The transition from classical to quantum field theory replaces deterministic, continuous-valued physical fields (governed by classical wave equations and carrying energy/information directly) with quantized fields whose complex-valued wave functions yield only probability distributions over observables, formalized via the generalization of the classical Lagrangian (a function of generalized coordinates and their time derivatives) to the Lagrangian density (a function of fields and their spacetime derivatives). The Euler-Lagrange equations of motion, derivable from either the Lagrangian or the Hamiltonian (related via the Legendre transformation using conjugate momenta), generalize correspondingly from particle mechanics to field theory, with the principle of least action remaining the unifying variational principle in both regimes. This body of theory belongs to theoretical/mathematical physics, specifically the Lagrangian and Hamiltonian formalisms as they extend from classical mechanics and classical field theory into relativistic quantum field theory.
The transition from classical to quantum field theory replaces deterministic, continuous-valued physical fields (governed by classical wave equations and carrying energy/information directly) with qu…