Poincaré gauge invariant spinor theory and the gravitational field. II

The nonlinear equation for an abstract noncanonical 2-component Weyl spinor field — as used with the inclusion of internal symmetries in Heisenberg's nonlinear spinor theory of elementary particles — which is invariant under scale, phase, and Poincaré transformations is modified in such a way as to become invariant under spacetime dependent phase gauge and Poincaré gauge transformations. In such an equation a phase gauge field B _m , six Lorentz gauge fields A[]m and four translation gauge fields g_m have to be introduced. It is demonstrated that all these fields can be identified as certain combinations of the Weyl spinor field, and hence should be considered in a rough sense as bound states of this spinor field. In particular the electromagnetic field B_m and the gravitational field g m appear as S-states and P-states of a spinor-antispinor system. The noncanonical property and the operator character of the spinor field is essential for this result. The relation between the translation gauge field and the spinor field involves a fundamental length. In a classical geometrical interpretation this relation leads to Einstein's equation of gravitation without cosmological term in a Riemannian space without torsion if the fundamental length is identified with Planck's length. It is shown that this equation is covariant under the larger symmetry group of phase gauge and Poincaré gauge transformations. The modified nonlinear equation constructed solely from a single 2-component Weyl field hence seems to incorporate in an extremely compact way electromagnetic and gravitational interaction in addition to non-mass-zero interactions. In this equation no arbitrary dimensionless constants enter. The considerations can be generalized to Dirac spinor fields and to spinor fields involving additional interior degress of freedom.An abridged version of this paper was presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.     

Induced geometry and confinement

A classical, Poincaré invariant dynamical system is developed which contains, besides the natural metric _v , an induced metricg _v that is generated by a real scalar dynamical field. It is shown that scalar fields whose dynamics are governed by the induced metric can be consistently introduced. Also, point particles which follow timelike quasi-geodesic trajectories can be introduced. The reaction forces acting ong _v due to the presence of these fields and particles are computed. A discussion of causality and geometrical confinement is given.     

Thes-f model with Coulomb repulsion: Thermodynamics of two atomic cluster. Spin 1/2

A cluster of two atoms described by thes-f model with Coulomb repulsion has been considered. The interaction between localized 4f electrons (S=1/2) is taken in the molecular field approximation. The thermodynamic quantities like magnetization, specific heat and correlation functions n , n , S ^z n , S ^z n , S ^z (n –n ), n n and S ⁺ a ⁺ a as functions of temperature are presented for different band fillingN=0, 0.5, 1, 1.5, 2. The dependence of Curie temperature onN is calculated. The phase diagram forN=1 (T=0K) shows the possibility of existence of two phases: paramagnetic and ferromagnetic.The Curie temperature and the specific heat as functions ofN exhibit similar trends as found in experiments on doped magnetic semiconductors.     

The two-dimensionalO(N) nonlinear σ-model: Renormalisation and effective actions

We establish the existence of the Wilson Renormalised trajectory of theO(N) sigma model in perturbation theory in the effective charge. This yields a proof of perturbative renormalisability, and is also relevant in the small-field analysis of the rigorous renormalisation group construction of the continuum theory.Laboratoire associé au CNRS UA 280On leave from the Tata Institute of Fundamental Research, Bombay, 400 005, IndiaResearch funded in part by NSF grant DMS 8601978     

Complete integrability of relativistic Calogero-Moser systems and elliptic function identities

Poincaré-invariant generalizations of the Galilei-invariant Calogero-MoserN-particle systems are studied. A quantization of the classical integralsS ₁, ...,S _N is presented such that the operators ₁, ..., _N mutually commute. As a corollary it follows thatS ₁, ...,S _N Poisson commute. These results hinge on functional equations satisfied by the Weierstrass - and -functions. A generalized Cauchy identity involving the -function leads to anN×N matrixL whose symmetric functions are proportional toS ₁, ...,S _N .     

Rate equations in the hopping transport

The usual kinetic equations for the site occupation probabilities in an external field are solved exactly in a simple one-dimensional periodic model with two kinds of atoms using a) free boundary conditions and order of limitsN, 0 needed for a proper treatment of the dc conductivity here b) boundary conditions with metallic contacts and order of limitsN, 0 and c) the same boundary conditions but reversed order of limiting processes 0,N typical of e.g. numerical and percolation treatments. (N and are the number of sites and frequency.) It is demonstrated that though the bulk dc conductivity is the same in all three cases, local bulk properties of the material are strongly dependent on the régime used. The role of the order of all three limiting processes 0,N+ andn+ (N–n+) for local shifts of the chemical potential _n in the dc limit is examined (n is the number of the relevant site calculated from a boundary of the chain). It is shown especially that the rate equation treatment (régime a) on the one hand and numerical or percolation treatments (régime c) on the other hand never yield the same bulk values of _r.     

Poincaré invariance for the master field on quantum chromodynamics

I construct allSU(N _c ) gauge fields with the property that Euclidean Poincaré transformations can be compensated by gauge transformations. Linear Abelian components are shown to be forbidden by Lorentz invariance. In a suitable gauge, the result is a set of constant potentials parametrized by Lorentz scalars. These scalars are constrained by the equation of motion atN _c =. A special solution is exhibited.Work supported in part by Schweizerischer Nationalfonds.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.I thank H. Leutwyler for drawing my attention to the configuration (35), and M. Lüscher, P. Schwab, P. Sorba and J. Stern for their comments.     

Some aspects of the geometry of first-Quantized theories. II

An alternative method of identifying base-space and fiber tensors is presented and is used to describe interacting Dirac and Yang-Mills fields with internal affine symmetry. In a separate construction, the cosmological constant is derived from the fiber bundle structure of a generalized theory of Poincaré gauge symmetry. Mathematical foundations of the computations are discussed.     

Direct gauging of the poincaré group. III. Interactions with internal symmetries

The problem of gauging matter fields with a Poincaré invariant action functional that admits anr parameter, semisimple groupG(r) of internal symmetries is considered. A minimal replacement operator for the total groupP ₁₀×G(r) is obtained, together with the requisite compensating 1-forms for both the Poincaré and theG(r) sectors. A basis forP ₁₀×G(r)-invariant Lagrangian densities for the free fields is obtained. Restriction to Lagrangian densities that are at most quadratic in the associated curvature and torsion fields eliminates active coupling between theP ₁₀ free field Lagrangian and theG(r) free field Lagrangian, although there is passive coupling that arises through the requirement of tensorial covariance under general coordinate transformations generated by the local action of the translation group. A superposition principle, modulo passive coupling, thus holds for quadratic free field Lagrangian for the total group:L _TOT=L _P +L _G(r) . Field equations for the matter fields and the compensating fields of theG(r) sector are obtained. Both share the passive coupling toP ₁₀ that is required in order to achieve tensorial covariance, but only the matter fields couple directly to the Poincaré fields and only to the Lorentz sector. For weak Poincaré fields, the field equations for the matter fields and the compensating fields of the internal symmetries go over into the standard field equations of gauge theory for an internal symmetry group.     

Tensor Gauge Fields in Arbitrary Representations of GL(D, ℝ). Duality and Poincaré Lemma

Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we have studied the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary irreducible representations of GL(D, ). We have proven a generalization of the Poincaré lemma which enables us to solve the above-mentioned problems in a systematic and unified way.     

Nontrivial extensions of a representation of the Poincaré group with mass and helicity zero by its tensorial product

Let U(a, ) be a representation of the Poincaré group with mass and helicity zero, realized in the space of C -functions with compact support on ³, without the origin. Let U ⁽²⁾(a, ) denote the tensorial product of U(a, ) by itself. We explicitly determine the cocycles of extension of U(a, ) by U ⁽²⁾(a, ) and we prove that the nontrivial cohomology is indexed by (u(),),u()D ¹ _]0,3\,.     

Asymptotic locality and the structure of local internal symmetries

Symmetries are investigated from the local viewpoint. Using the Haag-Ruelle construction, the action of a local internal symmetry on the asymptotic states is determined. A condition of asymptotic locality is derived and used to show that the symmetry acts linearly and locally on the asymptotic fields. Within a field theoretical framework it is shown that the internal symmetry must commute with the Poincaré group. The general structure of an internal symmetry is determined. The uniqueness of the representation of the Poincaré group is discussed, and a simple example of an infinite component field is given to indicate what occurs when there are infinitely degenerate particle multiplets.     

A discrete spectrum of solutions of the wave equation with strong cubic nonlinearity

The nonlinear wave equation, _tt –+³=0, has many solutions that are periodic in time and localized in space, all with infinte energies. The search for spherically symmetric solutions that are well represented by the simple approximation, (r, t)A(r) sin t, leads to a discrete spectrum of solutions{ _N (r, t; )}. The solutions are nonlinear wavepackets, and they can be regarded as particles. The asymptotic theory () of the motion of the guiding center of theNth wavepacket, in the presence of a specified potential, is characterized by an infinite mechanical mass and an infinte interaction mass, and they are compatible. The rest mass in the classical relativistic mechanics of guiding centers ism ₀ c ²= _N ; i.e. the spectrum { _N } determines a spectrum of Planck's constants.On leave (1972–73) Université de Paris VI, Département de Mécanique, 75 Paris 5^e, France.     

The asymmetric contact process on a finite set

The contact process onZ has one phase transition; let _c be the critical value at which the transition occurs. Let _N be the extinction time of the contact process on {0,...,N}. Durrett and Liu (1988), Durrett and Schonmann (1988), and Durrett, Schonmann, and Tanaka (1989) have respectively proved that the subcritical, supercritical, and critical phases can be characterized using a large finite system (instead ofZ) in the following way. There are constants ₁() and ₂() such that if < _c , lim _{N N} /logN = 1/₁(); if > _c , lim _N log _N /N = ₂(); if = _c , lim _{N N} /N= and lim _{N N} /N ⁴=0 in probability. In this paper we consider the asymmetric contact process onZ when it has two distinct critical values _c1< _c2. The arguments of Durrett and Liu and of Durrett and Schonmann hold for < _c1 and > _c2. We show that for [ _c1< _c2), lim _{N N} /N=-1/, (where _i is an edge speed) and for = _c2, lim _N log _N /logN=2 in probability.     

On the extreme relativistic limit of arbitrary spin wave equations

The extreme relativistic limit (E-representation) of the wave equation in the Schrödinger formi/t =H describing particles and anti-particles of spin s and non-zero rest mass m is presented here. As the wave function has just the minimum number of 2(2s+1) components, the necessity of avoiding redundant components by auxiliary conditions does not arise. Relevant expressions are given for the infinitesimal generators of the Poincaré group and for the operators representing the observables in this representation.     

Experimental and Real Coordinates in Space-Time Transformations

The experimental (apparent) space-time transformations connect coordinates altered by length contraction and clock retardation. When clocks are synchronized by means of light signals (Einstein–Poincaré procedure) or by slow clock transport, the experimental space-time. transformations assume the mathematical form of the Extended space-time transformations.⁽⁴⁾ These reduce to the Lorentz–Poincaré transformations when one of the frames they connect is the fundamental inertial frame. If the synchronization procedure were perfect, the experimental space-time transformations would assume the form of Selleris inertial transformations.⁽⁵⁾ The real space-time transformations are those which are disclosed when the systematic measurement distortions are corrected.     

Logically independent von Neumann lattices

Three definitions of logical independence of two von Neumann latticesP₁,P₂ of two sub-von Neumann algebras ₁, ₂ of a von Neumann algebra are given and the relations of the definitions clarified. It is shown that under weak assumptions the following notion, called logical independence is the strongest:A B 0 for any 0 A P₁, 0 B P₂. Propositions relating logical independence ofP₁,P₂ toC ^*-independence,W ^* independence, and strict locality of ₁, ₂ are presented.     

A finite size analysis of the structure factor in the antiferromagnetic Heisenberg (AFH) model

From a finite size analysis we extract the structure factorS(p, N=) of the one dimensional AFH-model in the groundstate: The gross structure is well described byL (p) = –ln(1– ^p ). The fine structure which only contributes a few percent reveals a pronounced non-linear behavior inL(p) with a maximum atp=0.20 and a minimum atp=0.82.     

SL(2,C)-gauge theory andN=1 supergravity

It is shown that as Riemannian space may be taken to give rise to a Poincaré gauge theory of gravitation, the superspace where the coordinates are given by (X, ), being a spinorial variable gives rise to anSL(2, C)-gauge theory and corresponds toN= 1 supergravity. It leads to a conserved current and the conserved quantity here corresponds to isospin, where the latter is taken to be generated from conformal reflection. Thus, supergravity plays a predominant role in the microlocal space-time.     

Unbounded derivations and invariant states

Let be a von Neumann algebra with a cyclic and separating vector . Let =i[H, ·] be the spatial derivation implemented by a selfadjoint operatorH, such thatH=0. Let be the modular operator associated with the pair (, ). We prove the equivalence of the following three conditions:1)H is essential selfadjoint onD(), andH commutes strongly with .2) The restriction ofH toD() is essential selfadjoint onD(^1/2) equipped with the inner product(|)_#=(|)+(^1/2|^1/2), , D(^1/2).3) exp (itH) exp (–itH)= for anyt.We show by an example, that the first part of 1),H is essential selfadjoint onD(), does not imply 3). This disproves a conjecture due to Bratteli and Robinson [3].Part of this work was done while O.B. was a member of Zentrum für interdisziplinäre Forschung der Universität Bielefeld