Schwinger-Dyson equation

From Physics wiki

< Quantum field theory | path integral methods

The Schwinger-Dyson equation is the quantum analog of the classical equations of motion.

Consider the vacuum expectation value of some polynomial function $\mathcal{F}(\phi)\,$ of the field variable:

$\left\langle 0| T \mathcal{F}(\phi) |0\right\rangle = \int\!\mathcal{D}\phi\, \mathcal{F}(\phi) e^{i S[\phi]}\,$ .

For a wide class of theories the action will be large (and positive) at the boundary of the space over which the functional integral is defined, if such a boundary exists. If we Wick rotate to Euclidean space to make the integral well-defined, the integrand is therefore of the form $\mathcal{F}(\phi) e^{-S_E[\phi]}\,$ , which dies off at "infinity" provided that $\mathcal{F}(\phi)\,$ is polynomial. Therefore

$\frac{\delta}{\delta \phi(x)} \int\!\mathcal{D}\phi\, \mathcal{F}(\phi) e^{i S[\phi]} = \int\!\mathcal{D}\phi\, \frac{\delta}{\delta \phi(x)} \left( \mathcal{F}(\phi) e^{i S[\phi]}\right) = 0\,$ ,

$0\,$	$= \int\!\mathcal{D}\phi\, \frac{\delta}{\delta \phi(x)} \left( \mathcal{F}(\phi) e^{i S[\phi]}\right)\,$ ,
	$= \int\!\mathcal{D}\phi\, \frac{\delta \mathcal{F}(\phi)}{\delta \phi(x)} e^{i S[\phi]} + i\mathcal{F}(\phi) \frac{\delta S[\phi]}{\delta \phi(x)} e^{i S[\phi]}\,$ ,
	$= \langle 0\|\frac{\delta \mathcal{F}(\phi)}{\delta \phi(x)} \|0\rangle + i\langle 0\|\mathcal{F}(\phi) \frac{\delta S[\phi]}{\delta \phi(x)} \|0\rangle\,$ ,

where we have dropped the time ordering symbol for brevity. In particular, in the simplest case of $\mathcal{F}(\phi) = 1\,$ , the fields are seen to obey an analog to their classical equations of motion:

$\langle 0|\frac{\delta S[\phi]}{\delta \phi(x)} |0\rangle = 0\,$ .

Actually, since we can pull ordinary derivatives outside the functional integral, they should not appear inside the time ordering symbol, since the effect of pulling them out (the appearance of contact terms) has already been taken care of by the functional integral.

If we apply the same argument to the generating functional $Z[J]\,$ , then

$\frac{\delta S}{\delta \phi(x)}\left[-i \frac{\delta}{\delta J(x)}\right]Z[J]+J(x)Z[J]=0$ .

φ⁴ theory

For $\phi^4\,$ theory, $\mathcal{L}=\frac{1}{2}\phi (-\partial^2 -m^2) \phi -\frac{\lambda}{4!}\phi^4$ . Take $\mathcal{F}(\phi) = \phi(y)\,$ so that, ignoring regularization issues for the moment, $\langle 0|\delta(x-y) |0\rangle - i (\partial_x^2 + m^2)\langle 0| \phi(x) \phi(y) |0\rangle - i \frac{\lambda}{3!}\langle 0| \phi^3(x) \phi(y) |0\rangle= 0\,$ , so the two point function satisfies

$(\partial_x^2 + m^2) G(x,y) = -i \delta(x-y) - \frac{\lambda}{3!}\langle 0| \phi^3(x) \phi(y) |0\rangle\,$ ,

which has formal solution

$G(x,y) = G_0(x,y) - \frac{i\lambda}{3!}\int\!d^dx_1\, G_0(x,x_1)\langle 0| \phi^3(x_1) \phi(y) |0\rangle\,$ ,

where $(\partial^2 + m^2)G_0(x,y) = -i \delta(x-y)\,$ . We can obtain similar equations for higher order n-point functions and solve these Schwinger-Dyson equations by iteration.

For instance, try $\mathcal{F}(\phi) = \phi(y_1)\phi(y_2)\phi(y_3)\,$ , so

$\delta(x-y_1)\left\langle 0 | \phi(y_2)\phi(y_3) | 0 \right\rangle + \delta(x-y_2)\left\langle 0 | \phi(y_1)\phi(y_3) | 0 \right\rangle + \delta(x-y_3)\left\langle 0 | \phi(y_1)\phi(y_2) | 0 \right\rangle - i (\partial_x^2 + m^2)\langle 0| \phi(x) \phi(y_1) \phi(y_2) \phi(y_3) |0\rangle - i \frac{\lambda}{3!}\langle 0| \phi^3(x) \phi(y_1)\phi(y_2)\phi(y_3) |0\rangle= 0\,$

so that the 4-point function satisfies

$(\partial_x^2 + m^2) G(x,y_1,y_2,y_3) = -i \delta(x-y_1) G(y_2,y_3) -i \delta(x-y_2) G(y_1,y_3) -i \delta(x-y_3) G(y_1,y_2)- \frac{\lambda}{3!}\langle 0| \phi^3(x) \phi(y_1)\phi(y_2)\phi(y_3) |0\rangle\,$

and

$G(x,y_1,y_2,y_3) = G_0(x,y_1)G(y_2,y_3) + G_0(x,y_2)G(y_1,y_3) + G_0(x,y_3)G(y_1,y_2) - \frac{i\lambda}{3!}\int\!d^dx_1\, G_0(x,x_1)\langle 0| \phi^3(x_1) \phi(y_1)\phi(y_2)\phi(y_3) |0\rangle\,$