Algebra of physical space

In physics, the name "algebra of physical space" (APS) originally stems from the use of the Clifford or geometric algebra Cl3,0(R), also written <math>\mathbb{G}_3</math> or <math>\mathbb{R}_3</math>, of three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-dimensional vector plus a 1-dimensional scalar). Although, recent research has adopted the name "APS" as a synonym for Cl3,0(R) in general contexts.

The Clifford algebra Cl3,0(R) has a faithful representation, generated by Pauli matrices, on the spin representation C2; further, Cl3,0(R) is isomorphic to the even subalgebra Cl(R) (also <math>\mathbb{G}_{3,1}^+</math>) of the Clifford algebra Cl3,1(R) (also <math>\mathbb{G}_{3,1}</math>), and to the even subalgebra Cl(R) (also <math>\mathbb{G}_{1,3}^+</math>) of the spacetime algebra Cl1,3(R) (also <math>\mathbb{G}_{1,3}</math>).

The APS can be used to construct a compact, unified, and geometrical formalism for both classical and quantum mechanics. This blurs the line between what is traditionally considered classical or quantum.

The APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cl1,3(R) of the four-dimensional Minkowski spacetime.

Involution notation

All Clifford or geometric algebras have three main involutions: grade involution, reversion, and Clifford conjugation.

If <math>g\in\mathbb{G}_3</math> is an arbitrary multivector and <math>\langle g\rangle_j</math> projects <math>g</math> onto its grade-j subspace <math>\mathbb{G}_3^j</math> of j-vectors, then the grade involution in the APS is defined as <math display="block">g^-=\langle g\rangle_0-\langle g\rangle_1+\langle g\rangle_2-\langle g\rangle_3.</math>

In the APS, grade involution may be called parity conjugation as it is generated by the STA's definition of parity conjugation <math>g^-=\gamma_0 g\gamma_0</math> in tandem with the isomorphism <math>\mathbb{G}_3\approx\mathbb{G}_{1,3}^+</math>. The notation for grade involution in the APS is not a settled matter, and is also denoted by <math>\hat{g}</math> or <math>\overline{g}^\dagger</math>.

For an additional multivector <math>h\in\mathbb{G}_3</math>, the reversion (also reverse conjugate) in the APS is defined by <math display="block">(gh)^\dagger=h^\dagger g^\dagger</math> and <math display="block">g^\dagger = \langle g\rangle_0+\langle g\rangle_1-\langle g\rangle_2-\langle g\rangle_3.</math>

In the APS, reversion may be called Hermitian conjugation as it is completely equivalent through the Pauli matrix representation of the APS, and is generated by the STA's definition of Hermitian conjugation <math>g^\dagger = \gamma_0\widetilde{g}\gamma_0</math> in tandem with the isomorphism <math>\mathbb{G}_3\approx\mathbb{G}_{1,3}^+</math>.

The final involution, Clifford conjugation, is defined by <math display="block">\widetilde{gh}=\widetilde{h}\widetilde{g}</math> and <math display="block">\widetilde{g}=(g^-)^\dagger=(g^\dagger)^-=\langle g\rangle_0-\langle g\rangle_1-\langle g\rangle_2+\langle g\rangle_3.</math>

In the APS, reversion may be called spacetime reversion as via the isomorphism <math>\mathbb{G}_3\approx\mathbb{G}_{1,3}^+</math>, reversion within the STA is identical to Clifford conjugation within the APS. The above tilde notation is more recent and was adopted to emphasize this relationship.

Special relativity

Spacetime position paravector

In the APS, the spacetime position is represented as the paravector

<math display="block">x = x^\mu\mathbf{e}_\mu = x^0 + x^1 \mathbf{e}_1 + x^2 \mathbf{e}_2 + x^3 \mathbf{e}_3,</math>

where the time is given by the scalar part , e0=1, and {e1, e2, e3} is the standard orthonormal basis for position space. Throughout the remainder of this article and unless stated otherwise, units such that 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

<math display="block">x \rightarrow \begin{pmatrix} x^0 + x^3 && x^1 - ix^2 \\ x^1 + ix^2 && x^0-x^3\end{pmatrix}</math>

Lorentz transformations and rotors

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

In the matrix representation, the Lorentz rotor is seen to form an instance of the 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

<math display="block">\Lambda\widetilde{\Lambda} = \widetilde{\Lambda} \Lambda = 1 .</math>

This Lorentz rotor can be always decomposed in two factors, one Hermitian (a Lorentz boost), and the other unitary (a 3-dimensional rotation), such that

<math display="block"> \Lambda = L R .</math>

The unitary element R is called a rotor because this encodes rotations, and the Hermitian element L encodes boosts. The total object <math>\Lambda</math> is called a Lorentz rotor.

Four-velocity paravector

The four-velocity (also proper velocity or spacetime velocity) is defined as the derivative of the spacetime 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].

</math>

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

<math display="block"> \mathbf{v} = \frac{d}{d x^0}(x^1 \mathbf{e}_1 + x^2 \mathbf{e}_2 + x^3 \mathbf{e}_3) ,</math>

and recalling the definition of the gamma factor:

<math display="block">\gamma(\mathbf{v}) = \frac{1}{\sqrt{1-\frac{|\mathbf{v}|^2}{c^2} ,</math>

so that the proper velocity is more compactly:

<math display="block">u = \gamma(\mathbf{v})(1 + \mathbf{v}).</math>

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

<math display="block">u \widetilde{u} = 1 .</math>

The proper velocity transforms under the action of the Lorentz rotor <math>\Lambda</math> as

<math display="block">u \rightarrow u^\prime = \Lambda u \Lambda^\dagger.</math>

This transformation law can be easily derived from the isomorphism between the APS and the even subalgebra of the STA.

Four-momentum paravector

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

with the mass shell condition translated into

<math display="block"> \widetilde{p}p = p\widetilde{p} = pp^-= p^-p= m^2 .</math>

The proper velocity u may be represented as the Lorentz transformation of the rest velocity 1: <math display="block">u = \Lambda\Lambda^\dagger.</math> This implies that the spacetime momentum can likewise be written as the Lorentz transformation of the rest momentum m, <math display="block">p = \Lambda m\Lambda^\dagger.</math> This trivial rewrite also connects the APS to other areas of Physics; namely helicity-spinor methods for scattering amplitudes and for the Constructive Standard Model (CSM).

Classical electrodynamics

Electromagnetic field, potential, and current

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

<math display="block"> F = \mathbf{E}+ i \mathbf{B} ,</math>

where the Hermitian part gives the electric field E, the anti-Hermitian part gives the magnetic field B, and <math>i=\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3</math> is the unit pseudoscalar. In the standard Pauli matrix representation, the electromagnetic field is:

<math display="block"> 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}\,.

</math>

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

<math display="block">J = \rho + \mathbf{J}\,,</math>

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:

<math display="block">A=\phi+\mathbf{A}\,,</math>

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

<math display="block">F = \partial \widetilde{A} .</math>

The field can be split into electric

<math display="block">E = \langle \partial \widetilde{A} \rangle_1 </math>

and magnetic

<math display="block">B = i \langle \partial \widetilde{A} \rangle_2 </math>

components. Here,

<math display="block"> \partial = \partial_t + \mathbf{e}_1 \, \partial_x + \mathbf{e}_2 \, \partial_y + \mathbf{e}_3 \, \partial_z</math>

and F is invariant under a gauge transformation of the form

<math display="block">A \rightarrow A + \partial \chi \,,</math>

where <math>\chi</math> is a scalar field.

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

<math display="block">F \rightarrow F^\prime = \Lambda F \widetilde{\Lambda}\,.</math>

This transformation law can be easily derived from the isomorphism between the APS and the even subalgebra of the STA.

Maxwell's equations and the Lorentz force

The Maxwell equations can be expressed in a single equation:

<math display="block">\widetilde{\partial} F = \frac{1}{ \varepsilon_0} \widetilde{J}\,.</math>

The Lorentz force equation takes the form

<math display="block">\frac{d p}{d \tau} = e \langle F u \rangle_{0\oplus1}=e \langle F u \rangle_{0}+e \langle F u \rangle_{1}\,.</math>

Electromagnetic Lagrangian

The electromagnetic Lagrangian is

<math display="block">L = \frac{1}{2} \langle F F \rangle_{0\oplus3} - \langle A \widetilde{J} \rangle_{0\oplus3}\,,</math>

which is a real scalar invariant.

Relativistic quantum mechanics

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

<math display="block"> i \widetilde{\partial} \Psi\mathbf{e}_3 + e \widetilde{A} \Psi = m \Psi^- , </math>

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

Lorentz rotor & velocity

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

<math display="block">\frac{d \Lambda}{ d \tau} = \frac{e}{2mc} F \Lambda,</math>

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

<math display="block">u = \Lambda \Lambda^\dagger,</math>

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