# When does the correlator of a string of fields and the current vanish "sufficiently fast" at infinity and Ward's identity?

We will have to shut down our server temporarily for maintenance. The downtime will start at Wednesday, 27. January 2021 at 12:00 GMT and have a duration of about two hours. Please save your edits before this time. Thanks for your patience and your understanding.

+ 4 like - 0 dislike
819 views

One consequence of the Ward identity (cf. Di Francesco et al) is that it means variation of correlators under infinitesimal transformation is zero. This can be seen by integrating the ward identity, and using the Gauss divergence theorem to integrate over the surface at infinity, and setting the surface term to zero.

$$\delta_{\omega} \langle j^{\mu}_a \Phi(x_1) \ldots \Phi(x_n) \rangle =\int \frac{\partial}{\partial x^{\mu}}\langle j^{\mu}_a \Phi(x_1) \ldots \Phi(x_n) \rangle = \int_{\Sigma} ds_{\mu} \langle j^{\mu}_a \Phi(x_1) \ldots \Phi(x_n) \rangle \\ =0$$.

DiFrancesco says the integrand goes to zero at infinity because the divergence of the correlator vanishes at away from the points $x_1 \ldots x_n$. How does the vanishing of divergence mean vanishing of the correlator? In general when can we assume a correlator to vanish at infinity "sufficiently fast"?

In this particular case, from familiar examples of the free boson and fermion, I know that $j$ is proportional to the gradient of the fields, so does this mean we are considering field solutions which attain a constant value? How is it different from solitonic solutions?

I have seen surface term to be vanishing sufficiently fast in many places in QFT, cannot seem to remember them now ,but this seems to be quite an adhoc assumption, is it because you want the drama of the theory to happen only in a finite range, and you cannot consider particles at inifinity?

This post imported from StackExchange Physics at 2014-06-21 09:00 (UCT), posted by SE-user ramanujan_dirac
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar\varnothing$sicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.