Lorentz Group - The Lie Algebra of The Lorentz Group

The Lie Algebra of The Lorentz Group

As with any Lie group, the best way to study many aspects of the Lorentz group is via its Lie algebra. Since the Lorentz group is SO+(1,3), its Lie algebra is reducible and can be decomposed to two copies of the Lie algebra of SL(2,R), as will be shown explicitly below (this is the Minkowski space analog of the SO(4) SU(2)SU(2) decomposition in a Euclidean space). In particle physics, a state that is invariant under one of these copies of SL(2,R) is said to have chirality, and is either left-handed or right-handed, according to which copy of SL(2,R) it is invariant under.

The Lorentz group is a subgroup of the diffeomorphism group of R4 and therefore its Lie algebra can be identified with vector fields on R4. In particular, the vectors which generate isometries on a space are its Killing vectors, which provides a convenient alternative to the left-invariant vector field for calculating the Lie algebra. We can immediately write down an obvious set of six generators:

vector fields on R4 generating three rotations

vector fields on R4 generating three boosts

It may be helpful to briefly recall here how to obtain a one-parameter group from a vector field, written in the form of a first order linear partial differential operator such as

The corresponding initial value problem is

The solution can be written

$\left = \left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos(\lambda) & -\sin(\lambda) & 0 \\ 0 & \sin(\lambda) & \cos(\lambda) & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right] \left$

where we easily recognize the one-parameter matrix group of rotations about the z axis. Differentiating with respect to the group parameter and setting in the result, we recover the matrix

which corresponds to the vector field we started with. This shows how to pass between matrix and vector field representations of elements of the Lie algebra.

Reversing the procedure in the previous section, we see that the Möbius transformations which correspond to our six generators arise from exponentiating respectively (for the three boosts) or (for the three rotations) times the three Pauli matrices

$\sigma_1 = \left, \; \; \sigma_2 = \left, \; \; \sigma_3 = \left.$

For our purposes, another generating set is more convenient. We list the six generators in the following table, in which

the first column gives a generator of the flow under the Möbius action (after stereographic projection from the Riemann sphere) as a real vector field on the Euclidean plane,
the second column gives the corresponding one-parameter subgroup of Möbius transformations,
the third column gives the corresponding one-parameter subgroup of Lorentz transformations (the image under our homomorphism of preceding one-parameter subgroup),
the fourth column gives the corresponding generator of the flow under the Lorentz action as a real vector field on Minkowski spacetime.

Notice that the generators consist of

two parabolics (null rotations),
one hyperbolic (boost in direction),
three elliptics (rotations about x,y,z axes respectively).

Vector field on R2	One-parameter subgroup of SL(2,C), representing Möbius transformations	One-parameter subgroup of SO+(1,3), representing Lorentz transformations
Parabolic
		$\left[ \begin{matrix} 1+\alpha^2/2 & \alpha & 0 & -\alpha^2/2 \\ \alpha & 1 & 0 & -\alpha \\ 0 & 0 & 1 & 0 \\ \alpha^2/2 & \alpha & 0 & 1-\alpha^2/2 \end{matrix} \right]$
		$\left[ \begin{matrix} 1+\alpha^2/2 & 0 & \alpha & -\alpha^2/2 \\ 0 & 1 & 0 & 0 \\ \alpha & 0 & 1 & -\alpha \\ \alpha^2/2 & 0 & \alpha & 1-\alpha^2/2 \end{matrix} \right]$
Hyperbolic
	$\left[ \begin{matrix} \exp \left(\frac{\beta}{2}\right) & 0 \\ 0 & \exp \left(-\frac{\beta}{2}\right) \end{matrix} \right]$	$\left[ \begin{matrix} \cosh(\beta) & 0 & 0 & \sinh(\beta) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \sinh(\beta) & 0 & 0 & \cosh(\beta) \end{matrix} \right]$
Elliptic
	$\left[ \begin{matrix} \exp \left( \frac{i \theta}{2} \right) & 0 \\ 0 & \exp \left( \frac{-i \theta}{2} \right) \end{matrix} \right]$	$\left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) & 0 \\ 0 & \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right]$
	$\left[ \begin{matrix} \cos \left( \frac{\theta}{2} \right) & -\sin \left( \frac{\theta}{2} \right) \\ \sin \left( \frac{\theta}{2} \right) & \cos \left( \frac{\theta}{2} \right) \end{matrix} \right]$	$\left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos(\theta) & 0 & \sin(\theta) \\ 0 & 0 & 1 & 0 \\ 0 & -\sin(\theta) & 0 & \cos(\theta) \end{matrix} \right]$
	$\left[ \begin{matrix} \cos \left( \frac{\theta}{2} \right) & i \sin \left( \frac{\theta}{2} \right) \\ i \sin \left( \frac{\theta}{2} \right) & \cos \left( \frac{\theta}{2} \right) \end{matrix} \right]$	$\left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos(\theta) & -\sin(\theta) \\ 0 & 0 & \sin(\theta) & \cos(\theta) \end{matrix} \right]$

Let's verify one line in this table. Start with

Exponentiate:

$\exp \left( \frac{ i \theta}{2} \, \sigma_2 \right) = \left.$

This element of SL(2,C) represents the one-parameter subgroup of (elliptic) Möbius transformations:

Next,

The corresponding vector field on C (thought of as the image of S2 under stereographic projection) is

Writing, this becomes the vector field on R2

Returning to our element of SL(2,C), writing out the action and collecting terms, we find that the image under the spinor map is the element of SO+(1,3)

$\left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos(\theta) & 0 & \sin(\theta) \\ 0 & 0 & 1 & 0 \\ 0 & -\sin(\theta) & 0 & \cos(\theta) \end{matrix} \right].$

Differentiating with respect to at, we find that the corresponding vector field on R4 is

This is evidently the generator of counterclockwise rotation about the axis.

Read more about this topic: Lorentz Group

Famous quotes containing the words lie, algebra and/or group:

“I could lie down like a tired child,
And weep away the life of care
Which I have borne and yet must bear,
Till death like sleep might steal on me,”
—Percy Bysshe Shelley (1792–1822)

“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)

“No other group in America has so had their identity socialized out of existence as have black women.... When black people are talked about the focus tends to be on black men; and when women are talked about the focus tends to be on white women.”
—bell hooks (b. c. 1955)