Advanced Quantum Mechanics, 2nd Edition by F.J. Dyson (lecture notes), Michael J. Moravcsik (editor)

By F.J. Dyson (lecture notes), Michael J. Moravcsik (editor)

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69 (1946) 195. This is worth reading. Essential is the fact that you cannot fix the time-dependence of the field (phase) with a fixed number of quanta (energy). We consider a more general problem. What is the mean-square fluctuation in the vacuum state of a field-quantity? We define 1 VT 1 H1 (V T ) = VT E1 (V T ) = E1 (x) dτ dt VT H1 (x) dτ dt averaged over some finite space-volume V and also over a time T . Let V (k) = H = ∇ × A, we have 2 {H1 (V T )} o 2 o V e−ik·r dτ .

This he does as follows: Let x be any space-time point inside Ω. Let O(x) be any field operator defined at x, for example φβ (x) or φβµ (x). Then O(x) is given a meaning by defining its matrix element between the states |φ′2α , σ2 and |φ′1α , σ1 , where σ2 and σ1 are any two surfaces to the past and future of x. This matrix element is φ′1α , σ1 O(x) φ′2α , σ2 = N OH (x) exp H i IH (Ω) (175) The number OH is just the value which the expression O takes when the φα are given the values which they have in the history H.

Hence the general rule: the differential cross-section does not have a factor 21 , the total cross section does because each final state may only be counted once in integrating over the angles. Relativistic Treatment of Several Particles The Møller treatment of the interaction of two electrons succeeds because the field of particle 1 is calculated for all time without taking any account of the effect of particle 2 on particle 1. How can one do a better calculation taking such reactions into account?

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