# Do the Ward identities contain contact terms in Euclidean QFT?

In derivations of the Ward identities, I have never seen the signature of spacetime explicitly specified, so I'd always assumed they hold regardless of signature. However, the argument below seems to imply that they cannot hold in Euclidean QFT.

Consider a Euclidean QFT with Noether current $$j_\mu$$. The Ward identites with operator insertions $$\mathcal{O}_i(x_i)$$ are usually written as:

$$\tag{1}\left< \partial^\mu j_\mu(x) \mathcal{O}_1(x_1)\cdots \mathcal{O}_n(x_n) \right>=\sum_{i=1}^n \delta(x-x_i) \left<\mathcal{O}_1(x_1) \cdots \delta \mathcal{O}_i(x_i)\cdots \mathcal{O}_n(x_n)\right>,$$

where the angle brackets denote the vacuum expectation. In particular, the $$n=0,1$$ identities are:

$$\tag{2}\left<\partial^\mu j_\mu\right>=0$$

$$\tag{3}\left< \partial^\mu j_\mu(x) \mathcal{O}(y)\right>= \delta(x-y)\left<\delta\mathcal{O(y)}\right>.$$

However, in the path integral derivation of Eq. $$(1)$$, the past and future BCs are arbitrary (e.g. see the derivation in Polchinski's String Theory Vol 1 Sec 2.3). Vacuum BCs are not required. So in fact we can strip off the angle brackets to get operator equations. Note that this is unlike in Lorentzian signature, where time ordering would prevent us from simply stripping off the angle brackets. Now, $$(2)$$ and $$(3)$$ become:

$$\tag{4}\partial^\mu j_\mu=0$$

$$\tag{5} \partial^\mu j_\mu(x) \mathcal{O}(y)= \delta(x-y)\cdot\delta\mathcal{O(y)}.$$

Polchinski stresses that these do indeed hold as operator equations.

But $$(4)$$ and $$(5)$$ clearly cannot both be true as operator equations, at least not in the usual sense. If $$(4)$$ were true then the LHS of $$(5)$$ would equal zero. Something has gone wrong,

Have I missed something, or are the usual Ward identities false in Euclidean signature?

• i haven't read Polchinski so I cannot comment, but do note that in euclidean CFT the passage from correlator equations to operator equations still involves an ordering prescription, typically radial ordering. So 4 and 5 are both correct, and there is an implicit $R[\cdots]$ symbol. Dec 30, 2021 at 2:54
• @AccidentalFourierTransform (1) I thought that all local operators commuted in Euclidean QFT, since everything is spacelike separated. (2) If my Eq. 5 is to be fixed via some ordering prescription, $\textit{which}$ ordering prescription? The LHS is ordering-dependent whereas the RHS is not, so it surely will only hold for certain prescriptions but not others. Dec 30, 2021 at 10:49
• My understanding of Polchinski's statement has always been that this means that it is valid within expectation values only. Dec 30, 2021 at 13:47
• @Oбжорoв Expectation values in the vacuum state? In any state? Are the expectations time-ordered? Note that Polchinski's Eq.s 2.3.6 and 2.3.7 (which are the curved space equivalents of my Eq.s 4 and 5) are stated to hold as operator equations. Dec 30, 2021 at 14:51
• By 'operator relation' Polchinski means a relation that is true within path integral correlation functions which may involve arbitrary insertions of other local operators as long as those operators are far away from $x$ and $y$. From the operator perspective itself, the relation involving the contact term still arises from the derivative of a (Euclidean) time ordering operator just like for Minkowski. Dec 30, 2021 at 15:02

Euclidean correlators do also have an operator ordering prescription, typically radial ordering $${\cal R}$$, which is analogous to the time ordering $$T$$ for Minkowskian correlators. The radial ordering $${\cal R}$$ becomes covariant radial ordering $${\cal R}_{\rm cov}$$ in the Schwinger-Dyson (SD) equations,
$$\left< \Omega \left| {\cal R}_{\rm cov}\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(x)}\right\}\right| \Omega \right>_J~=~i\hbar\left< \Omega \left| {\cal R}_{\rm cov}\left\{\frac{\delta F[\phi]}{\delta \phi(x)} \right\}\right| \Omega \right>_J ~, \tag{A}$$
i.e. radial-differentiations inside its argument should be taken after/outside the usual radial ordering $${\cal R}$$, cf. e.g. this Phys.SE post. This in turn induces contact terms in the Ward identities.