By Laurie M. Brown Richard Phillips Feynman
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Extra info for Feynman's Thesis: A New Approach to Quantum Theory
Exactly the same formulas result if it is that γ(t) depends on the function y(t), as any general functional. The action of the particles need not be the integral of a Lagrangian in the original form (23) either. e. if no two of the oscillators interact directly) the expressions for the action, with the oscillators eliminated are similar to (35), (37) and (38), except that there is a sum of interaction terms, one for each oscillator. Thus, if the frequency of the jth oscillator is ωj , its mass mj , the interaction with the particles given by γj and there were N oscillators, (38), for example, would read, A = ∞ −∞ N [Ly + Lz ]dt + × γj (t)γj (s)dsdt .
B(t2 , t1 )B(t1 , T ) . (23 ) The corresponding quantum equation, which follows from the composition law . . , is (qt |qT ) = ... (qt |qm )dqm (qm |qm−1 )dqm−1 . . × (q2 |q1 )dq1 (q1 |qT ) , (24 ) qk being written for qtk for brevity. At ﬁrst sight there does not seem to be any close correspondence between (23) and (24). We must, however, analyse the meaning of (23) rather more carefully. We must regard each factor B as a function of the q’s at the two ends of the time interval to which it refers.
We may obtain the fundamental relation of our quantum mechanics, analogous to √ equation (52), by integrating the formula for the 1 ∂( gF ) √ average of g ∂qk by parts, to obtain, χ| ∂( g(qk )F ) i ∂A |ψ = − χ|F · |ψ . ∂q ∂qk g(qk ) k 1 (69) As we have remarked above, this contains the analogue of the equations of motion as well as the quantum commutation rules. 6. Conservation of Energy. Constants of the Motion Because of the importance in ordinary quantum mechanics of operators which correspond to classical constants of motion, we shall mention brieﬂy the analogue of these operators in our generalized formulation.
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