Foldy-Wouthuysen transformation

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The Foldy-Wouthuysen (FW) transformation is a unitary transformation on a fermion wave function of the form:

:$psi\; o\; psi\; \text{'}=Upsi$ (1)

where the unitary operator is the 4x4 matrix:

:. (2)

Above, $hat\{p^i\}\; equiv\; p^i/|p|$ is the unit vector oriented in the direction of the fermion momentum. The above are related to the Dirac matrices by and $alpha^\{i\}\; equiv\; gamma^\{0\}\; gamma^\{i\}$, with i=1,2,3. A straightforward series expansion applying the commutativity properties of the Dirac matrices demonstrates that (2) above is true. The inverse , so it is clear that $U^\{-1\}\; U=I$, where $I$ is a 4x4 identity matrix.

1. Foldy-Wouthuysen Transformation of the Dirac Hamiltonian for a Free Fermion

This transformation is of particular interest when applied to the free-fermion Dirac Hamiltonian operator in bi-unitary fashion, in the form:

: (3)

Using the commutativity properties of the Dirac matrices, this can be massaged over into the double-angle expression:

: (4)

This factors out into:

: (5)

2. Choosing a Particular Representation: Newton-Wigner

Clearly, the FW transformation is a "continuous" transformation, that is, one may employ any value for $heta$ which one chooses. Now comes the distinct question of choosing a particular value for $heta$, which amounts to choosing a particular transformed representation.

One particularly important representation, is that in which the transformed Hamiltonian operator $hat\{H\}\text{'}\_0$ is diagonalized. Clearly, a completely diagonalized representation can be obtained by choosing $heta$ such that the $alpha\; cdot\; p$ term in (5) is made to vanish. Such a representation is specified by defining:

:$an\; 2\; heta\; equiv\; |p|/m$ (6)

so that (5) is reduced to the diagonalized (this presupposes that is taken in the Dirac-Pauli representation in which it is a diagonal matrix):

: (7)

By elementary trigonometry, (6) also implies that:

:$sin\; 2\; heta\; =\; |p|/\; sqrt\{m^2+|p|^2\}$ and $cos\; 2\; heta\; =\; m/\; sqrt\{m^2+|p|^2\}$ (8)

so that using (8) in (7) now leads following reduction to:

: (9)

This calculation can be examined in further detail in the following [http://www.physics.ucdavis.edu/~cheng/230A/RQM7.pdf link] .

Prior to Foldy and Wouthuysen publishing their transformation, it was already known that (9) is the Hamiltonian in the Newton-Wigner (NW) representation of the Dirac equation. What (9) 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 paramater $heta$ 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 (9). See this [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.27.3209&rep=rep1&type=pdf link]

If one considers an "on shell" mass -- fermion or otherwise -- given by $m^2=p^sigma\; p\_sigma$, and employs a Minkowski metric tensor for which $diag(eta\_\{mu\; u\}=(+1,-1,-1,-1)$, it should be apparent that the expression $p^0=sqrt\{m^2+|p|^2\}$ is equivalent to the $E\; equiv\; p^0$ component of the energy-momentum vector $p^mu$, so that (9) is alternatively specified rather simply by .

3. Correspondence Between the Dirac-Pauli and Newton-Wigner Representations, for an "At Rest" Fermion

Now let us consider a fermion "at rest," which we may define in this context as a fermion for which $|p|\; equiv\; 0$. From (6) or (8), this means that $cos\; 2\; heta=1$, so that $heta=0,pmpi,pm\; 2pi...$, and, from (2), that the unitary operator $U=pm\; I$. Therefore, any operator $O$ in the Dirac-Pauli representation upon which we perform a bi-unitary transformation, will be given, for an "at rest" fermion, by:

:$O\; o\; O\text{'}\; equiv\; U\; O\; U^\{-1\}\; =\; pm\; I\; (O)\; pm\; I\; =\; O$. (10)

Contrasting the original Dirac-Pauli Hamiltonian Operator with the NW Hamiltonian (9), we do indeed find the $|p|\; equiv\; 0$ "at rest" correspondence:

: (11)

4. The Velocity Operator in the Dirac-Pauli Representation

Now, let us consider the velocity operator. To obtain this operator, we must commute the Hamiltonian operator $hat\{H\}\_0$ with the canonical position operators $x\_i$, i.e., we must calculate $hat\{v\_i\}equiv\; i\; [hat\{H\}\_0,x\_i]$. One good way to approach this calculation, is to start by writing the scalar rest mass $m$ as $m=gamma^0hat\{H\}\_0+gamma^jp\_j$, and then to mandate that the scalar rest mass commute with the $x\_i$. Thus, we may write:

:$0=\; [m,x\_i]\; =\; [(gamma^0hat\{H\}\_0+gamma^jp\_j),x\_i]\; =\; [gamma^0hat\{H\}\_0,x\_i]\; +igamma\_i$ (12)

where we have made use of the Heisenberg canonical commutation relationship $[x\_i,p\_j]\; =-ieta\_\{ij\}$ to reduce terms. Then, multiplying from the left by $gamma^0$ and rearranging terms, we arrive at:

:$frac\{hat\{dx\_i\{dt\}=hat\{v\_i\}equiv\; i\; [hat\{H\}\_0,x\_i]\; =alpha\_i$ (13)

Because the canonical relationship $i\; [hat\{H\}\_0,hat\{v\}\_i]\; e\; 0$, the above provides the basis for computing an inherent, non-zero acceleration operator, which specifies the oscillatory motion known as [http://en.wikipedia.org/wiki/Zitterbewegung Zitterbewegung] .

deleted (14)

5. The Velocity Operator in the Newton-Wigner Representation

In the Newton-Wigner representation, we now wish to calculate $hat\{v\_i\}\text{'}equiv\; i\; [hat\{H\}\text{'}\_0,x\_i]$. If we use the result at the very end of section 2 above, , then this can be written instead as:

:. (15)

Using the above, we need simply to calculate $[p\_0,x\_i]$, then multiply by .

The canonical calculation proceeds similarly to the calculation in section 4 above, but because of the square root expression in $p^0=sqrt\{m^2+|p|^2\}$, one additional step is required.

First, to accommodate the square root, we will wish to require that the scalar "square" mass $m^2$ commute with the canonical coordinates $x\_i$, which we write as:

:$0\; equiv\; [m^2,x\_i]\; =\; [(p^0p\_0+p^jp\_j),x\_i]\; =\; [p^0p\_0,x\_i]\; +2ip\_i$ (16)

where we again use the Heisenberg canonical relationship $[x\_i,p\_j]\; =-ieta\_\{ij\}$. Then, we need an expression for $[p\_0,x\_i]$ which will satisfy (16). It is straightforward to verify that:

:$i\; [p\_0,x\_i]\; =frac\{p\_i\}\{p^0\}=v\_i$ (17)

will satisfy (16) when again employing $[x\_i,p\_j]\; =-ieta\_\{ij\}$. Now, we simply return the factor via (15), to arrive at:

:. (18)

This is understood to be the velocity operator in the Newton-Wigner representation. Because:

:, (19)

it is commonly thought that the [http://en.wikipedia.org/wiki/Zitterbewegung Zitterbewegung] motion arising out of (13), vanishes when a fermion is transformed into the Newton-Wigner representation.

deleted (20)

6. The Velocity Operators for an "At Rest" Fermion

Now, let us compare equations (13) and (18) for a fermion "at rest," defined earlier in section 3 as a fermion for which $|p|\; equiv\; 0$. Here, (13) remains:

:$hat\{v\_i\}equiv\; i\; [hat\{H\}\_0,x\_i]\; =alpha\_i$ (21)

while (18) becomes:

:. (22)

In equation (10) we found that for an "at rest" fermion, $O\text{'}\; =\; O$ for any operator. One would expect this to include:

:$hat\{v\_i\}\text{'}=hat\{v\_i\}$, (23)

however, equations (21) and (22) for a $|p|\; equiv\; 0$ fermion appear to contradict (23).