Geometric Mechanics, Vol. 2, No. 3 (2025), 337–383.
Sebastià Xambó (DMAT)
This work is dedicated to the memory of Miguel Carlos Muñoz Lecanda (MCML) (1946-2023), who was a Full Professor at UPC. Broadly speaking, his scientific interests were focused on the interactions between mathematics (mainly differential geometry) and physics (geometric mechanics, classical field theory), but also on engineering applications (control theory, for instance). For a biographical sketch of MCML (by María Barbero-Liñán, Manuel de León, Eduardo Martínez, and Narciso Román-Roy), see pages 221-223 of the journal cited above.
The paper under review surveys several topics about which the author and MCML exchanged ideas for over five decades, but in this occasion phrased in the geometric algebra formalism. The paper pretends to be the fruit of a pending conversation reminiscent of the fascination experienced when they studied texts such as George Mackey‘s celebrated book Mathematical Foundations of Quantum Mechanics (AMS review by Jacob Feldman).
For any real vector space \(E=E_{q}\) (its elements, denoted \(v,v’,u,\dotsc\), are called vectors) endowed with a non-degenerate quadratic form \(q\) (usually called the metric), the corresponding geometric algebra \(\mathcal{G}_q=\mathcal{G}(E_q)\) is defined in §1 to be Grassmann’s exterior algebra \(\mathcal{G}=\mathcal{G}(E)\) (its elements \(x,x’,\dotsc\) are called multivectors; its product, \(x\wedge x’\), is the wedge product) enriched with the geometric product \(xx’\) (no infix symbol is used for this product: factors \(x, x’\in\mathcal{G}\) are just juxtaposed). If the metric \(q\) has signature \((r,s)\), so that \(n=r+s=\dim\mathcal{G}\), the geometric algebra is also denoted by \(G_{r,s}=\mathcal{G}(E_{r,s})\). The power of the geometric product stems from the fact that it is the only series associative, bilinear and unital product satisfying two conditions: (1) Contraction, \(v^2=q(v)\) for any vector \(v\); (2) Transference, \(v(v_1\wedge\cdots\wedge v_k)=v\wedge v_1\wedge\cdots\wedge v_k\) whenever \(v\) is orthogonal to \(v_j\) for \(j=1,\dotsc,k\). In particular we see that a vector \(v\) is invertible provided \(q(v)\ne0\), which confers a flexibility to calculations in \(\mathcal{G}\) akin to numerical calculations, although with the precaution that the geometric product is not commutative. For example (see Eq. (11)), \(vv’=v\cdot v’+v\wedge v’\), which shows that \(v\) and \(v’\) commute precisely when \(v\wedge v’=0\) (i.e, when \(v\) and \(v’\) are parallel) and that they anticommute precisely when \(v\cdot v’=0\) (i.e., when \(v\) and \(v’\) are orthogonal; here \(v\cdot v’\) denotes the bilinear form, or scalar product, associated to \(q\)).
XXXII International Fall Workshop on Geometry and Physics (Portugal, Coimbra, 2-5 September, 2024).
§2 is devoted to \(\mathcal{G}_2=\mathcal{G}(\mathcal{E}_2)\), the geometric algebra of the Euclidean plane \(\mathcal{E}_2\), which for historical reasons deserves to be called Wessel’s algebra. Two examples illustrate its applications: the derivation of a formula for the orthocenter of a triangle in terms of its vertices (Eq. (21)), and the recasting of the Levi-Civita ‘spinor regularization’ for the Kepler problem in \(\mathcal{E}_2\) (Eq. (25) and Fig. 3). Remark that if we choose any two unit orthogonal vectors \(u,u’\), then the bivector \(\mathfrak{i}=u\wedge u’=uu’\) (which represents a unit area) satisfies \(\mathfrak{i}^2=uu’uu’=-uuu’u’=-1\), which implies that the even algebra \(G^+_2=\mathcal{G}^0\oplus\mathcal{G}^2=\mathbb{R}\oplus \mathbb{R}\mathfrak{i}=\mathbb{C}\) is a geometric model of the field of complex numbers, and the spinors involved it the Levi-Civita regularization are the unit complex numbers \(e^{\alpha\mathfrak{i}}\), \(\alpha\in\mathbb{R}\) (note also that while \(\mathbb{C}\) is commutative, its action by multiplication on vectors is not, as \(v\mathfrak{i}=-\mathfrak{i} v\), or, more generally, \(ve^{\alpha\mathfrak{i}}=e^{-\alpha\mathfrak{i}}v\)). In the expression for the orthocenter, a key point is that non-zero bivevtors are invertible (the inverse of \(\beta\mathfrak{i}\) is \(-\beta^{-1}\mathfrak{i}\)).
Similarly, §3 is devoted to \(\mathcal{G}_3=\mathcal{G}(\mathcal{E}_3)\), the geometric algebra of the Euclidean space \(\mathcal{E}_3\). It is called Pauli algebra because Wolfgang Pauli discovered it «under the guise of \(2\times2\) complex matrices in his research about the spin of the electron». In this case, one gets a geometric model \(\mathbb{H}=\mathcal{G}_3^{+}=\mathcal{G}^0\oplus\mathcal{G}^2=\mathbb{R}\oplus \mathcal{E}\mathfrak{i}\) of Hamilton’s quaternions (here \(\mathfrak{i}\) is a unit volume, say \(uu’u^{\prime\prime}\) with \(u,u’,u^{\prime\prime}\) unit mutually orthogonal vectors, so that we also have \(\mathfrak{i}^2=-1\)), which provides shrewd expressions for symmetries and rotations, and in particular for the composition of rotations (Olinde Rodrigues’ formulas); a treatment of Newtonian multi-particle mechanics, including Kepler’s orbits and Rutherford’s scattering formula; and the celebrated Kustaanheimo–Stiefel spinor regularization of the Kepler problem in \(\mathcal{E}_3\). In this case, spinors are the elements of \(\mathbb{H}^{\times}\).
In §4, we find «a brief overview of \(\mathcal{G}_{1,3}\) and its capacity to deal with core relativity topics, including Lorentz transformations, Maxwell’s electromagnetism and Dirac’s equation for the relativistic electron». This algebra «is named after P. Dirac because he discovered it in the guise of a 16-dimensional subalgebra of the algebra of \(4\times 4\) complex matrices in his endeavor to get a relativistic theory of the electron». The Dirac spinors are the invertible even multivectors and they provide a means for describing Lorentz transformations (Theorems on page 369) and establishing their fundamental properties. In particular, the composition of two Lorentz boosts is resolved explicitly into the composition of a Lorentz boost and a rotation. Electromagnetic fields are represented as bivector fields \(F: E_{1,3}\to G_{1,3}^2\), an electric current by a vector field \(J:E_{1,3}\to E_{1,3}\), and the Riesz form of Maxwell’s equations is \(\partial F=J\), where \(\partial\) is the gradient operator of \(E_{1,3}\) (Theorem on page 372). The section ends with the geometric algebra form of the Dirac equation, and some of its consequences, including how spin is predicted and expressed (Eq. (73)).
The paper includes an appendix collecting some background facts, as for example an intrinsic construction (i.e., coordinate-free) of the Grassmannian varieties, a formula for the inner product of two decomposable \(k\)-vectors, the behavior of the grade and reversion involutions of the Grassman algebra with respect to the inner and geometric products, and the analytics of Kepler orbits.
Although the geometric algebra overview of §1 deals with general spaces \(E_{r,s}\), the applications considered in the paper are limited to \(\mathcal{G}_2\) (Wessel), \(\mathcal{G}_3\) (Pauli), and \(G_{1,3}\) (Dirac). Nevertheless, the paper may serve as an entry gate to the geometric algebra realm (represented by the twenty references provided at the end) for anyone wishing to get acquainted with it. Knowledge of mathematics should ease the understanding of its structure and facilitate the grasping of its applications to geometry and physics. The other way around may also work: knowledge of physics can be a sure hold for a finer appreciation of the mathematics involved.