Schwinger's quantum action principle is a variational approach to quantum mechanics and quantum field theory. This theory was introduced by Julian Schwinger in a series of articles starting 1950.
Approach In Schwinger's approach, the action principle is targeted towards quantum mechanics. The action becomes a quantum action, i.e. an operator, S {\displaystyle S} . Although it is superficially different from the path integral formulation where the action is a classical function, the modern formulation of the two formalisms are identical. Suppose we have two states defined by the values of a complete set of commuting operators at two times. Let the early and late states be | A ⟩ {\displaystyle |A\rangle } and | B ⟩ {\displaystyle |B\rangle } , respectively. Suppose that there is a parameter in the Lagrangian which can be varied, usually a source for a field. The main equation of Schwinger's quantum action principle is:
δ ⟨ B | A ⟩ = i ⟨ B | δ S | A ⟩ , {\displaystyle \delta \langle B|A\rangle =i\langle B|\delta S|A\rangle ,\ }
where the derivative is with respect to small changes ( δ {\displaystyle \delta } ) in the parameter, and S = ∫ L d t {\displaystyle S=\int {\mathcal {L}}\,\mathrm {d} t} with L {\displaystyle {\mathcal {L}}} the Lagrange operator. In the path integral formulation, the transition amplitude is represented by the sum over all histories of exp ( i S ) {\displaystyle \exp(iS)} , with appropriate boundary conditions representing the states | A ⟩ {\displaystyle |A\rangle } and | B ⟩ {\displaystyle |B\rangle } . The infinitesimal change in the amplitude is clearly given by Schwinger's formula. Conversely, starting from Schwinger's formula, it is easy to show that the fields obey canonical commutation relations and the classical equations of motion, and so have a path integral representation. Schwinger's formulation was most significant because it could treat fermionic anticommuting fields with the same formalism as bose fields, thus implicitly introducing differentiation and integration with respect to anti-commuting coordinates.
Derivation of field equations and commutation relations The Schwinger quantum action principle provides a unified framework for deriving both the equations of motion and the quantization conditions (commutation relations) for a quantum field theory. Unlike the canonical formalism, which postulates commutation relations, or the path integral formalism, which relies on functional integration, Schwinger's method derives these properties directly from the variation of the action operator S ^ {\displaystyle {\hat {S}}} .
The variational principle The principle asserts that the variation of the transformation amplitude between two spacelike surfaces σ 1 {\displaystyle \sigma _{1}} and σ 2 {\displaystyle \sigma _{2}} is determined by the matrix element of the variation of the action operator:
δ ⟨ σ 2 | σ 1 ⟩ = i ⟨ σ 2 | δ S ^ 21 | σ 1 ⟩ . {\displaystyle \delta \langle \sigma _{2}|\sigma _{1}\rangle =i\langle \sigma _{2}|\delta {\hat {S}}_{21}|\sigma _{1}\rangle .}
For a field theory involving a generic field ϕ ^ {\displaystyle {\hat {\phi }}} and Lagrangian density L ^ ( ϕ ^ , ∂ μ ϕ ^ ) {\displaystyle {\hat {\mathcal {L}}}({\hat {\phi }},\partial _{\mu }{\hat {\phi }})} , the action is the integral over the spacetime volume Ω {\displaystyle \Omega } bounded by σ 1 {\displaystyle \sigma _{1}} and σ 2 {\displaystyle \sigma _{2}} :
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