By Zbigniew Haba (auth.)

The Feynman crucial 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 via a normal over numerous interfering paths. The classical photo steered via the Feynman crucial might be illusory. via so much physicists the trail crucial is mostly handled as a handy formal mathematical device for a fast derivation of valuable approximations in quantum mechanics. effects acquired within the formalism of Feynman integrals obtain a mathematical justification by way of different (usually a lot tougher) equipment. In one of these case the rigour is completed on the fee of wasting the intuitive classical perception. the purpose of this ebook is to formulate a mathematical thought of the Feynman indispensable actually within the approach it was once expressed through Feynman, on the expense of complexifying the configuration house. In one of these case the Feynman fundamental should be expressed via a chance degree. The equations of quantum mechanics should be formulated as equations of random classical mechanics on a fancy configuration house. the potential for machine simulations exhibits an instantaneous benefit of one of these formula. A mathematical formula of the Feynman quintessential shouldn't be thought of completely as a tutorial query of mathematical rigour in theoretical physics.

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37) is (here Xo = x) €-2 t (Xk - Xk-l + €2VW(Xk_l)) 2 k=l n = €-2 2)Xk k=l Xk_l)2 + €2 n L (VW(Xk_l))2 k=l n + 2 L(Xk - Xk-I)VW(Xk-l). s. have a clear meaning in the discrete approximation to the Feynman integral. 23)). The last term needs more careful elaboration before we connect it with the Feynman integral solution of the Schrodinger equation. It contributes to the potential energy. We shall discuss it in Chapter 7. 2. The discrete form of the path integral can be used for a derivation of the Cameron-Martin-Girsanov formula which we discuss in Chapter 3 and 24 CHAPTER 2 Chapter 5.

The spectrum and the stability of the system can be investigated by means of the correlation functions. 2,t+ m - 2k . k=O They are growing in time (a positive Lyapunov exponent). 60) are exponentially decaying in time (for m =1= 0) showing that the linear model on the circle has a continuous spectrum [261] (for a study of the spectrum of discrete non-linear systems in classical and quantum mechanics see [41]). CHAPTER 3 Stochastic differential equations The operator J(~IT of Chapter 2 determines the evolution obtained from a solution of the discrete stochastic equation.

1. 37) is analytic. 37) because the Jacobian of the transformation x 4- Y is equal to (€O')nd. 32) we obtain the path integral in the form 25 Markov chains J dX¢1 (x)E [