# Why can we not choose the stress tensor in a CFT to be identically symmetric?

+ 4 like - 0 dislike
944 views

The stress tensor for a conformal field theory (or any quantum field theory) can be derived from the action $S$ by the functional derivative

$$T^{\mu \nu} ~=~ -\frac{2}{\sqrt{|g|}}\frac{\delta S}{\delta g_{\mu \nu}},\tag{2.193}$$

where $g_{\mu \nu}$ is the background metric of signature $(+,-,-,-)$. This formula for the stress tensor appears to be identically symmetric, i.e. $$T^{\mu \nu} = T^{\nu \mu}$$ for any field configuration in the path integral, not just those obeying the classical equations of motion.

On the other hand, I don't see how this is consistent with the Ward identity (e.g. see Di Francesco et al, p. 107)

$$\langle (T^{\mu \nu} - T^{\nu \mu})X \rangle ~=~ -i \sum_i \delta(x - x_i) S_i^{\nu \mu} \langle X \rangle, \tag{4.66}$$

where $X$ is some product of fields $\phi(x_1) \cdots \phi(x_n)$, and the field $\phi$ transforms internally under an infinitesimal rotation $x^{\mu} \to x^{\mu} + \omega^{\mu} {\,}_{\nu} x^\nu$ as $\phi \to \phi + \omega_{\mu \nu} S^{\mu \nu} \phi$.

If $T^{\mu \nu}$ was identically symmetric, then both sides should equal zero.

This post imported from StackExchange Physics at 2015-05-04 13:53 (UTC), posted by SE-user Dominic Else

edited May 4, 2015

+ 1 like - 0 dislike
1. Yes, eq. (2.193) is a classical formula, and the symmetry of the (Hilbert) stress-energy-momentum tensor $T^{\mu\nu}$ is only valid classically.

2. Quantum mechanically, the symmetry of $T^{\mu\nu}(x)$ is broken by the presence of other fields in positions $x_1,x_2,\ldots$ in the (time-ordered) correlator
$$\langle T\left\{ (\hat{T}^{\mu \nu}(x) - \hat{T}^{\nu \mu}(x))\hat{X}(x_1,x_2,\ldots)\right\} \rangle$$ $$~=~ -i\hbar \sum_i \delta(x - x_i) ~S_i^{\nu \mu} \langle T\left\{\hat{X}(x_1,x_2,\ldots)\right\} \rangle. \tag{4.66}$$ From the point of view of the path integral, this can be traced back to the time-slicing prescription. Here $T$ denotes time-ordering.

References:

1. P. Di Francesco, P. Mathieu and D. Senechal, CFT, 1997.
This post imported from StackExchange Physics at 2015-05-04 13:53 (UTC), posted by SE-user Qmechanic
answered May 2, 2015 by (3,120 points)
Thanks for your answer. If you could elaborate on your statement that the asymmetry at the quantum level is due to the time slicing prescription it would be very useful to me.

This post imported from StackExchange Physics at 2015-05-04 13:53 (UTC), posted by SE-user Dominic Else

 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$\hbar$ysic$\varnothing$OverflowThen 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.