Chapter 12: The True Nabla — Clifford, Pauli, Dirac, and Hamilton

§12.0 Combining Two Equations into One

In a note within Chapter 10, we actually wrote something like this—"Why two equations? Can we not combine $dF=0$ and $d(\ast F)=\mu_0(\ast\mathcal{J})$ into one?"

We can. And quite easily. Bring out the imaginary unit $i$, and they collapse into a single line of complex equations. In this chapter we first look at that "trick," and then move on to the more unified viewpoint beyond it—Pauli matrices and the Dirac operator.

Note (the position of this chapter) This chapter is an advanced supplement and lies off the main line of the book. Chapter 9 completed the rewrite of vector analysis, and Chapter 10 finished the matrix expansion of Maxwell's equations. Here we do not intend to write a textbook of rigorous geometric algebra (Clifford algebra). The purpose is to see, as an entry point to calculation, how grad, curl, and div—which this book has dismantled—are integrated again into a single operator inside Pauli matrices and the Dirac operator.

§12.1 Combining Them with the Imaginary Unit — Unifying Maxwell's Equations

In Chapter 10 we consolidated Maxwell's equations into the two equations

Notice here that both $F$ and $\ast F$ are $2$-forms. In four-dimensional spacetime, the Hodge star $\ast$ maps $2$-forms to $2$-forms. So $F$ and $\ast F$ are forms of the same degree.

When the degrees match, there is no obstacle to addition. Using the imaginary unit $i$, we build a single complex $2$-form (in this chapter we follow the sign conventions of Chapter 10; on a Lorentzian metric the sign of $\ast^2$ depends on dimension and signature conventions, so here we prioritize consistency with the component expansion).

Apply the exterior derivative $d$ to $G$. Since $d$ is a real differential operator, $d(i\ast F) = i\,d(\ast F)$.

That is,

That is all. The two equations $dF=0$ and $d(\ast F)=\mu_0(\ast\mathcal{J})$ have been unified into a single complex equation. The real part $F$ and the imaginary part $\ast F$ share the source-free and source Maxwell equations between them.

But pause here and think. This trick works only because, by accident, in four dimensions $\ast$ maps $2$-forms to $2$-forms. In general $n$ dimensions, $\ast$ maps $k$-forms to $(n-k)$-forms. Being able to add $2$-form to $2$-form is a privilege of $n=4$; adding $1$-form to $2$-form directly is not allowed in the usual framework of differential forms.

When $d$ and $\ast$ are combined, conversions between different degrees appear—grad ($0\to1$), curl ($1\to1$), div ($1\to0$), and so on. If we could treat the origins of these in a single algebra, we should be able not to dismantle $\nabla$ but to unify it.

Adding forms of different degrees—is there no such "magic box"?

§12.2 Pauli Matrices — The Magic of Adding Different Degrees

There is, in fact, a "magic box." Tools that physicists discovered for quantum mechanics can be used directly as that box. The Pauli matrices. More precisely, what we look at here is the Clifford algebra of three-dimensional Euclidean space, represented by Pauli matrices. Not differential forms themselves, but each degree ($0$ through $3$) is matched (under this chapter's convention) to basis elements inside this algebra.

12.2.1 Definition and Multiplication of Pauli Matrices

The Pauli matrices $\sigma_1, \sigma_2, \sigma_3$ are the following $2\times2$ matrices.

Let us examine their multiplication rules. Individually, $\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = I$ (the identity matrix), and for distinct indices anticommutativity holds: $\sigma_1\sigma_2 = -\sigma_2\sigma_1$. There are nine products in all.

Let us decompose these nine products into symmetric and antisymmetric parts—the same idea as when we introduced the wedge product $\wedge$ by antisymmetrization in Chapter 2.

For equal indices ($\sigma_1\sigma_1$, $\sigma_2\sigma_2$, $\sigma_3\sigma_3$), the symmetric part is $I$ and the antisymmetric part is $0$. For distinct indices, for example with $\sigma_1\sigma_2 = i\sigma_3$,

and similarly, products with distinct indices are purely antisymmetric. The symmetric part is $I$ only when the indices match; otherwise it is $0$—nothing other than the metric introduced in Chapter 6 (the identity matrix in Cartesian coordinates) appearing as the symmetric part of Pauli-matrix products. The antisymmetric part flips sign when indices are swapped—exactly the same structure as the wedge product in Chapter 2, $dx\wedge dy = -dy\wedge dx$.

Pauli-matrix products always decompose into "inner product (symmetric part) $+$ wedge product (antisymmetric part)." That single fact is the source of all the magic of Pauli matrices.

12.2.2 Writing Vectors with Pauli Matrices

Let us represent a three-dimensional vector $\mathbf{v} = (v_1, v_2, v_3)$ with Pauli matrices.

Compute the product of the Pauli matrices $V = \mathbf{v}\!\cdot\!\bm{\sigma}$, $W = \mathbf{w}\!\cdot\!\bm{\sigma}$ corresponding to two vectors $\mathbf{v}, \mathbf{w}$.

Apply the nine products to each term. Writing only the nonzero replacements,

and the nine obtained by reversing the indices: $\sigma_2\sigma_1 = -i\sigma_3$, $\sigma_3\sigma_2 = -i\sigma_1$, $\sigma_1\sigma_3 = -i\sigma_2$. Collect the coefficients of $I$ (symmetric part = inner product) and of $\sigma_1,\sigma_2,\sigma_3$ (antisymmetric part = wedge product).

Under the three-dimensional orientation convention used throughout this book, the expressions in parentheses for the coefficients of $\sigma_1,\sigma_2,\sigma_3$ are the same components that are read back as the cross product $\mathbf{v}\times\mathbf{w}$—equivalently, the components of the wedge product $\mathbf{v}\wedge\mathbf{w}$ defined in Chapter 2. Therefore

and the result takes the form inner product $+$ wedge product. Inner product and cross product emerge simultaneously from a single product $VW$.

That is the true nature of the "magic box." More precisely, what we are looking at is the Clifford algebra of three-dimensional Euclidean space, represented by Pauli matrices. Its product at once produces the inner product (a scalar corresponding to a $0$-form) and the cross product (a wedge product corresponding to a $2$-form). What in ordinary differential forms belonged to separate degrees coexists in this matrix representation as a single $2\times2$ matrix.

Note (why $2\times2$?) Representing a vector by a $2\times2$ matrix may seem odd. But to express both inner and wedge products in one product requires at least a noncommutative algebra, and the minimal realization is $2\times2$ complex matrices. In this book's style, one may think of it as packing two objects of different degree—a $1$-form (row vector $1\times3$) and a $2$-form (antisymmetric $3\times3$ matrix)—into a single $2\times2$ matrix.

12.2.3 Mixing in Scalars Too

We have now seen that vectors ($1$-forms) and wedge products ($2$-forms) appear together from a single calculation $VW$. Take one step further and mix scalars ($0$-forms) themselves into the same algebra.

The most general $2\times2$ matrix expressible as a linear combination of Pauli matrices has the form

Here $\varphi$ is a scalar ($0$-form) and $A_1,A_2,A_3$ are vector components ($1$-form). Furthermore, multiplying $\psi$ on the left by $\sigma_i$ brings in terms $i\sigma_k$ corresponding to $2$-forms. Under this chapter's convention, basis elements for degrees $0,1,2,3$ can all be matched inside this algebra.

Note (what has become possible) In differential forms one can formally arrange forms of different degrees as a direct sum. But to treat inner product (contraction) and outer product (antisymmetrization) simultaneously as one product, the ordinary wedge product alone is not enough. In the Pauli-matrix algebra, they separate and coexist naturally inside a single product $VW$. That is the core of geometric algebra.

Note (matching all degrees) Let us organize the degree correspondence under this chapter's convention. $I$ corresponds to $0$-forms (scalars), $\sigma_1,\sigma_2,\sigma_3$ to $1$-forms (vectors), $i\sigma_1,i\sigma_2,i\sigma_3$ to $2$-forms (bivectors), and $iI$ to $3$-forms (pseudoscalar). This is not identification with differential forms themselves, but a convention for matching basis elements of each degree inside the matrix representation of the three-dimensional Euclidean Clifford algebra. The most general element has the form $\psi = aI + \mathbf{v}\!\cdot\!\bm{\sigma} + i\,\mathbf{w}\!\cdot\!\bm{\sigma} + ibI$, containing all four degrees with complex coefficients.

Note (quaternions—a magic box that already existed) In fact the idea of "mixing scalars and vectors into one number" is much older than Pauli matrices. Older than matrix algebra. The quaternion $q = a + bi + cj + dk$ discovered by Hamilton in 1843 is exactly that. Maxwell used quaternionic language in his formulation of electromagnetism. Later, Gibbs and Heaviside separated the scalar and vector parts from quaternions, and modern vector analysis (inner and cross products) was born. From this book's viewpoint, Pauli matrices also look like a tool for viewing, in the context of quantum mechanics, a structure once split into inner and outer products again as a single noncommutative product.

Note (vector analysis in the age before matrices) When Gibbs and Heaviside organized vector analysis in the 1880s, matrix algebra was not yet common. Determinants existed, but the algebra of matrices itself—noncommutativity of products, eigenvalues, the unified understanding as linear transformations—was not yet the common working language of physicists. Gibbs invented his own notation, the dyad $\mathbf{a}\mathbf{b}$, to represent linear transformations; this corresponds to the modern matrix $\mathbf{a}\mathbf{b}^T$. From this book's viewpoint, vector analysis can be viewed as having differentiated from quaternionic unification into tools of inner product, outer product, and dyads. That this book, titled "Dismantling Nabla," decomposed $\nabla$ into $d$ and $\ast$ and in the final chapter shows re-unification via Pauli matrices also has the meaning of revisiting this history of separation and reunification from another angle.

Note (when matrix algebra was not the standard language of physicists) Incidentally, that matrix algebra was not the standard language of physicists shows up clearly in matrix mechanics in 1925. Heisenberg himself did not initially know that his calculations were noncommutative matrix products; it was Born who read them as matrices.

§12.3 The Dirac Operator and the "True Nabla"

12.3.1 $\bm{\sigma}\!\cdot\!\nabla$ — A Unified Differential Operator

Combine Pauli matrices with nabla $\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})$ and define the operator

$D$ is a differential operator in the form of a $2\times2$ matrix. Apply it to a scalar field $\varphi$.

This is $\mathrm{grad}\,\varphi$ written with Pauli matrices.

Next, write a vector field $\mathbf{A} = (A_1, A_2, A_3)$ as a Pauli matrix $A = \mathbf{A}\!\cdot\!\bm{\sigma}$ and apply $D$. Recall the product formula from §12.2.2.

Apply the same replacements $\sigma_i\sigma_j$ as in §12.2.2 to the nine terms, and collect the coefficient of $I$ (inner product = divergence) and the coefficients of $\sigma_1,\sigma_2,\sigma_3$ (wedge product = curl).

The expressions in parentheses for the coefficients of $\sigma_k$ are the components of $\mathrm{curl}\,\mathbf{A}$—equivalently, under the three-dimensional dictionary, the components of the $2$-form obtained by applying $d$ to the corresponding $1$-form. Therefore

A single operator $D$ simultaneously yields $\mathrm{grad}$ ($D\varphi$), $\mathrm{div}$ (the real part of $DA$), and $\mathrm{curl}$ (the imaginary part of $DA$). This is none other than one endpoint of the journey on which we deepened our understanding by decomposing $\nabla$ into $d$ and $\ast$—unification.

Note (in the language of $d$ and $\ast$) $D$ can also be expressed in this book's language. Up to sign conventions, $D$ can be understood as an operator combining $d$ (raising degree) and $\ast d\ast$ (lowering degree). The latter is called the codifferential $\delta$, defined by $\delta = \pm\ast d\ast$ (the sign depends on dimension and degree). Here we do not enter the exact signs; it is enough to keep in mind the structure $D \sim d + \delta$. $d$ raises degree and $\delta$ lowers it—the sum $D$ of these two binds grad, curl, and div into one operator.

12.3.2 $D^2$ — The Laplacian

What happens if we apply $D$ twice?

Expand into nine terms.

Terms containing the antisymmetric part of $\sigma_i\sigma_j$ (terms with $i\neq j$, such as $\sigma_1\sigma_2 = i\sigma_3$) cancel in pairs of opposite sign because of the commutativity of partial derivatives, $\frac{\partial^2}{\partial x\partial y} = \frac{\partial^2}{\partial y\partial x}$. For example, $\sigma_1\sigma_2\,\frac{\partial^2}{\partial x\partial y}$ and $\sigma_2\sigma_1\,\frac{\partial^2}{\partial y\partial x}$ sum to zero because $\sigma_2\sigma_1 = -\sigma_1\sigma_2$ and $\frac{\partial^2}{\partial y\partial x} = \frac{\partial^2}{\partial x\partial y}$. What remains are only the diagonal terms with $\sigma_i\sigma_i = I$.

$D^2$ is the Laplacian itself. $D$ is the "square root" of the Laplacian. That fact—taking a square root of a differential operator—was the core insight when Dirac derived the equation for the electron in relativistic quantum mechanics.

12.3.3 Unifying Maxwell — Revisited

In §12.1 we unified Maxwell's equations in four dimensions via $F + i\ast F$. With $D$, the same thing can be written using only three-dimensional language.

Bundle the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ into a single complex vector. This is called the Riemann–Silberstein vector.

In Pauli matrices, $F = \mathbf{E}\!\cdot\!\bm{\sigma} + i\,\mathbf{B}\!\cdot\!\bm{\sigma}$. $F$ is a purely complex Pauli vector with no scalar part. Apply the operator $D + \frac{\partial}{\partial t}$ including the time derivative.

Apply the formula for $D(\mathbf{A}\!\cdot\!\bm{\sigma})$ from §12.3.1 to both $\mathbf{E}$ and $\mathbf{B}$.

Organize by real and imaginary parts, and by $I$ (scalar) and $\bm{\sigma}$ (vector) components.

Setting these equal to zero yields all four vacuum Maxwell equations. When current $\mathbf{J}$ and charge $\rho_{\mathrm e}$ are present, under the same normalization as Chapter 10 the right-hand side can be set to $\rho_{\mathrm e}/\varepsilon_0 - \mu_0 c \mathbf{J}\!\cdot\!\bm{\sigma}$, and Maxwell's equations with sources collapse into one line.

Note (normalization and signs in Chapter 12)

Here $t, \mathbf{B}, \rho_{\mathrm e}, \mathbf{J}$ are normalized quantities as in Chapter 10 §10.1–10.5. The coefficients and signs of the source terms on the right-hand side are chosen to be consistent with the expansion of $d(\ast F) = \mu_0(\ast\mathcal{J})$ from Chapter 10.

The two equations $dF=0$ and $d(\ast F)=\mu_0(\ast\mathcal{J})$ have been unified into a single complex equation in front of $D$. Whereas the four-dimensional trick of §12.1 depended on a special dimension, here the unified operator $D$ itself achieves the unification.

12.3.4 $\cancel{\partial} F = J$ — The Dirac Operator in Four-Dimensional Spacetime

In §12.3.3, $(D + \frac{\partial}{\partial t})F = \rho_{\mathrm e} + \mathbf{J}\!\cdot\!\bm{\sigma}$ still separates time and space. That is because it was built on the three-dimensional Pauli matrices $\sigma_1,\sigma_2,\sigma_3$. If we treat four-dimensional spacetime from the start, even this separation disappears.

Extending Pauli matrices to $4\times4$ gives the gamma matrices $\gamma^0,\gamma^1,\gamma^2,\gamma^3$. They satisfy anticommutativity $\gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu = 2g^{\mu\nu}I$ ($g^{\mu\nu}$ is the Minkowski metric). The signs of the real coefficients in this equation depend on the chosen metric signature and index conventions. As in Chapter 10, this chapter adopts the convention with spacetime metric signature $(-,+,+,+)$. Using these gamma matrices, define the four-dimensional Dirac operator by

Identifying the electromagnetic field $F$ from Chapter 10 ($4\times4$ antisymmetric matrix) with an expansion in gamma-matrix wedge products $\gamma^\mu\!\wedge\!\gamma^\nu$, Maxwell's equations consolidate into the single line

That is the standard geometric-algebra picture. $F$ is the electromagnetic field and $J$ is the four-current (expanded in $\gamma^\mu$). In vacuum, $\cancel{\partial} F = 0$.

Note (on the product $\cancel{\partial} F$) The product here is not ordinary matrix multiplication but the geometric-algebra product generated by gamma matrices. The antisymmetric matrix $F$ from Chapter 10 is reinterpreted here as a $2$-vector expanded in $\gamma^\mu\!\wedge\!\gamma^\nu$. Details are left to textbooks on geometric algebra; it is enough to think that the four-dimensional version of $\sigma_i\sigma_j = \delta_{ij}I + i\varepsilon_{ijk}\sigma_k$ (§12.2.1) holds here as well.

Both §12.1's $F + i\ast F$ and §12.3.3's $(D + \frac{\partial}{\partial t})F$ are absorbed into this single line. That is the farthest landscape on the journey that began with dismantling $\nabla$. The operator Dirac discovered in 1928 to derive the equation for the electron is the core of geometric algebra, which describes the geometry of spacetime and physical laws in one algebra; see Doran & Lasenby's Geometric Algebra for Physicists for details.

§12.4 The Operator $\nabla$

Now let us return to the first question. What was $\nabla$?

In vector analysis, $\nabla$ was introduced as a formal vector of partial derivatives and was a mysterious symbol with three faces: grad, curl, and div. This book dismantled it into $d$ and $\ast$, thereby revealing what each operator truly is.

And now, tracing this in reverse, the scattered parts $d$ and $\ast$ are unified in geometric algebra as the three-dimensional Dirac operator $D = \bm{\sigma}\!\cdot\!\nabla$, and further its four-dimensional version, the Dirac operator $\cancel{\partial}$, gives the ultimate Maxwell equation $\cancel{\partial} F = J$.

By the way, in the context of geometric algebra this operator is not necessarily written $\cancel{\partial}$. Rather, this kind of vector differential operator is written