Spacetime algebra

The spacetime algebra may be built up from an orthogonal basis of one time-like vector ${\displaystyle \gamma _{0}}$ and three space-like vectors, ${\displaystyle \{\gamma _{1},\gamma _{2},\gamma _{3}\}}$, with the multiplication rule

${\displaystyle \gamma _{\mu }\gamma _{\nu }+\gamma _{\nu }\gamma _{\mu }=2\eta _{\mu \nu }}$

where ${\displaystyle \eta _{\mu \nu }}$ is the Minkowski metric with signature (+ − − −).

Thus, ${\displaystyle \gamma _{0}^{2}={+1}}$, ${\displaystyle \gamma _{1}^{2}=\gamma _{2}^{2}=\gamma _{3}^{2}={-1}}$, otherwise ${\displaystyle \gamma _{\mu }\gamma _{\nu }=-\gamma _{\nu }\gamma _{\mu }}$.

The basis vectors ${\displaystyle \gamma _{k}}$ share these properties with the Dirac matrices, but no explicit matrix representation need be used in STA.

This generates a basis of 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 ${\displaystyle \{i\gamma _{0},i\gamma _{1},i\gamma _{2},i\gamma _{3}\}}$ and one pseudoscalar ${\displaystyle \{i\}}$, where ${\displaystyle i=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}$.

The spacetime algebra also contains a non-trivial sub-algebra containing only the even grade elements, i.e. scalars, bivectors, and pseudoscalars. In the even sub-algebra, scalars and pseudoscalars both commute with all elements, and act like complex numbers. However, the pseudoscalar anticommutes with all odd-grade elements of the spacetime algebra, corresponding to the fact that under parity transformations, vectors and pseudovectors become negated.

Associated with the orthogonal basis ${\displaystyle \{\gamma _{\mu }\}}$ is the reciprocal basis ${\displaystyle \{\gamma ^{\mu }={\gamma _{\mu }}^{-1}\}}$ for ${\displaystyle \mu =0,\dots ,3}$, satisfying the relation

${\displaystyle \gamma _{\mu }\cdot \gamma ^{\nu }={\delta _{\mu }}^{\nu }.}$

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

A vector may be represented in either upper or lower index coordinates ${\displaystyle a=a^{\mu }\gamma _{\mu }=a_{\mu }\gamma ^{\mu }}$ with summation over ${\displaystyle \mu =0,\dots ,3}$, according to the Einstein notation, where the coordinates may be extracted by taking dot products with the basis vectors or their reciprocals.

${\displaystyle {\begin{aligned}a\cdot \gamma ^{\nu }&=a^{\nu }\\a\cdot \gamma _{\nu }&=a_{\nu }.\end{aligned}}}$

Like in tensor calculus, a change of index position can be achieved using the metric and the use of index gymnastics:

${\displaystyle {\begin{aligned}\gamma _{\mu }&=\eta _{\mu \nu }\gamma ^{\nu }\\\gamma ^{\mu }&=\eta ^{\mu \nu }\gamma _{\nu }.\end{aligned}}}$

The spacetime algebra is not a division algebra, because it contains idempotent elements ${\displaystyle {\tfrac {1}{2}}(1\pm \gamma _{0}\gamma _{i})}$ and nonzero zero divisors: ${\displaystyle (1+\gamma _{0}\gamma _{i})(1-\gamma _{0}\gamma _{i})=0}$. These can be interpreted as projectors onto the light-cone and orthogonality relations for such projectors, respectively. But in some cases it is possible to divide one multivector quantity by another, and make sense of the result: so, for example, a directed area divided by a vector in the same plane gives another vector, orthogonal to the first.

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

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

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

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

Spacetime split – examples:

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

${\displaystyle p\gamma _{0}=E+\mathbf {p} }$[1]

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

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

${\displaystyle \nabla \gamma _{0}=\partial _{t}-{\vec {\nabla }}}$[2]

In spacetime algebra, 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 3D space spanned by bivectors, equivalent to the standard 3D basis vectors in the algebra of physical space and
• a projection of the 4D space onto the chosen time axis, yielding a 1D space of scalars, representing the scalar time.[3]

This is achieved by pre- or post-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

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

As these bivectors ${\displaystyle \gamma _{k}\gamma _{0}}$ square to unity, 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 ${\displaystyle \gamma _{0}}$-spacetime 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 spacetime 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.

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

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

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

For a given spacelike bivector, ${\displaystyle \beta ^{2}=-1}$, so Euler's formula applies, 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, it is easy to see that 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.

Spacetime algebra 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:[6]

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

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. In the spacetime algebra the Pauli particle is described by the real Pauli–Schrödinger equation:[6]

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

where now ${\displaystyle i}$ is the unit pseudoscalar ${\displaystyle i=\sigma _{1}\sigma _{2}\sigma _{3}}$, and ${\displaystyle \psi }$ and ${\displaystyle \sigma _{3}}$ are elements of the geometric algebra, with ${\displaystyle \psi }$ an even multi-vector; ${\displaystyle H_{S}}$ is again the Schrödinger Hamiltonian. 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. This equation is more suited for the algebra of physical space, as nothing essential to the spacetime algebra appears in this equation.

Spacetime algebra 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:[7]

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

where ${\displaystyle {\hat {\gamma }}}$ are the Dirac matrices. In the spacetime algebra, following Hestenes' derivation, the Dirac particle is described by the equation:[7]

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. This allows the same mathematical operator to describe the equations of motion for both electromagnetism and quantum mechanics, leading to a much simpler unification of the two.

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

• 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
• Doran, Chris; Lasenby, Anthony (2003), Geometric Algebra for Physicists, Cambridge University Press, ISBN 978-0-521-48022-2
• Hestenes, David (2015) [1966], Space–Time Algebra (2nd ed.), Birkhäuser, ISBN 9783319184135
• Hestenes, David; Sobczyk (1984), Clifford Algebra to Geometric Calculus, Springer Verlag, ISBN 978-90-277-1673-6
• Hestenes, David (1973), "Local observables in the Dirac theory", Journal of Mathematical Physics, 14 (7): 893–905, Bibcode:1973JMP....14..893H, CiteSeerX 10.1.1.412.7214, doi:10.1063/1.1666413
• Hestenes, David (1967), "Real Spinor Fields", Journal of Mathematical Physics, 8 (4): 798–808, Bibcode:1967JMP.....8..798H, doi:10.1063/1.1705279
• Joot, Peeter (6 March 2023). Exploring Physics with Geometric Algebra, Book II (PDF) (V0.1.5 ed.). Retrieved 16 May 2023.

• 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