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, <math> S </math>. 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 <math>| A \rang</math> and <math>| B \rang</math>, 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:
:<math> \delta \langle B|A\rangle = i \langle B| \delta S |A\rangle,\ </math>
where the derivative is with respect to small changes (<math>\delta</math>) in the parameter, and <math>S=\int \mathcal{L} \, \mathrm{d}t</math> with <math>\mathcal{L}</math> the Lagrange operator.
In the path integral formulation, the transition amplitude is represented by the sum over all histories of <math>\exp(iS)</math>, with appropriate boundary conditions representing the states <math>| A \rang</math> and <math>| B \rang</math>. 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 <math>\hat{S}</math>.
The variational principle
The principle asserts that the variation of the transformation amplitude between two spacelike surfaces <math>\sigma_1 </math> and <math>\sigma_2 </math> is determined by the matrix element of the variation of the action operator:
:<math>
\delta \langle \sigma_2 | \sigma_1 \rangle = i \langle \sigma_2 | \delta \hat{S}_{21} | \sigma_1 \rangle.
</math>
For a field theory involving a generic field <math>\hat{\phi} </math> and Lagrangian density <math>\hat{\mathcal{L(\hat{\phi}, \partial_\mu \hat{\phi}) </math>, the action is the integral over the spacetime volume <math>\Omega </math> bounded by <math>\sigma_1 </math> and <math>\sigma_2 </math>:
:<math>
\hat{S}_{21} = \int_{\sigma_1}^{\sigma_2} d^4x \, \hat{\mathcal{L(\hat{\phi}, \partial_\mu \hat{\phi}).
</math>
The variation <math>\delta \hat{S}</math> arises from infinitesimal changes in the field operator <math>\hat{\phi} \to \hat{\phi} + \delta \hat{\phi}</math>. Using the chain rule and integration by parts (generalized Green's identities), the variation separates into a "bulk" volume integral and a "boundary" surface integral:
:<math>
\delta \hat{S}_{21} = \int_{\Omega} d^4x \left[ \frac{\partial \hat{\mathcal{L}{\partial \hat{\phi - \partial_\mu \left( \frac{\partial \hat{\mathcal{L}{\partial (\partial_\mu \hat{\phi})} \right) \right] \delta \hat{\phi}
+ \int_{\partial \Omega} d\sigma_\mu \, \hat{\pi}^\mu \delta \hat{\phi},
</math>
where <math>\hat{\pi}^\mu = \frac{\partial \hat{\mathcal{L}{\partial (\partial_\mu \hat{\phi})}</math> is the conjugate momentum four-vector.
Euler–Lagrange equations
Schwinger postulated that for the state evolution to be physically meaningful, the transition amplitude must depend only on the boundary conditions at <math>\sigma_1 </math> and <math>\sigma_2 </math>, not on the arbitrary variations in the interior of the spacetime volume. Therefore, the volume integral term in the variation must vanish for arbitrary field variations <math>\delta \hat{\phi}</math> that vanish at the boundaries.
This requirement immediately yields the Euler–Lagrange equations as operator equations of motion:
:<math>
\partial_\mu \left( \frac{\partial \hat{\mathcal{L}{\partial (\partial_\mu \hat{\phi})} \right) - \frac{\partial \hat{\mathcal{L}{\partial \hat{\phi = 0.
</math>
Canonical commutation relations
The remaining non-vanishing part of the action variation is the boundary term. If the boundaries are constant-time surfaces <math>t_1</math> and <math>t_2</math>, the surface element is <math>d\sigma_\mu = (d^3x, 0, 0, 0)</math>, and the boundary term involves the canonical momentum <math>\hat{\pi} = \hat{\pi}^0</math>. The variation of the action becomes the difference of generators <math>\hat{G}</math> at the boundaries:
:<math>
\delta \hat{S}_{21} = \hat{G}(t_2) - \hat{G}(t_1), \quad \text{where} \quad \hat{G}(t) = \int d^3x \, \hat{\pi}(\mathbf{x}, t) \delta \hat{\phi}(\mathbf{x}, t).
</math>
According to the quantum action principle, this generator <math>\hat{G}</math> generates infinitesimal unitary transformations on the fields. For a generic operator <math>\hat{A}</math>, an infinitesimal transformation generated by <math>\hat{G}</math> is given by the commutator:
:<math>
\delta \hat{A} = i [\hat{G}, \hat{A}].
</math>
Considering the variation of the field operator <math>\hat{\phi}(\mathbf{y}, t)</math> itself, we substitute <math>\hat{A} = \hat{\phi}(\mathbf{y}, t)</math>:
:<math>
\delta \hat{\phi}(\mathbf{y}, t) = i \left[ \int d^3x \, \hat{\pi}(\mathbf{x}, t) \delta \hat{\phi}(\mathbf{x}, t), \, \hat{\phi}(\mathbf{y}, t) \right].
</math>
Assuming the variation <math>\delta \hat{\phi}</math> commutes with the field operators (c-number variation), it can be pulled out of the commutator:
:<math>
\delta \hat{\phi}(\mathbf{y}, t) = i \int d^3x \, \delta \hat{\phi}(\mathbf{x}, t) [\hat{\pi}(\mathbf{x}, t), \hat{\phi}(\mathbf{y}, t)].
</math>
For this equation to hold for an arbitrary variation function <math>\delta \hat{\phi}(\mathbf{x}, t)</math>, the kernel of the integral must be a Dirac delta function. This enforces the canonical commutation relation:
:<math>
[\hat{\phi}(\mathbf{y}, t), \hat{\pi}(\mathbf{x}, t)] = i \delta^{(3)}(\mathbf{y} - \mathbf{x}).
</math>
Thus, the quantization of the field is not an ad hoc assumption but a necessary consequence of the boundary terms in the Schwinger action principle.
Derivation of the Schwinger–Dyson equations
The Schwinger–Dyson equations (SDEs) can be derived directly from the Schwinger quantum action principle without recourse to the functional integral formalism. This approach, pioneered by Julian Schwinger, relies on the operator nature of the fields and the canonical commutation relations implied by the action principle.
The action principle and field equations
The Schwinger action principle states that the variation of the transition amplitude between two states <math>\langle \text{out} | \text{in} \rangle</math> under a variation of parameters or fields is proportional to the matrix element of the variation of the action operator <math>\hat{S}</math>:
:<math>
\delta \langle \text{out} | \text{in} \rangle = i \left\langle \text{out} \left| \delta \hat{S} \right| \text{in} \right\rangle.
</math>
For a scalar field theory with the action functional
:<math>
S[\phi] = \int d^4x \left( \tfrac{1}{2} \partial_\mu \phi \partial^\mu \phi - V(\phi) \right),
</math>
the principle implies the Euler–Lagrange equations holding as operator equations of motion (Heisenberg equations):
:<math>
\left[ \square_x \left( \frac{1}{i} \frac{\delta}{\delta J(x)} \right) + V'\left( \frac{1}{i} \frac{\delta}{\delta J(x)} \right) \right] Z[J] = J(x) Z[J].
</math>
This derivation demonstrates that the SDEs are a consequence of the equations of motion combined with the non-commutativity of time-ordered operators, derived strictly within the operator framework.
See also
- Source field