Canonical quantization of massive vector field in Schwarzschild black hole background

We perform a first-principles canonical quantization of a massive vector field, often referred to as the Proca field, in a Schwarzschild spacetime background. While scalar, fermionic, and electromagnetic fields are well studied in this context, the Proca field requires a more nuanced treatment because of the physical nature of the longitudinal polarization mode and the constrained dynamics of the field variables. By implementing the Dirac bracket formalism to treat the constraints inherent in the Proca action, we derive a consistent framework for the commutator algebra of creation and annihilation operators. Following this construction, we define the usual Boulware, Unruh, and Hartle-Hawking vacua. Using the Unruh vacuum, we derive and analyze the Hawking spectrum of the Proca field. Furthermore, we numerically evaluate the Proca condensate constructed from the two-point correlation function $\langle A_\mu(x) A_\nu(x') \rangle$, defined on all three vacuum states. We find that the condensate becomes significant near the boundary of the future horizon. Our results highlight the interplay among the different polarization modes and the significance of the Proca mass in quantum observables.