By Mazurov V. D.
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Extra info for 2-Signalizers of finite groups
We are therefore led to expand ψ(x + η, t) in a Taylor series around η = 0, obtaining, after rearranging the integral, ψ(x, t + ε) = e− im V (x) e A + ∞ iε i m 2 η 2ε ψ(x, t) + η ∂ψ(x, t) ∂x η 2 ∂ 2 ψ(x, t) + . . dη . 2 ∂x2 2 Now, −∞ e ·2ε η · dη = 2πmεi (see Pierces integral tables 487), and by diﬀerentiating both sides with respect to m, one may show ∞ −∞ η2 · e im 2 η ·2ε dη = 2π εi εi . m m 30 Feynman’s Thesis — A New Approach to Quantum Theory The integral with η in the integrand is zero since it is the integral of an odd function.
49) Let us try to express this in terms of the wave function at some more future time, t = tm+1 , by equation (48), and in terms of the wave function ψ(q−m , t−m ) at some earlier time t = t−m by an equation analogous to (47) (we shall let negative indices stand for times earlier than t0 ). We obtain × exp ψ ∗ (qm+1 , tm+1 ) · f (q0 ) = i m L i=−m qi+1 − qi , qi+1 · (ti+1 − ti ) ti+1 − ti · f (q0 ) √ √ √ √ g dqm+1 . . g dq0 g dq−1 . . gdq−m . ·ψ(q−m , t−m ) · A(tm+1 − tm ) . . A(t0 − t−1 ) .
For example, in the theory of action at a distance, the kinetic energy of the particles is not conserved. To ﬁnd a conserved quantity one must add a term corresponding to the “energy in the ﬁeld”. The ﬁeld, however, is a functional of the motion of the particles, so that it is possible to express this “ﬁeld energy” in terms of the motion of the particles. For our simple example (8), account of the equations of motion (9), the quantity, E(t) = 2 m(x(t)) ˙ + V (x(t)) − k 2 2 t+T0 x ¨(σ − T0 )x(σ)dσ ˙ t ˙ x(t ˙ + T0 ) , + k2 x(t) (10) has, indeed, a zero derivative with respect to time.
2-Signalizers of finite groups by Mazurov V. D.