Four-momentum

In standard quantum field theory, particles are identified as (positive frequency) solutions ijj of the Dirac equation (p — m) fj = 0, with p = y p, m is the rest mass and p the four-momentum operator, and antiparticles (the CP conjugates, where P is parity or spatial inversion) as positive energy (and frequency) solutions of the adjoint equation (p + m) fi = 0. This requires Cq to be linear e u must be transformed into itself. Indeed, the Dirac equation and its adjoint are unitarily equivalent, being linked by a unitary transformation (a sign reversal) of the y matrices. Hence Cq is unitary. 

Only the zeroth component of the total four momentum, i.e. the energy, is conserved for general time-independent external potentials,... 

The third basic element of the perturbation expansion, the vertex, which describes the emission or absorption of a photon by a fermion (in lowest order), is given by in our notation (as usual, four momentum conservation is automatically implied at the vertices). In diagrammar the free propagators and the simple... 

The four-dimensional delta distribution arises as a result of four-momentum conservation. If the momentum transfer g = p2 — P2 is small compared to mz, then... 

The four-momentum is defined by = id/dxi, and A is the classical four-potential, where... 

We now identify the generalized time appropriate to this problem by the relationship t = using as the basic scale with respect to which the four momentum is measured. The quantity 1/m is the scale of the propagator. 

It easily is demonstrated that the unperturbed (by radiative corrections) transverse propagator Px(k) is strictly self-similar with respect to scaling of the four momentum. Thus, with )° treated as a function of the group parameter t and the propagator amplitude 1/m, namely. 

As the value of the four momentum increases the propagator acquires a dependence on the fine structure constant and so evolves from D°(t 1/m ) into a more complicated function DJ f, a, 1/m ). This equation we write in a form... 

Although these results fall short of fully resolving the issue of the high four-momentum behavior of the transverse photon propagator, they do establish that if this propagator is asymptotically self-similar it will exhibit a fractal-like asymptotic form... 

The essential property of 5 is that if p = E, p) is the four-momentum of the nucleon then... 

Note the odd feature, that although q pointed along OZ, g increases as S°° moves faster along OZ. The reason is that q is not the four-momentum of a particle. Indeed it is a space-like four-vector. 

When P = P + q ) S> m, so that the states X) contain many hadrons, it is permissible to replace the sum over X) by a sum over all possible parton states i), i 2)j i 2K3)) with the same total four-momentum as X). This is equivalent to stating that there is unit probabiilty for the partons to transmute into hadrons. 

Integration over four-momentum of each loop [MASS] ... 

The most general form for the Lorentz invariant NN scattering operator which is manifestly local (i.e., contains no explicit dependence on four-momentum) is the MRW form [Me 83a] given by... 

In the approximation of the first-order self-consistent perturbation theory Z(p) = 1 and (p) is given by the equation above. The quantities C and in Eq. (4.29) are functions of the four-momentum. This generality is required to treat retardation effects. 

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