2.9 An explicit calculation

Consider an infinite (d − 2)-plane in d-dimensional spacetime. The calculation of the entanglement entropy for this plane can be done explicitly by means of the heat kernel method. In flat spacetime, if the operator 𝒟 is the Laplace operator,
𝒟 = − ∇2 ,
one can use the Fourier transform in order to solve the heat equation. In d spacetime dimensions one has
′ 1 ∫ d ipμ(Xμ− X′μ) −sF(p2) K (s,X, X ) = (2π-)d d pe e . (23 )
Putting zi = z′i, i = 1,..,d − 2 and choosing in the polar coordinate system (r,ϕ), that ϕ = ϕ′ + w we have that pμ(X − X ′)μ = 2pr sin w-cos𝜃 2, where p2 = p μpμ and 𝜃 is the angle between the d-vectors pμ and (X μ− X ′μ). The radial momentum p and angle 𝜃, together with the other (d− 2) angles form a spherical coordinate system in the space of momenta μ p. Thus, one has for the integration measure ∫ ∫ ∫ ddp = Ωd −2 ∞0 dppd−1 π0 d 𝜃 sind−2 𝜃, where (d−1)∕2 Ωd− 2 = 2Γπ ((d−1)∕2)- is the area of a unit radius sphere in d − 1 dimensions. Performing the integration in Eq. (23View Equation) in this coordinate system we find
√-- Ωd− 2 π Γ (d−2-1) ∫ ∞ d w −sp2 K (s,w, r) = -(2π-)d-(r-sin-w)(d−2)∕2 dpp 2Jd−22(2rp sin 2-)e . (24 ) 2 0
For the trace one finds
---s----π-α-- TrK (s,w ) = (4πs)d2 sin2 w-A(Σ ), (25 ) 2
where ∫ A(Σ ) = dd− 2z is the area of the surface Σ. One uses the integral ∫ 1−ν ∞0 dxx1 −νJν(x) = 2Γ (ν) for the derivation of Eq. (25View Equation). The integral over the contour Γ in the Sommerfeld formula (22View Equation) is calculated via residues ([69Jump To The Next Citation Point, 101Jump To The Next Citation Point])
∫ C2(α) ≡ -i-- cot-w- -dw---= -1-(1 − α2 ). (26 ) 8πα Γ 2α sin2 w2- 6α2
Collecting everything together one finds that in flat Minkowski spacetime
1 TrK α(s) = (4-πs)d∕2-(αV + 2π αC2 (α )s A(Σ )), (27 )
where ∫ V = dτ dd−1x is the volume of spacetime and ∫ A (Σ ) = dd−2x is the area of the surface Σ. Substituting Eq. (27View Equation) into Eq. (20View Equation) we obtain that the effective action contains two terms. The one proportional to the volume V reproduces the vacuum energy in the effective action. The second term proportional to the area A(Σ ) is responsible for the entropy. Applying formula (16View Equation) we obtain the entanglement entropy
S = ---------A(Σ-)-------- (28 ) 6(d − 2)(4π)(d−2)∕2𝜖d−2
of an infinite plane Σ in d spacetime dimensions. Since any surface, locally, looks like a plane, and a curved spacetime, locally, is approximated by Minkowski space, this result gives the leading contribution to the entanglement entropy of any surface Σ in flat or curved spacetime.
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