Need help with Weinberg's QFT-Space inversion and time-reversal

[qinglong.1397]

Need help with Weinberg's QFT---Space inversion and time-reversal

Weinberg's Quantum Theory of Fields VI.

Page 78. He gave the last equation but he didn't prove it, although he qualitatively explained it. I tried to prove it quantitatively but I failed.

Also, in page 80, he didn't prove the first equation either. I need help with those two equations.

The two pictures are from Google http://books.google.com/books?id=Nc...ce=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

 

[azntoon]

For the first equation, you need to consider the effects of space inversion and time reversal. One way to do this is to use the transformation properties of the fields. Under a space inversion, the fields transform as follows: • Scalar fields: remain unchanged • Vector fields: invert their direction • Spinor fields: transform according to the spinor representation of the Lorentz group. Under time reversal, the fields transform as follows: • Scalar fields: remain unchanged • Vector fields: invert their direction • Spinor fields: transform according to the time-reversal representation of the Lorentz group. By considering the effect of these transformations on the fields and using the appropriate symmetry transformations, you can derive the desired equation. For the last equation, you need to consider the effect of space inversion and time reversal on the energy-momentum tensor. This can be done by using the transformation properties of the fields, as above, and then computing the energy-momentum tensor for each case. This will give you the desired result.