Lorentz transformation principles

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... 

Figure 1. The frames of reference S and S in relative translational motion. In the frame of reference S, Lorentz transformation and special relativity principles are valid. In the frame of reference S, superluminal transformation and SLRT principles are valid.

Per-Olov was born in Uppsala on October 28, 1916 as the son of the musician Erik Wilhelm Lowdin and his wife Eva Kristina, nee Ostgren. He showed mathematical proficiency early in his school work, and when he entered Uppsala University in 1935 his plans were to major in mathematical physics. His first scientific paper in 1939 on the Lorentz-transformation and the kinematical principle of relativity was published in Swedish in the journal Elementa. The years of war that followed meant interruptions in communications and research opportunities and Per-Olov, like most young Swedish men, spent time in the military defending his country. 

The conservation of momentum is one of the basic principles of mechanics and it is to be expected that momentum remains invariant under Lorentz transformation. The fact that the velocity of a moving body is observed to be different in relatively moving coordinate systems therefore implies that,... 

In summary, the principle of local invariance in a curved Riemannian manifold leads to the appearance of compensating fields. The electromagnetic field is the compensating field of local phase transformation and the gravitational field is the compensating field of local Lorentz transformations. 

For the same reason it is not clear, how to modify the equation for the inclusion of external fields. The principle of minimal coupling p —> p — A, E E + V for the (scalar) square-root Klein-Gordon equation was critizised by J. Sucher [4], who states that there are solutions ip x) and electromagnetic potentials, such that the Lorentz transformed solution is not a solution of the equation with the Lorentz-transformed potentials. Moreover, the nonlocal nature of the equation means that the value of the potential at some point influences the wave function at other points and it is not clear at all how one can interpret this. 

This corresponds to the principle of minimal coupling, according to which the interaction with a magnetic field is described by replacing in the Hamiltonian operator the canonical momentum p by the kinetic momentum 11 = p — f A(x). Other types of external-field interactions include scalar or pseudoscalar fields and anomalous magnetic moment interactions. The classification of external fields rests on the behavior of the Dirac equation rmder Lorentz transformations. A brief description of these potential matrices will be given below. 

Transformations that preserve the relativity principle are called Lorentz transformations. The form of these looks complicated at first (see diagram). However, they arise from the simple requirement that there can be no experiment in dynamics or electromagnetism that will distinguish between two different Galilean frames of reference. 

A question open for discussion is whether the equations of motion or the invariance principles are the more basic first principles It was a choice of Einstein that he gave the Lorentz transformation primary importance, and so the equations of motion had to be modified. 

Lorentz transformation (p. 100) velocity addition law (p. 103) relativity principle (p. 104)... 

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. 

The relativity principle of Einstein or, more precisely, the principle of constant speed of light led us to the definition of the four-dimensional distance Sj2 given by Eq. (3.4), which we recognized as an invariant quantity under Lorentz transformations relating different inertial frames of reference. It is therefore advisible to adopt a more suitable notation in order to reflect the... 

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. 

After having derived a truly relativistic quantum mechanical equation for a freely moving electron (i.e., in the absence of external electromagnetic fields), we now derive its solutions. It is noteworthy from a conceptual point of view that the solution of the field-free Dirac equation can in principle be pursued in two ways (i) one could directly obtain the solution from the (full) Dirac equation (5.23) for the electron moving with constant velocity v or (ii) one could aim for the solution for an electron at rest — which is particularly easy to obtain — and then Lorentz transform the solution according to Eq. (5.56) to an inertial frame of reference which moves with constant velocity —v) with respect to the frame of reference that observes the electron at rest. 

In section 3.1.2 we found the invariance under Lorentz transformations of the squared space-time interval s 2 between two events connected by a light signal being solely based on the relativity principle of Einstein, i.e., the constant speed of light in all inertial frames, cf. Eq. (3.5),... 

It follows from these arguments that the equations of physics inside this continuum (aether) should be invariant under arbitrary diffeomorphisms of the coordinates. For weak disturbances we may restrict ourselves to the linear wave equations which are invariant under Lorentz transformations. From the action principle (4), and (5) we may then get the equation of electromagnetic... 

For physical objects and reference frames moving with speeds ixc or u< , Einstein s theory led to the results of classical nonrelativistic theory Lorentz transformations changed into Galileo s transforms and the Einstein relativity principle into Galileo s relativity principle. 

So far we have only exploited the principle of constant speed of light in all inertial frames, but not the (first part of the) relativity principle itself, cf. section 3.1.2. Due to the relativity principle the Lorentz back transformation from IS to IS must have the same form as given by Eq. (3.60) with v replaced by —v,... 

In order to preserve the principles mentioned above, it was necessary to proceed further than Galileo s transformations (1.3.1) and (1.3.2). These transformations have been replaced by the mathematical Lorentz equations already known in physics. 

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