The Schwinger's quantum action principle is
a variational approach to quantum mechanics
and quantum field theory. This theory was
introduced by Julian Schwinger. In this approach,
the quantum action is an operator. 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 in the parameter.
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.
== External 
links ==
[1] A brief (but very technical) description
of Schwinger's paper
