Ivancevic_Applied-Diff-Geom

990 Applied Differential Geometry: A Modern IntroductionNamely, in Dirac–Feynman quantum formalism, each possible routefrom the initial system state A to the final system state B is called ahistory. This history comprises any kind of a route (see Figure 6.1), rangingfrom continuous and smooth deterministic (mechanical–like) paths tocompletely discontinues and random Markov chains (see, e.g., [Gardiner(1985)]). Each history (labelled by index i) is quantitatively described bya complex number 1 z i called the ‘individual transition amplitude’. Its absolutesquare, |z i | 2 , is called the individual transition probability. Now,the total transition amplitude is the sum of all individual transition amplitudes,∑ i z i, called the sum–over–histories. The absolute square of thissum–over–histories, | ∑ i z i| 2 , is the total transition probability.In this way, the overall probability of the system’s transition from someinitial state A to some final state B is given not by adding up the probabilitiesfor each history–route, but by ‘head–to–tail’ adding up the sequenceof amplitudes making–up each route first (i.e., performing the sum–over–histories) – to get the total amplitude as a ‘resultant vector’, and thensquaring the total amplitude to get the overall transition probability.Fig. 6.1 Two ways of physical transition from an initial state A to the correspondingfinal state B. (a) Classical physics proposes a single deterministictrajectory, minimizing the total system’s energy. (b) Quantum physics proposesa family of Markov stochastic histories, namely all possible routes from A toB, both continuous–time and discrete–time Markov chains, each giving an equalcontribution to the total transition probability.1 Recall that a complex number z = x + iy, where i = √ −1 is the imaginary unit, xis the real part and y is the imaginary part, can be represented also in its polar form,z = r(cos θ + i sin θ), where the radius vector in the complex plane, r = |z| = p x 2 + y 2 ,is the modulus or amplitude, and angle θ is the phase; as well as in its exponential formz = re iθ . In this way, complex numbers actually represent 2D vectors with usual vector‘head–to–tail’ addition rule.