By Freeman J Dyson; David Derbes

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By this one formula the whole theory is quantized and the answer to any physical problem in principle given. The method applies not only to field theory but to ordinary non-relativistic quantum theory too. We do not try to derive or justify the Feynman formula here. We just show that it gives the same results as the usual quantum mechanics.

A state of motion is specified by specifying a space-time surface σ and a set of numerical values φ α for the eigenvalue which the operators φ α on σ 52 Advanced Quantum Mechanics have in this state. The state is denoted by the Dirac ket vector |φ α , σ . This is a special kind of state in which the φ α on σ have eigenvalues: the general state is a linear combination of |φ α , σ with various values of φ α . The physically observable quantities are expressions such as the matrix element φ1α , σ1 φβ (x) φ2α , σ2 (172) of the field operator φβ (x) between the two states specified by φ 1α on σ1 and by φ2α on σ2 .

So solutions (92) exist for n = 0, k = +1, +2, +3, . . (94) The principal quantum number N is N = n + |k| Expanding in powers of α E = mc2 1 − α4 1 α2 + 2 N2 N3 NR levels 1 3 − 8N 2|k| (95) fine structure There is exact degeneracy between the two states of a given |k|. 12 1/2 1/2 S3/2 j= + 1 → k = −( + 1) 2 j= − 1 →k=+ 2  1  , k=1   2 → degenerate 1  is j = , k = −1   2 1 is j = , k = −2 2 is j= Behaviour of an Electron in a Non-Relativistic Approximation Multiplying the Dirac equation (64) by γµ γν µ ν ∂ e + i Aµ ∂xµ c ν γν ∂ ∂xν e ∂ + i Aν ∂xν c mc we have ψ=0 (96) + i ec Aν − ψ− m2 c2 2 24 Advanced Quantum Mechanics Using γµ2 = 1, γµ γν + γν γµ = 0 this gives µ 2 ie ∂ + Aµ ∂xµ c ψ− m2 c2 2 ψ+ ie 2 c σµν Fµν ψ = 0 µ Here σµν = 1 2 (γµ γν − γν γµ ) Fµν = (97) ν ∂Aµ ∂Aν − ∂xµ ∂xν Thus F12 = H3 magnetic field component F14 = i i ∂A1 ∂Φ + = −iE1 ∂x1 c ∂t σ12 = iσ3 electric field spin component σ14 = iα1 velocity component Thus (97) becomes µ 2 ∂ ie + Aµ ∂xµ c ψ− m2 c2 2 ψ− e {σ · H − iα · E} ψ = 0 c This is still exact.

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