LAHLH
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Hi,
If I have a matrix valued field [itex]B(x)_i^{..j}=B^a (x) (T^a)_i^{..j}[/itex] and the relevant part of my Lagrangian is [itex]L=Tr(-\tfrac{1}{2}\partial^{\mu}B\partial_{\mu}B+..)[/itex] then how can I see that the propagator for the matrix field is [itex]\Delta_{i..k}^{..j..l}(k^2)=\tfrac{(T^a)_i^j(T^a)_k^l}{k^2-i\epsilon}[/itex] ?
I understand that if we expand the L in terms of the coefficient field we get [itex]L=-\tfrac{1}{2}\partial^{\mu}B^a\partial_{\mu}B^{a}[/itex] and this leads to the propagator for the coefficient field as [itex]\Delta^{ab}(k^2)=\tfrac{\delta^{ab}}{k^2-i\epsilon}[/itex], (just like usual for a massless scalar field) but not sure how to see the propagator of matrix field...
thanks for any help...
If I have a matrix valued field [itex]B(x)_i^{..j}=B^a (x) (T^a)_i^{..j}[/itex] and the relevant part of my Lagrangian is [itex]L=Tr(-\tfrac{1}{2}\partial^{\mu}B\partial_{\mu}B+..)[/itex] then how can I see that the propagator for the matrix field is [itex]\Delta_{i..k}^{..j..l}(k^2)=\tfrac{(T^a)_i^j(T^a)_k^l}{k^2-i\epsilon}[/itex] ?
I understand that if we expand the L in terms of the coefficient field we get [itex]L=-\tfrac{1}{2}\partial^{\mu}B^a\partial_{\mu}B^{a}[/itex] and this leads to the propagator for the coefficient field as [itex]\Delta^{ab}(k^2)=\tfrac{\delta^{ab}}{k^2-i\epsilon}[/itex], (just like usual for a massless scalar field) but not sure how to see the propagator of matrix field...
thanks for any help...