Foldy-Wouthuysen transformation

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 '=Upsi (1)

where the unitary operator is the 4x4 matrix:

:U=e^{eta mathbf{alpha} cdot hat{p} heta} = cos heta + eta mathbf{alpha} cdot hat{p} sin heta =e^{mathbf{gamma} cdot hat{p} heta} = cos heta + mathbf{gamma} cdot hat{p} sin heta . (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 eta equiv gamma^{0} 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 U=e^{-eta mathbf{alpha} cdot hat{p} heta} = cos heta - eta mathbf{alpha} cdot hat{p} sin heta , 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 hat{H}_0 equiv alpha cdot p + eta m in bi-unitary fashion, in the form:

:hat{H}_0 o hat{H}'_0 equiv U hat{H}_0 U^{-1} = U (alpha cdot p + eta m) U^{-1} = (cos heta + eta mathbf{alpha} cdot hat{p} sin heta ) (alpha cdot p + eta m) (cos heta - eta mathbf{alpha} cdot hat{p} sin heta ) (3)

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

:hat{H}'_0 = (alpha cdot p + eta m) (cos heta - eta mathbf{alpha} cdot hat{p} sin heta )^{2} = (alpha cdot p + eta m) e^{-2eta mathbf{alpha} cdot hat{p} heta} = (alpha cdot p + eta m) (cos 2 heta - eta mathbf{alpha} cdot hat{p} sin 2 heta ) (4)

This factors out into:

:hat{H}'_0= alpha cdot p (cos 2 heta - frac{m} sin 2 heta) + eta (m cos 2 heta + |p| sin 2 heta)) (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}'_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 eta is taken in the Dirac-Pauli representation in which it is a diagonal matrix):

:hat{H}'_0= eta (m cos 2 heta + |p| sin 2 heta)) (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:

:hat{H}'_0= eta sqrt{m^2+|p|^2} (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 hat{H}'_0= eta E.

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' equiv U O U^{-1} = pm I (O) pm I = O. (10)

Contrasting the original Dirac-Pauli Hamiltonian Operator hat{H}_0 equiv alpha cdot p + eta m with the NW Hamiltonian (9), we do indeed find the |p| equiv 0 "at rest" correspondence:

:hat{H}_0 = hat{H}'_0= eta m (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}'equiv i [hat{H}'_0,x_i] . If we use the result at the very end of section 2 above, hat{H}'_0= eta p_0, then this can be written instead as:

:hat{v_i}'equiv i [hat{H}'_0,x_i] =i eta [p_0,x_i] . (15)

Using the above, we need simply to calculate [p_0,x_i] , then multiply by ieta.

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 ieta factor via (15), to arrive at:

:frac{hat{dx_i}'}{dt}=hat{v_i}'equiv i [hat{H}'_0,x_i] = eta frac{p_i}{p^0} = eta v_i . (18)

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

:i [hat{H}'_0,hat{v}_i'] =i [eta p_0,eta v_i] =0, (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:

:hat{v_i}'equiv i [hat{H}'_0,x_i] = eta frac{p_i}{p^0} = 0 . (22)

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

:hat{v_i}'=hat{v_i}, (23)

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


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