Английская Википедия:Algebra of physical space
In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(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 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 ClШаблон:Su(R) of the Clifford algebra Cl3,1(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 Cl1,3(R) of the four-dimensional Minkowski spacetime.
Special relativity
Spacetime position paravector
In APS, the spacetime position is represented as the paravector <math display="block">x = 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 Шаблон:Nowrap, and e1, e2, e3 are the standard basis for position space. Throughout, units such that Шаблон:Nowrap 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 <math display="block"> L = e^{\frac{1}{2}W} .</math>
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 <math display="block">L\bar{L} = \bar{L} L = 1 .</math>
This Lorentz rotor can be always decomposed in two factors, one Hermitian Шаблон:Nowrap, and the other unitary Шаблон:Nowrap, such that <math display="block"> L = B R .</math>
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: <math display="block">
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 \bar{u} = 1 .</math>
The proper velocity transforms under the action of the Lorentz rotor L as <math display="block">u \rightarrow u^\prime = L u L^\dagger.</math>
Four-momentum paravector
The four-momentum in APS can be obtained by multiplying the proper velocity with the mass as <math display="block">p = m u,</math> with the mass shell condition translated into <math display="block"> \bar{p}p = m^2 .</math>
Classical electrodynamics
The 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> 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: <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 \bar{A} .</math> The field can be split into electric <math display="block">E = \langle \partial \bar{A} \rangle_V </math> and magnetic <math display="block">B = i \langle \partial \bar{A} \rangle_{BV} </math> components. Where <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 = L F \bar{L}\,.</math>
Maxwell's equations and the Lorentz force
The Maxwell equations can be expressed in a single equation: <math display="block">\bar{\partial} F = \frac{1}{ \epsilon} \bar{j}\,,</math> where the overbar represents the Clifford conjugation.
The Lorentz force equation takes the form <math display="block">\frac{d p}{d \tau} = e \langle F u \rangle_{R}\,.</math>
Electromagnetic Lagrangian
The electromagnetic Lagrangian is <math display="block">L = \frac{1}{2} \langle F F \rangle_S - \langle A \bar{j} \rangle_S\,,</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 \bar{\partial} \Psi\mathbf{e}_3 + e \bar{A} \Psi = m \bar{\Psi}^\dagger , </math> 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
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 <math display="block">\frac{d x}{ d \tau} = u .</math>
See also
- Paravector
- Multivector
- wikibooks:Physics Using Geometric Algebra
- Dirac equation in the algebra of physical space
- Algebra
References
Textbooks
Articles
Шаблон:Algebra of Physical Space Шаблон:Number systems Шаблон:Industrial and applied mathematics
