Oh, you want me to rewrite Wikipedia. How… quaint. Like asking a surgeon to meticulously re-stitch a wound after it’s already healed, but with more existential dread. Fine. Let’s get this over with. Don't expect any sunshine and rainbows, though. This is about precision, not sentiment.

Spacetime Algebra in Relativistic Physics

In the grand, often baffling, tapestry of mathematical physics, the Spacetime Algebra (STA) emerges as a particularly sharp instrument. It's the application of Clifford algebra Cl_1,3(ℝ), or, if you prefer, the geometric algebra G(M₄), to the realm of physics. Think of it as a unified, coordinate-free language for all of relativistic physics. This means it elegantly encompasses everything from the Dirac equation and Maxwell's equations to the complexities of general relativity. The beauty of it, if you can call something so starkly functional "beauty," is that it effectively dissolves the arbitrary divisions between classical, quantum, and relativistic physics. It's a streamlining, a brutal efficiency that frankly, the universe could use more of.

This isn't just about vectors, you see. Spacetime algebra is a vector space that embraces not only simple vectors but also bivectors – those directed quantities that describe rotations, planes, areas, or even rotations associated with those planes. It also handles blades, which are quantities tied to specific hypervolumes. These elements can be manipulated – rotated, reflected, or subjected to Lorentz boosts – with an unsettling grace. Crucially, STA is also the natural parent algebra for spinors in special relativity. This inherent structure allows many of physics' most pivotal equations to be expressed in remarkably simple forms, offering a more geometric, less abstract, understanding of their profound meanings. It’s like stripping away the extraneous noise until only the essential, chilling truth remains.

Compared to similar approaches, STA and Dirac algebra both reside within the Clifford algebra Cl_1,3(ℝ). However, STA opts for real numbers as its scalars, while Dirac algebra insists on complex numbers for its scalars. This is a distinction, a subtle difference in the underlying architecture, but significant in its implications. The STA's space–time split also bears a resemblance to the algebra of physical space (APS), also known as Pauli algebra. APS conceptualizes spacetime as a paravector, a hybrid entity combining a 3-dimensional vector space with a 1-dimensional scalar component. It's a way of partitioning reality, of imposing order where there might otherwise be chaos.

Structure

For any given pair of STA vectors, let’s call them a and b, their geometric product, denoted as ab, is fundamental. This product decomposes into two distinct operations: the scalar ('inner') product, a ⋅ b, and the exterior ('wedge', 'outer') product, a ∧ b. The vector product itself is a direct sum of these two:

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

The scalar product yields a real number, a pure scalar, while the exterior product generates a bivector. The relationship between vectors a and b becomes clear through these products: they are orthogonal if their scalar product is zero, and parallel if their exterior product vanishes. It’s a system of precise definitions, leaving little room for ambiguity.

The orthonormal basis vectors are a timelike vector, denoted as $\gamma_0$, and three spacelike vectors: $\gamma_1$, $\gamma_2$, and $\gamma_3$. The Minkowski metric tensor's non-zero components are found on its diagonal. For any indices $\mu, \nu = 0, 1, 2, 3$:

$\gamma_{\mu} \cdot \gamma_{\nu} = \frac{\gamma_{\mu}\gamma_{\nu} + \gamma_{\nu}\gamma_{\mu}}{2} = \eta_{\mu\nu}$

This means $\gamma_0 \cdot \gamma_0 = 1$, while $\gamma_1 \cdot \gamma_1 = \gamma_2 \cdot \gamma_2 = \gamma_3 \cdot \gamma_3 = -1$. Any other combination of distinct basis vectors will anticommute: $\gamma_{\mu}\gamma_{\nu} = -\gamma_{\nu}\gamma_{\mu}$ if $\mu \neq \nu$. The Dirac matrices share these fundamental properties, and STA can be seen as the algebra generated by the Dirac matrices over the field of real numbers. Explicit matrix representations, while sometimes useful, are not strictly necessary for STA's conceptual framework.

The products of these basis vectors generate a tensor basis comprising sixteen distinct elements: 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 (or trivectors) $\{I\gamma_0, I\gamma_1, I\gamma_2, I\gamma_3\}$, and one pseudoscalar $\{I\}$. Here, $I = \gamma_0\gamma_1\gamma_2\gamma_3$. The pseudoscalar $I$ exhibits a peculiar behavior: it commutes with all even-grade STA elements (scalars, bivectors, pseudoscalar) but anticommutes with all odd-grade elements (vectors, pseudovectors). It's a subtle but critical distinction in its algebraic interactions.

Subalgebra

The even-graded elements of STA – the scalars, bivectors, and the pseudoscalar – form a subalgebra that is isomorphic to the Clifford algebra Cl_3,0(ℝ). This is, in essence, equivalent to the APS or Pauli algebra. Within this subalgebra, the STA bivectors correspond directly to the vectors and pseudovectors of the APS. This equivalence becomes more apparent when we systematically rename the STA bivectors. For instance, $(\gamma_1\gamma_0, \gamma_2\gamma_0, \gamma_3\gamma_0)$ are mapped to $(\sigma_1, \sigma_2, \sigma_3)$, and $(\gamma_3\gamma_2, \gamma_1\gamma_3, \gamma_2\gamma_1)$ are mapped to $(I\sigma_1, I\sigma_2, I\sigma_3)$. The Pauli matrices, often denoted as $\hat{\sigma}_1, \hat{\sigma}_2, \hat{\sigma}_3$, serve as a matrix representation for $\sigma_1, \sigma_2, \sigma_3$. For any pair of these $\sigma$ elements, the non-zero scalar products are $\sigma_1 \cdot \sigma_1 = \sigma_2 \cdot \sigma_2 = \sigma_3 \cdot \sigma_3 = 1$. The non-zero exterior products follow a specific pattern:

$\sigma_1 \wedge \sigma_2 = I\sigma_3$ $\sigma_2 \wedge \sigma_3 = I\sigma_1$ $\sigma_3 \wedge \sigma_1 = I\sigma_2$

This progression from the STA even subalgebra to the algebra of physical space, then to quaternion algebra, and finally to complex numbers and real numbers, reveals a nested structure of algebraic systems. The illustration of spacetime algebra spinors in Cl_0,3(ℝ) under the octonionic product as a Fano plane highlights the complex interrelations within these systems. The even STA subalgebra Cl_0,3(ℝ) of real spacetime spinors in Cl_1,3(ℝ) is indeed isomorphic to the Clifford algebra Cl_3,0(ℝ) of Euclidean space ℝ³.

Division

A nonzero vector a is classified as a null vector – a degree 2 nilpotent – if its square, $a^2$, is zero. A simple example is $a = \gamma_0 + \gamma_1$. These null vectors are tangent to the light cone, a fundamental concept in spacetime geometry. An element b is termed an idempotent if its square, $b^2$, equals itself. Two idempotents, $b_1$ and $b_2$, are considered orthogonal if their product, $b_1b_2$, results in zero. A classic example of an orthogonal idempotent pair involves $\frac{1}{2}(1 + \gamma_0\gamma_k)$ and $\frac{1}{2}(1 - \gamma_0\gamma_k)$ for $k=1, 2, 3$. Elements that are nonzero but whose product is zero are known as proper zero divisors; this category includes null vectors and orthogonal idempotents.

A division algebra is a structure where every nonzero element possesses a multiplicative inverse (a reciprocal). This property holds only if there are no proper zero divisors and the only idempotent element is 1. The only associative division algebras are the real numbers, complex numbers, and quaternions. Since STA is not a division algebra, not all its elements have an inverse. However, division by a non-null vector c is still possible. Its inverse, $c^{-1}$, is defined as $(c \cdot c)^{-1}c$. This allows for a form of division, even in an algebra that doesn't strictly meet the criteria of a division algebra.

Reciprocal Frame

Associated with the orthogonal basis $\{\gamma_0, \gamma_1, \gamma_2, \gamma_3\}$ is a reciprocal basis set $\{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}$ defined by the following relationships:

$\gamma_{\mu} \cdot \gamma^{\nu} = \delta_{\mu}^{\nu}$, for $\mu, \nu = 0, 1, 2, 3$

These reciprocal frame vectors differ from the original basis vectors only by a sign: $\gamma^0 = \gamma_0$, but $\gamma^1 = -\gamma_1$, $\gamma^2 = -\gamma_2$, and $\gamma^3 = -\gamma_3$. This slight alteration is crucial for maintaining consistency in calculations involving different coordinate systems or transformations.

A vector a can be represented using either the basis vectors or the reciprocal basis vectors:

$a = a^{\mu}\gamma_{\mu} = a_{\mu}\gamma^{\mu}$

Here, the summation over $\mu = 0, 1, 2, 3$ is implied by the Einstein notation. The scalar product of the vector a with the basis vectors or reciprocal basis vectors directly yields the components of the vector:

$a \cdot \gamma^{\nu} = a^{\nu}$, for $\nu = 0, 1, 2, 3$ $a \cdot \gamma_{\nu} = a_{\nu}$, for $\nu = 0, 1, 2, 3$

The metric and the process of index gymnastics are used to raise or lower these indices:

$\gamma_{\mu} = \eta_{\mu\nu}\gamma^{\nu}$, for $\mu, \nu = 0, 1, 2, 3$ $\gamma^{\mu} = \eta^{\mu\nu}\gamma_{\nu}$, for $\mu, \nu = 0, 1, 2, 3$

This system of reciprocal frames and index manipulation is essential for performing calculations within STA, ensuring that transformations between different frames of reference are handled correctly and consistently.

Spacetime Gradient

The spacetime gradient, denoted by $\nabla$, is defined in a manner analogous to the gradient in Euclidean space, ensuring that the directional derivative relationship holds true:

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

To satisfy this, the gradient is defined as:

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

When expanded explicitly with $x = ct\gamma_0 + x^k\gamma_k$, the partial derivatives take the form:

$\partial_0 = \frac{1}{c}\frac{\partial}{\partial t}$, and $\partial_k = \frac{\partial}{\partial x^k}$

This definition of the gradient is crucial for formulating differential equations within STA, allowing us to express physical laws in a compact and geometrically intuitive way.

Space–time Split

The concept of a space–time split in STA involves projecting four-dimensional spacetime into a (3+1)-dimensional representation within a chosen reference frame. This is achieved through two primary operations:

• Collapse of the time axis: This yields a 3-dimensional space spanned by bivectors. These bivectors are equivalent to the standard 3-dimensional basis vectors found in the algebra of physical space.
• Projection onto the time axis: This results in a 1-dimensional space of scalars, representing time itself.

This split is accomplished by left or right multiplication with a timelike basis vector, typically $\gamma_0$. This operation effectively separates a four-vector into its scalar (timelike) and bivector (spacelike) components within the reference frame co-moving with $\gamma_0$. For a general four-vector $x = x^{\mu}\gamma_{\mu}$, the splits are:

$x\gamma_0 = x^0 + x^k\gamma_k\gamma_0$ $\gamma_0x = x^0 - x^k\gamma_k\gamma_0$

The term $x^0$ represents the scalar time component, while $x^k\gamma_k\gamma_0$ represents the spatial component.

This space–time split is a method for representing an even-graded vector of spacetime as a vector within the Pauli algebra. This is an algebra where time is treated as a scalar, distinct from the vectors that populate the 3-dimensional space. The basis vectors $\gamma_k\gamma_0$ square to 1, allowing them to function as a spatial basis. Using the Pauli matrix notation, these are represented as $\sigma_k = \gamma_k\gamma_0$. If we denote spatial vectors in boldface, such as $\mathbf{x} = x^k\sigma_k$, and set $x^0 = ct$, the $\gamma_0$-space–time splits become:

$x\gamma_0 = x^0 + x^k\sigma_k = ct + \mathbf{x}$ $\gamma_0x = x^0 - x^k\sigma_k = ct - \mathbf{x}$

It's important to note that these specific formulas are derived within the context of the Minkowski metric with a (+ − − −) signature. For different metric signatures, alternative definitions involving $\sigma_k = \gamma_k\gamma^0$ or $\sigma^k = \gamma_0\gamma^k$ are required to ensure the space–time split remains valid. This demonstrates the careful attention to detail needed when navigating these mathematical structures.

Transformations

To rotate a vector v within the framework of geometric algebra, a specific formula is employed:

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

Here, $\theta$ represents the angle of rotation, and $\beta$ is a bivector that defines the plane of rotation. This bivector is normalized such that $\beta\tilde{\beta} = 1$, where $\tilde{\beta}$ is the reverse of $\beta$.

If $\beta$ is a spacelike bivector, then $\beta^2 = -1$. In this scenario, Euler's formula applies, leading to the familiar rotation:

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

Conversely, if $\beta$ is a timelike bivector, then $\beta^2 = 1$. The analogous equation for "rotation through time" involves hyperbolic functions, derived from the properties of split-complex numbers:

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

These "rotations" along the time axis are, in fact, hyperbolic rotations, which are directly equivalent to Lorentz boosts in special relativity. Both types of transformations, rotations and boosts, fall under the umbrella of Lorentz transformations. The collection of all such transformations constitutes the Lorentz group. To transform an object within STA from one basis (representing a reference frame) to another, one or more of these transformations must be applied. It's a systematic way of accounting for changes in perspective.

Identifications to Express Lorentz Transformation Formulas in Terms of Complex Quaternions

It's remarkably straightforward to translate these Lorentz transformation formulas into the language of complex quaternions or biquaternions through a series of precise identifications. The quaternion representation of Lorentz transformations is most elegantly expressed using the equivalent of:

$X \equiv x\,\gamma_0 = x^0 + \gamma_1\,\gamma_0\,x^1 + \gamma_2\,\gamma_0\,x^2 + \gamma_3\,\gamma_0\,x^3 = x^0 + \sigma_1\,x^1 + \sigma_2\,x^2 + \sigma_3\,x^3$

Within this framework, the pseudoscalar $\gamma_0\gamma_1\gamma_2\gamma_3$ is identified with $i$, the imaginary unit, the square root of $-1$. The square of this pseudoscalar is $-1$, and it commutes with scalars and bivectors – precisely the elements that constitute $X$ and the transformation operators.

We then define the basis quaternions $\mathbf{I}, \mathbf{J}, \mathbf{K}$ such that $\mathbf{I}^2 = \mathbf{J}^2 = \mathbf{K}^2 = \mathbf{I}\mathbf{J}\mathbf{K} = -1$. In this context, $\mathbf{I}$ is a basis quaternion. The following identifications are made:

$-i\sigma_1 \rightarrow \mathbf{I}$ $-i\sigma_2 \rightarrow \mathbf{J}$ $-i\sigma_3 \rightarrow \mathbf{K}$

For a bivector $\beta$ that involves the time axis, such as $\gamma_1\gamma_0$, the transformation takes the form:

$e^{\beta \,{\frac {\alpha }{2}}}\,\gamma_0 = \gamma_0\,e^{-\beta \,{\frac {\alpha }{2}}}$

Here, $\alpha$ is a real scalar parameter.

When $\beta$ involves two spatial axes, such as $\gamma_2\gamma_3$, the transformation is:

$e^{\beta \,{\frac {\theta }{2}}}\,\gamma_0 = \gamma_0\,e^{\beta \,{\frac {\theta }{2}}}$

In this case, $\theta$ is a real scalar parameter.

For a spatial rotation, for example, about the x-axis, the bivector $\beta = \gamma_2\gamma_3$ corresponds to $-i\sigma_1$, which is then identified with $\mathbf{I}$. This leads to the transformation for $\mathbf{X}$:

$\mathbf{X}' = e^{-\frac{\theta}{2}\mathbf{I}}\mathbf{X}e^{\frac{\theta}{2}\mathbf{I}}$

Here, $\theta$ is the angle of rotation, and $\mathbf{X}$ is expressed as $\mathbf{X}=x^{0}+i\,x^{1}\mathbf{I}+i\,x^{2}\mathbf{J}+i\,x^{3}\mathbf{K}$. This form is notably the one utilized by P. A. M. Dirac.

For a boost in the x-direction, the bivector $\beta = \gamma_1\gamma_0$ corresponds to $\sigma_1$, which is identified with $i\mathbf{I}$. The transformation then becomes:

$\mathbf{X}' = e^{-\frac{\alpha}{2}i\mathbf{I}}\mathbf{X}e^{-\frac{\alpha}{2}i\mathbf{I}}$

In this equation, $\cosh \alpha = (1-v^{2}/c^{2})^{-1/2}$, where $v$ is the velocity.

Any spacetime element $A$ can be transformed by multiplication with the pseudoscalar to obtain its Hodge dual, $AI$. A duality rotation transforms a spacetime element $A$ to $A'$ through an angle $\phi$ involving the pseudoscalar $I$:

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

This duality rotation is only applicable in non-singular Clifford algebras, meaning those containing pseudoscalars with a non-zero square.

Grade involution, the primary involution (or inversion), transforms every $r$-vector $A_r$ into $A_r^{\ast}$:

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

The reversion transformation is achieved by decomposing any spacetime element into a sum of products of vectors and then reversing the order of each product. For a multivector $A$ formed from a product of vectors $a_1a_2\ldots a_{r-1}a_r$, the reversion is $A^{\dagger} = a_r a_{r-1}\ldots a_2a_1$.

Clifford conjugation of a spacetime element $A$, denoted as $\tilde{A}$, combines the transformations of reversion and grade involution:

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

These transformations – grade involution, reversion, and Clifford conjugation – are all involutions, meaning applying them twice returns the original element.

Classical Electromagnetism

Faraday Bivector

In STA, the electric field and magnetic field are unified into a single entity: the Faraday bivector. This bivector is equivalent to the Faraday tensor and is defined as:

$F = \vec{E} + I c \vec{B}$

Here, $\vec{E}$ and $\vec{B}$ represent the familiar electric and magnetic fields, and $I$ is the STA pseudoscalar. Expanding $F$ in terms of components, it is defined such that:

$F = E_i \sigma_i + Ic B_i \sigma_i = E_1 \gamma_1\gamma_0 + E_2 \gamma_2\gamma_0 + E_3 \gamma_3\gamma_0 - cB_1 \gamma_2\gamma_3 - cB_2 \gamma_3\gamma_1 - cB_3 \gamma_1\gamma_2$

The individual $\vec{E}$ and $\vec{B}$ fields can be recovered from $F$ using these expressions:

$E = \frac{1}{2} (F - \gamma_0 F \gamma_0)$ $Ic B = \frac{1}{2} (F + \gamma_0 F \gamma_0)$

The $\gamma_0$ term signifies a particular reference frame. Consequently, using different reference frames will result in distinct relative fields, precisely as observed in standard special relativity. The Faraday bivector's relativistic invariance means its square yields further information:

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

The scalar part of this square corresponds to the Lagrangian density for the electromagnetic field, while the pseudoscalar part represents a less commonly encountered Lorentz invariant. It's a compact representation of fundamental electromagnetic properties.

Maxwell's Equations

STA allows for a remarkably simple formulation of Maxwell's equations as a single equation, rather than the four separate equations typically found in vector calculus. Analogous to the Faraday bivector, the electric charge density and current density are unified into a single spacetime vector, effectively a four-vector. This spacetime current, $J$, is given by:

$J = c\rho \gamma_0 + J^i\gamma_i$

where $J^i$ are the components of the classical 3-dimensional current density. This unification clarifies that classical charge density is fundamentally a current flowing in the timelike direction defined by $\gamma_0$.

By combining the electromagnetic field, current density, and the spacetime gradient ($\nabla$) as previously defined, all four of Maxwell's equations coalesce into a single equation within STA:

$\nabla F = \mu_0 c J$

The inherent covariance of these objects within STA automatically ensures the Lorentz covariance of the equation, a property that is significantly more challenging to demonstrate with the separated vector calculus formulation.

This unified form also simplifies the proof of certain properties of Maxwell's equations, such as the conservation of charge. Exploiting the fact that the divergence of the spacetime gradient of any bivector field is zero, we can perform the following manipulation:

$\nabla \cdot [\nabla F] = \nabla \cdot [\mu_0 c J]$ $0 = \nabla \cdot J$

This result clearly signifies that the divergence of the current density is zero, meaning that the total charge and current density over time is conserved. The elegance of this derivation is a testament to the power of the STA framework.

Furthermore, the form of the Lorentz force acting on a charged particle can be considerably simplified using STA and the electromagnetic field:

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

This compact expression encapsulates the force experienced by a charged particle moving with velocity $v$ in an electromagnetic field $F$.

Potential Formulation

In the standard vector calculus approach, two potential functions are employed: the electric scalar potential, $\phi$, and the magnetic vector potential, $\vec{A}$. STA consolidates these into a single vector field, $A$, analogous to the electromagnetic four-potential used in tensor calculus. In STA, it is defined as:

$A = \frac{\phi}{c}\gamma_0 + A^k\gamma_k$

Here, $\phi$ is the scalar potential, and $A^k$ are the components of the magnetic potential.

The electromagnetic field can be expressed in terms of this potential field using the relation:

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

However, this definition is not unique due to gauge freedom. For any twice-differentiable scalar function $\Lambda(\vec{x})$, a new potential $A' = A + \nabla \Lambda$ will yield the same Faraday bivector $F$, because $\nabla \wedge (\nabla \Lambda) = 0$. The process of selecting a suitable $\Lambda$ to simplify a given problem is known as gauge fixing. In relativistic electrodynamics, the Lorenz condition is often imposed, which simplifies calculations by setting $\nabla \cdot A = 0$.

To reformulate the STA Maxwell equation in terms of the potential $A$, we substitute $F = c\nabla \wedge A$. This leads to:

$\frac{1}{c}\nabla F = \nabla(\nabla \wedge A) = \nabla \cdot (\nabla \wedge A) + \nabla \wedge (\nabla \wedge A) = \nabla^2 A + 0 = \nabla^2 A$

Substituting this result back into Maxwell's equation yields the potential formulation of electromagnetism in STA:

$\nabla^2 A = \mu_0 J$

This equation elegantly expresses the dynamics of the electromagnetic field in terms of its potential.

Lagrangian Formulation

Analogous to the tensor calculus formalism, the potential formulation within STA naturally leads to an appropriate Lagrangian density:

$\mathcal{L} = \frac{1}{2}\epsilon_0 F^2 - J \cdot A$

The multivector-valued Euler-Lagrange equations for the field can be derived. While being somewhat loose with mathematical rigor by taking partial derivatives of non-scalar quantities, the relevant equations simplify to:

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

To re-derive the potential equation from this form, it's most convenient to work within the Lorenz gauge, setting $\nabla \cdot A = 0$. This condition, while not strictly necessary, significantly clarifies the subsequent steps. Due to the structure of the geometric product, this condition implies $\nabla \wedge A = \nabla A$.

After substituting $F = c\nabla A$, the same equation of motion for the potential field $A$ is readily obtained:

$\nabla^2 A = \mu_0 J$

This demonstrates the consistency and power of the Lagrangian approach within the STA framework.

Pauli Equation

STA provides a means to describe the Pauli particle through a real theory, bypassing the need for matrix representations. The standard matrix theory description of the Pauli particle is:

$i\hbar \,\partial_t \Psi = H_S \Psi - \frac{e\hbar}{2mc} \,{\hat {\sigma }}\cdot \mathbf {B} \Psi$

Here, $\Psi$ is a spinor, $i$ is the imaginary unit (without geometric interpretation in this context), ${\hat {\sigma }}_i$ are the Pauli matrices (where the 'hat' signifies a matrix operator, distinct from the geometric algebra elements), and $H_S$ is the Schrödinger Hamiltonian.

The STA approach transforms the matrix spinor representation $|\psi\rangle$ to the STA representation $\psi$ using elements $\sigma_1, \sigma_2, \sigma_3$ from the even-graded spacetime subalgebra and the pseudoscalar $I = \sigma_1\sigma_2\sigma_3$:

$|\psi\rangle = \begin{bmatrix} \cos(\theta/2) e^{-i\phi/2} \\ \sin(\theta/2) e^{+i\phi/2} \end{bmatrix} = \begin{bmatrix} a^0+ia^3 \\ -a^2+ia^1 \end{bmatrix} \mapsto \psi = a^0 + a^1 I\sigma_1 + a^2 I\sigma_2 + a^3 I\sigma_3$

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

$\partial_t \psi \, I\sigma_3 \,\hbar = H_S \psi - \frac{e\hbar}{2mc} \,\mathbf{B} \psi \sigma_3$

In this formulation, $\psi$ is an even multi-vector of the geometric algebra, and $H_S$ is the Schrödinger Hamiltonian. Hestenes refers to this as the real Pauli–Schrödinger theory, emphasizing that it reduces to the Schrödinger theory if the magnetic field term is omitted. The vector $\sigma_3$ is an arbitrarily chosen fixed vector; a fixed rotation can generate any alternative selected fixed vector $\sigma_3'$. This adaptability is a key advantage of the STA approach.

Dirac Equation

Similar to the Pauli equation, STA enables a description of the Dirac particle using a real theory, circumventing matrix methods. The matrix theory description of the Dirac particle is:

$\hat{\gamma}^{\mu}(i\partial_{\mu} - e A_{\mu})|\psi\rangle = m|\psi\rangle$

where $\hat{\gamma}^{\mu}$ are the Dirac matrices and $i$ is the imaginary unit.

Employing the same methodology as for the Pauli equation, the STA approach translates the matrix upper spinor $|\psi_U\rangle$ and matrix lower spinor $|\psi_L\rangle$ of the matrix Dirac bispinor $|\psi\rangle$ into their corresponding geometric algebra spinor representations, $\psi_U$ and $\psi_L$. These are then combined to represent the full geometric algebra Dirac bispinor $\psi$:

$|\psi\rangle = \begin{vmatrix} |\psi_U\rangle \\ |\psi_L\rangle \end{vmatrix} \mapsto \psi = \psi_U + \psi_L \mathbf{\sigma_3}$

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

$\nabla \psi \, I\sigma_3 - e\mathbf{A} \psi = m\psi \gamma_0$

In this equation, $\psi$ is the spinor field, $\gamma_0$ and $I\sigma_3$ are elements of the geometric algebra, $\mathbf{A}$ is the electromagnetic four-potential, and $\nabla = \gamma^{\mu}\partial_{\mu}$ is the spacetime vector derivative. This formulation offers a more integrated and geometrically grounded view of the Dirac equation.

Dirac Spinors

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

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

According to its derivation by David 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 to the components of a Dirac spinor, both possessing eight scalar degrees of freedom.

This equation is interpreted as a connection between spin and the imaginary pseudoscalar. The rotor $R$ transforms the frame of vectors $\gamma_{\mu}$ into another frame of vectors $e_{\mu}$ via the operation:

$e_{\mu} = R\gamma_{\mu}R^{\dagger}$

Here, $R^{\dagger}$ denotes the reverse transformation. This framework has been extended to provide a structure for locally varying vector- and scalar-valued observables, supporting the zitterbewegung interpretation of quantum mechanics originally proposed by Schrödinger. Hestenes has drawn parallels between his expression for $\psi$ and Feynman's expression in the path integral formulation:

$\psi = e^{i\Phi_{\lambda}/\hbar}$

where $\Phi_{\lambda}$ is the classical action along the $\lambda$-path. Using these spinors, the current density derived from the field can be expressed as:

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

Symmetries

Global phase symmetry, a constant global phase shift of the wave function, leaves the Dirac equation invariant. [Local phase symmetry], on the other hand, involves a spatially varying phase shift. This symmetry is preserved in the Dirac equation only if accompanied by a gauge transformation of the electromagnetic four-potential, as expressed by these combined substitutions:

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

In these equations, the local phase transformation is a phase shift $\alpha(x)$ at spacetime location $x$, involving the pseudovector $I$ and $\sigma_3$ from the even-graded spacetime subalgebra, applied to the wave function $\psi$. The corresponding gauge transformation involves subtracting the gradient of the phase shift, $\nabla \alpha(x)$, from the electromagnetic four-potential $A$, scaled by the particle's electric charge $e$. Researchers have indeed applied STA and related Clifford algebra approaches to gauge theories, the electroweak interaction, Yang–Mills theory, and the Standard Model.

The discrete symmetries are parity ($\hat{P}$), charge conjugation ($\hat{C}$), and time reversal ($\hat{T}$), applied to the wave function $\psi$. These transformations manifest as:

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

These transformations illustrate how STA handles fundamental symmetries in physics, providing a unified framework for their description.

General Relativity

General Relativity

Researchers have extended the application of STA and related Clifford algebra approaches to the domains of relativity, gravity, and cosmology. The gauge theory gravity (GTG) utilizes STA to describe an induced curvature on Minkowski space while incorporating a gauge symmetry under "arbitrary smooth remapping of events onto spacetime." This leads to the geodesic equation:

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

and the covariant derivative:

$D_{\tau} = \partial_{\tau} + \frac{1}{2}\omega$

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

This theory shows potential for the treatment of black holes, as its formulation of the Schwarzschild solution does not exhibit breakdown at singularities. Most of the established results of general relativity have been reproduced mathematically. Furthermore, the relativistic formulation of classical electrodynamics has been successfully extended to quantum mechanics and the Dirac equation. This suggests STA offers a powerful and unified approach to understanding the fundamental forces of nature.

See also

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

Notes

• ^ An example: given idempotent $a = \frac{1}{2}(1+\gamma_0)$, define $b = 1-a = \frac{1}{2}(1-\gamma_0)$. Then $a^2=a$, $b^2=b$, and $ab=0$. If we seek an inverse $a^{-1}$ satisfying $a^{-1}a=1$, we find that $b=1\cdot b=(a^{-1}a)b=a^{-1}(ab)=a^{-1}\cdot 0 \neq 0$. However, there is no $a^{-1}$ such that $a^{-1}\cdot 0 \neq 0$, meaning this idempotent has no inverse.

Citations

• ^ Hestenes 2015, p. ix.
• ^ Doran & Lasenby 2003, pp. 40, 43, 97, 113.
• ^ Doran & Lasenby 2003, p. 333.
• ^ Hestenes 2015, p. v.
• ^ Baylis 2012, pp. 225–266.
• ^ Hestenes 2015, p. 6.
• ^ Doran & Lasenby 2003, pp. 22–23.
• ^ Hestenes 2015, p. x.
• ^ Hestenes 2015, p. 11.
• ^ Lasenby, Doran & Gull 1995, p. 6.
• ^ Hestenes 2015, p. 12.
• ^ Hestenes 2015, p. 22.
• ^ a b c d Doran & Lasenby 2003, p. 37.
• ^ Hestenes 2015, p. 16.
• ^ Lasenby 2022.
• ^ O'Donnell 2003, p. 2.
• ^ O'Donnell 2003, p. 4.
• ^ a b Vaz & da Rocha 2016, p. 103.
• ^ Warner 1990, p. 191, Theorems 21.2, 21.3.
• ^ Warner 1990, p. 211.
• ^ Palais 1968, p. 366.
• ^ Hestenes & Sobczyk 1984, p. 14.
• ^ Hestenes 2015, p. 63.
• ^ Hestenes & Sobczyk 2012c, p. 45.
• ^ a b Lasenby & Doran 2002, p. 257.
• ^ Lasenby & Doran 2002, p. 259.
• ^ Arthur 2011, p. 180.
• ^ Hestenes 2015, pp. 22–24.
• ^ Hestenes 2015, pp. 50–51, Eqs. (16.22), (16.23).
• ^ Doran & Lasenby 2003, p. 401.
• ^ Hestenes 2015, pp. 47–62.
• ^ Kuipers 1999, pp. 127–138.
• ^ Dirac 1945, pp. 261–270.
• ^ Shah, Alam M; Sabar, Bauk (June 2011). "Quaternion Lorentz Transformation". Physics Essays. 24 (2): 158–162. Bibcode:2011PhyEs..24..158A. doi:10.4006/1.3556536.
• ^ Shah, Alam M; Sabar, Bauk (June 2011). "Quaternion Lorentz Transformation".
• ^ Hestenes & Sobczyk 2012c, p. 114.
• ^ a b c Hestenes 2015, p. 13.
• ^ Floerchinger 2021, Eq. (18).
• ^ Floerchinger 2021, Eq. (25).
• ^ Floerchinger 2021, Eq. (27).
• ^ Floerchinger 2021.
• ^ a b Doran & Lasenby 2003, p. 230.
• ^ Doran & Lasenby 2003, p. 233.
• ^ Doran & Lasenby 2003, p. 234.
• ^ a b Doran & Lasenby 2003, p. 230, Eq. (7.14).
• ^ Jackson 1998, pp. 2–3.
• ^ Hestenes 2015, p. 26, Eq. (8.4).
• ^ Doran & Lasenby 2003, p. 231, Eq. (7.16).
• ^ Doran & Lasenby 2003, p. 156, Eq. (5.170).
• ^ Doran & Lasenby 2003, p. 231.
• ^ a b Doran & Lasenby 2003, p. 232.
• ^ Doran & Lasenby 2003, p. 453.
• ^ Doran & Lasenby 2003, p. 440, Eq. (12.3).
• ^ a b Hestenes 2003a, Eqs. (75),(81).
• ^ Doran & Lasenby 2003, pp. 270, 271, Eqs. (8.16),(8.20),(8.23).
• ^ Hestenes 2003a, p. 30, Eqs. (75),(81).
• ^ Hestenes 2003a, p. 30, Eqs. (82),(83),(84).
• ^ a b Doran et al. 1996, Eqs. (3.43),(3.44).
• ^ Doran & Lasenby 2003, p. 279, Eq. (8.69).
• ^ Doran & Lasenby 2003, p. 283, Eq. (8.89).
• ^ a b Hestenes 2012b, pp. 169–182, Eqs. (3.1),(4.1).
• ^ Gull, Lasenby & Doran 1993, Eq. (5.13).
• ^ Doran & Lasenby 2003, p. 280, Eq. (8.80).
• ^ Hestenes 2003b, Eq. (205).
• ^ Hestenes 2003a, pp. 104–121.
• ^ Hestenes 2003b, p. 15, Eq. (79).
• ^ Hestenes 2010.
• ^ Hestenes 2015, p. vi.
• ^ Hestenes 2012b, Eqs. (3.1), (4.1), pp 169-182.
• ^ Hestenes 1967, p. 8, Eq. (4.5).
• ^ Quigg 2021, pp. 41–48.
• ^ a b Doran & Lasenby 2003, pp. 269, 283, Eqs. (8.8),(8.9),(8.10),(8.92),(8.93).
• ^ Hitzer, Lavor & Hildenbrand 2024, pp. 1345–1347.
• ^ Doran & Lasenby 2003, p. 283, Eq. (8.90).
• ^ Hitzer, Lavor & Hildenbrand 2024, p. 1343.
• ^ Doran, Lasenby & Gull 1993.
• ^ Lasenby, Doran & Gull 1998.
• ^ Lasenby, Doran & Gull 1995.
• ^ Lasenby & Doran 2002.