Spacetime Dependence of Local Temperature in Relativistic Quantum Field Theory

The spacetime dependence of the inverse temperature four-vector $\boldsymbol{\beta}$ for certain states of the quantized Klein-Gordon field on (parts of) Minkowski spacetime is discussed. These states fulfill a recently proposed version of the Kubo-Martin-Schwinger (KMS) boundary value condition, the so-called "local KMS (LKMS) condition". It turns out that, depending on the mass parameter $m\geq 0$, any such state can be extended either (i) to a LKMS state on some forward or backward lightcone, with $\boldsymbol{\beta}$ depending linearily on spacetime, or (ii) to a thermal equilibrium (KMS) state on all of Minkowski space with constant $\boldsymbol{\beta}$. This parallels previously known results for local thermal equilibrium (LTE) states of the quantized Klein-Gordon field. Furthermore, in the case of a massless field our results point to a discrepancy with some classic results in general approaches to (non-quantum) relativistic thermodynamics.