Potential Lorentz invariant

It is to be expected that the equations relating electromagnetic fields and potentials to the charge current, should bear some resemblance to the Lorentz transformation. Stating that the equations for A and (j> are Lorentz invariant, means that they should have the same form for any observer, irrespective of relative velocity, as long as it s constant. This will be the case if the quantity (Ax, Ay, Az, i/c) = V is a Minkowski four-vector. Easiest would be to show that the dot product of V with another four-vector, e.g. the four-gradient, is Lorentz invariant, i.e. to show that... 

It is important to notice that in Equation 10.11, only the sum of the three terms is Lorentz invariant. The first term corresponds to the interaction of the charge density with the external Coulomb potential and the last term can be written in the form... 

Under specific assumptions which are met when Lorentz invariance is broken via the chemical potential. 

Figure 12. The diffusive modes of the periodic Yukawa-potential Lorentz gas represented by their cumulative function depicted in the complex plane ReFk,hnFk) for two different nonvanishing wavenumbers k. The horizontal straight line is the curve corresponding to the vanishing wavenumber k = 0 at which the mode reduces to the invariant microcanonical equilibrium state.

Since ip depends on space-time coordinates, the relative phase factor of ip at two different points would be completely arbitrary and accordingly, a must also be a function of space-time. To preserve invariance it is necessary to compensate the variation of the phase a (a ) by introducing the electromagnetic potentials (T4.5). In similar vein the gravitational field appears as the compensating gauge field under Lorentz invariant local isotopic gauge transformation [150]. 

One of the most recent theoretical advances has been to correlate almost all n-p and p-p data below 300 Mev in terms of boundary conditions on S, P and D states with only one energy dependent parameter (see Breit and Feshbach ). However, there is still no answer to whether it is possible to write an analytical expression as a function of energy for the law of force between two nucleons. It may be that the need of a Lorentz invariant expression for the interaction is the fundamental reason why potentials (even of the repulsive core type (Jastrow )] specifically have not been able to fit all of the data. 

The quasi-two-body t-matrices of either the Watson or KMT optical potentials are to be evaluated in the proton-nucleus COM system. For nonrelativistic theories the r-matrix element is Galilean invariant and can be immediately obtained from the r-matrix in the proton-nucleon COM system [see eq. (3.9)]. For relativistic momenta, however, Lorentz invariance of flux requires that the quantity... 

At any rate, the relativistic NN t-matrix, Vj, introduced in eq. (4.8) for the optical potential, can also be expanded in terms of the five Lorentz invariants, where for on-shell matrix elements... 

Because and j are invariant, it follows that the four-vector potential A is also Lorentz invariant. Note that this is really implied in the Lorentz condition (3.23), which is a product of the Lorentz-invariant four-gradient and the four-vector A and can be written... 

While the expression above for the potential between two charged particles is indeed symmetric in the particle labels, it is not Lorentz invariant. This can be demonstrated by carrying out the appropriate transformation. To make things simple, we consider only the case of two one-dimensional inertial frames, where S moves with velocity v relative to S. In the moving (one-dimensional) system, u and r are parallel, and if we discard terms of order 4 and higher in c, the interaction takes the form... 

However, if we are concerned about Lorentz invariance, we should at this point remember that the scalar potential is only one component of a four-vector A = (A, i/c). If the scalar potential modifies %, or equivalently E, then we would expect the vector potential to modify the momentum, which accounts for the remaining components of the four-vector. 

In the light of the chapter on special relativity (chapter 2), it is apparent that there is a possible problem in performing this separation of the space and time variables, because the Lorentz transformation mixes them. The separation would have to be performed in a particular frame of reference, and only be valid in this frame of reference. If we want results in another frame of reference, we must perform a Lorentz transformation to that frame, and there is no guarantee that we will still have a stationary state. However, if our Hamiltonian is Lorentz invariant, the choice of the frame of reference is arbitrary, and, as we saw above, the probability density is independent of time and of the frame of reference. We may therefore choose the frame that is most convenient. In molecules (and in atoms) the Born-Oppenheimer frame is the most convenient frame of reference for electronic stmcture calculations because the nuclear potential is then simply the static Coulomb potential. Regardless of whether the Hamiltonian is Lorentz invariant or not, it is this frame that we work in from here on. 

Having recovered the potential surface from the solutions of the Born-Oppenheimer electronic problem, we can now proceed to solve the equation for nuclear motion. The Dirac-type equation for the nuclei can easily be reduced to the corresponding nonrelativistic equation by following the same reduction as we did for taking the nonrelativistic limit of the Dirac equation in section 4.6. Doing this, we abandon all pretense of Lorentz invariance for this part of the system, but we know from experiment that the nuclear relative motion in molecules takes place at rather low energies where relativistic effects may safely be neglected. 

Because of the spontaneously broken U 1) x 0(3) symmetry in Eq. (3), for A / 0 there should be collective Nambu-Goldstone excitations in the spectrum. However, due to the Lorentz non-invariance of the system there can be subtleties [19, 28-30], The NG spectrum can be analyzed within an underlying effective Higgs potential... 

On the other hand, due to the choice of a particular Lorentz frame the gauge invariance of if with respect to gauge transformations of the external potential has been partially broken Only static gauge transformations,... 

Breit constructed a many-electron relativistic theory that takes into aceount sueh a retarded potential in an approximate way. Breit explicitly considered only the electrons of an atom its nucleus (similar to the Dirac theory) created only an external field for the electrons. This ambitious project was only partly successful because the resulting theory turned out to be approximate not only from the point of view of quantum theory (wifli some interactions not taken into account), but also from the point of view of relativity theory (an approximate Lorentz transformation invariance). 

Apart from Lorentz covariance, invariance under gauge transformations of the 4-potential (cf. sections 2.4 and 3.4),... 

The many-electron Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian is neither gauge invariant — it does not even contain vector potentials — nor Lorentz covariant as these symmetries have explicitly been broken in section 8.1. Moreover, the first-quantized relativistic many-particle Hamiltonians suffer from serious conceptual problems [217], which are solely related to the unbounded spectrum of the one-electron Dirac Hamiltonian h. ... 

The one-meson exchange, relativistic optical potential was evaluated from eq. (4.8) using in eq. (4.37) and the kinematic factor in eq. (4.18) (except that the S ( )/S (0) factor was not includ ). For the direct term, p, only the scalar, vector and tensor invariants contribute to the optical potential for even-even nuclei, just as for the RIA potential of the previous section. For the exchange term it is clear from eq. (4.41) that all Lorentz components of contribute to the optical potential. For example, the exchange amplitude contribution to the scalar part of the optical potential involves the sum of amplitudes given by (using eq. (4.41) and the Fierz matrix in ref. [Ho 85])... 

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