Spacetime algebra


In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4). According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime.

It is a vector space that allows not only vectors, but also bivectors (directed quantities associated with 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. It is also the natural parent algebra of spinors in 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.

Structure edit

The spacetime algebra may be built up from an orthogonal basis of one time-like vector   and three space-like vectors,  , with the multiplication rule


where   is the Minkowski metric with signature (+ − − −).

Thus,  ,  , otherwise  .

The basis vectors   share these properties with the Dirac matrices, but no explicit matrix representation need be used in STA.

This generates a basis of one scalar  , four vectors  , six bivectors  , four pseudovectors   and one pseudoscalar  , where  .

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.

Reciprocal frame edit

Associated with the orthogonal basis   is the reciprocal basis   for  , satisfying the relation


These reciprocal frame vectors differ only by a sign, with  , and   for  .

A vector may be represented in either upper or lower index coordinates   with summation over  , according to the Einstein notation, where the coordinates may be extracted by taking dot products with the basis vectors or their reciprocals.


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


Multivector division edit

The spacetime algebra is not a division algebra, because it contains idempotent elements   and nonzero zero divisors:  . 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.

Spacetime gradient edit

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


This requires the definition of the gradient to be


Written out explicitly with  , these partials are


Spacetime split edit

Spacetime split – examples:
where   is the Lorentz factor

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  , which serves to split a four vector into a scalar timelike and a bivector spacelike component, in the reference frame co-moving with  . With   we have


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


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   and   must be used.

Lorentz transformations edit

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


where   is the angle to rotate by, and   is the normalized bivector representing the plane of rotation so that  .

For a given spacelike bivector,  , so Euler's formula applies, giving the rotation


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


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.

Classical electromagnetism edit

The Faraday Bivector edit

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.[4] It is defined as


where   and   are the usual electric and magnetic fields, and   is the STA pseudoscalar.[4]: 232  Alternatively, expanding   in terms of components,   is defined that


The separate   and   fields are recovered from   using


The   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.[4]: 233 

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


The scalar part corresponds to the Lagrangian density for the electromagnetic field, and the pseudoscalar part is a less-often seen Lorentz invariant.[4]: 234 

Maxwell's Equation edit

Maxwell's equations can be formulated using spacetime algebra, in a simpler form than can be done with standard vector calculus.[citation needed] 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   is given by


where the components   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  .

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 spacetime algebra.[4]: 230 

Maxwell's equation:


The fact that these quantities are all covariant objects in the spacetime algebra 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  , one can perform the following manipulation:


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.

Also, the right-hand side, being the product of a vector[disambiguation needed] and a bivector, could have a pseudovector part, which would describe magnetic monopoles. Experiment suggests that these do not exist, which makes the equation somewhat asymmetric.

Lorentz Force edit

The form of the Lorentz force on a charged particle can also be considerably simplified using spacetime algebra.

Lorentz force on a charged particle:


Potential Formulation edit

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


where   is the scalar potential, and   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


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


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


This phenomenon is called gauge freedom. The process of choosing a suitable function   to make a given problem simplest is known as gauge fixing. However, in relativistic electrodynamics, the Lorenz condition is often imposed, where  .[4]: 231 

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


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

Potential equation:


Lagrangian formulation edit

Analogously to the tensor calculus formalism, the potential formulation in STA naturally leads to an appropriate Lagrangian density.[4]: 453 

Electromagnetic Lagrangian density:


The multivector-valued Euler-Lagrange equations for the field can be derived, and being loose with the mathematical rigour of taking the partial derivative with respect to something that is not a scalar, the relevant equations become[5]


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


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  .

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

The Pauli equation edit

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]


where   is a spinor,   is the imaginary unit with no geometric interpretation,   are the Pauli matrices (with the 'hat' notation indicating that   is a matrix operator and not an element in the geometric algebra), and   is the Schrödinger Hamiltonian. In the spacetime algebra the Pauli particle is described by the real Pauli–Schrödinger equation:[6]


where now   is the unit pseudoscalar  , and   and   are elements of the geometric algebra, with   an even multi-vector;   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.

The Dirac Equation edit

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]


where   are the Dirac matrices. In the spacetime algebra, following Hestenes' derivation, the Dirac particle is described by the equation:[7]

Dirac equation in STA:


Here,   is the spinor field,   and   are elements of the geometric algebra,   is the electromagnetic four-potential, and   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.[citation needed]

Dirac spinors edit

The relativistic quantum wavefunction is sometimes expressed as a spinor field, i.e.[citation needed]


where   is a bivector, and[8][9]


where, according to its derivation by David Hestenes,   is an even multivector-valued function on spacetime,   is a unimodular spinor (or “rotor”[10]), and   and   are scalar-valued functions.[8] In this construction, the components of   can be directly corresponded with the components of a Dirac spinor, which can be easily checked by the fact that both have 8 scalar degrees of freedom.

This equation is interpreted as connecting spin with the imaginary pseudoscalar.[11]   is viewed as a Lorentz rotation which a frame of vectors   into another frame of vectors   by the operation  ,[10] where the tilde symbol indicates the reverse (the reverse is often also denoted by the dagger symbol, see also Rotations in geometric algebra).

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.

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


where   is the classical action along the  -path.[8]

Physical observables edit

The current density from the field can be expressed by


U(1) gauge symmetry edit

The Dirac equation is symmetric under some global phase shift given by some constant phase shift  . Performing the transformation  , where   is the charge of the field and   is the unit pseudoscalar, the Dirac equation as shown above remains unchanged.

However, all physical observables from the equation are invariant under a stronger localized phase symmetry, where the phase shift can vary by an arbitrary amount over space. The localized phase shift is given by the scalar function  , and the transformation is given by


Where the fact that the transformation commutes with   comes from the fact that   decomposes into a scalar and pseudoscalar part, which both commute with elements of the even sub-algebra.

However, when using this stronger symmetry in the Dirac equation, the spacetime derivative   transforms into


using the product rule and chain rule. Since the transformed derivative does not have the same form as the original (i.e. it has an extra term that explicitly depends on the phase), the derivative prevents the equation from being locally phase invariant.

To fix this problem, one can introduce a gauge field  , which will be defined to transform in a way to remove the local phase dependency in the equation. If one defines the gauge field to transform under the same arbitrary phase shift   as  , and this is precisely where the   interaction term comes from.

At first, it may be unclear as to what this abstract gauge field actually represents, since it seems to have no observable effect other than changing the phase of the wave function. But when examining the consequences of its introduction, it becomes apparent that   is simply the electromagnetic four-potential, and that the constant   is simply the charge of the given particle in the field.[how?] As such, forcing the Dirac equation to be phase-invariant allows it to describe the electromagnetic interaction as a bonus.

Similar gauge fields exist in STA for the gauge transformations that govern the electroweak interaction, and the strong interaction, which has lead some to begin reformulating the standard model in this framework.[12]

General relativity edit

A new formulation of general relativity edit

Lasenby, Doran, and Gull of Cambridge University have proposed a new formulation of gravity, termed gauge theory gravity (GTG), wherein spacetime algebra is used to induce curvature on Minkowski space while admitting a gauge symmetry under "arbitrary smooth remapping of events onto spacetime" (Lasenby, et al.); a nontrivial derivation then leads to the geodesic equation,


and the covariant derivative


where   is the connection associated with the gravitational potential, and   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

References edit

  • 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, 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.
  1. ^ a b 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.
  2. ^ Lasenby & Doran 2002, p. 259
  3. ^ 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.
  4. ^ a b c d e f g Doran, Chris; Lasenby, Anthony (2003), Geometric Algebra for Physicists, Cambridge University Press, ISBN 978-0-521-48022-2
  5. ^ Joot, Peeter. "A multivector Lagrangian for Maxwell's equation" (PDF). Retrieved 4 June 2023.
  6. ^ a b See eqs. (75) and (81) in: Hestenes & Oersted Medal Lecture 2002
  7. ^ a b See eqs. (3.43) and (3.44) in: Doran, Chris; Lasenby, Anthony; Gull, Stephen; Somaroo, Shyamal; Challinor, Anthony (1996). Hawkes, Peter W. (ed.). Spacetime algebra and electron physics. Advances in Imaging and Electron Physics. Vol. 95. Academic Press. pp. 272–386, 292. ISBN 0-12-014737-8.
  8. ^ a b c See eq. (3.1) and similarly eq. (4.1), and subsequent pages, in: Hestenes, D. (2012) [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 Archived 2022-10-29 at the Wayback Machine)
  9. ^ See also eq. (5.13) of Gull, S.; Lasenby, A.; Doran, C. (1993). "Imaginary numbers are not real – the geometric algebra of spacetime" (PDF).
  10. ^ a b See eq. (205) in Hestenes, D. (June 2003). "Spacetime physics with geometric algebra" (PDF). American Journal of Physics. 71 (6): 691–714. Bibcode:2003AmJPh..71..691H. doi:10.1119/1.1571836. Archived from the original (PDF) on 2023-01-04. Retrieved 2012-02-24.
  11. ^ Hestenes, David (2003). "Oersted Medal Lecture 2002: Reforming the mathematical language of physics" (PDF). American Journal of Physics. 71 (2): 104. Bibcode:2003AmJPh..71..104H. CiteSeerX doi:10.1119/1.1522700. Archived from the original (PDF) on 2023-01-04. Retrieved 2012-02-25.
  12. ^ Brage Gording, Angnis Schmidt-May (18 Aug 2020). "The Unified Standard Model". Advances in Applied Clifford Algebras. 30 (4). arXiv:1909.05641. doi:10.1007/s00006-020-01082-8. S2CID 202565534.

External links edit

  • 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