From vacuum restriction to thermal time

3. Vacuum restriction and geometric thermality

Let $D=\{(t,x):|t|+r\lt R\}$ with $r=|x|$. Assume the vacuum is cyclic and separating for the relevant local von Neumann algebras, the net admits the conformal wedge-to-diamond map, and the vacuum wedge satisfies Bisognano–Wichmann 5, 6. On the right wedge $X^1>|X^0|$, a future boost sends $X^1\pm X^0$ to $e^{\pm\eta}(X^1\pm X^0)$. With Heisenberg evolution $\alpha_t(A)=e^{itH}Ae^{-itH}$ and modular convention $\sigma_s=\operatorname{Ad}\Delta^{is}$, the wedge modular group is the boost at rapidity $-2\pi s$.

The wedge-to-diamond map

Set $b=-e_1/(2R)$ and $D_b=1-2b\cdot X+b^2X^2$, using the Lorentzian inner product. The explicit map is

$$x=\frac{X-bX^2}{D_b}+2R^2b.$$

On the wedge, $4R^2D_b=4R^2+4RX^1+X^2>0$. The null-tip identities are

$$(R\pm t)^2-r^2=\frac{2R(X^1\pm X^0)}{D_b}.$$

The connected wedge contains $X=2Re_1$, which maps to the center, so the image lies in $D$. To invert, set $Y=x-2R^2b$ and use

$$X=\frac{Y+bY^2}{1+2b\cdot Y+b^2Y^2}.$$

The inverse denominator is $((R-x^1)^2+|x_\perp|^2-t^2)/(4R^2)>0$ on $D$. The null-tip identities imply $X^1\pm X^0>0$. Differentiation gives the metric pullback $D_b^{-2}\eta$.

The hyperbolic cylinder

On $\mathbb R\times H^{d-1}$ put $v=\tau/R$. Use the map

$$t=\frac{R\sinh v}{\cosh u+\cosh v},\qquad r=\frac{R\sinh u}{\cosh u+\cosh v},\qquad r\pm t=R\tanh\!\left(\frac{u\pm v}{2}\right).$$

The angular coordinates are unchanged. The inverse takes $\operatorname{artanh}((r\pm t)/R)$ and adds or subtracts the results. The Minkowski metric pulls back to

$$(\cosh u+\cosh v)^{-2} \left[-d\tau^2+R^2\left(du^2+\sinh^2u\,d\Omega_{d-2}^2\right)\right].$$

For $d=2$, two radial directions from the center give the full hyperbolic line. Conformal covariance supplies the net isomorphism, including Weyl weights in a field description. Transporting the state then gives the smeared correlation functions.

The modular flow and charge

The unit future boost pushes forward to $R\partial_\tau$. In Minkowski coordinates,

$$\partial_\tau=\frac{1}{2R^2}\left[(R^2-t^2-r^2)\partial_t-2tr\partial_r\right].$$

This follows by differentiating the wedge map or the diamond null-coordinate flow. On $t=0$, it fixes the formal one-sided dimensionless stress-tensor generator, up to scalar and regulator-dependent terms:

$$2\pi\int_{r\lt R}\frac{R^2-r^2}{2R}\,T_{00}(0,x)\,d^{d-1}x.$$

The continuum modular implementer is the two-sided operator on the vacuum representation.

Theorem 3.1. Physical inverse temperature

With these hypotheses, the transported state on the hyperbolic cylinder is Kubo–Martin–Schwinger (KMS) at $\beta=2\pi R$ for future $\tau$-translations.

Proof. The two algebra transports give $\sigma_s=\alpha_{-2\pi Rs}$. For analytic elements, the modular boundary identity is $\omega(A\sigma_{-i}(B))=\omega(BA)$. Rescaling the modular strip gives a KMS strip of width $2\pi R$ for $\omega(A\alpha_\tau(B))$, with the reversed product on the upper edge. The algebraic proof controls the combined boundary values. It requires no separate analytic continuation of individual Weyl factors in field coordinates.

What thermality does not supply

If the diamond algebra is type III, it has no intrinsic density matrix defined with a normal semifinite trace. A split inclusion, where available, can supply a type I intermediate algebra for a separated inner diamond. On a type I factor with a faithful reduced density matrix $\rho$, modular action reduces to $\rho^{is}A\rho^{-is}$. In a cyclic and separating purification with nonzero Schmidt weights $p_i$, the modular eigenvalues in the Schmidt matrix-unit basis are $p_i/p_j$. This gives the precise link to finite modular readouts in the Canonical Spectral-Curvature Field 4.

The choice of type I factor and finiteness of its entropy require separate control. The continuum KMS result alone supplies neither a thermal trace nor a finite partition function on the infinite hyperbolic volume.