Historical research guide · 0830 · 1975
Radiation reaction and vacuum fluctuations in spontaneous emission
Peter W. Milonni · Wallace Arden Smith
In one minute
An excited atom sitting alone in the dark still gives up its light. Why? Peter Milonni and Wallace Arden Smith showed in 1975 that physics has two answers and that both of them are right. One says the atom is shaken into emitting by the vacuum’s zero-point field, the residual jitter of the electromagnetic field that survives even at absolute zero. The other says the atom is pushed by its own radiated field coming back on itself, which is called radiation reaction. Milonni and Smith redid the calculation in the Heisenberg picture, where atom and field both carry operators, and found that which answer comes out depends on a bookkeeping choice — the order you write the atom’s operators and the field’s operators in before averaging. One ordering hands the whole line shift to radiation reaction, another hands it to the vacuum, and the rest split it between them. The measured number never moves. They then showed a purely classical zero-point field reproduces the same shift and width.
Connection
Chapter 2 argues that the vacuum is a real, structured medium rather than an absence, and this is the paper that fixed the terms of that argument inside mainstream quantum electrodynamics: the vacuum field is not an optional picture you can talk yourself out of, it is one half of a ledger whose two halves are formally interchangeable. Chapter 6 gains the bridge Milonni and Smith build at the end, where a classical random zero-point field, averaged over its phases, delivers the quantum answer exactly.
From the source
Radiation reaction and vacuum fluctuations provide complementary and equally valid conceptual bases for interpreting the radiative corrections — the shift and the width of an emitted spectral line. Neither picture is the true one and the other an approximation to it.Abstract, and the outline of Section V given at the end of Section I
Which picture you get is fixed by the order in which the atom’s and the field’s operators are written before averaging. With normal ordering the entire contribution to the line shift comes from radiation reaction; with antinormal ordering twice the shift comes from vacuum-field fluctuations and minus one times the shift from radiation reaction; with a symmetric ordering, half normal and half antinormal, the entire shift stems from vacuum-field fluctuations; for other orderings each contributes a portion complementary to the other. Milonni and Smith call the two readings merely two sides of the same quantum-mechanical coin.Section V, the paragraph following Equation 48
For an atom starting in a pure excited state, an antinormally ordered calculation gives the correct Einstein A coefficient for the decay entirely from the vacuum-field fluctuation term. That is the strongest statement of the vacuum’s role anywhere in the paper, and it is exact within the second-order treatment used.Section VI, Equation 57 and the sentence following it
The vacuum picture on its own is nevertheless incomplete. For any initial state other than the pure excited one the radiation-reaction terms contribute essentially, and without them the calculation would predict spontaneous absorption — a ground-state atom soaking up energy from the vacuum and falling further. With both terms kept, a ground-state atom stays in the ground state as it should. Milonni and Smith state that this is true regardless of the ordering scheme used.Section VI, Equation 58 and the two paragraphs following it
A classical zero-point field does the same work. Substituting Timothy Boyer’s random-phase classical zero-point radiation field into the semiclassical Bloch equations for a two-level atom, and averaging over the random phases instead of taking vacuum expectation values, yields an equation that is the exact analogue of the quantum-mechanical result — the same line shift and the same linewidth. The classical zero-point fluctuations supply exactly the effects of the quantum vacuum field that ordinary semiclassical calculations leave out.Section VII, Equations 59 to 63
What to watch: the same accounting extends beyond spontaneous emission. Milonni and Smith note that the quantum operator equations for the van der Waals force between neutral atoms are formally the same as Boyer’s classical equations with the fluctuating zero-point field standing in for the vacuum field, which puts the Casimir and van der Waals forces in the same ledger. The open question they leave open is where the split between vacuum term and source term can be pinned down by measurement rather than by choice of ordering — the question Dalibard, Dupont-Roc and Cohen-Tannoudji reopened in 1982 and modern separation experiments now attack directly.Section VII, the paragraphs on Boyer’s zero-point field
The way in
https://doi.org/10.1103/PhysRevA.11.814How to cite it
Peter W. Milonni, Wallace Arden Smith (1975) Radiation reaction and vacuum fluctuations in spontaneous emission doi:10.1103/PhysRevA.11.814