By Zbigniew Haba (auth.)

The Feynman necessary 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 by way of a regular over quite a few interfering paths. The classical photograph instructed by means of the Feynman fundamental should be illusory. through such a lot physicists the trail crucial is mostly handled as a handy formal mathematical software for a fast derivation of precious approximations in quantum mechanics. effects got within the formalism of Feynman integrals obtain a mathematical justification through different (usually a lot tougher) tools. In one of these case the rigour is completed on the price of wasting the intuitive classical perception. the purpose of this publication is to formulate a mathematical idea of the Feynman vital actually within the means it was once expressed by way of Feynman, on the expense of complexifying the configuration house. In any such case the Feynman vital could be expressed by means of a likelihood degree. The equations of quantum mechanics might be formulated as equations of random classical mechanics on a posh configuration house. the possibility of desktop simulations exhibits an instantaneous benefit of this sort of formula. A mathematical formula of the Feynman imperative shouldn't be thought of completely as an instructional query of mathematical rigour in theoretical physics.

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**Example text**

21). Hence in the limit of small f. 21) coincide (this is a slightly stronger statement than that their generators coincide). In order to obtain a relation between the Trotter formula and stochastic equations we can directly represent k on functions of the form 'l/J = exp(i/IiW) 4>. 8) (neglecting the o(lx - Y12) terms). 15) with the initial condition q(O) = x. 9) and its exact form in the continuum limit in Chapter 7. CHAPTER 5 The Feynman integral The Feynman integral [126] gives an intuitive picture of quantum mechanics as a classical mechanics with some quantum corrections describing interference of various classically inadmissible amplitudes.

Let n-l b(n) = L: r(k) k=l and Q(n) = q(n) - AO'fb(n) . 42) then reads (Q(n + 1) - Q(n)) exp (-Q(n)) = _,,(€2 exp (AcHb(n)) . s. 44) in the form (Q(n + 1) - Q(n)) exp (-Q(n)) exp (-Q(n)) - exp (-Q(n + 1)) - exp (-Q(n)) X (1- exp (Q(n) - Q(n + 1)) - Q(n + 1) + Q(n)) . s. 45) is of the order €4. We can treat this term in a systematic perturbation expansion. So we solve first the equation exp (-Q(n + 1)) - exp (-Q(n)) = €2"(exp (AO"€b(n)) . 44). 46) we can compute Q(n + 1)(1) - Q(n)(l) in order to see that this difference is of the order €2.

The explosion takes place at a random explosion time r(b). If there is an explosion then an analytic computation of correlation functions of exploding processes is rather complicated. In such a case we shall apply another method (we construct so called weak solution). Namely, we determine a measure corresponding to the process (related to the Feynman integral of subsequent sections) rather than studying solutions of the stochastic equations directly. In Chapter 20 we shall briefly discuss numerical methods for stochastic systems.