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In the literature on QFT there are a lot of different equations that are all called "Schwinger-Dyson equation" so I wanted to know how are they related and if they have proper names.

  1. The first equation can be obtained by making a change of variables \begin{equation} \phi \rightarrow \phi + \epsilon \end{equation} where $\epsilon$ is an arbitrary variation of the fields. Then you write down the generating functional $Z[J]$ and use the fact that it should be invariant under changes of variables to get \begin{equation} Z[J]\rightarrow\int\mathcal{D}\phi e^{iS[\phi+\epsilon]+\int J(\phi+\epsilon)}=Z[J]. \end{equation} Expanding in powers of $\epsilon$ you get the following equation \begin{equation} \frac{\delta S}{\delta \phi(x) }[\frac{1}{i}\frac{\delta }{\delta J(x)}]Z[J]+J(x)Z[J]=0.\tag{1} \end{equation}

  2. The second one is pretty similar but instead of seeing how the generating functional changes, you see how does the $n$-point correlation function change. After expanding in powers of $\epsilon$ you get \begin{equation} \big(\square_x +m^2 \big)\langle \phi(x)\phi(y_1)\dots \phi(y_n)\rangle=\langle\mathcal{L}'_{int}[\phi(x)]\phi(y_1)\dots\phi(y_n)\rangle-i\sum_i\delta(x-y_i)\langle \phi(y_1)\dots\phi(y_{i-1})\phi(y_{i+1})\dots \phi(y_n)\rangle.\tag{2} \end{equation} It is my understanding that the first equation is the "generating" equation for the $n$ equations on this second case. If you take $n$ derivatives $\frac{\delta}{\delta J(y_i)}$ on the first one you get the second one. However, I'm not sure about this so I'd like some insight.

  3. The third one is a bit different. Instead of making an arbitrary change of variables we do a symmetry transformation \begin{equation} \phi \rightarrow \phi + A\phi \epsilon \end{equation} that keeps the action unchanged. If we treat the transformation parameter $\epsilon$ as a function of spacetime we get \begin{equation} \partial_{\mu}\langle j^{\mu}(x)\phi(y_1)\dots \phi(y_n)\rangle-i\sum_i\delta(x-y_i)\langle A \phi(y_1)\dots \phi(y_n)\rangle.\tag{3} \end{equation} where $j^{\mu}$ is the conserved Noether current associated with the symmetry we are using. This equation is usually known as the Ward-Takahashi identity but many books call it "the Schwinger-Dyson equation for global symmetries" which is a big point of confusion.

So the question is: What is the name for each equation? How are they related? Are there more equations related to this approach? What is the intuition behind these different equations?

Qmechanic
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P. C. Spaniel
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  • Comment to the post (v4): The term $\frac{\delta S}{\delta \phi(x)}$ in eq. (1) should be inside the path integral $Z[J]$ in order to make sense. – Qmechanic May 15 '19 at 21:38

1 Answers1

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All OP's 3 versions of the Schwinger-Dyson (SD) equations are consequences of the following SD equation

$$\langle \delta_{\epsilon}F[\phi]\rangle + \frac{i}{\hbar} \langle F[\phi]\delta_{\epsilon}S[\phi]\rangle~=~0.\tag{A}$$

Here it is implicitly assumed that the path integral measure is invariant under the infinitesimal transformation $$\delta_{\epsilon}\phi^{\alpha}(x).\tag{B}$$

In OP's eqs. (1) & (2) the transformation (B) is a just shift. Eq. (3) is explained on the Wikipedia for the Ward-Takahashi identity.

Qmechanic
  • 201,751
  • Notes for later: In a 0+0D matrix model with action $S(M)={\rm tr}V(M)$ then (i) an integration by parts $\quad\int!dM\frac{\partial}{\partial M^i{}_j}\left[G(M)^i{}_j{\rm tr}F(M)\exp\left(-\frac{1}{\hbar}{\rm tr}V(M)\right)\right]=0$ ignoring boundary contributions, or (ii) an infinitesimal change $\delta M^i{}_j=G(M)^i{}_j$ of the integration variable $M^i{}_j$, both lead to the SD loop equation $\quad\langle\frac{\partial G(M)^i{}_j}{\partial M^i{}_j}{\rm tr}F(M)\rangle+\langle {\rm tr}[G(M)F^{\prime}(M)]\rangle=\frac{1}{\hbar}\langle{\rm tr}F(M){\rm tr}[G(M)V^{\prime}(M)]\rangle$. – Qmechanic Feb 08 '24 at 08:54
  • If $G(M)=\frac{1}{z-M}$ with $|z|\gg 1$ is the resolvent then $\quad\frac{\partial G(M)^i{}_j}{\partial M^i{}_j}=\left({\rm tr}\frac{1}{z-M}\right)^2$. If $G(M)=e^{zM}$ then $\quad\frac{\partial G(M)^i{}_j}{\partial M^i{}_j}=z\int_0^1!d\alpha~{\rm tr}e^{\alpha zM}{\rm tr}e^{(1-\alpha) zM}$. – Qmechanic Feb 08 '24 at 09:42
  • Next define the $\beta$-matrix model with integration measure $\quad dM=\Delta(\lambda)^{\beta}\left[\prod_kd\lambda_k\right]$, where $\Delta(\lambda)=\prod_{i<j}(\lambda_j-\lambda_i)$ is the Vandermonde matrix, cf. e.g. arXiv:1510.04430. – Qmechanic Feb 08 '24 at 12:53
  • The SD loop equation for the $\beta$-matrix model reads $\quad\left(1-\frac{\beta}{2}\right)\langle{\rm tr}\frac{1}{(z-M)^2}{\rm tr}F(M)\rangle+\frac{\beta}{2}\langle\left({\rm tr}\frac{1}{z-M}\right)^2{\rm tr}F(M)\rangle+\langle {\rm tr}[\frac{1}{z-M}F^{\prime}(M)]\rangle$ $=\frac{1}{\hbar}\langle{\rm tr}F(M){\rm tr}[\frac{1}{z-M}V^{\prime}(M)]\rangle=\frac{1}{\hbar}\langle{\rm tr}F(M){\rm tr}\frac{1}{z-M}\rangle V^{\prime}(z)-\frac{1}{\hbar}\langle{\rm tr}F(M){\rm tr}[\frac{V^{\prime}(z)-V^{\prime}(M)}{z-M}]\rangle.\tag{4.1.20}$ – Qmechanic Feb 08 '24 at 17:51
  • Sketched proof: $\quad\sum_kG(\lambda_k)\frac{\partial\ln\Delta(\lambda)}{\partial\lambda_k}=\sum_{i<j}\frac{G(\lambda_j)-G(\lambda_i)}{\lambda_j-\lambda_i}$. If $G(\lambda_k)=\frac{1}{z-\lambda_k}$ then $\quad 2\sum_kG(\lambda_k)\frac{\partial\ln\Delta(\lambda)}{\partial\lambda_k}=\sum_{i\neq j}\frac{1}{z-\lambda_j}\frac{1}{z-\lambda_i}$ $=\left({\rm tr}\frac{1}{z-M}\right)^2-{\rm tr}\frac{1}{(z-M)^2}$ $=\left({\rm tr}\frac{1}{z-M}\right)^2+\frac{\partial}{\partial z}{\rm tr}\frac{1}{z-M}$. $\Box$ – Qmechanic Feb 08 '24 at 18:35