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Register a new userCovariant Quantum Fields on Noncommutative Spacetimes
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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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In this paper, we further develop the analysis started in an earlier paper on the inequivalence of certain quantum field theories on noncommutative spacetimes constructed using twisted fields. The issue is of physical importance. Thus it is well known that the commutation relations among spacetime coordinates, which define a noncommutative spacetime, do not constrain the deformation induced on the algebra of functions uniquely. Such deformations are all mathematically equivalent in a very precise sense. Here we show how this freedom at the level of deformations of the algebra of functions can fail on the quantum field theory side. In particular, quantum field theory on the Wick-Voros and Moyal planes are shown to be inequivalent in a few different ways. Thus quantum field theory calculations on these planes will lead to different physics even though the classical theories are equivalent. This result is reminiscent of chiral anomaly in gauge theories and has obvious physical consequences. The construction of quantum field theories on the Wick-Voros plane has new features not encountered for quantum field theories on the Moyal plane. In fact it seems impossible to construct a quantum field theory on the Wick-Voros plane which satisfies all the properties needed of field theories on noncommutative spaces. The Moyal twist seems to have unique features which make it a preferred choice for the construction of a quantum field theory on a noncommutative spacetime.
The Moyal and Wick-Voros planes A^{M,V}_{theta} are *-isomorphic. On each of these planes the Poincare group acts as a Hopf algebra symmetry if its coproducts are deformed by twist factors. We show that the *-isomorphism T: A^M_{theta} to A^V_{theta} does not also map the corresponding twists of the Poincare group algebra. The quantum field theories on these planes with twisted Poincare-Hopf symmetries are thus inequivalent. We explicitly verify this result by showing that a non-trivial dependence on the non-commutative parameter is present for the Wick-Voros plane in a self-energy diagram whereas it is known to be absent on the Moyal plane (in the absence of gauge fields). Our results differ from these of (arXiv:0810.2095 [hep-th]) because of differences in the treatments of quantum field theories.
We review in simple terms the covariant approaches to the canonical formulation of classical relativistic field theories (in particular gauge field theories) and we discuss the relationships between these approaches as well as the relation with the standard (non-covariant) Hamiltonian formulation. Particular attention is paid to conservation laws related to Poincare invariance within the different approaches. To make the text accessible to a wider audience, we have included an outline of Poisson and symplectic geometry for both classical mechanics and field theory.
We review the covariant canonical formalism initiated by DAdda, Nelson and Regge in 1985, and extend it to include a definition of form-Poisson brackets (FPB) for geometric theories coupled to $p$-forms, gauging free differential algebras. The form-Legendre transformation and the form-Hamilton equations are derived from a $d$-form Lagrangian with $p$-form dynamical fields $phi$. Momenta are defined as derivatives of the Lagrangian with respect to the velocities $dphi$ and no preferred time direction is used. Action invariance under infinitesimal form-canonical transformations can be studied in this framework, and a generalized Noether theorem is derived, both for global and local symmetries. We apply the formalism to vielbein gravity in $d=3$ and $d=4$. In the $d=3$ theory we can define form-Dirac brackets, and use an algorithmic procedure to construct the canonical generators for local Lorentz rotations and diffeomorphisms. In $d=4$ the canonical analysis is carried out using FPB, since the definition of form-Dirac brackets is problematic. Lorentz generators are constructed, while diffeomorphisms are generated by the Lie derivative. A doubly covariant hamiltonian formalism is presented, allowing to maintain manifest Lorentz covariance at every stage of the Legendre transformation. The idea is to take curvatures as velocities in the definition of momenta.
There is significant recent work on coupling matter to Newton-Cartan spacetimes with the aim of investigating certain condensed matter phenomena. To this end, one needs to have a completely general spacetime consistent with local non-relativisitic symmetries which supports massive matter fields. In particular, one can not impose a priori restrictions on the geometric data if one wants to analyze matter response to a perturbed geometry. In this paper we construct such a Bargmann spacetime in complete generality without any prior restrictions on the fields specifying the geometry. The resulting spacetime structure includes the familiar Newton-Cartan structure with an additional gauge field which couples to mass. We illustrate the matter coupling with a few examples. The general spacetime we construct also includes as a special case the covariant description of Newtonian gravity, which has been thoroughly investigated in previous works. We also show how our Bargmann spacetimes arise from a suitable non-relativistic limit of Lorentzian spacetimes. In a companion paper [arXiv:1503.02680] we use this Bargmann spacetime structure to investigate the details of matter couplings, including the Noether-Ward identities, and transport phenomena and thermodynamics of non-relativistic fluids.
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