     

Extended Diffeomorphism Algebras¶and Trajectories in Jet Space

Let the DRO (Diffeomorphism, Reparametrization, Observer) algebra?DRO(N) be the extension of diff(N)⊕ diff(1) by its four inequivalent Virasoro-like cocycles. Here diff(N) is the diffeomorphism algebra in N-dimensional spacetime and diff(1) describes reparametrizations of trajectories in the space of tensor-valued p-jets. DRO(N) has a Fock module for each p and each representation of gl(N). Analogous representations for gauge algebras (higher-dimensional Kac–Moody algebras) are also given. The reparametrization symmetry can be eliminated by a gauge fixing procedure, resulting in previously discovered modules. In this process, two DRO(N) cocycles transmute into anisotropic cocycles for diff(N). Thus the Fock modules of toroidal Lie algebras and their derivation algebras are geometrically explained. Received: 29 October 1998 / Accepted: 2 May 2000     

A spectrum generating algebra for meson resonances

The algebraSO(6,1) is considered as a unification ofSO(6), which is isomorphic toSU(4) SU(3), and the de Sitter algebraSO(4,1). The latter replaces the Poincaré algebra as the algebra of the group of motions of physical space-time. A representation ofSO(6,1) is constructed, which, on restriction toSU(3), decomposes into the direct sum of allSU(3) representations, each occurring just once in the decomposition. The expectation values of the mass-squared operator, when evaluated in the octet, give accurate mass formulae for the octets of 1^– and 2⁺ meson resonances.     

Complex-dimensional invariant delta functions and lightcone singularities

Invariant delta functions (including imaginary-mass case) defined in a complexn-dimensional space-time are explicitly calculated in position space. It is proposed to define products of invariant delta functions in the ordinary Minkowski space by analytically continuing the correspondingn-dimensional ones ton=4. The (not only leading but also non-leading) lightcone singularities of [(x; m ²)]², (x; m ²)⁽¹⁾(x; m ²), and [⁽¹⁾(x; m ²)]² are shown to be unambiguously determined in this way.     

On a Purely Geometric Approach to the Dirac Matter Field and Its Quantum Properties

We consider the most general axial torsion completion of gravity with electrodynamics for $\frac{1}{2}$ -spin spinors in an 8-dimensional representation of the Dirac matter field: this theory will allow to define antimatter as matter with all quantum numbers reversed, where also the sign of the mass beside that of the charge is inverted: we shall see that matter and antimatter solutions of the Dirac field equations coincide with the known ones with respect to all observables, that despite the inversion of the sign of the mass term only positive-mass states are present and only positive-energy densities are given; the present and the common approach will be compared, and some experimental implications will be discussed.     

Cauchy's problem and Huygens' principle for relativistic higher spin wave equations in an arbitrary curved space-time

Relativistic spins (s >/ 1/2), nonzero mass equations are given which in an arbitrary curved space-time are internally consistent. By means of Riesz' integration method a representation theorem for the solution of Cauchy's problem, using the constraints of the Cauchy data on the initial hypersurface and suitable Green's formulas, is proved. Finally, a necessary and sufficient condition for the validity of Huygens' principle is stated from which it follows that only in space-times of constant curvature do the field equations satisfy Huygens' principle.     

Singular vectors and invariant equations for the Schrödinger algebra in <Emphasis Type="Italic">n</Emphasis> ≥ 3 space dimensions. The general case

The singular vectors in Verma modules over the Schrödinger algebra ?(n) in (n + 1)-dimensional space-time are found for the case of general representations. Using the singular vectors, hierarchies of equations invariant under Schrödinger algebras are constructed.     

On the mass spectrum of the 2+1 gauge-Higgs lattice quantum field theory

We investigate the mass spectrum of a 2+1 lattice gauge-Higgs quantum field theory with Wilson action A _P+A _H, whereA _P(A _H) is the gauge (gauge-Higgs) interaction. We determine the complete spectrum exactly for all , >0 by an explicit diagonalization of the gauge invariant transfer matrix in the approximation that the interaction terms in the spatial directions are omitted; all gauge invariant eigenfunctions are generated directly. For fixed momentum the energy spectrum is pure point and disjoint simple planar loops and strings are energy eigenfunctions. However, depending on the gauge group and Higgs representations, there are bound state energy eigenfunctions not of this form. The approximate model has a rich particle spectrum with level crossings and we expect that it provides an intuitive picture of the number and location of bound states and resonances in the full model for small , >0. We determine the mass spectrum, obtaining convergent expansions for the first two groups of masses above the vacuum, for small , and confirm our expectations.Research partially supported by CNPq, Brasil     

Representations of quantumso(8) and related quantum algebras

We study irreducible representations of the quantum groupU (so(8)) when ^* is a primitivel ^th root of unity. By a theorem of De Concini and Kac, there is a finite number of such representations associated to each point of a complex algebraic variety of dimension 28 and the generic representation has dimensionl ¹².We give explicit constructions of essentially all the irreducible representations whose dimension is divisible byl ⁸. In addition, we construct all cyclic representations of minimal dimension. This minimal dimension isl ⁵, in accordance with a conjecture of De Concini, Kac and Procesi.Partially supported by the NSF, DMS-9115984     

Statistical mechanics of probabilistic cellular automata

We investigate the behavior of discrete-time probabilistic cellular automata (PCA), which are Markov processes on spin configurations on ad-dimensional lattice, from a rigorous statistical mechanics point of view. In particular, we exploit, whenever possible, the correspondence between stationary measures on the space-time histories of PCAs on ^d and translation-invariant Gibbs states for a related Hamiltonian on ( ^d+1). This leads to a simple large-deviation formula for the space-time histories of the PCA and a proof that in a high-temperature regime the stationary states of the PCA are Gibbsian. We also obtain results about entropy, fluctuations, and correlation inequalities, and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high-noise regime. We discuss phase transitions in the low-noise (or low-temperature) regime and review Toom's proof of nonergodicity of a certain class of PCAs.     

Highest weight representations of the Neveu-Schwarz and Ramond algebras

We construct a family of representationsK ^,w of the Neveu-Schwarz and Ramond algebras, which generalize the Fock representations of the Virasoro algebra. We show that the representationsK ^,w are intertwined by a vertex operator.The above results are used to give the proof of the conjectured formulas for the determinant of the contravariant form on the highest weight representations of the Neveu-Schwarz and Ramond algebras. Further results on the representation theory of the latter are derived from the determinant formulas.Partially supported by the National Science Foundation through the Mathematical Sciences Research InstitutePartially supported by NSF grant MCS-8201260     

Structure of Malicious Singularities

In this paper, we investigate relativistic spacetimes, together with their singular boundaries (including the strongest singularities of the Big Bang type, called malicious singularities), as noncommutative spaces. Such a space is defined by a noncommutative algebra on the transformation groupoid = × G, where is the total space of the frame bundle over spacetime with its singular boundary, and G is its structural group. We show that there exists the bijective correspondence between unitary representations of the groupoid and the systems of imprimitivity of the group G. This allows us to apply the Mackey theorem to this case, and deduce from it some information concerning singular fibers of the groupoid . At regular points the group representation, which is a part of the corresponding system of imprimitivity, does not have discrete components, whereas at the malicious singularity such a group representation can be a single representation (in particular, an irreducible one) or a direct sum of such representations. A subgroup K G, from which—according to the Mackey theorem—the representation is induced to the whole of G, can be regarded as measuring the richness of the singularity structure. In this sense, the structure of malicious singularities is richer than those of milder ones.     

On the space-time interpretation of classical canonical systems II: Relativistic canonical systems

Relativistic canonical systems and their symmetries are defined and classified within the class of canonical systems treated in a previous paper. Their algebra of variables contains a subset of monotone variables which satisfy a certain uniqueness condition and are later shown to increase strictly in the course of the dynamical evolution of the system on all physically acceptable states. This leads to a unique space-time interpretation of relativistic canonical systems. Finally we study space-time factorizations of such systems and introduce the appropriate notion of states. For a certain simple class of states the theory is shown to describe the motion of relativistic matter in some external gravitational and electromagnetic field.     

Symmetry properties of a charged fluid in the general theory of relativity

The problem of specifying the symmetry properties of a charged fluid in space-time V₄, which has a definite group of motions, is considered. It is shown that if nonempty spacetime V₄ with an energy-momentum tensor for the charged fluid has symmetry (i.e., admits a group of motions), then the mass density, the pressure, and the four-velocity of the fluid inherits this symmetry, and for an electromagnetic field the relationL f _ij =7.*f _ij is satisfied. The necessary and sufficient conditions are found so that =0 in the case of a one-parameter group. Then additional relationships between and the structure constants are obtained in the case of an r-parameter group. It is shown that under certain conditions the symmetry properties obtained for charged matter are necessary and sufficient conditions for the symmetry of space-time V₄.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 113–117, November, 1978.     

Representations of the CCR in the (φ4)3 model: Independence of space cutoff

The algebra of observables for the renormalized ⁴ interaction in 3-dimensional space-time is constructed. It is shown that the von Neumann algebras associated with observables in a bounded regionB are independent of the space cutoff which is used in the construction of a Hamiltonian for this interaction. This result is shown to be useful in the construction of a translation invariant ⁴ theory in three dimensions. It also gives a physical criterion for the equivalence of non-Fock representations of the canonical commutation relations.     

2-Particle asymptotic completeness and bound states in weakly coupled quantum field theories

Rigorous results on poles of the 2- and 4-point functions, which yield 2-particle asymptotic completeness and give information on the presence or absence of 2-particle bound states and resonances, are presented for weakly coupled even and non-even-field theories with mass gap in space-time dimensiond=2, 3 (and for related hypothetical theories in dimension 4). Methods used are more convenient and more general than those used previously (with more limited results) forP()₂ theories.     

Higher Spin Conserved Currents in Sp(2M) Symmetric Spacetime

An infinite set of higher spin conserved charges is found for the sp(2M) symmetric dynamical systems in M(M+ 1)/2-dimensional generalized spacetime _M. Since the dynamics in _M is equivalent to the conformal dynamics of infinite towers of fields in d-dimensional Minkowski spacetime with d = 3, 4, 6, 10, ... for M = 2, 4, 8, 16, ..., respectively, the constructed currents in _M generate infinite towers of (mostly new) higher spin conformal currents in Minkowski spacetime. The charges have a form of integrals of M-forms which are bilinear in the field variables and are closed as a consequence of the field equations. Conservation implies independence of a value of charge of a local variation of a M-dimensional integration surface _M analogous to Cauchy surface in the usual spacetime. The scalar conserved charge provides an invariant bilinear form on the space of solutions of the field equations that gives rise to a positive-definite norm on the space of quantum states.