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I am reading this paper by Zamolodchikov about the expectation value of $T \bar{T}$ in $2d$ QFT and I don't understand how he uses the classical equations of motion. For instance, classically, in any translationally invariant field theory, you have the conservation of the energy-momentum tensor:

$\partial_{\mu} T^{\mu \nu} = 0$.

However, this only holds on shell, so in the quantum theory what you have is a Ward identity:

$ \frac{\partial }{\partial x^{\mu}} \langle T^{\mu \nu} (x) X \rangle = - \sum_{i=1}^{n} \delta( \mathbf{x} - \mathbf{x}_i ) \frac{\partial}{\partial x^{\nu}_i} \langle X \rangle $ ,

where $X$ is a string of fields that depend on the points $\mathbf{x}_1, ..., \mathbf{x}_n$.

My question is: why does he uses the classical type of equation instead of the quantum type of equation even inside correlation functions?

Comment: I think I have been able to get to Eq. (9) using the Ward identities, but I haven't been able to reproduce Eq. (11). In any case I think my doubt goes beyond the paper. It's just the example I have.

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  • $\begingroup$ How is it that T^mu^nu is not a field operator? Or is that not your question? $\endgroup$
    – user196418
    May 9 '19 at 14:20
  • $\begingroup$ Sorry I don't understand what you mean. $T^{\mu \nu}$ is a field operator as far as I know. My question is why it seems that you can forget about its quantum nature and compute as if fields were on shell. $\endgroup$
    – MBolin
    May 9 '19 at 14:22
  • $\begingroup$ Related: physics.stackexchange.com/q/476030/2451 $\endgroup$
    – Qmechanic
    May 9 '19 at 14:24
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    $\begingroup$ I have a suspicion that Zamolodchikov uses $\partial_{\mu} T^{\mu \nu} = 0$ with a caveat that the domain of correlation functions is $x_i \neq x_j$. The delta functions are by definition zero when their arguments aren't zero. $\endgroup$ May 9 '19 at 18:56
  • $\begingroup$ Yes, I think it's is simply that. And he can do that because he doesn't take the limit $\lim_{x \rightarrow x'} T(x) \bar{T} (x')$ (at least not naively). $\endgroup$
    – MBolin
    May 9 '19 at 19:16
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Quantum field theory is a quantum theory. In quantum theory we have operators which act on states. $T^{\mu\nu}(x)$ is an operator of this kind. For example, we can write $$ |\Psi\rangle =T^{\mu\nu}(x)|0\rangle, $$ and so on. (To be more precise, fields are operator-valued distributions, and for $|\Psi\rangle$ to be an honest state we need to smear $T$ as $\int d^dx f(x) T^{\mu\nu}(x)$ where $f$ is a test function. Also, we should interpret all equations below in distributional sense.)

This operator satisfies the equation $$ \partial_\mu T^{\mu\nu}(x)=0. $$ There is nothing in the right hand side. It is zero, no contact terms. No, it is not classical.

This means that matrix elements of $T$ satisfy the same equations, $$ \langle \Phi|\partial_\mu T^{\mu\nu}(x)|\Phi'\rangle=0. $$ The states $|\Phi\rangle$ and $|\Phi'\rangle$ can be created from vacuum or other states by acting with other operators, $$ \langle 0|O_1(x_1)\cdots O_k(x_k)\partial_\mu T^{\mu\nu}(x)O_{k+1}(x_{k+1})\cdots O_n(x_n)|0\rangle=0. $$ Zamolodchikov seems is talking about such expectation values in states, see his discussion below eq. (3).

The contact terms arise only when we start talking about time-ordered correlators. Time-ordered correlators are often denoted by $\langle O_1(x_1)\cdots O_n(x_n)\rangle$. (This appears to be not the notation used by Zamolodchikov.) Explicitly, time-ordered correlators are $$ \langle 0|\mathcal{T}\{O_1(x_1)\cdots O_n(x_n)\}|0\rangle. $$ Here time-ordering means (for simplicity, in case of two operators) $$ \mathcal{T}\{O_1(x_1)O_2(x_2)\}=\theta(x_1^0-x_2^0)O_1(x_1)O_2(x_2)+\theta(x_2^0-x_1^0)O_2(x_2)O_1(x_1). $$ You can see that it contains Heaviside step functions, so if you differentiate wrt, say, $x_1$, then you are differentiating not only the operators but also these step functions, which leads to contact terms. Note that if $x_1$ is spacelike from $x_2$, then there is no difference between the two orderings because $[O_1(x_1),O_2(x_2)]=0$ in this case, and time-ordering is trivial. So the contact terms can arise only when $x_1^0=x_2^0$ and operators are not spacelike separated. This is only when $x_1=x_2$, i.e. at coincident points.

Path integral computes time-ordered correlators, and textbooks which are based around path-integral formulation often do not even discuss expectation values in states, so it is easy to get confused about the origin of contact terms.

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  • $\begingroup$ See also equations in section 5 of the paper. $\endgroup$ May 10 '19 at 3:22
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I think it is because this conservation equation for $T^{\mu\nu}$ does not come from varying the action, but directly from the definition of the stress-energy tensor as: $$ T^{\mu\nu}\propto \frac{\partial S}{\partial g_{\mu\nu}} $$ and the fact that you can set the metric to $g^{\mu\nu} = \delta^{\mu\nu}$.

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    $\begingroup$ I don't understand what you mean... Could you elaborate a bit more? Do you mean you can get to $\partial_{\mu} T^{\mu \nu} = 0$ without using the equations of motion? $\endgroup$
    – MBolin
    May 9 '19 at 16:31
  • $\begingroup$ Not exactly, in this language conservation of stress-tensor follows from the definition and reparametrization invariance of the action, i.e. if $\delta g_{\mu\nu}=\partial_{(\mu} \xi_{\nu)}$ is an infinitesimal change of coordinates. The variation of the action is $\int d^dx \delta g_{\mu\nu} \frac{\delta S}{\delta g_{\mu\nu}}$. Using the form of $\delta g_{\mu\nu}$ and definition of $T$, one gets conservation after integrating by parts and using the freedom of choosing any $\xi_\mu$. $\endgroup$ May 10 '19 at 3:16
  • $\begingroup$ If you try to use this derivation in a path integral, you will have to apply reparametrization to other operator insertions, which will lead to the contact terms in the Ward identity. $\endgroup$ May 10 '19 at 3:17

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