cliffordAlgebras-altmann

Clifford Algebra, Symmetries, andVectors S. L. ALTMANN” Brasenose College, University of Oxford, Oxford OX1 4AJ, United Kingdom Received April 27, 1995; accepted for publication October 23, 1995 ABSTRACT The realization of a Clifford algebra in laboratory space is considered and it is demonstrated that the elements of the algebra cannot, as often assumed, be directly identified with vectors in this space, but, rather, that they form the parametric space of the symmetry operations of the Euclidean group as performed in the laboratory space. Details of this parametrization are established and expressions are given that determine the action of the Euclidean-group operations (screws included) on laboratory-space vectors in terms of the elements of the Clifford algebra. A discussion of Clifford vectors, bivectors, and pseudoscalars and their relation to the Gibbs vectors is provided. The correct definition of axial and polar vectors within the Clifford algebra is carefully discussed. It is shown how simple it is to generate finite point groups in 4-dimensional space by means of the Clifford algebra. 0 1996 John Wiley & Sons, Inc. rn Introduction n early work about molecular and solid-state studies, the necessary symmetries were treated by geometrical or matrix methods, the latter being based on the clumsy Euler-angle parametrization of rotations, subject to various singularities in its parameter space. It is desirable in modern work to move over to algebraic methods, which are far more precise and are easier to program. Progress with algebraic methods, however, have been hampered by problems of interpretation. Algebras have *It is my pleasure to dedicate this article to Roy McWeeny, in warm recognition for his extensive contributions to the study of symmetry. InternationalJournal of Quantum Chemistry, Vol. 60, 359-372 (1996) 0 1996 John Wiley & Sons, Inc. I usually been defined in relation to some abstract linear vector spaces and it has often been the practice in the past to identify in an axiomatic way the elements of such spaces with physical objects such as physical vectors. These realizations of the abstract vector spaces are frequently regarded as mappings which require no further justification than the verification of some formal rules. The inventor of the first algebra that had direct relevance to the study of the rotation group S0(3), Hamilton [ 11, thus identified the algebra spanned by the three quaternion units with ordinary vectors in 3-space. Riesz [2] pointed out that such identification was wrong since the vectors so defined are axial and not polar vectors. Altmann [3] went further and showed that Hamilton’s “vectors’’ were nothing else than binary rotations (rota- CCC 0020-7608 I 9 6 I 0 10359-14 (1) Given e. if one wishes to deal with laboratory-space vectors by means of the Clifford algebra. as follows by repeated The applications of Clifford algebra in physics have been much studied. although point groups in higher-dimensional spaces will be discussed in a final section. e. e. and in this article. are identified with vectors in 3-space. 11-24) but not necessarily by all proponents of the multivector algebra.e.e. It should be stressed that whatever the status of the above interpretation might be the value and importance of the multivector algebra cannot be ignored.3) such that application of () I: e2ez= 1. position vectors must first be mapped by poles of symmetry operations and it is only then that carefully defined rules permit the recovery of the ordinary vector algebra in laboratory space. although its practical application to the study of symmetry has not been extensive. e2..e. = 26.eke. general results along this line valid for spaces of 360 VOL. disagrees with current practice in physics. When more detail is required and especially if symmetry operations in spaces of symmetry higher than three are to be studied. (See also [8.. (See also [lo]. [121. of course. where 75 point groups and their double groups have consistently been tabulated by this method. as fully demonstrated in [41.e. thus regarding the poles of rotations as vectors. (Of course. respectively. which will give us all symmetries in three-dimensional space. since Cartan [13].ek= - = el el ek ek -1. the inversion has to be treated in an ad hoc manner since it cannot be mapped within the algebra. that the elements of the Clifford algebra must be recognized as mappings of symmetry operations of O(3) forming the parametric space of this group and that therein lies their primary geometrical meaning. we have three units el ( i = 1. namely. Hestenes [5] concluded that a vector product is a polar and not an axial vector. which is the point of the unit sphere left invariant by the rotation and such that from outside that point the Dtation is seen as counterclockwise..e. albeit from a different point of view.ALTMANN tions by T). Such poiltLaare called poles of the rotations. In pure mathematics. is most adequate for this purpose. + e. For simplicity.e. which.e.e. we shall discuss only C. be uniquely identified by poles of rotations that coincide with them. It is well known that the Clifford algebra.) Position vectors of points of the unit sphere can. 1111. e~e e e 1 = -ee. a point that appears to have been glossed over in the past. Thus. Although the quaternion parametrization allows for a very compact presentation of the results.e. however. The connection between the Clifford algebra and symmetry reflections is. Handling a position vector via a pole must be done very carefully. well known. The geometrical interpretation to be discussed in this article.el ~ 1 = (2) -1.e. In C. if necessary. as this article will demonstrate.) Such interpretation was also followed by Greider and his school (see. in most of this article.9]. 1 .e.e. starts from a different point of view. but this identification has not always been done without some uncomfortable results.. further developed by Hestenes and Sobczyk [61.2. The elements of Hestenes’ multivector algebra. e. As in the case of quaternions.e. Hestenes [51 constructed a very elegant and powerful multivector algebra. C. can be formed and all these terms are units in the sense that their square is of unit modulus. The study of the symmetry operations in SO(3) and O(3) can be done by means of the quaternion algebra. = -e.e. After Marcel Riesz’s pioneering work. Numerous examples of this important technique will be given in this article.. (4) e. This work has been continued and extended by Greider [7] and his school at Davis.e. That a position vector and the pole of a rotation coincide does not alter the fact that the position vector and the symmetry operation used to map the vector via its pole are entirely different geometrical objects (because their transformation properties are different). terms of second and third rank. = (3) e. a more general algebra is required. and e. of course. a principle which will be followed very strictly in the present article.g. and e.He showed that binary rotations could masquerade as vectors for the following reason: To each rotation of SO(3).e.e. a unique point of the unit sphere can be associated with it.) A very important contribution of Hestenes to the subject is his insistence in giving paramount importance to the geometrical meaning of any vector algebra used. p.e. 60.. the pole of a rotation can be identified with the vector corresponding to it. Riesz 121 devoted a section of his book to this subject. we shall do so..NO. for which the quaternion method fails.e. It must be emphasized that the relation between polar and axial vectors is still most important in physical applications and that it cannot be ignored or tampered with without significant consequences. and which is the most natural way to deal with rotations. the Clifford units will be identified not with vectors but with orthogonal reflections in O(3). which. since the parametrization of SO(3) by the quaternion algebra. a point that requires a little care. One other important preoccupation in this article is in relation to another aspect of the multivector algebras. and these distinctions appear to be correlated with those between polar and axial vectors and pseudoscalars. is fundamental in much of physics. which is a great deal simpler. Before embarking into our main INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 361 . Like Hestenes. whereas the bivectors (which are written in terms of products e. Thus. which is one of the main problems addressed in this article. we shall construct a Clifford space out of the standard Clifford algebra and we shall make the important connection between the two spaces by proposing that the Clifford space be used as the parametric space of the symmetry operations in the laboratory space. We shall not define. as defined in the laboratory space. must be made not in terms of the inversion of an arbitrarily defined vector space but rather in terms of the inversion operation in the physical configuration space (laboratory space). One of the main results of this article. SYMMETRIES.15]. the symmetry operations that are used must transform correctly all the physical Gibbs vectors required. whatever the terminology used in mathematics.e. It does not seem to be widely appreciated that the distinction between polar and axial vectors which. The provision of an algebra along the lines described. any new type of vector in the laboratory space: Our vectors there will be the standard Gibbs vectors with all their standard properties. in such a way as to ensure that the correct multiplication rules are obtained and. For this reason. As can be seen from the above. however. affects profoundly the study of the point groups so much used in crystallography and in the study of molecular structure. by means of the Clifford algebra. the language that we shall adopt will be as nearly as possible that used in laboratory-space geometry. leaving it to one of the last sections to make contact with the properties of vectors and bivectors. which as we shall see will allow us to transform vectors in the laboratory space under all the operations of the Euclidean group. that however well known theoretically these results were they had never been systematically exploited to parametrize the operations of O(3) in terms of Clifford-algebra elements. of course. Fundamental distinctions are often made between the vectors and bivectors on the grounds that when the units e. is of some importance in many practical applications.) do not. will be the obtention of a parametrization of the Euclidean group. AND VECTORS all dimensions have been obtained [14. bivectors and pseudoscalars.] There is a wide gap in practice between the existential results obtained in pure mathematics as regards the mapping of Clifford vectors onto reflections and the formulation of a workable parametrization of these symmetry operations. of the full Euclidean group. It will be shown that this proposal is entirely in agreement with the geometry of O(3) and that the Clifford algebra spanned by the units so-defined maps all the operations of O(3) and. in molecular applications for instance. most importantly. On the other hand. was practically ignored in the study of point groups until some 10 years ago or so. that one can operate on the (Gibbs) vectors of the laboratory space correctly. change sign then the vectors change sign. similarly. but also in such varied subjects as robotics and ray-tracing in optical systems. not only in molecular and solid-state structure. the general approach of this article is firmly based on the premise that geometry in the laboratory space is paramount and that no assertion about any mapping between the Clifford-space and the laboratory-space vectors should be made until rigorous tests are made to ensure that all the properties of the laboratory-space geometry are correctly given. nevertheless.CLIFFORD ALGEBRA. [This is not surprising. we shall strongly emphasize at each stage the geometrical meaning in laboratory space of the objects defined. therefore. It often appears in these algebras that vectors are defined in terms of the Clifford units and then. The main purpose of this article was to present an alternative point of view to that of the multivector algebras of the Hestenes type. It cannot be stressed too strongly that up to the present day the parametrization of the operations of O(3)is carried out in the literature by ad hoc heuristic procedures based on S 0 ( 3 ) . It is self-evident that. a point that will be guaranteed by the present method. It should be stressed. It is for this reason that in the main body of the article we shall stick to the now rather old-fashioned terminology of polar and axial vectors. indeed. kl (11) are the usual quaternion units which satisfy Hamilton’s well-known rules: i2 = -1. sin$+. k = [O. jl. axial vector. rather. sin$+. Symmetry operations will always be understood in the active picture: They transform all position vectors in the laboratory space but they do not change the axes. The righthanded orthogonal axes i. The positive hemisphere need not be defined as a connected area on the unit sphere and its choice must be carefully done if one is dealing with finite subgroups of SO(3) (see [3]). (10) R(+. (5) All rotation angles are defined as positive (counterclockwise) as seen from outside the sphere when looking at the head of the pole. and therefore also their poles. in those axes. (9) n3 sin+(b3= n 2 cos++.n) is defined by an angle of rotation 0 2 2 T and an axial vector n from the fixed origin of the unit sphere. The distinction between positive and negative rotations is done through the pole: All positive rotations and all binary rotations are given poles in a conventionally defined positive hemisphere H and all negative rotations are given poles in the negative hemisphere.n)n = n. A rotation R(+. which is the problem that we have in mind. Also.n) by the unique positive quaternion in ( 6 ) by means of the following device: All poles n are chosen as stated above. 1 . but. by addition of a multiple of 2 ~ and any r negative factor thus appearing is disregarded (although it is. a + A’. and most of the literature that followed on quaternions. (12) ijk = - 1. binary rotations (rotations by T ) . the mapping is isomorphic: If Conventions Before we go any further.ALTMANN discussion. k..) For the purpose of multiplying rotations. A] (a. of course). and all rotation angles are chosen in the range given in (6). 60. 362 VOL. and purely for this purpose. j. it is possible to parametrize each rotation R(+. All symmetry operations are fixed in i. j = [IO. if necessary. the rotation angle in the product rotation is always brought into the range stated.sin+(b. If a vector such as n is written in terms of the usual orthogonal unit vectors i.) (Notice that the vector n is axial. A (Gibbs) vector such as r is given by components rl. are never changed when symmetry operations are performed in the laboratory space. with cos++. r2. + n1cos++. of course.n) of SO(3) as follows: Equations (9) and (10) are the well-known quaternion-multiplication rules. real scalar. which means that the symmetry elements. and k are fixed in the laboratory and never transformed. say. it is evident that the quaterinion units are not unit vectors at all. then i = LO. essential if one wishes to construct projective or spinor representations of the rotation group). equal to unity) can be ’ mapped onto the rotations R(+. This vector is invariant under the rotation and it is called its pole: = Rotations and Quaternions 4cos++. and cyclic permutations. (See [3] for further discussion of the results above. A. made no distinction between i and i. j.. it will be useful to review the simpler work that is done in SO(3) by means of the quaternion algebra.NO. ill. we shall clarify a few conventions that we shall use in this article and which are absolutely necessary to avoid such concepts as those of an axial vector to be ridden by fallacies thLat easily result from mixing conventions or from the use of loose terminology. + n1 x n 2 sin:+. and k (axial. respectiveliy. ij = k . - + n1 * n 2 sin$+. sin++. Normalized quaternions [a. It must be noted. and r3. When two rotations are multiplied. j. on comparing (11) with (6). cos++. that Hamilton. Under these restrictive conditions. however. (These are statements that can cause confusion: For them to be absolutely clear in the passive picture. k. thus. of the inversion i with a binary rotation C. b. k for the identity but. the instructions to construct a vector product must read as follows: Choose a basis of three orthogonal unit vectors i. the reversal of direction of a and b under inversion does not entail a reversal of direction of a X b. k. The main property of P. allegedly behaving like a polar vector). Although we do not use the passive picture in this article. K must be defined that coincide with i. This is identically the same. j. k are inverted. thus resulting in the cross-product changing sign (and. within our conventions. j. SYMMETRIES. j. are never transformed.. statements which may or may not be equivalent. the inversion reverses the direction of the i. It is thus obvious that in the active picture. and a x b must have the same chirality as that of i. vectors. k. (13) INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 363 . It is sometimes stated that when inverting a X b not only the vectors must be inverted but also that the cross-product rule must be changed. with the axis of rotation perpendicular to the plane of u and it can be uniquely mapped to an axial vector P. ”reverse direction” of a vector is equivalent to ”change the sign of its components. in the active picture but not necessarily otherwise. which is therefore axial. j. k.” Reflections and the Clifford Units Quaternion algebra is an excellent tool for dealing with all the operations of SO(3) but the inversion and reflections that appear in O(3) are not tractable within this algebra. but does not change the sign of the components of axial vectors. as chosen by the rules stated.CLIFFORD ALGEBRA. k of arbitrary chirality.” meaning either that the vector reverses its direction or that its components change sign. Polar vectors reverse direction under inversion (and their components change sign). within the conventions defined (always using active transformations). a and b remain invariant in the sense just explained.but it will now be done in a more convenient and general way. as a difference from i. instead. since there is no change of basis and thus no change of chirality.. as saying that the inversion operation reverses the direction of polar but not of axial vectors. we shall restate the last result in that picture. In the passive picture. but axial vectors remain invariant. but that it can also be extended to cover all the operations of the Euclidean group. j. Axial vectors. . When we say that a polar vector does not change direction. we mean that it is invariant with respect to the fixed axes I. j. This is a unit axial vector normal to the reflection plane and chosen so that it coincides with the pole of the binary rotation C. An axial vector reverses direction when reflected through a parallel plane but remains invariant when this plane is perpendicular. in the sense that one must be able to use such statements without knowing which is the chirality that one is using. it has to be very carefully formulated to avoid the fallacious result stated. Polar vectors do not change direction but their components change sign. for any basis i. It is thus a major principle that all statements involving chirality must be neutral. The virtue of the Clifford algebra is not only that it deals with all the operations of O(3) equally well. There is no doubt whatever that at least in low-energy physics it is not possible to establish a difference between the two possible chiralities (handedness). It will be useful for later reference to summarize the distinctions between polar and axial vectors. A polar vector reverses direction when reflected through a perpendicular reflection plane but remains invariant when this plane is parallel. a. The relation between reflections and the Clifford units was already sketched in [ 3 ] . given two polar vectors a and b. although it is possible to deal with them by ad hoc techniques. Thus. a x b must reverse its direction and it is thus axial. but because the set a. (A simple geometrical treatment of these results may be found in [16]. = P. K and contrarywise for an axial vector. A reflection u is given as the product iC. Although there is an element of truth in this statement. b.) When i. reverse their direction and thus their components are invariant. j. Then. J. is that it is invariant under u. Notice also that the positive sense of rotation should be defined as that which takes i into j when looking from the head of k. three unit vectors I. AND VECTORS The inversion operation changes the sign of the components of all position vectors in laboratory space and of all polar vectors. u P. J. Notice that endless confusion can be created by loosely referring to ”vectors changing sign.) Also. a x b must change chirality (owing to the change in chirality of the basis). which we shall call the pole of the reflection. uz. u. (19) (20) (21) (22) = [O. from the quaternion multiplication rules (9) and (101. to avoid confusion with the quaternion unit i. [P..P. C.u2= CZy= [O.i + n.e2e3ele2e3 = -1.i + n2 j + n. (26) It follows that for i4 5 i n . to P. uy. are the units of C.) uzuY C.. it has been demonstrated [171 that the Cartan gauge cannot be ignored: its use. (Although the details of this mapping differ from those given in [3]. j + n.i] = = i. on using the rule stated for the rotation products.. As regards reflections. = = (23) (24) e2e1 = c-) k .P11 = = = i 2 [ . c-) my. They are i2 = 1 or i2 = -1.and cyclic permutations. )From (16) and (17). the orthogonal reflections a. .. Whereas most of the literature on the symmetry operations of O(3) is based on the Pauli gauge. then n is in the opposite direction. Given two reflections u2 and u. in fact. much developed recently by Biedenharn (see [ 181).. is where C. We write u for this purpose. that uY uX = C2*. k. respectively. X P. (17) ii * z2 = e. Rather. the sign of the quaternion requires change.which has been chosen so as to agree without change with the products in the Cartan gauge as given in [ 3 ] .kJ (27) (28) n.P. is a binary rotation about the z axis. the present choice is more adequate for the purposes of this article. (15) = 1 = eIe2e3 [cosi4. e. (16) - uxuyu. as we shall see.j] uYux = j. although. (18) (Notice that an upper-case letter has to be used for the image of the inversion in the Clifford algebra. - e3e2e3e2 -e3e3e2e2 = -1.) We can verify that this mapping is isomorphic: ii ij * e. NO. is. We shall now identify a general reflection v with pole P as an element of the Clifford algebra C. we find. P2 x PI] x P2il (14) in agreement with (12).P. since he never parametrized rotations by half-angles. respectively perpendicular to the orthogonal directions x . because I behaves like the imaginary unit i. but if i4 > i n . it has been shown [3.. then.IliCO. (2611 and will henceforth be adopted.the operations that multiply all operands by +1 and -I. 364 VOL.e. x P21. from (6). If we call the identity E and the anti-identity 2.n).nsin$c. 1 . z . = [O. e. as iC2. = Hamilton's turns.)/IP.k. since. with poles P. respectively. It will be important in what follows to keep in mind some properties of the inversion and reflection operations.ALTMANN because an axial vector is invariant under a reflection perpendicular to it. will be useful. the construction of a rotation by 4 as the product of two reflections separated by i+was the basis o f C2 = LO. we propose the following mapping: el whereupon e3e2 c. u2u.e3 e2 - u2. with pole n which.el and the quaternion units.17] that there are two multiplication rules for the inversion which are both fully compatible with the conventions used in physics. k] = We have thus established a mapping between the bivectors e. = eIe3 e2e1 - ax.e2e. Relation (16) was obtained by Hamilton [l]. ~~O~~. separated by the angle i4 from P. it is often identified with it. 60. unavoidable in the present work [see Eq. y. The geometrical relation (15) will be important for us as a check of the validity of the identification that we shall now provide of reflections within the Clifford algebra. n. which is not entirely satisfactory: There are many objects in this work that share this form of behavior. the vector n is in the direction of P.h] R(+. = (25) with and that we are unavoidably led to the Cartan gauge: n = (PI x P.. and P. he did not give it explicitly. If el.. = nl = [O..) It will presently be useful to consider the form of a general binary rotation C. as before. i.P.. (There is an element of convention here for the case i4 = in-.. which have been called by Altmann the Pauli and Cartan gauges.. It should be noticed that within this mapping the inversion is a trivector: . in particular. Notice also that. from (301. In fact.. 3 of the Clifford vector (or bivector) Transformation of a Vector It will first of all be necessary to make a notational distinction between a polar vector r and an axial vector with the same components.3e3f (31) (32) P. (39) The operator "Ve" here means: "find the components 1. For a reflection. + P3e3. (Two such vectors will be called associated. in the active picture of symmetry operations.) We have seen in (5) and (13) that for any rotation or reflection h of O(3) a unique axial vector f h can be given. We assert that g z h coincides with the pole of the operation ghg-l of O(3). as here.e. h (see Fig. respectively. the "invariance" of a pole under a symmetry operation. (37) e. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 365 . On the other hand. we must assume that we superimpose on each pole a position vector which is a copy of the pole and that it is this copy which is then transformed or otherwise left invariant.. it will be given in terms of its corresponding positive quaternion in (6). Consider the reflections u. 8 i . Therefore. + P.. as we have discussed in the section on conventions.CLIFFORD ALGEBRA. we have u= = h f h= Zh. their poles are never affected by any of them. it will be given by the right-hand side of (301. to regain (151. and ul.e. verifying (so far) the parametrization of reflections given by (30). so that we write gzh = Ve(ghg-'). 2. (axial vector transformation). SYMMETRIES.i P. (Strictly speaking.ze2 + '1. h will be the element of the Clifford algebra which parametrizes the operation h.Ze2 + P2. Similarly.e2e3(P. When we discuss the "transformation" property or. and for a rotation.e. + P. This equation coincides with (15) and thus relation (16) is established. AND VECTORS with P as the pole of C2. and which is called the pole of the corresponding operation. and write.3e3. no arbitrary factors are allowed in (31) and (321. thus supporting the consistency of our parametrization. = '2. and PI. it follows from (38) that if we want to transform a vector f h under g it is sufficient to find the pole of ghg-'. On using (25) and (281. which is correct in the Cartan gauge. with poles P. which is invariant under that operation FIGURE 1. We shall verify that the multiplication rule (15) is recovered with this identification. (29) (30) as can readily be derived on using (20) to (22) and (1). As is clear from (6) and (30).. Conjugate poles. Let us call g the element of the Clifford algebra that parametrizes the operation g. to be denoted with i. j + P 3 k ) + P. Notice that from (36) it follows that u 2 = + E . 1). each g and h will entail a vector part which is the pole of the corresponding operation. which is the vector part of ghg-'. Also. except for a factor of -1 in both equations. which merely entails a trivial redefinition of the positive hemisphere H.) Consider now an arbitrary operation g of O(3) that transforms the vector f k into another one.lel Ul = + '2. of an axial vector spanned by the unit vectors i. under the operation g and this axial vector will transform identically with r if and only if g is a proper rotation: Axial vectors transform identically with polar vectors under proper operations but change sign with respect to the transform of a polar vector under an improper operation (reflection or inversion). Therefore. on changing at the same time axial into polar vectors. Although (46) can be used heuristically and although it is undoubtedly convenient to take a in (42) to vanish.belong to the same class of conjugation.. must be left invariant. we shall most often want to transform a polar vector r. since the days of the original conical transformation of Cayley and Hamilton.e.Psin~+1Ca. in the right order. Thus. g proper. he never made a distinction between i and r. second. P ) = ve+{Ccost+. This. (40) = . we can take h to be a reflection. formally equal to r and." (If h is a rotation and h is not a pure quaternion. i']. (41) We shall illustrate the first choice for mapping the vector r in the case when it is desired to transform this vector under g .. (44) is written in the Hamilton sense as This is the Cayley-Hamilton conical trunsformation. the norms of the parameters.i~ ~ X [ . ( polar vector transformation). now appealring on the right-hand side of (441. (It should have been recognized instead that h could be the parametric im- (44) (45) Ve+[a. Equation (40) then gives R ( + . hence. the identification of the pure quater- 366 VOL. with the vector r. we are transforming the axial vector I. taken to "prove" that the "vector" h must be obtained from ghg-' by a symmetry operation. NO. We do this. as we have assumed. a proper rotation with pole P and rotation angle +. (42) The scalar part a of the quaternion in this equation can be quite arbitrary and we have now dropped thle subscript of the pole. if we now understand V e + to mean "apply V e on the expression on its right and replace axial vectors by their associated polar vectors. Thus. in which case h must be a quaternion. if h is a rotation.. + F. h = ?. One important observation must be made about these equations: The transformed vector g? in the source Eq. is not all that is required. he identified the so-called pure quaternion 60. 60. we can take h to be a rotation.e.V e + Cghg-' 1. by taking the vector part of ghg-'. Third. we have now dropped the subscript h from the vector r on the left of these equations.) In practice.1 . because symmetry operations must be norm-preserving. Second.e. This is the reason why the inner automorphism ghg-' of the Clifford space is one that must preserve the norm of h. it must be remembered that h is the Clifford-algebra element that maps the operation of which r is the pole. Because h and ghg. he took a to vanish (a point about which far too much has been made in the literature to this day). by finding a pole i. (39) is the pole of the operation ghg-' conjugate to h. a well-known result that can be directly tested and that it is often. in which case ghg-' is the pararnetric image of another operation in the same class of conjugation. when using them. both operations must be of the same point-group type (i. it must be remembered that the vector part of the quaternion contains the vector in the form n sin$+.. for the following reason: In ghg-'. C O S -Psin++]} +~. = For simplicity.) and the norms of their parametric elements must be preserved. g improper. ( polar vector transformation). from (301. First. however. first. We have two choices as regards the operation h whose pole is going to map the desired vector r. (43) ve+(ghg-'). however." we have gr gr = age of a syinmetry operation. ' where the quaternion on the right-hand side is the product of the three quaternions in (44). etc. r]. in which case h = l a . ill. or both reflections. in which case.) Thus. k in laboratory space.ALTMANN on its right and use them as components. ghg-' is a rotation of the same angle so that the quaternions h and ghg-' have both the same scalar part.el + ?. as well as those of any vector parts that they might contain. Eq. both rotations by the same angle. j. which allows us to interpret the transforma tion entailed in these equations as one that transforms only the underlying vectors. Second. Hamilton wrote this equation differently. First. First. once one is absolutely sure of what one is doing.}.P)P. thus. that PwP. = -PurP. (47) + f . ( + F2e2 + P 3 e 3 ) x(F.e. we obtain crr = = -(r . by its associated polar vector. cancels Equation (56) agrees with the geometrical result (47). (48) -{P x -(P X (i x P))2 j (54) (? X P)}. On introducing (54) into (49). whose parametric Clifford-algebra element will be written. We shall first of all use the transformation entailed in (491.i F2eie3 (52) but its undoubted validity will provide a good check on our method. = F. (51) but it must be understood that this is mere shorthand: On the right-hand side of this expression.) (53) = (I. is its own inverse.ZJ r . bearing in mind that P is a unit vector. as we have done. a simplified notation in the style of (46) may be adopted. Obviously. + P. as in the quaternion algebra.. we shall first of all rewrite (41) for the two choices for h given by (42) and (43). although the Ve operator in (39) to (41) may appear as somewhat clumsy.} (49) vr r .e2 + + F3e2el.k = Fle. as follows: P . This property allows the latter. We must take g in (41) to be a reflection cr with pole P.Z(r P)P. Moreover. Thus. P. hence. + f. the axial property of the vectors used is established beyond INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 367 . equals unity. We thus write crr = -PuyPu.e. = r . (55) which. from (51). P . It is very easy to prove.] To do this work. Thus.CLIFFORD ALGEBRA.e.2(r * P)P.k. the symbol " V e + " can now be dispensed with. We shall now use the transformation method given by (41) in order to find the transform v r of a (polar) position vector r under a reflection v on a plane through the origin 0 with pole P. Therefore.?]q.e2e. it does precisely the operations that are left dangerously implicit in the more standard notation.r] with the vector r is imprudent. (56) This last expression will be much simplified so as to adopt a notation more familiar in the literature.Z(r (50) . its recovery in (56) validates our operating rules and." Thus.e.) x ( F l e l + F2e2+ P3e. AND VECTORS nion [O. + f 2 e 2 + F. + f2e. I should like to stress that the object of this test is the following: Equation (47) is correct beyond any reasonable doubt. to be regarded as "vectors. the curly bracket in (501. = P. from (30).P)P + {P x (r x PI} * -Ve+{P.[O. and Pu as in (48).)P. x P)}li - - + P.e.e. [Notice that in this section we identify explicitly axial vectors with a tilde and that in (47) P must be replaced. [ Y o . SYMMETRIES. we have (first choice for h) crr = as it can easily be proved after some simple algebra. our contention that the elements of the Clifford algebra cannot be identified with vectors but that they are the parametric images of reflections and that here. vectors can only be manipulated by means of products of their image symmetry operations. we can write as or (second choice for h ) vr = -Ve+(P.(?. since from comparison with (6) this quaternion maps a rotation by r r (binary rotation) and does not have the transformation properties of a vector. one is simply multiplying symmetry operations rather than operating in any strict sense with vectors and one must also remember that the Clifford components must be properly assembled into a laboratory-axes vector. for which purpose we write [O. and. in particular. because the axial nature of the vector in P . as will later on be done in this article. This is a rather trivial geometrical result: vr = with that of Pgl.e. we notice that as a difference with (49) there is no scalar part in the Clifford algebra product and therefore that the vector to be extracted from the right-hand side of (50) is obtained from the whole of the Clifford algebra element. hence.P)P {P x (I. and all that we require in order to apply the symbol is to remember to replace Y by its corresponding polar vector r. j + F. ~ ~= PFlel ~ .e. . + 2v. On introducing (51) in that equation. we have another copy of O(3) with point-group operation g ’ with fixed point 0’ and another set of poles (Pgr] (see Fig. We propose to express all the point-group operations of the Euclidean group in terms of the pointgroup operations at the origin 0 of the laboratory space. (60) where v is the perpendicular vector from 0 to the plane u ’. it is clear that the pole Px8 of an operation g’ must be given by upg. It must be remembered that in the active picture the translation. the work is very much the same and it will not be reproduced here. We have seen in (40) and (41) how such point-group operations transform a position (polar) vector r. however. Although (50) is more compact than is (49). Eq.e. if convenient. cannot be free vectors. otherwise. the vital sign in (49) would not be obtained. does not clhange the symmetry elements at all. we obtain v’r = = -P. Call for this purpose u the position vector of 0’ (with respect.u) + u.(Y .rP. which may be taken. (61) 368 VOL.. 60. This is quite legitimate because all that we need is a relation between Pg. to the origin 0 of the laboratory space) and consider a translation u by the vector u.. to be also a continuous function of 7 and. We assume that these operations have all been parametrized as described above and. we get FIGURE 2. Neither can they be translated as the position vectors of the space are. in the same manner as in (38). however. Thus. We shall illustrate the general procedure given by (58) for a reflection u ’ (see Fig. where Pg is the pole of some operation g at 0. In the Euclidean group. the fixed-point of which has position vector u with respect to 0: The Euclidean Group: Translations We want to consider now the Euclidean group. that their poles have all been determined and that they form a set (Pg}. at any point 0’ of the space. 3). are not transformed by the translation in (57). PUuPu= -U +2 ( ~ PIP . + u. of course. + 2v. by using (50) instead of (49). (58) r + u. since. To relate Pxt and Px under the translation u.NO. in general. We can thus obtain the action on a vector r of any point-group operation of the Euclidean group. e. On introducing this result into (59).u) = g(r . The pointgroup operation g will in this case be a rotation. Under the irule just given. Parametrization of symmetry operations centered at different points. +u + P. which transforms all position vectors of the space as follows: ur = g’r = ugu-lr = u(r .. as will be discussed in a later section on vectors and bivectors. to be an infinitesimal rotation. if the vector u is taken to be a continuous function of a real parameter T (which might be the time). that upg is the pole of ugu-l so that g‘ must coincide with this product. dummy copies of these vectors must be superimposed on them with a view to transforming these dummies. not without significance. = -U (59) From (56). (57) It is useful to recognize that. and Pg.u ) P . We must therefore be able to express an operation g ‘ in terms of the point-group operations { g } at 0. i. We must therefore define the translation by u of the pole Pg (strictly speaking.. in particular. since they would then cease to be of unit length. It is very easy to prove. They. like all symmetry operations. (58) describes a general screw in laboratory space.g.. the poles at 0 and 0’.ALTMANN reasonable doubt. however. 1 . Let us call ( g } the set of all point-group operations g E 0(3). The fact that the two different approaches agree entirely is. -PurP. that of its dummy copy) as merely translating its point of application. since their points of application are vital. u ’ r = -P. g’. such that their fixed points coincide with the origin 0 of the space. The geometrical result (55) can be identically reproduced on imaging the vector r by (43) instead of (42). in (571.uP. 2). AND VECTORS In’ o FIGURE 3. i. Before this important point is discussed in detail. not on f those of an arbitrarily defined vector space. and it is. (56) that both of them behave beyond any reasonable doubt in identically the same manner as axial vectors.P u ( . instead. whereas this property is significant for the vectors r and s in laboratory space that can be extracted from r and s in (65) and (66). + 2v. When using Clifford multivector algebras. SYMMETRIES. with the distinction between Gibbs polar and axial vectors which must be done (in the passive picture) by applying the inversion on the unit vectors i. o the laboratory space. obvious that Is does indeed give r.) Since we assert. the bivector in (52) should not behave in the same manner as does the vector in (431. the following remark is useful: In accordance with the argument just given.i + r2 j r3k. As always. where only the space coordinates can be inverted [19]. whereas we have seen after Eq. Vectors and Bivectors I shall now deal with a troublesome fallacy. presumably. in fact. r. we perform this operation within the active picture which requires. k. the use of the rule given in (41) for applying an improper rotation on a vector.e. This result shows that. P:rP: (63) (64) = + 2v = r + 2v. A reflection contain the origin. From (611. is in the same manner identified with a pseudoscalar. rg in (65) behave like the components of the axial vector r = r.(ar)P. P . it follows that u’ur = .P g r P u ) ~ + 2v . in the space spanned by the Clifford units. To hammer this last point home. in principle. unfortunately. that they do not change sign under the (active) inversion in the + INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 369 . However convincing this argument appears. It must be strongly stressed that axiality or polarity is a property that can only be discussed for vectors defined in the laboratory space. that the components rl. although a distinction between these two objects should be made [12]). It is useful to note that the careful consideration of the inversion of the laboratory-space coordinates is very important in the Lorentz group. because the pseudoscalar multiplied by the pseudovector should give a vector. it is sometimes asserted that the old-fashioned concepts of polar and axial vectors are superseded in these algebras. Consider two successive parallel reflections u ’u where the perpendicular vector from 0 (which is on the plane a ) to the plane u ’ is v. however. can be called a vector because on changing the sign of the Clifford units (this.el (65) (66) (67) I = e1e2e3. This appears to work very well. is thought to be the passive picture of the inversion) its compcnents do change signs as vectors should do. does not change sign under such ”operation” and it is thus said to behave like an axial vector (or a pseudovector. it makes little sense for the vectors in the Clifford space. y 2 . (62) On using (56) here.. V CT’ at a plane that does not We shall now show that translations may also be expressed within this algebra. since it changes sign under the so-called inversion. r = r. whether active or passive. thus completing the description of all the operations of the Euclidean group within the Clifford algebra. let us transform under the inversion the vectors r and s associated with r and s in (65) and (66). I . two successive reflections on two parallel planes such that the perpendicular distance from the first to the second plane is given by the vector v is a translation t by 2v. equals unity. a‘ar = -P. it has nothing whatever to do. s = rle3e2 + r2ele3 + Y3e2el. It is asserted that the first of these elements. as is well known.ZLIFFORD ALGEBRA. The reasoning behind this assertion appears to go roughly as follows: Define the following elements of the algebra: + r2e2 + r3e3. j. Thus. (Notice that this operation has nothing whatever to do with a transformation. on the other hand. since. as already remarked. The bivector s. ] This proof not only corroborates once more that the work on the transformation of vectors that we have given is correct. respectively.. Because the spinor representations are projective ones. will be isomorphic to spinor representations of O(3) and are capable of generating all of them.k = r. like e3e2 before. since the inversion commutes with all symmetry operations: This is another clear example of how transparent the meaning of products in the Clifford space is. and 3 of the expression on its right and that we apply them. + r2e2 + r3e3)e3e2el} (68) (69) = Ve{r. it must be remembered that "Ve" requires that we extract the components 1. It is thus easy to verify that (71) and (73) give the dihedral group D. we have seen in (26) that the mapping of the trivector onto the inversion entails the Cartan gauge. 2. It is. is a binary rotation about the hyperline yz. Clearly. j. is that we obtain precisely the same transformation rule for a bivector. C. When this is done for e3e2. with their identification (in the same order as given in the first column). in the correct order. we list the eight Clifford algebra elements and. on the unit vectors i. e4e. once their nature in terms of symmetry operations is understood. merely demonstrates the consistency of our work. We must. The crucially important result. Likewise. 60. of course.i + r.In the same manner. we shall revert to the use of the transformation rule for axial vectors (39). It should be noted that the Clifford bivectors are the three orthogonal binary rotations that are the generators of the spinor representations of S0(3). h must be replaced by Y from (65). instead. The elements The identification of the symmetry operations is very simply done by plugging them into an equation like (68). accordingly. but it reinforces our statement that the property of axiality has nothing whatever to do with the distinction between vectors and bivectors in the Clifford space. Finally. it means that it must be a binary rotation about the hyperline xw.ALTMANN space spanned by i. It is easy to prove that I-' equals e3e2e1.. if it is desired to generate the corresponding groups. under this rule. as shown in [3]. as asserted. the elements of which. it is found that the x and w components are invariant and the other two change sign. are listed below: Point Groups in Three and Four Dimensions I shall first review briefly how point groups in three dimensions are generated by the Clifford algebra C. In the first column here. projective factors of tl must be ignored when obtaining cllosure. Also. for (72) from (19) and for (73) from (20) to (22).(rle. and k. All this means that the elements of C. j + r.el + r2e2 + r. ir as can easily be verified. to be expected. k of the laboratory space on which r has been defined. NO. the operations of the full rotation group that they map. We can now deal with the Clifford algebra in four dimensions. r is an axial vector. f. in fact. whereby g must be taken to be I from (67). 1 .e. on their right.e2e. of course. of course.}.. which are listed below. Since this operation. The 370 VOL. thus giving the lie to the assumption that Clifford vectors and bivectors behave differently under the inversion. p. j. the right-hand side of (69) gives = r.g. must be a binary rotation under which all vectors operate in the same manner. change the notation a little. We therefore write ir = are IYI-' = Ve{e. 117. extremely easy to prove that (70) is still valid when s is substituted for Y in (68). [Both this result and (70) were. (70) thus verifying that. First. e. the operation g will be chosen to be the inversion. will be replaced by r and. This result. is a reflection on a plane normal to the hyperplane xyz which is parallel to zu-hence. are The latter group is a dihedral group with a binary rotation C. The very simple geometrical tests here provided show clearly that no element of a Clifford algebra can be used to directly map a vector. strictly speaking. in fact. with a rotation Cix. Most importantly. The superiority of the Clifford over the quaternion algebra must be noticed since. It is thus sensible to identify e4e. it follows from the present work that both Clifford vectors and bivectors may be associated with laboratory-space axial vectors and that much care must be exercised in defining axiality or polarity when using Clifford-algebra methods. This formalism will thus be useful in any work that requires the tracking of motions in Euclidean space. these operations being the entities directly mapped by elements of the Clifford algebra.g. orthogonal to the six binary rotations in (86).) It is very easy from (75) to (79) to deduce finite point subgroups of 0(4). C.e. Conclusions The main object of this article was that of providing a number of simple geometrical tests in the laboratory space in order to understand the properties of Clifford-space vectors and to construct a workable parametrization of O(3). which is a binary rotation of the whole space (it changes all four components and behaves like a binary rotation around the center of a strictly two-dimensional plane). which will be designated with a notation similar to that used for those in 0(3). in (781.e. The Clifford space.Vectors in the laboratory space must first be mapped by poles of rotations or reflections. Just like in 0(3). the inversion. but it must be remembered that closure is only satisfied on disregarding the projective factors since. and they likewise commute with the primed operations with the same label. These examples will suffice to show how easily it is to generate point groups in four dimensions by means of the Clifford algebra. It is useful to notice in (77) that all the unprimed operations commute. shows that e. The operation e. is easily proved to be e1e2e3e4. although he lists them in a different way (see also [21]. in analogy to the fact that. remembering now that under reflection axial components parallel to the reflection plane change sign. e. the utility of Clifford algebras in forming point groups in 4dimensional space was demonstrated and further applications along this line are in preparation. the binary rotations C 2 x . CZy. AND VECTORS product of e3e2 with e4e. the operation in (79) when multiplied by the binary rotations in (77) transforms them in other binary rotations of the same list. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 371 . reflections. and thus all improper operations. the symbol adopted. is the parametric space of O(3). in such a way that the unprimed rotations transform into their orthogonal primed ones and vice versa. whereas in the quaternion algebra the inversion. The first few. orthogonal to CZx. These tests allow us to obtain practical rules of transformation of laboratory-space vectors under the most general transformations of the Euclidean group (such as screws). the standard rules o transf formation of the Gibbs vectors are preserved with no trouble. of the whole space. Going back to (76). is a reflection on the hyperplane yzw. and translations are all correctly mapped by the elements of the Clifford algebra. including the projective factor. must be handled on an ad hoc basis. are the group generators and the six operations in (77) are the generators of 0(4). the product of two perpendicular binaries is a binary rotation perpendicular to the plane of the two factors. The primed operations commute only if projective factors are disregarded. but with a distinguishing superscript denoting the dimension of the space. SYMMETRIES. the Clifford algebra will always generate the covering or double group. instead. in three dimensions. Finally..CLIFFORD ALGEBRA. the same test. When this is done with some care. which agrees with Wulfman [20]. Also. normal to the x direction.. for which help I am most grateful. Urbantke. and J. 11. Dublin. Paris. Hamilton. J. Phys. Coxeter and W. 18. Ed. 1013 (1971). M. H. Baltimore. E. Pezzaglia for providing me with copies of their theses and to Professor Pezzaglia for very useful correspondence. Moser. London. Dordrecht. in Group Theory and Applications E. Gruppen. as I am for a discussion with Professor J. 8. Hestenes. S. 151. 2. J. Generators and Relations for Discrete Groups. 3rd ed. M. M. Theory and Application (Addison Wesley. Herzig. I am also grateful to Dr. 1984). D. M. Founds. Angular Momentum in Quantum Physics.. (Springer. 4th ed. Riesz. MD. R. K. 1992). 11. Vol. 39. Spinors and Calibrations (Academic Press. Doctoral Thesis (University of California. K. A preliminary version of this article was presented at the Second International School of Theoretical Physics at Poznafi. J. Ross and Professor W. Topology 3 (Suppl. Bayliss. Common. M. Altmann. J. and Double Groups (Clarendon Press. 6. 3 (1964). (Academic Press. S. 10. Eds. (Springer. S. R. F. S. Vienna. Atiyah. and A. Altmann and P. D. 1990). 1981). Am. Oxford. Professor H. Legons sur la Th&oorie des Spineurs (Hermann. Lounesto. Rotations. Greider. 13. Davis. J. Phys. This work was supported by a travel grant of the Royal Society and by the Austrian Ministry of Science under Project GZ 49. 1958). Lectures on Quaternions (Hodges & Smith. L. Sobczyk. Point-Group Theory Tables (Clarendon Press. A: Math. W. F. C. 788 (1992). Harvey. Quaternions. Clifford Numbers and Spinors. for which I am grateful. E. 721 (1981). Shapiro. 20. 60. Am. Chisholm and A. M. 16. 2. 14. in August 1992. New York. 15.NO. 1992).467 (1984). 1984). Hestenes. M. p. Wulfman. L. 1986). 60. 20. Relativitut. K. Teilchen. C. 1986). 17. Morzymas from Wroclaw. 14. W. D. 1853). Urbantke from Vienna very kindly read a draft of this article and contributed useful comments. Sex1 and H. R. K. L. Clifford Algebra to Geometric Calculus. D. 7. 12. Phys. Huschilt. Phys. S. Bott.731/224/91. 9. Lecture Series No. Loebl. Space-Time Algebra (Gordon and Breach. Phys. 1966). Biedcnharn and J. Cartan. Altmann. A Unified Language for Mathematics and Physics (Reidel. P. Wei. Altmann. L. Pezzaglia.ALTMANN ACKNOWLEDGMENTS I should like to acknowledge stimulating correspondence with the late Professor Ken Greider. Clzfford Algebras and Their Applications in Mathematical Physics (Reidel. W. Berlin. 19. 3. Oxford. Louck. 5. 1 . 1938). 11. Icons and Symmetries (Clarendon Press. 1975). References 1. 0. Founds. 38 (Institute for Fluid Dynamics and Applied Mathematics. 21. Reading. R. J. E. Oxford. Gen. U. 1971). 4587 (1987). 1994). MA. 4. 372 VOL. New York. S. Hestenes and G. Dordrecht. L. R. 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