Schwinger–Dyson Identities

A Schwinger–Dyson identity is the exact integration-by-parts statement obtained by changing a regulated field variable. For the chapter’s normalized Lorentzian in–out functional, a bosonic translation gives

$$\left\langle F[\phi]\left( \frac{\delta S}{\delta\phi(x)}+J(x) \right) \right\rangle_J = i\left\langle \frac{\delta F}{\delta\phi(x)} \right\rangle_J.$$

The derivative of $F$ produces contact terms. Choosing successively larger products for $F$ therefore relates every full time-ordered correlator to higher ones. The relation is exact at the declared regulator when the integration cycle can be shifted, its boundary contribution vanishes, and the measure variation is included. It is an infinite hierarchy, not by itself a closure, a unique solution, or a causal evolution equation.

Required background. The Generating Functional supplies the normalized source functional, source-insertion rule, in–out state, and Feynman boundary prescription used below.

Helpful background. The 1PI Effective Action and Mean-Field Equations relates the source to a mean field on a locally invertible branch. Changes of Variables and Regulated Jacobians develops the finite-dimensional Jacobian calculation used to qualify a general field redefinition.

Regulated integration by parts

Replace the field temporarily by $N$ regulated bosonic coordinates $\phi_a$ with a flat measure on an integration domain or cycle $\mathcal C_\Lambda$. Retain the same source sign and normalization as on the generating-functional page:

$$\begin{aligned} \mathcal Z_\Lambda[J] &= \int_{\mathcal C_\Lambda}\mathrm d^N\phi\, e^{iS_\Lambda(\phi)+iJ\cdot\phi}, \\ Z_\Lambda[J] &= \frac{\mathcal Z_\Lambda[J]}{\mathcal Z_\Lambda[0]}, \qquad Z_\Lambda[0]=1, \\ \langle\mathcal O\rangle_J &= \frac{1}{\mathcal Z_\Lambda[J]} \int_{\mathcal C_\Lambda}\mathrm d^N\phi\, \mathcal O(\phi)e^{iS_\Lambda+iJ\cdot\phi}. \end{aligned}$$

For one fixed coordinate $a$, define the normalized total derivative

$$\mathcal B_{F,a}[J] \equiv \frac{1}{\mathcal Z_\Lambda[J]} \int_{\mathcal C_\Lambda}\mathrm d^N\phi\, \frac{\partial}{\partial\phi_a} \left(Fe^{iS_\Lambda+iJ\cdot\phi}\right).$$

Ordinary differentiation gives

$$\mathcal B_{F,a} = \left\langle\partial_aF\right\rangle_J +i\left\langle F\left(\partial_aS_\Lambda+J_a\right) \right\rangle_J.$$

If the boundary flux or contour-end contribution vanishes, $\mathcal B_{F,a}=0$, and hence

$$\left\langle F\left(\partial_aS_\Lambda+J_a\right) \right\rangle_J =i\left\langle\partial_aF\right\rangle_J.$$

If it does not vanish, the right-hand side is instead $i\langle\partial_aF\rangle_J-i\mathcal B_{F,a}$. Thus a hard field boundary, a forbidden contour deformation, or inadequate falloff changes the identity rather than merely weakening its proof.

Continuum notation abbreviates this finite-regulator result. Let $\delta_\Lambda(x,y)$ be the identity kernel on the regulated field space. For a permitted localized translation, the result becomes the identity displayed in the lead. Its Lorentzian factor follows directly from differentiating $e^{iS+iJ\cdot\phi}$; the Euclidean weight $e^{-S_E+J\cdot\phi}$ has different signs. Zinn-Justin states the regularization assumptions and Euclidean integration-by-parts equation in Zinn-Justin 2021, §§ 7.5–7.5.1, pp. 133–134.

Two immediate choices are useful. With $F=1$,

$$\left\langle \frac{\delta S_\Lambda}{\delta\phi(x)} \right\rangle_J +J(x)=0.$$

This is an equation for an expectation value of the Euler–Lagrange insertion. It does not say that every field configuration in the integral satisfies the classical equation of motion.

Contact terms generate the hierarchy

Choose a product of $n$ scalar fields,

$$F_n[\phi]=\prod_{r=1}^{n}\phi(x_r).$$

At the fixed regulator,

$$\frac{\delta F_n}{\delta\phi(x)} = \sum_{r=1}^{n} \delta_\Lambda(x,x_r) \prod_{s\ne r}\phi(x_s).$$

The translation identity therefore reads

$$\begin{aligned} &\left\langle \left( \frac{\delta S_\Lambda}{\delta\phi(x)}+J(x) \right) \prod_{r=1}^{n}\phi(x_r) \right\rangle_J \\ &\qquad= i\sum_{r=1}^{n}\delta_\Lambda(x,x_r) \left\langle \prod_{s\ne r}\phi(x_s) \right\rangle_J. \end{aligned}$$

At zero source these brackets are the full in–out time-ordered correlators generated by $Z$, not the connected correlators generated by $W$. Each delta function is a contact term created when the field variation hits an insertion. In continuum language the statement is distributional and should be smeared against test functions; products with coincident arguments remain regulated composite insertions. Srednicki gives the all-orders Lorentzian hierarchy and identifies these delta-function contacts in Srednicki 2007, § 22, pp. 147–148.

The contact terms are exactly why one cannot apply the classical equation of motion inside a time-ordered product and set the result to zero. Away from every insertion point the right-hand side vanishes, but at coincidence it is essential.

The source-functional equation

One source derivative inserts $i\phi$, so multiplication by a field is represented on $Z_\Lambda[J]$ by

$$\phi(x)\longmapsto -i\frac{\delta}{\delta J(x)}.$$

The $F=1$ identity can consequently be written as a differential equation for the complete generating functional:

$$\left[ \left. \frac{\delta S_\Lambda}{\delta\phi(x)} \right|_{\phi\,\mapsto\,-i\delta/\delta J} +J(x) \right]Z_\Lambda[J]=0.$$

All source derivatives act to the right. The equation is unchanged by division of $\mathcal Z_\Lambda[J]$ by the source-independent factor $\mathcal Z_\Lambda[0]$. Differentiating it with respect to $J$ and then setting $J=0$ reproduces the contact hierarchy.

The source-functional identity and the operator equation of motion must nevertheless be handled with their ordering prescription intact. Moving a spacetime differential operator naively through a time-ordering symbol would erase the very contacts generated above. Schwartz derives the Lorentzian source equation and explains this distinction in Schwartz 2014, § 14.7.2, pp. 275–276.

The exact scalar two-point identity

Consider the regulated quartic scalar model

$$S_{F,\Lambda}[\phi] = -\frac12\phi\mathbin{\cdot}P_{F,\Lambda} \mathbin{\cdot}\phi -\frac{\lambda}{4!} \int_\Lambda\mathrm d^dx\,\phi(x)^4.$$

The symmetric kernel $P_{F,\Lambda}$ includes the regulator and the deformation selecting the Feynman boundary value. Its field derivative is

$$\frac{\delta S_{F,\Lambda}}{\delta\phi(x)} = -\bigl(P_{F,\Lambda}\phi\bigr)(x) -\frac{\lambda}{3!}\phi(x)^3.$$

Writing $\delta_J(x)=\delta/\delta J(x)$, the source-functional equation becomes

$$\left[ i\bigl(P_{F,\Lambda}\delta_J\bigr)(x) -\frac{i\lambda}{3!}\delta_J(x)^3 +J(x) \right]Z_\Lambda[J]=0.$$

Define the source-dependent full moments by

$$G_{\Lambda,J}^{(n)}(x_1,\ldots,x_n) \equiv \left\langle \mathrm T\phi(x_1)\cdots\phi(x_n) \right\rangle_J, \qquad G_{\Lambda,J}^{(0)}=1.$$

Substitution into the product identity gives, for $n\ge1$,

$$\begin{aligned} &P_{F,\Lambda,x} G_{\Lambda,J}^{(n+1)}(x,x_1,\ldots,x_n) +\frac{\lambda}{3!} G_{\Lambda,J}^{(n+3)}(x,x,x,x_1,\ldots,x_n) \\ &\qquad -J(x)G_{\Lambda,J}^{(n)}(x_1,\ldots,x_n) \\ &= -i\sum_{r=1}^{n}\delta_\Lambda(x,x_r) G_{\Lambda,J}^{(n-1)} (x_1,\ldots,\widehat{x_r},\ldots,x_n). \end{aligned}$$

Here $P_{F,\Lambda,x}$ acts on the first argument, and a hat means that the argument is omitted. The $n=1$, $J=0$ member is the required two-point application:

$$\boxed{ P_{F,\Lambda,x}G_\Lambda^{(2)}(x,y) +\frac{\lambda}{3!} G_\Lambda^{(4)}(x,x,x,y) =-i\delta_\Lambda(x,y) }.$$

The repeated $x$ arguments are meaningful here because the regulator is still present. In a continuum renormalized theory, $\phi^3(x)$ is a composite operator whose mixing and renormalization prescription must be specified; the bare coincident notation cannot simply be retained unchanged. The Lorentzian two-point contact term and its interacting extension are derived in Schwartz 2014, § 14.7.1, pp. 273–274. Zinn-Justin gives the Euclidean scalar hierarchy at a fixed regularized level in Zinn-Justin 2021, § 7.5.1, p. 135.

The free limit is the sharp sign check. Setting $\lambda=0$ and identifying $G_\Lambda^{(2)}=D_{F,\Lambda}$ gives

$$P_{F,\Lambda}D_{F,\Lambda} =-iI_\Lambda,$$

in agreement with the generating-functional and 1PI pages. A result with $+iI_\Lambda$ or $I_\Lambda$ has mixed either the action sign or the physical propagator $D_F$ with the inverse kernel $G_F=iD_F$.

The $n=0$ equation also makes the lack of closure visible. With $\bar\phi_J=\langle\phi\rangle_J$,

$$P_{F,\Lambda}\bar\phi_J +\frac{\lambda}{3!} \left\langle\phi^3\right\rangle_J =J,$$

while the coincident third moment decomposes as

$$\left\langle\phi(x)^3\right\rangle_J = \bar\phi_J(x)^3 +3\bar\phi_J(x)G_{c,J}^{(2)}(x,x) +G_{c,J}^{(3)}(x,x,x).$$

Thus even the one-point equation couples the mean field to higher connected data. On a locally invertible 1PI branch, the $F=1$ identity and $\Gamma^{(1)}[\bar\phi_J]=-J$ combine to give

$$\Gamma_x^{(1)}[\bar\phi_J] = \left\langle \frac{\delta S_{F,\Lambda}}{\delta\phi(x)} \right\rangle_J.$$

This reorganizes the same information; it neither replaces the expectation of a nonlinear operator by that operator evaluated at $\bar\phi_J$ nor makes the hierarchy close.

Field-dependent changes and the Jacobian

The translation used above has unit Jacobian for a flat regulated measure. A general infinitesimal bosonic change

$$\phi_a\longmapsto\phi_a+\varepsilon R_a(\phi)$$

does not. For a measure $\mathrm d\mu_\Lambda=\mu_\Lambda(\phi)\mathrm d^N\phi$, understand $\mathcal Z_\Lambda$ and the normalized brackets with this measure and define

$$\operatorname{div}_{\mu_\Lambda}R \equiv \frac{1}{\mu_\Lambda} \partial_a\!\left(\mu_\Lambda R_a\right),$$

with $a$ summed. Let the corresponding normalized boundary flux be

$$\mathcal B_R[F] \equiv \frac{1}{\mathcal Z_\Lambda[J]} \int_{\mathcal C_\Lambda}\mathrm d^N\phi\, \partial_a\!\left( \mu_\Lambda F R_a e^{iS_\Lambda+iJ\cdot\phi} \right).$$

Repeating the regulated calculation gives

$$\begin{aligned} \left\langle F R_a\left(\partial_aS_\Lambda+J_a\right) \right\rangle_J &= i\left\langle R_a\partial_aF +F\operatorname{div}_{\mu_\Lambda}R \right\rangle_J \\ &\quad-i\mathcal B_R[F]. \end{aligned}$$

The divergence is the infinitesimal regulated Jacobian contribution. It vanishes for a constant translation in a flat measure, but not for an arbitrary field-dependent transformation. Zinn-Justin derives the general finite-regulator Jacobian identity in Zinn-Justin 2021, § 7.5.3, pp. 136–137.

A nonzero regulated Jacobian is not automatically an anomaly. An anomaly requires a nominal symmetry whose measure variation survives the regulator and renormalization analysis. Fermionic transformations also require Berezin measures, left or right derivatives, and graded contact signs; a perturbative gauge-theory treatment generally requires gauge fixing and ghost or BRST/BV structure. Those qualifications are not supplied by the scalar translation alone.

What the regulated change assumes

Ingredient	Role in the identity	What changes when it fails
Finite regulator	Turns the field integral into defined coordinates, kernels, and Jacobians	The continuum symbols remain formal until a controlled limit is given
Shiftable domain or cycle	Permits the chosen infinitesimal change of variables	A forbidden deformation invalidates that change
Vanishing boundary flux	Sets $\mathcal B_R[F]=0$	Endpoint or boundary terms appear explicitly
Declared measure behavior	Makes the translation Jacobian unit and retains other divergences	A field-dependent or anomalous measure term must be added
Feynman state and contour	Fixes which time-ordered correlators solve the relations	Different boundary data select different solutions
Regulated composite insertions	Gives meaning to coincident products such as $\phi^3(x)$	The continuum requires operator renormalization and mixing

The Lorentzian in–out hierarchy is a set of Feynman boundary-value identities. It is not a retarded response equation. Real causal expectation-value evolution requires the closed-time-path construction developed on In–Out versus In–In Expectation Values.

Why exact does not mean solved

The hierarchy is exact because no perturbative expansion or moment factorization was used at the fixed regulator. It remains unclosed because the equation for an $n$-point function contains higher-point functions. Solving a theory requires information not created by integration by parts:

• a state, contour, boundary or initial data, and normalization;
• renormalized parameters and prescriptions for composite operators;
• symmetry, spectral, positivity, or phase information appropriate to the formulation;
• a justified closure, truncation, expansion, or numerical representation when the full hierarchy is not solved;
• existence, uniqueness, convergence, branch-selection, and validation arguments for the proposed solution.

Different approximations may satisfy some Schwinger–Dyson equations while violating others. The equations are therefore powerful consistency conditions, but satisfying a selected subset is not by itself evidence that a truncation is controlled.

Common pitfalls

Using the classical equation inside a time-ordered product. The insertion $\delta S/\delta\phi$ produces delta-function contacts whenever it meets another field. Dropping them gives the wrong inverse-propagator equation.

Calling an oscillatory continuum integral a proof. The derivation is elementary at a declared finite regulator with a permitted contour and controlled boundary term. Removing that regulator, defining coincident composites, and preserving the measure identity are additional steps.

Dropping a Jacobian by analogy with a translation. A constant flat-measure shift has unit Jacobian. A field-dependent change carries the regulated divergence term, and a nontrivial measure may contribute even before a continuum limit is taken.

Equating exact with closed. The quartic two-point equation contains a four-point insertion, and the next equation contains still higher moments. A closure changes the problem and needs its own error and symmetry analysis.

Reading an in–out identity causally. The correlators retain Feynman boundary data. Retarded evolution requires an in–in generating functional rather than a relabeling of the same $Z[J]$.

Check your understanding

Use the success criteria and repair links to trace any mismatch to regulated integration by parts, contact terms, or signs.

Check	A successful response	Repair route
Recover the translation identity	Expands one regulated total derivative and obtains the plus source and the factor $+i$	Regulated integration by parts
Find the contact term	Uses $F=\phi(y)$ and obtains $i\delta_\Lambda(x,y)$ before multiplying by the scalar action sign	Contact terms
Check the scalar sign	Reduces the free two-point equation to $P_FD_F=-iI$	Exact scalar two-point identity
Diagnose nonclosure	Identifies the higher correlator introduced at each level and names the additional input needed by a proposed approximation	Why exact does not mean solved
Retain a measure variation	Includes $\operatorname{div}_{\mu_\Lambda}R$ and any nonzero boundary flux for a field-dependent change	Field-dependent changes

Where the hierarchy continues

• Schwinger–Dyson Hierarchies and Renormalization Inputs develops renormalized propagator and vertex hierarchies, closure choices, boundary inputs, and continuum solution methods.
• 2PI and nPI Effective Actions develops bilocal sources and self-consistent variational equations used in controlled truncation programs.
• Coincident Products and Contact Terms treats the distributional and operator-renormalization questions hidden by repeated regulated arguments.
• Renormalization and Effective Field Theory treats regulator removal, renormalized parameters, and composite-operator mixing.
• In–Out versus In–In Expectation Values changes the contour when the target is causal expectation-value dynamics.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Srednicki, Mark. Quantum Field Theory. Cambridge: Cambridge University Press, 2007. DOI.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford: Oxford University Press, 2021. DOI.