Interacting Dirac Particle and the Helicity Operator

The Dirac Hamiltonian for free particle commutes with the helicity operator Sigma.p. But the Dirac Hamiltonian with interaction term will not commute with the Helicity operator.They will not have simultaneous eigen states. The corresponding observables will be connected by the uncertainty relation As long as the interaction is going on, helicity becomes uncertain.The point is, it is no more the old helicity.

After the interaction is over the Dirac particle again becomes a free particle and the helicity operator produces the usual eigen value by acting  on the Dirac solution.Change of helicity is connected with change of weak hyper charge. For conservation of weak hyper charge some pother particle has to figure in...The important point is that the psi function is no more the free Dirac solution  while the interaction is going on.Helicity operator or any other operator will not produce the same result as for the free Dirac solution.

Weak hyper charge for left handed electron:-1

Weak hypercharge for right handed electron:-2

But in this case there is a perturbative change in helicity and not a large significant change that occurs when a right handed electron flips to a left handed one and vice versa[Higg's mechanism is formally linked with helicity flip between left handed and right handed helicity states].

But this idea of perturbative change can undergo serious modification f the interaction state be expressed as a  linear superposition of the usual helicity states using the completeness property of eigen functions :

Psi-function under interaction=c_1*psi with helicity(+1)+c_2*psi function with helicity(-1) ;c_1^2+c_2^2=1; p;ropability of the two states are |c_1|^2  and |c_2|^2 respectively.

So long as the interaction is going on there will be flipping between the two helicities. At this stage we are closer to Higg's mechanism.

Since our concern is with the Dirac particle, we may include the strong interactions into our consideration. For strong interactions the interaction cannot be removed because of confinement property.The quarks carry electrical as well as color charges.Two types of interaction are always present. The QCD Lagrangian contains the Dirac part but the psi function is not the free Dirac solution .It is not an eigen function of the helicity operator.

There will be flipping of helicity.

Weak hyper charge of quarks:

u(L):1/3,u(R):4/3,;d(L):1/3,d(R):-2/3

[L: left handed;R:right handed;u:up quark;d:down quark

There is some sort of symmetry breaking which might  relate to the axion.

[U(1) axial symmetry is theoretically preserved but is not observed in nature. This is at the root of the strong CP problem. U(1) axial symmetry is broken by the axion field]

Clarification o n the relation:

Psi-function under interaction=c_1*psi with helicity(+1)+c_2*psi function with helicity(-1) ;c_1^2+c_2^2=1; p;ropability of the two states are |c_1|^2  and |c_2|^2 respectively.

It would be appropriate to re write the above formula

Interaction state function=Sigma c_i positive helicity ,psi free Dirac solution +Sigma d_j negative helicity , free Dirac psi solution (A)

The free Dirac psi solution contains a 4*1 column  matrix with an exponential factor ,which is a function of various (E_k,p_k)

On the left side of (A) we have a 4*1 column matrix , each component being a function of (x,,z,t,E,p)In effect each such component is undergoing a complex Fourier expansion.

The ratio of probabilities of the two helicities given by Sigma |c_i|^2/Sigma |d_j|^2.The stated ratio has to be time dependent as usual of any perturbation theory .This time dependence in effect is proportional to the amount of a periodicity present in the temporal part of the expansion over the duration of the perturbation [which will be indefinitely long for strong interactions because of confinement]. The same set of constant coefficients will not serve for different time segments of the perturbation stretching over a finite or infinite duration of time.

Further Points:

Fourier expansion for one variable  on the interval: E or p_i=(alpha,alpha+2c)=

f(E 0r p_i)=a_0/2+Sigma a_nCos n*pi*(E orp_i)/c+Sigma b_nCos n*pi*(E or p_i)/c

a_0=1/c Integral [alpha to alpha+2c]f(E orp_i)dE or p_i

a_n=1/c Integral[alpha to alpha+2c]f(E or p_i)Cos n*pi*E or p_i/cd(Eor pi)

b_n=1/c Integral[alpha to alpha+2c]f(E or p_i)Sin n*pi*(Eor p_i)/cd(E or p_i)

Let n*E=E_n

By the formula n*E we have to sieve out the appropriate values of E_n from the free particle solutions. 

One set of E_n would be sufficient

In fact several sets would be supportive to the time dependence since f(E/p) will have several possible expansions.

NB: |exp[-1/h_bar Et|^2=1: it will not contribute to changes in probabities involved in the expansion

We may have improved considerations through the Fourier Transforms

Now for argument's sake let us assume time independence of the constants in the Fourier expansion. To analyze the situation we consider a simple analogy

There are n unbiased coins: probability half for both head up and tail up

What are the chances that you have two heads up and n-2 tails up if all are tossed simultaneously [or one coin is tossed n times]: nC2*(1/2)^2(1/2)^(n-2)  

What are the chances that you have p heads up and q tails up  if all are tossed simultaneously [or one coin is tossed n times]: nCp*(1/2)^p (1/2)^q; p+q=1

p=0,1,n...

In each case yo have non zero probability . Head up and tail up may be compared with the two helicity states. Here we have fixed probabilities half for each  

NB: In any type of perturbation theory we express the perturbed psi function as a liner superposition of free particle solutions which are simple the components of a Fourier expansion over various possible(E,p) states. The completness property goes n rhyme with Fourier expansion

Even in non relativistic theory we use expansion psi=Sigma a_n(t) phi(x) exp[i/h_bar E_nt]

psi is the wave function for the perturbed Hamiltonian while phi are free particle solutions for the Schrodinger equation