Algebra of physical space

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^[0]
_3,1(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}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3},

where the time is given by the scalar part x⁰ = t, and e₁, e₂, e₃ are the standard basis for position space. Throughout, units such that 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

x\rightarrow {\begin{pmatrix}x^{0}+x^{3}&&x^{1}-ix^{2}\\x^{1}+ix^{2}&&x^{0}-x^{3}\end{pmatrix}}

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

L=e^{{\frac {1}{2}}W}.

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

L{\bar {L}}={\bar {L}}L=1.

This Lorentz rotor can be always decomposed in two factors, one Hermitian B = B^†, and the other unitary R^† = R⁻¹, such that

L=BR.

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={\frac {dx}{d\tau }}={\frac {dx^{0}}{d\tau }}+{\frac {d}{d\tau }}(x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3})={\frac {dx^{0}}{d\tau }}\left[1+{\frac {d}{dx^{0}}}(x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3})\right].

This expression can be brought to a more compact form by defining the ordinary velocity as

\mathbf {v} ={\frac {d}{dx^{0}}}(x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3}),

and recalling the definition of the gamma factor:

\gamma (\mathbf {v} )={\frac {1}{\sqrt {1-{\frac { \mathbf {v}  ^{2}}{c^{2}}}}}},

so that the proper velocity is more compactly:

u=\gamma (\mathbf {v} )(1+\mathbf {v} ).

The proper velocity is a positive unimodular paravector, which implies the following condition in terms of the Clifford conjugation

u{\bar {u}}=1.

The proper velocity transforms under the action of the Lorentz rotor L as

u\rightarrow u^{\prime }=LuL^{\dagger }.

Four-momentum paravector

The four-momentum in APS can be obtained by multiplying the proper velocity with the mass as

p=mu,

질량 포탄 조건을 로 환산하여.

{\bar {p}}p=m^{2}.

Classical electrodynamics

The electromagnetic field, potential and current

The electromagnetic field is represented as a bi-paravector F:

F=\mathbf {E} +i\mathbf {B} ,

전기장E를 나타내는 은둔자 부분과 자기장B를 나타내는 반헤르미티아 부분. 표준 Pauli 행렬 표현에서 전자기장은 다음과 같다.

F\rightarrow {\begin{pmatrix}E_{3}&E_{1}-iE_{2}\\E_{1}+iE_{2}&-E_{3}\end{pmatrix}}+i{\begin{pmatrix}B_{3}&B_{1}-iB_{2}\\B_{1}+iB_{2}&-B_{3}\end{pmatrix}}\,.

The source of the field F is the electromagnetic four-current:

j=\rho +\mathbf {j} \,,

여기서 스칼라 부분은 전하 밀도와 같고, 벡터 부분은 전류 밀도j. 다음과 같이 정의된 전자파 전위파라벡터 도입:

A=\phi +\mathbf {A} \,,

스칼라 부분이 전기 전위와 같고 벡터 부분은 자기 전위A와 같다. 그러면 전자기장도 다음과 같다.

F=\partial {\bar {A}}.

그 밭은 전기로 나눌 수 있다.

E=\langle \partial {\bar {A}}\rangle _{V}

자석의

B=i\langle \partial {\bar {A}}\rangle _{BV}

구성 요소들 어디에

\partial =\partial _{t}+\mathbf {e} _{1}\,\partial _{x}+\mathbf {e} _{2}\,\partial _{y}+\mathbf {e} _{3}\,\partial _{z}

F는 폼의 게이지 변환에 따라 불변함

A\rightarrow A+\partial \chi \,,

여기서

\chi

\chi

은

\chi

(는) 스칼라 필드다.

The electromagnetic field is covariant under Lorentz transformations according to the law

F\rightarrow F^{\prime }=LF{\bar {L}}\,.

Maxwell's equations and the Lorentz force

The Maxwell equations can be expressed in a single equation:

{\bar {\partial }}F={\frac {1}{\epsilon }}{\bar {j}}\,,

여기서 막대는 클리포드 결합을 나타낸다.

The Lorentz force equation takes the form

{\frac {dp}{d\tau }}=e\langle Fu\rangle _{R}\,.

Electromagnetic Lagrangian

The electromagnetic Lagrangian is

L={\frac {1}{2}}\langle FF\rangle _{S}-\langle A{\bar {j}}\rangle _{S}\,,

진짜 스칼라 불변제야

Relativistic quantum mechanics

The Dirac equation, for an electrically charged particle of mass m and charge e, takes the form:

i{\bar {\partial }}\Psi \mathbf {e} _{3}+e{\bar {A}}\Psi =m{\bar {\Psi }}^{\dagger },

여기서 e는₃ 임의의 단일 벡터, A는 위와 같은 전자기 파라벡터 전위다. 전자기 상호작용은 잠재적 A 측면에서 최소 결합을 통해 포함되었다.

Classical spinor

The differential equation of the Lorentz rotor that is consistent with the Lorentz force is

{\frac {d\Lambda }{d\tau }}={\frac {e}{2mc}}F\Lambda ,

적절한 속도가 정지 상태에서 적절한 속도의 로렌츠 변환으로 계산되는 경우

u=\Lambda \Lambda ^{\dagger },

시간 궤적

x(\tau )

(

x(\tau )

)

{\displaystyle

x

(\tau )}

을

x(\tau)

(를) 찾기 위해 통합될 수 있다.

{\frac {dx}{d\tau }}=u.

References

Textbooks

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

Articles

Baylis, W E (2004). "Relativity in introductory physics". Canadian Journal of Physics. 82 (11): 853–873. arXiv:physics/0406158. Bibcode:2004CaJPh..82..853B. doi:10.1139/p04-058. S2CID 35027499.
Baylis, W E; Jones, G (7 January 1989). "The Pauli algebra approach to special relativity". Journal of Physics A: Mathematical and General. 22 (1): 1–15. Bibcode:1989JPhA...22....1B. doi:10.1088/0305-4470/22/1/008.
Baylis, W. E. (1 March 1992). "Classical eigenspinors and the Dirac equation". Physical Review A. 45 (7): 4293–4302. Bibcode:1992PhRvA..45.4293B. doi:10.1103/physreva.45.4293. PMID 9907503.
Baylis, W. E.; Yao, Y. (1 July 1999). "Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach". Physical Review A. 60 (2): 785–795. Bibcode:1999PhRvA..60..785B. doi:10.1103/physreva.60.785.

v t Algebra of physical space (APS)
Mathematical concepts	Paravector algebra
Applications in physics	Special relativity with APS Dirac equation with APS

v t Number systems
Countable sets	Natural numbers ( $\mathbb {N}$ ) Integers ( $\mathbb {Z}$ ) Rational numbers ( $\mathbb {Q}$ ) Constructible numbers Algebraic numbers ( $\mathbb {A}$ ) Periods Computable numbers Arithmetical numbers Set-theoretically definable numbers Gaussian integers
Composition algebras	Division algebras: Real numbers ( $\mathbb {R}$ ) Complex numbers ( $\mathbb {C}$ ) Quaternions ( $\mathbb {H}$ ) 옥토니언( $\mathbb {O}$ ${\$ $\mathbb {O}$ )
분할 종류들	$\mathbb {R}$ ${\$ 초과 $\mathbb {R}$ 분할복수 스플릿쿼터니온 스플릿옥톤 $\mathbb {C}$ 이상 ${\$ 바이콤플렉스 수 바이쿼터니온스 바이오크토니언스
기타 하이퍼 복합체	이중수 이중 쿼터니온 이중 복합수 쌍곡선 쿼터니언 Sedenion( $\mathbb {S}$ ${\$ $\mathbb {S}$ ) 스플릿 바이쿼터니온 멀티콤플렉스 수 기하 대수학 물리적 공간의 대수 스페이스타임 대수
기타유형	기수 연장된 자연수 비이성수 퍼지 수 초현실수 레비시비타 필드 초현실수 초월수 서수 p-adic 번호(p-adic 솔레노이드) 초자연적 수 초현실수
분류 리스트

Search