Dirac equation Lorentz invariance

The Dirac equation is invariant to Lorentz transformations [8], a necessary requirement of a relativistic equation. In the limit of large quantum numbers the Dirac equation reduces to the Klein-Gordon equation [9,10]. The time-independent form of Dirac s Hamiltonian is given by... 

The group of Poincare transformations consists of coordinate transformations (rotations, translations, proper Lorentz transformations...) linking the different inertial frames that are supposed to be equivalent for the description of nature. The free Dirac equation is invariant under these Poincare transformations. More precisely, the free Dirac equation is invariant under (the covering group of) the proper orthochronous Poincare group, which excludes the time reversal and the space-time inversion, but does include the parity transformation (space reflection). 

In other words, it is the distance, ct, travelled by light in a given time interval which fulfills the role of the fourth coordinate, rather than the time interval itself. The special theory of relativity requires that after a Lorentz transformation the new form of all laws of physics is the same as the old form. The Dirac equation, for example, is invariant under a Lorentz transformation. 

In this section we briefly review the main properties of the Dirac equation that is the basic equation to start with to build a relativistic effective Hamiltonian for atomic and molecular calculations. This single particle equation, as already stated in the introduction, was established in 1928 by P.A.M Dirac [1] as the Lorentz invariant counterpart of the Schrodinger equation. On a note let us recall that the first attempts to replace the Schrodinger equation by an equation fulfilling the requirements of special relativity started just after quantum... 

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). 

Levy-Leblond [16] has realized that not only the Lorentz group (or rather the homomorphic group SL(2) [32, 7]), but also the Galilei group has spinor-field representations. While the simplest possible spinor field with s = I and m 0 in the Lorentz framework is described by the Dirac equation, the corresponding field in a Galilei-invariant theory satisfies the Levy-Leblond equation (LLE)... 

In his detailed analysis of Dirac s theory [6], de Broglie pointed out that, in spite of his equation being Lorentz invariant and its four-component wave function providing tensorial forms for all physical properties in space-time, it does not have space and time playing full symmetrical roles, in part because the condition of hermiticity for quantum operators is defined in the space domain while time appears only as a parameter. In addition, space-time relativistic symmetry requires that Heisenberg s uncertainty relations. 

This equation has at least one advantage over the Schrodinger equation ct and x, y, z enter the equation on equal footing, whieh is required by special relativity. Moreover, the Fock-Klein-Gordon equation is invariant with respect to the Lorentz transformation, whereas the Schrodinger equation is not. This is a prerequisite of any relativity-consistent theory, and it is remarkable that such a simple derivation made the theory invariant. The invariance, however, does not mean that the equation is accurate. The Fock-Klein-Gordon equation describes a boson particle because vk is a usual scalar-type function, in contrast to what we will see shortly in the Dirac equation. 

Paul Dirac used the Fock-Klein-Gordon equation to derive a Lorentz transformation invariant equation for a single fermion particle. The Dirac equation is solvable only for several very simple cases. One of them is the free particle (Dirac), and the other is an electron in the electrostatic field of a nucleus (Charles Darwin-but not the one you are thinking of). 

Despite the glorious invariance with respect to the Lorentz transformation and despite spectacular successes, the Dirac equation has some serious drawbacks, including a lack of clear physical interpretation. These drawbacks are removed by a more advanced theory-quantum electrodynamics. 

The Dirac equation is rigorously invariant with respect to the Lorentz transformation, which is certainly the most important requirement for a relativistic theory. Therefore, it would seem to be a logically sound approximation for a relativistic description of a single quantum particle. Unfortunately, this is not true. Recall that the Dirac Hamiltonian spectrum contains a... 

The Dirac equation makes the kinetic energy part of the SchrOdinger equation invariant for the Lorentz transformation (Dirac 1928) ... 

The most important requirement for truly fundamental physical equations is their invariance in form under Lorentz transformations (principle of relativity). To investigate the behavior of the Dirac equation in Eq. (5.23) under Lorentz transformations, we rewrite it as... 

In 1982-1983 a relativistic scattering model was introduced [Me 83b, Sh 83, Cl 83b] which used the MRW five-term, Lorentz invariant NN operator, the scalar and vector densities of the Walecka model for finite nuclei [Ho 81d], and the Dirac equation for the scattered proton. The model could be used to calculate proton-nucleus scattering observables throughout the medium energy range. The first predictions [Sh 83, Cl 83b] at 500 and 800 MeV were startlingly successful. This model, which is now called the relativistic impulse approximation model, or the RIA model, has been used extensively. 

When properly interpreted, the Klein-Gordon equation gives quite satisfactory results for bosonic particles. However, there are reasons for rejecting it for the description of an electron. For instance, it does not accommodate the spin i nature of the electron. Furthermore, the occurrence of a second derivative with respect to time makes it difficult to introduce the notion of stationary states. To derive an alternative equation, Dirac" tried to find a Lorentz invariant equation of the form... 

It should be noted that the two-electron term in this Dirac—Coulomb equation is based on a classical (i.e., nonrelativistic) picture of the interaction, and it is therefore not Lorentz invariant. In many applications this is a problem of minor importance, and results that are in good agreement with experiment often are obtained with the Dirac-Coulomb equation (Eqs. [71]—[74]) or with theory derived from it through further simplifications. For situations calling for a more accurate model, a relativistic picture of the electron—electron interaction is given by the Breit operator ... 

Dirac was seeking an equation that is analogous to the Schrodinger equation, with a first-order derivative in time, and simultaneously satisfying the Lorentz invariance. He took a bold step If the relativistic energy, the expression under the radical in the energy expression (Eq. [27]) or in the square bracket (Eq. [28]), is factored into a perfect square such that... 

Here Hd, is the Dirac Hamiltonian for a single particle, given by Eq. [30]. Recall from above that the Coulomb interaction shown is not strictly Lorentz invariant therefore, Eq. [59] is only approximate. The right-hand side of the equation gives the relativistic interactions between two electrons, and is called the Breit interaction. Here a, and a, denote Dirac matrices (Eq. [31]) for electrons i and /. Equation [59] can be cast into equations similar to Eq. [36] for the Foldy-Wouthuysen transformation. After a sequence of unitary transformations on the Hamiltonian (similar to Eqs. [37]-[58]) is applied to reduce the off-diagonal contributions, one obtains the Hamiltonian in terms of commutators, similar to Eq. [58]. When each term of the commutators are expanded explicitly, one arrives at the Breit-Pauli Hamiltonian, for a many-electron system " ... 

We will not prove the Lorentz invariance of the Dirac equation. However, we will point to a few features that at least suggest Lorentz invariance. The first thing to note is that the rest mass is obviously a relativistic invariant. If we multiply (4.29) by p and define y = )3a, we have... 

It is fairly obvious that if the operator on the left has as its eigenvalue a Lorentz invariant quantity, the operator itself must be Lorentz invariant. This, of course, does not constitute a proof. Such proofs may be found in most standard textbooks in physics that cover the Dirac equation. 

We have previously commented on the Lorentz invariance of the Dirac equation. Considering that this places time and space coordinates on an equal footing, it may seem inconsistent to discuss transformations in spin space and only. We therefore now turn our attention to time transformations. With only one coordinate, there are only two possible transformations translation and reversal. Translation will be treated in connection with a discussion of the Lorentz transformations in the next section. Here, we will consider the symmetry of the Dirac equation under time reversal. 

While we still have no intention of proving the invariance of the Dirac equation under Lorentz transformations, we do want to conclude this discussion of relativistic symmetry by demonstrating the relationship between the Lorentz transformations and the spatial transformations already discussed. To make the syuunetry between space and time more explicit we introduce the variable r, defined as... 

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. 

It is ironic that spin, which is the only non-classical attribute of quantum mechanics, is absent from the pioneering formulations of Heisenberg and Schrodinger. Even in Dirac s equation, the appearance of spin is ascribed by fiat to Lorentz invariance, without further elucidation. In reality, both Lorentz invariance and spin, representing relativity and quantum mechanics, respectively, are properties of the quaternion field that underpins both theories. 

I returned to the University of Toronto in the summer of 1940, having completed a Master s degree at Princeton, to enroll in a Ph.D. program under Leopold Infeld for which I wrote a thesis entitled A Study in Relativistic Quantum Mechanics Based on Sir A.S. Eddington s Relativity Theory of Protons and Electrons. This book summarized his thought about the constants of Nature to which he had been led by his shock that Dirac s equation demonstrated that a theory which was invariant under Lorentz transformation need not be expressed in terms of tensors. 

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