Question

some useful examples

the 1st one is the question where x is a vector while the second are examples.

(d) Now consider the N-dimensional vector x an the integral ((22%)lejbTA-lb. (23.46) det A By differentiating with respect to components of the vector b, and then setting b 0, show that (r,a) (23.47) (e) Using these results, argue that +(A)u(A (23.48) j k. (f) Write down an expression for the general case Ti.z) This is the basis of Wick's theorem in the path in- tegral approach. 23 Example 23.3 the integral we make an orthogonal transformation A OT DO, where evaluat ur matrices O in such a way as to make D a diagonal matrix. Our integral is then make the transformation Ox y, and since the uatrix Q is orthogonal, the Next ian of this transformation is 1 so Jd (23.21) -dN y, and we get (23.22) D(), which allows us to separate out our . D is diagonal, we have yT Dy multidimensional integral. d ye dyN e 2n (2T) N (23.23) where we've used Dii-det D-det A. 2318 Step 4: We're finally in a position to generalize our important result, eqn 23.19, to the multidimensional case. We want to evaluate the inte- (23.24) where b is an N-dimensional vector. Example 23.4 By analogy with Example 23.2 we consider P(x)-xAx+bTx. We can easily ind its minimun (remembering that Ay is symmetric and using Oa 6y) (23.25) s gives us a minimum of PiblA b at x eqn 23.17, we can write A-1b. So again, just as in x)min -1(x-A-1b)TA(x-A-ib), (23.26) = This all looks familiar, of course. Again, we can perform a transformation and ebA-er our new variable y (x - A-b). We'll obtain a factor of the form emerging at the front The previous example leads us to 23.27 det A Again we note that we've integrated out x, and while A touches twox' on the left-hand side, A-1 takes b to bT.

Answer 1

mthe LH.S 2 actor rings down a a bi R-H-S 으 gives detA」 乙 terms enuolving b4 abi deiA so setting 6=0 a, e obtain .as claim e) This next result roma com pa rison oith arts (a) and (b) - the structure is the same Note that this is just hat we have n bast ( q ll Purs o indice ths tomes fom Endu cion using(e)