A8 Scaling of zero restmass energy tensors

It should be noted that for an energy tensor Tab that is trace-free (Taa=0), we find that the scaling (P&R 5.9.2)

Tab = n-2 Tab preserves the conservation equation VaTab=0, since we find

In Maxwell theory, we have an expression for the energy tensor in terms of Fab that translates into spinor form as (P&R 5.2.4)

Tab = 2ÏÏ VAB ty A B '. In the case of Yang-Mills theory, we simply have extra indices

For a massless scalar field, subject to the equation (□ + f)0 = 0 considered earlier (P&R 6.8.30), we have the conformal invariance (P&R 6.8.32)

<$ = n-10, and then its (sometimes called 'new improved')[A3] energy tensor (P&R 6.8.36)

Tab = C{2Va(a<Vb')< - <Va(a-Vb-)< + <p2 Oaba-b-}

= § C{4Va< Va< - gabV<V< - 2<VaVb< + ftf<2gab - <2tfab }, C being a positive constant, satisfies the required conditions Taa = 0, VaTab=0, and "Tab=H-2Tab.

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