Algebra of physical space

Algebra of physical space (APS)

Mathematical concepts	Paravector algebra

Applications in physics	Special relativity with APS Dirac equation with APS

In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cℓ₃ of the three-dimensional Euclidean space as a model for (3+1)-dimensional space-time, representing a point in space-time via a paravector (3-dimensional vector plus a 1-dimensional scalar).

The Clifford algebra Cℓ₃ has a faithful representation, generated by Pauli matrices, on the spin representation C²; further, Cℓ₃ is isomorphic to the even subalgebra of the 3+1 Clifford algebra, Cℓ0
3,1.

APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics.

APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cℓ_1,3(R) of the four-dimensional Minkowski spacetime.

Special relativity

Space-time position paravector

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 x⁰ = t, and e₁, e₂, e₃ are the standard basis for position space. Throughout, units such that c = 1 are used, called natural units. 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}

Lorentz transformations and rotors

Main articles: Lorentz transformation and Rotor (mathematics)

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 W

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 (special linear group of degree 2 over the complex numbers), 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 B = B^†, and the other unitary R^† = R⁻¹, such that

L = B R

The unitary element R is called a rotor because this encodes rotations, and the Hermitian element B encodes boosts.

Four-velocity paravector

The four-velocity also called proper velocity is defined as the derivative of the space-time position paravector with respect to proper time τ:

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}\left[1 + \frac{d}{d x^0}(x^1 \mathbf{e}_1 + x^2 \mathbf{e}_2 + x^3 \mathbf{e}_3)\right].

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

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

and recalling the definition of the gamma factor:

\gamma(\mathbf{v}) = \frac{1}{\sqrt{1-\frac{|\mathbf{v}|^2}{c^2}}}

so that the proper velocity is more compactly:

u = \gamma(\mathbf{v})(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 L as

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

Four-momentum paravector

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

Classical electrodynamics

Main article: Classical electrodynamics

The electromagnetic field, potential and current

The electromagnetic field is represented as a bi-paravector F:

F = \mathbf{E}+ i \mathbf{B}

with the Hermitian part representing the electric field E and the anti-Hermitian part representing the magnetic field B. In the standard Pauli matrix representation, the electromagnetic field is:

F \rightarrow \begin{pmatrix} E_3 & E_1 -i E_2 \\ E_1 +i E_2 & -E_3 \end{pmatrix} + i \begin{pmatrix} B_3 & B_1 -i B_2 \\ B_1 +i B_2 & -B_3 \end{pmatrix}\,.

The source of the field F is the electromagnetic four-current:

j = \rho + \mathbf{j}\,,

where the scalar part equals the electric charge density ρ, and the vector part the electric current density j. Introducing the electromagnetic potential paravector defined as:

A=\phi+\mathbf{A}\,,

in which the scalar part equals the electric potential ϕ, and the vector part the magnetic potential A. The electromagnetic field is then also:

F=\partial {\bar {A}}.

The field can be split into electric

E=\langle \partial {\bar {A}}\rangle _{V}

and magnetic

B=i\langle \partial {\bar {A}}\rangle _{BV}

components. Where

\partial =\partial _{t}+\mathbf {e} _{1}\,\partial _{x}+\mathbf {e} _{2}\,\partial _{y}+\mathbf {e} _{3}\,\partial _{z}\,.

and F is invariant under a gauge transformation of the form

A \rightarrow A + \partial \chi \,,

where $\chi$ is a scalar field.

The electromagnetic field is covariant under Lorentz transformations according to the law

F \rightarrow F^\prime = L F \bar{L}\,.

Maxwell's equations and the Lorentz force

The Maxwell equations can be expressed in a single equation:

\bar{\partial} F = \frac{1}{ \epsilon} \bar{j}\,,

where the overbar represents the Clifford conjugation.

The Lorentz force equation takes the form

{\frac {dp}{d\tau }}=e\langle Fu\rangle _{V}\,.

Electromagnetic Lagrangian

The electromagnetic Lagrangian is

L = \frac{1}{2} \langle F F \rangle_S - \langle A \bar{j} \rangle_S\,,

which is a real scalar invariant.

Relativistic quantum mechanics

Main article: Relativistic quantum mechanics

The Dirac equation, for an electrically charged particle of mass m and charge e, takes the form:

i \bar{\partial} \Psi\mathbf{e}_3 + e \bar{A} \Psi = m \bar{\Psi}^\dagger

where e₃ is an arbitrary unitary vector, and A is the electromagnetic paravector potential as above. The electromagnetic interaction has been included via minimal coupling in terms of the potential A.

Classical spinor

Main article: spinor

The differential equation of the Lorentz rotor that is consistent with the Lorentz force is

\frac{d \Lambda}{ d \tau} = \frac{e}{2mc} F \Lambda,

such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest

u = \Lambda \Lambda^\dagger,

which can be integrated to find the space-time trajectory $x(\tau)$ with the additional use of

\frac{d x}{ d \tau} = u

References

Textbooks

Baylis, William (2002). Electrodynamics: A Modern Geometric Approach (2nd ed.). Birkhäuser. ISBN 0-8176-4025-8
W. E. Baylis, editor, Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering, Birkhäuser, Boston 1996.
Chris Doran and Anthony Lasenby, Geometric Algebra for Physicists, Cambridge University Press (2003)
David Hestenes: New Foundations for Classical Mechanics (Second Edition). ISBN 0-7923-5514-8, Kluwer Academic Publishers (1999)

Articles

Baylis, William (2002). Relativity in Introductory Physics, Can. J. Phys. 82 (11), 853—873 (2004). (ArXiv:physics/0406158)
W. E. Baylis and G. Jones, The Pauli-Algebra Approach to Special Relativity, J. Phys. A22, 1-16 (1989)
W. E. Baylis, Classical eigenspinors and the Dirac equation, Phys Rev. A, Vol 45, number 7 (1992)
W. E. Baylis, Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach, Phys Rev. A, Vol 60, number 2 (1999)

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