Schwinger's quantum action principle

A formulation of quantum dynamics in which variations of the action generate equations of motion and relations among quantum states and operators. It extends the classical action principle to quantum transition amplitudes.

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Action functional: A functional that assigns a numerical action to each possible history of a physical system. The principle varies this quantity to derive quantum dynamics.
Hamiltonian mechanics: A formulation of classical mechanics based on generalized coordinates, conjugate momenta, and a Hamiltonian. Its canonical variables and equations provide a starting point for quantum dynamics.
Julian Schwinger: An American theoretical physicist who helped develop quantum electrodynamics and formulated a variational principle for quantum dynamics. He introduced and developed the action principle bearing his name.
Quantum field theory: A framework that describes particles and interactions using quantized fields. The principle is widely applied to derive field-theoretic equations and identities.
Feynman path integral: A formulation of quantum mechanics that computes amplitudes by summing contributions from possible paths. It foregrounds a sum over histories, unlike Schwinger’s variation of transition amplitudes.
Transition amplitude: A complex number whose squared magnitude gives the probability of a specified quantum transition. Variations of the action determine changes in transition amplitudes.
Lagrangian mechanics: A formulation of mechanics in which motion follows from a Lagrangian, usually kinetic minus potential energy. Its action integral is the quantity varied in the quantum principle.
Quantum electrodynamics: The quantum field theory of electromagnetic interactions between charged particles and photons. Schwinger applied his formulation to derive relations in this theory.
Green's function: A function that describes a system’s response to a localized source or encodes propagation between points. Variational equations for quantized fields often determine relations among Green's functions.
Canonical quantization: A procedure that turns classical canonical variables into operators with prescribed commutation relations. It begins with operator rules that the action principle can instead derive from variations.