Algebra of Physical Space - Special Relativity

Special Relativity

In APS, the space-time position is represented as a paravector

$x = x^0 + x^1 \mathbf{e}_1 + x^2 \mathbf{e}_2 + x^3 \mathbf{e}_3,$

where the time is given by the scalar part with . In the Pauli matrix representation the unit basis vectors are replaced by the Pauli matrices and the scalar part by the identity matrix. This means that the Pauli matrix representation of the space-time position is

$x \rightarrow \begin{pmatrix} x^0 + x^3 && x^1 - ix^2 \\ x^1 + ix^2 && x^0-x^3 \end{pmatrix}$

The four-velocity also called proper velocity is paravector defined as the proper time derivative of the space-time position

$u = \frac{d x }{d \tau} = \frac{d x^0}{d\tau} + \frac{d}{d\tau}(x^1 \mathbf{e}_1 + x^2 \mathbf{e}_2 + x^3 \mathbf{e}_3) = \frac{d x^0}{d\tau}(1 + \frac{d}{d x^0}(x^1 \mathbf{e}_1 + x^2 \mathbf{e}_2 + x^3 \mathbf{e}_3)).$

This expression can be brought to a more compact form by defining the ordinary velocity as

and recalling the definition of the gamma factor, so that the proper velocity becomes

$u = \gamma(1+ \mathbf{v})$

The proper velocity is a positive unimodular paravector, which implies the following condition in terms of the Clifford conjugation

$u \bar{u} = 1$

The proper velocity transforms under the action of the Lorentz rotor as

$u \rightarrow u^\prime = L u L^\dagger.$

The restricted Lorentz transformations that preserve the direction of time and include rotations and boosts can be performed by an exponentiation of the space-time rotation biparavector

$L = e^{\frac{1}{2}W}$

In the matrix representation the Lorentz rotor is seen to form an instance of the SL(2,C) group, which is the double cover of the Lorentz group. The unimodularity of the Lorentz rotor is translated in the following condition in terms of the product of the Lorentz rotor with its Clifford conjugation

$L\bar{L} = \bar{L} L = 1$

This Lorentz rotor can be always decomposed in two factors, one Hermitian, and the other unitary, such that

$L = B R^{\,}$

The unitary element is called rotor because encodes rotations and the Hermitian element is called boost.

The four-momentum in APS can be obtained by multiplying the proper velocity with the mass as

$p = m u^{\,},$

with the mass shell condition translated into

$\bar{p}p = m^2$

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