PACS: 03.65.Fd; 11.80.Cr

1. Introduction

The use of the helicity, *i.e*. the projection of the spin along the direction of the momentum, to describe the polarization of Dirac particles in collision problems became common as a result of the pioneering work by Jacob and Wick ^{1}. Obviously, the reason is that the energy eigenstates of the Hamiltonian are also helicity eigenstates. In particular the plane wave solutions of the free Dirac equation which are used to represent the incident and outgoing particles in the first order *S*-matrix are simultaneous eigenstates of the helicity operator

Among the interactions that conserve helicity, probably, the interaction with a static magnetic field is the most popular. As is well-known, the helicity of a Dirac particle in an electromagnetic potential is conserved given that there is no electric field acting on the particle ^{2}. Indeed, the Heisenberg equation of motion for the helicity operator

Here, *H* is the Hamiltonian of a Dirac particle in an electromagnetic field. Thus, the helicity of a particle in a static magnetic field is conserved. In physical terms, conservation of helicity is described as the invariance of the component of the spin of the particle along its momentum. In the perturbative expansion of a helicity-conserving theory, helicity is conserved at each order of the perturbation series. For example, in the first order S-matrix element of the elastic scattering of a particle in some helicity-conserving vector potential, the conservation of helicity manifests itself through the fact that if the incident state is in an eigen state of the helicity operator ^{2} (**p**
_{
i
} and **p**
_{
f
} are, respectively, the incident and outgoing momenta). This work focuses on the conservation of helicity for the scattering of a Dirac particle in a static magnetic field *at this order*. It is shown that, by formulating the whole spin dynamics in terms of the three operators **k = p**
_{f}
**+ p**
_{i}, the momentum transfer **q = p**
_{f}
**- p**
_{i}, and **l = k × q**, one gets a more symmetric and intuitive picture of the dynamics that lead to the conservation of the helicity in the transition. It is also demonstrated that one can, within this framework, express the helicity sector of the matrix element in a form that is independent of the specific form of the vector potential.

2. The Spin Interaction

Consider a Dirac particle in a given magnetic field whose vector potential is the static vector function **A** (**x**) and such that there is no scalar potential. The first order S-matrix element for the elastic scattering of a particle in this potential is:

Carrying out the time integral, we get this as

which can be casted in the form

Where **A** (**q**) is the Fourier transform of the vector potential with respect to the momentum transfer vector **q = p**
_{
f
}
**- p**
_{
i
} and *N* is a normalization constant. Recalling that

and

where we have introduced the unit vector

where

We now note that the two unit vectors;

along the total momentum and

along the momentum transfer are orthonormal; see Fig. 1. This is, of course, true for the scattering in any potential field.

Introducing a third unit vector

Thus, the consequences:

and,

The above relations says that the newly introduced *SU*(2) algebra, and are generators of rotation in the spin space. We will now express all the spin operators and the SI in terms of these generators. We will thus, demonstrate that the description of the helicity-conserving first order transition in the spin space becomes more symmetric. To start with, express

The symmetry in the above expression between the helicity operators of the initial and final particles - which goes with the symmetry in figure - is obvious. One can actually go further and check that - as the figure also suggests- **l**-axis:

The above equation makes explicit the intuitive picture that the spin of the incident particle gets rotated by an angle

3. The Transition in the

In this section we will express the SI in terms of the newly introduced generators and investigate the interesting consequences of this. We will then write the scattering states in terms of the

Note how the above two equations go with the symmetry in Fig. 2. Now, from Fig. 3, we have the unit vector

The spin interaction operator will then take the form:

with *A*, *B* and *C* defined in Eq. (16) above. The transition matrix element, Eq. (7), upon employing the expansion given by Eq. (17) above can be further reduced. To do this, consider first the matrix element of

where we have used Eq. (15) to write the second line. Letting operators act on their eigenstates and noting that

with the obvious consequence:

The **
f
** (

**q**) arbitrary. So, if the matrix element is to be gauge-invariant, which is indeed so, then the contribution of

This can be reduced (see the appendix) to:

The matrix element of the

The transition is induced solely by *i.e* the component of the spin interaction operator along the direction of the total momentum vector **k!**. To see what is special with this direction, look again at Fig. 2. The helicity-conserving transition is a transition that leaves the component of the spin along **k**, see Fig. 4. The transition, however, takes place in the spin space, and the relevant quantity is the orientation of the spin of the particle.

This picture can be enhanced by expanding the initial and final helicity states in terms of the eigenstates of

and,

Investigating the above equations it is obvious that

while,

In fact, one can check directly that the SI interaction connects initial and final

Thus,

Similarly,

where in the last line we noted that

These results support our earlier arguments regarding the conservation of the

Finally, one can, by expanding the initial and final states in terms of the

One can easily check that

Combining this result with Eq. (23) we get

Acting with

In the above equation, the only reference to the initial and final states is through the kinematical/geometrical factors *A* and *B*. So, to calculate the transition matrix element for any vector potential, just find these factors - which is a trivial task- and plug them into the above expression. Things can be even further simplified if we use the explicit forms of the spinors:

where *p* being the conserved magnitude of the initial and the final momenta. Plugging this expression into Eq. (34) and using

we have:

This is just a “plug and play" formula, where one just fixes the geometrical factors *A* and *B* for the specific vector potential present, and then gets the spin sector of the matrix element immediately. The following two examples illustrate this explicitly.

4. Examples

In this section we consider two concrete examples of vector potentials whose field configurations conserve helicity, and we bring the first order transition matrix elements of Dirac particles in these potentials to the form given by Eq. (34). Consider first the Ahronov-Bohm (AB) potential ^{3} which gives rise to a

where *z*-axis; the *z*-component of the incident momentum doe not change during the scattering process. Therefore, we consider normal scattering, *i.e.* take the incident, and consequently, the outgoing momenta to be in the *x* - *y* plane. In such a geometry,

So,

with

For the purpose of applying the formula (34), we need to find the geometrical factors *A* and *B*. Obviously, *A* = 0. As for B, we note that we can without any loss of generality, take the incident momentum to be along the *x*-axis;

Straight forward algebra shows that with such a choice of the incident momentum, we get *B* = 1. The matrix element for the AB potential then becomes ^{4}:

We can even move to calculate the scattering cross section. The unpolarized scattering cross section of a Dirac particle in the AB field is given as ^{5}^{,}^{6}^{,}^{7}:

where the summation is over the initial and final particles’ helicities. As a consequence of Eqs. (41) we have

So, using Eq. (33), and taking the normalization constant

we get

which is the well-known AB scatttering cross section of a Dirac particle at first order ^{5}.

The second example is the vector potential of a magnetic dipole, and is less symmetric as the resulting field is not, contrary to the AB one, axial. The vector potential of the dipole is given by ^{8}

where

Therefore

The kinematical factors of Eq. (33) are just

and

which are straight forward to calculate; just specify **p**
_{i}. Therefore, the transition matrix element reads now:

The cross section can be calculated straight forwardly from the above amplitude.

5. Conclusions

The spin interaction in the first order *S*-matrix of a Dirac particle in a static magnetic field was investigated. Noting that the total momentum vector **k = p**
_{f}
**+ p**
_{i} and the momentum transfer vector **q = p**
_{f}
**- p**
_{i} are always perpendicular, we suggested that the three unit vectors; *SU*(2) algebra. When the spin interaction operator

Expressing