Lorentz boosts

Consider, for example, a Lorentz boost in the Z direction using Jackson s notation [5], and start with the 4-derivative... 

Consideration of the symmetry of the Poincare group also shows that the cyclic theorem is independent of Lorentz boosts in any direction, and also reveals the physical meaning of the E(2) little group of Wigner. This group is unphysical for a photon without mass, but is physical for a photon with mass. This proves that Poincare symmetry leads to a photon with identically nonzero mass. The proof is as follows. Consider in the particle interpretation the PL vector... 

If we assume that the Lie algebra (804) is independent of Lorentz boosts in any direction, we obtain the Lie algebra ... 

The latter is therefore independent of Lorentz boosts of any kind, and independent of spacetime translations of any kind. As demonstrated previously in this chapter, this result can be arrived at independently and self-consistently by considering the following definition ... 

Figure 3. The normalized electron four velocity distribution function downstream of the shock. The dot-dashed line is a power law fit to the non-thermal high energy tail, while the dashed curve is a Lorentz boosted thermal electron population.

The self-adjoint generators of Lorentz boosts are the three operators... 

It is perhaps interesting to see the Lorentz boost of a wave packet. Consider a wave packet with positive energy that is at rest in some intertial frame. That means, it has the average velocity zero and the Fourier transform is localized around the origin in momentum space. In Figure 5 we show the wave packet with positive energy and its Lorentz transformation. After the transformation. 

Figure 6. A superposition of positive and negative energies before and after a Lorentz boost. Apart from the Lorentz contraction, the relative motion between observer and wave packet produces interference effects. The interference is caused by the separation of the negative and positive-energy parts in momentum space.

So far, only general properties of Lorentz transformations have been investigated but no explicit expression for the transformation matrix A has yet been given. We are now going to derive the transformation matrix A for a Lorentz boost in x-direction in a very clear and elementary fashion. For t = t = 0 the two inertial frames IS and IS shall coincide, and the constant motion of IS relative to IS shall be described by the velocity vector v = vCx, cf. Figure 3.2. Since the y- and z-directions are not affected by this transformation, we explicitly write down the transformation given by Eq. (3.12) (for a = 0) for the relevant subspace... 

Figure 3.2 Illustration of a Lorentz boost in x-direction. For f = f = 0 the two inertial frames IS and IS coincide, and for later times the coordinates of the event E with respect to IS and IS are related to each other according to Eq. (3.67).

For V = 0 Eq. (3.67) for a Lorentz boost in x-direction reduces to the Galilei boost as given by Eq. (2.17). But for nonvanishing velocities v the relativistic transformation is more involved and mixes space- and time-coordinates. 

It is sometimes convenient to have another representation of the Lorentz boost in x-direction at hand. This will, by the way, yield yet another (more compact) derivation of the transformation given by Eq. (3.67). As a direct consequence of the fundamental relation for Lorentz transformations as given by Eq. (3.17) we find for the relevant f-x-subspace the following three relations,... 

We now consider a general Lorentz boost from the inertial system IS to the inertial system IS, which moves with constant velocity v = relative to... 

In order to finally determine the desired transformation matrix A(v) for a general Lorentz boost, we note that we can always first rotate IS to IS" by application of A(K) as given by Eq. (3.19) such that the new coordinate axis x is parallel to v, then apply the familiar Lorentz boost A(t>) in -direction as given by Eq. (3.66), and finally rotate the new system IS " back to IS by the inverse rotation A(E ),... 

The effect of the rotations as described by A R) and A RJ) on the Lorentz boost matrix A v) is as follows the element A°q remains unaffected, the elements M g and AP I transform like a 3-vector, and the elements A transform like a 3-tensor of second rank. Since the transformation matrix A( ) must reduce to the familiar Lorentz boost in c-direction as given by Eq. (3.66) for V = vei, we must have... 

The derivation above was focused on a Lorentz boost in the direction of the stick, i.e., the relative motion of the two frames IS and IS, described by the velocity vector v, and the stick were parallel. Due to the Lorentz boost given by Eq. (3.67) no length contraction will occur in directions perpendicular to v. This immediately yields the general expression for the length contraction. 

Similar to the effect on lengths, moving clocks show different times as compared to those at rest. In order to exhibit this peculiarity of special relativity, we again consider the situation of a Lorentz boost as illustrated in Figure 3.2. Now a clock is placed at the origin of IS which shall be at rest in IS. With this clock we may measure the time between any two events Ei and E2 in IS. ... 

For general, i.e., non-parallel velocities V and vz we cannot simplify the problem by application of a suitable rotation of the coordinate axes. Now all three Lorentz boost matrices occurring in Eq. (3.95) are of the most general form given by Eq. (3.81) with a yet undetermined resulting velocity V. We thus have to evaluate Eq. (3.95) directly. In order to achieve this task, it is convenient to introduce the abbreviations... 

General Lorentz boosts A(z ) are therefore in general not commutative, whereas boosts for parallel velocities commute, cf. Eq. (3.98). 

For the simple case of a Lorentz boost in x-direction with v = ve the relation between the Newtonian force F = (fi, F , fs) and the Minkowski force is by virtue of Eq. (3.66) given by... 

For the general case of an arbitrary Lorentz boost A( ) we note that the inverse transformation A(—i ) is easily obtained from Eq. (3.81) by switching the signs of the velocity components V (cf. Eq. (3.82)). The Minkowski force / in IS is therefore given by... 

Figure 3.5 Two electromagnetically interacting moving charges in an inertial frame of reference IS. The frame IS is chosen to move with the same speed as particle 2,v = f2- For the sake of simplicity a Lorentz boost in x-direction as in section 3.2.1.1 is depicted, a restriction solely made for the readability of the figure but not imposed onto the derivations in the text.

In order to calculate the interaction energy of particles 1 and 2 we need to transform these potentials to the original frame of reference IS by the inverse Lorentz boost as given by Eqs. (3.81) and (3.82),... 

We now proceed to derive the Dirac states for a freely moving electron of mass Me- Note that the charge of the fermion does not enter the Dirac equation for this fermion being at rest or moving freely with constant velocity v. The solutions in Eqs. (5.72) and (5.73) may now be subjected to a general Lorentz boost as given by Eq. (3.81) into an inertial frame of reference moving relatively to the previous one, in which the fermion is at rest, with velocity (— ). This option, namely that the solutions for a complicated kinematic problem can be obtained from those of a simple kinematic problem in a suitably chosen frame of reference by a Lorentz transformation, cannot be overemphasized from a conceptual point of view. However, instead of this Lorentz transformation a direct solution of the Dirac Eq. (5.23) is easier. For this purpose we choose an ansatz of plane waves,... 

It is useful to clarify the various ways in which relativity enters the nucleon-nucleus scattering problem. With respect to kinematics, the nucleon-nucleus center-of-momentum (COM) system wave number and reduced mass, which appear in the Schrodinger equation, are replaced by corresponding relativistic quantities. Also the transformation of the NN scattering amplitude in the NN COM system (where it is known) to the nucleon-nucleus COM frame (where it is needed for the calculation) is done using a proper Lorentz boost [Me 83a]. Both of these procedures for accounting for relativistic kinematics are included in the nonrelativistic scattering calculations done here and shown in this work. 

We have provided a pedagogical derivation of the traditional, nonrelativistic form of multiple scattering theory based on the optical potential formalism. We have also discussed in detail each of the important advances made over the past ten years in the numerical application of the NR formalism. These include the full-folding calculation of the first-order optical potential, off-shell NN t-matrix contributions, relativistic kinematics and Lorentz boost of the NN t-matrix, electromagnetic effects, medium corrections arising from Pauli blocking and binding potentials in intermediate states, nucleon... 

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