Lorentz covariant 4-vector

We recall that we can saturate the t Hooft anomaly conditions either with massless fermionic degrees of freedom or with gapless bosonic excitations. However in absence of Lorentz covariance the bosonic excitations are not restricted to be fluctuations related to scalar condensates but may be associated, for example, to vector condensates [51]. 

Non-Abelian electrodynamics has been presented in considerable detail in a nonrelativistic setting. However, all gauge fields exist in spacetime and thus exhibits Poincare transformation. In flat spacetime these transformations are global symmetries that act to transform the electric and magnetic components of a gauge field into each other. The same is the case for non-Abelian electrodynamics. Further, the electromagnetic vector potential is written according to absorption and emission operators that act on element of a Fock space of states. It is then reasonable to require that the theory be treated in a manifestly Lorentz covariant manner. 

Requiring these order parameters to transform in a Lorentz-covariant way, we are led to a particular basis of 4 x 4 matrices , which was recently derived in detail (Capelle and Gross 1999a). The resulting order parameters represent a Lorentz scalar (one component), a four vector (four components), a pseudo scalar (one component), an axial four vector (four components), and an antisymmetric tensor of rank two (six independent components). This set of 4 x 4 matrices is different from the usual Dirac y matrices. The latter only lead to a Lorentz scalar, a four vector, etc., when combined with one creation and one annihilation operator, whereas the order parameter consists of two annihilation operators. 

In contrast to the three-dimensional situation of nonrelativistic mechanics there are now two kinds of vectors within the four-dimensional Minkowski space. Contravariant vectors transform according to Eq. (3.36) whereas covariant vectors transform acoording to Eq. (3.38) by the transition from IS to IS. The reason for this crucial feature of Minkowski space is solely rooted in the structure of the metric g given by Eq. (3.8) which has been shown to be responsible for the central structure of Lorentz transformations as given by Eq. (3.17). As a consequence, the transposed Lorentz transformation A no longer represents the inverse transformation A . As we have seen, the inverse Lorentz transformation is now more involved and given by Eq. (3.25). 

The 4-gradient has been written as a row vector above solely for our convenience it still is to be interpreted mathematically as a column vector, of course. Being defined as the derivative with respect to the contravariant components x, the 4-gradient dpi is naturally a covariant vector since its transformation property under Lorentz transformations is given by... 

Consequently, the transformation property of a covariant Lorentz tensor under Lorentz transformations is therefore given as the one of an n-fold product of covariant vectors. 

Here, it is important to understand that is a four-component object consisting of (4x4)-matrices rather than a Lorentz scalar, since is not a Lorentz 4-vector. Moreover, although the equation already seems to be in covariant form, this still needs to be shown because we do not yet know how Y transforms under Lorentz transformations. In the following, we must determine the transformation properties of Y so that Eq. (5.54) is covariant under Lorentz transformations, which is a mandatory constraint for any true law of Nature. 

The components of the 4-potential are given by AT (, A). Note that the vector potential A = A, A, A ) contains the contravariant components of the 4-potential. According to what follows Eq. (5.54), we need to add a Lorentz scalar to the (scalar) Dirac Hamiltonian in order to preserve Lorentz covariance. This Lorentz scalar shall depend on the 4-potential. The simplest choice is a linear dependence on the 4-potential and by multiplication with 7H we obtain the desired Lorentz scalar. Minimal coupling thus means the following substitution for the 4-momentum operator... 

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 only common factor is that the charge-current 4-tensor transforms in the same way. The vector representation develops a time-like component under Lorentz transformation, while the tensor representation does not. However, the underlying equations in both cases are the Maxwell-Heaviside equations, which transform covariantly in both cases and obviously in the same way for both vector and tensor representations. 

We can arrive at this result from the consideration that the velocity defined by the components dxjdt, dyjdt, dzjdt (or the momentum obtained from this velocity on multiplication by the mass) cannot, in view of Lorentz s transformation, be regarded as a vector, since the differential dt in the denominator is also transformed. We obtain a serviceable covariant definition if we replace dt by dt, where dt is the element of the proper time of the particle, i.e. the time measured in the system of reference in which the particle is at rest. The relation between dt and dt is found by taking the derivative of t y... 

From 7q = I4 we see immediately that V = cjoVcov- The equation (83) is called the Dirac equation in covariant form. It is best suited for investigations concerning relativistic invariance, because it me is a scalar (which by definition of a scalar is invariant under Lorentz transformations) and the term (7,5) is written in the form of a Minkowski scalar product (if 7 and d were ordinary vectors in Minkowski space, the invariance of this term would be already guaranteed by (81). 

Einstein s theory of special relativity relying on a modified principle cf relativity is presented and the Lorentz transformations are identified as the natural coordinate transformations of physics. This necessarily leads to a modification cf our perception of space and time and to the concept of a four-dimensional unified space-time. Its basic Mnematic and dynamical implications on classical mechanics are discussed. Maxwell s gauge theory of electrodynamics is presented in its natural covariant 4-vector form. 

IS. Note that the velocity n is a 3-vector and therefore its components are labeled with subscript indices, e.g., Vi = Vx. This must not be confused with the covariant components of a 4-vector. The velocity vector v may point in any direction, but the coordinate axes of IS and IS must be parallel to each other, i.e., there is no constant rotation between the axes of the two systems. For a particle at rest in IS (for example at the origin r = 0), we have dr = 0 and can therefore immediately write down the Lorentz transformation given by Eq. (3.13) as... 

So far we have just defined another four-component quantity Af, but by now it is not clear whether it properly transforms under Lorentz transformations in order to justify the phrase 4-vector. In order to prove the transformation property of the gauge field, we re-express the inhomogeneous Maxwell equations in Lorenz gauge as given by Eq. (2.138) in explicitly covariant form by employment of the charge-current density and the gauge field A, ... 

Similarly to the nonrelativistic situation [cf. Eq. (2.29)], the components of the Lorentz transformation matrix A may be expressed as derivatives of the new coordinates with respect to the old ones or vice versa. However, since we have to distinguish contra- and covariant components of vectors in the relativistic framework, there are now four different possibilities to express these derivatives ... 

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