The Helicity States

Given a Hilbert Space H of all possible initial and final states, we may define an -matrix with elements consisting of the probability amplitude for scattering from an intitial state to a final state These elements define the scattering matrix for this Hilbert space representation. The matrix may therefore be written in terms of the operator.

The Hilbert Space of initial and final states that will be used in this treatment are the relativistic helicity states. Firstly, what is the justification of using this formalism and what are it's advantages? For the case of spin interactions we may uniquely define a particles quantum state in a frame S, in terms of its momentum, and its spin orientation as measured in its helicity rest frame. This frame is constructed by rotating the z-axis of S along the direction of the particle's momentum, and then boosting this S' frame so that the particle is now at rest in the new frame S''. The state of the particle is then defined by its momentum in S and its spin observed in S''.

For example, the helicity state indicates a particle with momentum in the frame S, and spin in the frame

the helicity rest frame of the particle. The rotation brings the z-axis of the frame in line with the momentum of the particle and the Lorentz transformation with the speed

boosts the new frame, to one in which the particle is stationary. The helicity of a state is invariant under rotations. Applying a rotation to the state , with the rotation operator U(r),

where [20] . the helicity is unchanged. It is clear that,

for the angular momentum operator, and so the helicity states are eigenstates of this ``Helicity Operator.'' This gives a physical interpretation of helicity as the projection of the total angular momentum onto the direction of motion which indeed allows the formalism to be extended to massless particles which have no rest frame.

In the S-Matrix approach, incoming and outgoing states can be treated as those of non-interacting particles, or a direct product of single particle states. Using the helicity formalism in the centre of mass frame of an interaction, the total angular momentum along the line of motion of the particles is given by their helicities and so the angular dependence about the z-axis is simple in form. A two particle Centre of Mass(CMS) state may be defined as,

A question of great importance is how these states translate to other frames, typically a CMS to laboratory frame translation. Considering a particle in a state in a frame S, a frame S' which has been boosted by l with respect to S will describe the state of the particle as,

where l is a Lorentz transform boosting S to S'. Clearly the values of p in the two different frames only differ by the Lorentz transformation [Appendix 4].

and is denoted simply as . In order to assess the effect of the transformation we first express the state as the product of transforms on the rest state, ,

again with v related to as in Eqn 2.3. It will be better if the two unitary transformation operators are compressed to a single operator as they are fully parameterized by . Conventionally is chosen as the product of the two transformations and the corresponding operator is .
Then applying the Lorentz transformation yields,

Then introducing the transformation defined by,

we may then multiply the transformation equation by,

yielding,

The combination of operators in parenthesis are seen to effect a physical rotation. gives a particle a boost to a momentum of ,
boosts this to the momentum state ,
takes a particle in a state of momentum to a particle in a momentum state , therefore these operators combine to give no more than a rotation. Defining the operator effecting this rotation as R it is known that a stationary state will be transformed in the manner  [3],

an admixture of spin up and down states dictated by the angle of rotation. will commute with and so

R and therefore are fully parameterized by l and alone, not being independent of these.
This rotation known as the Wick Helicity Rotation, may not simply be reduced to the sum of the rotations in and . The three Lorentz transformations in R effect but this is not the identity transformation. This becomes clear on looking at the important CMS to Lab frame transformation. In this case it is assumed that both frames have common x,y, and z directions and only differ by a Lorentz boost along the z-axis.

Supposing a particle A scatters in the CMS frame with momentum , at an angle to the z-axis and with zero azimuthal angle, thus defining the xz scattering plane, in the Lab frame, most likely defined by the rest frame of the target particle, the kinematic state of motion of the particle will be described by the quantities and . A helicity state , defined in the CMS frame will be seen in the Lab frame as a superposition . The matrix is generated by the operator R(r), where the rotation

for , and v is the speed of the CMS system as seen from the Lab frame. Given that r is simply a rotation, effected by rotations and boosts in the xz plane, it must be a rotation about the y-axis. The effect on a unit vector in the z direction is given by,

where is the CMS scattering angle and the LAB scattering angle. Regarding the x and z components as and respectively, this provides the definition of the Wick angle for this transformation.

Using standard relativistic kinematics, the centre of mass scattering angle may be eliminated from these expressions in favor of initial and final CMS energies and momenta. In the case of elastic scattering where and , the Wick angle for particle D is , where is the Lab recoil angle of D, and further for the scattering of identical particles the Wick angle for C is .