Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1011Generalization to many degrees of freedom is straightforward:∫{ ∫ [t′ N] }∑〈q ′ 1 ...q ′ N , t ′ |q 1 ...q N , t〉 = D[p]D[q] exp i p n ˙q n − H(p n , q n ) dτ ,with∫∫D[p]D[q] =N∏n=1dq n dp n.2πHere, q n (t) = q n and q n (t ′ ) = q n ′ for all n = 1, ..., N, and we are allowingfor the full Hamiltonian of the system to depend upon all the N momentaand coordinates collectively.tn=16.3.2 Application: Particles, Sources, Fields and Gauges6.3.2.1 Particles(i) Consider first〈q ′ , t ′ |Q(t 0 )|q, t〉∫ ∏= dqi (t i ) 〈q ′ , t ′ |q n , t n 〉 ... 〈q i0 , t i0 |Q(t 0 )|q i−1 , t i−1 〉 ... 〈q 1 , t 1 |q, t〉 ,where we choose one of the time interval ends to coincide with t 0 , i.e.,t i0 = t 0 . If we operate Q(t 0 ) to the left, then it is replaced by its eigenvalueq i0 = q(t 0 ). Aside from this one addition, everything else is evaluated justas before and we will obviously get∫{ ∫ }t′〈q ′ , t ′ |Q(t 0 )|q, t〉 = D[p]D[q] q(t 0 ) exp i [p ˙q − H(p, q)]dτ .(ii) Next, suppose we want a path–integral expression for〈q ′ , t ′ |Q(t 1 )Q(t 2 )|q, t〉 in the case where t 1 > t 2 . For this, we have to insertas intermediate states |q i1 , t i1 〉 〈q i1 , t i1 | with t i1 = t 1 and |q i2 , t i2 〉 〈q i2 , t i2 |with t i2 = t 2 and since we have ordered the times at which we do theinsertions we must have the first insertion to the left of the 2nd insertionwhen t 1 > t 2 . Once these insertions are done, we evaluate 〈q i1 , t i1 | Q(t 1 ) =〈q i1 , t i1 | q(t 1 ) and 〈q i2 , t i2 | Q(t 2 ) = 〈q i2 , t i2 | q(t 2 ) and then proceed as beforeand get∫{ ∫ }t′〈q ′ , t ′ |Q(t 1 )Q(t 2 )|q, t〉 = D[p]D[q] q(t 1 ) q(t 2 ) exp i [p ˙q − H(p, q)]dτ .tt