By Zbigniew Haba (auth.)

The Feynman vital is taken into account as an intuitive illustration of quantum mechanics displaying the complicated quantum phenomena in a language understandable at a classical point. It means that the quantum transition amplitude arises from classical mechanics through a regular over quite a few interfering paths. The classical photograph instructed by means of the Feynman vital might be illusory. by way of such a lot physicists the trail essential is mostly handled as a handy formal mathematical instrument for a fast derivation of priceless approximations in quantum mechanics. effects received within the formalism of Feynman integrals obtain a mathematical justification through different (usually a lot more durable) equipment. In this type of case the rigour is accomplished on the fee of wasting the intuitive classical perception. the purpose of this e-book is to formulate a mathematical conception of the Feynman indispensable actually within the method it used to be expressed via Feynman, on the rate of complexifying the configuration house. In any such case the Feynman critical will be expressed through a chance degree. The equations of quantum mechanics will be formulated as equations of random classical mechanics on a fancy configuration area. the potential of laptop simulations exhibits an instantaneous good thing about this sort of formula. A mathematical formula of the Feynman indispensable shouldn't be thought of completely as a tutorial query of mathematical rigour in theoretical physics.

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6) Then the differential of f(q(t)) is df = ojfdqj + ! dt. 2) (called the Stratonovitch integral). We rewrite it as follows J g(b(s)) (S,n) = 0 ~ (g (b _ g (b J (l,n) db(s) - g(b(s))db(s) V: 1) [b (H~ + k: 1) ) -bV: 1)]) (l : 1) )) (b (t~) _b (t k : We expand g (b (tk;;:l) + 8b) 1)) . 2. Then in the limit Stochastic differential equations 35 n -+ 00 we find the following relation between the Stratonovitch and Ito integrals J t J t g(b(s)) 0 db(s) - g(b(s))db(s) =! J t div g(b(s))ds. 7) the following basic property of the Stratonovitch differential can be obtained df = Vfodb.

1. Assume that 4> E L2(m,d) is analytic in the region z c cd = (x + (1 + i)y : x E m,d, Y E m,d) . 3) with the initial condition 4> has a unique square integrable solution which is also analytic in the region Z. For t ~ 0 it can be represented by the Brownian motion itHo) 4>) (x) = E [4> (x + Aubt)] . 5) Proof: 4> can be represented in the form of the Fourier transform 4>(x) = J dp exp(ipx)~(p) . s. 5). We obtain 4>t(x)= J ( ilit dpexp i px- 2m P 2)-4>(p). From this representation of the solution there follow the statements of the lemma.

8). 8) implies that q(n) depend only on r(k) with k = 0,1, .. , n - 1. 12) Eo"Y) . Let us define the operator (K€,a¢» (x) = EO"Y) . 2) is well explored in mathematics [261] [240] [54] as well as in physics [119][230][282][202][201]. It does not describe any unitary evolution (we shall discuss a degenerate case of the real perturbations in Chapter 14). 5)) . The Markov chains discussed in this section have an application to dissipative quantum mechanics (Chapter 16). We can treat the dissipative quantum mechanics as a random perturbation of the discrete classical dynamical system discussed in [50] + E2p(n) , q(n + 1) q(n) p(n + 1) p(n) - E2VV (q(n) + E2p(n)) + E(1r(n) .