Mathematical Methods for Physicists: A concise introduction - Site Map

ELEMENTS OF GROUP THEORY
Figure 12.6.
The strong isospin space.
Homogeneous Lorentz group
Before we describe the homogeneous Lorentz group, we need to know the
Lorentz transformation. This will bring us back to the origin of special theory
of relativity. In classical mechanics, time is absolute and the Galilean transformation
(the principle of Newtonian relativity) asserts that all inertial frames
are equivalent for describing the laws of classical mechanics. But physicists in
the nineteenth century found that electromagnetic theory did not seem to obey
the principle of Newtonian relativity. Classical electromagnetic theory is summarized
in Maxwell's equations, and one of the consequences of Maxwell's
equations is that the speed of light (electromagnetic waves) is independent of
the motion of the source. However, under the Galilean transformation, in a
frame of reference moving uniformly with respect to the light source the light
wave is no longer spherical and the speed of light is also di€erent. Hence, for
electromagnetic phenomena, inertial frames are not equivalent and Maxwell's
equations are not invariant under Galilean transformation. A number of
experiments were proposed to resolve this con¯ict. After the Michelson±
Morley experiment failed to detect ether, physicists ®nally accepted that
Maxwell's equations are correct and have the same form in all inertial frames.
There had to be some transformation other than the Galilean transformation
that would make both electromagnetic theory and classical mechanical invariant.
This desired new transformation is the Lorentz transformation, worked out by
H. Lorentz. But it was not until 1905 that Einstein realized its full implications
and took the epoch-making step involved. In his paper, `On the Electrodynamics
of Moving Bodies' (The Principle of Relativity, Dover, New York, 1952), he
developed the Special Theory of Relativity from two fundamental postulates,
which are rephrased as follows:
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