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In matrix representation, the special unitary group is distinguished from the more general unitary group by the sign of the matrix determinant. However, this presupposes that the special unitary group is formulated in matrix representation. For a unitary group action NOT formulated in matrix...
I just need a hint to get started, and then I reckon the rest will follow...
We consider a theory where matter is a covector field ##\omega_a## which is described by a diffeomorphism-invariant action ##S_m##. Define:$$E^{a} = \frac{1}{\sqrt{-g}} \frac{\delta S_m}{\delta \omega_a}$$Also, ##T^{ab}...
Dirac derives Einstein's field equations from the action principle ##\delta I=0## where $$I=\int R\sqrt{-g} \, d^4x$$ (##R## is the Ricci scalar). Using partial integration, he shows that $$I=\int L\sqrt{-g} \, d^4x$$ where ##L## involves only ##g_{\mu\nu}## and its first derivatives, unlike...
Here is an action for a theory which couples gravity to a field in this way:$$S = \int d^4 x \ \sqrt{-g} e^{\Phi} (R + g^{ab} \Phi_{;a} \Phi_{;b})$$I determine\begin{align*}
\frac{\partial L}{\partial \phi} &= \sqrt{-g} e^{\Phi} (R + g^{ab} \Phi_{;a} \Phi_{;b}) \\
\nabla_a \frac{\partial...
I am stuck at the final part where one is supposed to show that the derivative of the second term of the action gives the mass term in the Majorana equation. For $$\chi^T\sigma^2\chi = -(\chi^\dagger\sigma^2\chi^*)^*$$ we get $$\frac{\delta}{\delta\chi^\dagger}(\chi^\dagger\sigma^2\chi^*)^*$$...
Here is the action:
##S = \frac{1}{16\pi} \int d^4 x \sqrt{-g} (R\phi - \frac{\omega}{\phi} g^{ab} \phi_{,a} \phi_{,b} + 16\pi L_m)##
the ordinary matter is included via ##L_m##. Zeroing the variation ##\delta/\delta g^{\mu \nu}## in the usual way gives
##\frac{\delta}{\delta g^{\mu \nu}}[R\phi...