Operator Ordering in the Relativistic Quantization: Specific Heat in the Rindler Frame

We introduce a covariant canonical quantization for a particle in curved spacetime that tracks operator-ordering ambiguities. Parameterizing spatial and temporal ordering, we derive a Hermitian Hamiltonian with leading quantum-relativistic corrections. In a uniformly accelerated frame, we show the semiclassical heat-capacity approximation misses these effects and then develop a perturbative quantum treatment using Airy-function modes to obtain analytical first- and second-order energy shifts. Including these shifts in the partition function yields nontrivial, ordering-dependent specific-heat corrections. Numerical studies for electrons in extreme electric fields and ultra-light particles in strong gravitational fields demonstrate that these corrections become significant at intermediate temperatures. Enforcing the Tolman-Ehrenfest relation for spatial temperature variation further modulates the heat-capacity profile. Our results suggest that precision calorimetry in laser-acceleration or analogue gravity setups could probe quantum-ordering effects in relativistic regimes.