Lorentz equations relativity theory

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. 

Two equivalent forms of Maxwell s field equations in terms of the standard vector formalism are Eq. (5), or (6) with the Lorentz gauge 9 4 = 0. The former is in terms of the antisymmetric second-rank tensor solution F, which is a combination of the electric and magnetic field variables. The latter is in terms of the vector potential, A, shown in Eq. (6) [as well as Eq. (7) in terms of the pseudo vector potential Bassuming that the parameter E, is nonzero]. (Experimental results to this point in time indicate that indeed this parameter is zero to within experimental accuracy [15]—even though the symmetry of relativity theory has no reason to exclude it. Henceforth, we will assume that this parameter is zero.)... 

A consequent 5-dimensional treatment would require Unified Theory of Quantum Mechanics and General Relativity. This unified theory is not available now, and we know evidences that present QM is incompatible with present GR. The well-known demonstrative examples are generally between QFT and GR (e.g. the notion of Quantum Field Theory vacua is only Lorentz-invariant and hence come ambiguities about the existence of cosmological Hawking radiations [19]). But also, it is a fundamental problem that the lhs of Einstein equation is c-number, while the rhs should be a quantum object. 

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. 

The theory of relativity and quantum mechanics constitute the two basic foundations of theoretical physics. It is also well known that quantum mechanics based upon the Schrodinger equation has been used for decades to investigate atomic and molecular structure by physicists and chemists. However, the Schrddinger equation is non-relativistic i.e., it is not Lorentz-invariant as it does not obey the special theory of relativity. 

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. 

It is the fundamental Eq. (4.16), which has to obey the principle of Einstein s theory of special relativity, namely of being invariant in form in different inertial frames of reference. Hence, the choices for the Hamiltonian operator H are further limited by the requirement of form invariance of the whole equation under Lorentz transformations which will be discussed in detail in chapter 5. 

One postulate that has not explicitly been formulated as a basic axiom of quantum mechanics in the last chapter, because this postulate is valid for any physical theory, is that the equations of quantum mechanics have to be valid and invariant in form in all intertial reference frames. In this chapter, we take the first step toward a relativistic electronic structure theory and start to derive the basic quantum mechanical equation of motion for a single, freely moving electron, which shall obey the principles of relativity outlined in chapter 3. We are looking for a Hamiltonian which keeps Eq. (4.16) invariant in form under Lorentz transformations. 

If the relative velocity V of the Galilean ttansformation (2.4) approaches the speed of light c, the uniformity of time is not applicable, and the Newtonian framework is no longer valid. That is, the Galilean transformation is changed into the Lorentz transformation under invariance of Maxwell s electtomagnetic equations, and the equation of motion is now described in relation to Einstein s theory of relativity. ... 

According to the special theory of relativity, all physical laws are postulated to be invariant in any inertial frame of reference. Furthermore, the equations of motion must be invariant under a Lorentz transformation. 

Another requirement in satisfying the special theory of relativity is to have the spatial and temporal variables being treated on the same footing. The differential operators in Lorentz invariant form are QIQx, 6/0y, didz, and (l/ic)0/0t, giving the magnitude for spatial and temporal dimensions as x + y + z — c t. This means that the order of the differentials for the coordinate and time in the equation of motion must be the same. The time-dependent Schrodinger equation exhibits time derivatives in the first order and coordinate derivatives in the second order therefore it is not Lorentz invariant. 

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. 

The theory of special relativity is conveniently summarized by a set of equations, known as a Lorentz transformation, which describes all relative motion, including that of electromagnetic signals, observed to propagate with constant speed c, irrespective of the observer s state of motion. This transformation. 

With a wave model in mind as a chemical theory it is helpful to first examine wave motion in fewer dimensions. In all cases periodic motion is associated with harmonic functions, best known of which are defined by Laplace s equation in three dimensions. It occurs embedded in Schrodinger s equation of wave mechanics, where it generates the complex surface-harmonic operators which produce the orbital angular momentum eigenvectors of the hydrogen electron. If the harmonic solutions of the four-dimensional analogue of Laplace s equation are to be valid in the Minkowski space-time of special relativity, they need to be Lorentz invariant. This means that they should not be separable in the normal sense of Sturm-Liouville problems. In standard wave mechanics this is exactly the way in which space and time variables are separated to produce a three-dimensional wave equation. 

Maxwell s theory of electromagnetism was actually the first theory that fulfilled the requirements of special relativity (i.e. the equations are invariant under a Lorentz transformation), even before special relativity was formulated by Einstein. 

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