Light-cone thermodynamics: purification of the Minkowski vacuum

Abstract

We explicitly express the Minkowski vacuum of a massless scalar field in terms of the particle notion associated with suitable spherical conformal killing fields. These fields are orthogonal to the light wavefronts originating from a sphere with a radius of $r_{H}$ in flat spacetime: a bifurcate conformal killing horizon that exhibits semiclassical features similar to those of black hole horizons and Cauchy horizons of spherically symmetric black holes. Our result highlights the quantum aspects of this analogy and extends the well-known decomposition of the Minkowski vacuum in terms of Rindler modes, which are associated with the boost Killing field normal to a pair of null planes in Minkowski spacetime (the basis of the Unruh effect). While some features of our result have been established by Kay and Wald's theorems in the 90s -- on quantum field theory in stationary spacetimes with bifurcate Killing horizons -- the added value we provide here lies in the explicit expression of the vacuum.

Figures (2)

• Figure 1: Integral lines of the spherical conformal Killing field \ref{['kikin']}. The vector field becomes null on the light wave-fronts emanating from a sphere of radius $r_H$ (by choice here at $t=0$). These light fronts are bifurcate conformal Killing horizons with bifurcation surface given by the same sphere.
• Figure 2: The causal character of the spherical conformal Minkowski Killing vector field divides flat spacetime in six different regions. The shaded regions correspond to those where the conformal Killing vector field is spacelike. Fock spaces can be constructed according to the positive frequency notion associated with the Killing time in the four white regions where the vector field is timelike. We call such Hilbert spaces ${\mathfs {F}}_I$, ${\mathfs {F}}_{II}$, ${\mathfs {F}}_{III}$, and ${\mathfs {F}}_{-III}$ . The Fock space constructed from positive frequency solutions in inertial time will be called ${\mathfs {F}}$. Solutions of the massless Klein-Gordon equation in Minkowski spacetime can be fully characterized by their value on the portions of null surfaces emphasized in red. This is the key for the different ways one can express the Minkowski vacuum presented in equations \ref{['UNO']}, \ref{['CUATRO']}, \ref{['DOS']}, and \ref{['TRES']}.