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Register a new userConvergence of quantum electrodynamics on the Poincare group
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Authors
V. V. Varlamov
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Extended particles are considered in terms of the fields on the Poincar{e} group. Dirac like wave equations for extended particles of any spin are defined on the various homogeneous spaces of the Poincar{e} group. Free fields of the spin 1/2 and 1 (Dirac and Maxwell fields) are considered in detail on the eight-dimensional homogeneous space, which is equivalent to a direct product of Minkowski spacetime and two-dimensional complex sphere. It is shown that a massless spin-1 field, corresponding to a photon field, should be defined within principal series representations of the Lorentz group. Interaction between spin-1/2 and spin-1 fields is studied in terms of a trilinear form. An analogue of the Dyson formula for $S$-matrix is introduced on the eight-dimensional homogeneous space. It is shown that in this case elements of the $S$-matrix are defined by convergent integrals.
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We show how to get a non-commutative product for functions on space-time starting from the deformation of the coproduct of the Poincare group using the Drinfeld twist. Thus it is easy to see that the commutative algebra of functions on space-time (R^4) can be identified as the set of functions on the Poincare group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincare group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and involves cohomological features. As is known, spacetime algebra fixes the coproduct on the dffeomorphism group of the manifold. We now see that the influence is reciprocal: they are strongly tied.
We study Yang-Baxter sigma models with deformed 4D Minkowski spacetimes arising from classical $r$-matrices associated with $kappa$-deformations of the Poincare algebra. These classical $kappa$-Poincare $r$-matrices describe three kinds of deformations: 1) the standard deformation, 2) the tachyonic deformation, and 3) the light-cone deformation. For each deformation, the metric and two-form $B$-field are computed from the associated $r$-matrix. The first two deformations, related to the modified classical Yang-Baxter equation, lead to T-duals of dS$_4$ and AdS$_4$,, respectively. The third deformation, associated with the homogeneous classical Yang-Baxter equation, leads to a time-dependent pp-wave background. Finally, we construct a Lax pair for the generalized $kappa$-Poincare $r$-matrix that unifies the three kinds of deformations mentioned above as special cases.
Relativistic spin states are convention dependent. In this work we prove that the zero momentum-transfer limits of the leading two form factors in the decomposition of the energy-momentum tensor matrix elements are independent of this choice. In particular, we demonstrate that these constraints are insensitive to whether the corresponding states are massive or not, and that they arise purely due to the Poincare covariance of the states.
In this work we analyse the constraints imposed by Poincare symmetry on the gravitational form factors appearing in the Lorentz decomposition of the energy-momentum tensor matrix elements for massive states with arbitrary spin. By adopting a distributional approach, we prove for the first time non-perturbatively that the zero momentum transfer limit of the leading two form factors in the decomposition are completely independent of the spin of the states. It turns out that these constraints arise due to the general Poincare transformation and on-shell properties of the states, as opposed to the specific characteristics of the individual Poincare generators themselves. By expressing these leading form factors in terms of generalised parton distributions, we subsequently derive the linear and angular momentum sum rules for states with arbitrary spin.
A spinless covariant field $phi$ on Minkowski spacetime $M^{d+1}$ obeys the relation $U(a,Lambda)phi(x)U(a,Lambda)^{-1}=phi(Lambda x+a)$ where $(a,Lambda)$ is an element of the Poincare group $Pg$ and $U:(a,Lambda)to U(a,Lambda)$ is its unitary representation on quantum vector states. It expresses the fact that Poincare transformations are being unitary implemented. It has a classical analogy where field covariance shows that Poincare transformations are canonically implemented. Covariance is self-reproducing: products of covariant fields are covariant. We recall these properties and use them to formulate the notion of covariant quantum fields on noncommutative spacetimes. In this way all our earlier results on dressing, statistics, etc. for Moyal spacetimes are derived transparently. For the Voros algebra, covariance and the *-operation are in conflict so that there are no covariant Voros fields compatible with *, a result we found earlier. The notion of Drinfeld twist underlying much of the preceding discussion is extended to discrete abelian and nonabelian groups such as the mapping class groups of topological geons. For twists involving nonabelian groups the emergent spacetimes are nonassociative.
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