Next: Thomas Precession
Up: Generators of the Lorentz
Previous: Generators of the Lorentz
Contents
We seek (Lie) groups of continous linear transformations,
|
(16.41) |
or
|
(16.42) |
for
. We require that the
are
real numbers (parameters) that characterize the transformation. must be
minimal (``essential'').
Examples of transformations of importance in physics (that you should
already be familiar with) include
where
. This is the ( parameter) translation
group in dimensions. Also,
|
(16.44) |
where
|
(16.45) |
is the (three parameter) rotation group.
An infinitesimal transformation in one of the parameters is defined by
|
(16.46) |
In this definition, are the (-parameter) values associated with
the identity transformation . These can be chosen to be zero by suitably
choosing the parameter coordinates. The infinitesimal parameters
are taken to zero, so that
(summed) is neglible. Thus
|
(16.47) |
where
|
(16.48) |
and
|
(16.49) |
Putting this all together,
(summed over
in four dimensional space-time and
). Thus (unsurprisingly)
|
(16.51) |
which has the form of the first two terms of a Taylor series. This is
characteristic of infinitesimal linear transformations.
One can easily verify that
|
(16.52) |
(infinitesimal transformations commute) and that
|
(16.53) |
(to order ). They thus have an identity, an inverse, and can be
shown to be associative.
The continuous transformation group (mentioned above) follows
immediately from making (the displacement of coordinates)
infinitesimal and finding finite displacements by integration. The
rotation group (matrices) are a little trickier. They are
|
(16.54) |
where
|
(16.55) |
The infinitesimal are antisymmetric and traceless (in 3D), so they have
only three independent parameters (that are thus ``essential''). We can write
them generally as
|
(16.56) |
where the is the infinitesimal parameter and where
is the antisymmetric unit tensor. Thus, if
|
(16.57) |
we see that
|
(16.58) |
A moment of thought should convince you that
is the
infinitesimal (vector) rotation angle, with direction that points along
the axis of rotation.
To obtain the rotation group we must show that every
rotation can be obtained by integrating . This follows by
writing an arbitrary rotation or product of rotations as a single
rotation about a fixed axis. For
parallel to this axis
, this is obviously true, as I show next. Since any
rotation can be written this way, the rotations indeed form a group.
The integration proceeds like:
|
(16.59) |
where
and
. We can parameterize this as
|
(16.60) |
where
|
(16.61) |
Believe it or not, this was one of the primary things we wanted to show in
this aside. What it shows is that rotations about an arbitrary axis can
be written as an exponential that can be thought of as the infinite product of a series of infinitesimal transformations
where each transformation has various nice properties.
With these known results from simpler days recalled to mind, we return
to the homogeneous, proper Lorentz group. Here we seek the
infinitesimal linear transformations, etc. in four dimensions.
Algebraically one proceeds almost identically to the case of rotation,
but now in four dimensions and with the goal of preserving length in a
different metric. A general infinitesimal transformation can be written
compactly as:
|
(16.62) |
where (as before)
(and hence is traceless), is
infinitesimal, and where is the usual metric tensor (that follows
from all the annoying derivatives with respect to the parameters and
coordinates).
Thus
|
(16.63) |
defines the form of a general transformation matrix associated
with a given ``direction'' in the parameter space constructed from an
infinite product of infinitesimal transformations, each of which is
basically the leading term of a Taylor series of the underlying
coordinate function transformation in terms of the parameters. This
justifies the ``ansatz'' made by Jackson. The matrices are called
the generators of the linear transformation.
Thus, whenever we write
|
(16.64) |
where the 's are (to be) the generators of the Lorentz group
transformations we should remember what it stands for. Let's find the
distinct . Each one is a real, traceless matrix that is
(as we shall see) antisymmetric in the spatial part (since is
antisymmetric from the above).
To construct (and find the distinct components of ) we make use
of its properties. Its determinant is
|
(16.65) |
(This follows from doing a similarity transformation to put in diagonal
form. is necessarily then diagonal. Similarity transformations do not
alter the determinant, because
|
(16.66) |
If is diagonal, then the last equation follows from the usual properties
of the exponential and the definition of the exponential of a matrix.)
If is real then
is excluded by this result. If is
traceless (and only if, given that it is real), then
|
(16.67) |
which is required to be true for proper Lorentz transformations (recall
from last time). Making a traceless 4x4 matrix therefore suffices
to ensure that we will find only proper Lorentz transformations.
Think back to the requirement that:
|
(16.68) |
in order to preserve the invariant interval where
|
(16.69) |
and is a real, traceless, matrix.
If we multiply from the right by and the left by , this equation
is equivalent also to
|
(16.70) |
Since
,
, and :
|
(16.71) |
or
|
(16.72) |
(This can also easily be proven by considering the ``power series'' or
product expansions of the exponentials of the associated matrices above,
changing the sign/direction of the infinitesimal series.)
Finally, if we multiply both sides from the left by and express the
left hand side as a transpose, we get
|
(16.73) |
From this we see that the matrix is traceless and antisymmetric as
noted/expected from above. If we mentally factor out the , we can
without loss of generality write as:
|
(16.74) |
This matrix form satisfies all the constraints we deduced above for the
generators. Any of this form will make an that preserves the
invariant interval (length) of a four vector. There are exactly six
essential parameters as expected. Finally, if we use our
intuition, we would expect that the for
form the rotation subgroup and describe physical rotations.
So this is just great. Let us now separate out the individual couplings
for our appreciation and easy manipulation. To do that we define six
fundamental matrices (called the generators of the group from
which we can construct an arbitrary and hence . They are
basically the individual matrices with unit or zero components that can
be scaled by the six parameters . The particular choices
for the signs make certain relations work out nicely:
|
(16.75) |
|
(16.76) |
|
(16.77) |
|
(16.78) |
|
(16.79) |
|
(16.80) |
The matrices generate rotations in the spatial part and the
matrices generate boosts. Note that the squares of these matrices
are diagonal and either or in the submatrix involved:
|
(16.81) |
and
|
(16.82) |
etc. From this we can deduce that
Note that these relations are very similar to the multiplication
rules for unit pure complex or pure real numbers.
The reason this is important is that if we form the dot product of a
vector of these generators with a spatial vector (effectively
decomposing a vector parameter in terms of these matrices) in the
exponential expansion, the following relations can be used to reduce
powers of the generators.
|
(16.85) |
and
|
(16.86) |
In these expressions,
an arbitrary unit vector, and
these expressions effectively match up the generator axes (which were
arbitrary) with the direction of the parameter vector for rotation or
boost respectively. After the reduction (as we shall see below) the
exponential is, in fact, a well-behaved and easily understood matrix!
It is easy (and important!) to determine the commutation relations of these
generators. They are:
The first set are immediately recognizable. They tells us that ``two
rotations performed in both orders differ by a rotation''. The second and
third show that ``a boost and a rotation differ by a boost'' and ``two boosts
differ by a rotation'', respectively. In quotes because that is somewhat
oversimplified, but it gets some of the idea across.
These are the generators for the groups or . The latter is
the group of relativity as we are currently studying it.
A question that has been brought up in class is ``where is the factor
in the generators of rotation'' so that
as we might expect from considering spin and angular momentum
in other contexts. It is there, but subtly hidden, in the fact that
in the projective block of the rotation matrices
only. Matrices appear to be a way to represent geometric
algebras, as most readers of this text should already know from their
study of the (quaternionic) Pauli spin matrices. We won't dwell on this
here, but note well that the Pauli matrices
are isomorphic to the unit quaternions
via the mapping ,
,
,
as the reader can
easily verify16.6 Note well that:
|
(16.90) |
is both real and, not at all coincidentally, the structure of an
sub-block.
With these definitions in hand, we can easily decompose in terms of the
and the
matrices. We get:
|
(16.91) |
where
is a (finite) rotation around an axis in direction
and where
is a (finite) boost in direction
. Thus the completely general form of is
|
(16.92) |
The (cartesian) components of
and
are now the six
free parameters of the transformation.
Let us see that these are indeed the familiar boosts and rotations we are used
to. After all, this exponential notation is not transparent. Suppose that
and
. Then and
or (in matrix form)
|
(16.94) |
which (ohmygosh!) is our old friend the Lorentz transformation, just like we
derived it a la kiddy-physics-wise. As an exercise, show that the
result is a rotation around the
axis. Note that the step of ``adding and subtracting'' is
essential to reconstructing the series of the sine and cosine, just like the
was above for cosh and sinh.
Now, a boost in an arbitrary direction is just
|
(16.95) |
We can certainly parameterize it by
|
(16.96) |
(since we know that
, inverting our former
reasoning for
. Then
|
(16.97) |
I can do no better than quote Jackson on the remainder:
``It is left as an exercise to verify that ...''
|
(16.98) |
(etc.) which is just the explicit full matrix form of
from before.
Now, we have enough information to construct the exact form of a simultaneous
boost and rotation, but this presents a dual problem. When we go to factorize
the results (like before) the components of independent boosts and rotations
do not commute! If you like,
|
(16.101) |
and we cannot say anything trivial like
|
(16.102) |
since it depends on the order they were performed in! Even worse, the product
of two boosts is equal to a single boost and a rotation (if the boosts are not
in the same direction)!
The worst part, of course, is the algebra itself. A useful exercise for the
algebraically inclined might be for someone to construct the general solution
using, e.g. - mathematica.
This suggests that for rotating relativistic systems (such as atoms or
orbits around neutron stars) we may need a kinematic correction to account for
the successive frame changes as the system rotates.
The atom perceives itself as being ``elliptically deformed''. The
consequences of this are observable. This is known as ``Thomas precession''.
Next: Thomas Precession
Up: Generators of the Lorentz
Previous: Generators of the Lorentz
Contents
Robert G. Brown
2007-12-28