Hadamard renormalization of the stress-energy tensor for a quantized scalar field in a general spacetime of arbitrary dimension

Abstract

We develop the Hadamard renormalization of the stress-energy tensor for a massive scalar field theory defined on a general spacetime of arbitrary dimension. Our formalism could be helpful in treating some aspects of the quantum physics of extra spatial dimensions. More precisely, for spacetime dimension up to six, we explicitly describe the Hadamard renormalization procedure and for spacetime dimension from seven to eleven, we provide the framework permitting the interested reader to perform this procedure explicitly in a given spacetime. We complete our study (i) by considering the ambiguities of the Hadamard renormalization of the stress-energy tensor and the corresponding ambiguities for the trace anomaly, (ii) by providing the expressions of the gravitational counterterms involved in the renormalization process (iii) by discussing the connections between Hadamard renormalization and renormalization in the effective action. All our results are expanded on standard bases for Riemann polynomials constructed from group theoretical considerations and thus given on irreducible forms.

Static self-forces in a five-dimensional black hole spacetime

“…We present only a brief overview here and we refer the reader to the detailed description of Hadamard renormalization in arbitrary dimensions given in Ref. [32].…”

“…Differentiating with respect to γ gives 32) which permits us to sum the modes in our electrostatic potential to obtain the integral representation…”

Dirac fermions on an anti-de Sitter background

“…The Hadamard renormalization technique has been extensively studied for scalar fields [12]. Since the Dirac equation is a first order differential equation, it is convenient to introduce the auxiliary bi-spinor G F , defined by analogy with flat space-time by [4]:…”

Hawking Radiation from Higher-Dimensional Black Holes

“…One method of renormalization is covariant geodesic point separation, in which the two operators whose products are taken are evaluated at different space-time points x and x ′ , yielding a finite bitensor stress-energy tensor, whose expectation value is written Tµν (x, x ′ ). This expectation value is renormalized by the subtraction of purely geometric, state independent renormalization terms T div µν (x, x ′ ) (see for example [48,49] for expressions for these geometric terms for fields of spin 0, 1/2 and 1 in four dimensions, and [50] for a scalar field in higher-dimensional spacetime). The points x and x ′ are then brought together and a finite renormalized expectation value for the stress-energy tensor is yielded.…”