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I have a very technical question on deriving a Ward identity directly from a given explicit form of the "conserved current". Let me emphasize that I do not start with an apriori knowledge on the symmetry transformation that corresponds to the given "conserved current". I am given an explicit expression for the "conserved current" from the beginning. Just for clarity, let me define a "conserved current" $\Theta_{\mu\nu}$ given as \begin{equation} \Theta_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-g_{\mu\nu}\cal{L}\,, \end{equation} where $\phi$ is a scalar field and I have a simple massive $\lambda\phi^4$ theory in mind. This is nothing but the conventional energy-momentum tensor, and we already know that the corresponding symmetry transformation is space-time translation. However, I don't want to use any additional information except the explicit expression for the current and the canonical commutation relation given as \begin{equation} [\partial_t\phi(\boldsymbol{x},t),\phi(\boldsymbol{y},t)]=-i\delta^3({\boldsymbol{x},\boldsymbol{y}})\,, \end{equation} to arrive at the result: \begin{equation} k^{\mu}\Gamma^n_{\mu\nu}(k;p_1,...,p_n)=-i\underset{P}{\sum}(p_{1}+k)_{\nu}G^{n}\left(p_{1}+k,p_{2},...,p_{n}\right)\,, \end{equation} where the sum is on cyclic permutations of the indices $1$ to $n$. This kind of procedure seems to be done in Eqs.(2.8-2.13) of this renowned reference: http://inspirehep.net/record/61135, although they do not show it explicitly.

Define a function given by \begin{equation} i\Gamma^n_{\mu\nu}(x;x_1,...,x_n)=\frac{\delta}{\delta \cal{J}^{\mu\nu}(x)}G^n(x_1,...,x_n)\big|_{\cal{J}^{\mu\nu}=0}\,, \end{equation} where $G^n$ is the $n$-point Green's function. In the above equation, I am thinking of the following replacement of the Lagrangian inside the $e^{iS}$ of the path integral: \begin{equation} \cal{L}\rightarrow\cal{L}+\Theta_{\mu\nu}\cal{J}^{\mu\nu}\,. \end{equation} Now let's say I want to find an expression for $k^\mu\Gamma^n_{\mu\nu}$ ($\Gamma^n_{\mu\nu}$ here is Fourier transformed into momentum space), i.e., a Ward identity. In principle, I should be able to explicitly do this using only the equal-time canonical commutation relation.

Now here is my question. When explicitly taking $\partial^{\mu}\Gamma^n_{\mu\nu}$ in position space, I need to evaluate an expression like \begin{equation} \left\langle T_\ast(\square_x\phi(x)\partial^x_\nu\phi(x)\phi(x_1)...\phi(x_n))\right\rangle\,\,\,\,\,(\star), \end{equation} where $T_*$ is the "covariant" time ordering that is different from the usual time ordering $T$. For example, the difference is highlighted as $$T_*(\partial_x^\mu\phi(x)\partial_y^\nu\phi(y))=\partial^\mu_x\partial^\nu_y T(\phi(x)\phi(y)).$$ (Note that $T_*$ ordering appears because the definition of $\Gamma^n_{\mu\nu}$ is through a path integral.) To evaluate this $T_*$ ordered object using canonical commutation relation given above, I must first take out all the derivatives out of the $T_*$ ordering to make it into a $T$ ordering, e.g., \begin{equation} (\textrm{differential operator})\times\langle T(\phi(x)^{m}\phi(x_1)...\phi(x_n))\rangle\,. \end{equation} in position space. Then using the definition of $T$-ordering, I could differentiate the $\theta$-functions and continue.

It is not so clear to me how to do this for the expression $(\star)$, since stripping out the derivatives out of the $T_*$-ordering seems hard to do. I feel like this is a very technical question in the sense that I may be just not coming up with the relevant algebraic manipulations.

Any comments are appreciated.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – rob Dec 11 '18 at 17:52

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