Lorentz Transformations

Now we begin to introduce a little physics. (The form of the multiplication law is already hiding a bit of physics.)

We are going to derive a Lorentz transformation. To be specific, we are going to derive a proper Lorentz transformation, in other words, the transformation can include boosts and rotations, but not inversions.

We may expect that a general transformation can be obtained by the following rule

A' = S A R

We require two elements, S and R, since multiplication on the left is a distinct operation from multiplication on the right.

Using this general form, we first require that the transformation does not change the real-ness, or imaginary-ness of A. We can do this by assuming that A is real, and setting A' equal to the Hermitian conjugate of A'. This condition is stated as

R^† A^† S^† = S A R

Since A is entirely real, this condition can only be satisfied if

R = S^†

Thus, the form of the proper Lorentz transformation must be

A' = S A S^†

The other condition on a Lorentz transformation is that it leaves the space-time interval invariant. The space-time interval is the length of the displacement in space-time. We know how to find such a length - we use a dot product, which has previously been defined for our algebra. In other words, a Lorentz transformation must be invariant to the dot product defined for our algebra.

<A'B'>_S = <AB>_S

We can expand the left hand side of the equation as follows

(1/2) (S A S^†S^† B S + S B S^†S^† A S) = S S(S S)^†<AB>_S

Thus the dot product remains invariant only if the factor involving S is equal to 1. For a proper Lorentz transformation, this is accomplished if

S S = 1

We give the symbol L to such a quantity that satisfies these conditions. The components of L can be parametrized in terms of a direction N and an angle θ.

L = ( cosh(θ/2), N sinh(θ/2) )

In this parametrization θ is allowed to be real, imaginary, or complex. If θ is purely real then it represents the rapidity of a Lorentz boost in the direction of N. If θ is purely imaginary then it represents the rotation angle of a spatial rotation around an axis N. If θ is complex, then there is a combination of boost and rotation, in a screw-like motion.

We can also use an exponential form to describe L. For instance, a general Lorentz transformation can be written as

L = e^{(1/2) (Γ + iΘ)}

Here Γ and Θ are pure vectors that are entirely real, and they represent the 6 generators of Lorentz transformations. Using this form, we can represent the Lorentz transformation as

A' = e^{(1/2) (Γ + iΘ)} A e^{(1/2) (Γ - iΘ)}

This form is reminiscent of the group theoretic operator approach that is used in quantum mechanics. This will not be the last time we are reminded of quantum mechanics...