The Foldy–Wouthuysen transformation was historically significant and was formulated by Leslie Lawrance Foldy and Siegfried Adolf Wouthuysen in 1949 to understand the nonrelativistic limit of the Dirac equation, the equation for spin-½ particles. A detailed general discussion of the Foldy–Wouthuysen-type transformations in particle interpretation of relativistic wave equations is in Acharya and Sudarshan (1960). Its utility in high energy physics is now limited due to the primary applications being in the ultra-relativistic domain where the Dirac field is treated as a quantised field.
A canonical transform
The FW transformation is a unitary transformation of the orthonormal basis in which both the Hamiltonian and the state are represented. The eigenvalues do not change under such a unitary transformation, that is, the physics does not change under such a unitary basis transformation. Therefore, such a unitary transformation can always be applied: in particular a unitary basis transformation may be picked which will put the Hamiltonian in a more pleasant form, at the expense of a change in the state function, which then represents something else. See for example the Bogoliubov transformation, which is an orthogonal basis transform for the same purpose. The suggestion that the FW transform is applicable to the state or the Hamiltonian is thus not correct.
Foldy and Wouthuysen made use of a canonical transform that has now come to be known as the Foldy–Wouthuysen transformation. A brief account of the history of the transformation is to be found in the obituaries of Foldy and Wouthuysen and the biographical memoir of Foldy. 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. The Foldy–Wouthuysen transform was extended to the physically important cases of spin-0 and spin-1 particles, and even generalized to the case of arbitrary spins.
Description
The Foldy–Wouthuysen (FW) transformation is a unitary transformation on a fermion wave function of the form:
where the unitary operator is the 4 × 4 matrix:
Above,
is the unit vector oriented in the direction of the fermion momentum. The above are related to the Dirac matrices by and , with . A straightforward series expansion applying the commutativity properties of the Dirac matrices demonstrates that above is true. The inverse
so it is clear that , where is a 4 × 4 identity matrix.
Transforming the Dirac Hamiltonian for a free fermion
This transformation is of particular interest when applied to the free-fermion Dirac Hamiltonian operator
in biunitary fashion, in the form:
Using the commutativity properties of the Dirac matrices, this can be massaged over into the double-angle expression:
This factors out into:
Choosing a particular representation: Newton–Wigner
Clearly, the FW transformation is a continuous transformation, that is, one may employ any value for which one chooses. Now comes the distinct question of choosing a particular value for , which amounts to choosing a particular transformed representation.
One particularly important representation, is that in which the transformed Hamiltonian operator is diagonalized. Clearly, a completely diagonalized representation can be obtained by choosing such that the term in is made to vanish. Such a representation is specified by defining:
so that is reduced to the diagonalized (this presupposes that is taken in the Dirac–Pauli representation (after Paul Dirac and Wolfgang Pauli) in which it is a diagonal matrix):
By elementary trigonometry, also implies that:
so that using in now leads following reduction to:
Prior to Foldy and Wouthuysen publishing their transformation, it was already known that is the Hamiltonian in the Newton–Wigner (NW) representation (named after Theodore Duddell Newton and Eugene Wigner) of the Dirac equation. What therefore tells us, is that by applying a FW transformation to the Dirac–Pauli representation of Dirac's equation, and then selecting the continuous transformation parameter so as to diagonalize the Hamiltonian, one arrives at the NW representation of Dirac's equation, because NW itself already contains the Hamiltonian specified in (). See this link.
If one considers an on-shell mass—fermion or otherwise—given by , and employs a Minkowski metric tensor for which , it should be apparent that the expression