In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl_{1,3}(R), or equivalently the geometric algebra G(M^{4}). 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.
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.
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:
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.
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 – examples: |
^{[1]} |
^{[1]} |
where is the Lorentz factor |
^{[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:
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.
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.
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 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 }
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.
The form of the Lorentz force on a charged particle can also be considerably simplified using spacetime algebra.
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:
Analogously to the tensor calculus formalism, the potential formulation in STA naturally leads to an appropriate Lagrangian density.^{[4]}^{: 453 }
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.
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.
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]}
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]}
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]}
The current density from the field can be expressed by
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]}
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.