Nontrivial extensions of a massless representation with definite helicity of the Poincaré group in 3+1 dimensions by the tensor product of two other massless representations with definite helicity

Let U _n (a, ) be a massless, helicity n/2, representation of the Poincaré group in 3+1 dimensions. U _n (a, ) is realized in an adapted nuclear space D _n. We explicitly determine the various classes of cohomology for the extension of U _n (a, ) by U _n1 (a, ) U _n2 (a, ).     

Direct gauging of the Poincaré group

Realization of the Poincaré group as a subgroup ofGL(5,R) that maps an affine set into itself is shown to lead to a well-defined minimal replacement operator when the Poincaré group is allowed to act locally. The minimal replacement operator is obtained by direct application of the Yang-Mills procedure without the explicit introduction of fiber bundle techniques. Its application gives rise to compensating 1-formsW , 1 6, for the local action of the Lorentz groupL(4,R), and to compensating 1-forms ^k , 1k4, for the translation groupT(4). When applied to the basis 1-formsdx ⁱ of Minkowski space, distortion 1-formsB ^k result that define a canonical anholonomic coframe that contains both theT(4) and theL(4,R) compensating fields. When the canonical coframe is considered as a differential system onM ₄, it gives rise to gauge curvature expressions and Cartan torsion, but the latter has important differences from that usually encountered in the associated literature in view of the inclusion of the compensating fields forL(4,R). The standard Yang-Mills minimal coupling construct is used to obtain a total Lagrangian. This leads to a system of field equations for the matter fields, theT(4) compensating fields, and theL(4,R) compensating fields. Part of the current that drives theT(4) compensating fields is the 3-form of gauge momentum energy that obtains directly from the momentum-energy tensor of the matter fields onM ₄ under minimal replacement. Introduction of the Cartan torsion in the free-field Lagrangian is shown to lead to a direct spin decoupling in the sense that the gauge momentum energy (orbital) contribution of the matter fields to the spin current is eliminated. Explicit conservation laws for total momentum energy current and total spin current are obtained.     

Quantum fields in 1+1-dimension carrying a true ray representation of the Poincaré group

In 1+1 spacetime dimensions there are genuine ray representations of the Poincaré group P ₊ ^sup . We shall construct a free relativistic quantum field theory, such that the fields _d are covariant under P ₊ ^sup and transform with a nontrivial infinitesimal exponent d.     

Geometrical meaning of the Poincaré group gauge theory

We find a condition (6) under which a gauge theory of the Poincaré group is equivalent to the Einstein-Cartan theory of gravitation.     

Direct gauging of the Poincaré group. II

This paper continues the study of direct gauge theory of the Poincaré groupP ₁₀. The meanings and implications of transformations induced by the local action ofP ₁₀ are studied, and transformation rules for all field quantities are derived for the local action ofP ₁₀ in a sufficiently small neighborhood of the identity. These results lead directly to a system of fundamental partial differential equations that are both necessary and sufficient for invariance of the free field Lagrangian density. Homogeneity arguments and the classical theory of invariants are used to obtain the most general free field Lagrangian density. Gauge conditions are shown to imply coordinate conditions, and an algebraic system of antiexact gauge conditions is implemented. The underlying Minkowski space,M ₄, and the resulting Riemann-Cartan space,U ₄, become attached at their centers, as do their respective frame and coframe bundles. Weak constraints of vanishing torsion are studied. All field quantities are shown to be determined in terms of the compensating l-forms for the Lorentz sector alone provided an explicit system of integrability conditions is satisfied. Field equations of the Einstein type are shown to result.     

The Poincaré group as the symmetry group of canonical general relativity

This work reconsiders the formulation, due to Regge and Teitelboim, of the phase space approach to general relativity in the asymptotically flat context, phrasing it in the language of symplectic geometry. The necessary boundary conditions at spatial infinity are spelled out in detail. Precise meaning is given to the statement that, as a result of these boundary conditions, the Poincaré group acts as a symmetry group on the phase space of general relativity. This situation is compared with the spi-picture of Ashtekar and Hansen, where a larger asymptotic symmetry group is obtained.     

The geometry of minimal replacement for the Poincaré group

The constructs of this paper rest on two elementary facts: (1) the Poincaré group P₁₀ is the maximal group of isometries of Minkowski space-time M₄; (2) P₁₀ has a faithful matrix representation as a subgroup of GL(5, R) that maps an affine set into itself. Local action of P₁₀ and Yang-Mills minimal replacement are shown to induce a well-defined minimal replacement operator that maps the tensor algebra over M₄ onto the tensor algebra over a new space-time U₄. The natural frame and coframe fields of M₄ go over into a canonical system of frame and coframe fields of U₄ with both translation and Lorentz-rotation parts. The coframe fields define soldering 1-form fields for U₄ that give rise to the standard geometric quantities through the Cartan equations of structure. This leads to unique determinations of all relevant connection coefficients and the associated 2-forms of curvature and torsion that involve the compensating 1-forms for local action of both the translation and the Lorentz-rotation sectors. The metric tensor of U₄, that is induced by the minimal replacement operator, is shown to satisfy the Ricci lemma; U₄ is necessarily a Riemann-Cartan space. This space admits gauge covariant constant basis fields for the Lie algebra of the Lorentz group and for the Dirac algebra. The induced basis for the Dirac algebra evaluates the images of Dirac operators under minimal replacement, while the induced basis for the Lie algebra of L(4, R) serves to show that the holonomy group of U₄ is the Lorentz group. The minimal replacement operator is extended to include the case of a total gauge group that is the direct product of the Poincaré group and a Lie group of internal symmetries of matter fields. This provides a precise method of lifting any action integral of the matter fields from M₄ up to U₄ so that invariance properties are retained when the total group acts locally. The natural representations afforded by minimal replacement result in curvature being evaluated in terms of first order derivatives of the compensating fields that share many properties in common with the Dirac derivation algebra for spin fields. Direct interpretations of the compensating fields are obtained from the geodesic equations.     

Synchronization of Huygens' clocks and the Poincaré method

We study two models of connected pendulum clocks synchronizing their oscillations, a phenomenon originally observed by Huygens. The oscillation angles are assumed to be small so that the pendulums are modeled by harmonic oscillators, clock escapements are modeled by the van der Pol terms. The mass ratio of the pendulum bobs to their casings is taken as a small parameter. Analytic conditions for existence and stability of synchronization regimes, and analytic expressions for their stable amplitudes and period corrections are derived using the Poincaré theorem on existence of periodic solutions in autonomous quasi-linear systems. The anti-phase regime always exists and is stable under variation of the system parameters. The in-phase regime may exist and be stable, exist and be unstable, or not exist at all depending on parameter values. As the damping in the frame connecting the clocks is increased the in-phase stable amplitude and period are decreasing until the regime first destabilizes and then disappears. The results are most complete for the traditional three degrees of freedom model, where the clock casings and the frame are consolidated into a single mass.     

Gravity as an internal Yang-Mills gauge field theory of the Poincaré group

In the framework of affine bundles we present gravity as an internal gauge field theory of the Poincaré group. The resulting geometry is a Riemann-Cartan space-time carrying torsion and curvature. In order to admit a nontrivial action of the translation group we formally extend the matter Lagrangian to affine field variables. Finally, we establish the relation of our approach with the formalism of Hehl et al.     

On the representation of long-range potentials by a [1, 0] Padé approximant

The utility of the [1, 0] Padé approximant in the determination of long-range interaction constants is investigated using the recently calculated Rydberg-Klein-Rees potentials for the B³Π_Ou ⁺ states of I₂, Br₂ and Cl₂. The conditions under which the [1, 0] Padé approximant is a valid representation of the long-range potential outside the electron-overlap region are studied in detail, using as model systems the B states of I₂, Br₂ and Cl₂, and the X¹Σg⁺, B¹Σu⁺ and B′¹Σu⁺ states of H₂. In general, the validity of the representation depends on the nature of the potential, and is governed by

if the long-range potential can be adequately represented by a maximum number of three terms (n, m, m1) in the inverse-power expansion.     

Statistics of Poincaré recurrences for maps with integrable and ergodic components

Recurrence gives powerful tools to investigate the statistical properties of dynamical systems. We present in this paper some applications of the statistics of first return times to characterize the mixed behavior of dynamical systems in which chaotic and regular motion coexist. Our analysis is local: we take a neighborhood A of a point x and consider the conditional distribution of the points leaving A and for which the first return to A, suitably normalized, is bigger than t. When the measure of A shrinks to zero the distribution converges to the exponential e(-t) for almost any point x, if the system is mixing and the set A is a ball or a cylinder. We consider instead a system, a skew integrable map of the cylinder, which is not ergodic and has zero entropy. This map describes a shear flow and has a local mixing property. We rigorously prove that the statistics of first return is of polynomial type around the fixed points and we generalize around other points with numerical computations. The result could be extended to quasi-integrable area preserving maps such as the standard map for small coupling. We then analyze the distribution of return times in a region which is composed by two invariants subdomains: one with a mixing dynamics and the other with an integrable dynamics given by our shear flow. We show that the statistics of first return in this mixed region is asymptotically given by the exponential law, but this limit is attained by an intermediate regime where exponential and polynomial laws are linearly superposed and weighted by some factors which are proportional to the relative sizes of the chaotic and regular regions. The result on the statistics of first return times for mixed regions in the phase space can provide a basis to analyze such a property for area preserving maps in mixed regions even when a rigorous result is not available. To this end we present numerical investigations on the standard map which confirm the results of the model.     

On the Poincaré and Gauge Symmetry of a Model where Vector and Axial Vector Interaction get Mixed up with Different Weight

A (1+1) dimensional model where vector and axial vector interaction get mixed up with different weight is considered with a generalized masslike term for gauge field. Through Poincaré algebra it has been made confirm that only a Lorentz covariant masslike term leads to a physically sensible theory as long as the number of constraints in the phase space is two. With that admissible masslike term, phase space structure of this model with proper identification of physical canonical pair has been determined using Diracs’ scheme of quantization of constrained system. The bosonized version of the model remains gauge non-invariant to start with. Therefore, with the inclusion of appropriate Wess-Zumino term it is made gauge symmetric. An alternative quantization has been carried out over this gauge symmetric version to determine the phase space structure in this situation. To establish that the Wess-Zumino fields allocates themselves in the un-physical sector of the theory an attempts has been made to get back the usual theory from the gauge symmetric theory of the extended phase-space without hampering any physical principle. It has been found that the role of gauge fixing is crucial for this transmutation.     

Parameter optimization for a reflective bistable twisted nematic display by use of the Poincar sphereé method

Although reflective bistable twisted nematic (RBTN) displays have potential in low-power-consumption applications, to achieve the optimum conditions for both bistable states simultaneously remains a challenge. We use a geometrical method based on the Poincaré sphere representation to obtain the optimum conditions that can simultaneously satisfy both bistable states for a RBTN structure. With this method, the optimum conditions can be obtained analytically and the operation modes can be clearly visualized and better understood.     

Direct gauging of the Poincaré group V. Group scaling,classical gauge theory,and gravitational corrections

Homogeneous scaling of the group space of the Poincaré group,P ₁₀, is shown to induce scalings of all geometric quantities associated with the local action ofP ₁₀. The field equations for both the translation and the Lorentz rotation compensating fields reduce toO(1) equations if the scaling parameter is set equal to the general relativistic gravitational coupling constant 8Gc ^–4. Standard expansions of all field variables in power series in the scaling parameter give the following results. The zeroth-order field equations are exactly the classical field equations for matter fields on Minkowski space subject to local action of an internal symmetry group (classical gauge theory). The expansion process is shown to breakP ₁₀-gauge covariance of the theory, and hence solving the zeroth-order field equations imposes an implicit system ofP ₁₀-gauge conditions. Explicit systems of field equations are obtained for the first- and higher-order approximations. The first-order translation field equations are driven by the momentum-energy tensor of the matter and internal compensating fields in the zeroth order (classical gauge theory), while the first-order Lorentz rotation field equations are driven by the spin currents of the same classical gauge theory. Field equations for the first-order gravitational corrections to the matter fields and the gauge fields for the internal symmetry group are obtained. Direct Poincaré gauge theory is thus shown to satisfy the first two of the three-part acid test of any unified field theory. Satisfaction of the third part of the test, at least for finite neighborhoods, seems probable.