Spacetime algebra

Inmathematical physics,spacetime algebra(STA) is the application ofClifford algebraCl_1,3(R), or equivalently thegeometric algebraG(M⁴)to physics.Spacetimealgebra provides a "unified, coordinate-free formulation for all ofrelativistic physics,including theDirac equation,Maxwell equationandGeneral Relativity"and" reduces the mathematical divide betweenclassical,quantumandrelativistic physics."

Spacetime algebra is avector spacethat allows not onlyvectors,but alsobivectors(directed quantities describing rotations associated with rotations or particular planes, such as areas, or rotations) orblades(quantities associated with particular hyper-volumes) to be combined, as well asrotated,reflected,orLorentz boosted.It is also the natural parent algebra ofspinorsin special relativity.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.

In comparison to related methods, STA andDirac algebraare both Clifford Cl_1,3algebras, but STA usesreal numberscalarswhile Dirac algebra usescomplex numberscalars. The STA spacetime split is similar to thealgebra of physical space(APS, Pauli algebra)approach. APS represents spacetime as aparavector,a combined 3-dimensional vector space and a 1-dimensional scalar.

Structure

For any pair of STA vectors,${\textstyle a,b}$,there is avector (geometric) product${\textstyle ab}$,inner (dot) product${\textstyle a\cdot b}$andouter (exterior, wedge) product${\textstyle a\wedge b}$.The vector product is a sum of an inner and outer product:

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

The inner product generates a real number (scalar), and the outer product generates a bivector. The vectors${\textstyle a}$and${\textstyle b}$are orthogonal if their inner product is zero; vectors${\textstyle a}$and${\textstyle b}$are parallel if their outer product is zero.

Theorthonormal basisvectors are atimelikevector${\textstyle \gamma _{0}}$and 3spacelikevectors${\textstyle \gamma _{1},\gamma _{2},\gamma _{3}}$.TheMinkowski metrictensor's nonzero terms are the diagonal terms,${\textstyle (\eta _{00},\eta _{11},\eta _{22},\eta _{33})=(1,-1,-1,-1)}$.For${\textstyle \mu,\nu =0,1,2,3}$:

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

TheDirac matricesshare these properties, and STA is equivalent to the algebra generated by the Dirac matrices over the field of real numbers;explicit matrix representation is unnecessary for STA.

Products of the basis vectors generate atensor basiscontaining one scalar$\{1\}$,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 onepseudoscalar$\{I\}$with${\textstyle I=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}$.The pseudoscalarcommuteswith alleven-gradeSTA elements, butanticommuteswith allodd-gradeSTA elements.

Subalgebra

STA'seven-graded elements(scalars, bivectors, pseudoscalar) form a Clifford Cl_3,0(R)even subalgebraequivalent to the APS or Pauli algebra.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${\textstyle (I\sigma _{1},I\sigma _{2},I\sigma _{3})}$.The Pauli matrices,${\textstyle {\hat {\sigma }}_{1},{\hat {\sigma }}_{2},{\hat {\sigma }}_{3}}$,are a matrix representation for${\textstyle \sigma _{1},\sigma _{2},\sigma _{3}}$.For any pair of${\textstyle (\sigma _{1},\sigma _{2},\sigma _{3})}$,the nonzero inner products are${\textstyle \sigma _{1}\cdot \sigma _{1}=\sigma _{2}\cdot \sigma _{2}=\sigma _{3}\cdot \sigma _{3}=1}$,and the nonzero outer products are:

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

The sequence of algebra to even subalgebra continues as algebra of physical space, quaternion algebra, complex numbers and real numbers.

Division

A nonzero vector${\textstyle a}$is anull vector(degree 2nilpotent) if${\textstyle a^{2}=0}$.An example is${\textstyle a=\gamma ^{0}+\gamma ^{1}}$.Null vectors are tangent to thelight cone(null cone).An element${\textstyle b}$is anidempotentif${\textstyle b^{2}=b}$.Two idempotents${\textstyle b_{1}}$and${\textstyle b_{2}}$are orthogonal idempotents if${\textstyle b_{1}b_{2}=0}$.An example of an orthogonal idempotent pair is${\tfrac {1}{2}}(1+\gamma _{0}\gamma _{k})$and${\tfrac {1}{2}}(1-\gamma _{0}\gamma _{k})$with${\textstyle k=1,2,3}$.Proper zero divisors are nonzero elements whose product is zero such as null vectors or orthogonal idempotents.Adivision algebrais 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. The only associative division algebras are the real numbers, complex numbers and quaternions.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$c^{-1}=(c\cdot c)^{-1}c$.

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:

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

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

A vector${\textstyle a}$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$\mu =0,1,2,3$,according to theEinstein notation.The inner product of vector and basis vectors or reciprocal basis vectors generates the vector components.

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

Themetricandindex gymnasticsraise or lower indices:

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

Spacetime gradient

The spacetime gradient, like the gradient in a Euclidean space, is defined such that thedirectional derivativerelationship is satisfied:

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

This requires the definition of the gradient to be

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

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

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

Spacetime split

Spacetime split – examples:

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

$p\gamma _{0}=E+\mathbf {p}$

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

where$\gamma$is theLorentz factor

$\nabla \gamma _{0}=\partial _{t}-{\vec {\nabla }}$

In STA, aspacetime splitis 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 thealgebra of physical spaceand
• a projection of the 4D space onto the chosen time axis, yielding a 1-dimensional space of scalars, representing the scalar time.

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

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

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${\textstyle (\gamma )}$

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

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

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$\sigma _{k}=\gamma _{k}\gamma ^{0}$and$\sigma ^{k}=\gamma _{0}\gamma ^{k}$must be used.

Transformations

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

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

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

For a given spacelike bivector,$\beta ^{2}=-1$,soEuler's formulaapplies,giving the rotation

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

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

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

Interpreting this equation, these rotations along the time direction are simplyhyperbolic rotations.These are equivalent toLorentz boostsin special relativity.

Both of these transformations are known asLorentz transformations,and the combined set of all of them is theLorentz 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.

Any spacetime element${\textstyle A}$is transformed by multiplication with the pseudoscalar to form itsdualelement${\textstyle AI}$.Duality rotationtransforms spacetime element${\textstyle A}$to element${\textstyle A^{\prime }}$through angle${\textstyle \phi }$with pseudoscalar${\textstyle I}$is:

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

Duality rotation occurs only fornon-singularClifford algebra, non-singular meaning a Clifford algebra containing pseudoscalars with a non-zero square.

Grade involution (main involution, inversion)transforms every r-vector${\textstyle A_{r}}$to${\textstyle A_{r}^{\ast }}$:

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

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

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

Clifford conjugationof a spacetime element${\textstyle A}$combines reversion and grade involution transformations, indicated as${\textstyle {\tilde {A}}}$:

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

The grade involution, reversion and Clifford conjugation transformations areinvolutions.

Classical electromagnetism

The Faraday bivector

In STA, theelectric fieldandmagnetic fieldcan be unified into a single bivector field, known as the Faraday bivector, equivalent to theFaraday tensor.It is defined as:

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

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

$F=E^{i}\sigma _{i}+IcB^{i}\sigma _{i}=E^{1}\gamma _{1}\gamma _{0}+E^{2}\gamma _{2}\gamma _{0}+E^{3}\gamma _{3}\gamma _{0}-cB^{1}\gamma _{2}\gamma _{3}-cB^{2}\gamma _{3}\gamma _{1}-cB^{3}\gamma _{1}\gamma _{2}.$

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

${\begin{aligned}E={\frac {1}{2}}\left(F-\gamma _{0}F\gamma _{0}\right),\\IcB={\frac {1}{2}}\left(F+\gamma _{0}F\gamma _{0}\right).\end{aligned}}$

The$\gamma _{0}$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.

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:

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

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

Maxwell's equation

STA formulatesMaxwell's equationsin a simpler form as one equation,rather than the 4 equations ofvector calculus.Similarly to the above field bivector, the electriccharge densityandcurrent densitycan be unified into a single spacetime vector, equivalent to afour-vector.As such, the spacetime current$J$is given by

$J=c\rho \gamma _{0}+J^{i}\gamma _{i},$

where the components$J^{i}$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$\gamma _{0}$.

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.

Maxwell's equation:

$\nabla F=\mu _{0}cJ$

The fact that these quantities are all covariant objects in the STA automatically guaranteesLorentz covarianceof 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 theconservation of charge.Using the fact that for any bivector field, thedivergenceof its spacetime gradient is$0$,one can perform the following manipulation:

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

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 theLorentz forceon a charged particle can also be considerably simplified using STA.

Lorentz force on a charged particle:

${\mathcal {F}}=qF\cdot v$

Potential formulation

In the standard vector calculus formulation, two potential functions are used: theelectric scalar potential,and themagnetic vector potential.Using the tools of STA, these two objects are combined into a single vector field$A$,analogous to theelectromagnetic four-potentialin tensor calculus. In STA, it is defined as

$A={\frac {\phi }{c}}\gamma _{0}+A^{k}\gamma _{k}$

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

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

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

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

$A'=A+\nabla \Lambda$

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

$\nabla \wedge \left(A+\nabla \Lambda \right)=\nabla \wedge A+\nabla \wedge \nabla \Lambda =\nabla \wedge A.$

This phenomenon is calledgauge freedom.The process of choosing a suitable function$\Lambda$to make a given problem simplest is known asgauge fixing.However, in relativistic electrodynamics, theLorenz conditionis often imposed, where$\nabla \cdot {\vec {A}}=0$.

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

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

Substituting in this result, one arrives at the potential formulation of electromagnetism in STA:

Potential equation:

$\nabla ^{2}A=\mu _{0}J$

Lagrangian formulation

Analogously to the tensor calculus formalism, the potential formulation in STA naturally leads to an appropriateLagrangian density.

Electromagnetic Lagrangian density:

${\mathcal {L}}={\frac {1}{2}}\epsilon _{0}F^{2}-J\cdot A$

The multivector-valuedEuler-Lagrange equationsfor 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:

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

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

$\nabla \cdot A=0.$

This process can be done regardless of the chosen gauge, but this makes the resulting process considerably clearer. Due to the structure of thegeometric product,using this condition results in$\nabla \wedge A=\nabla A$.

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

The Pauli equation

STA allows the description of thePauli particlein terms of arealtheory in place of a matrix theory. The matrix theory description of the Pauli particle is:

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

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

The STA approach transforms the matrix spinor representation${\textstyle |\psi \rangle }$to the STA representation${\textstyle \psi }$using elements,${\textstyle \mathbf {\sigma _{1},\sigma _{2},\sigma _{3}} }$,of the even-graded spacetime subalgebra and the pseudoscalar$I=\sigma _{1}\sigma _{2}\sigma _{3}$:

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

The Pauli particle is described by thereal Pauli–Schrödinger equation:

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

where now$\psi$is an even multi-vector of the geometric algebra, and the Schrödinger Hamiltonian is$H_{S}$.Hestenes refers to this as thereal Pauli–Schrödinger theoryto emphasize that this theory reduces to the Schrödinger theory if the term that includes the magnetic field is dropped.The vector${\textstyle \sigma _{3}}$is an arbitrarily selected fixed vector; a fixed rotation can generate any alternative selected fixed vector${\textstyle \sigma _{3}^{\prime }}$.

The Dirac equation

STA enables a description of theDirac particlein terms of arealtheory in place of a matrix theory. The matrix theory description of the Dirac particle is:

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

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

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

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

Following Hestenes' derivation, the Dirac particle is described by the equation:

Dirac equation in STA:

$\nabla \psi \,I\sigma _{3}-e\mathbf {A} \psi =m\psi \gamma _{0}$

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

Dirac spinors

A relativisticDirac spinor${\textstyle \psi }$can be expressed as:

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

where, according to its derivation byDavid Hestenes,$\psi =\psi (x)$is an even multivector-valued function on spacetime,$R=R(x)$is a unimodular spinor or "rotor",and$\rho =\rho (x)$and$\beta =\beta (x)$are scalar-valued functions.In this construction, the components of$\psi$directly correspond with the components of aDirac spinor,both having 8 scalar degrees of freedom.

This equation is interpreted as connecting spin with the imaginary pseudoscalar.

The rotor,$R$,Lorentz transforms the frame of vectors$\gamma _{\mu }$into another frame of vectors$e_{\mu }$by the operation$e_{\mu }=R\gamma _{\mu }R^{\dagger }$;note that${\textstyle R^{\dagger }}$indicates thereverse transformation.

This has been extended to provide a framework for locally varying vector- and scalar-valued observables and support for theZitterbewegunginterpretation of quantum mechanics originally proposed bySchrödinger.

Hestenes has compared his expression for$\psi$with Feynman's expression for it in the path integral formulation:

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

where$\Phi _{\lambda }$is the classical action along the$\lambda$-path.

Using the spinors, the current density from the field can be expressed by

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

Symmetries

Global phase symmetryis a constant global phase shift of the wave function that leaves the Dirac equation unchanged.Local phase symmetryis a spatially varying phase shift that leaves the Dirac equation unchanged if accompanied by agauge transformationof theelectromagnetic four-potentialas expressed by these combined substitutions.

$\psi \mapsto \psi e^{\alpha (x)I\sigma _{3}},\quad eA\mapsto eA-\nabla \alpha (x)$

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

Researchers have applied STA and related Clifford algebra approaches to gauge theories,electroweakinteraction,Yang–Mills theory,and thestandard model.

The discrete symmetries areparity${\textstyle ({\hat {P}})}$,charge conjugation${\textstyle ({\hat {C}})}$andtime reversal${\textstyle ({\hat {T}})}$applied to wave function${\textstyle \psi }$.These effects are:

${\begin{aligned}{\hat {P}}|\psi \rangle &\mapsto \gamma _{0}\psi (\gamma _{0}x\gamma _{0})\gamma _{0}\\{\hat {C}}|\psi \rangle &\mapsto \psi \sigma _{1}\\{\hat {T}}|\psi \rangle &\mapsto I\gamma _{0}\psi (\gamma _{0}x\gamma _{0})\gamma _{1}\end{aligned}}$

General relativity

General relativity

Main article:Gauge theory gravity

Researchers have applied STA and related Clifford algebra approaches to relativity, gravity and cosmology.Thegauge theory gravity(GTG) uses STA to describe an induced curvature onMinkowski spacewhile admitting agauge symmetryunder "arbitrary smooth remapping of events onto spacetime" leading to this geodesic equation.

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

and the covariant derivative

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

where$\omega$is the connection associated with the gravitational potential, and$\Omega$is an external interaction such as an electromagnetic field.

The theory shows some promise for the treatment of black holes, as its form of theSchwarzschild solutiondoes not break down at singularities; most of the results ofgeneral relativityhave been mathematically reproduced, and the relativistic formulation ofclassical electrodynamicshas been extended toquantum mechanicsand theDirac equation.

See also

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

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• 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,S2CID119389813
• 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.ISBN978-1-4020-0593-0.
• O'Donnell, Peter J. (2003).Introduction to 2-spinors in General Relativity.World Scientific.ISBN978-981-279-531-1.
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• Vaz, Jayme; da Rocha, Roldão (2016).An Introduction to Clifford Algebras and Spinors.Oxford University Press.ISBN978-0-19-108578-9.
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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 Algebracourse-notes, see especially part 2.
• Cambridge University Geometric Algebra group
• Geometric Calculus research and development