Schwinger–Dyson Identities and the Classical Limit

The previous page showed how perturbation theory can be generated by functional derivatives with respect to a source. This page extracts a deeper consequence of the same formalism: the path integral is invariant under changes of integration variables. An infinitesimal change of the field variable gives exact identities among correlation functions. These are the Schwinger–Dyson identities.

They are sometimes called the quantum equations of motion. That phrase is good, but only if it is understood carefully. The classical Euler–Lagrange expression can indeed be inserted into a quantum correlation function. Away from coincident insertion points it vanishes in the same way the classical equation of motion vanishes. At coincident points, however, time ordering produces delta-function contact terms. Those contact terms are not optional decoration; they are what make the Feynman propagator a Green function rather than an ordinary solution of the homogeneous Klein–Gordon equation.

This page also explains how the classical limit emerges. When $\hbar$ is restored, the Schwinger–Dyson identities show that the leading source-dependent expectation value obeys the classical sourced field equation, while quantum fluctuations appear as higher powers of $\hbar$, equivalently as loops.

Total derivatives in field space

Equation-of-motion sign and restored hbar

We use a real Lorentzian scalar field with

$$S[\phi]=\int d^dx\left[\frac12\partial_\mu\phi\partial^\mu\phi-\frac12m^2\phi^2-V(\phi)\right].$$

Thus

$$\frac{\delta S}{\delta\phi(x)}=-\left[(\Box+m^2)\phi(x)+V'(\phi(x))\right],$$

and the classical equation of motion is

$$(\Box+m^2)\phi+V'(\phi)=0.$$

We temporarily restore $\hbar$ in the section on the classical limit. Elsewhere, setting $\hbar=1$ recovers the course convention.

Begin with a normalized path-integral expectation value

$$\langle\mathcal O[\phi]\rangle =\frac{\int\mathcal D\phi\,\mathcal O[\phi]e^{iS[\phi]}} {\int\mathcal D\phi\,e^{iS[\phi]}}.$$

The field $\phi$ is an integration variable. A dummy change of variables cannot change the value of the integral. This is the same idea as shifting the variable in an ordinary integral; the only new ingredient is that the variable is now a function. Formally, this gives

$$0=\int\mathcal D\phi\, \frac{\delta}{\delta\phi(x)} \left[\mathcal O[\phi]e^{iS[\phi]}\right].$$

Expanding the derivative gives

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

After dividing by the vacuum functional, the basic Schwinger–Dyson identity is

$$\boxed{ \left\langle\mathcal O[\phi]\frac{\delta S}{\delta\phi(x)}\right\rangle =i\left\langle\frac{\delta\mathcal O}{\delta\phi(x)}\right\rangle. }$$

The identity is the field-theory analogue of the elementary integration-by-parts formula

$$0=\int dq\,\frac{d}{dq}\left[f(q)e^{iS(q)}\right].$$

The sign in the boxed identity is worth checking once. From $0=\langle\delta\mathcal O/\delta\phi\rangle+i\langle\mathcal O\,\delta S/\delta\phi\rangle$, division by $i$ gives $\langle\mathcal O\,\delta S/\delta\phi\rangle=i\langle\delta\mathcal O/\delta\phi\rangle$, because $1/i=-i$. This small algebraic point prevents many wrong signs in contact-term formulas.

The formal step must be handled with care in a continuum quantum field theory. The composite operator $\delta S/\delta\phi(x)$ and products of fields at coincident points need a regulator. Once the theory is regulated, the identity is the statement that the regulated functional measure is invariant under the chosen change of variables. An anomalous symmetry is precisely a case where the corresponding measure statement fails or acquires an extra local term.

A Schwinger–Dyson identity is integration by parts in the infinite-dimensional space of fields. Differentiating the exponential gives the equation-of-motion insertion, while differentiating the operator insertion gives contact terms.

Contact terms from field insertions

Now choose the operator insertion to be a product of fields,

$$\mathcal O[\phi]=\phi(x_1)\phi(x_2)\cdots\phi(x_n).$$

Its functional derivative is

$$\frac{\delta\mathcal O}{\delta\phi(x)} =\sum_{j=1}^n\delta^{(d)}(x-x_j) \phi(x_1)\cdots\widehat{\phi(x_j)}\cdots\phi(x_n),$$

where the hat means that the factor is omitted. Therefore

$$\boxed{ \left\langle\mathcal T\frac{\delta S}{\delta\phi(x)} \phi(x_1)\cdots\phi(x_n)\right\rangle =i\sum_{j=1}^n\delta^{(d)}(x-x_j) G_{n-1}(x_1,\ldots,\widehat{x_j},\ldots,x_n). }$$

Substituting the scalar action gives the Green-function equation

$$\boxed{ \begin{aligned} &(\Box_x+m^2)G_{n+1}(x,x_1,\ldots,x_n) +\left\langle\mathcal T V'(\phi(x))\phi(x_1)\cdots\phi(x_n)\right\rangle \\ &\qquad =-i\sum_{j=1}^n\delta^{(d)}(x-x_j) G_{n-1}(x_1,\ldots,\widehat{x_j},\ldots,x_n). \end{aligned} }$$

For a free scalar field, $V=0$, so

$$(\Box_x+m^2)G_{n+1}(x,x_1,\ldots,x_n) =-i\sum_{j=1}^n\delta^{(d)}(x-x_j) G_{n-1}(x_1,\ldots,\widehat{x_j},\ldots,x_n).$$

The two-point case is the familiar propagator equation

$$(\Box_x+m^2)G_F(x-y)=-i\delta^{(d)}(x-y).$$

This is why the propagator is not annihilated by the Klein–Gordon operator. It is annihilated away from $x=y$, but it has a delta-function source at coincidence.

The equation-of-motion insertion at $x$ is ordinary away from other insertions. When $x$ collides with one of the points $x_j$, the functional derivative of the insertion produces a contact term proportional to $\delta^{(d)}(x-x_j)$.

For an interacting theory, the same equation becomes a hierarchy. If

$$V(\phi)=\frac{\lambda}{4!}\phi^4,$$

then

$$V'(\phi)=\frac{\lambda}{3!}\phi^3,$$

and the exact two-point equation is

$$\boxed{ (\Box_x+m^2)G_2(x,y) +\frac{\lambda}{3!}\langle\mathcal T\phi^3(x)\phi(y)\rangle =-i\delta^{(d)}(x-y). }$$

The two-point function is therefore coupled to a four-field expectation value. Applying the identity to that four-field expectation value produces six-field expectations, and so on. The hierarchy is exact but infinite. Perturbation theory solves it order by order by evaluating the higher correlators with Wick contractions. Conceptually, this is the quantum version of a nonlinear equation: the equation for an average field or two-point function does not close because fluctuations of all orders feed back into it.

Source-functional form

The same identity can be packed into one functional differential equation. Define the unnormalized source functional

$$Z[J]=\int\mathcal D\phi\, \exp\left(iS[\phi]+i\int d^dx\,J(x)\phi(x)\right).$$

The total derivative with no extra insertion gives

$$0=\int\mathcal D\phi\, \left(\frac{\delta S}{\delta\phi(x)}+J(x)\right) \exp\left(iS[\phi]+i\int J\phi\right).$$

Inside $Z[J]$, multiplication by $\phi(x)$ is represented by

$$\phi(x)\longleftrightarrow \frac1i\frac{\delta}{\delta J(x)}.$$

Thus

$$\boxed{ \left[ \frac{\delta S}{\delta\phi(x)}\bigg|_{\phi\to (1/i)\delta/\delta J} +J(x) \right]Z[J]=0. }$$

For the scalar action above this becomes

$$\boxed{ \left[ (\Box_x+m^2)\frac1i\frac{\delta}{\delta J(x)} +V'\left(\frac1i\frac{\delta}{\delta J(x)}\right) -J(x) \right]Z[J]=0. }$$

The minus sign in front of $J$ is a consequence of the action convention $\delta S/\delta\phi=-[(\Box+m^2)\phi+V'(\phi)]$. If one writes the Schwinger–Dyson equation directly in terms of $\delta S/\delta\phi+J$, the same statement appears with the opposite overall sign. Multiplying the whole functional equation by $-1$ changes nothing, but changing only the $J$ sign would be wrong.

For $\lambda\phi^4/4!$ theory,

$$\left[ (\Box_x+m^2)\frac1i\frac{\delta}{\delta J(x)} +\frac{\lambda}{3!}\left(\frac1i\frac{\delta}{\delta J(x)}\right)^3 -J(x) \right]Z[J]=0.$$

Taking derivatives of this equation with respect to $J$ and then setting $J=0$ reproduces the whole Schwinger–Dyson hierarchy. This is the same hierarchy that appears in operator language when one applies the equation of motion to time-ordered products and keeps track of contact terms.

The tadpole from the Schwinger–Dyson equation

The first-order tadpole correction to the two-point function can be derived directly from the exact identity. Write

$$G_2(x,y)=G_0(x,y)+G_2^{(1)}(x,y)+O(\lambda^2),$$

where $G_2^{(1)}$ is the order-$\lambda$ correction to the two-point function. This notation keeps it distinct from the one-point function $G_1(x)$. The free propagator satisfies

$$(\Box_x+m^2)G_0(x,y)=-i\delta^{(d)}(x-y).$$

The order-$\lambda$ part of the two-point Schwinger–Dyson equation is

$$(\Box_x+m^2)G_2^{(1)}(x,y) =-\frac{\lambda}{3!}\langle\mathcal T\phi^3(x)\phi(y)\rangle_0.$$

By Wick theorem,

$$\langle\mathcal T\phi^3(x)\phi(y)\rangle_0 =3G_0(x,x)G_0(x,y).$$

Thus

$$(\Box_x+m^2)G_2^{(1)}(x,y) =-\frac{\lambda}{2}G_0(x,x)G_0(x,y).$$

Since $(\Box_x+m^2)G_0(x,z)=-i\delta^{(d)}(x-z)$, the inverse of $\Box+m^2$ with Feynman boundary conditions is $iG_0$. Therefore

$$\boxed{ G_2^{(1)}(x,y) =-\frac{i\lambda}{2}\int d^dz\, G_0(x,z)G_0(z,z)G_0(z,y). }$$

The Schwinger–Dyson equation turns the interaction term $\lambda\phi^3/3!$ into the free contraction $3G_0(x,x)G_0(x,y)$. Inverting the free Klein–Gordon operator attaches another propagator and gives the tadpole correction to $G_2$.

The coincident propagator $G_0(z,z)$ is the important warning sign. In continuum field theory it is generally ultraviolet divergent. The local divergence has the form of a correction to the mass term, which is why tadpoles become part of mass renormalization.

Classical limit from stationary phase

Now restore $\hbar$:

$$Z[J]=\int\mathcal D\phi\, \exp\left[\frac{i}{\hbar}\left(S[\phi]+\int d^dx\,J\phi\right)\right].$$

The source-functional Schwinger–Dyson equation becomes

$$\boxed{ \left[ \frac{\delta S}{\delta\phi(x)}\bigg|_{\phi\to (\hbar/i)\delta/\delta J} +J(x) \right]Z[J]=0. }$$

Let

$$Z[J]=e^{iW[J]/\hbar}, \qquad \varphi_J(x)=\frac{\delta W[J]}{\delta J(x)}.$$

The field $\varphi_J$ is the source-dependent expectation value of $\phi$. Since

$$\frac{\hbar}{i}\frac{\delta Z}{\delta J(x)}=\varphi_J(x)Z[J],$$

the leading small-$\hbar$ approximation replaces every $(\hbar/i)\delta/\delta J$ by $\varphi_J$. Extra derivatives acting on $\varphi_J$ bring explicit powers of $\hbar$ and represent quantum fluctuations. Therefore the leading equation is

$$\boxed{ \frac{\delta S}{\delta\varphi_J(x)}+J(x)=0. }$$

With the action convention of this page,

$$(\Box+m^2)\varphi_J(x)+V'(\varphi_J(x))=J(x).$$

This is the classical sourced field equation. Notice that this is not the same as the free theory. A classical field can be nonlinear and interacting. The classical limit suppresses loops, not nonlinearities.

The same result follows from stationary phase. Define the source-dependent phase functional

$$S_J[\phi] =S[\phi]+\int d^dx\,J(x)\phi(x),$$

and expand around its saddle:

$$\phi(x)=\phi_{\mathrm{cl},J}(x)+\sqrt{\hbar}\,\eta(x), \qquad \left.\frac{\delta S_J}{\delta\phi(x)} \right|_{\phi=\phi_{\mathrm{cl},J}}=0.$$

Then

$$S_J[\phi_{\mathrm{cl},J}+\sqrt{\hbar}\eta] =S_J[\phi_{\mathrm{cl},J}] +\frac{\hbar}{2}\int d^dx\,d^dy\,\eta(x)S^{(2)}(x,y)\eta(y) +O(\hbar^{3/2}).$$

The linear term vanishes by the sourced classical equation of motion. The source is linear in $\phi$, so the quadratic kernel is still $S^{(2)}$, now evaluated on $\phi_{\mathrm{cl},J}$. At $J=0$ this reduces to the ordinary zero-source saddle. The quadratic term gives the Gaussian fluctuation determinant. Higher terms generate interactions among fluctuations. In diagrammatic language, tree diagrams survive at leading order, while each loop carries a power of $\hbar$.

As $\hbar\to0$, stationary configurations dominate the path integral. Tree-level propagation on the classical background is leading, while loop corrections are suppressed by powers of $\hbar$.

Connected functions and the quantum equation of motion

The connected generator $W[J]$ makes the quantum corrections more explicit. In $\lambda\phi^4/4!$ theory, the source equation can be rewritten schematically as

$$(\Box+m^2)\varphi_J(x) +\frac{\lambda}{3!}\left[ \varphi_J^3(x)+3\varphi_J(x)G_{J,c}(x,x)+G_{J,c}^{(3)}(x,x,x) \right] =J(x),$$

where $G_{J,c}$ and $G_{J,c}^{(3)}$ are connected two- and three-point functions in the presence of the source, with conventional Lorentzian factors absorbed into their definition. This formula should be read structurally, not as an invitation to multiply unregulated coincident distributions. The classical term $V'(\varphi_J)$ is corrected by connected fluctuations at coincident points, and those local fluctuation terms require the same regulator and renormalization as ordinary loop diagrams.

Later this idea is sharpened using the effective action $\Gamma[\varphi]$, the Legendre transform of $W[J]$. Its equation of motion is

$$\frac{\delta\Gamma[\varphi]}{\delta\varphi(x)}=-J(x).$$

At tree level, $\Gamma=S$. Loop corrections replace the classical action by the quantum effective action. Schwinger–Dyson identities are the exact identities behind that replacement.

Related nonperturbative treatment

This lesson gives a first Lorentzian derivation, including contact terms, the tadpole, and the classical limit. Schwinger–Dyson Hierarchies and Renormalization Inputs continues to renormalized propagator and vertex hierarchies, closure choices, solution branches, and continuum methods.

Summary

Schwinger–Dyson identities come from the invariance of the path integral under changes of integration variables. They say that inserting $\delta S/\delta\phi$ into a correlator is equivalent to differentiating the other insertions. For products of fields, differentiating the insertions gives contact terms supported at coincident points.

For a free scalar, the identity says that the Feynman propagator is a Green function of $\Box+m^2$. For an interacting scalar, it generates an infinite hierarchy: the two-point function is coupled to higher correlators through $V'(\phi)$. Perturbation theory solves this hierarchy order by order, and the first nontrivial example is the tadpole correction in $\phi^4$ theory.

Restoring $\hbar$ shows how the classical limit emerges. The leading Schwinger–Dyson equation is the classical sourced equation of motion, while connected fluctuations and loops are suppressed by powers of $\hbar$.

Common pitfalls

Do not drop contact terms when differentiating time-ordered products. The equation of motion holds inside a correlator only away from coincident insertion points.

Do not treat the Schwinger–Dyson identity as a perturbative statement. The identity is exact; perturbation theory is one method for approximately solving it.

Do not forget the sign of $\delta S/\delta\phi$. With the convention used here, $\delta S/\delta\phi=-[(\Box+m^2)\phi+V'(\phi)]$.

Do not assume that coincident quantities such as $G_0(x,x)$ are finite. They are usually ultraviolet divergent and must be regulated.

Do not confuse the classical limit with the free limit. Classical interacting field equations can be nonlinear; what disappears in the classical limit are loop corrections.

Exercises

Exercise 1

Derive the Schwinger–Dyson identity

$$\left\langle\mathcal T\frac{\delta S}{\delta\phi(x)} \phi(x_1)\cdots\phi(x_n)\right\rangle =i\sum_{j=1}^n\delta^{(d)}(x-x_j) G_{n-1}(x_1,\ldots,\widehat{x_j},\ldots,x_n).$$

Solution

Start from

$$0=\int\mathcal D\phi\, \frac{\delta}{\delta\phi(x)} \left[\phi(x_1)\cdots\phi(x_n)e^{iS[\phi]}\right].$$

Expanding the derivative gives

$$0=\int\mathcal D\phi\,e^{iS[\phi]} \left[ \sum_{j=1}^n\delta^{(d)}(x-x_j) \phi(x_1)\cdots\widehat{\phi(x_j)}\cdots\phi(x_n) +i\frac{\delta S}{\delta\phi(x)}\phi(x_1)\cdots\phi(x_n) \right].$$

Divide by the vacuum functional and rearrange:

$$\left\langle\mathcal T\frac{\delta S}{\delta\phi(x)} \phi(x_1)\cdots\phi(x_n)\right\rangle =i\sum_{j=1}^n\delta^{(d)}(x-x_j) G_{n-1}(x_1,\ldots,\widehat{x_j},\ldots,x_n).$$

Exercise 2

Use Wick theorem to verify the free identity

$$(\Box_x+m^2)G_4(x,x_1,x_2,x_3) =-i\sum_{j=1}^3\delta^{(d)}(x-x_j) G_2(x_1,\ldots,\widehat{x_j},\ldots,x_3).$$

Solution

For a free real scalar field,

$$\begin{aligned} G_4(x,x_1,x_2,x_3) &=G_0(x,x_1)G_0(x_2,x_3) +G_0(x,x_2)G_0(x_1,x_3)\\ &\quad+G_0(x,x_3)G_0(x_1,x_2). \end{aligned}$$

Acting with $\Box_x+m^2$ only affects the propagator containing $x$. Since

$$(\Box_x+m^2)G_0(x,x_j)=-i\delta^{(d)}(x-x_j),$$

we obtain

$$\begin{aligned} (\Box_x+m^2)G_4 &=-i\delta^{(d)}(x-x_1)G_0(x_2,x_3)\\ &\quad -i\delta^{(d)}(x-x_2)G_0(x_1,x_3)\\ &\quad -i\delta^{(d)}(x-x_3)G_0(x_1,x_2), \end{aligned}$$

which is the desired contact-term identity.

Exercise 3

For $V(\phi)=\lambda\phi^4/4!$, derive

$$G_2^{(1)}(x,y)=-\frac{i\lambda}{2}\int d^dz\,G_0(x,z)G_0(z,z)G_0(z,y)$$

from the two-point Schwinger–Dyson equation.

Solution

The exact two-point equation is

$$(\Box_x+m^2)G_2(x,y) +\frac{\lambda}{3!}\langle\mathcal T\phi^3(x)\phi(y)\rangle =-i\delta^{(d)}(x-y).$$

Set $G_2=G_0+G_2^{(1)}+O(\lambda^2)$. The free part satisfies

$$(\Box_x+m^2)G_0(x,y)=-i\delta^{(d)}(x-y).$$

At order $\lambda$,

$$(\Box_x+m^2)G_2^{(1)}(x,y) =-\frac{\lambda}{3!}\langle\mathcal T\phi^3(x)\phi(y)\rangle_0.$$

Wick theorem gives

$$\langle\mathcal T\phi^3(x)\phi(y)\rangle_0 =3G_0(x,x)G_0(x,y).$$

Therefore

$$(\Box_x+m^2)G_2^{(1)}(x,y) =-\frac{\lambda}{2}G_0(x,x)G_0(x,y).$$

Using the inverse relation $(\Box_x+m^2)G_0(x,z)=-i\delta^{(d)}(x-z)$ gives

$$G_2^{(1)}(x,y)=-\frac{i\lambda}{2}\int d^dz\,G_0(x,z)G_0(z,z)G_0(z,y).$$

Exercise 4

Restore $\hbar$ and show that the leading equation for $\varphi_J=\delta W/\delta J$, with $Z=e^{iW/\hbar}$, is

$$\frac{\delta S}{\delta\varphi_J(x)}+J(x)=0.$$

Solution

With $\hbar$ restored,

$$\left[ \frac{\delta S}{\delta\phi(x)}\bigg|_{\phi\to (\hbar/i)\delta/\delta J} +J(x) \right]Z[J]=0.$$

Because $Z=e^{iW/\hbar}$,

$$\frac{\hbar}{i}\frac{\delta Z}{\delta J(x)} =\frac{\delta W}{\delta J(x)}Z =\varphi_J(x)Z.$$

When several source derivatives act on $Z$, the leading term is obtained by replacing each $(\hbar/i)\delta_J$ by $\varphi_J$. Derivatives that act on $\varphi_J$ itself carry explicit powers of $\hbar$. Hence the leading equation is

$$\left[\frac{\delta S}{\delta\phi(x)}\bigg|_{\phi=\varphi_J}+J(x)\right]Z[J]=0,$$

which implies

$$\frac{\delta S}{\delta\varphi_J(x)}+J(x)=0.$$

Exercise 5

Explain why the Schwinger–Dyson identity does not simply say that the interacting quantum field obeys the classical equation of motion pointwise.

Solution

The identity contains contact terms:

$$\left\langle\mathcal T\frac{\delta S}{\delta\phi(x)} \phi(x_1)\cdots\phi(x_n)\right\rangle =i\sum_{j=1}^n\delta^{(d)}(x-x_j) G_{n-1}(x_1,\ldots,\widehat{x_j},\ldots,x_n).$$

Away from coincident points the right-hand side vanishes, so the classical equation holds inside the correlator in that restricted sense. At coincident points the contact terms are essential. In addition, composite operators such as $V'(\phi(x))$ need regularization and renormalization. Thus the Schwinger–Dyson equation is an exact identity for regulated correlation functions, not a naive pointwise operator equation with all short-distance subtleties ignored.

References and further reading

• Mark Srednicki, Quantum Field Theory, Sections 9 and 22, for path integrals, source functionals, and Schwinger–Dyson equations.
• Sidney Coleman, Lectures of Sidney Coleman on Quantum Field Theory, Chapters 4, 7–8, and 28, for equations of motion, Dyson expansion, Wick diagrams, and functional integration.
• Steven Weinberg, The Quantum Theory of Fields, Volume I, Chapters 7 and 9, for canonical equations, path-integral methods, and source-functional reasoning.
• A. Zee, Quantum Field Theory in a Nutshell, Appendix A and Chapter I.12, for Gaussian identities and diagrammatic intuition from source functionals.