- #1
LayMuon
- 149
- 1
I am reading my textbook of QFT (Maggiore, Modern Introduction in QFT), and there is this statement:
"If [itex] T^a_R [/itex] is a representation of the algebra and V a unitary matrix of the same dimension as [itex] T^a_R [/itex] , then [itex] V T^a_R V^\dagger [/itex] is still a solution o the Lie algebra and therefore provides an equivalent representation. We can fix V requiring that it diagonalizes the matrix [itex] D^{ab}(R) ≡ Tr (T^a_R T^b_R) [/itex], so that [itex] Tr (T^a_R T^b_R) = C(R) \delta^{ab} [/itex]."
I can't understand the second sentence, matrix D has different dimensions than V, how can it be used to diagonalize? Putting V within the trace doesn't make any sense, it would give unit matrix.
Thanks.
"If [itex] T^a_R [/itex] is a representation of the algebra and V a unitary matrix of the same dimension as [itex] T^a_R [/itex] , then [itex] V T^a_R V^\dagger [/itex] is still a solution o the Lie algebra and therefore provides an equivalent representation. We can fix V requiring that it diagonalizes the matrix [itex] D^{ab}(R) ≡ Tr (T^a_R T^b_R) [/itex], so that [itex] Tr (T^a_R T^b_R) = C(R) \delta^{ab} [/itex]."
I can't understand the second sentence, matrix D has different dimensions than V, how can it be used to diagonalize? Putting V within the trace doesn't make any sense, it would give unit matrix.
Thanks.