Question

What is Hawking Hartle vacuum state and why does the following Euclidean path integral gives the wave functional of it?

Answer 1

I am studying the wave function of black hole via the paper by Sergey Solodukhkin, Entanglement entropy of black holes,arXiv:hep-th: 1104.3712. In the paper, equation (53) is as follows: Ψ[ψ−(x),ψ+(x)]=∫ψ(x)|ϕ=0=ψ+(x)ψ(x)|ϕ=0=ψ−(x)Dψe−W[ψ] where W[ψ]=12∫ψD^ψ is the action of the quantum field ψ, and the path integral is defined over field configuration on the half-period Euclidean instanton defined by the metric: ds2=ρ2dϕ2+dρ2+γijdθidθj , where ϕ varies from 0 to 2π. Then the author claims that the path integral is the Hartle Hawking vacuum state.  What confuse me is that what on earth is Hartle Hawking state and how can a path integral over finite imaginary time project out all the state other than vacuum? Wish somebody can give some reference related to Hartle Hawking state and give me some explanation on my confusion

Answer 2

Hello mate here is your answer.

where W[ψ]=12∫ψD^ψW[ψ]=12∫ψD^ψ is the action of the quantum field ψψ, and the path integral is defined over field configuration on the half-period Euclidean instanton defined by the metric:

ds2=ρ2dϕ2+dρ2+γijdθidθjds2=ρ2dϕ2+dρ2+γijdθidθj

, where ϕϕ varies from 0 to 2π2π. Then the author claims that the path integral is the Hartle Hawking vacuum state.

Hope it helps you.