Spacetime algebra

In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl_1,3(R), or equivalently the geometric algebra G(M⁴) of 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]

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] It is also the natural parent algebra of spinors in special relativity.^[3] 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.^[4]

In comparison to related methods, STA and Dirac algebra are both Clifford Cl_1,3(R) algebras, but STA uses real number scalars while Dirac algebra uses complex number scalars^{[citation needed]}. The STA space–time 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.^[5]

Structure

For any pair of STA vectors, ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠, there is a geometric product ⁠${\displaystyle ab}$⁠, scalar ('inner') product ⁠${\displaystyle a\cdot b}$⁠ and exterior ('wedge', 'outer') product ⁠${\displaystyle a\wedge b}$⁠. The vector product is a sum of a scalar and exterior product:^[6]

${\displaystyle a\cdot b={\frac {ab+ba}{2}}=b\cdot a,\quad a\wedge b={\frac {ab-ba}{2}}=-b\wedge a,\quad ab=a\cdot b+a\wedge b.}$

The scalar product generates a real number (scalar), and the exterior product generates a bivector. The vectors ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are orthogonal if their scalar product is zero; vectors ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are parallel if their exterior product is zero.^[7]

The orthonormal basis vectors are a timelike vector ⁠${\displaystyle \gamma _{0}}$⁠ and 3 spacelike vectors ${\textstyle \gamma _{1},\gamma _{2},\gamma _{3}}$. The Minkowski metric tensor's nonzero terms are the diagonal terms, ⁠${\displaystyle {1}}$⁠. For ⁠${\displaystyle \mu ,\nu =0,1,2,3}$⁠:

${\displaystyle \gamma _{\mu }\cdot \gamma _{\nu }={\frac {\gamma _{\mu }\gamma _{\nu }+\gamma _{\nu }\gamma _{\mu }}{2}}=\eta _{\mu \nu },\quad \gamma _{0}\cdot \gamma _{0}=1,\ \gamma _{1}\cdot \gamma _{1}=\gamma _{2}\cdot \gamma _{2}=\gamma _{3}\cdot \gamma _{3}=-1,\quad {\text{ otherwise }}\ \gamma _{\mu }\gamma _{\nu }=-\gamma _{\nu }\gamma _{\mu }}$

The Dirac matrices share these properties, and STA is equivalent to the algebra generated by the Dirac matrices over the field of real numbers;^[8] explicit matrix representation is unnecessary for STA.

Products of the basis vectors generate a tensor basis containing one scalar ⁠${\displaystyle \{1\}}$⁠, four vectors ${\displaystyle \{\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}\}}$, six bivectors ${\displaystyle \{\gamma _{0}\gamma _{1},\gamma _{0}\gamma _{2},\gamma _{0}\gamma _{3},\gamma _{1}\gamma _{2},\gamma _{2}\gamma _{3},\gamma _{3}\gamma _{1}\}}$, four pseudovectors (trivectors) ${\displaystyle \{I\gamma _{0},I\gamma _{1},I\gamma _{2},I\gamma _{3}\}}$ and one pseudoscalar ${\displaystyle \{I\}}$ with ⁠${\displaystyle I=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}$⁠.^[9] The pseudoscalar commutes with all even-grade STA elements, but anticommutes with all odd-grade STA elements.^[10]

Subalgebra

STA's even-graded elements (scalars, bivectors, pseudoscalar) form a subalgebra isomorphic to Clifford algebra Cl_3,0(R), which is equivalent to the APS or Pauli algebra.^[11] The STA bivectors are equivalent to the APS vectors and pseudovectors. The STA subalgebra becomes more explicit by renaming the STA bivectors ${\textstyle (\gamma _{1}\gamma _{0},\gamma _{2}\gamma _{0},\gamma _{3}\gamma _{0})}$ as ${\textstyle (\sigma _{1},\sigma _{2},\sigma _{3})}$ and the STA bivectors ${\textstyle (\gamma _{3}\gamma _{2},\gamma _{1}\gamma _{3},\gamma _{2}\gamma _{1})}$ as ${\textstyle (I\sigma _{1},I\sigma _{2},I\sigma _{3})}$.^[12][13] The Pauli matrices, ${\textstyle {\hat {\sigma }}_{1},{\hat {\sigma }}_{2},{\hat {\sigma }}_{3}}$, are a matrix representation for ${\textstyle \sigma _{1},\sigma _{2},\sigma _{3}}$.^[13] For any pair of ${\textstyle (\sigma _{1},\sigma _{2},\sigma _{3})}$, the nonzero scalar products are ${\textstyle \sigma _{1}\cdot \sigma _{1}=\sigma _{2}\cdot \sigma _{2}=\sigma _{3}\cdot \sigma _{3}=1}$, and the nonzero exterior products are:^[13][14]

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

The sequence of algebra to even subalgebra continues as algebra of physical space, quaternion algebra, complex numbers and real numbers. The even STA subalgebra Cl^[0]
_(1,3)(R) of real space-time spinors in Cl_1,3(R) is isomorphic to the Clifford algebra Cl_3,0(R) of Euclidean space R³ with basis elements. See the illustration of space-time algebra spinors in Cl^[0]
_(1,3)(R) under the octonionic product as a Fano plane.^[15]

Division

A nonzero vector ⁠${\displaystyle a}$⁠ is a null vector (degree 2 nilpotent) if ⁠${\displaystyle a^{2}=0}$⁠.^[16] An example is ⁠${\displaystyle a=\gamma ^{0}+\gamma ^{1}}$⁠. Null vectors are tangent to the light cone (null cone).^[17] An element ⁠${\displaystyle b}$⁠ is an idempotent if ⁠${\displaystyle b^{2}=b}$⁠.^[18] Two idempotents ⁠${\displaystyle b_{1}}$⁠ and ⁠${\displaystyle b_{2}}$⁠ are orthogonal idempotents if ⁠${\displaystyle b_{1}b_{2}=0}$⁠.^[18] An example of an orthogonal idempotent pair is ${\displaystyle {\tfrac {1}{2}}(1+\gamma _{0}\gamma _{k})}$ and ${\displaystyle {\tfrac {1}{2}}(1-\gamma _{0}\gamma _{k})}$ with ⁠${\displaystyle k=1,2,3}$⁠. Proper zero divisors are nonzero elements whose product is zero such as null vectors or orthogonal idempotents.^[19] 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.^[18][20][a] The only associative division algebras are the real numbers, complex numbers and quaternions.^[21] As STA is not a division algebra, some STA elements may lack an inverse; however, division by the non-null vector ${\textstyle c}$ may be possible by multiplication by its inverse, defined as ⁠${\displaystyle c^{-1}=(c\cdot c)^{-1}c}$⁠.^[22]

Reciprocal frame

Associated with the orthogonal basis ${\displaystyle \{\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}\}}$ is the reciprocal basis set ${\displaystyle \{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\}}$ satisfying these equations:^[23]

${\displaystyle \gamma _{\mu }\cdot \gamma ^{\nu }=\delta _{\mu }^{\nu },\quad \mu ,\nu =0,1,2,3}$

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

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

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

The metric and index gymnastics raise or lower indices:

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

Spacetime gradient

The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied:^[24]

${\displaystyle a\cdot \nabla F(x)=\lim _{\tau \rightarrow 0}{\frac {F(x+a\tau )-F(x)}{\tau }}.}$

This requires the definition of the gradient to be

${\displaystyle \nabla =\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}=\gamma ^{\mu }\partial _{\mu }.}$

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

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

Space–time split

Space–time split – examples:

${\displaystyle x\gamma _{0}=x^{0}+\mathbf {x} }$

${\displaystyle p\gamma _{0}=E+\mathbf {p} }$[25]

${\displaystyle v\gamma _{0}=\gamma (1+\mathbf {v} )}$[25]

where ${\displaystyle \gamma }$ is the Lorentz factor

${\displaystyle \nabla \gamma _{0}=\partial _{t}-{\vec {\nabla }}}$[26]

In STA, a space–time 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.^[27]

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

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

Space–time 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 (⁠${\displaystyle \gamma }$⁠)^[28]

As these bivectors ⁠${\displaystyle \gamma _{k}\gamma _{0}}$⁠ square to ⁠${\displaystyle 1}$⁠, they serve as a spatial basis. Utilizing the Pauli matrix notation, these are written ⁠${\displaystyle \sigma _{k}=\gamma _{k}\gamma _{0}}$⁠. Spatial vectors in STA are denoted in boldface; then with ⁠${\displaystyle \mathbf {x} =x^{k}\sigma _{k}}$⁠ and ⁠${\displaystyle x^{0}=ct}$⁠, the ⁠${\displaystyle \gamma _{0}}$⁠-space–time split ⁠${\displaystyle x\gamma _{0}}$⁠, and its reverse ⁠${\displaystyle \gamma _{0}x}$⁠ are:

${\displaystyle {\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\sigma _{k}=ct+\mathbf {x} \\\gamma _{0}x&=x^{0}-x^{k}\sigma _{k}=ct-\mathbf {x} \end{aligned}}}$

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

Transformations

To rotate a vector ${\displaystyle v}$ in geometric algebra, the following formula is used:^[29]

${\displaystyle v'=e^{-\beta {\frac {\theta }{2}}}\ v\ e^{\beta {\frac {\theta }{2}}}}$,

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

For a given spacelike bivector, ⁠${\displaystyle \beta ^{2}=-1}$⁠, so Euler's formula applies,^[30] giving the rotation

${\displaystyle v'=\left(\cos \left({\frac {\theta }{2}}\right)-\beta \sin \left({\frac {\theta }{2}}\right)\right)\ v\ \left(\cos \left({\frac {\theta }{2}}\right)+\beta \sin \left({\frac {\theta }{2}}\right)\right)}$.

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

${\displaystyle v'=\left(\cosh \left({\frac {\theta }{2}}\right)-\beta \sinh \left({\frac {\theta }{2}}\right)\right)\ v\ \left(\cosh \left({\frac {\theta }{2}}\right)+\beta \sinh \left({\frac {\theta }{2}}\right)\right)}$.

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.^[31]

Identifications to express Lorentz transformation formulas in terms of complex quaternions

It is easy to put these Lorentz transformation formulas in terms of complex quaternions or biquaternions by making some simple identifications. The quaternion Lorentz transformations are most naturally expressed in terms of the equivalent of ${\displaystyle X\equiv x\,\gamma _{0}=x^{0}+\gamma _{1}\,\gamma _{0}\,x^{1}+\gamma _{2}\,\gamma _{0}\,x^{2}+\gamma _{3}\,\gamma _{0}\,x^{3}=x^{0}+\sigma _{1}\,x^{1}+\sigma _{2}\,x^{2}+\sigma _{3}\,x^{3}}$ Identify the pseudoscalar ${\displaystyle \gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}$ as ⁠${\displaystyle i}$⁠, the square root of ⁠${\displaystyle -1}$⁠. The square of the pseudoscalar ${\displaystyle \gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}\,}$ is -1 and it commutes with scalars and bivectors, which is what ${\displaystyle X}$ is and what the transformation operators are.

Define ${\displaystyle {\textbf {I}},\,{\textbf {J}},\,{\textbf {K}}}$ to be the basis quaternions such that ${\displaystyle {\textbf {I}}\,{\textbf {I}}={\textbf {J}}\,{\textbf {J}}={\textbf {K}}\,{\textbf {K}}={\textbf {I}}\,{\textbf {J}}\,{\textbf {K}}=-1\;}$ In this discussion ${\displaystyle {\textbf {I}}}$ is a basis quaternion. Identify ${\displaystyle -i\sigma _{1}\rightarrow {\textbf {I}}\quad -i\sigma _{2}\rightarrow {\textbf {J}}\quad -i\sigma _{3}\rightarrow {\textbf {K}}\quad }$.

For the bivector ⁠${\displaystyle \beta }$⁠ having a time axis, such as ${\displaystyle \gamma _{1}\gamma _{0},}$ ${\displaystyle e^{\beta \,{\frac {\alpha }{2}}}\,\gamma _{0}=\gamma _{0}\,e^{-\beta \,{\frac {\alpha }{2}}}}$. Here, ⁠${\displaystyle \alpha }$⁠ is a real scalar parameter.

For the bivector ⁠${\displaystyle \beta }$⁠ having two space axes, such as ${\displaystyle \gamma _{2}\,\gamma _{3},}$ ${\displaystyle e^{\beta \,{\frac {\theta }{2}}}\,\gamma _{0}=\gamma _{0}\,e^{\beta \,{\frac {\theta }{2}}}}$. Here, ⁠${\displaystyle \theta }$⁠ is a real scalar parameter.

For a spatial rotation^[32] about the x-axis ${\displaystyle \beta =\gamma _{2}\,\gamma _{3}=-(\gamma _{0}\,\gamma _{1}\,\gamma _{2}\,\gamma _{3})\,\gamma _{1}\,\gamma _{0}\rightarrow -i\,\sigma _{1}\rightarrow {\textbf {I}}}$, giving ${\displaystyle {\textbf {X}}'\,=\,e^{-{\frac {\theta }{2}}\,{\textbf {I}}}\,{\textbf {X}}\,e^{{\frac {\theta }{2}}\,{\textbf {I}}}\quad }$ Here, ⁠${\displaystyle \theta }$⁠ is the angle of rotation and ${\displaystyle {\textbf {X}}=x^{0}+i\,x^{1}{\textbf {I}}+i\,x^{2}{\textbf {J}}+i\,x^{3}{\textbf {K}}}$. This is the form used by P. A. M. Dirac.^[33]

For a boost^[34][35] in the x-direction, the bivector ${\displaystyle \beta =\gamma _{1}\,\gamma _{0}\rightarrow \sigma _{1}\rightarrow i\,{\textbf {I}}}$, giving ${\displaystyle {\textbf {X}}'\,=\,e^{-{\frac {\alpha }{2}}\,i\,{\textbf {I}}}\,{\textbf {X}}\,e^{-{\frac {\alpha }{2}}\,i\,{\textbf {I}}}\quad }$ Here ⁠${\displaystyle \cosh \alpha =(1-v^{2}/c^{2})^{-1/2}}$⁠, where ⁠${\displaystyle v}$⁠ is the velocity.

Any spacetime element ⁠${\displaystyle A}$⁠ is transformed by multiplication with the pseudoscalar to form its Hodge dual ⁠${\displaystyle AI}$⁠.^[36] Duality rotation transforms spacetime element ⁠${\displaystyle A}$⁠ to element ⁠${\displaystyle A^{\prime }}$⁠ through angle ⁠${\displaystyle \phi }$⁠ with pseudoscalar ⁠${\displaystyle I}$⁠ is:^[37]

${\displaystyle A^{\prime }=e^{I\phi }A.}$

Duality rotation occurs only for non-singular Clifford algebra, non-singular meaning a Clifford algebra containing pseudoscalars with a non-zero square.^[37]

Grade involution (main involution, inversion) transforms every ⁠${\displaystyle r}$⁠-vector ⁠${\displaystyle A_{r}}$⁠ to ⁠${\displaystyle A_{r}^{\ast }}$⁠:^[37][38]

${\displaystyle A_{r}^{\ast }=(-1)^{r}\ A_{r}.}$

Reversion transformation occurs by decomposing any spacetime element as a sum of products of vectors and then reversing the order of each product.^[37][39] For multivector ${\textstyle A}$ arising from a product of vectors, ${\textstyle a_{1}a_{2}\ldots a_{r-1}a_{r}}$ the reversion is ${\textstyle A^{\dagger }}$:

${\displaystyle A=a_{1}a_{2}\ldots a_{r-1}a_{r},\quad A^{\dagger }=a_{r}a_{r-1}\ldots a_{2}a_{1}.}$

Clifford conjugation of a spacetime element ⁠${\displaystyle A}$⁠ combines reversion and grade involution transformations, indicated as ⁠${\displaystyle {\tilde {A}}}$⁠:^[40]

${\displaystyle {\tilde {A}}=A^{\ast \dagger }}$

The grade involution, reversion and Clifford conjugation transformations are involutions.^[41]

Classical electromagnetism

Pauli equation

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:^[54]

${\displaystyle i\hbar \,\partial _{t}\Psi =H_{S}\Psi -{\frac {e\hbar }{2mc}}\,{\hat {\sigma }}\cdot \mathbf {B} \Psi ,}$

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

The STA approach transforms the matrix spinor representation ⁠${\displaystyle \vert \psi \rangle }$⁠ to the STA representation ⁠${\displaystyle \psi }$⁠ using elements, ${\textstyle \sigma _{1},\sigma _{2},\sigma _{3}}$, of the even-graded spacetime subalgebra and the pseudoscalar ⁠${\displaystyle I=\sigma _{1}\sigma _{2}\sigma _{3}}$⁠:^[13][55]

${\displaystyle |\psi \rangle ={\begin{bmatrix}\operatorname {cos(\theta /2)\ e^{-i\phi /2}} \\\operatorname {sin(\theta /2)\ e^{+i\phi /2}} \end{bmatrix}}={\begin{bmatrix}a^{0}+ia^{3}\\-a^{2}+ia^{1}\end{bmatrix}}\mapsto \psi =a^{0}+a^{1}\mathbf {I\sigma _{1}} +a^{2}\mathbf {I\sigma _{2}} +a^{3}\mathbf {I\sigma _{3}} }$

The Pauli particle is described by the real Pauli–Schrödinger equation:^[54]

${\displaystyle \partial _{t}\psi \,I\sigma _{3}\,\hbar =H_{S}\psi -{\frac {e\hbar }{2mc}}\,\mathbf {B} \psi \sigma _{3},}$

where now ⁠${\displaystyle \psi }$⁠ is an even multi-vector of the geometric algebra, and the Schrödinger Hamiltonian is ⁠${\displaystyle H_{S}}$⁠. 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.^[56] The vector ⁠${\displaystyle \sigma _{3}}$⁠ is an arbitrarily selected fixed vector; a fixed rotation can generate any alternative selected fixed vector ⁠${\displaystyle \sigma _{3}^{\prime }}$⁠.^[57]

Dirac equation

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:^[58]

${\displaystyle {\hat {\gamma }}^{\mu }(i\partial _{\mu }-e\mathbf {A} _{\mu })|\psi \rangle =m|\psi \rangle ,}$

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

Using the same approach as for Pauli equation, the STA approach transforms the matrix upper spinor ⁠${\displaystyle \vert \psi _{U}\rangle }$⁠ and matrix lower spinor ⁠${\displaystyle \vert \psi _{L}\rangle }$⁠ of the matrix Dirac spinor⁠${\displaystyle \vert \psi \rangle }$⁠ to the corresponding geometric algebra spinor representations ⁠${\displaystyle \psi _{U}}$⁠ and ⁠${\displaystyle \psi _{L}}$⁠. These are then combined to represent the full geometric algebra Dirac spinor ⁠${\displaystyle \psi }$⁠.^[59]

${\displaystyle |\psi \rangle ={\begin{vmatrix}|\psi _{U}\rangle \\|\psi _{L}\rangle \end{vmatrix}}\mapsto \psi =\psi _{U}+\psi _{L}\mathbf {\sigma _{3}} }$

Following Hestenes' derivation, the Dirac particle is described by the equation:^[58][60]

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

General relativity

See also

• Geometric algebra
• Dirac algebra
• Maxwell's equations
• Dirac equation
• General relativity
• Quaternion Lorentz Transformations

Notes

Citations

References

External links

• Exploring Physics with Geometric Algebra, book I
• Exploring Physics with Geometric Algebra, book II
• 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
• Physical Applications of Geometric Algebra course-notes, see especially part 2.
• Cambridge University Geometric Algebra group
• Geometric Calculus research and development