Spectral Functions in Mathematics and Physics

On the Casimir energy for a massive quantum scalar field and the cosmological constant

“…Indeed, the only possible covariant form for the energy of the quantum vacuum is (12b), which is mathematically equivalent to the Cosmological constant Λ by Λ = 8πGρ vac (28) (12b) takes the form of a perfect fluid with energy density ρ vac and isotropic pressure p vac = −ρ vac which corresponds to (1b) of the introduction. Note that (1a) is universally true as an intrinsic property of the quantum vacuum ( [9], [10]), which, together with (1b), implies (1c).…”

“…In going over from (8) to a compact region K with boundary ∂K, it is convenient to rewrite the Wick product in (8) using the point-splitting technique ( [4],p.316; [6], appendix) and reexpressing the field in terms of different creation and annihilation operatorsã k associated to the eigenfunctions given below in (11d),(11e)(see (10) of [4] for details). These fields, denoted byφ, define a new representation, corresponding to a different observable algebra ( the Hilbert space of test functions is no longer L 2 ((R) 3 ), see ([5], (2.13) et ff), a new density T 00 (x), which results from writing (6) in the new representation, and a new vacuum Ω, annihilated by the new absorption operators associated to the eigenfunctions.…”

“…It should be stressed that there exists an immense literature on the Casimir effect. In particular, zeta function methods have been studied during several decades and are also, in principle, applicable to the problem treated in section III, see [9], [10] and references given there, in particular, reference [10] for rigorous results.…”

“…However, a.) is included because it is universally true as a well-known intrinsic property of the vacuum state ( [9], [10]) and we wish to preserve it for our class of states.…”

Zeta functions in brane world cosmology

“…The first is to compute a zeta function where the operator spectrum is not known, for which we adopt the technique developed in Refs. [26,27,28,29,30,31,32]. This general technique, elaborated in various forms, allows one to obtain the zeta function using only the knowledge of the basis functions.…”