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PubMed1970

Dynamics of systems with large number of degrees of freedom and generalized transformation theory.

A generalized transformation theory which leads to a non-Hamiltonian description of dynamics is introduced. The transformation is such that all averages of observables remain invariant. However, the time evolution of the density matrix can no longer be expressed in terms of a commutator with the Hamiltonian. Therefore such transformations are not canonical in the usual sense. An explicit "two components" representation of the equations of motion is given which has the following properties: (a) each of the components satisfies a separate equation of motion, and (b) one component satisfies a kinetic equation of a generalized Boltzmann type.WE OBTAIN, THEREFORE, THE MOST REMARKABLE RESULT THAT THE RELATION BETWEEN DYNAMICS AND STATISTICAL MECHANICS (OR THERMODYNAMICS) TAKES A SPECIALLY TRANSPARENT AND SIMPLE FORM: thermodynamics appears in a precise sense as the random phase approximation of dynamics.Other problems such as the meaning of diagonalization of the Hamiltonian and definition of excitations will be treated in a forthcoming paper.

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PubMed1972

Cohomology of various completions of quasicoherent sheaves on affines.

Let O be a complete discrete valuation ring and let A be a commutative O-algebra. Let M be any A-module. In this paper, a class of completions M on the affine X corresponding to A, which includes, e.g., the Washnitzer-Monsky completion [1], and the full completion is studied. We then prove that for all of these completions we have, H(i)(X,M(+)) = O for i >/= 1, H degrees (X,M(+)) = M(+).

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PubMed1973

p-Algebras of Arbitrary Exponents.

To a purely inseparable Galois field extension C over A we associate a certain differential polynomial ring D. We show that all central simple A-algebras that contain C as a maximal commutative subalgebra can be obtained from D by factoring out certain ideals determined by Witt vectors.

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PubMed1974

L-multipliers for Noncompact Symmetric Spaces.

Let G be a real noncompact semi-simple Lie group with finite center and K a maximal compact sub-group. The symmetric space M = G/K carries a measure invariant under the action of G. The operators which map L(p)(M) continuously into itself and commute with the action of G, can be easily characterized when p = 2 or p = 1. This note gives some results on "singular integrals" which map L(p) into itself (1 < p < + infinity).

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PubMed1974

An invariant for certain operator algebras.

It is shown that the so-called principal function invariant, which is associated in a unitarily invariant way to operators with trace class self commutator TT(*) - T(*)T, is invariant under trace class perturbations of T and is an extension of the index of T-z to the whole plane. The connection of the principal function, under additional hypothesis, with the determination of the maximal ideal space of the C(*) algebra generated by T is discussed, and it is shown that the principal function, even when it takes noninteger values, plays a role in establishing the existence of invariant subspaces for T and in determining the point spectrum of T.

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PubMed1975

An algebraic description of stereochemical correspondence.

It is shown that a stereochemical correspondence between two molecular systems can be represented by a commuting diagram of the point groups and permutation groups involved. The effect of the diagrammatic condition on the mappings of the cosets, double cosets, subgroup lattice, and double coset algebra determined for the two molecular systems by point group to permutation group homomorphisms is detailed. Chemical similarities implied by a stereochemical correspondence are indicated, and six examples are provided and discussed.

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PubMed1975

Singular Integrals and the Principal Series, IV.

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.

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PubMed1976

Classification of irreducible tempered representations of semisimple Lie groups.

For each connected real semisimple matrix group, one obtains a constructive list of the irreducible tempered unitary representations and their characters. These irreducible representations all turn out to be instances of a more general kind of representation, here called basic. The result completes Langland's classification of all irreducible admissible representations for such groups. Since not all basic representations are irreducible, a study is made of character identities relating different basic representations and of the commuting algebra for each basic representation.

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PubMed1977

Cauchy integrals on Lipschitz curves and related operators.

In this note, we establish certain properties of the Cauchy integral on Lipschitz curves and prove the L(p)-boundedness of some related operators. In particular, we obtain the recent results of R. R. Coifman and Y. Meyer [(1976) "Commutateurs d'intégrales singulières:" Analyse harmonique d'Orsay n(o) 211, Université Paris XI] on the continuity of the so-called commutator operators.

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PubMed1978

Nonequilibrium entropy, Lyapounov variables, and ergodic properties of classical systems.

We discuss the problem of defining (nonequilibrium) entropy in terms of the concepts of mechanics and of reconciling its monotonic increase with the Hamiltonian evolution of the dynamical system. This leads to investigating necessary and sufficient conditions for the existence of monotonically increasing quantities or the so-called Lyapounov variables of classical systems. It is found that the condition of "mixing" is necessary and the property of being K-flow is sufficient for the existence of a Lyapounov variable. The significance of the study of Lyapounov variables for the elucidation of the fundamental questions of statistical mechanics is briefly discussed. It is seen that every Lyapounov variable must fail to commute with at least some of the operators of multiplication by phase space functions. The uncertainty relations implied by this necessary noncommutativity would then set a limit on the simultaneous determination of entropy and trajectories in phase space. These considerations thus support and sharpen the view that the thermodynamical and the (microscopic) dynamical descriptions of classical systems could be consistently reconciled as being complementary descriptions analogous to the complementary descriptions encountered in quantum mechanics.

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PubMed1979

Characterization of parallel subtraction.

Parallel subtraction is an operation defined on pairs of positive operators. In terms of electrical networks, one may pose the following problem: Given an electrical network, represented by a specified positive operator, determine the set of positive operators which when connected in parallel with the specified operator yield another prescribed operator. The set of solutions of this electrical network problem is shown to have a minimum. The minimum is termed "the parallel difference of the fixed operators," and the operation is termed "parallel subtraction." The parallel difference is used to obtain explicit error estimates for an iteration procedure which approximates the geometric mean of positive operators. This concept of the geometric mean reduces to the square root of the product of the operators if the operators commute. Finally, by using the geometric mean, an operator version of the Gaussian mean is presented.

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PubMed1980

Geometrical gauge theory of ghost and Goldstone fields and of ghost symmetries.

We provide a geometrical identification of the ghost fields, essential to the renormalization procedure in the non-Abelian (Yang-Mills) case. These are some of the local components of a connection on a principal bundle. They multiply the differentials of coordinates spanning directions orthogonal to those of a given section, whereas the Yang-Mills potential multiplies the coordinates in the section itself. In the case of a supergroup, the ghosts become commutative for the odd directions, and represent Nambu-Goldstone fields. We apply the results to chiral "flavor" SU(3)(L) x SU(3)(R) and to SU(2/1). The latter reproduces a highly constrained Weinberg-Salam model.

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PubMed1980

Spectral theory for domains in R of finite measure.

Let Omega be a measurable subset of R(n) of finite positive Lebesgue measures. The following two problems are considered: (i) Find commuting self-adjoint extensions of the minimal operators -i( partial differential)/ partial differentialx(k), k = 1,..., n (Omega open). (ii) Find a set Lambda subset R(n) such that the functions e(lambda) = exp(ilambda(1)x(1) +... + ilambda(n)x(n)) for lambda in Lambda form an orthonormal basis for L(2)(Omega). The problems are known to be equivalent under mild regularity conditions on Omega, and existence holds in two cases: (i) there is a connected open set Omega' such that the symmetric difference OmegaDeltaOmega' is a null set and Omega' is a fundamental domain for a discrete total subgroup; and (ii)Omega = [unk](ainR) (a + [unk]), disjoint union neglecting null sets, in which [unk] is a fundamental domain and R is a "set of representors" for a finite group of translations. Case i is equivalent to a function theoretic condition of Forelli, and case ii is established when the existence of a discrete covariance group is assumed. Generalizations of the geometric results i and ii for spectral sets in arbitrary Lie groups are indicated.

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PubMed1981

A construction of F(1) as automorphisms of a 196,883-dimensional algebra.

In this note, I announce the construction of the finite simple group F(1), whose existence was predicted independently in 1973 by Bernd Fischer and by me. The group has order 2(46)3(20)5(9)7(6)11(2)13(3)17.19.23.29.31.41. 47.59.71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 and is realized as a group of automorphisms of a 196,883-dimensional commutative nonassociative algebra over the rational numbers, which has an associative form. Equivalently, it is a group of automorphisms of a cubic form in 196,883 variables. It turns out that all the relevant arguments and calculations may be done by hand. Furthermore, existence of the group F(1) implies the existence of a number of other sporadic simple groups for which existence proofs formerly depended on work with computers. We are beginning to look upon this group as a "friendly giant."

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PubMed1981

Equipartition of energy for higher-order hyperbolic equations.

Let A(0), A(1),...,A(2(N)-1) be commuting skew-adjoint operators on a Hilbert space [unk]. Then the equation Pi(j=0) (2(N)-1) (d/dt - A(j))v(t) = 0 (t real) admits equipartition of energy [in the sense that the jth partial energy E(j)(t) of any solution at time t satisfies lim(t-->+/-infinity)E(j)(t) = 2(-N).(total energy) for each of the 2(N) values of j] if and only if the closure B(jk) of A(j) - A(k) satisfies weak-operator-limit exp(tB(jk)) = 0 as t --> +/-infinity whenever j not equal k.

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PubMed1984

Superintegrable systems.

Families of matrix differential superintegrable systems of Lax type are constructed. Each family is a commutative Lie superalgebra with an infinite common set of conservation laws.

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PubMed1985

Calculus of twisted vertex operators.

Starting from an arbitrary isometry of an arbitrary even lattice, twisted and shifted vertex operators are introduced. Under commutators, these operators provide realizations of twisted affine Lie algebras. This construction, generalizing a number of known ones, is based on a self-contained "calculus."

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PubMed1987

Explanation of parity nonconservation.

Space inversion and other discrete symmetries are treated within the frame of a theory of fundamental forces based only on general considerations of causality, symmetry, and stability, without ad hoc differential equations. The basic space-time M is the Einstein universe R(1) x S(3) as a causal (or conformal) rather than a pseudo-Riemannian manifold. Its connected symmetry group is then a 15-parameter group G locally equivalent to SO(2, 4), while the isometry group K of the Einstein universe is a 7-parameter subgroup. Correlation with conventional relativistic theory is based on a canonical imbedding of Minkowski space M(0) into M, together with the unique extendability of all transformations of the scaling-extended Poincaré group P from M(0) to global transformations on M. The fundamental fermion field F and boson field B are here restricted to be real and are fully invariant under G(e), where the superscript e denotes the inclusion of space and time inversions. The role of C on F is taken over by a real matrix having the eigenvalues +/-i, that commutes with G but anticommutes with space inversion. The spin space for B consists of the real linear transformations on that for F. There is a corresponding natural total Lagrangian that is both G(e) and O(2)-gauge invariant, the latter leading to lepton and baryon number conservation, and which is nonparametric except for scale. The Weyl and Maxwell equations are deduced, and compelling identifications made for neutrinos and the photon. The e and mu neutrino pairs occur in strikingly inequivalent positions in F, appearing symmetric only in the conventional relativistic limit R --> infinity, where R is the ( G-invariant) fundamental length interpretable as the radius of the space S(3). The photon occurs as the lowest member of a coherent subfamily of B that includes natural candidates for bare versions of the W and Z particles. In the relativistic limit the interaction Lagrangian becomes a sum over all elementary processes, one of which appears as quantum electrodynamics with Majorana-type electrons.

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