Question

Show that Tr | П Ai-Tr | Cyclic Permutation

I need someone to answer this for me please its for Mathematical Methods for Physicist and it also relates to Linear Algebra. Thanks

Answer 1

Baker haisdorff formula Given T_r[product^n_i = 1 A_i^barbar] = T_r[cyclic permutation] If can be written has e^-T Ae^T = A + [A_1 T] + 1/2![A_1 T] T + ... Let us consider a real parameter t A(t) = e^-tT A e^-tT = sigma^infinity_n = 0 A_n t^n/n! Differential above equation A_1 = dA(t)/dt = e^-tT AT e^tT + (-T) e^-tT Ae^tT = e^-tT (AT - TA)e^tT = [T, A(t)] = [A(t), T] A_2 = d^2 A(t)/dt^2 = T^2 e^tT Ae^-tT - 2 Te^tT T Ae^-tT + e^tT T^2 Ae^-tT for t = 0 d^2 A(t)/dt^2 = T^2 A - 2 TAT + AT^2 = [(A, T), T] A(t) = e^tT A e^tT = sigma^infinity_n = 0 A_n(0)/n! t^n A(t) = A_0 + A_1 t^1 + A_2 t^2/2! + A_3 t^3/3! \r\n e^-tT A e^tT = A + [A_1 T]t + [A_1 T], T/2! t^2 + ... at t = 0 e^-T Ae^T = A + [A_1 T] + [A, T, T]/2!