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CAF123
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Homework Statement
Consider a theory which is translation and rotation invariant. This implies the stress energy tensor arising from the symmetry is conserved and may be made symmetric. Define the (Schwinger) function by ##S_{\mu \nu \rho \sigma}(x) = \langle T_{\mu \nu}(x)T_{\rho \sigma}(0)\rangle##. By the above restrictions on the theory, we have that $$S_{\mu \nu \rho \sigma} = S_{\nu \mu \rho \sigma} = S_{\mu \nu \sigma \rho} = S_{\nu \mu \sigma \rho}\,\,\,(1),$$ $$S_{\mu \nu \rho \sigma}(x) = S_{\rho \sigma \mu \nu}(-x)\,\,\,\,(2).$$ If we now impose parity invariance and scale invariance with ##T_{\mu \nu}## of dimension 2, we have further $$S_{\mu \nu \rho \sigma}(x) = S_{\rho \sigma \mu \nu}(x)\,\,\,(3)$$ and $$S_{\mu \nu \rho \sigma}(\lambda x) = \lambda^{-4}S_{\mu \nu \rho \sigma}(x)\,\,\,(4)$$
All these contraints restrict the form that ##S_{\mu \nu \rho \sigma}## can take. What I want to do is to write out the most generic form first and then build in the constraints given by (1),(2),(3) and (4) above.
Homework Equations
Problem from Di Francesco et al, 'Conformal Field Theory' P.108
The Attempt at a Solution
We can build the 4th rank tensor from only x and delta's (or the metric g). We cannot use the levi-civita tensor because that is antisymmetric in any two indices so fails (1) immediately. The most general 4th rank tensor I was getting is $$S_{\mu \nu \rho \sigma}(x) = a_1x_{\mu}x_{\nu}x_{\rho}x_{\sigma}f_1(x^2) + a_2g_{\mu \nu}g_{\rho \sigma}f_{2,1}(x^2) + a_3g_{\mu \rho}g_{\nu \sigma}f_{2,2}(x^2) + a_4g_{\mu \sigma}g_{\rho \nu}f_{2,3}(x^2) + a_5g_{\mu \nu}x_{\rho}x_{\sigma}f_{3,1}(x^2) $$$$+ a_6g_{\rho \sigma}x_{\mu}x_{\nu}f_{3,2}(x^2) + a_7g_{\mu \sigma}x_{\rho}x_{\nu}f_{3,3}(x^2) + a_8g_{\mu \rho}x_{\sigma}x_{\nu}f_{3,4}(x^2) + a_9 g_{\nu \rho}x_{\sigma}x_{\mu}f_{3,5}(x^2) + a_{10}g_{\nu \sigma}x_{\rho}x_{\mu}f_{3,6}(x^2)$$ where ##a_i## are constants and ##f_i(x^2)## are Lorentz invariant functions.
Consider the first term above. By (4), this becomes ##S_{\mu \nu \rho \sigma}(\lambda x) = \lambda^4x_{\mu}x_{\nu}x_{\rho}x_{\sigma}f_1(\lambda^2 x^2) = \lambda^{-4}S_{\mu \nu \rho \sigma}(x) \Rightarrow f_1(x^2) = \lambda^8f_1(\lambda^2x^2)##. The answer in the book given above has ##f_1(x^2) = A_4/(x^2)^4##. How did they obtain this? Once I have the general idea about how to correctly impose the constraints, I should be able to continue.
Thanks!