Lorentz covariance

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]. 

The precise correspondence between field and photon interpretation developed here indicates that E(2) symmetry does not imply that Ii(3) is zero, any more than it implies that J<3> = 0. The assertion B(3> = 0 is counterindicated by a range of data reviewed here and in Ref. 44, and the B cyclic theorem is Lorentz-covariant, as it is part of a Lorentz-covariant Lie algebra. If we assume the particular solutions (809) and (810) and use in it the particular solution (803), we obtain the cyclics (809) from the three cyclics Eq. (810) thus we obtain... 

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. 

Apart from Lorentz covariance the quantum mechanical state equation must obey certain mathematical criteria (i) it must be homogeneous in order to fulfill Eq. (4.7) for all times, and (ii) it must be a linear equation so that linear combinations of solutions are also solutions. The latter requirement is often denoted as the superposition principle, which is required for the description of interference phenomena. However, it is equally well justified to regard these requirements as the consequences of the equation of motion in accordance with experiment if the equation of motion and the form of the Hamiltonian operator are postulated. 

In 1928, Dirac proposed a new quantum mechanical equation for the electron [99,100], which solved two problems at once, namely the Lorentz-covariance requirement and the duplexity of atomic states, which was accounted for by Goudsmit and Uhlenbeck s phenomenological introduction of spin. In fact, he showed how the dynamic spin variable is connected to Lorentz covariance — a connection that will become clear in the following. To derive this fundamental quantum mechanical equation for the electron, which features relativistic covariance, we set out with a basic ansatz for this equation based on the results of the preceding section. 

Lorentz Covariance of the Field-Free Dirac Equation... 

To demonstrate the Lorentz covariance of Eq. (5.54) we introduce the coordinate tranformation step by step and start in analogy to Eq. (3.41) with the... 

Thus it has been shown that the Dirac equation for a freely moving electron obviously fulfills the principle of relativity and is Lorentz covariant. 

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... 

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

Note that this ansatz does not violate the Lorentz covariance of Eq. (6.4) because after having performed a Lorentz transformation to a new set of coordinates [ct, r ) one would choose an ansatz as in Eq. (6.5) but for the new coordinates t and r. Of course, the new time coordinate t may then be expressed by the old time and spatial coordinates t and r as governed by the Lorentz transformation. 

By following the ideas of canonical quantization, both manifest Lorentz co-variance and gauge invariance of the theoretical description have been sacrificed. All physical quantities such as transition amplitudes and S-matrix elements, however, will be Lorentz covariant and independent of the chosen... 

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. ... 

On grounds of Lorentz covariance the forward nucleonic matrix elements have a similar structure ... 

The relativistic Hohenberg-Kohn theorem was first formulated by Rajagopal and Callaway [5,6] and by McDonald and Vosko [7]. As expected for a Lorentz covariant situation it states that the ground-state energy is a rniique functional of the ground-state four-current... 

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