Ivancevic_Applied-Diff-Geom

998 Applied Differential Geometry: A Modern Introductionputer simulation we can use 10 7 ≤ n ≤ 10 8 , approaching the number of neuronsin the brain. Such equations are usually solved using Markov–ChainMonte–Carlo methods on parallel (cluster) computers (see, e.g., [Wehnerand Wolfer (1983a); Wehner and Wolfer (1983b)]).6.2 Path Integral History6.2.1 Extract from Feynman’s Nobel LectureIn his Nobel Lecture, December 11, 1965, Richard (Dick) Feynman saidthat he and his PhD supervisor, John Wheeler, had found the action A =A[x; t i , t j ], directly involving the motions of the charges only, 9∫A[x; t i , t j ] = m i (ẋ i µẋ i µ) 1 2 dti + 1 ∫ ∫2 e ie j δ(Iij) 2 ẋ i µ(t i )ẋ j µ(t j ) dt i dt jwith (i ≠ j) (6.8)I 2 ij = [ x i µ(t i ) − x j µ(t j ) ] [ x i µ(t i ) − x j µ(t j ) ] ,where x i µ = x i µ(t i ) is the four–vector position of the ith particle as a functionof the proper time t i , while ẋ i µ(t i ) = dx i µ(t i )/dt i is the velocity four–vector.The first term in the action A[x; t i , t j ] (6.8) is the integral of the propertime t i , the ordinary action of relativistic mechanics of free particles ofmass m i (summation over µ). The second term in the action A[x; t i , t j ](6.8) represents the electrical interaction of the charges. It is summed overeach pair of charges (the factor 1 2is to count each pair once, the term i = jis omitted to avoid self–action). The interaction is a double integral over adelta function of the square of space–time interval I 2 between two points onthe paths. Thus, interaction occurs only when this interval vanishes, thatis, along light cones (see [Wheeler and Feynman (1949)]).Feynman comments here: “The fact that the interaction is exactly one–half advanced and half–retarded meant that we could write such a principleof least action, whereas interaction via retarded waves alone cannot bewritten in such a way. So, all of classical electrodynamics was contained inthis very simple form.”“...The problem is only to make a quantum theory, which has as its classicalanalog, this expression (6.8). Now, there is no unique way to make aquantum theory from classical mechanics, although all the textbooks make9 Wheeler–Feynman Idea [Wheeler and Feynman (1949)] “The energy tensor can beregarded only as a provisional means of representing matter. In reality, matter consistsof electrically charged particles.”