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J. S. Dowker

Publications and source records attributed to J. S. Dowker.

At least 55 records · Page 3Linked to original sources
arXiv2015

Conformal weights of charged Renyi entropy twist operators for free Dirac fields in arbitrary dimensions

The conformal weights of spherical twist operators at non--zero (Euclidean) chemical potential are computed for free Dirac fields in arbitrary dimensions. An image technique, equivalent to replicas, is again used to obtain the $n$--fold cover quantities. The proof of a conjecture made by Bueno, Myers and Witczac--Krempa regarding the relation between the conformal weights and a corner coefficient in the Rényi entropy, given before for scalar fields, is extended to the fermion case. The variation of the weights with chemical potential indicates phase changes.

hep-th↗
arXiv2015

On sums of powers of cosecs

The finite sums of powers of cosecs occur in numerous situations, both physical and mathematical, examples being the Casimir effect, Renyi entropy, Verlinde's formula and Dedekind sums. I here present some further discussion which consists mainly of a reprise of early work by H.M.Jeffery in 1862-64 which has fallen by the wayside and whose results are being reproduced up to the present day. The motivation is partly historical justice and partly that, because of the continuing appearance of the sums, his particular methods deserve re--exposure. For example, simple trigonometric generating functions are found and these have a field theoretic, Green function significance and I make a few comments in the topic of Rényi entropies.

hep-th↗
arXiv2015

Computation of the derivative of the Hurwitz zeta-function and the higher Kinkelin constants

I use the numerical values of the generalised Glaisher-Kinkelin-Bendersky (GKB) constants to give numerical values for the derivatives of the Hurwitz zeta function at negative integers, rather than the other way round. I point out that both Glaisher's numerical approach and Bendersky's recursion for the generalised gamma function were anticipated by Jeffery in 1862 who gave the value of the second constant as an example. I therefore propose that GKB become GKBJ.

math.NT↗
arXiv2015

Effective action for conformal spins on spheres with the multiplicative and conformal anomalies

Two multiplicative anomalies are evaluated for the determinant of the conformal higher spin propagating operator on spheres given by Tseytlin. One holds for the decomposition of the higher derivative product into its individual second order factors and the other applies to its complete linear factorisation. Using this last factorisation, I also calculate the determinant explicitly in terms of the Riemann zeta function, for both even and odd dimensions. In the latter case there is no anomaly of course. The conformal anomaly is also found for arbitrary spin and dimension.

hep-th↗
arXiv2014

Expansion of Rényi entropy for free scalar fields

An expression for the effective action of a conformal scalar on odd spheres allows a relatively simple computation of the expansion coefficients of the Rényi entropy for any odd dimension, d. Explicit values are listed for d=3,5 and 7. The alternative method, using a mapping to a flat conical manifold, is also employed, again for any odd dimension, and some mathematical details are presented on the computation of certain integrals.

hep-th↗
arXiv2014

The boundary F-theorem for free fields

The boundary free energy, as defined by Gaiotto, is further analysed for free scalars on a hemisphere and shown to be the same as the N-D determinant that earlier occurred in a treatment of GJMS operators. It is also shown to be identical, up to spin degeneracy, to the free energy for a spin-half field on the hemisphere boundary. This is also true if the hemisphere is replaced by a lune. The calculations are carried out in arbitrary dimensions.

hep-th↗
arXiv2014

Evaluation of spherical GJMS determinants

An expression in the form of an easily computed integral is given for the determinant of the scalar GJMS operator on an odd--dimensional sphere. Manipulation yields a sum formula for the logdet in terms of the logdets of the ordinary conformal Laplacian for other dimensions. This is formalised and expanded by an analytical treatment of the integral which produces an explicit combinatorial expression directly in terms of the Riemann zeta function, and $\log2$. An incidental byproduct is a (known) expression for the central factorial coefficients in terms of higher Bernoulli numbers.

math-ph↗
arXiv2014

Spherical Dirac GJMS operator determinants

Motivated by AdS/CFT, the extension is made to spin-half of a scalar calculation of the conformal anomalies and functional determinants of GJMS operators. The formal aspects are heuristic but sufficient. A Barnes zeta function representation again proves effective. The determinants are calculated for the two factorisations of the gamma-function (intertwiner) form of the GJMS operator, and shown to be equal, even including the multiplicative anomaly. A comment is made on the general eigenvalue problem and a few numerical results are presented. An alternative approach is detailed for odd dimensions and it is shown that the scalar determinants are expressed in terms of the spinor ones, and vice versa. An explicit, general form is given.

hep-th↗
arXiv2014

Imaginary mass lune determinants

Functional determinants for a scalar field with negative mass squared are numerically evaluated on an orbifolded three-sphere, in particular on a lune and on a regular 4--polytope fundamental domain. Graphs are provided of the logdets and some Hartle-Hawking probabilities, on the basis of the dS/CFT correspondence.

hep-th↗
arXiv2014

Imaginary mass lens space determinants

Functional determinants for a single scalar field with negative mass squared are evaluated on homogeneous lens spaces. For example, on even order spaces, the Hartle--Hawking wave function oscillates about its zeros with increasing amplitude as the (imaginary) mass increases. I also present results for the binary tetrahedral, octahedral and icosahedral factors of the three--sphere. The final answer is given as a quadrature and some graphs are drawn. In the technical evaluation of the infinite sums, the explicit form of the degeneracies is not needed.

hep-th↗
arXiv2014

Massive sphere determinants

An expression for the functional determinant on a sphere for a massive (scalar) field derived by Denef, Hartnoll and Sachdev using quasinormal modes is shown to exist already in the literature together with the multiplicative anomaly interpretation. The relevant expressions are outlined and several equivalent versions are given. The variation with mass is determined numerically. As an application of the derived formulae, the Hartle--Hawking probability of the Universe (via the dS/CFT correspondance) is recomputed. Agreement is found with Anninos, Denef and Harlow. The calculation is extended to all (odd) dimensions. I also compute the wave function which reveals an interesting feature.

hep-th↗
arXiv2013

Numerical evaluation of spherical GJMS determinants for even dimensions

The functional determinants of the GJMS scalar operators, P_{2k}, on even-dimensional spheres are computed via Barnes multiple gamma functions relying on the numerical availability of the digamma function. For the critical k=d/2 case, it is necessary to calculate the Stirling moduli. The multiplicative anomalies are given as odd polynomials in $k$ and it is emphasised that that the Dirichlet--to--Robin factorisation, P_{2l+1}, gives the same results as P_{2k} if k=l+1/2.The results are presented as graphs and show a series of extrema in the effective action as k is varied in the reals. For odd dimensions these extrema occur at integer k.

hep-th↗
arXiv2013

Numerical evaluation of spherical GJMS determinants

A numerical expression in the form of an integral is given for the determinant of the scalar GJMS operator on an odd--dimensional sphere. Manipulation yields a curious sum formula for the logdet in terms of the logdets of the ordinary conformal Laplacian for other dimensions. A few graphs are drawn.

math-ph↗
arXiv2013

Remarks on Lubbock's summation formulae

The coefficients occurring in summation formulae of the Lubbock type are shown to be generalised Bernoulli polynomials which turn up in subdivision questions such as quantum field theory around a conical singularity and on spherical lunes. An image interpretation is made, generating functions are brought in and some trigonometric summations encountered. A formal Lubbock formula is introduced for a general delta operator.

math.NA↗
arXiv2013

Poweroids revisited - an old symbolic approach

Jeffery's 1861 computations using finite difference calculus are resurrected and extended from forward differences to general delta operators and used to neatly prove theorems in the Rota--Mullins theory of polynomials of binomial type (Steffensen's poweroids) allowing, for example, compact treatments of umbral composition, the binomial property and the connection constants. It is shown that it forms a legitimate alternative to the usual umbral device and also anticipates a number of results obtained more recently.

math.CO↗