2.8 Heat kernel and the Sommerfeld formula

Consider for concreteness a quantum bosonic field described by a field operator 𝒟 so that the partition function is Z = det− 1∕2 𝒟. Then, the effective action defined as
1 ∫ ∞ ds W = − 2- 2 -s-TrK (s), (20 ) 𝜖
where parameter 𝜖 is a UV cutoff, is expressed in terms of the trace of the heat kernel K (s,X, X ′) = < X |e−s𝒟|X ′ >. The latter is defined as a solution to the heat equation
{ (∂s + 𝒟 )K (s,X, X ′) = 0, K (s=0, X, X ′) = δ(X,X ′). (21 )
In order to calculate the effective action W (α ) we use the heat kernel method. In the context of manifolds with conical singularities this method was developed in great detail in [69Jump To The Next Citation Point, 101Jump To The Next Citation Point]. In the Lorentz invariant case the invariance under the abelian symmetry ϕ → ϕ + w plays an important role. The heat kernel ′ K (s,ϕ, ϕ) (where we omit the coordinates other than the angle ϕ) on regular flat space then depends on the difference (ϕ − ϕ ′). This function is 2π periodic with respect to (ϕ − ϕ ′). The heat kernel K α(s,ϕ,ϕ ′) on a space with a conical singularity is supposed to be 2πα periodic. It is constructed from the 2 π periodic quantity by applying the Sommerfeld formula [207]
′ ′ i ∫ w ′ K α(s,ϕ,ϕ ) = K (s,ϕ − ϕ ) + ---- cot---K (s,ϕ − ϕ + w )dw . (22 ) 4πα Γ 2α
That this quantity still satisfies the heat kernel equation is a consequence of the invariance under the abelian isometry ϕ → ϕ + w. The contour Γ consists of two vertical lines, going from (− π + i∞ ) to (− π − i∞ ) and from (π − i∞ ) to (π − +i ∞ ) and intersecting the real axis between the poles of the cot w- 2α: − 2π α, 0 and 0, +2 πα, respectively. For α = 1 the integrand in Eq. (22View Equation) is a 2π-periodic function and the contributions of these two vertical lines cancel each other. Thus, for a small angle deficit the contribution of the integral in Eq. (22View Equation) is proportional to (1 − α).
  Go to previous page Go up Go to next page