Spacetime algebra

Setting of relativistic physics in geometric algebra

In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) to physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell equation and General Relativity" and "reduces the mathematical divide between classical, quantum and relativistic physics."[1]: ix

Spacetime algebra is a vector space that allows not only vectors, but also bivectors (directed quantities describing rotations associated with rotations or particular planes, such as areas, or rotations) or blades (quantities associated with particular hyper-volumes) to be combined, as well as rotated, reflected, or Lorentz boosted.[2]: 40, 43, 97, 113 It is also the natural parent algebra of spinors in special relativity.[2]: 333 These properties allow many of the most important equations in physics to be expressed in particularly simple forms, and can be very helpful towards a more geometric understanding of their meanings.[1]: v

In comparison to related methods, STA and Dirac algebra are both Clifford Cl1,3 algebras, but STA uses real number scalars while Dirac algebra uses complex number scalars. The STA spacetime split is similar to the algebra of physical space (APS, Pauli algebra) approach. APS represents spacetime as a paravector, a combined 3-dimensional vector space and a 1-dimensional scalar.[3]: 225–266

## Structure

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For any pair of STA vectors,  a , b {\textstyle a,b} , there is a vector (geometric) product a b {\textstyle ab} , inner (dot) product a ⋅ b {\textstyle a\cdot b} and outer (exterior, wedge) product a ∧ b {\textstyle a\wedge b} . The vector product is a sum of an inner and outer product:[1]: 6

a ⋅ b = a b + b a 2 = b ⋅ a , a ∧ b = a b − b a 2 = − b ∧ a , a b = a ⋅ b + a ∧ b {\displaystyle a\cdot b={\frac {ab+ba}{2}}=b\cdot a,\quad a\wedge b={\frac {ab-ba}{2}}=-b\wedge a,\quad ab=a\cdot b+a\wedge b}

The inner product generates a real number (scalar), and the outer product generates a bivector. The vectors  a {\textstyle a} and  b {\textstyle b} are orthogonal if their inner product is zero; vectors  a {\textstyle a} and  b {\textstyle b} are parallel if their outer product is zero.[2]: 22–23

The orthonormal basis vectors are a timelike vector  γ 0 {\textstyle \gamma _{0}} and 3 spacelike vectors  γ 1 , γ 2 , γ 3 {\textstyle \gamma _{1},\gamma _{2},\gamma _{3}} . The Minkowski metric tensor's nonzero terms are the diagonal terms,  ( η 00 , η 11 , η 22 , η 33 ) = ( 1 , − 1 , − 1 , − 1 ) {\textstyle (\eta _{00},\eta _{11},\eta _{22},\eta _{33})=(1,-1,-1,-1)} . For  μ , ν = 0 , 1 , 2 , 3 {\textstyle \mu ,\nu =0,1,2,3} :

γ μ ⋅ γ ν = γ μ γ ν + γ ν γ μ 2 = η μ ν , γ 0 ⋅ γ 0 = 1 , γ 1 ⋅ γ 1 = γ 2 ⋅ γ 2 = γ 3 ⋅ γ 3 = − 1 , otherwise γ μ γ ν = − γ ν γ μ {\displaystyle \gamma _{\mu }\cdot \gamma _{\nu }={\frac {\gamma _{\mu }\gamma _{\nu }+\gamma _{\nu }\gamma _{\mu }}{2}}=\eta _{\mu \nu },\quad \gamma _{0}\cdot \gamma _{0}=1,\ \gamma _{1}\cdot \gamma _{1}=\gamma _{2}\cdot \gamma _{2}=\gamma _{3}\cdot \gamma _{3}=-1,\quad {\text{ otherwise }}\ \gamma _{\mu }\gamma _{\nu }=-\gamma _{\nu }\gamma _{\mu }}

The Dirac matrices share these properties, and STA is equivalent to the algebra generated by the Dirac matrices over the field of real numbers;[1]: x explicit matrix representation is unnecessary for STA.

Products of the basis vectors generate a tensor basis containing one scalar  { 1 } {\displaystyle \\{1\\}} , four vectors  { γ 0 , γ 1 , γ 2 , γ 3 } {\displaystyle \\{\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}\\}} , six bivectors  { γ 0 γ 1 , γ 0 γ 2 , γ 0 γ 3 , γ 1 γ 2 , γ 2 γ 3 , γ 3 γ 1 } {\displaystyle \\{\gamma _{0}\gamma _{1},\,\gamma _{0}\gamma _{2},\,\gamma _{0}\gamma _{3},\,\gamma _{1}\gamma _{2},\,\gamma _{2}\gamma _{3},\,\gamma _{3}\gamma _{1}\\}} , four pseudovectors (trivectors)  { I γ 0 , I γ 1 , I γ 2 , I γ 3 } {\displaystyle \\{I\gamma _{0},I\gamma _{1},I\gamma _{2},I\gamma _{3}\\}} and one pseudoscalar { I } {\displaystyle \\{I\\}} with  I = γ 0 γ 1 γ 2 γ 3 {\textstyle I=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}} .[1]: 11 The pseudoscalar commutes with all even-grade STA elements, but anticommutes with all odd-grade STA elements.[4]: 6

## Subalgebra

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STA's even-graded elements (scalars, bivectors, pseudoscalar) form a Clifford Cl3,0(R) even subalgebra equivalent to the APS or Pauli algebra.[1]: 12 The STA bivectors are equivalent to the APS vectors and pseudovectors. The STA subalgebra becomes more explicit by renaming the STA bivectors  ( γ 1 γ 0 , γ 2 γ 0 , γ 3 γ 0 ) {\textstyle (\gamma _{1}\gamma _{0},\gamma _{2}\gamma _{0},\gamma _{3}\gamma _{0})} as  ( σ 1 , σ 2 , σ 3 ) {\textstyle (\sigma _{1},\sigma _{2},\sigma _{3})} and the STA bivectors  ( γ 3 γ 2 , γ 1 γ 3 , γ 2 γ 1 ) {\textstyle (\gamma _{3}\gamma _{2},\gamma _{1}\gamma _{3},\gamma _{2}\gamma _{1})} as  ( I σ 1 , I σ 2 , I σ 3 ) {\textstyle (I\sigma _{1},I\sigma _{2},I\sigma _{3})} .[1]: 22 [2]: 37 The Pauli matrices,  σ ^ 1 , σ ^ 2 , σ ^ 3 {\textstyle {\hat {\sigma }}_{1},{\hat {\sigma }}_{2},{\hat {\sigma }}_{3}} , are a matrix representation for  σ 1 , σ 2 , σ 3 {\textstyle \sigma _{1},\sigma _{2},\sigma _{3}} .[2]: 37 For any pair of  ( σ 1 , σ 2 , σ 3 ) {\textstyle (\sigma _{1},\sigma _{2},\sigma _{3})} , the nonzero inner products are  σ 1 ⋅ σ 1 = σ 2 ⋅ σ 2 = σ 3 ⋅ σ 3 = 1 {\textstyle \sigma _{1}\cdot \sigma _{1}=\sigma _{2}\cdot \sigma _{2}=\sigma _{3}\cdot \sigma _{3}=1} , and the nonzero outer products are:[2]: 37 [1]: 16

σ 1 ∧ σ 2 = I σ 3 σ 2 ∧ σ 3 = I σ 1 σ 3 ∧ σ 1 = I σ 2 {\displaystyle {\begin{aligned}\sigma _{1}\wedge \sigma _{2}&=I\sigma _{3}\\\\\sigma _{2}\wedge \sigma _{3}&=I\sigma _{1}\\\\\sigma _{3}\wedge \sigma _{1}&=I\sigma _{2}\\\\\end{aligned}}}

The sequence of algebra to even subalgebra continues as algebra of physical space, quaternion algebra, complex numbers and real numbers.[1]: 12

## Division

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A nonzero vector  a {\textstyle a} is a null vector (degree 2 nilpotent) if  a 2 = 0 {\textstyle a^{2}=0} .[5]: 2 An example is  a = γ 0 + γ 1 {\textstyle a=\gamma ^{0}+\gamma ^{1}} . Null vectors are tangent to the light cone (null cone).[5]: 4 An element  b {\textstyle b} is an idempotent if  b 2 = b {\textstyle b^{2}=b} .[6]: 103 Two idempotents  b 1 {\textstyle b_{1}} and  b 2 {\textstyle b_{2}} are orthogonal idempotents if  b 1 b 2 = 0 {\textstyle b_{1}b_{2}=0} .[6]: 103 An example of an orthogonal idempotent pair is  1 2 ( 1 + γ 0 γ k ) {\displaystyle {\tfrac {1}{2}}(1+\gamma _{0}\gamma _{k})} and  1 2 ( 1 − γ 0 γ k ) {\displaystyle {\tfrac {1}{2}}(1-\gamma _{0}\gamma _{k})} with  k = 1 , 2 , 3 {\textstyle k=1,2,3} . Proper zero divisors are nonzero elements whose product is zero such as null vectors or orthogonal idempotents.[7]: 191 A division algebra is an algebra that contains multiplicative inverse (reciprocal) elements for every element, but this occurs if there are no proper zero divisors and if the only idempotent is 1.[6]: 103 [8]: 211 [a] The only associative division algebras are the real numbers, complex numbers and quaternions.[9]: 366 As STA is not a division algebra, some STA elements may lack an inverse; however, division by the non-null vector  c {\textstyle c} may be possible by multiplication by its inverse, defined as  c − 1 = ( c ⋅ c ) − 1 c {\displaystyle c^{-1}=(c\cdot c)^{-1}c} .[10]: 14

## Reciprocal frame

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Associated with the orthogonal basis  { γ 0 , γ 1 , γ 2 , γ 3 } {\displaystyle \\{\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}\\}} is the reciprocal basis set  { γ 0 , γ 1 , γ 2 , γ 3 } {\displaystyle \\{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\\}} satisfying these equations: [1]: 63

γ μ ⋅ γ ν = δ μ ν , μ , ν = 0 , 1 , 2 , 3 {\displaystyle \gamma _{\mu }\cdot \gamma ^{\nu }=\delta _{\mu }^{\nu },\quad \mu ,\nu =0,1,2,3}

These reciprocal frame vectors differ only by a sign, with  γ 0 = γ 0 {\displaystyle \gamma ^{0}=\gamma _{0}} , but  γ 1 = − γ 1 , γ 2 = − γ 2 , γ 3 = − γ 3 {\displaystyle \gamma ^{1}=-\gamma _{1},\ \ \gamma ^{2}=-\gamma _{2},\ \ \gamma ^{3}=-\gamma _{3}} .

A vector  a {\textstyle a} may be represented using either the basis vectors or the reciprocal basis vectors  a = a μ γ μ = a μ γ μ {\displaystyle a=a^{\mu }\gamma _{\mu }=a_{\mu }\gamma ^{\mu }} with summation over  μ = 0 , 1 , 2 , 3 {\displaystyle \mu =0,1,2,3} , according to the Einstein notation. The inner product of vector and basis vectors or reciprocal basis vectors generates the vector components.

a ⋅ γ ν = a ν , ν = 0 , 1 , 2 , 3 a ⋅ γ ν = a ν , ν = 0 , 1 , 2 , 3 {\displaystyle {\begin{aligned}a\cdot \gamma ^{\nu }&=a^{\nu },\quad \nu =0,1,2,3\\\a\cdot \gamma _{\nu }&=a_{\nu },\quad \nu =0,1,2,3\end{aligned}}}

The metric and index gymnastics raise or lower indices:

γ μ = η μ ν γ ν , μ , ν = 0 , 1 , 2 , 3 γ μ = η μ ν γ ν , μ , ν = 0 , 1 , 2 , 3 {\displaystyle {\begin{aligned}\gamma _{\mu }&=\eta _{\mu \nu }\gamma ^{\nu },\quad \mu ,\nu =0,1,2,3\\\\\gamma ^{\mu }&=\eta ^{\mu \nu }\gamma _{\nu },\quad \mu ,\nu =0,1,2,3\end{aligned}}}

## Spacetime gradient

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The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied:[11]: 45

a ⋅ ∇ F ( x ) = lim τ -> 0 F ( x + a τ ) − F ( x ) τ . {\displaystyle a\cdot \nabla F(x)=\lim _{\tau \rightarrow 0}{\frac {F(x+a\tau )-F(x)}{\tau }}.}

This requires the definition of the gradient to be

∇ = γ μ ∂ ∂ x μ = γ μ ∂ μ . {\displaystyle \nabla =\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}=\gamma ^{\mu }\partial _{\mu }.}

Written out explicitly with  x = c t γ 0 + x k γ k {\displaystyle x=ct\gamma _{0}+x^{k}\gamma _{k}} , these partials are

∂ 0 = 1 c ∂ ∂ t , ∂ k = ∂ ∂ x k {\displaystyle \partial _{0}={\frac {1}{c}}{\frac {\partial }{\partial t}},\quad \partial _{k}={\frac {\partial }{\partial {x^{k}}}}}

## Spacetime split

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Spacetime split – examples:

x γ 0 = x 0 + x {\displaystyle x\gamma _{0}=x^{0}+\mathbf {x} }
p γ 0 = E + p {\displaystyle p\gamma _{0}=E+\mathbf {p} } [12]: 257
v γ 0 = γ ( 1 + v ) {\displaystyle v\gamma _{0}=\gamma (1+\mathbf {v} )} [12]: 257
where  γ {\displaystyle \gamma } is the Lorentz factor
∇ γ 0 = ∂ t − ∇ -> {\displaystyle \nabla \gamma _{0}=\partial _{t}-{\vec {\nabla }}} [12]: 259

In STA, a spacetime split is a projection from four-dimensional space into (3+1)-dimensional space in a chosen reference frame by means of the following two operations:

* a collapse of the chosen time axis, yielding a 3-dimensional space spanned by bivectors, equivalent to the standard 3-dimensional basis vectors in the algebra of physical space and
* a projection of the 4D space onto the chosen time axis, yielding a 1-dimensional space of scalars, representing the scalar time.[13]: 180

This is achieved by pre-multiplication or post-multiplication by a timelike basis vector  γ 0 {\displaystyle \gamma _{0}} , which serves to split a four vector into a scalar timelike and a bivector spacelike component, in the reference frame co-moving with  γ 0 {\displaystyle \gamma _{0}} . With  x = x μ γ μ {\displaystyle x=x^{\mu }\gamma _{\mu }} we have

x γ 0 = x 0 + x k γ k γ 0 γ 0 x = x 0 − x k γ k γ 0 {\displaystyle {\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\gamma _{k}\gamma _{0}\\\\\gamma _{0}x&=x^{0}-x^{k}\gamma _{k}\gamma _{0}\end{aligned}}}

Spacetime split is a method for representing an even-graded vector of spacetime as a vector in the Pauli algebra, an algebra where time is a scalar separated from vectors that occur in 3 dimensional space. The method replaces these spacetime vectors  ( γ ) {\textstyle (\gamma )} [1]: 22–24

As these bivectors  γ k γ 0 {\displaystyle \gamma _{k}\gamma _{0}} square to unity, they serve as a spatial basis. Utilizing the Pauli matrix notation, these are written  σ k = γ k γ 0 {\displaystyle \sigma _{k}=\gamma _{k}\gamma _{0}} . Spatial vectors in STA are denoted in boldface; then with  x = x k σ k {\displaystyle \mathbf {x} =x^{k}\sigma _{k}} and  x 0 = c t {\displaystyle x^{0}=ct} , the  γ 0 {\displaystyle \gamma _{0}} -spacetime split  x γ 0 {\displaystyle x\gamma _{0}} , and its reverse  γ 0 x {\displaystyle \gamma _{0}x} are:

x γ 0 = x 0 + x k σ k = c t + x γ 0 x = x 0 − x k σ k = c t − x {\displaystyle {\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\sigma _{k}=ct+\mathbf {x} \\\\\gamma _{0}x&=x^{0}-x^{k}\sigma _{k}=ct-\mathbf {x} \end{aligned}}}

However, the above formulas only work in the Minkowski metric with signature (+ - - -). For forms of the spacetime split that work in either signature, alternate definitions in which  σ k = γ k γ 0 {\displaystyle \sigma _{k}=\gamma _{k}\gamma ^{0}} and  σ k = γ 0 γ k {\displaystyle \sigma ^{k}=\gamma _{0}\gamma ^{k}} must be used.

## Transformations

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To rotate a vector  v {\displaystyle v} in geometric algebra, the following formula is used:[14]: 50–51

v ′ = e − β θ 2 v e β θ 2 {\displaystyle v'=e^{-\beta {\frac {\theta }{2}}}\ v\ e^{\beta {\frac {\theta }{2}}}} ,

where  θ {\displaystyle \theta } is the angle to rotate by, and  β {\displaystyle \beta } is the normalized bivector representing the plane of rotation so that  β β ~ = 1 {\displaystyle \beta {\tilde {\beta }}=1} .

For a given spacelike bivector,  β 2 = − 1 {\displaystyle \beta ^{2}=-1} , so Euler's formula applies,[2]: 401 giving the rotation

v ′ = ( cos ⁡ ( θ 2 ) − β sin ⁡ ( θ 2 ) ) v ( cos ⁡ ( θ 2 ) + β sin ⁡ ( θ 2 ) ) {\displaystyle v'=\left(\cos \left({\frac {\theta }{2}}\right)-\beta \sin \left({\frac {\theta }{2}}\right)\right)\ v\ \left(\cos \left({\frac {\theta }{2}}\right)+\beta \sin \left({\frac {\theta }{2}}\right)\right)} .

For a given timelike bivector,  β 2 = 1 {\displaystyle \beta ^{2}=1} , so a "rotation through time" uses the analogous equation for the split-complex numbers:

v ′ = ( cosh ⁡ ( θ 2 ) − β sinh ⁡ ( θ 2 ) ) v ( cosh ⁡ ( θ 2 ) + β sinh ⁡ ( θ 2 ) ) {\displaystyle v'=\left(\cosh \left({\frac {\theta }{2}}\right)-\beta \sinh \left({\frac {\theta }{2}}\right)\right)\ v\ \left(\cosh \left({\frac {\theta }{2}}\right)+\beta \sinh \left({\frac {\theta }{2}}\right)\right)} .

Interpreting this equation, these rotations along the time direction are simply hyperbolic rotations. These are equivalent to Lorentz boosts in special relativity.

Both of these transformations are known as Lorentz transformations, and the combined set of all of them is the Lorentz group. To transform an object in STA from any basis (corresponding to a reference frame) to another, one or more of these transformations must be used.[1]: 47–62

Any spacetime element  A {\textstyle A} is transformed by multiplication with the pseudoscalar to form its dual element  A I {\textstyle AI} .[11]: 114 Duality rotation transforms spacetime element  A {\textstyle A} to element  A ′ {\textstyle A^{\prime }} through angle  ϕ {\textstyle \phi } with pseudoscalar  I {\textstyle I} is:[1]: 13

A ′ = e I ϕ A {\displaystyle A^{\prime }=e^{I\phi }A}

Duality rotation occurs only for non-singular Clifford algebra, non-singular meaning a Clifford algebra containing pseudoscalars with a non-zero square.[1]: 13

Grade involution (main involution, inversion) transforms every r-vector  A r {\textstyle A_{r}} to  A r ∗ {\textstyle A_{r}^{\ast }} :[1]: 13 [15]

A r ∗ = ( − 1 ) r A r {\displaystyle A_{r}^{\ast }=(-1)^{r}\ A_{r}}

Reversion transformation occurs by decomposing any spacetime element as a sum of products of vectors and then reversing the order of each product.[1]: 13 [16] For multivector  A {\textstyle A} arising from a product of vectors,  a 1 a 2 … a r − 1 a r {\textstyle a_{1}a_{2}\ldots a_{r-1}a_{r}} the reversion is  A † {\textstyle A^{\dagger }} :

A = a 1 a 2 … a r − 1 a r , A † = a r a r − 1 … a 2 a 1 {\displaystyle A=a_{1}a_{2}\ldots a_{r-1}a_{r},\quad A^{\dagger }=a_{r}a_{r-1}\ldots a_{2}a_{1}}

Clifford conjugation of a spacetime element  A {\textstyle A} combines reversion and grade involution transformations, indicated as  A ~ {\textstyle {\tilde {A}}} :[17]

A ~ = A ∗ † {\displaystyle {\tilde {A}}=A^{\ast \dagger }}

The grade involution, reversion and Clifford conjugation transformations are involutions.[18]

## Classical electromagnetism

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### The Faraday bivector

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In STA, the electric field and magnetic field can be unified into a single bivector field, known as the Faraday bivector, equivalent to the Faraday tensor.[2]: 230 It is defined as:

F = E -> + I c B -> , {\displaystyle F={\vec {E}}+Ic{\vec {B}},}

where  E {\displaystyle E} and  B {\displaystyle B} are the usual electric and magnetic fields, and  I {\displaystyle I} is the STA pseudoscalar.[2]: 230 Alternatively, expanding  F {\displaystyle F} in terms of components,  F {\displaystyle F} is defined that

F = E i σ i + I c B i σ i = E 1 γ 1 γ 0 + E 2 γ 2 γ 0 + E 3 γ 3 γ 0 − c B 1 γ 2 γ 3 − c B 2 γ 3 γ 1 − c B 3 γ 1 γ 2 . {\displaystyle F=E^{i}\sigma _{i}+IcB^{i}\sigma _{i}=E^{1}\gamma _{1}\gamma _{0}+E^{2}\gamma _{2}\gamma _{0}+E^{3}\gamma _{3}\gamma _{0}-cB^{1}\gamma _{2}\gamma _{3}-cB^{2}\gamma _{3}\gamma _{1}-cB^{3}\gamma _{1}\gamma _{2}.}

The separate  E -> {\displaystyle {\vec {E}}} and  B -> {\displaystyle {\vec {B}}} fields are recovered from  F {\displaystyle F} using

E = 1 2 ( F − γ 0 F γ 0 ) , I c B = 1 2 ( F + γ 0 F γ 0 ) . {\displaystyle {\begin{aligned}E={\frac {1}{2}}\left(F-\gamma _{0}F\gamma _{0}\right),\\\IcB={\frac {1}{2}}\left(F+\gamma _{0}F\gamma _{0}\right).\end{aligned}}}

The  γ 0 {\displaystyle \gamma _{0}} term represents a given reference frame, and as such, using different reference frames will result in apparently different relative fields, exactly as in standard special relativity.[2]: 233

Since the Faraday bivector is a relativistic invariant, further information can be found in its square, giving two new Lorentz-invariant quantities, one scalar, and one pseudoscalar:

F 2 = E 2 − c 2 B 2 + 2 I c E -> ⋅ B -> . {\displaystyle F^{2}=E^{2}-c^{2}B^{2}+2Ic{\vec {E}}\cdot {\vec {B}}.}

The scalar part corresponds to the Lagrangian density for the electromagnetic field, and the pseudoscalar part is a less-often seen Lorentz invariant.[2]: 234

### Maxwell's equation

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STA formulates Maxwell's equations in a simpler form as one equation,[19]: 230 rather than the 4 equations of vector calculus.[20]: 2–3 Similarly to the above field bivector, the electric charge density and current density can be unified into a single spacetime vector, equivalent to a four-vector. As such, the spacetime current  J {\displaystyle J} is given by[21]: 26

J = c ρ γ 0 + J i γ i , {\displaystyle J=c\rho \gamma _{0}+J^{i}\gamma _{i},}

where the components  J i {\displaystyle J^{i}} are the components of the classical 3-dimensional current density. When combining these quantities in this way, it makes it particularly clear that the classical charge density is nothing more than a current travelling in the timelike direction given by  γ 0 {\displaystyle \gamma _{0}} .

Combining the electromagnetic field and current density together with the spacetime gradient as defined earlier, we can combine all four of Maxwell's equations into a single equation in STA. [19]: 230

Maxwell's equation:

∇ F = μ 0 c J {\displaystyle \nabla F=\mu _{0}cJ}

The fact that these quantities are all covariant objects in the STA automatically guarantees Lorentz covariance of the equation, which is much easier to show than when separated into four separate equations.

In this form, it is also much simpler to prove certain properties of Maxwell's equations, such as the conservation of charge. Using the fact that for any bivector field, the divergence of its spacetime gradient is  0 {\displaystyle 0} , one can perform the following manipulation:[22]: 231

∇ ⋅ [ ∇ F ] = ∇ ⋅ [ μ 0 c J ] 0 = ∇ ⋅ J . {\displaystyle {\begin{aligned}\nabla \cdot \left[\nabla F\right]&=\nabla \cdot \left[\mu _{0}cJ\right]\\\0&=\nabla \cdot J.\end{aligned}}}

This equation has the clear meaning that the divergence of the current density is zero, i.e. the total charge and current density over time is conserved.

Using the electromagnetic field, the form of the Lorentz force on a charged particle can also be considerably simplified using STA.[23]: 156

Lorentz force on a charged particle:

F = q F ⋅ v {\displaystyle {\mathcal {F}}=qF\cdot v}

### Potential formulation

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In the standard vector calculus formulation, two potential functions are used: the electric scalar potential, and the magnetic vector potential. Using the tools of STA, these two objects are combined into a single vector field  A {\displaystyle A} , analogous to the electromagnetic four-potential in tensor calculus. In STA, it is defined as

A = ϕ c γ 0 + A k γ k {\displaystyle A={\frac {\phi }{c}}\gamma _{0}+A^{k}\gamma _{k}}

where  ϕ {\displaystyle \phi } is the scalar potential, and  A k {\displaystyle A^{k}} are the components of the magnetic potential. As defined, this field has SI units of webers per meter (V⋅s⋅m−1).

The electromagnetic field can also be expressed in terms of this potential field, using

1 c F = ∇ ∧ A . {\displaystyle {\frac {1}{c}}F=\nabla \wedge A.}

However, this definition is not unique. For any twice-differentiable scalar function  Λ ( x -> ) {\displaystyle \Lambda ({\vec {x}})} , the potential given by

A ′ = A + ∇ Λ {\displaystyle A'=A+\nabla \Lambda }

will also give the same  F {\displaystyle F} as the original, due to the fact that

∇ ∧ ( A + ∇ Λ ) = ∇ ∧ A + ∇ ∧ ∇ Λ = ∇ ∧ A . {\displaystyle \nabla \wedge \left(A+\nabla \Lambda \right)=\nabla \wedge A+\nabla \wedge \nabla \Lambda =\nabla \wedge A.}

This phenomenon is called gauge freedom. The process of choosing a suitable function  Λ {\displaystyle \Lambda } to make a given problem simplest is known as gauge fixing. However, in relativistic electrodynamics, the Lorenz condition is often imposed, where  ∇ ⋅ A -> = 0 {\displaystyle \nabla \cdot {\vec {A}}=0} .[2]: 231

To reformulate the STA Maxwell equation in terms of the potential  A {\displaystyle A} ,  F {\displaystyle F} is first replaced with the above definition.

1 c ∇ F = ∇ ( ∇ ∧ A ) = ∇ ⋅ ( ∇ ∧ A ) + ∇ ∧ ( ∇ ∧ A ) = ∇ 2 A + ( ∇ ∧ ∇ ) A = ∇ 2 A + 0 = ∇ 2 A {\displaystyle {\begin{aligned}{\frac {1}{c}}\nabla F&=\nabla \left(\nabla \wedge A\right)\\\&=\nabla \cdot \left(\nabla \wedge A\right)+\nabla \wedge \left(\nabla \wedge A\right)\\\&=\nabla ^{2}A+\left(\nabla \wedge \nabla \right)A=\nabla ^{2}A+0\\\&=\nabla ^{2}A\end{aligned}}}

Substituting in this result, one arrives at the potential formulation of electromagnetism in STA:[2]: 232

Potential equation:

∇ 2 A = μ 0 J {\displaystyle \nabla ^{2}A=\mu _{0}J}

### Lagrangian formulation

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Analogously to the tensor calculus formalism, the potential formulation in STA naturally leads to an appropriate Lagrangian density.[2]: 453

Electromagnetic Lagrangian density:

L = 1 2 ϵ 0 F 2 − J ⋅ A {\displaystyle {\mathcal {L}}={\frac {1}{2}}\epsilon _{0}F^{2}-J\cdot A}

The multivector-valued Euler-Lagrange equations for the field can be derived, and being loose with the mathematical rigor of taking the partial derivative with respect to something that is not a scalar, the relevant equations become:[24]: 440

∇ ∂ L ∂ ( ∇ A ) − ∂ L ∂ A = 0. {\displaystyle \nabla {\frac {\partial {\mathcal {L}}}{\partial \left(\nabla A\right)}}-{\frac {\partial {\mathcal {L}}}{\partial A}}=0.}

To begin to re-derive the potential equation from this form, it is simplest to work in the Lorenz gauge, setting[2]: 232

∇ ⋅ A = 0. {\displaystyle \nabla \cdot A=0.}

This process can be done regardless of the chosen gauge, but this makes the resulting process considerably clearer. Due to the structure of the geometric product, using this condition results in  ∇ ∧ A = ∇ A {\displaystyle \nabla \wedge A=\nabla A} .

After substituting in  F = c ∇ A {\displaystyle F=c\nabla A} , the same equation of motion as above for the potential field  A {\displaystyle A} is easily obtained.

## The Pauli equation

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STA allows the description of the Pauli particle in terms of a real theory in place of a matrix theory. The matrix theory description of the Pauli particle is:[25]

i ℏ ∂ t Ψ = H S Ψ − e ℏ 2 m c σ ^ ⋅ B Ψ , {\displaystyle i\hbar \,\partial _{t}\Psi =H_{S}\Psi -{\frac {e\hbar }{2mc}}\,{\hat {\sigma }}\cdot \mathbf {B} \Psi ,}

where  Ψ {\displaystyle \Psi } is a spinor,  i {\displaystyle i} is the imaginary unit with no geometric interpretation,  σ ^ i {\displaystyle {\hat {\sigma }}_{i}} are the Pauli matrices (with the 'hat' notation indicating that  σ ^ {\displaystyle {\hat {\sigma }}} is a matrix operator and not an element in the geometric algebra), and  H S {\displaystyle H_{S}} is the Schrödinger Hamiltonian.

The STA approach transforms the matrix spinor representation  | ψ ⟩ {\textstyle |\psi \rangle } to the STA representation  ψ {\textstyle \psi } using elements,  σ 1 , σ 2 , σ 3 {\textstyle \mathbf {\sigma _{1},\sigma _{2},\sigma _{3}} } , of the even-graded spacetime subalgebra and the pseudoscalar  I = σ 1 σ 2 σ 3 {\displaystyle I=\sigma _{1}\sigma _{2}\sigma _{3}} :[2]: 37 [26]: 270, 271

| ψ ⟩ = | c o s ( θ / 2 ) e - i ϕ / 2 s i n ( θ / 2 ) e + i ϕ / 2 | = | a 0 + i a 3 − a 2 + i a 1 | ↦ ψ = a 0 + a 1 I σ 1 + a 2 I σ 2 + a 3 I σ 3 {\displaystyle |\psi \rangle ={\begin{vmatrix}\operatorname {cos(\theta /2)\ e^{-i\phi /2}} \\\\\operatorname {sin(\theta /2)\ e^{+i\phi /2}} \end{vmatrix}}={\begin{vmatrix}a^{0}+ia^{3}\\\\-a^{2}+ia^{1}\end{vmatrix}}\mapsto \psi =a^{0}+a^{1}\mathbf {I\sigma _{1}} +a^{2}\mathbf {I\sigma _{2}} +a^{3}\mathbf {I\sigma _{3}} }

The Pauli particle is described by the real Pauli–Schrödinger equation:[25]

∂ t ψ I σ 3 ℏ = H S ψ − e ℏ 2 m c B ψ σ 3 , {\displaystyle \partial _{t}\psi \,I\sigma _{3}\,\hbar =H_{S}\psi -{\frac {e\hbar }{2mc}}\,\mathbf {B} \psi \sigma _{3},}

where now  ψ {\displaystyle \psi } is an even multi-vector of the geometric algebra, and the Schrödinger Hamiltonian is  H S {\displaystyle H_{S}} . Hestenes refers to this as the real Pauli–Schrödinger theory to emphasize that this theory reduces to the Schrödinger theory if the term that includes the magnetic field is dropped.[25]: 30 The vector  σ 3 {\textstyle \sigma _{3}} is an arbitrarily selected fixed vector; a fixed rotation can generate any alternative selected fixed vector  σ 3 ′ {\textstyle \sigma _{3}^{\prime }} .[27]: 30

## The Dirac equation

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STA enables a description of the Dirac particle in terms of a real theory in place of a matrix theory. The matrix theory description of the Dirac particle is:[28]

γ ^ μ ( i ∂ μ − e A μ ) | ψ ⟩ = m | ψ ⟩ , {\displaystyle {\hat {\gamma }}^{\mu }(i\partial _{\mu }-e\mathbf {A} _{\mu })|\psi \rangle =m|\psi \rangle ,}

where  γ ^ {\displaystyle {\hat {\gamma }}} are the Dirac matrices and  i {\textstyle i} is the imaginary unit with no geometric interpretation.

Using the same approach as for Pauli equation, the STA approach transforms the matrix upper spinor  | ψ U ⟩ {\textstyle |\psi _{U}\rangle } and matrix lower spinor  | ψ L ⟩ {\textstyle |\psi _{L}\rangle } of the matrix Dirac bispinor  | ψ ⟩ {\textstyle |\psi \rangle } to the corresponding geometric algebra spinor representations  ψ U {\textstyle \psi _{U}} and  ψ L {\textstyle \psi _{L}} . These are then combined to represent the full geometric algebra Dirac bispinor  ψ {\textstyle \psi } .[29]: 279

| ψ ⟩ = | | ψ U ⟩ | ψ L ⟩ | ↦ ψ = ψ U + ψ L σ 3 {\displaystyle |\psi \rangle ={\begin{vmatrix}|\psi _{U}\rangle \\\|\psi _{L}\rangle \end{vmatrix}}\mapsto \psi =\psi _{U}+\psi _{L}\mathbf {\sigma _{3}} }

Following Hestenes' derivation, the Dirac particle is described by the equation:[28][30]: 283

Dirac equation in STA:

∇ ψ I σ 3 − e A ψ = m ψ γ 0 {\displaystyle \nabla \psi \,I\sigma _{3}-e\mathbf {A} \psi =m\psi \gamma _{0}}

Here,  ψ {\displaystyle \psi } is the spinor field,  γ 0 {\displaystyle \gamma _{0}} and  I σ 3 {\displaystyle I\sigma _{3}} are elements of the geometric algebra,  A {\displaystyle \mathbf {A} } is the electromagnetic four-potential, and  ∇ = γ μ ∂ μ {\displaystyle \nabla =\gamma ^{\mu }\partial _{\mu }} is the spacetime vector derivative.

### Dirac spinors

[edit]

A relativistic Dirac spinor ψ {\textstyle \psi } can be expressed as:[31][32][33]: 280

ψ = R ( ρ e i β ) 1 2 {\displaystyle \psi =R(\rho e^{i\beta })^{\frac {1}{2}}}

where, according to its derivation by David Hestenes,  ψ = ψ ( x ) {\displaystyle \psi =\psi (x)} is an even multivector-valued function on spacetime,  R = R ( x ) {\displaystyle R=R(x)} is a unimodular spinor or "rotor",[34] and  ρ = ρ ( x ) {\displaystyle \rho =\rho (x)} and  β = β ( x ) {\displaystyle \beta =\beta (x)} are scalar-valued functions.[31] In this construction, the components of  ψ {\displaystyle \psi } directly correspond with the components of a Dirac spinor, both having 8 scalar degrees of freedom.

This equation is interpreted as connecting spin with the imaginary pseudoscalar.[35]: 104–121

The rotor,  R {\displaystyle R} , Lorentz transforms the frame of vectors  γ μ {\displaystyle \gamma _{\mu }} into another frame of vectors  e μ {\displaystyle e_{\mu }} by the operation  e μ = R γ μ R † {\displaystyle e_{\mu }=R\gamma _{\mu }R^{\dagger }} ;[36]: 15 note that  R † {\textstyle R^{\dagger }} indicates the reverse transformation.

This has been extended to provide a framework for locally varying vector- and scalar-valued observables and support for the Zitterbewegung interpretation of quantum mechanics originally proposed by Schrödinger.[37][1]: vi

Hestenes has compared his expression for  ψ {\displaystyle \psi } with Feynman's expression for it in the path integral formulation:

ψ = e i Φ λ / ℏ , {\displaystyle \psi =e^{i\Phi _{\lambda }/\hbar },}

where  Φ λ {\displaystyle \Phi _{\lambda }} is the classical action along the  λ {\displaystyle \lambda } -path.[31]

Using the spinors, the current density from the field can be expressed by[38]: 8

J μ = ψ ¯ γ μ ψ {\displaystyle J^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi }

### Symmetries

[edit]

Global phase symmetry is a constant global phase shift of the wave function that leaves the Dirac equation unchanged.[39]: 41–48 Local phase symmetry is a spatially varying phase shift that leaves the Dirac equation unchanged if accompanied by a gauge transformation of the electromagnetic four-potential as expressed by these combined substitutions.[40]: 269, 283

ψ ↦ ψ e α ( x ) I σ 3 , e A ↦ e A − ∇ α ( x ) {\displaystyle \psi \mapsto \psi e^{\alpha (x)I\sigma _{3}},\quad eA\mapsto eA-\nabla \alpha (x)}

In these equations, the local phase transformation is a phase shift  α ( x ) {\displaystyle \alpha (x)} at spacetime location  x {\textstyle x} with pseudovector  I {\textstyle I} and  σ 3 {\textstyle \sigma _{3}} of even-graded spacetime subalgebra applied to wave function  ψ {\textstyle \psi } ; the gauge transformation is a subtraction of the gradient of the phase shift  ∇ α ( x ) {\textstyle \nabla \alpha (x)} from the electromagnetic four-potential  A {\textstyle A} with particle electric charge  e {\textstyle e} .[40]: 269, 283

Researchers have applied STA and related Clifford algebra approaches to gauge theories, electroweak interaction, Yang–Mills theory, and the standard model.[41]: 1345–1347

The discrete symmetries are parity ( P ^ ) {\textstyle ({\hat {P}})} , charge conjugation ( C ^ ) {\textstyle ({\hat {C}})} and time reversal ( T ^ ) {\textstyle ({\hat {T}})} applied to wave function  ψ {\textstyle \psi } . These effects are:[42]: 283

P ^ | ψ ⟩ ↦ γ 0 ψ ( γ 0 x γ 0 ) γ 0 C ^ | ψ ⟩ ↦ ψ σ 1 T ^ | ψ ⟩ ↦ I γ 0 ψ ( γ 0 x γ 0 ) γ 1 {\displaystyle {\begin{aligned}{\hat {P}}|\psi \rangle &\mapsto \gamma _{0}\psi (\gamma _{0}x\gamma _{0})\gamma _{0}\\\\{\hat {C}}|\psi \rangle &\mapsto \psi \sigma _{1}\\\\{\hat {T}}|\psi \rangle &\mapsto I\gamma _{0}\psi (\gamma _{0}x\gamma _{0})\gamma _{1}\end{aligned}}}

## General relativity

[edit]

### General relativity

[edit]

Main article: Gauge theory gravity

Researchers have applied STA and related Clifford algebra approaches to relativity, gravity and cosmology.[41]: 1343 The gauge theory gravity (GTG) uses STA to describe an induced curvature on Minkowski space while admitting a gauge symmetry under "arbitrary smooth remapping of events onto spacetime" leading to this geodesic equation.[43][44][4][12]

d d τ R = 1 2 ( Ω − ω ) R {\displaystyle {\frac {d}{d\tau }}R={\frac {1}{2}}(\Omega -\omega )R}

and the covariant derivative

D τ = ∂ τ + 1 2 ω , {\displaystyle D_{\tau }=\partial _{\tau }+{\frac {1}{2}}\omega ,}

where  ω {\displaystyle \omega } is the connection associated with the gravitational potential, and  Ω {\displaystyle \Omega } is an external interaction such as an electromagnetic field.

The theory shows some promise for the treatment of black holes, as its form of the Schwarzschild solution does not break down at singularities; most of the results of general relativity have been mathematically reproduced, and the relativistic formulation of classical electrodynamics has been extended to quantum mechanics and the Dirac equation.

## See also

[edit]

* Geometric algebra
* Dirac algebra
* Maxwell's equations
* Dirac equation
* General relativity

## Notes

[edit]

1. ^ An example: given idempotent  a = 1 2 ( 1 + γ 0 ) {\textstyle a={\tfrac {1}{2}}(1+\gamma _{0})} , define  b = 1 − a = 1 2 ( 1 − γ 0 ) {\textstyle b=1-a={\tfrac {1}{2}}(1-\gamma _{0})} , then  a 2 = a {\textstyle a^{2}=a} ,  b 2 = b {\textstyle b^{2}=b} , and  a b = 0 {\textstyle ab=0} . Find the inverse  a − 1 {\textstyle a^{-1}} satisfying  a − 1 a = 1 {\textstyle a^{-1}a=1} . Thus,  b = 1 ⋅ b = ( a − 1 a ) b = a − 1 ( a b ) = a − 1 ⋅ 0 ≠ 0 {\textstyle b=1\cdot b=(a^{-1}a)b=a^{-1}(ab)=a^{-1}\cdot 0\neq 0} . However, there is no  a − 1 {\textstyle a^{-1}} satisfying  a − 1 ⋅ 0 ≠ 0 {\textstyle a^{-1}\cdot 0\neq 0} , so this idempotent has no inverse.

## Citations

[edit]

1. ^ a b c d e f g h i j k l m n o p q Hestenes 2015.
2. ^ a b c d e f g h i j k l m n o p Doran & Lasenby 2003.
3. ^ Baylis 2012.
4. ^ a b Lasenby, Doran & Gull 1995.
5. ^ a b O'Donnell 2003.
6. ^ a b c Vaz & da Rocha 2016.
7. ^ Warner 1990, Theorems 21.2, 21.3.
8. ^ Warner 1990.
9. ^ Palais 1968.
10. ^ Hestenes & Sobczyk 1984.
11. ^ a b Hestenes & Sobczyk 2012c.
12. ^ a b c d Lasenby & Doran 2002.
13. ^ Arthur 2011.
14. ^ Hestenes 2015, Eqs. (16.22),(16.23).
15. ^ Floerchinger 2021, Eq. (18).
16. ^ Floerchinger 2021, Eq. (25).
17. ^ Floerchinger 2021, Eq. (27).
18. ^ Floerchinger 2021.
19. ^ a b Doran & Lasenby 2003, Eq. (7.14).
20. ^ Jackson 1998.
21. ^ Hestenes 2015, Eq. (8.4).
22. ^ Doran & Lasenby 2003, Eq. (7.16).
23. ^ Doran & Lasenby 2003, Eq. (5.170).
24. ^ Doran & Lasenby 2003, Eq. (12.3).
25. ^ a b c Hestenes 2003a, Eqs. (75),(81).
26. ^ Doran & Lasenby 2003, Eqs. (8.16),(8.20),(8.23).
27. ^ Hestenes 2003a, Eqs. (82),(83),(84).
28. ^ a b Doran et al. 1996, Eqs. (3.43),(3.44).
29. ^ Doran & Lasenby 2003, Eq. (8.69).
30. ^ Doran & Lasenby 2003, Eq. (8.89).
31. ^ a b c Hestenes 2012b, Eqs. (3.1),(4.1),pp 169-182.
32. ^ Gull, Lasenby & Doran 1993, Eq. (5.13).
33. ^ Doran & Lasenby 2003, Eq. (8.80).
34. ^ Hestenes 2003b, Eq. (205).
35. ^ Hestenes 2003a.
36. ^ Hestenes 2003b, Eq. (79).
37. ^ Hestenes 2010.
38. ^ Hestenes 1967, Eq. (4.5).
39. ^ Quigg 2021.
40. ^ a b Doran & Lasenby 2003, Eqs. (8.8),(8.9),(8.10),(8.92),(8.93).
41. ^ a b Hitzer, Lavor & Hildenbrand 2024.
42. ^ Doran & Lasenby 2003, Eq. (8.90).
43. ^ Doran, Lasenby & Gull 1993.
44. ^ Lasenby, Doran & Gull 1998.

## References

[edit]

* Arthur, John W. (2011). Understanding Geometric Algebra for Electromagnetic Theory. IEEE Press Series on Electromagnetic Wave Theory. Wiley. p. 180\. ISBN 978-0-470-94163-8.
* Baylis, William E. (2012). "Vector Algebra of Physical Space". Theoretical Methods in the Physical Sciences: An introduction to problem solving using Maple V. Birkhäuser. p. 225-266. ISBN 978-1-4612-0275-2.
* Doran, Chris; Lasenby, Anthony; Gull, Stephen (1993). "Gravity as a Gauge Theory in the STA". In Brackx, F.; Delanghe, R.; Serras, H. (eds.). Clifford Algebras and their Applications in Mathematical Physics. Fundamental Theories of Physics. Vol. 55\. Springer Netherlands. pp. 375–385. doi:10.1007/978-94-011-2006-7_42. ISBN 978-94-011-2006-7.
* Doran, Chris; Lasenby, Anthony; Gull, Stephen; Somaroo, Shyamal; Challinor, Anthony (1996). Hawkes, Peter W. (ed.). STA and electron physics. Advances in Imaging and Electron Physics. Vol. 95\. Academic Press. pp. 272–386, 292. ISBN 0-12-014737-8.
* Doran, Chris; Lasenby, Anthony (2003). Geometric Algebra for Physicists. Cambridge University Press. ISBN 0-521-48022-1.
* Floerchinger, Stefan (2021). "Real Clifford Algebras and Their Spinors for Relativistic Fermions". Universe. 7 (6): 168.
* Gull, S.; Lasenby, A.; Doran, C. (1993). "Imaginary numbers are not real—the geometric algebra of spacetime" (PDF). Foundations of Physics. 23: 1175–1201.
* Hestenes, David (1967), "Real Spinor Fields" (PDF), Journal of Mathematical Physics, 8 (4): 798–808, Bibcode:1967JMP.....8..798H, doi:10.1063/1.1705279
* Hestenes, David; Sobczyk (1984), Clifford Algebra to Geometric Calculus, Springer Verlag, ISBN 978-90-277-1673-6
* Hestenes, David (2003a). "Oersted Medal Lecture 2002: Reforming the mathematical language of physics". American Journal of Physics. 71 (2): 104–121. Bibcode:2003AmJPh..71..104H. CiteSeerX 10.1.1.649.7506. doi:10.1119/1.1522700.
* Hestenes, D. (2003b). "Spacetime physics with geometric algebra" (PDF). American Journal of Physics. 71 (6): 691–714. Bibcode:2003AmJPh..71..691H. doi:10.1119/1.1571836. Retrieved 2012-02-24.
* Hestenes, David (2010). "Zitterbewegung in Quantum Mechanics" (PDF). Foundations of Physics. 40.
* Hestenes, D. (2012b) [1990]. "On decoupling probability from kinematics in quantum mechanics". In Fougère, P.F. (ed.). Maximum Entropy and Bayesian Methods. Springer. pp. 161–183. ISBN 978-94-009-0683-9. PDF
* Hestenes, D.; Sobczyk, Garret (2012c). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Springer Science & Business Media. ISBN 978-94-009-6292-7.
* Hestenes, David (2015). Space-Time Algebra. Springer International Publishing. ISBN 978-3-319-18412-8.
* Hitzer, Eckhard; Lavor, Carlile; Hildenbrand, Dietmar (2024). "Current survey of Clifford geometric algebra applications". Mathematical Methods in the Applied Sciences. 47 (3): 1331–1361. doi:10.1002/mma.8316. ISSN 0170-4214.
* Jackson, John David (1998). Classical Electrodynamics. John Wiley & Sons. ISBN 978-0-471-30932-1.
* Lasenby, Anthony; Doran, Chris; Gull, Stephen (1995). "Astrophysical and Cosmological Consequences of a Gauge Theory of Gravity". In Sanchez, Norma; Zichichi, Antonino (eds.). Advances In Astrofundamental Physics: International School Of Astrophysics "D. Chalonge". World Scientific. pp. 359–401. ISBN 978-981-4548-78-6.Reprint
* Lasenby, A.; Doran, C.; Gull, S. (1998), "Gravity, gauge theories and geometric algebra", Phil. Trans. R. Soc. Lond. A, 356 (1737): 487–582, arXiv:gr-qc/0405033, Bibcode:1998RSPTA.356..487L, doi:10.1098/rsta.1998.0178, S2CID 119389813
* Lasenby, A.N.; Doran, C.J.L. (2002). "Geometric algebra, Dirac wavefunctions and black holes". In Bergmann, P.G.; De Sabbata, Venzo (eds.). Advances in the interplay between quantum and gravity physics. Springer. pp. 256–283, See p. 257. ISBN 978-1-4020-0593-0.
* O'Donnell, Peter J. (2003). Introduction to 2-spinors in General Relativity. World Scientific. ISBN 978-981-279-531-1.
* Palais, R. S. (1968). "The Classification of Real Division Algebras". The American Mathematical Monthly. 75 (4): 366–368. doi:10.2307/2313414. ISSN 0002-9890. JSTOR 2313414.
* Quigg, Chris (29 November 2021). Gauge Theories Of Strong, Weak, And Electromagnetic Interactions. CRC Press. ISBN 978-0-429-68902-4.
* Vaz, Jayme; da Rocha, Roldão (2016). An Introduction to Clifford Algebras and Spinors. Oxford University Press. ISBN 978-0-19-108578-9.
* Warner, Seth (1990). Modern algebra. New York : Dover Publications. pp. 191, 211. ISBN 978-0-486-66341-8.

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* A multivector Lagrangian for Maxwell's equation
* Imaginary numbers are not real – the geometric algebra of spacetime, a tutorial introduction to the ideas of geometric algebra, by S. Gull, A. Lasenby, C. Doran
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