can define the determinant using zeta function regularization. See "analytic torsion." suggested using this idea to evaluate path integrals in curved spacetimes. He studied zeta function regularization in order to calculate the partition functions for thermal graviton and matter's quanta in curved background such as on the horizon of black holes and on de Sitter background using the relation by the inverse Mellin transformation to the trace of the kernel of heat equations. Example The first example in which zeta function regularization is available appears in the Casimir effect, which is in a flat space with the bulk contributions of the quantum field in three space dimensions. In this case we must calculate the value of Riemann zeta function at -3, which diverges explicitly. However, it can be analytically continued to s=-3 where hopefully there is no pole, thus giving a finite value to the expression. A detailed example of this regularization at work is given in the article on the detail example of the Casimir effect, where the resulting sum is very explicitly the Riemann zeta-function (and where the seemingly legerdemain analytic continuation removes an additive infinity, leaving a physically significant finite number). An example of zeta-function regularization is the calculation of the vacuum expectation value of the energy of a particle field in quantum field theory. More generally, the zeta-function approach can be used to regularize the whole energy–momentum tensor in curved spacetime. The unregulated value of the energy is given by a summation over the zero-point energy of all of the excitation modes of the vacuum: Here, is the zeroth component of the energy–momentum tensor and the sum (which may be an integral) is understood to extend over all (positive and negative) energy modes ; the absolute value reminding us that the energy is taken to be positive. This sum, as written, is usually infinite ( is typically linear in n). The sum may be regularized by writing it as where s is some parameter, taken to be a complex number. For large, real s greater than 4 (for three-dimensional space), the sum is manifestly finite, and thus may often be evaluated theoretically. The zeta-regularization is useful as it can often be used in a way such that the various symmetries of the physical system are preserved. Zeta-function regularization is used in conformal field theory, renormalization and in fixing the critical spacetime dimension of string theory. Relation