Quantum energy inequalities and local covariance. I. Globally hyperbolic spacetimes

By a stationary state, we mean one whose $n$-point functions are invariant under translations along the Killing flow: $wn(ψt(x1),…,ψt(xn))=wn(x1,…,xn)$⁠, where $ψt$ is the group of isometries associated with the Killing field.

The proof employed in Refs. 18 and 16 proceeds by defining a function $V:suppg→R$ with $V′(v)=1∕g(v)2$⁠, and therefore only applies in the first instance to the case where $g$ has connected support and no zeros in the interior thereof. We extend the result to more general $g∊C0∞(I;R)$ by choosing a nonnegative $χ∊C0∞(I)$ with no zeros in the interior of its support, assumed to be connected, and which is equal to unity on the support of $g$⁠. Applying Flanagan’s result to $gε(τ)=g(τ)2+ε2χ(τ)2$⁠, we may take the limit $ε→0$ to obtain Eq. (25) (cf. Cor. A.2 in Ref. 20, where the notation $G=g2$ is used).

We have $g̃(τ)2=g(τ+τ0)2$⁠, and want to show that $g̃(τ)2=g′(τ+τ0)2$ for all $τ$⁠. Differentiating and squaring yields $4g̃(τ)2g̃(τ)2=4g(τ+τ0)2g′(τ+τ0)2$ from which it follows that $g̃(τ)2=g′(τ+τ0)2$ except perhaps at zeros of $g̃(τ)$⁠. If $g̃$ vanishes in a neighborhood of $τ$ then so must $g̃$ and the result holds trivially. For the remaining case, choose a sequence $τn→τ$ with $g̃(τn)≠0$⁠; since $g̃(τn)2=g′(τn+τ0)2$⁠, we conclude the required result by continuity as $n→∞$⁠.

To see this, we note that $Q2(coshα)=tanhα+α(1−tanh2α)$ and that the maximum of this expression on $R+$ occurs at the (unique) solution to $αtanhα=1$⁠, which is numerically $α0=1.199679$⁠. Now $Q2(coshα0)=α0$⁠, so we find that $Q2(x)⩽α0<1.2$ on [$1,∞$⁠) as claimed.

An operator $A$ is symmetric on a domain $D$ if $⟨ψ∣Aφ⟩=⟨Aψ∣φ⟩$ for all $ψ,φ∊D$⁠, which shows that the adjoint $A*$ agrees with $A$ on $D$⁠, but does not exclude the possibility that $A*$ has a strictly larger domain of definition than $A$⁠.

The contour involved is the rectangle with “long” sides given by the interval $[−R,R]$ of the real axis, and its translate $[−R,R]+2πξoi$⁠. The contour encloses a single pole, of fourth order, at $z=2iϵ$ and the contribution of the “short” sides vanishes as $R→∞$⁠. One also exploits the fact that the contributions from the two “long” sides are equal up to a factor of $−e−2πξo$⁠.

The Friedrichs extension has a domain contained in the closure of $C0∞(I)$ in the norm $∥g∥+12=⟨g∣g⟩+⟨g∣Lg⟩$⁠, which is equivalent to the norm of the Sobolev space $W02(I)$ since $τ−4$ is bounded and bounded away from zero on $I$⁠. Accordingly, the closure of $C0∞(I)$ is precisely $W02(I)$ and the desired domain lies in this Sobolev space, all elements of which obey $g=g′=0$ on $∂I$⁠.

For a scalar field with arbitrary curvature coupling constant $ε$⁠, replaced the numerical coefficient $K(a)$ in the stress tensor with $Kε(a)=−∑n=1∞[cosech4(na∕2)+4εcosech2(na∕2)]$⁠.

To generalize the result to a scalar field with arbitrary curvature coupling constant $ζ$⁠, replace the 11 in the numerator with $(11–60ζ)$⁠.

Note that the weight function in Ref. 21 was parametrized in terms of $η$⁠, rather than proper time $τ$⁠: our $g(τ)$ is related to the $f(η)$ of Ref. 21 by $g(τ)=f(τ∕ξo)∕ξo$⁠.