Dirac Lorentz covariance

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 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... 

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... 

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. ... 

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). 

The most obvious objection to the Dirac-Coulomb operator is that the instemt-aneous Coulomb interaction, l/r -, is manifestly not Lorentz invariant. It is, however, the leading term of the covariant interaction... 

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