FROM LEIBNIZ' CHARACTERISTICA GEOMETRICA TO CONTEMPORARY

Sebastià Xambó-Descamps

Abstract. After a few remarks on the significance of Leibniz’ characteristica geometrica, seen as a subsystem of his characteristica univeralis,1 we provide a brief historical perspective on the devel- opment of geometric algebra, one of its archetypal incarnations, and illustrate how it works in geom- etry and by a few chosen examples. In our considerations, Leibniz’ characteristica univer- salis is not seen as real possibility, but rather as an ideal horizon that fosters productive thinking moods and values, in line with his ars inveniendi and ars judicandi. These roles are underlined by inserting comments on ways in which they are reflected in contemporary developments.

FOREWORD

The view advanced in this paper is that Geometric algebra (GA) is a mathematical structure that allows the representation of geometrical concepts, and operations on them, in an intrinsic manner (it is a coordinate-free system), and that its syntax and semantics are logically rigor- ous and yet not inimical to intuition nor to sophisticated applications. In our view, these fea- tures are in tune with Leibniz’ ideas on the subject, even though, as we shall argue, their scope was necessarily relatively limited at his time. Many researchers have contributed to the rather slow unfolding of GA, often in scientific and engineering applications in which plays a key role. Here is a sample of a few land- mark texts produced in the last half a century: CHEVALLEY (1956), RIESZ (1958), HESTENES (1966, 2015), HESTENES (1990, 1999), HESTENES-SOBCZYK (1984), LOUNESTO (1997), DORAN-LASENBY (2003), DORST-FONTIJNE-MANN (2007), SNYGG (2012), RODRIGUES- OLIVEIRA (2016). However, any newcomer to this rich field will learn, sooner or later, that GA has a much longer history. Fundamental and systematic works were published since the beginning of the XIX century, but, most importantly for our purposes here, it was Leibniz who put forward the seminal ideas, in what he called characteristica geometrica, as early as the beginning of the last quarter of the XVII century. This was well one century and a half before the issue resurfaced again. It is natural, therefore, to ask about the impact of Leibniz ideas in those later developments. It is also natural to consider to what degree have they been fulfilled today.

1 In today’s terms, characteristica amounts to symbolism, or symbolic calculus. The main purpose of this paper is to try to shed some light on both issues. This is not as straightforward as it might seem. Regarding the first question, we can document only one key influence, which happens to be only an indirect one (GRASSMANN 1847), but we cannot refrain from conjecturing that there must have been others, most likely caused by the power- ful irradiation of Leibniz’s thinking, and writing, on all sorts of topics. For BURALI-FORTI 1897b, for example, “Leibniz’s idea was destined to propagate and to produce great results”, and mentions names that did important work before Grassmann, such as Caspar Vessel and Giusto Bellavitis. One likely propagation process may have been through philosophy and theology. Grassmann was a theologian and a linguist, with no university education in math- ematics, and to a good extent, he received his ideas about mathematical innovation from Friedrich D. E. Schleiermacher (1768-1834), who was a philosopher, classical philologist and theologian (see ACHTNER 2016). Fundamentally, it came about through the study of Schleiermacher’s teachings, among which it is worth singling out not only his extensive writ- ings on dialectics, but also his 1831 lecture on Leibniz's idea of a universal language (FOR- STER 2015). Those teachings “were picked up by Grassmann and operationalized in his phil- osophical-mathematical treatise Ausdehnungslehre in 1844” (ACHTNER 2016). The main obstacle to assess the second question is that there is not a clear consensus on what the geometric algebra structure is. The main source of tension is the unfortunate divide be- tween abstract and general mathematical presentations, that tend to eschew any sort of appli- cations, and those by authors whose main motivation lies in physics or engineering, that tend to be regarded by the other side as lacking rigor. Our approach will be to outline a structure that retains the positive aspects of both sides. Our intention is none other than to facilitate, whatever the background of the reader, to hook on it to understand the other backgrounds. Thus, a mathematician can get a good understanding of physical theories based on her math- ematical knowledge and a physicist can get a better appreciation of the by stand- ing on his understanding of the physics. One may even indulge in the fancy that Leibniz himself would have judged the system as a flexible embodiment of his characteristica geo- metrica.

ON THE CHARACTERISTICA UNIVERSALIS

In RUSSELL 1946, chapter on Leibniz, we find (pp. 572-3) an authoritative and general view about the pioneering ideas of Leibniz on logic and mathematics: Leibniz was a firm believer in the importance of logic, not only in its own sphere, but [also] as a of metaphysics. He did work on mathematical logic which would have been enormously important if he had published it; he would, in that case, have been the founder of mathematical logic, which would have become known a century and a half sooner than it did in fact. He ab- stained from publishing, because he kept on finding evidence that Aristotle's doctrine of the syllogism was wrong on some points; respect for Aristotle made it impossible for him to believe this, so he mistakenly supposed that the errors must be his own. Nevertheless he cherished throughout his life the hope of discovering a kind of generalized math- ematics, which he called characteritica universalis (CU), by means of which thinking could be replaced by calculation. ‘If we had it,’ he says, ‘we should be able to reason in metaphysics and morals in much the same way as in geometry and analysis.’ ‘If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, to sit down to their slates, and to say to each other (with a friend as witness, if they liked): Let us calculate (Calculemus).

The closest idea in today’s world that has an appearance of a CU is what may be loosely called pattern theory. Although it appears in many guises in all sorts of developments oc- curred in the last decades (cf. GRENANDER 1996, 2012), my interest here is the general views expressed by David Mumford and which I regard as representative of those held by most workers in that discipline. In MUMFORD 1992 we find the following view (boldface and italic emphasis not in the original): In summary, my belief is that pattern theory contains the germs of a universal theory of thought itself, one which stands in opposition to the accepted analysis of thought in terms of logic. The successes to date of the theory are certainly insufficient to justify such a grandiose dream, but no other theory has been more successful. The extraordinary similarity of the structure of all parts of the human cortex to each other and of human cortex with the cortex of the most primitive mammals suggests that a relatively simple universal principal governs its operation, even in complex processes like language: pattern theory is a proposal for what these principles may be.

The paper MUMFORD 1996 is an edited version of MUMFORD 1992, and the italic text in the quotation was deleted. In my biographical paper XAMBÓ-DESCAMPS 2013, I asked Mumford whether he still held, two decades after that paper, that pattern “contains the germs of a uni- versal theory of thought” and how did this relate to “other universal languages that have recurrently been proposed in the past”. His answer was this: “At the time of that lecture, I didn't appreciate fully the importance of graphical structures underlying thought. I feel the identification of feedback in cortex with the prior in Bayes’ formula is quite correct but I have since pursued the relevance of grammar like graphs, e.g. in my monograph, with S-C Zhu, A stochastic grammar of images: Foundations and Trends in Computer Graphics and Vision (2007, 259-362). But I continue to argue strongly that logic based models are inade- quate”. Asked about the progress in the understanding of the neocortex, he said that the basic cortical “algorithms” remain a mystery and that the difficulty of recording the operation of something like a million neurons is a great practical obstacle. This, and other developments related to and artificial intelligence, lead me to re- gard Leibniz’ phrasings of his characteristica universalis as fundamentally utopic, which does not mean that they cannot be useful as an unreachable horizon in the sense explained in the abstract.

ON THE CHARACTERISTICA GEOMETRICA

Leibniz’ idea of a characteristica geometrica (CG), a (mathematical) branch of the CU, was advanced to Christian Huygens in a letter, sent together with an essay (both in French), dated September 8, 1679. The first paragraph of next quotation is from the letter and the second and third from the essay (LEIBNIZ 1850, 1956; cf. CROWE 1994, pages 3-4, that quotes the English translation): But after all the progress I have made in these matters [in going beyond Viète and Descartes], I am not yet satisfied with Algebra, because it does not give the shortest methods or the most beautiful constructions in geometry. This is why I believe that, as far as geometry is concerned, we need still another analysis, which is distinctly geometrical or linear and which will express situation directly as algebra expresses magnitude directly. I believe that I have found the way, and that we can represent figures and even machines and movements by characters, as algebra represents numbers or magnitudes. I believe that by this method one could treat al- most like geometry. I am sending you an essay, which seems to me to be important. […] I have discovered certain elements of a new characteristic which is entirely different from alge- bra and which will have great advantages in representing to the mind, exactly and in a way faithful to its nature, even without figures, everything which depends on sense perception. Alge- bra is the characteristic for undetermined numbers or magnitudes only, but it does not express situation, angles, and motion directly. Hence it is often difficult to analyze the properties of a figure by calculation, and still more difficult to find very convenient geometrical demonstrations and constructions, even when the algebraic calculation is completed. But this new characteristic, which follows the visual figures, cannot fail to give the solution, the construction, and the geo- metric demonstration all at the same time, and in a natural way in one analysis, that is, through determined procedure. But its chief value lies in the reasoning which can be done and the conclusions that can be drawn by operations with its characters, which could not be expressed in figures, and still less in models, without multiplying these too greatly or without confusing them with too many points and lines in the course of the many futile attempts one is forced to make. This method, by contrast, will guide us surely and without effort. I believe that by this method one could treat mechanics almost like geometry, and one could even test the qualities of materials, because this ordinarily depends on certain figures in their sensible parts. Finally, I have no hope that we can get very far in physics until we have found some such method of abridgement to lighten its burden of imagina- tion. All those claims must have appeared as too ambitious, or even groundless, to contemporaries, and in particular to Huygens. In retrospect, however, they are natural thoughts of a mind in possession of a powerful model of logic, construed (see ANTOGNAZZA 2009, 233 ff.) as tools of valid reasoning, to be used by all other sciences (scientia generalis), to discover (ars in- veniendi) and to demonstrate from sufficient data (ars judicandi), all driven by a universal symbolism (characteristica universlis). It seems clear that his transforming vision in the case of mathematics was not limited to what is usually recognized (, Arith- metic, Vieta’s symbolic algebra/analysis (1591), Descartes’ analytical geometry (1637)), but that it included the basic ideas on projective geometry (initiated as a study of perspective, in which sight perception is the guiding sense). Introduced by Desargues (1639), it is plausible that Leibniz could foresee not only a CG encompassing all those domains, but also the deep relationships between Euclidean and projective geometry that were discovered in the XIX century. But I have no factual evidence for this theoretical plausibility beyond the consider- ations on perspective in the report CORTESE 2016, the references quoted there related to this issue, and the trust in the great generality of Leibniz’ thought. Let us also comment that the CG is not as utopic as the CU. For one thing, it refers to math- ematics, where the language of logic is in principle sufficient. But it is not completely free from that unreal character, basically because it seems to implicitly assume that there is a single geometric world to be captured by ‘the’ CG, a scenario propounded later by Kant. As is well known, however, there are many geometric worlds and some adjustment of any CG has to possible if it is to be relevant for all those worlds. To that regard, it is useful to quote Poincaré: “I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect...geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics”. Unfortunately the impact that the CG could have had on the development of GA, or on other more general mathematical systems, was hindered by the fact that public notice about it had to wait until the publication of Huygens’ correspondence in 1833 (by Uylenbroek). Even then, it took a while until some interest arouse, and then the effect was somewhat indirect. For our concerns here, it will suffice to recall a momentous episode of the Princely Jablo- nowski Society Prize for the Sciences and to trace its consequences. In 1845, the prize (48 gold ducats) was for “the restoration and further development of the geometrical calculus invented by Leibniz, or the construction of one equivalent to it”. The prize was awarded (in 1846, 200 years after Leibniz’ birthday) to Herrmann Grassmann (1809-1877), the author of the book GRASSMANN 1844 (Ausdehnungslehre), for the work GRASSMANN 1847 (Geome- trisches Analyse). It was the only one submitted and the report was published in MÖBIUS 1847. As it turns out, the true winner of the Jablonowsky prize was the Ausdehnungslehre, for its genesis and unfolding clearly show that Grassmann was on the right track for constructing a CG and thus that he needed not more than to push a bit forward his ideas to meet the prize requirements. In the words of COUTURAT 1901, “Grassmann took advantage of this oppor- tunity to explain his Extension Calculus, and to link it to Leibniz's project, while criticizing the philosopher’s rather formless essay”. Indeed, in the introduction of GRASSMANN 1847, the author stresses that his main tool was the ‘geometric analysis’ he had developed, and that with further development provided a mature fulfillment of Leibniz’ embryonic idea. In retrospect, the fundamental value of the Ausdehnungslehre and the prize memoir, is that it shifted the focus of mathematics from the study of magnitudes to the study of (abstract) structures and relations. In particular, he essentially completed a conceptual building of , including Grassmann’s and the insight that quadratic forms were the natural entity to express metrical notions in linear spaces. Thus his decisive steps in the directions envisioned by Leibniz’s CG were based on remarkably general mathematical ideas that have stood very well the passage of time. But the recognition of its deep significance also took many years, even after the publication of the much more readable second edition GRASSMANN 1862. This is a convenient point to state the purposes of this paper in a more specific form. After the considerations so far, it makes sense to regard the Ausdehnungslehre as a natural and suitable ground on which to graft a GA that meets the aspirations of Leibniz’s CG. Among the many important and interesting developments in this direction, our interest will be in the form that yields a comprehensive view with a rather moderate effort and we will show with a few selected examples its bearing on geometry and physics. For those interested in the analysis of Leibniz’ CG in terms of the historical documents, we refer to the text included in LEIBNIZ 1858, or to later studies, such as COUTURAT 1901 (chap- ter 9), LEIBNIZ 1995 (particularly the introductory study, ECHEVERRÍA 1995). This text re- produces the original Latin documents on left pages and the French translations on the right ones. It is also worth mentioning LEIBNIZ 2011 (particularly the introduction ECHEVERRÍA 2011), in which a good sample of Leibniz’ writings are translated onto Spanish. The master- ful ANTOGNAZZA 2009 is also very helpful.

DEVELOPMENT OF GEOMETRIC ALGEBRA

William K. Clifford (1845-1879) coined the name geometric algebra (GA) in CLIFFORD 1878. This landmark work advanced a synthesis of Grassmann’s Ausdehnungslehre and Hamilton’s algebra (HAMILTON 1843). It is worth noting that Clifford assessed Grassmann’s work as follows:

Until recently I was unacquainted with the Ausdehnungslehre [...]. I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work and my conviction that its principles will exercise a vast influence upon the future of science.

The main innovation introduced by Clifford was the geometric product, which is an associ- ative bilinear product on the Grassmann algebra of (exterior algebra). The cru- cial structure discovered by Clifford, called geometric algebra, or Clifford algebra for many later authors, is what we regard as a CG in our current understanding and will be described later in this paper along the lines advanced in the foreword. Here let us just anticipate that the complex numbers and the are very special cases of geometric algebras and that all their properties appear in a natural way, with its full geometric meaning, when ℂ ℍ seen in that light. Clifford’s idea of geometric algebra was independently discovered by Rudolf Lipschitz (1832-1903) and in some important respects he went further than Clifford, particularly with the introduction of what today are called Lipschitz groups, which appear in a natural way in the analysis of how geometric algebra encodes geometric transformations (LIPSCHITZ 1880). Let us also take notice that at the end of the XIX century there were a number of contributions about the logical and philosophical foundations of the system of geometric algebra, and which pay ample homage to Grassmann’s innovations, and indirectly to those of Leibniz. Two important works in that direction are PEANO 1896 and WHITEHEAD 1898. According to Peano, “The first step in geometrical calculus was taken be Leibniz, whose vast mind opened several new paths to mathematics”, stating that that calculus “differs from Cartesian geome- try in that whereas the latter operates analytically with coordinates, the former operates di- rectly on geometrical entities”. He also recognizes GRASSMANN 1844 as a landmark that was “little read and not appreciated by his contemporaries, but later it was found admirable by many scientists”. According to Whitehead, “The greatness of my obligations to Grassmann will be understood by those who have mastered his two Ausdehnungslehres. The technical development of the subject is inspired chiefly by his work of 1862, but the underlying ideas follow the work of 1844”.

It is a historical pity that Clifford’s or Lipschitz’ syntheses had to wait over six decades for their flourishing. Clifford died in Funchal (Madeira) in 1879, aged 33, precisely two centuries after Leibniz’ communication to Huygens. Then Gibbs eclipsed those ideas with his vector calculus. The immediate and lasting popularity of the Gibbs’ system it is due to the fact that it is a no frills calculus restricted to the three-dimensional Euclidean space and that it can be used to phrase classical mechanics and electromagnetism. Other new and important develop- ments were unaware of the value of GA and because of this, they appeared as isolated ad- vances, and it was not until decades later that the GA symbolism brought unity and strength. The most visible cases were Minkowsk’s space-time geometry (1908), Pauli’s quantum the- ory of spin (1926), Dirac’s equation for the relativistic electron (1928), Cartan’s theory of (1938), its algebraic presentation by Chevalley (1954), and, closer to the mark, but almost unknown at the time, Riesz (1958).

The founding document of the new era of GA is HESTENES 1966. It uncovers the full geo- metrical meaning of the Pauli algebra (as the GA of the Euclidean three-dimensional space), of the Dirac algebra (as the GA of the Minkowsky space), clarifying the deep relationship between the two, and finds a neat interpretation of the Pauli and Dirac spinors. In this and later works, particularly HESTENES-SOBCZYK 1984, Hestenes sets up a fully developed geo- metric calculus which allow him to write Maxwell’s equations as a single equation in the Dirac algebra (this had already been observed in RIESZ 1958) and uncover a deep geometric meaning of Dirac’s equation (DIRAC 1928). All the references cited in the Foreword posterior to 1966, share Hestenes’ vision, which quite likely Clifford would have accomplished had he not died so young. In the last years, GA has been applied or connected to a great variety of fields, including solid mechanics, robotics, electromagnetism and wave propagation, , cosmology, computer graphics, computer vision, pattern theory, molecular design, symbolic algebra, automated theorem proving, quantum computing …

AN ELEMENTARY VIEW OF GA

One of the driving ideas in mathematics has been to extend a given structure in order to include some new desirable features. The successive extensions of the notion of number pro- vide a good illustration. Starting from the natural numbers, = {1,2,3, … }, the inclusion of 0 and the negative numbers leads to the integers, = {… , 3, 2, 1, 0, 1,2,3, … }, in which ℕ the difference of any two numbers is always defined. Now division is not always possible in ℤ − − − , but one can introduce fractions or rational numbers, = { : , , 0}, in which division by a non-zero rational number is always possible. The real numbers are the ℤ ℚ 𝑚𝑚⁄𝑛𝑛 𝑚𝑚 𝑛𝑛 ∈ ℤ 𝑛𝑛 ≠ natural extension of that makes possible to take the upper bound of bounded sets, and is ℝ the natural extension of in which negative real numbers have roots, in particular, = 1. ℚ ℂ Similarly, GA arises outℝ of the desire to multiply vectors with the usual rules of multiplying𝑖𝑖 √− numbers, including the usual rules for taking inverses. Geometric algebra of the Euclidean plane. For definiteness, let us start with an Euclidean plane, .2 So is a real of 2 endowed with an Euclidean metric (a symmetric bilinear map 𝐸𝐸2 𝐸𝐸2 𝑞𝑞 ×

3 such that𝑞𝑞 ∶ (𝐸𝐸2, )𝐸𝐸>2 →0 for𝐸𝐸2 any 0). We want to enlarge to a system in which vectors can be multiplied, and non-zero vectors inverted, with the usual rules. To elicit the expected 𝑞𝑞 𝑎𝑎 𝑎𝑎 𝑎𝑎 ≠ 𝐸𝐸2 𝒢𝒢2 goods that is going to bring us, let us first make a few remarks. By we denote the product of , (simple juxtaposition of the factors) and we say that it is the geometric 𝒢𝒢2 𝑥𝑥𝑥𝑥 product of and . 𝑥𝑥 𝑦𝑦 ∈ 𝒢𝒢2 Technically,𝑥𝑥 the structure𝑦𝑦 of with the geometric product is supposed to be an associative and unital -algebra. This means that is a real vector space and that the geometric product 𝒢𝒢2 is bilinear, associative, with unit 1 . The map , 1, allows us to regard as ℝ 𝒢𝒢2 embedded in , and so we will not distinguish between and 1 . Such elements ∈ 𝒢𝒢2 ℝ → 𝒢𝒢2 𝜆𝜆 ↦ 𝜆𝜆 ℝ are the scalars of . We also have , and its elements are the vectors of . We 𝒢𝒢2 𝜆𝜆 ∈ ℝ 𝜆𝜆 ∈ 𝒢𝒢2 assume that = {0}. 𝒢𝒢2 𝐸𝐸2 ⊂ 𝒢𝒢2 𝒢𝒢2 ℝ ∩ 𝐸𝐸2

2 Many authors take = . Our notation is meant to stress that no basis is special. 3 Instead of ( , ), we will 2also write . As we will see, however, the two notations usually mean differ- 2 ent things when or 𝐸𝐸 are ℝnot vectors. 𝑞𝑞 𝑎𝑎 𝑏𝑏 𝑎𝑎 ⋅ 𝑏𝑏 𝑎𝑎 𝑏𝑏 Let . If is to have an inverse with respect to the geometric product, it is natural to assume that = { }, because is the only subset of that can be con- 𝑎𝑎 ∈ 𝐸𝐸2 𝑎𝑎 𝑎𝑎′ structed out of′ using the linear structure. But then 1 = = ( ) = , which implies 𝑎𝑎 ∈ 〈𝑎𝑎〉 𝜆𝜆𝜆𝜆 ∶ 𝜆𝜆 ∈ ℝ 〈𝑎𝑎〉 𝐸𝐸2 that must be a non-zero scalar. If now , , then 2 2 𝑎𝑎 𝑎𝑎′𝑎𝑎 𝜆𝜆𝜆𝜆 𝑎𝑎 𝜆𝜆𝑎𝑎 𝑎𝑎 ( + ) = + + + 𝑎𝑎 𝑏𝑏 ∈+𝐸𝐸2 . 2 2 2 The main𝑎𝑎 insight𝑏𝑏 of 𝑎𝑎Clifford𝑏𝑏 was𝑎𝑎𝑎𝑎 to postulate𝑏𝑏𝑏𝑏 ⇒ 𝑎𝑎𝑎𝑎 that𝑏𝑏 𝑏𝑏 ∈ ℝ + = ( , ) + ( , ) = 2 ( , ), because 𝑎𝑎𝑎𝑎 is symmetric,𝑏𝑏𝑏𝑏 𝑞𝑞 𝑎𝑎 and𝑏𝑏 in𝑞𝑞 particular𝑏𝑏 𝑎𝑎 that𝑞𝑞 𝑎𝑎 𝑏𝑏 𝑞𝑞 = ( ), 2 where for𝑎𝑎 simplicity𝑞𝑞 𝑎𝑎 we write ( ) = ( , ). In any case, the expression + defines a symmetric bilinear form of , and if 0, then = / ( ) is the inverse of . 𝑞𝑞 𝑎𝑎 𝑞𝑞 𝑎𝑎 𝑎𝑎 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 Notice also that = if and only if ( , ) =−01, that is, if and only if and are 𝐸𝐸2 𝑎𝑎 ≠ 𝑎𝑎 𝑎𝑎 𝑞𝑞 𝑎𝑎 𝑎𝑎 ∈ 𝐸𝐸2 orthogonal. 𝑎𝑎𝑎𝑎 −𝑏𝑏𝑏𝑏 𝑞𝑞 𝑎𝑎 𝑏𝑏 𝑎𝑎 𝑏𝑏 To go further, let us take an arbitrary orthonormal basis , ,4 which means that

( ) = ( ) = 1 and ( , ) = 0 = 𝑢𝑢1 =𝑢𝑢21∈ and𝐸𝐸2 = . 2 2 These rules𝑞𝑞 𝑢𝑢1 imply,𝑞𝑞 owing𝑢𝑢2 to the bilinearity𝑞𝑞 𝑢𝑢1 𝑢𝑢2 of the⇔ geometric𝑢𝑢1 𝑢𝑢2 product, 𝑢𝑢that2𝑢𝑢1 all geometric−𝑢𝑢1𝑢𝑢2 prod- ucts of (any number of) vectors belong to the subspace 1, , , . Since we want that is a minimal solution to our problem, it is thus natural to assume that 〈 𝑢𝑢1 𝑢𝑢2 𝑢𝑢1𝑢𝑢2〉 ⊆ 𝒢𝒢2 1, , , = . It is important to recognize that 1, , , are necessarily linearly 𝒢𝒢2 independent. Indeed, if 〈 𝑢𝑢1 𝑢𝑢2 𝑢𝑢1𝑢𝑢2〉 𝒢𝒢2 𝑢𝑢1 𝑢𝑢2 𝑢𝑢1𝑢𝑢2 + + + = 0 is a linear𝜆𝜆 relation,𝜆𝜆1𝑢𝑢1 then𝜆𝜆2𝑢𝑢 2we can𝜆𝜆12 𝑢𝑢multiply1𝑢𝑢2 by from the left and from the right and we obtain + = 0. 𝑢𝑢1 Adding𝜆𝜆 the 𝜆𝜆two1𝑢𝑢1 last− 𝜆𝜆 relations,2𝑢𝑢2 − 𝜆𝜆12 we𝑢𝑢1 𝑢𝑢conclude2 that + = 0 and hence = = 0. So we are left with + = 0, and this relation leads, after multiplying on the right by 𝜆𝜆 𝜆𝜆1𝑢𝑢1 𝜆𝜆 𝜆𝜆1 , to + = 0, and hence to = = 0 as well. This proves the claim.5 𝜆𝜆2𝑢𝑢2 𝜆𝜆12𝑢𝑢1𝑢𝑢2 Consequently,𝑢𝑢2 𝜆𝜆2 𝜆𝜆12 we𝑢𝑢1 have a decomposition𝜆𝜆2 𝜆𝜆12= + + , with 0 1 2 = 1 = , = , = 𝒢𝒢2, and𝒢𝒢 2 =𝒢𝒢2 𝒢𝒢2 , 0 1 2 N1 and it is𝒢𝒢 2not hard〈 〉 toℝ see𝒢𝒢 that2 it〈𝑢𝑢 is1 independen𝑢𝑢2〉 𝐸𝐸2 t of𝒢𝒢 the2 orthonormal〈𝑢𝑢1𝑢𝑢2〉 basis used (to ease the reading we collect some of the mathematical deductions with labels N1,… in the Notes sec- tion at the end). The elements of are called or pseudoscalars. To get a better 2 𝒢𝒢2 4 Grassmann, and many others ever since, use the notation , , … Here we avoid it to prevent potential con- fusions in important expressions involving exponentials. 1 2 5 This proof is a particular case of the general method introduced𝑒𝑒 𝑒𝑒 in RIESZ 1958 (Riesz method in this paper). appreciation of this component of , consider the map , ( , ) ( ). 1 Since this map is bilinear and skew-symmetric, it gives a linear2 map such that 𝒢𝒢2 𝐸𝐸2 → 𝒢𝒢2 𝑎𝑎 𝑏𝑏 ↦ 2 𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏 2 ( ). N2 Λ 𝐸𝐸2 → 𝒢𝒢2 1 In particular,𝑎𝑎 ∧ 𝑏𝑏 ↦ 2 𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏 , and so we have a canonical linear isomorphism . Since = = and = = , we actually have a canonical linear2 isomor-2 𝑢𝑢1 ∧ 𝑢𝑢2 ↦ 𝑢𝑢1𝑢𝑢2 Λ 𝐸𝐸2 ≃ 𝒢𝒢2 phism 0 0 1 1 Λ 𝐸𝐸2 ℝ 𝒢𝒢2 Λ 𝐸𝐸2 𝐸𝐸2 𝒢𝒢2 = + + + + = . 0 1 2 0 1 2 This allowsΛ𝐸𝐸 usΛ to𝐸𝐸 copy2 Λ onto𝐸𝐸2 Λ the𝐸𝐸2 exterior≃ 𝒢𝒢2 product𝒢𝒢2 𝒢𝒢2 of 𝒢𝒢2. The result is the exterior product in coexisting, from now on, with the geometric product. The basic relation between the 𝒢𝒢2 Λ𝐸𝐸 two products is the following key relation, also discovered by Clifford: 𝒢𝒢2 = ( + ) + ( ) = + . 1 1 From the𝑎𝑎𝑎𝑎 geometric2 𝑎𝑎𝑎𝑎 point𝑏𝑏𝑏𝑏 of view,2 𝑎𝑎𝑎𝑎 −the𝑏𝑏 elements𝑏𝑏 𝑎𝑎 ⋅ 𝑏𝑏of 𝑎𝑎 ∧=𝑏𝑏 represent, following the stand- ard interpretation of the Grassmann algebra, oriented2 areas2 . 𝒢𝒢2 Λ 𝐸𝐸2 Among these, the unit area = = plays a very important role. It anticommutes with vectors and satisfies = = = 1, so that is a geometric root of 𝒊𝒊 𝑢𝑢1 ∧ 𝑢𝑢2 𝑢𝑢1𝑢𝑢2 1. This entails several more2 marvels. If we set 2 2 = + = 1, , which is called the 𝒊𝒊 𝑢𝑢1𝑢𝑢2𝑢𝑢1𝑢𝑢2 −𝑢𝑢1𝑢𝑢2 − 𝒊𝒊 even geometric algebra, we see that , via + + 0 2+ , but of course the geo- − 𝒢𝒢2 𝒢𝒢2 𝒢𝒢2 〈 𝒊𝒊〉 metric meaning of the left hand side (which+ henceforth will be denoted and called complex 𝒢𝒢2 ≃ ℂ 𝛼𝛼 𝛽𝛽𝒊𝒊 ↦ 𝛼𝛼 𝛽𝛽𝛽𝛽 scalars) is lost when we move to , the formal square root of 1. On the other hand, the map 𝐂𝐂 = , , is a linear isomorphism, with inverse the map = , 𝑖𝑖 − (since0 2= 1). Because of this, we say that the area elements are the2 pseudoscalars0 , ℝ 𝒢𝒢2 → 𝒢𝒢2 𝜆𝜆 ↦ 𝜆𝜆𝒊𝒊 𝒢𝒢2 → 𝒢𝒢2 ℝ 𝑠𝑠 ↦ and in particular2 that is the unit pseudoscalar (associated to the basis , ). If we change −𝑠𝑠𝒊𝒊 𝒊𝒊 − the orthonormal basis, we have seen that the corresponding pseudoscalar is ± . Still another 𝒊𝒊 𝑢𝑢1 𝑢𝑢2 marvel is that = is a -vector space, because = and = . Note that these 𝒊𝒊 relations show that1 the linear isomorphism , , is the counterclockwise rotation 𝒢𝒢2 𝐸𝐸2 𝐂𝐂 𝑢𝑢1𝒊𝒊 𝑢𝑢2 𝑢𝑢2𝒊𝒊 −𝑢𝑢1 by /2. More generally, the map , 2 2 , is the rotation of by in the orien- 𝐸𝐸 → 𝐸𝐸 𝜃𝜃𝑎𝑎𝒊𝒊 ↦ 𝑎𝑎𝒊𝒊 tation. Here expands, as usual, 2to cos2 + sin , and so 𝜋𝜋 𝜃𝜃𝒊𝒊 𝐸𝐸 → 𝐸𝐸 𝑎𝑎 ↦ 𝑎𝑎𝑒𝑒 𝑎𝑎 𝜃𝜃 𝒊𝒊 𝑒𝑒= cos + sin , 𝜃𝜃 =𝒊𝒊 𝜃𝜃cos sin (see Figure 1). N3 𝜃𝜃𝒊𝒊 𝜃𝜃𝒊𝒊 𝑢𝑢1𝑒𝑒 𝑢𝑢1 𝜃𝜃 𝑢𝑢2 𝜃𝜃 𝑢𝑢2𝑒𝑒 𝑢𝑢2 𝜃𝜃 − 𝑢𝑢1 𝜃𝜃

Figure 1. , , and the action of on them. 𝜃𝜃𝒊𝒊 𝑢𝑢1 𝑢𝑢2 𝒊𝒊 𝑒𝑒 Geometric algebra of the Euclidean space. To fully appreciate GA, even in the case of dimension 2, we need to explore the geometric algebra of , the three dimensional Eu- clidean space. The development is similar to that of and we only need to summarize the 𝒢𝒢3 𝐸𝐸3 main points. In this case we have the decomposition 𝒢𝒢2 = + + + , with 0 1 2 3 𝒢𝒢3 = 𝒢𝒢13 =𝒢𝒢3 , 𝒢𝒢3= 𝒢𝒢3, , = , = , , , = , 0 1 2 3 where 𝒢𝒢3, ,〈 〉 ℝ is𝒢𝒢3 an orthonormal〈𝑢𝑢1 𝑢𝑢2 𝑢𝑢3〉 basis.𝐸𝐸3 𝒢𝒢T3he decomposition〈𝑢𝑢1𝑢𝑢2 𝑢𝑢1𝑢𝑢3 𝑢𝑢 2does𝑢𝑢3〉 not𝒢𝒢3 depend〈𝑢𝑢1𝑢𝑢2 on𝑢𝑢3 〉the basis used, as follows from the fact that there is a canonical linear isomorphism 𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 ∈ 𝐸𝐸3 for = 0,1,2,3. For = 0,1, it is the identity. For = 2 it is defined as for : 𝑘𝑘 𝑘𝑘 Λ 𝐸𝐸 ≃ 𝒢𝒢3 𝑘𝑘 ( 𝑘𝑘 ). 𝑘𝑘 𝐸𝐸2 1 And for𝑎𝑎 ∧=𝑏𝑏 3↦, 2 𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏

𝑘𝑘 ( + + ). 1 So we get𝑎𝑎 ∧ an𝑏𝑏 ∧exterior𝑐𝑐 ↦ 6 product𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏 in𝑏𝑏𝑏𝑏 by𝑐𝑐𝑐𝑐𝑐𝑐 copying− 𝑎𝑎𝑎𝑎𝑎𝑎 −the𝑏𝑏 exterior𝑏𝑏𝑏𝑏 − 𝑐𝑐𝑐𝑐𝑐𝑐 product of via the described isomorphisms. Notice that we have = and = .N4 𝒢𝒢3 Λ𝐸𝐸 The pseudoscalar = (the unit𝑢𝑢𝑖𝑖 ∧ volume𝑢𝑢𝑗𝑗 𝑢𝑢𝑖𝑖)𝑢𝑢 plays𝑗𝑗 𝑢𝑢a 𝑖𝑖special∧ 𝑢𝑢𝑗𝑗 ∧ 𝑢𝑢role:𝑘𝑘 it𝑢𝑢 𝑖𝑖commutes𝑢𝑢𝑗𝑗𝑢𝑢𝑘𝑘 with all vectors and = 1. Thus we have a linear isomorphism , ( , , of 𝐢𝐢 𝑢𝑢1𝑢𝑢2𝑢𝑢3 are mapped 2to the basis = , = , = 1 of 2 ), with inverse 1 𝐢𝐢 − 𝒢𝒢3 ≃ 𝒢𝒢3 𝑎𝑎 ↦ 𝑎𝑎𝐢𝐢 𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 𝒢𝒢3 given by . And of ∗course there∗ is a linear ∗isomorphism 2 , , with2 in-1 𝑢𝑢1 𝑢𝑢2𝑢𝑢3 𝑢𝑢2 𝑢𝑢3𝑢𝑢1 𝑢𝑢3 𝑢𝑢1𝑢𝑢2 𝒢𝒢3 𝒢𝒢3 ≃ 𝒢𝒢3 verse . In general, we have an isomorphism ( 0 =3 ) which is usu- 𝑏𝑏 ↦ −𝑏𝑏𝐢𝐢 𝒢𝒢3 ≃ 𝒢𝒢3 𝜆𝜆 ↦ 𝜆𝜆𝐢𝐢 ally called Hodge duality (see Figure 2). 𝑘𝑘 3−𝑘𝑘 ∗ 𝑠𝑠 ↦ −𝑠𝑠𝐢𝐢 𝒢𝒢3 ≃ 𝒢𝒢3 𝑥𝑥 ↦ 𝑥𝑥 𝑥𝑥𝐢𝐢

Scalars 1

Vectors: oriented segments , , (polar vectors) 𝑢𝑢1 𝑢𝑢 2 𝑢𝑢3 Bivectors: oriented areas , , (axial vectors) ∗ ∗ ∗ 𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 Pseudoscalars: oriented volumes

𝐢𝐢 Hodge duality = ∗ 𝑢𝑢𝑘𝑘 𝑢𝑢𝑘𝑘𝐢𝐢 Figure 2. Geometric algebra units of .

3 Rotations and rotors. We are now ready to describe rotations 𝐸𝐸of in terms of and use it = ( ) = 1 to establish some interesting consequences. Let be a unit vector 3(that is, 3 ). 𝐸𝐸 2 𝒢𝒢 𝑛𝑛 𝑛𝑛 𝑞𝑞 𝑛𝑛 Then it is easy to see that is the unit area of the perpendicular plane and hence rotates any by in the sense of . The important observation⊥ now is that the 𝑛𝑛𝐢𝐢 𝑛𝑛 𝑎𝑎 ↦ rotation𝜃𝜃𝜃𝜃𝐢𝐢 of any vector ⊥ by about the axis is given by what may be called Euler’s 𝑎𝑎𝑒𝑒 𝑎𝑎 ∈ 𝑛𝑛 𝜃𝜃 𝑛𝑛𝐢𝐢 formula: 𝑎𝑎 ∈ 𝐸𝐸3 𝜃𝜃 𝑛𝑛 / / . −𝜃𝜃𝜃𝜃𝐢𝐢 2 𝜃𝜃𝜃𝜃𝐢𝐢 2 To prove𝑎𝑎 ↦this,𝑒𝑒 it is enough𝑎𝑎 𝑒𝑒 to show that = is fixed and that if is orthogonal to , then it is rotated by in (the plane orthogonal to ). Indeed, since commutes with / , 𝑎𝑎 𝑛𝑛 𝑎𝑎 𝑛𝑛 the value of the formula⊥ for = is / / = ; and if , then anticom-−𝜃𝜃𝜃𝜃𝐢𝐢 2 𝜃𝜃 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑒𝑒 mutes with , hence also with / , and−𝜃𝜃𝜃𝜃 the𝐢𝐢 2 formula𝜃𝜃𝜃𝜃𝐢𝐢 2 yields , which⊥ is, as we have 𝑎𝑎 𝑛𝑛 𝑛𝑛𝑒𝑒 𝑒𝑒 𝑛𝑛 𝑎𝑎 ∈ 𝑛𝑛 𝑎𝑎 remarked, the rotation of in the− plane𝜃𝜃𝜃𝜃𝐢𝐢 2 by in the sense of . 𝜃𝜃𝜃𝜃 𝐢𝐢 𝑛𝑛 𝑒𝑒 𝑎𝑎𝑒𝑒 / ⊥ Let us say that , = 𝑎𝑎 is the rotor𝑛𝑛 of the𝜃𝜃 rotation , about𝑛𝑛𝐢𝐢 by . Thus 𝜃𝜃𝜃𝜃𝐢𝐢 2 𝑛𝑛 𝜃𝜃 𝑛𝑛 𝜃𝜃 , ( )𝑅𝑅= , 𝑒𝑒 , . 𝜌𝜌 𝑛𝑛 𝜃𝜃 −1 𝑛𝑛 𝜃𝜃 𝑛𝑛 𝜃𝜃 𝑛𝑛 𝜃𝜃 If , 𝜌𝜌 is another𝑎𝑎 𝑅𝑅rotation,𝑎𝑎 𝑅𝑅 and , its rotor, then the product , , is clearly the rotor ′ ′ of the composition ′ ′. Therefore we have the equation ′ ′ 𝜌𝜌𝑛𝑛, 𝜃𝜃 , 𝑅𝑅𝑛𝑛 ,𝜃𝜃 𝑅𝑅𝑛𝑛 𝜃𝜃𝑅𝑅𝑛𝑛 𝜃𝜃 ′′ ′′ ′ ′ 𝑅𝑅𝑛𝑛 𝜃𝜃 cos + sin 𝜌𝜌=𝑛𝑛 𝜃𝜃cos𝜌𝜌𝑛𝑛 𝜃𝜃 + sin cos + sin ′′ ′′ ′ ′ 𝜃𝜃 ′′ 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 which itself 2is equivalent𝑛𝑛 𝐢𝐢 2(equating� 2scalar𝑛𝑛𝐢𝐢 and2 � � 2 parts)𝑛𝑛′𝐢𝐢 to the2 �equations

cos = cos cos ( ) sin sin ′′ ′ ′ 𝜃𝜃 𝜃𝜃 𝜃𝜃 ′ 𝜃𝜃 𝜃𝜃 sin2 = 2sin 2cos− 𝑛𝑛+⋅ 𝑛𝑛 cos 2sin 2 + ( × ) sin sin ′′ ′ ′ ′, ′′ 𝜃𝜃 𝜃𝜃 𝜃𝜃 ′ 𝜃𝜃 𝜃𝜃 𝜃𝜃 𝜃𝜃 where 𝑛𝑛× =2 ( 𝑛𝑛 ) 2 is the2 cross𝑛𝑛 product2 (the2 dual𝑛𝑛 of 𝑛𝑛the′ wedge2 product).2 These are the famous Olinde′ Rodrigues’s formulas obtained in 1843 with a remarkably long and involved 𝑛𝑛 𝑛𝑛 𝑛𝑛 ∧ 𝑛𝑛′ 𝐢𝐢 computation using Cartesian coordinates.

The algebra is the main tool used in the excellent treatise HESTENES 1990 for the formu- lation of classical mechanics in for the first time. Aside from the many advantages that this 𝒢𝒢3 approach provides, including the use of rotors as dynamic variables, plays a crucial role, as we will see, as the relative algebra of an observer in the geometric algebra approach to 𝒢𝒢3 (space-time algebra). Quaternions. provides a geometric realization of Hamilton’s quaternions . Let

= 𝒢𝒢3, = , = . 𝐇𝐇 ℍ Then we𝒊𝒊1 have𝑢𝑢1 𝐢𝐢 𝒊𝒊2= 1𝑢𝑢,2𝐢𝐢 ,𝒊𝒊3 , 𝑢𝑢,3 which𝐢𝐢 we will denote . This even geometric algebra is isomorphic to ,+ because the unit areas , , satisfy Hamilton’s famous relations:N5 𝒢𝒢3 〈 𝒊𝒊1 𝒊𝒊2 𝒊𝒊3〉 𝐇𝐇 = ℍ= = 1, = 𝒊𝒊1 =𝒊𝒊2 𝒊𝒊, 3 = = , = = . 2 2 2 𝒊𝒊1 𝒊𝒊2 𝒊𝒊3 − 𝒊𝒊1𝒊𝒊2 −𝒊𝒊2𝒊𝒊1 𝒊𝒊3 𝒊𝒊2𝒊𝒊3 −𝒊𝒊3𝒊𝒊3 𝒊𝒊1 𝒊𝒊3𝒊𝒊1 −𝒊𝒊1𝒊𝒊3 𝒊𝒊2 The advantage of over is the direct relation to the geometry of . For example, in Euler’s spinor formula, = + + is a bivector (a pure quaternion in the ha- 𝐇𝐇 ℍ 𝐸𝐸3 bitual terminology) and Euler’s1 spinor1 2formula2 3 coincides3 with the way quaternions are used 𝑛𝑛𝐢𝐢 𝑛𝑛 𝒊𝒊 𝑛𝑛 𝒊𝒊 𝑛𝑛 𝒊𝒊 / / to produce rotations. Note that the rotation given by the rotor = = is , , that is, the axial symmetry about . 𝜋𝜋𝒊𝒊𝑘𝑘 2 𝜋𝜋𝑢𝑢𝑘𝑘𝐢𝐢 2 𝒊𝒊𝑘𝑘 𝑒𝑒 𝑒𝑒 𝜌𝜌𝑢𝑢𝑘𝑘 𝜋𝜋 Geometric covariance. Another advantage𝑢𝑢𝑘𝑘 of is that Euler’s spinor formula can be applied to any . Actually the map , = / / is an automorphism 𝒢𝒢3 of (if = / , ( ) = ( ) = ′ −𝜃𝜃𝜃𝜃=𝐢𝐢 2 𝜃𝜃𝜃𝜃). 𝐢𝐢This2 is what may be 𝒢𝒢3 → 𝒢𝒢3 𝑥𝑥 ↦ 𝑥𝑥 𝑒𝑒 𝑥𝑥𝑒𝑒 called the principle𝜃𝜃𝜃𝜃𝐢𝐢 of2 geometric′ covariance−1 . The−1 simplest−1 illustration is that ( ) = 𝒢𝒢3 𝑅𝑅 𝑒𝑒 𝑥𝑥𝑥𝑥 𝑅𝑅 𝑥𝑥𝑥𝑥 𝑅𝑅 𝑅𝑅 𝑥𝑥𝑥𝑥 𝑅𝑅 𝑦𝑦𝑦𝑦 𝑥𝑥′𝑦𝑦′ (this relation is true because ( ) = ( ) = ( ) = ). Now 1 1 𝑎𝑎 ∧ 𝑏𝑏 ′ 𝑎𝑎′ ∧ notice that the right hand side constructs′ the oriented area′ over the′ ′ rotated′ ′ vectors′ and , 𝑏𝑏′ 𝑎𝑎 ∧ 𝑏𝑏 2 𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏 2 𝑎𝑎 𝑏𝑏 − 𝑏𝑏 𝑎𝑎 𝑎𝑎 ∧ 𝑏𝑏′ which is the rotated oriented area, while the left hands side gives the good news that we can 𝑎𝑎′ 𝑏𝑏′ arrive at the same result by directly ‘rotating’ the oriented area expressed as a bivector. Here is another illuminating example. Let us apply the transformation to a rotor = / ( a unit vector, real), giving = . Now the geometric meaning of is easy𝛼𝛼𝛼𝛼 𝐢𝐢to2 𝑆𝑆 𝑒𝑒 ascertain: from ′ −1 𝑝𝑝 𝛼𝛼 𝑆𝑆 𝑅𝑅 𝑆𝑆𝑆𝑆 𝑆𝑆′ = cos + sin = cos + sin ′ ′ 𝛼𝛼 𝛼𝛼 𝛼𝛼 𝛼𝛼 we see that𝑆𝑆 � is the2 rotor𝑝𝑝𝐢𝐢 of the2� rotation2 about𝑝𝑝′𝐢𝐢 , 2which is the rotation of about by . In other words, to ‘rotate’ a rotation . to , , it is enough to ‘rotate’ the rotor to , for 𝑆𝑆′ 𝑝𝑝′ 𝑝𝑝 𝑛𝑛 𝜃𝜃 is the rotor of . ′ ′ , 𝜌𝜌𝑝𝑝 𝛼𝛼 𝜌𝜌𝑝𝑝 𝛼𝛼 𝑆𝑆 𝑆𝑆 𝑆𝑆′ ′ The Pauli representation.𝜌𝜌𝑝𝑝 𝛼𝛼 The algebra is isomorphic to Pauli’s spin algebra. In technical terms, the latter is a representation of the former. In this representation, , , are 𝒢𝒢3 mapped to Pauli’s matrices 𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 1 1 = , = , = , 1 1 1 2 −𝑖𝑖 3 which satisfy𝜎𝜎 � Clifford’s� relations𝜎𝜎 � � 𝜎𝜎 � � 𝑖𝑖 − + = 2 , ( , is 0 if and 1 if = ).

The Pauli𝜎𝜎𝑗𝑗𝜎𝜎 matrices𝑘𝑘 𝜎𝜎𝑘𝑘𝜎𝜎𝑗𝑗 belong𝛿𝛿𝑗𝑗 𝑘𝑘to 𝛿𝛿𝑗𝑗(𝑘𝑘2), the 2𝑗𝑗 ×≠ 2𝑘𝑘 complex 𝑗𝑗matrices,𝑘𝑘 and it is easy to check that , , , , , , , form a basis of (2) as a real vector space ( denotes ℂ the identity matrix). We conclude that we have an isomorphism of real algebras (2). 𝜎𝜎0 𝜎𝜎1 𝜎𝜎2 𝜎𝜎3 𝜎𝜎1𝜎𝜎2 𝜎𝜎1𝜎𝜎3 𝜎𝜎2𝜎𝜎3 𝜎𝜎1𝜎𝜎2𝜎𝜎3 ℂ 𝜎𝜎0 Note, however, that the rich geometric structure of is invisible in the algebra (2).6 𝒢𝒢3 ≃ ℂ 𝒢𝒢3 ℂ

6 In the presentation of , many authors use the symbols , , (or , , ) instead of , , . For reasons that we shall explain, we will revert to this convention in the presentation of the space-time algebra 3 1 2 3 1 2 3 1 2 3 below. Notice also that 𝒢𝒢the Pauli representation provides a𝜎𝜎 proof𝜎𝜎 𝜎𝜎of the 𝝈𝝈existence𝝈𝝈 𝝈𝝈 of . 𝑢𝑢 𝑢𝑢 𝑢𝑢

𝒢𝒢3 The indices in the Pauli matrices are explained as follows. Those matrices have eigenvalues ±1, and the corresponding eigenvectors in (this is called Pauli’s spinor space) are [1, ±1], [1, ± ], and [1,0] and [0,1], respectively. These2 eigenvalues become the unit points on the ℂ axes , , under the spinor map (Hopf fibration for mathematics), where is 𝑖𝑖 the unit sphere in ([ , ] 3 if and2 only if + = 1) and is the unit3 𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 𝑆𝑆 → 𝑆𝑆 𝑆𝑆 sphere in (2 = 4 + +3 if and only if + + =2 1). By the ℂ ≃ ℝ 𝜉𝜉0 𝜉𝜉1 ∈ 𝑆𝑆 𝜉𝜉0𝜉𝜉0̅ 𝜉𝜉1𝜉𝜉1̅ 𝑆𝑆 3 = [ , ] 2 2 2 2 spinor map,3 the image 1 1 of 2 2 3 3 is given by 1 2 3 𝐸𝐸 ≃ ℝ 𝑎𝑎 𝑎𝑎 𝑢𝑢 2 𝑎𝑎 𝑢𝑢 𝑎𝑎 𝑢𝑢 ∈ 𝑆𝑆 𝑎𝑎 𝑎𝑎 𝑎𝑎 = + 𝑎𝑎 ∈, 𝑆𝑆 = 𝜓𝜓( 𝜉𝜉0 𝜉𝜉1 ), = . With this,𝑎𝑎1 the𝜉𝜉 claim0𝜉𝜉1̅ follows𝜉𝜉0̅ 𝜉𝜉1 𝑎𝑎 immediately.2 𝑖𝑖 𝜉𝜉0𝜉𝜉1̅ − 𝜉𝜉0̅ 𝜉𝜉1 𝑎𝑎3 𝜉𝜉1𝜉𝜉1̅ − 𝜉𝜉0𝜉𝜉0̅

Space-tima algebra

The final example we will examine is the geometric algebra = , of the Minkowsky 7 space-time , (the Dirac algebra). The metric of , will be denoted (instead of ). It 𝒟𝒟 𝒢𝒢1 3 has signature (1,3), which means that there exist bases = , , , of , such that 𝐸𝐸1 3 𝐸𝐸1 3 𝜂𝜂 𝑞𝑞 , = , where 𝜸𝜸 𝛾𝛾0 𝛾𝛾1 𝛾𝛾2 𝛾𝛾3 𝐸𝐸1 3 8 𝜂𝜂�𝛾𝛾𝜇𝜇 𝛾𝛾𝜈𝜈� =𝜂𝜂𝜇𝜇0𝜇𝜇 for , = 1 , = 1 for = 1,2,3. Such base𝜂𝜂𝜇𝜇𝜇𝜇s , which𝜇𝜇 are≠ 𝜈𝜈said𝜂𝜂 to00 be orthonormal𝜂𝜂𝑘𝑘𝑘𝑘 − in mathematics,𝑘𝑘 are called inertial frames in special relativity texts. The axis is the temporal axis and , , the spatial axes. In 𝜸𝜸 general, a vector is timelike (spacelike) if ( ) > 0 ( ( ) < 0). Vectors such that ( ) = 𝛾𝛾0 𝛾𝛾1 𝛾𝛾2 𝛾𝛾3 0 are called null vectors (isotropic in mathematics). Figure 3 summarizes the correspondence 𝑎𝑎 𝜂𝜂 𝑎𝑎 𝜂𝜂 𝑎𝑎 𝜂𝜂 𝑎𝑎 between a short list of mathematics and special relativity terms. Henceforth we will not bother about such language differences.

Symbols Mathematics Special relativity

, Vectors Events or vectors

𝑎𝑎(∈)𝐸𝐸>1 30 Positive vector Time-like vector

𝜂𝜂(𝑎𝑎) < 0 Negative vector Space-like vector

𝜂𝜂(𝑎𝑎) = 0 Isotropic vector Null or light-line vector

( ) 𝜂𝜂= 𝑎𝑎( ), linear Isometry Lorentz transformation

𝜂𝜂 𝐿𝐿𝐿𝐿 , 𝜂𝜂 𝑎𝑎= 𝐿𝐿 , Orthonormal basis Inertial frame

𝜇𝜇 𝜈𝜈 𝜇𝜇 𝜈𝜈 Figure𝜂𝜂�𝛾𝛾 3.𝛾𝛾 Translating� 𝜂𝜂 between symbolics, mathematics and physics.

7 This is the algebra whose detailed study in HESTENES 1966 best signals the beginning of the contemporary understanding of geometric algebra. 8 As we will see, the symbols correspond to the so-called Dirac -matrices much in the same way as the reference symbols for correspond to Pauli’s -matrices. 𝛾𝛾𝜇𝜇 Γ 𝒢𝒢3 𝜎𝜎 Now we proceed to construct the geometric algebra = , of , by analogy with the case . Starting with a frame , let us write = for = , … , = {0,1,2,3,4}. 𝒟𝒟 𝒢𝒢1 3 𝐸𝐸1 3 Then we find that the 2 products with < 1 < 𝑘𝑘, = 0, … ,4, form a basis of and 𝒢𝒢3 𝜸𝜸 𝛾𝛾𝐽𝐽 𝛾𝛾𝑗𝑗 ⋯ 𝛾𝛾𝑗𝑗 𝐽𝐽 𝑗𝑗1 𝑗𝑗𝑘𝑘 ∈ 𝑁𝑁 that by grouping the products4 for the different we have a grading 𝛾𝛾𝐽𝐽 𝑗𝑗1 ⋯ 𝑗𝑗𝑘𝑘 𝑘𝑘 𝒟𝒟 = + + + + . 𝑘𝑘 0 1 2 3 4 Thus 𝒟𝒟 = 𝒟𝒟1 = 𝒟𝒟 (as 𝒟𝒟 =𝒟𝒟1) and𝒟𝒟 = , , , = , . The other terms can be convenient0 ly described by introducing the1 bivectors = ( = 1,2,3), the pseudoesca- 𝒟𝒟 〈 〉 ℝ 𝛾𝛾∅ 𝒟𝒟 〈𝛾𝛾0 𝛾𝛾1 𝛾𝛾2 𝛾𝛾3〉 𝐸𝐸1 3 = = lar and the notation (Hodge dual𝑘𝑘 of 𝑘𝑘). Then0 ∗ 𝜎𝜎 𝛾𝛾 𝛾𝛾 𝑘𝑘 𝐢𝐢 𝛾𝛾0𝛾𝛾=1𝛾𝛾2𝛾𝛾3 , = ,𝑥𝑥 =𝑥𝑥𝐢𝐢 and 𝑥𝑥= , , , , , . ∗ ∗ ∗ 2 ∗ ∗ ∗ Similarly,𝜎𝜎1 −=𝛾𝛾2𝛾𝛾3 𝜎𝜎2, −=𝛾𝛾3𝛾𝛾1 𝜎𝜎3, −=𝛾𝛾1𝛾𝛾2 , 𝒟𝒟= 〈𝜎𝜎1 ,𝜎𝜎 and2 𝜎𝜎3 𝜎𝜎1 𝜎𝜎2 𝜎𝜎3〉 ∗ ∗ ∗ ∗ Similarly, 𝛾𝛾0 = 𝛾𝛾1𝛾𝛾2,𝛾𝛾 3 𝛾𝛾=1 𝛾𝛾0, 𝛾𝛾2𝛾𝛾3= 𝛾𝛾2 , 𝛾𝛾0𝛾𝛾=3𝛾𝛾1 𝛾𝛾3, and𝛾𝛾0 𝛾𝛾 1𝛾𝛾2= , , , . ∗ ∗ ∗ ∗ 3 ∗ ∗ ∗ ∗ Finally it is𝛾𝛾 0clear𝛾𝛾 123that 𝛾𝛾1 =𝛾𝛾023 =𝛾𝛾21 .𝛾𝛾 031Figure𝛾𝛾3 4 summarizes𝛾𝛾012 𝒟𝒟 all these〈𝛾𝛾0 relations𝛾𝛾1 𝛾𝛾2 𝛾𝛾 3at〉 a glance. 4 ∗ We still have canonical linear isomorphisms , , and hence a canonical linear iso- 〈 〉 〈 〉 𝒟𝒟 𝐢𝐢 𝑘𝑘 𝑘𝑘 morphism , , that allows us to copy the exterior product of , to an exterior prod- Λ 𝐸𝐸1 3 ≃ 𝒟𝒟 uct of . In particular, the metric can be extended canonically to a metric of N6 and Λ𝐸𝐸1 3 ≃ 𝒟𝒟 Λ𝐸𝐸1 3 hence to a metric of . For the basis elements the value of is just the product of the values 𝒟𝒟 𝜂𝜂 Λ𝐸𝐸 of the factors. For example, ( ) = ( ) = ( ) ( ) = 1, ( ) = ( 1)( 1) = 𝒟𝒟 𝜂𝜂 𝜂𝜂 1, and ( ) = (+1)( 1)( 1)( 1) = 1. The scheme also reflects the∗ Hodge dualities 𝜂𝜂 𝜎𝜎𝑘𝑘 𝜂𝜂 𝛾𝛾𝑘𝑘𝛾𝛾0 𝜂𝜂 𝛾𝛾𝑘𝑘 𝛾𝛾 𝛾𝛾0 − 𝜂𝜂 𝜎𝜎𝑘𝑘 − − , (they are anti-isometries: ( ) = ( ) = ( )). 𝑘𝑘 𝜂𝜂4−𝐢𝐢𝑘𝑘 ∗ − − − − ∗ 𝒟𝒟 ≃ 𝒟𝒟 𝑥𝑥 ↦ 𝑥𝑥 𝜂𝜂 𝑥𝑥 𝜂𝜂 𝑥𝑥𝐢𝐢 −𝜂𝜂 𝑥𝑥 Grade Names Bases Notations

0 Scalars 1 =

1 Vectors 𝐢𝐢 𝛾𝛾0=𝛾𝛾1𝛾𝛾2 𝛾𝛾3 ∗ 2 Bivectors 𝛾𝛾0 𝛾𝛾1̅ 𝛾𝛾̅2 𝛾𝛾̅3 𝑥𝑥( ) =𝑥𝑥𝐢𝐢1 ∗ ∗ ∗ 3 Pseudovectors 𝜎𝜎�1 𝜎𝜎�2 𝜎𝜎�3 𝜎𝜎1 𝜎𝜎2 𝜎𝜎3 𝜂𝜂( 𝑥𝑥) = 1 ∗ ∗ ∗ ∗ 4 Pseudoscalars 𝛾𝛾̅0 𝛾𝛾11 𝛾𝛾 2 𝛾𝛾3 𝜂𝜂 𝑥𝑥̅ = −1 ∗ 2 � Figure 4. Basis of . Products for which is 1 are distinguished with an𝐢𝐢 overbar,− and otherwise ( ) = +1. Note that if is a basis element, then = ( ) only happens for scalars, vectors and 𝒟𝒟 𝜂𝜂 − pseudoscalars. For bivectors and pseudovectors,2 = ( ). 𝜂𝜂 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝜂𝜂 𝑥𝑥 2 𝑥𝑥 −𝜂𝜂 𝑥𝑥 Relative space. The space = , , , which depends on the frame , is called the rela- tive space. It generates the even subalgebra = and since = ( ) = 1, it turns ℰ 〈𝜎𝜎1 𝜎𝜎2 𝜎𝜎3〉 𝜸𝜸 out that is isomorphic to the Pauli algebra. The pseudoscalar+ of 2 coincides with , for 𝒫𝒫 𝒟𝒟 𝜎𝜎𝑘𝑘 −𝜂𝜂 𝜎𝜎𝑘𝑘 𝒫𝒫 𝒫𝒫 𝐢𝐢 = = = .

The grading𝜎𝜎1𝜎𝜎2 𝜎𝜎3 = 𝛾𝛾1𝛾𝛾0+𝛾𝛾2𝛾𝛾0𝛾𝛾+3𝛾𝛾0 +𝛾𝛾0𝛾𝛾1 is𝛾𝛾2 easy𝛾𝛾3 to𝐢𝐢 describe in terms of : 0 1 2 3 =𝒫𝒫 , 𝒫𝒫 = 𝒫𝒫, 𝒫𝒫= ,𝒫𝒫 = = . 𝒟𝒟 0 1 2 3 4 Notice that𝒫𝒫 ℝ+ 𝒫𝒫 = ℰ +𝒫𝒫 =ℰ𝐢𝐢 .𝒫𝒫 〈𝐢𝐢〉 𝒟𝒟 1 2 2 If , 𝒫𝒫and we𝒫𝒫 writeℰ =ℰ𝐢𝐢 𝒟𝒟 + + + , it is customary to also use the symbols , , , such that =0 , =1 , =2 , 3= (or = if units are chosen 𝑎𝑎 ∈ 𝐸𝐸1 3 𝑎𝑎 𝑎𝑎 𝛾𝛾0 𝑎𝑎 𝛾𝛾1 𝑎𝑎 𝛾𝛾2 𝑎𝑎 𝛾𝛾3 so that , the speed of light in0 empty space,1 is 1). The2 parameter3 0is the time assigned to 𝑡𝑡 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑎𝑎 𝑐𝑐𝑐𝑐 𝑥𝑥 𝑎𝑎 𝑦𝑦 𝑎𝑎 𝑧𝑧 𝑎𝑎 𝑎𝑎 𝑡𝑡 in the frame , ( , , ) are the space coordinates, and , , , are often referred to as lab 𝑐𝑐 𝑡𝑡 𝑎𝑎 coordinates. In these coordinates, ( ) agrees with the familiar Lorentz quadratic form: 𝜸𝜸 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑡𝑡 𝑥𝑥 𝑦𝑦 𝑧𝑧 ( ) = ( + +𝜂𝜂 𝑎𝑎). 2 2 2 2 2 The time𝜂𝜂 coordinate𝑎𝑎 𝑐𝑐 𝑡𝑡 is− 𝑥𝑥= (𝑦𝑦, )𝑧𝑧= . On the other hand, it defines the relative vector = = + + , where , , are the space coordinates of . Then = 𝑡𝑡 𝜂𝜂 𝑎𝑎 𝛾𝛾0 𝑎𝑎 ⋅ 𝛾𝛾0 + + 2 0 and1 therefore2 3 𝒂𝒂2 𝑎𝑎 ∧2 𝛾𝛾 2 𝑥𝑥𝜎𝜎 𝑦𝑦𝜎𝜎 𝑧𝑧𝜎𝜎 ∈ ℰ 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑎𝑎 𝒂𝒂 𝑥𝑥 𝑦𝑦 ( 𝑧𝑧) = ( + + ) = . 2 2 2 2 2 2 Remark𝜂𝜂 that𝑎𝑎 if we𝑡𝑡 −let 𝑥𝑥 denote𝑦𝑦 the𝑧𝑧 Euclidean𝑡𝑡 − 𝒂𝒂metric of such that , , is orthonormal, then ( ) = ( ). 𝑞𝑞 ℰ 𝜎𝜎1 𝜎𝜎2 𝜎𝜎3 𝑘𝑘 𝑘𝑘 Innner𝑞𝑞 𝜎𝜎product.−𝜂𝜂 Now𝜎𝜎 we need to introduce the inner product of two multivectors , . Since we want it to be bilinear, we only need to define it for basis elements, say . 𝑥𝑥 ⋅ 𝑦𝑦 𝑥𝑥 𝑦𝑦 ∈ To that end, first notice that 𝒟𝒟 𝛾𝛾𝐽𝐽 ⋅ 𝛾𝛾𝐾𝐾 = ( 1) ( , ) (Artin’s formula), 𝜄𝜄 𝐽𝐽 𝐾𝐾 where 𝛾𝛾(𝐽𝐽𝛾𝛾, 𝐾𝐾 ) denotes− the𝜂𝜂 number�𝛾𝛾𝐽𝐽∩𝐾𝐾�𝛾𝛾 𝐽𝐽ofΔ𝐾𝐾 inversions in the joint sequence , and is the symmetric difference of and . For example, it is clear that = = 𝜄𝜄 𝐽𝐽 𝐾𝐾 𝐽𝐽 𝐾𝐾 𝐽𝐽 Δ 𝐾𝐾 = , while (1, 013) = 1 (hence ( 1) ( , ) = 1), ( ) = ( ) = 𝐽𝐽 𝐾𝐾 𝛾𝛾1𝛾𝛾013 𝛾𝛾1𝛾𝛾0𝛾𝛾1𝛾𝛾3 and2 = .N7 𝜄𝜄 1 013 2 −𝛾𝛾1 𝛾𝛾03 𝛾𝛾03 𝜄𝜄 − − 𝜂𝜂 𝛾𝛾1∩013 𝜂𝜂 𝛾𝛾1 𝛾𝛾1 Thus𝛾𝛾 1the∆013 grade𝛾𝛾 03of is | | = | | + | | 2| |. Fixing = | | and = | |, we see that this grade has the form + 2 , where = | |. Consequently, the maximum pos- 𝛾𝛾𝐽𝐽𝛾𝛾𝐾𝐾 𝐽𝐽∆𝐾𝐾 𝐽𝐽 𝐾𝐾 − 𝐽𝐽 ∩ 𝐾𝐾 𝑟𝑟 𝐽𝐽 𝑠𝑠 𝐾𝐾 sible grade is + , which occurs precisely when = , and then = . Sim- 𝑟𝑟 𝑠𝑠 − 𝑙𝑙 𝑙𝑙 𝐽𝐽 ∩ 𝐾𝐾 ilarly, the minimum possible grade is | | and it occurs precisely when either (grade 𝑟𝑟 𝑠𝑠 𝐽𝐽 ∩ 𝐾𝐾 ∅ 𝛾𝛾𝐽𝐽𝛾𝛾𝐾𝐾 𝛾𝛾𝐽𝐽 ∧ 𝛾𝛾𝐾𝐾 ) or (grade ). Now we can proceed to the definition of the inner (or interior) 𝑟𝑟 − 𝑠𝑠 𝐽𝐽 ⊆ 𝐾𝐾 product . If or , or if = 0 or = 0, set = 0 (the reason for latter 𝑠𝑠 − 𝑟𝑟 𝐾𝐾 ⊆ 𝐽𝐽 𝑟𝑟 − 𝑠𝑠 rule will be seen in a moment). Otherwise (so , 1 and either or ), set 𝛾𝛾𝐽𝐽 ⋅ 𝛾𝛾𝐾𝐾 𝐽𝐽 ⊈ 𝐾𝐾 𝐾𝐾 ⊊ 𝐽𝐽 𝑟𝑟 𝑠𝑠 𝛾𝛾𝐽𝐽 ⋅ 𝛾𝛾𝐾𝐾 ( , ) = , which is ( 1) 𝑟𝑟 𝑠𝑠 ≥ if , and𝐽𝐽 ⊆ (𝐾𝐾 ) 𝐾𝐾 ⊆ if𝐽𝐽 . 𝜄𝜄 𝐽𝐽 𝐾𝐾 In particular,𝛾𝛾𝐽𝐽 ⋅ 𝛾𝛾𝐾𝐾 𝛾𝛾𝐽𝐽𝛾𝛾=𝐾𝐾 if −. 𝜂𝜂�𝛾𝛾𝐽𝐽�𝛾𝛾𝐾𝐾−𝐽𝐽 𝐽𝐽 ⊆ 𝐾𝐾 𝜂𝜂 𝛾𝛾𝐾𝐾 𝛾𝛾𝐽𝐽−𝐾𝐾 𝐾𝐾 ⊆ 𝐽𝐽 2 𝛾𝛾𝐽𝐽 ⋅ 𝛾𝛾𝐽𝐽 𝛾𝛾𝐽𝐽 𝐽𝐽 ≠ ∅ Key formula. If is a vector and a multivector, then

= 𝑎𝑎 + and 𝑥𝑥 = + . This is 𝑎𝑎true𝑎𝑎 when𝑎𝑎 ⋅ 𝑥𝑥 𝑎𝑎is ∧a 𝑥𝑥scalar, 𝑥𝑥because𝑥𝑥 𝑥𝑥 ⋅ in𝑎𝑎 that𝑥𝑥 ∧ case𝑎𝑎 the interior product vanishes and the exterior product agrees with the product. By bilinearity, we can assume that = , = , 𝑥𝑥 | | 1. If , then = = 0, while = and = . And if , 𝑎𝑎 𝛾𝛾𝑗𝑗 𝑥𝑥 𝛾𝛾𝐾𝐾 then = = 0, while = and = . 𝐾𝐾 ≥ 𝑗𝑗 ∈ 𝐾𝐾 𝑎𝑎 ∧ 𝑥𝑥 𝑥𝑥 ∧ 𝑎𝑎 𝑎𝑎 ⋅ 𝑥𝑥 𝑎𝑎𝑎𝑎 𝑥𝑥 ⋅ 𝑎𝑎 𝑥𝑥𝑥𝑥 𝑗𝑗 ∉ 𝐾𝐾 Involutions.𝑎𝑎 ⋅ 𝑥𝑥 𝑥𝑥The⋅ 𝑎𝑎 linear involution𝑎𝑎𝑎𝑎 𝑎𝑎 ∧ 𝑥𝑥 , 𝑥𝑥𝑥𝑥 , where𝑥𝑥 ∧ 𝑎𝑎 = ( 1) for , turns out to be an automorphism of (parity involution), in the sense that 𝑟𝑟 𝑟𝑟 𝒟𝒟 → 𝒟𝒟 𝑥𝑥 ↦ 𝑥𝑥� 𝑥𝑥� − 𝑥𝑥 𝑥𝑥 ∈ 𝒟𝒟 = , =𝒟𝒟 and = . Similarly,𝑥𝑥𝑥𝑥� the𝑥𝑥 �linear𝑦𝑦� 𝑥𝑥� involution∧ 𝑦𝑦 𝑥𝑥� ∧ 𝑦𝑦� , 𝑥𝑥�⋅ 𝑦𝑦 , 𝑥𝑥where� ⋅ 𝑦𝑦� = ( 1) , is an antiautomorphism 𝑟𝑟 of (reversal involution), in the sense that �2� 𝒟𝒟 → 𝒟𝒟 𝑥𝑥 ↦ 𝑥𝑥� 𝑥𝑥� − 𝑥𝑥 𝒟𝒟 = , = and = . For both𝑥𝑥𝑥𝑥� assertions𝑦𝑦�𝑥𝑥� 𝑥𝑥� it∧ is𝑦𝑦 enough𝑦𝑦� ∧ 𝑥𝑥� to check𝑥𝑥�⋅ 𝑦𝑦the identities𝑦𝑦� ⋅ 𝑥𝑥� for two basis elements, = and = , say of grades and , respectively. For the parity involution, note that the grades of 𝑥𝑥 𝛾𝛾𝐽𝐽 , , and are + 2 ( the cardinal of ), + and | |, respec- 𝑦𝑦 𝛾𝛾𝐾𝐾 𝑟𝑟 𝑠𝑠 tively, and that all are congruent to + mod 2. In the case of the reversion involution, the 𝛾𝛾𝐽𝐽𝛾𝛾𝐾𝐾 𝛾𝛾𝐽𝐽∧𝛾𝛾𝐾𝐾 𝛾𝛾𝐽𝐽 ⋅ 𝛾𝛾𝐾𝐾 𝑟𝑟 𝑠𝑠 − 𝑙𝑙 𝑙𝑙 𝐽𝐽 ∩ 𝐾𝐾 𝑟𝑟 𝑠𝑠 𝑟𝑟 − 𝑠𝑠 argument is similar if we take into account that = , where is the reversal of the se- 𝑟𝑟 𝑠𝑠 quence (note that reordering involves sign changes).̃ 𝛾𝛾�𝐽𝐽 𝛾𝛾𝐽𝐽 𝐽𝐽̃ 𝑟𝑟 We will𝐽𝐽 need the following property:𝐽𝐽̃ an �2� is a vector if and only if = and = . Indeed, the first equality says that can have only odd components ( = + ) and the 𝑥𝑥 ∈ 𝒟𝒟 𝑥𝑥� −𝑥𝑥 𝑥𝑥� 𝑥𝑥 second equality says that = 0, because = . 𝑥𝑥 𝑥𝑥 𝑥𝑥1 𝑥𝑥3 3 1 3 Complex structure. The complex𝑥𝑥 scalars of𝑥𝑥� , 𝑥𝑥=−1𝑥𝑥, = + , coincide with the com- plex scalars of , because + = + . The space 0 + 3 = + is closed 𝒫𝒫 𝐂𝐂 〈 𝐢𝐢〉 𝒫𝒫 𝒫𝒫 under multiplic-ation by , and0 hence4 by0 complex3 scalars, and1 will3 be called1 the1 space of 𝒟𝒟 𝒟𝒟 𝒟𝒟 𝒫𝒫 𝒫𝒫 𝒟𝒟 𝒟𝒟 𝒟𝒟 𝒟𝒟 𝐢𝐢 complex vectors. A typical complex vector has the form + , , . The space of 𝐢𝐢 bivectors is closed under multiplication by and hence it is a -space 1as well. A typical2 𝑎𝑎 𝑏𝑏𝐢𝐢 𝑎𝑎 𝑏𝑏 ∈ 𝒟𝒟 𝒟𝒟 bivector has the form + , with , . Thus a complex multivector has the form 𝐢𝐢 𝐂𝐂 ( + ) + ( 𝒙𝒙+ 𝒚𝒚)𝐢𝐢+ ( +𝒙𝒙 𝒚𝒚)∈ ℰ 𝛼𝛼, 𝛽𝛽𝐢𝐢 , , 𝑎𝑎 𝑏𝑏𝐢𝐢 = 𝒙𝒙, , 𝒚𝒚,𝐢𝐢 . 1 1 3 Lorentz𝛼𝛼 transformations.𝛽𝛽 ∈ ℝ 𝑎𝑎 𝑏𝑏 ∈ 𝒟𝒟 Given𝐸𝐸 𝒙𝒙= 𝒚𝒚 +∈ ℰ , we have = + 2( ) (we have used that commutes with bivectors, that + =22( 2 ), where2 the inner product 𝒛𝒛 𝒙𝒙 𝒚𝒚𝐢𝐢 𝒛𝒛 𝒙𝒙 − 𝒚𝒚 𝒙𝒙 ⋅ 𝒚𝒚 𝐢𝐢 ∈ 𝐂𝐂 is relative to , and that , , · ). We will say that is a Lorentz vector if = 𝐢𝐢 𝒙𝒙𝒙𝒙 𝒚𝒚𝒚𝒚 𝒙𝒙 ⋅ 𝒚𝒚 ±1 = and in this case we2 define2 the -rotor of amplitude as 2 ℰ 𝒙𝒙 𝒚𝒚 𝒙𝒙 𝒚𝒚 ∈ 𝐑𝐑 𝒛𝒛 𝒛𝒛 𝜖𝜖 𝒛𝒛 𝛼𝛼 ∈ 𝐑𝐑 / = , = = cos ( /2) + sin ( /2), 𝛼𝛼𝒛𝒛 2 where cos𝑅𝑅 𝑅𝑅and𝒛𝒛 𝛼𝛼 sin𝑒𝑒 denote cosh𝜖𝜖 𝛼𝛼 and sinh𝒛𝒛 if𝜖𝜖 𝛼𝛼 = 1, cos and sin if = 1. Note that / , , =𝜖𝜖 1 and hence𝜖𝜖 = , . Since = , this also shows that = . −𝛼𝛼𝒛𝒛 2 −1 𝜖𝜖 𝜖𝜖 −−1 Let𝑅𝑅𝒛𝒛 𝛼𝛼 𝑅𝑅−=𝒛𝒛 𝛼𝛼 , be the automorphism𝑒𝑒 of𝑅𝑅 𝒛𝒛 𝛼𝛼 defined 𝒛𝒛�by −𝒛𝒛 𝑅𝑅 𝑅𝑅� 𝐿𝐿 𝐿𝐿(𝒛𝒛 )𝛼𝛼 = = . 𝒟𝒟 −1 The map𝐿𝐿 𝑥𝑥 has𝑅𝑅 the𝑅𝑅𝑅𝑅 property𝑅𝑅𝑅𝑅𝑅𝑅 �that = , for if , then = = and = = = , and so we can1 apply1 the observation1 at the end of the Involutions 𝐿𝐿 𝐿𝐿𝒟𝒟 𝒟𝒟 𝑎𝑎 ∈ 𝒟𝒟 𝐿𝐿�𝐿𝐿 𝑅𝑅�𝑎𝑎�𝑅𝑅� −𝑅𝑅𝑅𝑅𝑅𝑅� subsection. 𝐿𝐿�𝐿𝐿 𝑅𝑅�𝑎𝑎�𝑅𝑅� 𝑅𝑅𝑅𝑅𝑅𝑅� 𝐿𝐿𝐿𝐿 Furthermore, is a proper Lorentz isometry (in symbols, ), for + ( )𝐿𝐿= ( ) = = = 𝐿𝐿=∈ 𝑂𝑂(𝜂𝜂 ), and 2 −1 −1 2 −1 2 𝜂𝜂det𝐿𝐿(𝐿𝐿 ) =𝐿𝐿𝐿𝐿( ) = 𝑅𝑅𝑅𝑅𝑅𝑅 =𝑅𝑅𝑅𝑅𝑅𝑅, which𝑅𝑅 implies𝑎𝑎 𝑅𝑅 det𝑎𝑎( ) =𝜂𝜂1𝑎𝑎. −1 For practical𝐿𝐿 computations,𝐢𝐢 𝐿𝐿 𝐢𝐢 𝑅𝑅 𝐢𝐢note𝑅𝑅 that𝐢𝐢 𝐿𝐿

( ) = = (cos ( /2) + sin ( /2)) (cos ( /2) sin ( /2)). −1 𝜖𝜖 𝜖𝜖 𝜖𝜖 𝜖𝜖 Example.𝐿𝐿 𝑎𝑎Let 𝑅𝑅be𝑅𝑅 𝑅𝑅a unit vector of𝛼𝛼 , so that𝒛𝒛 𝛼𝛼= 1, and𝑎𝑎 write 𝛼𝛼 = − 𝒛𝒛 𝛼𝛼, , . Note that is the relative vector of , for = 2 = . Since is a Lorentz vector, we can 𝒖𝒖 ℰ 𝒖𝒖 𝑢𝑢 𝒖𝒖𝛾𝛾0 ∈ 〈𝛾𝛾1 𝛾𝛾2 𝛾𝛾3〉 consider = , . We will see that is the Lorentz boost in the direction of rapidity 𝒖𝒖 𝑢𝑢 𝑢𝑢 ∧ 𝛾𝛾0 𝑢𝑢𝛾𝛾0 𝒖𝒖 𝒖𝒖 (these terms are explained in the reasoning that follows). First, let us find . Since an- 𝐿𝐿 𝐿𝐿𝒖𝒖 𝛼𝛼 𝐿𝐿 𝑢𝑢 𝛼𝛼 ticommutes with , 𝐿𝐿𝛾𝛾0 𝛾𝛾0

( ) = 𝒖𝒖 , = = cosh( ) + sinh( ) . 2 𝛼𝛼𝒖𝒖 Now we𝐿𝐿 have𝛾𝛾0 𝑅𝑅𝒖𝒖 𝛼𝛼𝛾𝛾0 𝑒𝑒 𝛾𝛾0 𝛼𝛼 𝛾𝛾0 𝛼𝛼 𝑢𝑢 = / / = / / = = (cosh( ) + sinh( ) ) = sinh𝛼𝛼𝒖𝒖 2( )−𝛼𝛼𝒖𝒖+2cosh𝛼𝛼(𝒖𝒖 )2 . 𝛼𝛼𝒖𝒖 2 𝛼𝛼𝒖𝒖 𝐿𝐿𝐿𝐿 𝑒𝑒 𝑢𝑢𝑒𝑒 𝑒𝑒 𝒖𝒖𝑒𝑒 𝛾𝛾0 𝑒𝑒 𝑢𝑢 𝛼𝛼 𝛼𝛼 𝒖𝒖 𝑢𝑢 Since vectors 𝛼𝛼 ,𝛾𝛾0 commute𝛼𝛼 𝑢𝑢with , these vectors satisfy = . ⊥ Finally letting𝑎𝑎 ∈=〈𝛾𝛾tanh0 𝑢𝑢〉( ), then 𝒖𝒖 𝐿𝐿𝐿𝐿 𝑎𝑎 cosh sinh 1 𝛼𝛼 𝛽𝛽= , where = cosh = (1 ) , sinh cosh 1 𝛼𝛼 𝛼𝛼 𝛽𝛽 2 −1⁄2 � � 𝛾𝛾 � � 𝛾𝛾 𝛼𝛼 − 𝛽𝛽 which agrees 𝛼𝛼with the so𝛼𝛼 called Lorentz𝛽𝛽 boost in the direction and relativistic speed . In this context, is usually called the rapidity parameter. 𝑢𝑢 𝛽𝛽 Example. Let𝛼𝛼 be a unit vector of , and consider = . Since = = 1, is a Lorentz vector with = 1. If we let = , , then commutes with2 and2 hence 𝒖𝒖 ℰ 𝒛𝒛 𝒖𝒖𝐢𝐢 𝒛𝒛 −𝒖𝒖 − 𝒛𝒛 ( ) = 𝜖𝜖 − = 𝐿𝐿 𝐿𝐿𝒛𝒛 𝛼𝛼 = 𝛾𝛾0. 𝒛𝒛 𝛼𝛼𝒛𝒛⁄2 −𝛼𝛼𝒛𝒛⁄2 𝛼𝛼𝒛𝒛⁄2 −𝛼𝛼𝒛𝒛⁄2 𝐿𝐿 𝛾𝛾0 𝑒𝑒 𝛾𝛾0𝑒𝑒 𝑒𝑒 𝑒𝑒 𝛾𝛾0 𝛾𝛾0 It follows that is a rotation of the space , , . Since commutes with , is fixed by and hence is the axis of the rotation. And if , , is orthogonal to , then 𝐿𝐿 〈𝛾𝛾1 𝛾𝛾2 𝛾𝛾3〉 𝑢𝑢 𝒛𝒛 𝑢𝑢 anticommutes with and 𝐿𝐿 〈𝑢𝑢〉 𝑣𝑣 ∈ 〈𝛾𝛾1 𝛾𝛾2 𝛾𝛾3〉 𝑢𝑢 𝑣𝑣 ( ) = 𝒛𝒛= = cos( ) + sin( ). 𝛼𝛼𝒛𝒛 𝛼𝛼𝒖𝒖𝐢𝐢 This shows𝐿𝐿 𝑣𝑣 that𝑒𝑒 the𝑣𝑣 amplitude𝑒𝑒 𝑣𝑣 of the rotation𝛼𝛼 𝑣𝑣 𝒖𝒖 𝐢𝐢 𝑣𝑣is , because𝛼𝛼 is orthogonal to and to (it is a plain computation to see that anticommutes with and with ). 𝐿𝐿 𝛼𝛼 𝒖𝒖𝐢𝐢𝑣𝑣 𝑢𝑢 𝑣𝑣 The gradient operator. For application𝒖𝒖𝐢𝐢𝑣𝑣s of , particularly to𝑢𝑢 physics, we𝑣𝑣 also need the gra- dient operator = .9 It is defined as .10 Owing to the Schwarz rule, the behave as 𝒟𝒟 scalars and so behaves as a vector. Thus𝜇𝜇 we have, for any multivector , 𝜕𝜕 𝜕𝜕𝒟𝒟 𝛾𝛾 𝜕𝜕𝜇𝜇 𝜕𝜕𝜇𝜇 = 𝜕𝜕 + . 𝑥𝑥 The relative𝜕𝜕𝜕𝜕 version𝜕𝜕 ⋅ 𝑥𝑥 is𝜕𝜕 ∧ 𝑥𝑥 = = = = , 𝑘𝑘 where 𝝏𝝏 = 𝜕𝜕 ∧ 𝛾𝛾 is0 the𝛾𝛾 gradient∧ 𝛾𝛾0𝜕𝜕𝑘𝑘 operator−𝜎𝜎𝑘𝑘𝜕𝜕 in𝑘𝑘 the− (Euclidean)∇ relative space. Then we have: ∇ 𝜎𝜎=𝑘𝑘𝜕𝜕𝑘𝑘 + = and = = + . The Maxwell𝜕𝜕𝛾𝛾0 -Riesz𝜕𝜕0 𝝏𝝏equation.𝜕𝜕0 − ∇ In , the𝛾𝛾0 𝜕𝜕electromagnetic𝜕𝜕0 − 𝝏𝝏 𝜕𝜕0 field∇ is represented by a bivector and the charge-current density by a vector . The Maxwell-Riesz equation (MR) for is 𝒟𝒟 𝐹𝐹 = . 𝑗𝑗 𝐹𝐹 Our next𝜕𝜕𝜕𝜕 task 𝑗𝑗is to indicate how this equation is equivalent to the textbook Maxwell’s equa- tions. For the advantages of this formulation, see DORAN-LASENBY 2003. Using the split = + , we can write in relative terms: 2 = +𝒟𝒟 , ℰ, 𝐢𝐢𝐢𝐢 . 𝐹𝐹 Thus 𝐹𝐹appears𝑬𝑬 to𝐢𝐢𝑩𝑩 be 𝑬𝑬constituted,𝑩𝑩 ∈ ℰ in the frame , of the electric and magnetic (relative) vec- tor fields and . We also have, writing = (charge density) and = (rela- 𝐹𝐹 𝜸𝜸 tive current density), that 𝑬𝑬 𝑩𝑩 𝜌𝜌 𝑗𝑗 ⋅ 𝛾𝛾0 𝒋𝒋 𝑗𝑗 ∧ 𝛾𝛾0 = + = + .

From this𝑗𝑗𝛾𝛾0 we get𝑗𝑗 ⋅ 𝛾𝛾 that0 𝑗𝑗 ∧ 𝛾𝛾0 𝜌𝜌 𝒋𝒋 = ( + ) and = .

Now we𝑗𝑗 can 𝜌𝜌proceed𝒋𝒋 𝛾𝛾0 to show𝛾𝛾0 𝑗𝑗that 𝜌𝜌the− MR𝒋𝒋 equation is equivalent to the Maxwell’s equa- tions for , , and . Multiply the MR by on the left to obtain:

𝑬𝑬 𝑩𝑩 𝜌𝜌 𝒋𝒋 𝛾𝛾0

9 There is not a generally accepted notation for this operator. Among the symbols used, there are and . 10 We follow Einstein’s convention of assuming a summation over a repeated index, and that Greek indices vary from 0 to 4. In contrast, indices that vary from 1 to 3 are denoted by Latin characters. ∇ □ ( + )( + ) = .

This expands𝜕𝜕0 ∇to 𝑬𝑬 𝐢𝐢𝑩𝑩 𝜌𝜌 − 𝒋𝒋 + + ( + ) = .

But we 𝜕𝜕also0𝑬𝑬 have∇𝑬𝑬 𝐢𝐢=𝜕𝜕0𝑩𝑩 +∇𝑩𝑩 𝜌𝜌 and− 𝒋𝒋 = + , and so the last equation gives + ∇𝑬𝑬+ ∇ ⋅ 𝑬𝑬+ ∇( ∧ 𝑬𝑬 + ∇𝑩𝑩 + ∇ ⋅ 𝑩𝑩 ) =∇ ∧ 𝑩𝑩 . This equation𝜕𝜕0𝑬𝑬 ∇is ⋅equivalent𝑬𝑬 ∇ ∧ 𝑬𝑬 to 𝐢𝐢the𝜕𝜕 0four𝑩𝑩 equations∇ ⋅ 𝑩𝑩 ∇ ∧obtained𝑩𝑩 𝜌𝜌 by− 𝒋𝒋equating the components for the grades 0 to 3, which are: = + ( ) = ∇ ⋅ 𝑬𝑬 𝜌𝜌 + = 0 𝜕𝜕0𝑬𝑬 𝐢𝐢 ∇ ∧ 𝑩𝑩 −𝒋𝒋 ( ) = 0 ∇ ∧ 𝑬𝑬 𝐢𝐢 ∂0𝑩𝑩 Using that𝐢𝐢 ∇ ⋅(𝑩𝑩 ) = × and ( ) = × , the four equations can be written as: 𝐢𝐢 ∇=∧ 𝑩𝑩 −∇ 𝑩𝑩 𝐢𝐢 (Gauss∇ ∧ 𝑬𝑬 law− for∇ 𝑬𝑬) × = (Ampère-Maxwell equation) ∇ ⋅ 𝑬𝑬 𝜌𝜌 𝑬𝑬 + × = 0 (Faraday induction law) ∇ 𝑩𝑩 − 𝜕𝜕0𝑬𝑬 𝒋𝒋 = 0 (Gauss law for ) ∂0𝑩𝑩 ∇ 𝑬𝑬 Electromagnetic∇ ⋅ 𝑩𝑩 potential. An electromagnetic potential𝑩𝑩 is a vector field such that = . 𝐴𝐴 Let = (scalar potential) and = (vector potential). Then = + and 𝜕𝜕𝜕𝜕 𝐹𝐹 = ( + ) = . So we have: 𝜙𝜙 𝐴𝐴 ⋅ 𝛾𝛾0 𝑨𝑨 𝐴𝐴 ∧ 𝛾𝛾0 𝐴𝐴𝛾𝛾0 𝜙𝜙 𝑨𝑨 𝛾𝛾0𝐴𝐴 𝛾𝛾0+𝜙𝜙 =𝑨𝑨 𝛾𝛾0= 𝜙𝜙 −=𝑨𝑨 = ( )( ). Equating𝑬𝑬 components𝐢𝐢𝑩𝑩 𝑭𝑭 𝜕𝜕of𝜕𝜕 the 𝜕𝜕same𝛾𝛾0𝛾𝛾0 𝐴𝐴grade𝜕𝜕, 0we− ∇find𝜙𝜙 that− 𝑨𝑨 this expression is equivalent to the following three equations (the pseudoscalar components vanish): + = 0 (Lorentz gauge equation) = ( + ) (expression of in terms of the potentials) 𝜕𝜕0𝜙𝜙 ∇ ⋅ 𝐴𝐴 = × (expression of in terms of the vector potential) 𝑬𝑬 − ∇𝜙𝜙 𝜕𝜕0𝑨𝑨 𝐸𝐸 These formulas𝑩𝑩 ∇ agree𝑨𝑨 with the textbook relations𝑩𝑩 giving the electric and magnetic fields in terms of the scalar and vector potentials (in the Lorentz gauge). . In the original setting, Dirac spinors are elements of . The careful study of the Dirac equation with a geometric algebra perspective (we refer to4 the excellent paper 𝐂𝐂 Hestenes 2003 for details) leads to the conclusion that a Dirac spinor is best represented as an element . Moreover, in this approach the Dirac equation for the electron in an electromagnetic field+ of potential takes the form 𝜓𝜓 ∈ 𝒟𝒟 = ( + ) 𝐴𝐴, 𝜕𝜕𝜕𝜕 𝑚𝑚𝑚𝑚𝛾𝛾0 𝑒𝑒𝑒𝑒𝑒𝑒 𝛾𝛾2𝛾𝛾1 where and are the mass and charge of the electron, respectively. It is written purely in terms of GA and its study leads to its deep geometrical and physical significance. 𝑚𝑚 𝑒𝑒

CONCLUSION Reading again the long quotation of Leibniz’ correspondence with Huygens included in the second section of this paper, we realize that Leibniz vision is realized in GA not only for the geometry of ordinary space (the algebra ), but also for other spaces of fundamental rele- vance for mathematics and physics, as illustrated here with the study of the Dirac algebra . 𝒢𝒢3 “I believe that by this method one could treat mechanics almost like geometry”, says Leibniz, 𝒟𝒟 and it this is found to be amply achieved in HESTENES 1990 (we can even drop the ‘almost’). The treatment of rotations in ordinary space using , particularly the determination of the composition of two rotations, confirm the foresight that “this new characteristic cannot fail 𝒢𝒢3 to give the solution, the construction, and the geometric demonstration all at the same time, and in a natural way in one analysis, that is, through determined procedure”. It is important to remark that GA comes with the big bonus that an analogous procedure works in many other situations in geometry, physics and engineering (illustrated here with the description of Lorentz transformations, but widely confirmed by the available literature). A final remark is that the language of mathematical structures initiated by Grassmann plays the role of Leibniz’ characteristica universalis. It is in that realm that GA is grounded as a most remarkable jewel.

NOTES

N1. Notice that if = + and = + is another orthonormal ba- , = , = sis, then 1 11 1 , 12and2 since 2 21 1 22 2 𝑣𝑣 𝛼𝛼 𝑢𝑢 1𝛼𝛼 𝑢𝑢 𝑣𝑣 𝛼𝛼 𝑢𝑢 𝛼𝛼 𝑢𝑢 〈𝑣𝑣1 =𝑣𝑣2〉 〈𝑢𝑢1 +𝑢𝑢2〉 𝒢𝒢2 + ( ) = ± (for 𝑣𝑣1𝑣𝑣2+ 𝛼𝛼11𝛼𝛼21= 𝛼𝛼(12𝛼𝛼, 22 ) =𝛼𝛼011 and𝛼𝛼22 − 𝛼𝛼12𝛼𝛼21 𝑢𝑢1𝑢𝑢2 = det𝑢𝑢1𝑢𝑢2, ( , ) = ±1), we also have = = . 1 2 𝛼𝛼11𝛼𝛼21 𝛼𝛼12𝛼𝛼22 𝑞𝑞 𝑣𝑣1 𝑣𝑣2 𝛼𝛼11𝛼𝛼22 − 𝛼𝛼12𝛼𝛼21 𝑢𝑢 𝑢𝑢 𝑣𝑣1 𝑣𝑣2 2 N2. Here is〈𝑣𝑣 1a 𝑣𝑣brief2〉 〈summary𝑢𝑢1𝑢𝑢2〉 of𝒢𝒢2 the Grassmann exterior algebra. Let is a real vector space of finite dimension . The ingredients of the -th exterior power of ( a non-negative inte- 𝐸𝐸 ger) are a vector space and a multilinear alternating map : . The defining 𝑛𝑛 𝑗𝑗 𝐸𝐸 𝑗𝑗 property of these ingredients𝑗𝑗 is that for any multilinear alternating map𝑗𝑗 : 𝑗𝑗 , where Λ 𝐸𝐸 ∧𝑗𝑗 𝐸𝐸 → Λ 𝐸𝐸 is any vector space, there exists a unique : such that 𝑗𝑗 𝑗𝑗 𝑓𝑓 𝐸𝐸 → 𝐹𝐹 𝐹𝐹 ( , … , ) = , … , . 𝑓𝑓̅ Λ 𝐸𝐸 → 𝐹𝐹

From this𝑓𝑓̅ � it∧ 𝑗𝑗is 𝑎𝑎straightforward1 𝑎𝑎𝑘𝑘 � 𝑓𝑓 �to𝑎𝑎 1see that𝑎𝑎𝑗𝑗� there is a unique bilinear map , : × 𝑗𝑗 𝑘𝑘 𝑗𝑗+𝑘𝑘 such that∧ 𝑗𝑗 𝑘𝑘 Λ 𝐸𝐸 Λ 𝐸𝐸 → Λ 𝐸𝐸

, , … , , , … , = , … , , , … ,

Let ∧=𝑗𝑗 𝑘𝑘 �∧𝑗𝑗 �𝑎𝑎1 𝑎𝑎𝑗𝑗� ∧𝑘𝑘 �𝑎𝑎𝑗𝑗+1 (note𝑎𝑎 𝑗𝑗that+𝑘𝑘� � ∧𝑗𝑗=+𝑘𝑘0� if𝑎𝑎1 > 𝑎𝑎𝑗𝑗).𝑎𝑎 Given𝑗𝑗+1 𝑎𝑎, 𝑗𝑗+𝑘𝑘� , its ex- terior product0 1is defined as 𝑛𝑛 , , ( , 𝑘𝑘). It is bilinear, unital, and associative, and Λ𝐸𝐸 Λ 𝐸𝐸 ⊕ Λ 𝐸𝐸 ⊕ ⋯ ⊕ Λ 𝐸𝐸 Λ 𝐸𝐸 𝑘𝑘 𝑛𝑛 𝑥𝑥 𝑦𝑦 ∈ Λ𝐸𝐸 the (Grassmann) exterior algebra of is endowed with the exterior product . 𝑥𝑥 ∧ 𝑦𝑦 ∑𝑗𝑗 𝑘𝑘∧𝑗𝑗 𝑘𝑘 𝑥𝑥𝑗𝑗 𝑦𝑦𝑘𝑘 The elements of are called multivectors𝐸𝐸 Λ𝐸𝐸. The multivectors of are called∧ -vectors. For = 0, = and its elements are called scalars. For = 1, 𝑘𝑘 = and its elements Λ𝐸𝐸 Λ 𝐸𝐸 𝑘𝑘 are called vectors0 . Instead of 2-vectors or 3-vectors we usually say bivectors1 and trivectors, 𝑘𝑘 Λ 𝐸𝐸 ℝ 𝑘𝑘 Λ 𝐸𝐸 𝐸𝐸 respectively. If and , then 𝑘𝑘 𝑙𝑙 𝑥𝑥𝑘𝑘 ∈ Λ 𝐸𝐸 and 𝑥𝑥𝑙𝑙 ∈ Λ=𝐸𝐸( 1) . 𝑘𝑘+𝑙𝑙 𝑘𝑘𝑘𝑘 If , …𝑥𝑥,𝑘𝑘 ∧ are𝑥𝑥𝑙𝑙 ∈ vectors,Λ 𝐸𝐸 then𝑥𝑥 𝑘𝑘 ∧ 𝑥𝑥𝑙𝑙 − = 𝑥𝑥(𝑙𝑙 ∧ 𝑥𝑥, .𝑘𝑘. , ) and a key property of the exterior algebra is that = 0 if and only , … , are𝑘𝑘 linearly dependent. It fol- 𝑎𝑎1 𝑎𝑎𝑘𝑘 𝑎𝑎1 ∧ ⋯ ∧ 𝑎𝑎𝑘𝑘 ∧𝑗𝑗 𝑎𝑎1 𝑎𝑎𝑘𝑘 ∈ Λ 𝐸𝐸 lows that if 0 (in which case we say that is a -blade), then it 𝑎𝑎1 ∧ ⋯ ∧ 𝑎𝑎 𝑎𝑎1 𝑎𝑎𝑘𝑘 determines the , … , as the vector subspace of formed by the vectors 𝑎𝑎1 ∧ ⋯ ∧ 𝑎𝑎𝑘𝑘 ≠ 𝑎𝑎 ∧ ⋯ ∧ 𝑎𝑎𝑘𝑘 𝑘𝑘 such that = 0. This leads to the fundamental representation of linear sub- 〈𝑎𝑎1 𝑎𝑎𝑘𝑘〉 ⊆ 𝐸𝐸 𝐸𝐸 spaces of dimension of as -blades, up to a non-zero scalar multiple. 𝑎𝑎 𝑎𝑎 ∧ 𝑎𝑎1 ∧ ⋯ ∧ 𝑎𝑎𝑘𝑘 Let , … , be a any𝑘𝑘 basis𝐸𝐸 of 𝑘𝑘 . Given a multiindex = { < < } {1, … , }, let 𝑒𝑒1 =𝑒𝑒𝑛𝑛 𝐸𝐸 . 𝐼𝐼 𝑖𝑖1 ⋯ 𝑖𝑖𝑘𝑘 ⊆ 𝑛𝑛 𝑘𝑘 These blades𝑒𝑒̂𝐼𝐼 𝑒𝑒 form𝑖𝑖1 ∧ ⋯ a ∧basis𝑒𝑒𝑖𝑖𝑘𝑘 ∈ofΛ 𝐸𝐸 and consequently dim = and dim = 2 . 𝑘𝑘 𝑘𝑘 𝑛𝑛 𝑛𝑛 N3. Strictly speaking, so far Λwe𝐸𝐸 have only shown that ifΛ 𝐸𝐸 exists,�𝑘𝑘� then it is Λunique𝐸𝐸 up to a canonical isomorphism, and that it has dimension 4. The existence is a consequence of the 𝒢𝒢2 Pauli representation, but it can be proved in general without resorting to matrices. This is done in many references, and in particular in XAMBÓ-DESCAMPS 2016. For , see next note.

N4. The of 1, , , , , , , can𝒢𝒢 3be seen by the method used for (Riesz’ method). Suppose we have a linear relation 𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 𝑢𝑢1𝑢𝑢2 𝑢𝑢1𝑢𝑢3 𝑢𝑢2𝑢𝑢3 𝑢𝑢1𝑢𝑢2𝑢𝑢3 + 𝐸𝐸+2 + + + + + = 0. Multiplying𝜆𝜆 𝜆𝜆 by1𝑢𝑢 1 from𝜆𝜆2𝑢𝑢2 the 𝜆𝜆left3𝑢𝑢 3and𝜇𝜇 from1𝑢𝑢2𝑢𝑢 the3 right,𝜇𝜇2𝑢𝑢1 𝑢𝑢we3 get𝜇𝜇 3𝑢𝑢1𝑢𝑢2 𝜇𝜇 𝑢𝑢1𝑢𝑢2𝑢𝑢3 + 𝑢𝑢1 + + = 0. Adding𝜆𝜆 the 𝜆𝜆two,1𝑢𝑢1 we− 𝜆𝜆 obtain2𝑢𝑢2 − 𝜆𝜆3𝑢𝑢3 𝜇𝜇1𝑢𝑢2𝑢𝑢3 − 𝜇𝜇2𝑢𝑢1𝑢𝑢3 − 𝜇𝜇3𝑢𝑢1𝑢𝑢2 𝜇𝜇 𝑢𝑢1𝑢𝑢2𝑢𝑢3 + + + = 0.

Now multiply𝜆𝜆 𝜆𝜆1 𝑢𝑢by1 𝜇𝜇 1from𝑢𝑢2𝑢𝑢 3the left𝜇𝜇 𝑢𝑢 1and𝑢𝑢2𝑢𝑢 from3 the right, we get 𝑢𝑢2 + = 0, and hence𝜆𝜆 − 𝜆𝜆+1𝑢𝑢1 − 𝜇𝜇1𝑢𝑢2𝑢𝑢=3 0. 𝜇𝜇Since𝑢𝑢1𝑢𝑢2 (𝑢𝑢3 ) = 1, and , are real, we must have = = 0. Thus + = 0. Multiplying 2by from the left, we conclude, as in the 𝜆𝜆 𝜇𝜇 𝑢𝑢1𝑢𝑢2𝑢𝑢3 𝑢𝑢1𝑢𝑢2𝑢𝑢3 − 𝜆𝜆 𝜇𝜇 𝜇𝜇 previous step, that = = 0. So we are left with the relation 𝜆𝜆 𝜆𝜆1𝑢𝑢1 𝜇𝜇1𝑢𝑢2𝑢𝑢3 𝑢𝑢1 + 𝜆𝜆1+ 𝜇𝜇1 + = 0. Repeat 𝜆𝜆the2𝑢𝑢 game:2 𝜆𝜆3 multiplying𝑢𝑢3 𝜇𝜇2𝑢𝑢1𝑢𝑢 3by 𝜇𝜇3 from𝑢𝑢1𝑢𝑢2 the left and the right, we easily conclude that = = 0 and then = = 0 follows readily. 𝑢𝑢2 𝜆𝜆2 N5𝜇𝜇2 . Indeed, =𝜆𝜆3 𝜇𝜇3 = = 1 ( = 1,2,3) and if ( , , ) is a cyclic permutation of (1,2,3), then 2 = =2 2 = = . 𝒊𝒊𝑘𝑘 𝑢𝑢𝑘𝑘𝒊𝒊𝑢𝑢𝑘𝑘𝒊𝒊 𝑢𝑢𝑘𝑘𝒊𝒊 − 𝑘𝑘 𝑗𝑗 𝑘𝑘 𝑙𝑙 N6. With the 𝒊𝒊same𝑗𝑗𝒊𝒊𝑘𝑘 notation𝑢𝑢𝑗𝑗𝒊𝒊𝑢𝑢𝑘𝑘𝒊𝒊 s as𝒊𝒊𝑢𝑢 𝑗𝑗in𝑢𝑢 𝑘𝑘N2𝒊𝒊 , the𝑢𝑢𝑙𝑙𝒊𝒊 extension𝒊𝒊𝑙𝑙 of a metric to is determined by requiring that ( , ) = 0 is , and and that ( , ) = ( , ) if 𝑞𝑞 Λ𝐸𝐸 , are blades, where ( , 𝑗𝑗) denoted𝑘𝑘 the Gram determinant of and : 𝑘𝑘 𝑞𝑞 𝑥𝑥 𝑦𝑦 𝑥𝑥 ∈ Λ 𝐸𝐸 𝑦𝑦 ∈ Λ 𝐸𝐸 𝑗𝑗 ≠ 𝑘𝑘 𝑞𝑞 𝐴𝐴 𝐵𝐵 𝐺𝐺 𝐴𝐴 𝐵𝐵 𝐴𝐴 𝐵𝐵 ∈ Λ (𝐸𝐸 , 𝐺𝐺 𝐴𝐴 )𝐵𝐵= det , . 𝐴𝐴 𝐵𝐵 , 1 𝑘𝑘 1 𝑘𝑘 𝑖𝑖 𝑗𝑗 In particular,𝐺𝐺 𝑎𝑎 ∧ (⋯)∧=𝑎𝑎 𝑏𝑏( ∧), ⋯where∧ 𝑏𝑏 ( ) =�𝑞𝑞(�𝑎𝑎, 𝑏𝑏).� For�1≤ 𝑖𝑖example,𝑗𝑗≤𝑘𝑘

( 𝑞𝑞 𝐴𝐴) = 𝐺𝐺( 𝐴𝐴 ) ( ) 𝐺𝐺 𝐴𝐴( , 𝐺𝐺) 𝐴𝐴. 𝐴𝐴 2 N7. Let𝑞𝑞 𝑎𝑎 be1 ∧ a 𝑎𝑎metric2 𝑞𝑞 of𝑎𝑎 a1 real𝑞𝑞 𝑎𝑎 2space− 𝑞𝑞 𝑎𝑎 of1 𝑎𝑎dimension2 . Let , … , be an -orthonormal basis of . Let be the geometric algebra of ( , ). For any non-negative integer and any 𝑞𝑞 𝐸𝐸 𝑛𝑛 𝑢𝑢1 𝑢𝑢𝑛𝑛 𝑞𝑞 sequence = , … , {1, … , }, write ,…, = . Then the with < < , 𝐸𝐸 𝒢𝒢 𝐸𝐸 𝑞𝑞 𝑟𝑟 0 , form a basis of (of for a fixed ), and Artin’s formula holds for this basis: 𝐽𝐽 𝑗𝑗1 𝑗𝑗𝑟𝑟 ∈ 𝑛𝑛 𝑢𝑢𝑗𝑗1 𝑗𝑗𝑟𝑟 𝑢𝑢𝑗𝑗1 ⋯ 𝑢𝑢𝑗𝑗𝑟𝑟 𝑢𝑢𝐽𝐽 𝑗𝑗1 ⋯ 𝑗𝑗𝑟𝑟 𝑟𝑟 ( , ) ≤ 𝑟𝑟 ≤ 𝑛𝑛 = ( 1) 𝒢𝒢 𝒢𝒢 . 𝑟𝑟 𝜄𝜄 𝐽𝐽 𝐾𝐾 The proof𝑢𝑢𝐽𝐽𝑢𝑢 is𝐾𝐾 straightforward.− 𝑞𝑞�𝑢𝑢 We𝐽𝐽∩𝐾𝐾 can�𝑢𝑢𝐽𝐽 ∆rearrange𝐾𝐾 the product until the sequence of indices is in non-decreasing order. This produces ( , ) sign changes, hence the sign ( 1) ( , ); for 𝑢𝑢𝐽𝐽𝑢𝑢𝐾𝐾 each , we get a factor = ( ), hence altogether a factor , and𝜄𝜄 𝐽𝐽 the𝐾𝐾 re- 𝜄𝜄 𝐽𝐽 𝐾𝐾 − maining indices form in increasing2 order, to which corresponds . 𝑙𝑙 ∈ 𝐽𝐽 ∩ 𝐾𝐾 𝑢𝑢𝑙𝑙 𝑞𝑞 𝑢𝑢𝑙𝑙 𝑞𝑞�𝑢𝑢𝐽𝐽∩𝐾𝐾� 𝐽𝐽∆𝐾𝐾 Acknowledgements. It 𝐽𝐽is∆ my𝐾𝐾 pleasure to thank Josep Manuel Parra i Serra𝑢𝑢 for many discussions on geometric algebra along the years, and in particular for having provided me insights into historical texts such as COUTURAT 1901. I am also grateful to Christoph Spiegel for having helped me to better understand the very long sentence in GRASSMANN 1847 in which Grassmann describes his views about Leibniz’ characteristica geometrica and how he is going to proceed, armed with his “geometric analysis” (i.e., the Ausdehnungslehre), to frame Leibniz’s ideas.

REFERENCES

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