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Feb 2013

Volume 54, Issue 2, Articles (02xxxx)

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J. Math. Phys. 54, 021506 (2013); http://dx.doi.org/10.1063/1.4790887 (24 pages)

Gui-Qiang Chen, Vaibhav Kukreja, and Hairong Yuan
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back to top Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

A fully covariant information-theoretic ultraviolet cutoff for scalar fields in expanding Friedmann Robertson Walker spacetimes

A. Kempf, A. Chatwin-Davies, and R. T. W. Martin

J. Math. Phys. 54, 022301 (2013); http://dx.doi.org/10.1063/1.4790482 (22 pages)

Online Publication Date: 13 February 2013

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While a natural ultraviolet cutoff, presumably at the Planck length, is widely assumed to exist in nature, it is nontrivial to implement a minimum length scale covariantly. This is because the presence of a fixed minimum length needs to be reconciled with the ability of Lorentz transformations to contract lengths. In this paper, we implement a fully covariant Planck scale cutoff by cutting off the spectrum of the d’Alembertian. In this scenario, consistent with Lorentz contractions, wavelengths that are arbitrarily smaller than the Planck length continue to exist. However, the dynamics of modes of wavelengths that are significantly smaller than the Planck length possess a very small bandwidth. This has the effect of freezing the dynamics of such modes. While both wavelengths and bandwidths are frame dependent, Lorentz contraction and time dilation conspire to make the freezing of modes of trans-Planckian wavelengths covariant. In particular, we show that this ultraviolet cutoff can be implemented covariantly also in curved spacetimes. We focus on Friedmann Robertson Walker spacetimes and their much-discussed trans-Planckian question: The physical wavelength of each comoving mode was smaller than the Planck scale at sufficiently early times. What was the mode's dynamics then? Here, we show that in the presence of the covariant UV cutoff, the dynamical bandwidth of a comoving mode is essentially zero up until its physical wavelength starts exceeding the Planck length. In particular, we show that under general assumptions, the number of dynamical degrees of freedom of each comoving mode all the way up to some arbitrary finite time is actually finite. Our results also open the way to calculating the impact of this natural UV cutoff on inflationary predictions for the cosmic microwave background.
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04.20.Jb Exact solutions
98.80.Jk Mathematical and relativistic aspects of cosmology
04.20.Gz Spacetime topology, causal structure, spinor structure

Spectral action for a one-parameter family of Dirac operators on SU(2) and SU(3)

Alan Lai and Kevin Teh

J. Math. Phys. 54, 022302 (2013); http://dx.doi.org/10.1063/1.4790484 (21 pages)

Online Publication Date: 13 February 2013

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The one-parameter family of Dirac operators containing the Levi-Civita, cubic, and the trivial Dirac operators on a compact Lie group is analyzed. The spectra for the one-parameter family of Dirac Laplacians on SU(2) and SU(3) are computed by considering a diagonally embedded Casimir operator. Then the asymptotic expansions of the spectral actions for SU(2) and SU(3) are computed, using the Poisson summation formula and the two-dimensional Euler-Maclaurin formula, respectively. The inflation potential and slow-roll parameters for the corresponding pure gravity inflationary theory are generated, using the full asymptotic expansion of the spectral action on SU(2).
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11.30.Ly Other internal and higher symmetries
98.80.Bp Origin and formation of the Universe
98.80.Jk Mathematical and relativistic aspects of cosmology
04.50.-h Higher-dimensional gravity and other theories of gravity

Conformal field theories with infinitely many conservation laws

Ivan Todorov

J. Math. Phys. 54, 022303 (2013); http://dx.doi.org/10.1063/1.4790408 (14 pages)

Online Publication Date: 19 February 2013

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Globally conformal invariant quantum field theories in a D-dimensional space-time (D even) have rational correlation functions and admit an infinite number of conserved (symmetric traceless) tensor currents. In a theory of a scalar field of dimension D-2 they were demonstrated to be generated by bilocal normal products of free massless scalar fields with an O(N), U(N), or Sp(2N) (global) gauge symmetry [B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Unitary positive energy representations of scalar bilocal fields,” Commun. Math. Phys. 271, 223–246 (2007)10.1007/s00220-006-0182-2; e-print arXiv:math-ph/0604069v3; B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Infinite dimensional Lie algebras in 4D conformal quantum field theory,” J. Phys. A Math Theor. 41, 194002 (2008)10.1088/1751-8113/41/19/194002; e-print arXiv:0711.0627v2 [hep-th]]. Recently, conformal field theories “with higher spin symmetry” were considered for D = 3 by Maldacena and Zhiboedov [“Constraining conformal field theories with higher spin symmetry,” e-print arXiv:1112.1016 [hep-th]] where a similar result was obtained (exploiting earlier study of CFT correlators). We suggest that the proper generalization of the notion of a 2D chiral algebra to arbitrary (even or odd) dimension is precisely a conformal field theory (CFT) with an infinite series of conserved currents. We recast and complement (part of) the argument of Maldacena and Zhiboedov into the framework of our earlier work. We extend to D = 4 the auxiliary Weyl-spinor formalism developed by Giombi et al. [“A note on CFT correlators in three dimensions,” e-print arXiv:1104.4317v3 [hep-th]] for D = 3. The free field construction only follows for D > 3 under additional assumptions about the operator product algebra. The problem of whether a rational CFT in 4D Minkowski space is necessarily trivial remains open.
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11.15.-q Gauge field theories
11.25.Hf Conformal field theory, algebraic structures
11.30.Ly Other internal and higher symmetries
11.30.Rd Chiral symmetries
02.10.Ud Linear algebra
02.20.-a Group theory
02.30.Lt Sequences, series, and summability
04.62.+v Quantum fields in curved spacetime

Conformally invariant formalism for the electromagnetic field with currents in Robertson-Walker spaces

E. Huguet and J. Renaud

J. Math. Phys. 54, 022304 (2013); http://dx.doi.org/10.1063/1.4791688 (18 pages)

Online Publication Date: 22 February 2013

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We show that the Laplace-Beltrami equation □6a = j in (math6,η), η ≔ diag(+ − − − − +), leads under very moderate assumptions to both the Maxwell equations and the conformal Eastwood-Singer gauge condition on conformally flat spaces including the spaces with a Robertson-Walker metric. This result is obtained through a geometric formalism which gives, as byproduct, simplified calculations. In particular, we build an atlas for all the conformally flat spaces considered which allows us to fully exploit the Weyl rescalling to Minkowski space.
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03.50.De Classical electromagnetism, Maxwell equations
02.30.-f Function theory, analysis
02.40.-k Geometry, differential geometry, and topology

Higher dimensional abelian Chern-Simons theories and their link invariants

L. Gallot, E. Pilon, and F. Thuillier

J. Math. Phys. 54, 022305 (2013); http://dx.doi.org/10.1063/1.4791677 (27 pages)

Online Publication Date: 25 February 2013

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The role played by Deligne-Beilinson cohomology in establishing the relation between Chern-Simons theory and link invariants in dimensions higher than three is investigated. Deligne-Beilinson cohomology classes provide a natural abelian Chern-Simons action, non trivial only in dimensions 4l + 3, whose parameter k is quantized. The generalized Wilson (2l + 1)-loops are observables of the theory and their charges are quantized. The Chern-Simons action is then used to compute invariants for links of (2l + 1)-loops, first on closed (4l + 3)-manifolds through a novel geometric computation, then on math4l+3 through an unconventional field theoretic computation.
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11.15.Yc Chern-Simons gauge theory
02.40.-k Geometry, differential geometry, and topology

Seiberg-Witten equations and non-commutative spectral curves in Liouville theory

Leonid Chekhov, Bertrand Eynard, and Sylvain Ribault

J. Math. Phys. 54, 022306 (2013); http://dx.doi.org/10.1063/1.4792241 (21 pages)

Online Publication Date: 25 February 2013

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We propose that there exist generalized Seiberg-Witten equations in the Liouville conformal field theory, which allow the computation of correlation functions from the resolution of certain Ward identities. These identities involve a multivalued spin one chiral field, which is built from the energy-momentum tensor. We solve the Ward identities perturbatively in an expansion around the heavy asymptotic limit, and check that the first two terms of the Liouville three-point function agree with the known result of Dorn, Otto, Zamolodchikov, and Zamolodchikov. We argue that such calculations can be interpreted in terms of the geometry of non-commutative spectral curves.
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11.10.Nx Noncommutative field theory
11.30.Rd Chiral symmetries
02.40.-k Geometry, differential geometry, and topology
11.25.Hf Conformal field theory, algebraic structures
02.30.Rz Integral equations

Wick rotation for quantum field theories on degenerate Moyal space(-time)

Harald Grosse, Gandalf Lechner, Thomas Ludwig, and Rainer Verch

J. Math. Phys. 54, 022307 (2013); http://dx.doi.org/10.1063/1.4790886 (21 pages)

Online Publication Date: 28 February 2013

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In this paper the connection between quantum field theories on flat noncommutative space(-times) in Euclidean and Lorentzian signature is studied for the case that time is still commutative. By making use of the algebraic framework of quantum field theory and an analytic continuation of the symmetry groups which are compatible with the structure of Moyal space, a general correspondence between field theories on Euclidean space satisfying a time zero condition and quantum field theories on Moyal Minkowski space is presented (“Wick rotation”). It is then shown that field theories transferred to Moyal space(-time) by Rieffel deformation and warped convolution fit into this framework, and that the processes of Wick rotation and deformation commute.
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11.10.Nx Noncommutative field theory
02.10.-v Logic, set theory, and algebra
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