View source for Algebra of physical space

{{No footnotes|date=March 2021}}

In [[physics]], the '''algebra of physical space''' ('''APS''') is the use of the [[Clifford algebra|Clifford]] or [[geometric algebra]] Cl<sub>3,0</sub>('''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<sub>3,0</sub>('''R''') has a [[faithful representation]], generated by [[Pauli matrices]], on the [[spin representation]] '''C'''<sup>2</sup>; further, Cl<sub>3,0</sub>('''R''') is isomorphic to the even subalgebra Cl{{su|p=[0]|b=3,1|lh=1em}}('''R''') of the Clifford algebra Cl<sub>3,1</sub>('''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<sub>1,3</sub>('''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 {{nowrap|1=''x''<sup>0</sup> = ''t''}}, and '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> is a [[Basis (linear algebra)|basis]] for position space. Throughout, units such that {{nowrap|1=''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
<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 ===

{{main|Lorentz transformation|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 [[paravector#Biparavectors|biparavector]] ''W'' 
<math display="block"> L = e^{W/2} .</math>

In the matrix representation, the Lorentz rotor is seen to form an instance of the {{nowrap|SL(2, '''C''')}} group ([[special linear group]] of degree 2 over the [[complex number]]s), 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 operator|Hermitian]] {{nowrap|1=''B'' = ''B''<sup>†</sup>}}, and the other [[unitary operator|unitary]] {{nowrap|1=''R''<sup>†</sup> = ''R''<sup>−1</sup>}}, such that
<math display="block"> L = B R .</math>

The unitary element ''R'' is called a [[Rotor (mathematics)|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 [[Lorentz factor|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 matrix|unimodular]] paravector, which implies the following condition in terms of the [[Paravector#Clifford conjugation|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 ==

{{main|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>
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 vector potential|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. 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 [[Lorentz covariance|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 [[paravector#Clifford conjugation|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 (field theory)|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 ==

{{main|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 '''e'''<sub>3</sub> 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|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 ===

* {{cite book |first=William |last=Baylis |title=Electrodynamics: A Modern Geometric Approach |edition=2nd |year=2002 |publisher=Springer |isbn=0-8176-4025-8 }}
* {{cite book |editor-first=William |editor-last=Baylis |title=Clifford (Geometric) Algebras: with applications to physics, mathematics, and engineering |url=https://books.google.com/books?id=0Nji78YQKfQC |date=1999 |orig-year=1996 |publisher=Springer |isbn=978-0-8176-3868-9}}
* {{cite book |first1=Chris |last1=Doran |first2=Anthony |last2=Lasenby |title=Geometric Algebra for Physicists |url=https://books.google.com/books?id=iskgAwAAQBAJ |date=2007 |orig-year=2003 |publisher=Cambridge University Press |isbn=978-1-139-64314-6}}
* {{cite book |author-link=David Hestenes |first=David |last=Hestenes |title=New Foundations for Classical Mechanics |publisher=Kluwer |edition=2nd |year=1999 |isbn=0-7923-5514-8 }}

===Articles===

* {{cite journal | last=Baylis | first=W E | title=Relativity in introductory physics | journal=Canadian Journal of Physics | volume=82 | issue=11 | year=2004 | doi=10.1139/p04-058 | pages=853–873 |arxiv=physics/0406158| bibcode=2004CaJPh..82..853B | s2cid=35027499 }}
* {{cite journal | last1=Baylis | first1=W E | last2=Jones | first2=G | title=The Pauli algebra approach to special relativity | journal=Journal of Physics A: Mathematical and General | volume=22 | issue=1 | date=7 January 1989 | doi=10.1088/0305-4470/22/1/008 | pages=1–15| bibcode=1989JPhA...22....1B }}
* {{cite journal | last=Baylis | first=W. E. | title=Classical eigenspinors and the Dirac equation | journal=Physical Review A | volume=45 | issue=7 | date=1 March 1992 | doi=10.1103/physreva.45.4293 | pages=4293–4302| pmid=9907503 | bibcode=1992PhRvA..45.4293B }}
* {{cite journal | last1=Baylis | first1=W. E. | last2=Yao | first2=Y. | title=Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach | journal=Physical Review A | volume=60 | issue=2 | date=1 July 1999 | doi=10.1103/physreva.60.785 | pages=785–795| bibcode=1999PhRvA..60..785B }}

{{Algebra of Physical Space}}
{{Number systems}}
{{Industrial and applied mathematics}}

[[Category:Mathematical physics]]
[[Category:Geometric algebra]]
[[Category:Clifford algebras]]
[[Category:Special relativity]]
[[Category:Electromagnetism]]