Editing Quantization (physics) (section)

== Covariant canonical quantization ==

There is a way to perform a canonical quantization without having to resort to the non covariant approach of [[foliation|foliating]] spacetime and choosing a [[Hamiltonian (quantum mechanics)|Hamiltonian]]. This method is based upon a classical action, but is different from the functional integral approach.

The method does not apply to all possible actions (for instance, actions with a noncausal structure or actions with [[analysis of flows|gauge "flows"]]). It starts with the classical algebra of all (smooth) functionals over the configuration space. This algebra is quotiented over by the ideal generated by the [[Euler–Lagrange equation]]s. Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called the [[Peierls bracket]]. This Poisson algebra is then ℏ -deformed in the same way as in canonical quantization.

In [[quantum field theory]], there is also a way to quantize actions with [[analysis of flows|gauge "flows"]]. It involves the [[Batalin–Vilkovisky formalism]], an extension of the [[BRST formalism]].