Spacetime algebra

In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl_1,3(R), or equivalently the geometric algebra Template:Nowrap 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".Template:Sfn

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.Template:Sfn It is also the natural parent algebra of spinors in special relativity.Template:Sfn 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.Template:Sfn

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 scalarsTemplate: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.Template:Sfn

Structure

For any pair of STA vectors, Template:Tmath and Template:Tmath, there is a geometric product Template:Tmath, scalar ('inner') product Template:Tmath and exterior ('wedge', 'outer') product Template:Tmath. The vector product is a sum of a scalar and exterior product:Template:Sfn

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

The scalar product generates a real number (scalar), and the exterior product generates a bivector. The vectors Template:Tmath and Template:Tmath are orthogonal if their scalar product is zero; vectors Template:Tmath and Template:Tmath are parallel if their exterior product is zero.Template:Sfn

The orthonormal basis vectors are a timelike vector Template:Tmath and 3 spacelike vectors ${\gamma }_1,{\gamma }_2,{\gamma }_3$. The Minkowski metric tensor's nonzero terms are the diagonal terms, Template:Tmath. For Template:Tmath:

${\gamma }_{\mu }\cdot {\gamma }_{\nu }=\frac{{\gamma }_{\mu }{\gamma }_{\nu }+{\gamma }_{\nu }{\gamma }_{\mu }}{2}={\eta }_{\mu \nu },{\gamma }_0\cdot {\gamma }_0=1,{\gamma }_1\cdot {\gamma }_1={\gamma }_2\cdot {\gamma }_2={\gamma }_3\cdot {\gamma }_3=-1,\text{ otherwise }{\gamma }_{\mu }{\gamma }_{\nu }=-{\gamma }_{\nu }{\gamma }_{\mu }$

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

Products of the basis vectors generate a tensor basis containing one scalar Template:Tmath, four vectors $\{{\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3\}$, six bivectors $\{{\gamma }_0{\gamma }_1,{\gamma }_0{\gamma }_2,{\gamma }_0{\gamma }_3,{\gamma }_1{\gamma }_2,{\gamma }_2{\gamma }_3,{\gamma }_3{\gamma }_1\}$, four pseudovectors (trivectors) $\{I{\gamma }_0,I{\gamma }_1,I{\gamma }_2,I{\gamma }_3\}$ and one pseudoscalar $\{I\}$ with Template:Tmath.Template:Sfn The pseudoscalar commutes with all even-grade STA elements, but anticommutes with all odd-grade STA elements.Template:Sfn

Subalgebra

This is an illustration of space-time algebra spinors in ClTemplate:Su(R) under the octonionic product as a Fano plane

The associated octonion multiplication tables in e_n and STA form.

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.Template:Sfn The STA bivectors are equivalent to the APS vectors and pseudovectors. The STA subalgebra becomes more explicit by renaming the STA bivectors $({\gamma }_1{\gamma }_0,{\gamma }_2{\gamma }_0,{\gamma }_3{\gamma }_0)$ as $({\sigma }_1,{\sigma }_2,{\sigma }_3)$ and the STA bivectors $({\gamma }_3{\gamma }_2,{\gamma }_1{\gamma }_3,{\gamma }_2{\gamma }_1)$ as $(I{\sigma }_1,I{\sigma }_2,I{\sigma }_3)$.Template:SfnTemplate:Sfn The Pauli matrices, ${\hat{\sigma }}_1,{\hat{\sigma }}_2,{\hat{\sigma }}_3$, are a matrix representation for ${\sigma }_1,{\sigma }_2,{\sigma }_3$.Template:Sfn For any pair of $({\sigma }_1,{\sigma }_2,{\sigma }_3)$, the nonzero scalar products are ${\sigma }_1\cdot {\sigma }_1={\sigma }_2\cdot {\sigma }_2={\sigma }_3\cdot {\sigma }_3=1$, and the nonzero exterior products are:Template:SfnTemplate:Sfn

$\begin{array}{cc}{\sigma }_1\wedge {\sigma }_2& =I{\sigma }_3\\ {\sigma }_2\wedge {\sigma }_3& =I{\sigma }_1\\ {\sigma }_3\wedge {\sigma }_1& =I{\sigma }_2\end{array}$

The sequence of algebra to even subalgebra continues as algebra of physical space, quaternion algebra, complex numbers and real numbers. The even STA subalgebra ClTemplate:Su(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 ClTemplate:Su(R) under the octonionic product as a Fano plane.Template:Sfn

Division

A nonzero vector Template:Tmath is a null vector (degree 2 nilpotent) if Template:Tmath.Template:Sfn An example is Template:Tmath. Null vectors are tangent to the light cone (null cone).Template:Sfn An element Template:Tmath is an idempotent if Template:Tmath.Template:Sfn Two idempotents Template:Tmath and Template:Tmath are orthogonal idempotents if Template:Tmath.Template:Sfn An example of an orthogonal idempotent pair is $\frac{1}{2}(1+{\gamma }_0{\gamma }_k)$ and $\frac{1}{2}(1-{\gamma }_0{\gamma }_k)$ with Template:Tmath. Proper zero divisors are nonzero elements whose product is zero such as null vectors or orthogonal idempotents.Template:Sfn 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.Template:SfnTemplate:SfnTemplate:Efn The only associative division algebras are the real numbers, complex numbers and quaternions.Template:Sfn As STA is not a division algebra, some STA elements may lack an inverse; however, division by the non-null vector $c$ may be possible by multiplication by its inverse, defined as Template:Tmath.Template:Sfn

Reciprocal frame

Associated with the orthogonal basis $\{{\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3\}$ is the reciprocal basis set $\{{\gamma }^0,{\gamma }^1,{\gamma }^2,{\gamma }^3\}$ satisfying these equations:Template:Sfn

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

These reciprocal frame vectors differ only by a sign, with Template:Tmath, but Template:Tmath, Template:Tmath, Template:Tmath.

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

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

The metric and index gymnastics raise or lower indices:

$\begin{array}{cc}{\gamma }_{\mu }& ={\eta }_{\mu \nu }{\gamma }^{\nu },\mu ,\nu =0,1,2,3\\ {\gamma }^{\mu }& ={\eta }^{\mu \nu }{\gamma }_{\nu },\mu ,\nu =0,1,2,3\end{array}$

Spacetime gradient

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

$a\cdot \mathrm{\nabla }F(x)=\underset{\tau \to 0}{lim}\frac{F(x+a\tau )-F(x)}{\tau }.$

This requires the definition of the gradient to be

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

Written out explicitly with Template:Tmath, these partials are

${\partial }_0=\frac{1}{c}\frac{\partial }{\partial t},{\partial }_k=\frac{\partial }{\partial x^k}.$

Space–time split

Space–time split – examples:

$x{\gamma }_0=x^0+𝐱$

$p{\gamma }_0=E+𝐩$Template:Sfn

$v{\gamma }_0=\gamma (1+𝐯)$Template:Sfn

where $\gamma$ is the Lorentz factor

$\mathrm{\nabla }{\gamma }_0={\partial }_t-\overrightarrow{\mathrm{\nabla }}$Template:Sfn

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.Template:Sfn

This is achieved by left-multiplication or right-multiplication by a timelike basis vector Template:Tmath, which serves to split a four vector into a scalar timelike and a bivector spacelike component, in the reference frame co-moving with Template:Tmath. With Template:Tmath, we have

$\begin{array}{cc}x{\gamma }_0& =x^0+x^k{\gamma }_k{\gamma }_0\\ {\gamma }_{0}x& =x^0-x^k{\gamma }_k{\gamma }_0\end{array}$

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 (Template:Tmath)Template:Sfn

As these bivectors Template:Tmath square to Template:Tmath, they serve as a spatial basis. Utilizing the Pauli matrix notation, these are written Template:Tmath. Spatial vectors in STA are denoted in boldface; then with Template:Tmath and Template:Tmath, the Template:Tmath-space–time split Template:Tmath, and its reverse Template:Tmath are:

$\begin{array}{cc}x{\gamma }_0& =x^0+x^k{\sigma }_k=ct+𝐱\\ {\gamma }_{0}x& =x^0-x^k{\sigma }_k=ct-𝐱\end{array}$

However, the above formulas only work in the Minkowski metric with signature Template:Nowrap. For forms of the space–time split that work in either signature, alternate definitions in which Template:Tmath and Template:Tmath must be used.

Transformations

To rotate a vector $v$ in geometric algebra, the following formula is used:Template:Sfn

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

where $\theta$ is the angle to rotate by, and Template:Tmath is the bivector representing the plane of rotation normalized so that Template:Tmath.

For a given spacelike bivector, Template:Tmath, so Euler's formula applies,Template:Sfn giving the rotation

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

For a given timelike bivector, Template:Tmath, so a "rotation through time" uses the analogous equation for the split-complex numbers:

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

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.Template:Sfn

Any spacetime element Template:Tmath is transformed by multiplication with the pseudoscalar to form its Hodge dual Template:Tmath.Template:Sfn Duality rotation transforms spacetime element Template:Tmath to element Template:Tmath through angle Template:Tmath with pseudoscalar Template:Tmath is:Template:Sfn

$A^{\prime }=e^{I\varphi }A.$

Duality rotation occurs only for non-singular Clifford algebra, non-singular meaning a Clifford algebra containing pseudoscalars with a non-zero square.Template:Sfn

Grade involution (main involution, inversion) transforms every Template:Tmath-vector Template:Tmath to Template:Tmath:Template:SfnTemplate:Sfn

$A_r^{\ast }=(-1{)}^r{A}_r.$

Reversion transformation occurs by decomposing any spacetime element as a sum of products of vectors and then reversing the order of each product.Template:SfnTemplate:Sfn For multivector $A$ arising from a product of vectors, $a_1{a}_2\dots a_{r-1}{a}_r$ the reversion is $A^{†}$:

$A=a_1{a}_2\dots a_{r-1}{a}_r,A^{†}=a_r{a}_{r-1}\dots a_2{a}_1.$

Clifford conjugation of a spacetime element Template:Tmath combines reversion and grade involution transformations, indicated as Template:Tmath:Template:Sfn

$\tilde{A}=A^{\ast †}$

The grade involution, reversion and Clifford conjugation transformations are involutions.Template:Sfn

Classical electromagnetism

Faraday bivector

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.Template:Sfn It is defined as:

$F=\overrightarrow{E}+Ic\overrightarrow{B},$

where $E$ and $B$ are the usual electric and magnetic fields, and $I$ is the STA pseudoscalar.Template:Sfn Alternatively, expanding $F$ in terms of components, $F$ is defined that

$F=E^i{\sigma }_i+Ic{B}^i{\sigma }_i=E^1{\gamma }_1{\gamma }_0+E^2{\gamma }_2{\gamma }_0+E^3{\gamma }_3{\gamma }_0-c{B}^1{\gamma }_2{\gamma }_3-c{B}^2{\gamma }_3{\gamma }_1-c{B}^3{\gamma }_1{\gamma }_2.$

The separate $\overrightarrow{E}$ and $\overrightarrow{B}$ fields are recovered from $F$ using

$\begin{array}{c}E=\frac{1}{2}(F-{\gamma }_{0}F{\gamma }_0),\\ IcB=\frac{1}{2}(F+{\gamma }_{0}F{\gamma }_0).\end{array}$

The Template:Tmath 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.Template:Sfn

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:

$F^2=E^2-c^2{B}^2+2Ic\overrightarrow{E}\cdot \overrightarrow{B}.$

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

Maxwell's equation

STA formulates Maxwell's equations in a simpler form as one equation,Template:Sfn rather than the 4 equations of vector calculus.Template:Sfn 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 Template:Tmath is given byTemplate:Sfn

$J=c\rho {\gamma }_0+J^i{\gamma }_i,$

where the components Template:Tmath 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 Template:Tmath.

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. Template:Sfn Template:Equation box 1

The fact that these quantities are all covariant objects in the STA automatically ensures 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 Template:Tmath, one can perform the following manipulation:Template:Sfn

$\begin{array}{cc}\mathrm{\nabla }\cdot \left[\mathrm{\nabla }F\right]& =\mathrm{\nabla }\cdot \left[{\mu }_{0}cJ\right]\\ 0& =\mathrm{\nabla }\cdot J.\end{array}$

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.Template:Sfn

Potential formulation

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 $A$, analogous to the electromagnetic four-potential in tensor calculus. In STA, it is defined as

$A=\frac{\varphi }{c}{\gamma }_0+A^k{\gamma }_k,$

where Template:Tmath is the scalar potential, and Template:Tmath are the components of the magnetic potential.

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

$\frac{1}{c}F=\mathrm{\nabla }\wedge A.$

However, this definition is not unique. For any twice-differentiable scalar function Template:Tmath, the potential given by

$A^{\prime }=A+\mathrm{\nabla }\mathrm{\Lambda }$

will also give the same $F$ as the original, due to the fact that

$\mathrm{\nabla }\wedge (A+\mathrm{\nabla }\mathrm{\Lambda })=\mathrm{\nabla }\wedge A+\mathrm{\nabla }\wedge \mathrm{\nabla }\mathrm{\Lambda }=\mathrm{\nabla }\wedge A.$

This phenomenon is called gauge freedom. The process of choosing a suitable function Template:Tmath to make a given problem simplest is known as gauge fixing. However, in relativistic electrodynamics, the Lorenz condition is often imposed, where Template:Tmath.Template:Sfn

To reformulate the STA Maxwell equation in terms of the potential Template:Tmath, Template:Tmath is first replaced with the above definition.

$\begin{array}{cc}\frac{1}{c}\mathrm{\nabla }F& =\mathrm{\nabla }(\mathrm{\nabla }\wedge A)\\ & =\mathrm{\nabla }\cdot (\mathrm{\nabla }\wedge A)+\mathrm{\nabla }\wedge (\mathrm{\nabla }\wedge A)\\ & ={\mathrm{\nabla }}^{2}A+(\mathrm{\nabla }\wedge \mathrm{\nabla })A={\mathrm{\nabla }}^{2}A+0\\ & ={\mathrm{\nabla }}^{2}A\end{array}$

Substituting in this result, one arrives at the potential formulation of electromagnetism in STA:Template:Sfn Template:Equation box 1

Lagrangian formulation

Analogously to the tensor calculus formalism, the potential formulation in STA naturally leads to an appropriate Lagrangian density.Template:Sfn

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:Template:Sfn

$\mathrm{\nabla }\frac{\partial ℒ}{\partial \left(\mathrm{\nabla }A\right)}-\frac{\partial ℒ}{\partial A}=0.$

To begin to re-derive the potential equation from this form, it is simplest to work in the Lorenz gauge, settingTemplate:Sfn

$\mathrm{\nabla }\cdot A=0.$

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 Template:Tmath.

After substituting in Template:Tmath, the same equation of motion as above for the potential field Template:Tmath is easily obtained.

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:Template:Sfn

$i\hslash {\partial }_t\mathrm{\Psi }=H_S\mathrm{\Psi }-\frac{e\hslash }{2mc}\hat{\sigma }\cdot 𝐁\mathrm{\Psi },$

where Template:Tmath is a spinor, Template:Tmath is the imaginary unit with no geometric interpretation, Template:Tmath are the Pauli matrices (with the 'hat' notation indicating that Template:Tmath is a matrix operator and not an element in the geometric algebra), and Template:Tmath is the Schrödinger Hamiltonian.

The STA approach transforms the matrix spinor representation Template:Tmath to the STA representation Template:Tmath using elements, ${\sigma }_1,{\sigma }_2,{\sigma }_3$, of the even-graded spacetime subalgebra and the pseudoscalar Template:Tmath:Template:SfnTemplate:Sfn

$|\psi ⟩=\left[\begin{array}{c}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{(}\mathrm{\theta }/\mathrm{2}\mathrm{)}{\mathrm{e}}^{\mathrm{-}\mathrm{i}\mathrm{\varphi }/\mathrm{2}}\\ \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{(}\mathrm{\theta }/\mathrm{2}\mathrm{)}{\mathrm{e}}^{\mathrm{+}\mathrm{i}\mathrm{\varphi }/\mathrm{2}}\end{array}\right]=\left[\begin{array}{c}{a}^0+i{a}^3\\ -a^2+i{a}^1\end{array}\right]\mapsto \psi =a^0+a^1𝐈{𝝈}_{\mathrm{𝟏}}+a^2𝐈{𝝈}_{\mathrm{𝟐}}+a^3𝐈{𝝈}_{\mathrm{𝟑}}$

The Pauli particle is described by the real Pauli–Schrödinger equation:Template:Sfn

${\partial }_t\psi I{\sigma }_3\hslash =H_S\psi -\frac{e\hslash }{2mc}𝐁\psi {\sigma }_3,$

where now Template:Tmath is an even multi-vector of the geometric algebra, and the Schrödinger Hamiltonian is Template:Tmath. 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.Template:Sfn The vector Template:Tmath is an arbitrarily selected fixed vector; a fixed rotation can generate any alternative selected fixed vector Template:Tmath.Template:Sfn

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:Template:Sfn

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

where Template:Tmath are the Dirac matrices and Template:Tmath is the imaginary unit with no geometric interpretation.

Using the same approach as for Pauli equation, the STA approach transforms the matrix upper spinor Template:Tmath and matrix lower spinor Template:Tmath of the matrix Dirac spinorTemplate:Tmath to the corresponding geometric algebra spinor representations Template:Tmath and Template:Tmath. These are then combined to represent the full geometric algebra Dirac spinor Template:Tmath.Template:Sfn

$|\psi ⟩=\left|\begin{array}{c}|{\psi }_U⟩\\ |{\psi }_L⟩\end{array}\right|\mapsto \psi ={\psi }_U+{\psi }_L{𝝈}_{\mathrm{𝟑}}$

Following Hestenes' derivation, the Dirac particle is described by the equation:Template:SfnTemplate:Sfn Template:Equation box 1 Here, Template:Tmath is the spinor field, Template:Tmath and Template:Tmath are elements of the geometric algebra, Template:Tmath is the electromagnetic four-potential, and Template:Tmath is the spacetime vector derivative.

Dirac spinors

A relativistic Dirac spinor $\psi$ can be expressed as:Template:SfnTemplate:SfnTemplate:Sfn

$\psi =R(\rho e^{i\beta }{)}^{\frac{1}{2}}$

where, according to its derivation by David Hestenes, Template:Tmath is an even multivector-valued function on spacetime, Template:Tmath is a unimodular spinor or "rotor",Template:Sfn and Template:Tmath and Template:Tmath are scalar-valued functions.Template:Sfn In this construction, the components of Template:Tmath 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.Template:Sfn

The rotor, Template:Tmath, Lorentz transforms the frame of vectors Template:Tmath into another frame of vectors Template:Tmath by the operation Template:Tmath;Template:Sfn note that Template:Tmath 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.Template:SfnTemplate:Sfn

Hestenes has compared his expression for Template:Tmath with Feynman's expression for it in the path integral formulation:

$\psi =e^{i{\mathrm{\Phi }}_{\lambda }/\hslash },$

where Template:Tmath is the classical action along the Template:Tmath-path.Template:Sfn

Using the spinors, the current density from the field can be expressed byTemplate:Sfn

$J^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi$

Symmetries

Global phase symmetry is a constant global phase shift of the wave function that leaves the Dirac equation unchanged.Template:Sfn 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.Template:Sfn

$\psi \mapsto \psi e^{\alpha (x)I{\sigma }_3},eA\mapsto eA-\mathrm{\nabla }\alpha (x)$

In these equations, the local phase transformation is a phase shift Template:Tmath at spacetime location Template:Tmath with pseudovector Template:Tmath and Template:Tmath of even-graded spacetime subalgebra applied to wave function Template:Tmath; the gauge transformation is a subtraction of the gradient of the phase shift Template:Tmath from the electromagnetic four-potential Template:Tmath with particle electric charge Template:Tmath.Template:Sfn

Researchers have applied STA and related Clifford algebra approaches to gauge theories, electroweak interaction, Yang–Mills theory, and the Standard Model.Template:Sfn

The discrete symmetries are parity (Template:Tmath), charge conjugation (Template:Tmath) and time reversal (Template:Tmath) applied to wave function Template:Tmath. These effects are:Template:Sfn

$\begin{array}{cc}\hat{P}|\psi ⟩& \mapsto {\gamma }_0\psi ({\gamma }_{0}x{\gamma }_0){\gamma }_0\\ \hat{C}|\psi ⟩& \mapsto \psi {\sigma }_1\\ \hat{T}|\psi ⟩& \mapsto I{\gamma }_0\psi ({\gamma }_{0}x{\gamma }_0){\gamma }_1\end{array}$

General relativity

General relativity

Template:Main Researchers have applied STA and related Clifford algebra approaches to relativity, gravity and cosmology.Template:Sfn 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.Template:SfnTemplate:SfnTemplate:SfnTemplate:Sfn

$\frac{d}{d\tau }R=\frac{1}{2}(\mathrm{\Omega }-\omega )R$

and the covariant derivative

$D_{\tau }={\partial }_{\tau }+\frac{1}{2}\omega ,$

where Template:Tmath is the connection associated with the gravitational potential, and Template:Tmath 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

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

References

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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