  Classical physics is reformulated as a constrained Hamiltonian system in the
history phase space. Dynamics, i.e. the Euler-Lagrange equations, play the role
of first-class constraints. This allows us to apply standard methods from the
theory of constrained Hamiltonian systems, e.g. Dirac brackets and
cohomological methods. In analogy with BRST quantization, we quantize in the
history phase space first and impose dynamics afterwards. To obtain a truly
covariant formulation, all fields must be expanded in a Taylor series around
the observer's trajectory, which acquires the status of a quantized physical
field. The formalism is applied to the harmonic oscillator and to the free
scalar field. Standard results are recovered, but only in the approximation
that the observer's trajectory is treated as a classical curve.
