[1][2][3][4] A detailed general discussion of the Foldy–Wouthuysen-type transformations in particle interpretation of relativistic wave equations is in Acharya and Sudarshan (1960).
[8] Before their work, there was some difficulty in understanding and gathering all the interaction terms of a given order, such as those for a Dirac particle immersed in an external field.
With their procedure the physical interpretation of the terms was clear, and it became possible to apply their work in a systematic way to a number of problems that had previously defied solution.
This is arranged by choosing: In the Dirac-Pauli representation where β is a diagonal matrix, 5 is then reduced to a diagonal matrix: By elementary trigonometry, 6 also implies that: so that using 8 in 7 and then simplifying now leads to: Prior to Foldy and Wouthuysen publishing their transformation, it was already known that 9 is the Hamiltonian in the Newton–Wigner (NW) representation (named after Theodore Duddell Newton and Eugene Wigner) of the Dirac equation.
If one considers an on-shell mass—fermion or otherwise—given by m2 = pσpσ, and employs a Minkowski metric tensor for which diag(η) = (+1, −1, −1, −1), it should be apparent that the expression is equivalent to the E ≡ p0 component of the energy-momentum vector pμ, so that 9 is alternatively specified rather simply by Ĥ′0 = βE.
Thus, we may write: where we have made use of the Heisenberg canonical commutation relationship [xi,pj] = −iηij to reduce terms.
Then, multiplying from the left by γ0 and rearranging terms, we arrive at: Because the canonical relationship the above provides the basis for computing an inherent, non-zero acceleration operator, which specifies the oscillatory motion known as zitterbewegung.
The powerful machinery of the Foldy–Wouthuysen transform originally developed for the Dirac equation has found applications in many situations such as acoustics, and optics.
It has found applications in very diverse areas such as atomic systems[13][14] synchrotron radiation[15] and derivation of the Bloch equation for polarized beams.
For the Dirac equation (which is first-order in time) this is done most conveniently using the Foldy–Wouthuysen transformation leading to an iterative diagonalization technique.
The suggestion to employ the Foldy–Wouthuysen Transformation technique in the case of the Helmholtz equation was mentioned in the literature as a remark.
[26] It was only in the recent works, that this idea was exploited to analyze the quasiparaxial approximations for specific beam optical system.