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| {{Algebra of Physical Space}}
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| In [[physics]], the '''algebra of physical space''' (APS) is the use of the [[Clifford algebra|Clifford]] or [[geometric algebra]] ''C''ℓ<sub>3</sub> 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).
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| The Clifford algebra ''C''ℓ<sub>3</sub> has a [[faithful representation]], generated by [[Pauli matrices]], on the [[spin representation]] '''C'''<sup>2</sup>; further, ''C''ℓ<sub>3</sub> is isomorphic to the ''even'' subalgebra of the 3+1 Clifford algebra, ''C''ℓ{{su|p=0|b=3,1}}.
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| APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics.
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| APS should not be confused with [[spacetime algebra]] (STA), which concerns the [[Clifford algebra]] ''C''ℓ<sub>1,3</sub>('''R''') of the four dimensional [[Minkowski spacetime]].
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| ==Special relativity==
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| ===Space-time position paravector===
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| In APS, the [[space-time]] position is represented as a [[paravector]]
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| :<math>
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| x = x^0 + x^1 \mathbf{e}_1 + x^2 \mathbf{e}_2 + x^3 \mathbf{e}_3,
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| </math>
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| where the time is given by the scalar part {{nowrap|''x''<sup>0</sup> {{=}} ''t''}}, and '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> are the [[standard basis]] for position space. Throughout, units such that {{nowrap|''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
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| :<math>
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| x \rightarrow \begin{pmatrix} x^0 + x^3 && x^1 - ix^2 \\ x^1 + ix^2 && x^0-x^3
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| \end{pmatrix}
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| </math>
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| ===Lorentz transformations and rotors===
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| {{main|Lorentz transformation|Rotor (mathematics)}}
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| 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 [[paravector#Biparavectors|biparavector]] ''W''
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| :<math>
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| L = e^{\frac{1}{2}W}
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| </math>
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| 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 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
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| :<math>
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| L\bar{L} = \bar{L} L = 1
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| </math>
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| This Lorentz rotor can be always decomposed in two factors, one [[Hermitian operator|Hermitian]] {{nowrap|''B'' {{=}} ''B''<sup>†</sup>}}, and the other [[unitary operator|unitary]] {{nowrap|''R''<sup>†</sup> {{=}} ''R''<sup>−1</sup>}}, such that
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| :<math>
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| L = B R
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| </math>
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| The unitary element ''R'' is called a [[Rotor (mathematics)|rotor]] because this encodes rotations, and the Hermitian element ''B'' encodes boosts.
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| ===Four-velocity paravector===
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| The [[four-velocity]] also called '''proper velocity''' is defined as the [[derivative]] of the space-time position paravector with respect to [[proper time]] ''τ'':
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| :<math>
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| u = \frac{d x }{d \tau} = \frac{d x^0}{d\tau} +
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| \frac{d}{d\tau}(x^1 \mathbf{e}_1 + x^2 \mathbf{e}_2 + x^3 \mathbf{e}_3) =
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| \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].
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| </math>
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| This expression can be brought to a more compact form by defining the ordinary velocity as
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| :<math> \mathbf{v} = \frac{d}{d x^0}(x^1 \mathbf{e}_1 + x^2 \mathbf{e}_2 + x^3 \mathbf{e}_3) </math>
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| and recalling the definition of the [[Lorentz factor|gamma factor]]:
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| :<math>
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| \gamma(\mathbf{v}) = \frac{1}{\sqrt{1-\frac{|\mathbf{v}|^2}{c^2}}}
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| </math>
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| so that the proper velocity is more compactly:
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| :<math>
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| u = \gamma(\mathbf{v})(1 + \mathbf{v})
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| </math>
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| The proper velocity is a positive [[Unimodular matrix|unimodular]] paravector, which implies the following condition in terms of the [[paravector#Clifford conjugation|Clifford conjugation]]
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| :<math>
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| u \bar{u} = 1
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| </math>
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| The proper velocity transforms under the action of the '''Lorentz rotor''' ''L'' as
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| :<math>
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| u \rightarrow u^\prime = L u L^\dagger.
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| </math> | |
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| ===Four-momentum paravector===
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| The [[four-momentum]] in APS can be obtained by multiplying the proper velocity with the mass as
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| :<math>
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| p = m u,
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| </math> | |
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| with the [[mass shell]] condition translated into
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| :<math>
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| \bar{p}p = m^2
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| </math>
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| ==Classical electrodynamics==
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| {{main|Classical electrodynamics}}
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| ===The electromagnetic field, potential and current===
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| The [[electromagnetic field]] is represented as a bi-paravector ''F'':
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| :<math> F = \mathbf{E}+ i \mathbf{B} </math>
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| 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:
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| :<math> F \rightarrow
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| \begin{pmatrix}
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| E_3 & E_1 -i E_2 \\ E_1 +i E_2 & -E_3
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| \end{pmatrix} + i \begin{pmatrix}
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| B_3 & B_1 -i B_2 \\ B_1 +i B_2 & -B_3
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| \end{pmatrix}\,.
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| </math>
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| The source of the field ''F'' is the electromagnetic [[four-current]]:
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| :<math> | |
| j = \rho + \mathbf{j}\,,
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| </math>
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| 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:
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| :<math>A=\phi+\mathbf{A}\,,</math>
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| in which the scalar part equals the [[electric potential]] ''ϕ'', and the vector part the [[magnetic potential]] '''A'''. The electromagnetic field is then also:
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| :<math>
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| F = \langle \partial \bar{A} \rangle_V \,.
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| </math>
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| and ''F'' is invariant under a [[gauge transformation]] of the form
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| :<math>
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| A \rightarrow A + \partial \chi \,,
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| </math>
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| where <math>\chi</math> is a [[scalar field]].
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| The electromagnetic field is [[Lorentz covariance|covariant]] under Lorentz transformations according to the law
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| :<math>
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| F \rightarrow F^\prime = L F \bar{L}\,.
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| </math>
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| ===Maxwell's equations and the Lorentz force===
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| The [[Maxwell equations]] can be expressed in a single equation:
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| :<math>
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| \bar{\partial} F = \frac{1}{ \epsilon} \bar{j}\,,
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| </math>
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| where the overbar represents the [[paravector#Clifford conjugation|Clifford conjugation]].
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| The [[Lorentz force]] equation takes the form
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| :<math>
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| \frac{d p}{d \tau} = e \langle F u \rangle_{R}\,.
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| </math>
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| ===Electromagnetic Lagrangian=== | |
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| The electromagnetic [[Lagrangian]] is
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| :<math>
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| L = \frac{1}{2} \langle F F \rangle_S - \langle A \bar{j} \rangle_S\,,
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| </math>
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| which is a real scalar invariant.
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| ==Relativistic quantum mechanics==
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| {{main|Relativistic quantum mechanics}}
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| The [[Dirac equation]], for an electrically [[charged particle]] of mass ''m'' and charge ''e'', takes the form:
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| :<math> i \bar{\partial} \Psi\mathbf{e}_3 + e \bar{A} \Psi = m \bar{\Psi}^\dagger </math>,
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| 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''.
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| ==Classical spinor==
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| {{main|spinor}}
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| The [[differential equation]] of the Lorentz rotor that is consistent with the Lorentz force is
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| :<math>
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| \frac{d \Lambda}{ d \tau} = \frac{e}{2mc} F \Lambda,
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| </math>
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| such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest
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| :<math>
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| u = \Lambda \Lambda^\dagger,
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| </math>
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| which can be integrated to find the space-time trajectory <math>x(\tau)</math> with the additional use of
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| :<math>
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| \frac{d x}{ d \tau} = u
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| </math>
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| ==See also==
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| * [[Paravector]]
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| * [[Multivector]]
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| * [[wikibooks:Physics in the Language of Geometric Algebra. An Approach with the Algebra of Physical Space]]
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| * [[Dirac equation in the algebra of physical space]]
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| ==References==
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| ===Textbooks===
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| * Baylis, William (2002). ''Electrodynamics: A Modern Geometric Approach'' (2nd ed.). Birkhäuser. ISBN 0-8176-4025-8
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| * W. E. Baylis, editor, ''Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering'', Birkhäuser, Boston 1996.
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| * Chris Doran and Anthony Lasenby, ''Geometric Algebra for Physicists'', Cambridge University Press (2003)
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| * [[David Hestenes]]: New Foundations for Classical Mechanics (Second Edition). ISBN 0-7923-5514-8, Kluwer Academic Publishers (1999)
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| ===Articles===
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| *Baylis, William (2002). ''Relativity in Introductory Physics'', Can. J. Phys. 82 (11), 853—873 (2004). ([http://arxiv.org/pdf/physics/0406158 ArXiv:physics/0406158])
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| *W. E. Baylis and G. Jones, ''The Pauli-Algebra Approach to Special Relativity'', J. Phys. A22, 1-16 (1989)
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| *W. E. Baylis, ''Classical eigenspinors and the Dirac equation'', Phys Rev. A, Vol 45, number 7 (1992)
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| *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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| {{DEFAULTSORT:Algebra Of Physical Space}}
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| [[Category:Geometric algebra]]
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| [[Category:Clifford algebras]]
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| [[Category:Special relativity]]
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| [[Category:Electromagnetism]]
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