Reference request for a treatment of Schwinger–Dyson equations

Question

Is there a treatment of Schwinger–Dyson equations with no mention of Green's functions? Is there perhaps a purely algebraic analog?

Answer 1

I will differ somewhat with the answers by Carlo and Zurab. Just because the identity now known as the Schwinger-Dyson equation was first discovered in the context of Green functions, does not mean that it is most naturally expressed in terms of Green functions (aka QFT correlation functions).

Let me refer to the excellent treatment of the Schwinger-Dyson equation in Chater 15 of Quantization of Gauge Systems by Henneaux & Teitelboim (PUP, 1994). Consider functionals $A[\phi]$ on the space of (off-shell) field configurations $\phi$. Let us suppose that all of these functionals are sufficiently regular so that we can use the variation formula $A[\phi + t\zeta] = A[\phi] + t \int \alpha_i[\phi](x) \zeta^i(x)\, dx + O(t^2)$, for some $\alpha_i[\phi](x)$ which we also denote by $\frac{\delta A[\phi]}{\delta \phi^i(x)}$. One of these functionals is the action functional $S[\phi]$ and its variation $E_i[\phi](x) = \frac{\delta S[\phi]}{\delta \phi^i(x)}$ gives the classical equations of motion of the field theory.

For any given off-shell functional, there is a map $A[\phi] \mapsto T(A[\phi])$ which stands for the time ordered quantization of $A[\phi]$, so that $T(A[\phi])$ is an element of the quantum algebra of (on-shell) observables (or operators). The map $T$ is linear, but obviously not an algebraic homomorphism because it maps a classical (commutative) algebra into a (quantum) non-commutative one. Also, since it maps (classical) off-shell functionals to (quantum) on-shell operators, it must have a kernel that is somehow generated by the equations of motion.

The Schwinger-Dyson equation precisely specifies the kernel of $T$ within the space of classical off-shell functionals: $$ T\left(A[\phi] \int \lambda^i(x) \frac{\delta S[\phi]}{\delta \phi^i(x)}\, dx + \frac{\hbar}{i} \int \frac{\delta A[\phi]}{\delta \phi^i(x)} \lambda^i(x)\, dx \right) = 0 , $$ for any $A[\phi]$ and $\lambda^i(x)$, where $\lambda^i$ needs to be $\phi$-independent. Note that the treatment in Henneaux & Teitelboim uses equations like $\langle T(-) \rangle = 0$ instead of $T(-) = 0$. However, they use $\langle \hat{O} \rangle$ to stand for an arbitrary matrix element of the operator $\hat{O}$ (evaluated using a path integral), so the two kinds of equalities are equivalent. As can be seen from the above formula, the kernel of $T$ is an $\hbar$-deformation of the subspace of off-shell functionals generated by $\int \lambda^i(x) E_i[\phi](x)\, dx$ for arbitrary $\lambda^i(x)$, which all vanish when restricted on-shell, to the solutions of $E_i[\phi](x) = 0$ among all possible field configurations. To get the full classical kernel of the restriction on-shell, we would need to allow $\lambda^i(x) = \lambda^i[\phi](x)$ to be $\phi$-dependent. But doing that naively breaks the above formula. So, when possible, arbitrary $\lambda^i[\phi](x)$ should be approximated by linear combinations of appropriate choices of $A[\phi]$ and $\lambda^i(x)$.

One final remark about a more geometric way of rewriting the above version of the Schwinger-Dyson equation. Let us interpret $\lambda^i(x)$ as a vector field on the field configuration space and denote the corresponding Lie derivative by $\mathcal{L}_\lambda$. The Schwinger-Dyson equation then reads $$ T\left( \frac{\hbar}{i} e^{-iS[\phi]/\hbar} \mathcal{L}_\lambda ( A[\phi] e^{iS[\phi]/\hbar} ) \right) = 0. $$

Answer 2

A quick and clear and rigorous derivation of the Schwinger-Dyson equation from BV-theory in causal perturbation theory is offered in remark 7.7 of Rejzner 16, it's spelled out at nLab:BV-operator -- Schwinger-Dyson equation.

The upshot is the the SD-equation is nothing but the incarnation of the free field theory quantum BV-differential, schematically of the form $\{S,-\}_{BV} + i \hbar \Delta_{BV}$; hence the quantum correction in the SD-equation originates in the BV-operator $\Delta_{BV}$.

In the traditional path-integral heuristics this is highlighted already in section 15.5.3 of Henneaux-Teitelboim 92 (which Igor Khavkine is pointing to in his reply), where therefore it is suggested to call the free quantum BV-differential the "Schwinger-Dyson operator".

Answer 3

I'm not sure why you would want to avoid Green's functions, which provide the most natural formulation of the Schwinger-Dyson equations, but indeed, algebraic approaches do exist. In the context of random Riemann surfaces the Schwinger-Dyson equations appear as a set of algebraic constraints on the partition function, and they can be given a geometric interpretation. See for example Section III of The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders.

Answer 4

In the formulation of QFT using formal functional integrals, as mentioned by Igor in his answer, the Schwinger-Dyson equation becomes an infinite-dimensional differential equation for the partition function. As such, since the partition function is the generating functional for the Green functions, all you are doing is simply grouping the Schwinger-Dyson equations for all Green functions together in a single formula. I also share Igor's viewpoint that the Schwinger-Dyson equations written in terms of the partition function is more natural, because it makes the structure of the equation much more evident.

That being said, there are two nice references that have not been mentioned by the answers given so far. One is the book Path Integral Methods in Quantum Field Theory by R. J. Rivers (Cambridge University Press, 1987), where the Schwinger-Dyson equations are a central concept and are formulated in many different ways. Another, which I strongly recommend, is the paper by M. Dütsch and K. Fredenhagen The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory, Commun. Math. Phys. 243 (2003) 275-314, arXiv:hep-th/0211242. In this second reference, it is shown that the Schwinger-Dyson equation is actually a particular case of the quantum correction (due to perturbative renormalization) to all identities that follow from the classical equations of motion, called the Master Ward Identity (MWI for short). The quantum BV-equation mentioned in Urs's answer is also a particular case of the MWI, which has a nice algebraic interpretation - to wit, one can see the classical limit of the MWI as the simple fact that the (left-hand sides of) all identities that follow from the classical equations of motion are elements of the ideal of off-shell functionals of field configurations (satisfying certain conditions which we do not need to recall here) which is generated by the Euler-Lagrange operator. At the quantum level, not all identities survive due to renormalization - the ensuing violations are known as anomalies. This means that the "classical" ideal of formal power series in $\hbar$ of functionals generated by the Euler-Lagrange operator is no longer an ideal with respect to the quantum product. However, the most general form for the anomalies is dictated precisely by the MWI. In the case of local gauge (or BRS) symmetries, this leads through the quantum BV equation to the Wess-Zumino consistency conditions and so on.

Just a final word of warning: it must be said that all of the above can be made rigorous only at the formal perturbative level. In constructive QFT, the Schwinger-Dyson equations are usually not used as a tool to construct a model but are rather derived rigorously as a consequence after the model has been constructed, since only then can they be made sense of. Analysis of QFT models using the Schwinger-Dyson equations is usually deemed as "non-perturbative" for the equations make no explicit mention of perturbation theory, but making sense of this equations outside perturbation theory is a messy business.

Answer 5

In addition to Carlo Beenakker's answer, in the context of the theory of random matrices the Schwinger-Dyson equations are discussed (without any mention of Green's functions) in the book http://link.springer.com/book/10.1007%2F978-3-540-69897-5 (Large Random Matrices: Lectures on Macroscopic Asymptotics, by Alice Guionnet). See also http://arxiv.org/abs/1307.1806 (Schwinger-Dyson equations: classical and quantum, by James A. Mingo and Roland Speicher).

However, I agree to Carlo Beenakker that the most natural language for Schwinger-Dyson equations in quantum field theory is the language of Green's functions. Even when the Schwinger-Dyson equations are discussed in the very abstract mathematical context, it is useful to use the analogy with Green's functions to get an insight of what is going on. See, for example, http://arxiv.org/abs/hep-th/0404090 (The residues of quantum field theory - numbers we should know, by Dirk Kreimer) and http://arxiv.org/abs/hep-th/0407016 (What is the trouble with Dyson--Schwinger equations? by the same author).