Algebra of physical space explained

In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl_3,0(R) of the 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).

The Clifford algebra Cl_3,0(R) has a faithful representation, generated by Pauli matrices, on the spin representation C²; further, Cl_3,0(R) is isomorphic to the even subalgebra Cl(R) of the Clifford algebra Cl_3,1(R).

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

Spacetime position paravector

In APS, the spacetime position is represented as the paravector$$x\; =\; x^0\; +\; x^1\; \backslash mathbf\_1\; +\; x^2\; \backslash mathbf\_2\; +\; x^3\; \backslash mathbf\_3,$$where the time is given by the scalar part, and e₁, e₂, e₃ are the standard basis for position space. Throughout, 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

Lorentz transformations and rotors

See main article: 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 spacetime rotation biparavector W $$L\; =\; e^\; .$$

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$$L\backslash bar\; =\; \backslash bar\; L\; =\; 1\; .$$

This Lorentz rotor can be always decomposed in two factors, one Hermitian, and the other unitary, such that$$L\; =\; B\; R\; .$$

The unitary element 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 spacetime position paravector with respect to proper time τ:$$u\; =\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac(x^1\; \backslash mathbf\_1\; +\; x^2\; \backslash mathbf\_2\; +\; x^3\; \backslash mathbf\_3)\; =\; \backslash frac\backslash left[1\; +\; \backslash frac\{d\}\{d\; x^0\}(x^1\; \backslash mathbf\{e\}\_1\; +\; x^2\; \backslash mathbf\{e\}\_2\; +\; x^3\; \backslash mathbf\{e\}\_3)\backslash right].$$

This expression can be brought to a more compact form by defining the ordinary velocity as $$\backslash mathbf\; =\; \backslash frac(x^1\; \backslash mathbf\_1\; +\; x^2\; \backslash mathbf\_2\; +\; x^3\; \backslash mathbf\_3),$$and recalling the definition of the gamma factor:$$\backslash gamma(\backslash mathbf)\; =\; \backslash frac,$$so that the proper velocity is more compactly:$$u\; =\; \backslash gamma(\backslash mathbf)(1\; +\; \backslash mathbf).$$

The proper velocity is a positive unimodular paravector, which implies the following condition in terms of the Clifford conjugation$$u\; \backslash bar\; =\; 1\; .$$

The proper velocity transforms under the action of the Lorentz rotor L as$$u\; \backslash rightarrow\; u^\backslash prime\; =\; L\; u\; L^\backslash dagger.$$

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$$\backslash barp\; =\; m^2\; .$$

Classical electrodynamics

See main article: Classical electrodynamics.

Electromagnetic field, potential, and current

The electromagnetic field is represented as a bi-paravector F: $$F\; =\; \backslash mathbf+\; i\; \backslash mathbf,$$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:

The source of the field F is the electromagnetic four-current:$$j\; =\; \backslash rho\; +\; \backslash mathbf\backslash ,,$$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:$$A=\backslash phi+\backslash mathbf\backslash ,,$$in which the scalar part equals the electric potential ϕ, and the vector part the magnetic potential A. The electromagnetic field is then also:$$F\; =\; \backslash partial\; \backslash bar\; .$$The field can be split into electric$$E\; =\; \backslash langle\; \backslash partial\; \backslash bar\; \backslash rangle\_V$$and magnetic $$B\; =\; i\; \backslash langle\; \backslash partial\; \backslash bar\; \backslash rangle\_$$components. Here,$$\backslash partial\; =\; \backslash partial\_t\; +\; \backslash mathbf\_1\; \backslash ,\; \backslash partial\_x\; +\; \backslash mathbf\_2\; \backslash ,\; \backslash partial\_y\; +\; \backslash mathbf\_3\; \backslash ,\; \backslash partial\_z$$and F is invariant under a gauge transformation of the form$$A\; \backslash rightarrow\; A\; +\; \backslash partial\; \backslash chi\; \backslash ,,$$where

\chi is a scalar field.

The electromagnetic field is covariant under Lorentz transformations according to the law$$F\; \backslash rightarrow\; F^\backslash prime\; =\; L\; F\; \backslash bar\backslash ,.$$

Maxwell's equations and the Lorentz force

The Maxwell equations can be expressed in a single equation:$$\backslash bar\; F\; =\; \backslash frac\; \backslash bar\backslash ,,$$where the overbar represents the Clifford conjugation.

The Lorentz force equation takes the form$$\backslash frac\; =\; e\; \backslash langle\; F\; u\; \backslash rangle\_\backslash ,.$$

Electromagnetic Lagrangian

The electromagnetic Lagrangian is$$L\; =\; \backslash frac\; \backslash langle\; F\; F\; \backslash rangle\_S\; -\; \backslash langle\; A\; \backslash bar\; \backslash rangle\_S\backslash ,,$$which is a real scalar invariant.

Relativistic quantum mechanics

See main article: Relativistic quantum mechanics.

The Dirac equation, for an electrically charged particle of mass m and charge e, takes the form:$$i\; \backslash bar\; \backslash Psi\backslash mathbf\_3\; +\; e\; \backslash bar\; \backslash Psi\; =\; m\; \backslash bar^\backslash dagger,$$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

See main article: Spinor.

The differential equation of the Lorentz rotor that is consistent with the Lorentz force is$$\backslash frac\; =\; \backslash frac\; F\; \backslash Lambda,$$such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest$$u\; =\; \backslash Lambda\; \backslash Lambda^\backslash dagger,$$which can be integrated to find the space-time trajectory

x(\tau) with the additional use of $$\backslash frac\; =\; u\; .$$

See also

• Paravector
• Multivector
  • Dirac equation in the algebra of physical space
• Algebra

References

Textbooks

• Book: Baylis, William . Electrodynamics: A Modern Geometric Approach . 2nd . 2002 . 0-8176-4025-8 .
• Book: William . Baylis . Clifford (Geometric) Algebras: with applications to physics, mathematics, and engineering . 1999 . 1996 . Springer . 978-0-8176-3868-9.
• Book: Chris . Doran . Anthony . Lasenby . Geometric Algebra for Physicists . 2007 . 2003 . Cambridge University Press . 978-1-139-64314-6.
• Book: Hestenes, David . David Hestenes . New Foundations for Classical Mechanics . Kluwer . 2nd . 1999 . 0-7923-5514-8 .

Articles=

• Baylis . W E . Relativity in introductory physics . Canadian Journal of Physics . 82 . 11 . 2004 . 10.1139/p04-058 . 853–873 . physics/0406158. 2004CaJPh..82..853B . 35027499 .
• Baylis . W E . Jones . G . The Pauli algebra approach to special relativity . Journal of Physics A: Mathematical and General . 22 . 1 . 7 January 1989 . 10.1088/0305-4470/22/1/008 . 1–15. 1989JPhA...22....1B .
• Baylis . W. E. . Classical eigenspinors and the Dirac equation . Physical Review A . 45 . 7 . 1 March 1992 . 10.1103/physreva.45.4293 . 4293–4302. 9907503 . 1992PhRvA..45.4293B .
• Baylis . W. E. . Yao . Y. . Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach . Physical Review A . 60 . 2 . 1 July 1999 . 10.1103/physreva.60.785 . 785–795. 1999PhRvA..60..785B .