From the principle of relativity follows that the matrices which perform
the transformation of coordinates between inertial frames constitute
a group. The group postulates (and the finite speed of light) lead to
the following form of the coordinate transformation under velosity boost
( tīzī) =
1/√(1-v²/c²)
[ 1 -v -v/c21 ]
( tz)
These matrices conserve the interval
c2t2-z2 = c'2t'2-z'2
Lie groups and Lie algebras
Lie group
A continuous group G whose elements g are
differentiable functions of a set of continuous parameters α:
g=g(α1,...,αn). We assume g(α=0)=1.
Generators
An element close to 1 can be expessed in terms of the generators Ik
g(dα) = 1 + i Ijdαj
Lie algebras
The commutators of the generators constitute an algebra
Ij Im - Im Ij = i Ckjm Ik,
where Ckjm is the so called structure constant.
Exercises
Identify all groups of different numbers under operations of addition
and multiplication.
A group parameter φ is called additive if the the additivity law
is satisfied
g(φ1+φ2) = g(φ1)g(φ2).
Find the additive parameter for the one-parameter Lorenz group (velocity
boosts along the z-axis). Find the generator for this group.
(Mulders, Exercise 3.1) Show that if U=exp(-iJkαk) is a
unitary operator with real αk, the operators Jk are hermitian.
Show that if det(U)=1 then the trace of the operators Jk is zero.