Spacetime algebra
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.
[2]: 333 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.
APS represents spacetime as a paravector, a combined 3-dimensional vector space and a 1-dimensional scalar.
: The Dirac matrices share these properties, and STA is equivalent to the algebra generated by the Dirac matrices over the field of real numbers;[1]: x explicit matrix representation is unnecessary for STA.
[4]: 6 STA's even-graded elements (scalars, bivectors, pseudoscalar) form a Clifford Cl3,0(R) even subalgebra equivalent to the APS or Pauli algebra.
See the illustration of space-time algebra spinors in Cl+(1,3) under the octonionic product as a Fano plane.
Proper zero divisors are nonzero elements whose product is zero such as null vectors or orthogonal idempotents.
satisfying these equations: [1]: 63 These reciprocal frame vectors differ only by a sign, with
, these partials are In STA, 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: This is achieved by pre-multiplication 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
For forms of the spacetime split that work in either signature, alternate definitions in which
, so Euler's formula applies,[2]: 401 giving the rotation For a given timelike bivector,
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.
[2]: 233 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: The scalar part corresponds to the Lagrangian density for the electromagnetic field, and the pseudoscalar part is a less-often seen Lorentz invariant.
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
The fact that these quantities are all covariant objects in the STA 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 electromagnetic field, the form of the Lorentz force on a charged particle can also be considerably simplified using STA.
Using the tools of STA, these two objects are combined into a single vector field
as the original, due to the fact that This phenomenon is called gauge freedom.
Substituting in this result, one arrives at the potential formulation of electromagnetism in STA:[2]: 232
Analogously to the tensor calculus formalism, the potential formulation in STA naturally leads to an appropriate Lagrangian density.
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:[25]: 440 To begin to re-derive the potential equation from this form, it is simplest to work in the Lorenz gauge, setting[2]: 232 This process can be done regardless of the chosen gauge, but this makes the resulting process considerably clearer.
These are then combined to represent the full geometric algebra Dirac bispinor
directly correspond with the components of a Dirac spinor, both having 8 scalar degrees of freedom.
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.
[39] Using the spinors, the current density from the field can be expressed by[40]: 8 Global phase symmetry is a constant global phase shift of the wave function that leaves the Dirac equation unchanged.
[41]: 41–48 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.
[43]: 1343 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.
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.
