|
|
| Line 1: |
Line 1: |
| {{Algebra of Physical Space}}
| | She is recognized by the name of Myrtle Shryock. My working day job is a meter reader. The thing she adores most is body building and now she is attempting to make money with it. Years ago we moved to Puerto Rico and my family members loves it.<br><br>my web-site - at home std test ([http://nuvem.tk/altergalactica/NumbersdbEstepvu this website]) |
| | |
| 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).
| |
| | |
| 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}}.
| |
| | |
| 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]] ''C''ℓ<sub>1,3</sub>('''R''') of the four dimensional [[Minkowski spacetime]].
| |
| | |
| ==Special relativity==
| |
| | |
| ===Space-time position paravector===
| |
| | |
| In APS, the [[space-time]] position is represented as a [[paravector]]
| |
| | |
| :<math>
| |
| 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|''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
| |
| | |
| :<math>
| |
| 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 space-time rotation [[paravector#Biparavectors|biparavector]] ''W''
| |
| | |
| :<math>
| |
| 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 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>
| |
| L\bar{L} = \bar{L} L = 1
| |
| </math>
| |
| | |
| 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
| |
| | |
| :<math>
| |
| 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 space-time position paravector with respect to [[proper time]] ''τ'':
| |
| | |
| :<math>
| |
| 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> \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>
| |
| \gamma(\mathbf{v}) = \frac{1}{\sqrt{1-\frac{|\mathbf{v}|^2}{c^2}}}
| |
| </math>
| |
| | |
| so that the proper velocity is more compactly:
| |
| | |
| :<math>
| |
| 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>
| |
| u \bar{u} = 1
| |
| </math>
| |
| | |
| The proper velocity transforms under the action of the '''Lorentz rotor''' ''L'' as
| |
| | |
| :<math>
| |
| 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>
| |
| p = m u,
| |
| </math> | |
| | |
| with the [[mass shell]] condition translated into
| |
| | |
| :<math>
| |
| \bar{p}p = m^2
| |
| </math>
| |
| | |
| ==Classical electrodynamics==
| |
| | |
| {{main|Classical electrodynamics}}
| |
| | |
| ===The electromagnetic field, potential and current===
| |
| | |
| The [[electromagnetic field]] is represented as a bi-paravector ''F'':
| |
| | |
| :<math> 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> 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>
| |
| 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>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>
| |
| F = \langle \partial \bar{A} \rangle_V \,.
| |
| </math>
| |
| | |
| and ''F'' is invariant under a [[gauge transformation]] of the form
| |
| | |
| :<math>
| |
| 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>
| |
| 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>
| |
| \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>
| |
| \frac{d p}{d \tau} = e \langle F u \rangle_{R}\,.
| |
| </math>
| |
| | |
| ===Electromagnetic Lagrangian===
| |
| | |
| The electromagnetic [[Lagrangian]] is
| |
| | |
| :<math>
| |
| 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> 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>
| |
| \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>
| |
| 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>
| |
| \frac{d x}{ d \tau} = u
| |
| </math>
| |
| | |
| ==See also==
| |
| | |
| * [[Paravector]]
| |
| * [[Multivector]]
| |
| * [[wikibooks:Physics in the Language of Geometric Algebra. An Approach with the Algebra of Physical Space]]
| |
| * [[Dirac equation in the algebra of physical space]]
| |
| | |
| ==References==
| |
| | |
| ===Textbooks===
| |
| | |
| * Baylis, William (2002). ''Electrodynamics: A Modern Geometric Approach'' (2nd ed.). Birkhäuser. ISBN 0-8176-4025-8
| |
| * W. E. Baylis, editor, ''Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering'', Birkhäuser, Boston 1996.
| |
| * Chris Doran and Anthony Lasenby, ''Geometric Algebra for Physicists'', Cambridge University Press (2003)
| |
| * [[David Hestenes]]: New Foundations for Classical Mechanics (Second Edition). ISBN 0-7923-5514-8, Kluwer Academic Publishers (1999)
| |
| | |
| ===Articles===
| |
| | |
| *Baylis, William (2002). ''Relativity in Introductory Physics'', Can. J. Phys. 82 (11), 853—873 (2004). ([http://arxiv.org/pdf/physics/0406158 ArXiv:physics/0406158])
| |
| *W. E. Baylis and G. Jones, ''The Pauli-Algebra Approach to Special Relativity'', J. Phys. A22, 1-16 (1989)
| |
| *W. E. Baylis, ''Classical eigenspinors and the Dirac equation'', Phys Rev. A, Vol 45, number 7 (1992)
| |
| *W. E. Baylis, ''Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach'', Phys Rev. A, Vol 60, number 2 (1999)
| |
| | |
| {{DEFAULTSORT:Algebra Of Physical Space}}
| |
| [[Category:Geometric algebra]]
| |
| [[Category:Clifford algebras]]
| |
| [[Category:Special relativity]]
| |
| [[Category:Electromagnetism]]
| |