Exploring the Rindler vacuum and the Euclidean plane

If we consider a massive scalar field, then the corresponding kernel will involve the mass m and is related to the massless heat kernel by the simple relation: $Km≠0(x,x0;s)=e−im2⁡sKm=0(x,x0;s)$ so that it is enough to concentrate on the massless case, most of the time.

Defining the Schwinger kernel through an explicit solution of Eq. (10) is somewhat different—conceptually and technically—from defining the kernel simply as a solution to the partial differential equation [Eq. (7)]. In the latter approach, one has a certain level of ambiguity in the solution since one could add any solution to the homogeneous part of the equation. This ambiguity is removed when we define the kernel directly through Eq. (10). Therefore, one cannot claim that the difference between the Rindler vacuum and the Minkowski vacuum can be taken care of by adding a suitable solution to the homogeneous part of the differential equation in Eq. (7).

In fact, the result in Eq. (4) holds for a wider class of functions, viz., all even functions $FR(τ)$⁠. Such a function can be expressed as a superposition of exp(−iω|τ|) with positive and negative ω with an weightage factor that is even in ω. We will comment on this generalization toward the end.

We mention the following point to avoid possible misunderstanding. Since $FM(r)(x)≡FM(x+iπ)$⁠, these two functions are related by a shift through a purely imaginary quantity, viz., iπ. For shifts by real ℓ, the Fourier transforms of $FM(x)$ and $fM(x)≡FM(x+ℓ)$⁠, with respect to x, are related by $f̃M(ν)=e−iℓνF̃M(ν)$⁠, where the tilde over a function indicates the Fourier transform. In general, one cannot generalize this result to the case of a purely imaginary value of ℓ, say ℓ = iy, where $y∈R$⁠, and write $f̃M(ν)=eyνF̃M(ν)$⁠. This is because shifting the contour in the complex plane may not, in general, be legitimate. In that case, $F̃M(r)(ω)$ has to be computed by a careful, independent, calculation. This happens to be the situation here, and we will discuss later an explicit example in which this fact will become clearer.

If one approaches the problem via standard canonical quantization, this question has a seemingly natural answer. We first obtain the relevant positive frequency modes in Rindler and Minkowski coordinates and define the corresponding Rindler and Minkowski vacua. The propagators are then defined as the expectation values of products of field operators in these two vacua. However, we are approaching it from the point of view of solutions to a particular differential equation. It is then necessary to understand how the distinction between the two vacua is encoded in these solutions.

In the context of quantum mechanics (QM) in a two-dimensional plane, one would not dream of writing the Dirac delta function with just δ(θ − θ₀). In QM, the kernel evolves an initial wave function to a final wave function; in fact, K(x, x₀; s) is indeed the wave function for a particle at time s if it started out as a delta function, δ(x, x₀) at time s = 0. Hence, to justify the choice of the delta function with a factor δ(θ − θ₀), one should be able to interpret it as the initial wave function of a particle in the plane. This is tricky because one would expect the wave functions to be single-valued. Since θ and θ + 2π represent the same physical point, the function δ(θ − θ₀) cannot be an acceptable choice for the initial wave function. Therefore, the kernel K_n=0 never arises in the context of, say, normal QM.

Put y = cos θ, where 0 ≤ θ ≤ π, so that $iy+1−y2=ie−iθ$⁠. Therefore, $cos−1(y)=π2−i⁡log(ie−iθ)=θ$⁠.

The consideration of this condition also appears in another important context. This also concerns the interrelationship between $G̃M(ν)$ and $G̃M(r)(ν)$⁠. Note that since the right-hand side of Eq. (91) is symmetric under ρ ↔ ρ^′, we could as well make the replacement ρ → ρ_< and ρ^′ → ρ_> in this equation. Now, we could have arrived at the expression hence obtained, by replacing ρ_< → e^−iπρ_< in Eq. (90). Since ρ_< → e^−iπρ_< is also essentially “reflection,” this provides another independent route of arriving at Eq. (91). However, even though ρ_> → e^−iπρ_> also implies a “reflection,” we cannot make this replacement in Eq. (90) to derive $G̃M(r)$⁠. This is, once again, because such a replacement would imply the choice of parameters ξ_R = (τ − τ^′), ξ_I = 0, α = e^i(π−ϵ)μρ_<, and β = μρ_>, which violates the necessary condition for the convergence of Eq. (86).

For large |u|, the leading order asympotic expansion for I_u(z) is given by $Iu(z)∼[Γ(1+u)]−1z2u$⁠. This implies that $Iiν(μρ<)Kiν(μρ>)∼12νiρ>ρ<−Im[ν]+iRe[ν];Im[ν]→−∞.$ Clearly, the RHS vanishes as Im[ν] → −∞ when ρ_> > ρ_<.