New PDF release: On the calculation of time correlation functions

By Berne B.J., Harp G.D.

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18) To find the equation satisfied by K, consider the time t = tf + , when the position is xn+2 . 20) where xn+2 = xn+1 − η. 28) This has justified our choice of normalization N( ) in Eq. 17). 29) as required. 30) In a scattering problem, the initial wavefunction is a plane wave. In a bound-state problem, the initial wavefunction is unknown, so one has an eigenvalue problem (solving ψf = ψi ), which is very difficult, in general. In a problem where the potential term is small, it is most convenient to calculate K in perturbation theory.

191) Now the little group of this vector is clearly O(n − 1). 194) which do not leave φ invariant. We then reparametrize the n-component field φk as  0  ..  n−1 ki   . 195) v  0  i=1 v+η in terms of the (n − 1) fields ξi , and the field η. The action of ki on the vector vi = vδin is given by (ki v)j = v(Lin )j l δin = −ivδij Thus to lowest order, one has   ξ1  ξ2     .  φ =  .. 200) and the (n − 1) fields ξi are massless Goldstone bosons. Note that the number of Nambu–Goldstone bosons is equal to the difference in the number of generators of the original symmetry O(n) and the final symmetry O(n − 1): 1 1 n(n − 1) − (n − 1)(n − 2) = (n − 1) 2 2 This is an example of a general theorem that we now prove.

We then reparametrize the n-component field φk as  0  ..  n−1 ki   . 195) v  0  i=1 v+η in terms of the (n − 1) fields ξi , and the field η. The action of ki on the vector vi = vδin is given by (ki v)j = v(Lin )j l δin = −ivδij Thus to lowest order, one has   ξ1  ξ2     .  φ =  .. 200) and the (n − 1) fields ξi are massless Goldstone bosons. Note that the number of Nambu–Goldstone bosons is equal to the difference in the number of generators of the original symmetry O(n) and the final symmetry O(n − 1): 1 1 n(n − 1) − (n − 1)(n − 2) = (n − 1) 2 2 This is an example of a general theorem that we now prove.

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