CAF123

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**1. Homework Statement**

Find the infinitesimal dilation and conformal transformations and thereby show they are generated by ##D = ix^{\nu}\partial_{\nu}## and ##K_{\mu} = i(2x_{\mu}x^{\nu}\partial_{\nu} - x^2\partial_{\mu})##

The conformal algebra is generated via commutation relations of elements of the set ##\left\{P_{\mu}, D, L_{\mu \nu}, K_{\mu}\right\}##. Write down the non vanishing commutation relations for the comformal group given that ##P_{\mu} = i\partial_{\mu}## and ##L_{\mu \nu} = i(x_{\nu}\partial_{\mu} - x_{\mu}\partial_{\nu})##, Hence show that ##P^2## is no longer a Casimir operator and thus that only massless theories can be conformally invariant.

2. Homework Equations

2. Homework Equations

##x'^{\mu} = \alpha x^{\mu}## for finite dilations and ##x'^{\mu} = \frac{x^{\mu} - x^2b_{\mu}}{1-2x_{\mu}b^{\mu} +b^2x^2}## for finite CT's.

**3. The Attempt at a Solution**

Write an infinitesimal dilation transformation as ##x'^{\mu} = e^{\Lambda}x^{\mu} = (1+i\alpha D)x^{\mu}## where D is the dilation generator and alpha some infinitesimal scaling. Alternatively, by taylor expanding to first order, ##x'^{\nu} = x^{\nu} + x^{\mu}\partial_{\mu}x^{\nu}##. Comparison with the previous result gives ##D = ix^{\mu}\partial_{\mu}##. Is that ok?

Similarly, for CT's, ##x'^{\mu} = (1+i\alpha^{\rho} K_{\rho})x^{\mu} = x^{\mu} + 2x^{\mu}(b \cdot x) - x^2b^{\mu}##. But if I identify lhs and rhs here I don't get any derivative terms appearing?

And to check my reasoning for the last part: P^2 is not a casimir operator for the conformal algebra anymore because it doesn't commute with D. (A casimir must commute with all the algebra generators). I am not really sure how to show that massless theories must be conformally invariant given the fact that ##P^2## is not a casimir, so a hint here too would be great. Thanks!