arXiv:0711.3220v1 [math-ph] 20 Nov 2007 oretragba10 7 algebra Fourvector 2 3 fourvectors The 2 Introduction 1 Contents words Key PACS . h om...... 13 13 13 ...... 12 . . . inverse . Multiplicative . . . 8 . fourvector 3.4 . Identity . . 9 norm . 3.3 The . matrices . with 3.2 Product . . . . 3.1 . . 6 ...... fourvectors 2 . Complex . . . 6 . 2.2 Discussion . . . 2.1 ...... 5 . . . . . matrices . . Dirac . . . Pauli 1.4 . The . . quaternions Hamilton 1.3 . The . 1.2 General 1.1 oain ihteueo oretr,a ela hi s in use their as rotations. excel well simple the as as shown understood is fourvectors, are It which of boosts, use form. Lorentz natural the a sciences with in the rotations vectors and Physics 3D in the use embrace its for quaternions Hamilton 02.10.Vr : h ler ffuvcosi ecie.Tefuvcosar fourvectors The described. is fourvectors of algebra The oretr,dvso ler,3-oain,4D-rotations 3D-rotations, algebra, division fourvectors, : 1 sul oi´ciaNcoa.Qio–Ecuador. – Quito Polit´ecnica Nacional. Escuela (1) oretrAlgebra Fourvector -al [email protected] e-mail: ig Sa´aDiego Abstract 1 ocuin 22 17 Conclusions 5 rotation Fourvector 4 . eetos...... 21 ...... 20 ...... Reflections rotations of 4.2 Composition 4.1 15 . 14 14 14 . fourvec- quadratic fourvector of . . a . Solution of . . factor 3.10 fourvector Left . . a 14 . . of factor 3.9 . . Right . . . . division 3.8 . Fourvector . . . fourvector 3.7 Unit . multiplication 3.6 Scalar 3.5 1 o oyoil 16 ...... polynomials tor oeaporaeta the than appropriate more e ngnrl h fourvectors The general. in eaiiyfrproducing for relativity etaiiyt perform to ability lent 1 Introduction Heaviside was expressing when he wrote that there is much more thinking to be done [to set 1.1 General up equations]. In principle, most everything done with the new system of vec- In this paper it is suggested the use of tors could be done with quaternions, but the fourvectors with the purpose of replacing the operations required to make the quaternions 3D vectors and the quaternions. Because behave like vectors added difficulty to using the fourvectors contain the three dimensional them and provided little benefit to the physi- vectors and can be considered a formalization cist.” of them. “Gibbs was acutely aware that quater- The discovery of the quaternions is attributed nionic methods contained the most important to the Irish mathematician William Rowan pieces of his vector methods.” [32] Hamilton in 1843 and they have been used for After a heated debate, “by 1894 the debate the study of several areas of Physics, such as had largely been settled in favor of modern mechanics, electromagnetism, rotations and vectors” [32]. relativity [26], [20], [6], [2], [23], [13]. James was one of the Clerk Maxwell used the quaternion calculus debaters and seems to have been one of the in his Treatise on Electricity and Magnetism, few in realizing what the real problem with published in 1873 [22]. An extensive bibliog- the quaternions was. “MacFarlane’s attitude raphy of more than one thousand references was intermediate - between the position of about Quaternions in mathematical physics the defenders of the Gibbs–Heaviside system has been compiled by Gsponer and Hurni and that of the quaternionists. He supported [14]. the use of the complete quaternionic product The modern vectors were discovered by of two vectors, but he accepted that the the Americans Gibbs and Heaviside between scalar part of this product should have a 1888 and 1894. Their work may be consid- positive sign. According to MacFarlane the ered a sort of combination of quaternions and equation j k = i was a convention that ideas developed around 1840 by the German should be interpreted in a geometrical way, Hermann Grassman. The notation was pri- but he did not accept that it implied the marily borrowed from quaternions but the negative sign of the scalar product”. [25] geometric interpretation was borrowed from (The emphases are mine). Grassman’s system. He incorrectly attributed the problem to a By the end of the nineteenth century secondary and superficial matter of repre- the mathematicians and physicists were hav- sentation of symbols, instead of blaming to ing difficulty in applying the quaternions to the more profound definition of the quater- Physics. nion product. “MacFarlane credited the Ryan J. Wisnesky [32] explains that “The controversy concerning the sign of the scalar difficulty was a purely pragmatic one, which product to the conceptual mixture done by

2 Hamilton and Tait. He made clear that the ford. According to Gaston Casanova [4] “It negative sign came from the use of the same was the English Clifford who carried out the symbol to represent both a quadrantal decisive path of abandoning all the commu- and a unitary vector. His view was that tativity for the vectors but conserving their different symbols should be used to represent associativity.” “This algebra absorbs the those different entities.” [25] (The emphasis Hamilton quaternions, the Girard’s complex is mine). quaternions, the and the com- plex , the hyperbolic numbers and At the beginning of the twenty century, the dual numbers.” [4]. Also the Hestenes’ Physics in general, and relativity theory “geometric product” conserves associativity in particular, was lacking an appropriate [18]. In this sense, the associativity of the mathematical formalism to represent the product is finally abandoned in the fourvector new physical quantities that were being algebra proposed in the present paper. This discovered. But, despite the fact that all means that the fourvectors do not constitute physical variables such as space-time points, a Clifford Algebra [2] or a Geometric Alge- velocities, potentials, currents, etc., were rec- bra [1]. This is a collateral effect of the pro- ognized that must be represented with four posed algebra, and constitutes a hint about values, the quaternions were not being used the form the fourvectors should handle, for to represent and manipulate them. It was example, a sequence of rotations; besides, the necessary to develop some new mathematical complex numbers are not handled as in the devices to manipulate such variables. Be- Hamilton quaternions, where the real sides vectors, other systems such as tensors, is put in the scalar part and the imaginary in spinors, matrices and were the vector part, but a whole developed or used to handle the physical can be put in each component, so it is pos- variables. sible to have up to four complex numbers in each fourvector. But, what is more impor- During the twenty century we have wit- tant, it is known that in quantum mechanics, nessed further efforts to overcome the diffi- observables do not form an associative alge- culties remaining, with the development of bra, so this could be the natural algebra for other algebras, which recast several of the Physics. ideas of Grassman, Hamilton and Clifford in The proposed algebra could have been a slightly different framework. An example in already developed, around 1900, under this direction is Hestenes’ Geometric Algebra the name of hyperbolic quaternion, which in three and Space Time Algebra is a mathematical concept introduced by in four dimensions. [16], [17], [21], [8] [18] Alexander MacFarlane of Chatham, Ontario. The commutativity of the product was The idea was dismissed for its failure to abandoned in all the previous quaternions conform to associativity of multiplication, and in some algebras, such as the one of Clif- but it has a legacy in and

3 as an extension of “split-complex numbers”. Physics using quaternions. But this attempt Like the quaternions, it is a has been plagued with recurring pitfalls for over the real numbers of 4. There reasons until now unknown to both physi- is only the mention to such quaternions but cists and mathematicians. The quaternions no accessible references to confirm if those have not been making problem solving easier quaternions satisfy the same algebraic rules or simplifying the equations. Very often the given in the following for the fourvectors. Hamilton quaternions require an extreme ha- However our intent is to convince the reader bility to guess when and where a quaternion that the presented here is one of the most needs to be conjugated, in order to obtain important mathematical tools for Physics. some particular result. I believe that this has been due to an The fourvector representation is without a internal problem in the mathematical struc- doubt a more unified theory in comparison to ture of the Hamilton quaternions, which I the classical vector or matrix representations. will try to reveal in the present paper. The correction of such problem constitutes a new The use of fourvectors allows discerning non-commutative, non-associative normed constants, variables and relations, previously algebraic structure with which it is possible unknown to Physics, which are needed to to work with fourvectors in an improved complete and make coherent the theory. way relative to the Hamilton, Pauli or Dirac quaternions, geometric algebra, space–time The vectors have lost some ground in algebra and other formalisms. favor of the Hamilton quaternions due to the lack of an appropriate 4D-algebra. For In the present paper, in particular, the example, Douglas Sweetser, who has worked present author exhibits the application of the extensively in the application of Hamilton fourvectors to 3D and 4D rotations, which quaternions to many possible physical areas, requires a reformulation of the Hamiltonian in general with very little success, sustains mathematics. these opinions: “Today, quaternions are of interest to historians of mathematics. Vector The powerful Mathematica R package analysis performs the daily mathematical includes a standard algebra package for the routine that could also be done with quater- manipulation of the Hamilton quaternions. nions. I personally think that there may be I have borrowed from that package the 4D roads in physics that can be efficiently symbol, as double asterisk, to represent the traveled only by quaternions.” [29] fourvector product. It is easy to modify the cited package to handle the fourvectors, as In fact those 4D roads should be traveled well as to permit not only their numeric but only by properly handled fourvectors. It has also symbolic and complex manipulation. been an old dream to express the laws of Though I have still not been able to figure

4 out a simple way to reproduce the trigono- are non-commutative, we have in general metric and exponential functions for the A**B = B**A, where the double aster- 6 fourvectors (if at all possible), which that isk represents quaternion multiplication. Un- package allows to compute for the Hamilton der these conditions, quaternion multiplica- quaternions. tion is associative, so that (A**B)**C = A**(B**C) for any three quaternions A, B, In the following three subsections, a cur- C. sory revision is made of the Hamilton, Pauli The sum of two quaternions is and Dirac quaternions, for an easy compari- son with fourvectors. In section 2 the fourvec- A + B =(a + b)+ i(ax + bx)+ tors are presented. Section 3 can be skimmed (6) by the mathematician, since it is mostly clas- j(ay + by)+ k(az + bz), sic algebra. Finally, in section 4 the formulae needed to perform rotations and reflections and, using relations (2) and (3)-(5), the prod- with fourvectors is presented. uct is given by:

1.2 The Hamilton quaternions A B =(ab axbx ayby azbz)+ ∗∗ − − − i (abx + axb + aybz azby)+ Quaternions are “four-dimensional numbers” − (7) of the form [31]: j (aby axbz + ayb + azbx)+ − k (abz + axby aybx + azb). − A = a + i ax + j ay + k az, (1) The notation of three-dimensional vec- B = b + i bx + j by + k bz tor analysis furnish a useful shorthand for quaternion operations. Regarding i, j, k as where the basis elements 1, i, j, k satisfy the unit vectors in a Cartesian coordinate system, relations: we interpret the quaternion A as comprising 2 2 2 a i = j = k = ij k = 1 (2) the scalar part and the vector part − a = i ax + j ay + k az. Then we write in the and also: simplified form A = (a, a). With this nota- tion, the sum (6) and the product (7) may i j = j i = k, (3) more compactly be expressed as: − j k = k j = i, (4) − a b k i = i k = j. (5) A + B =( + , a + b) (8) − Here 1 is the usual real unit; its product A B =(a b a b, a b + b a + a b) with i, j or k leaves them unchanged. ∗∗ − · × Thus, since the products of the basis elements (9)

5 where the usual rules for vector sum and for i 1, 2, 3, 4 . dot and cross products are being invoked. ∈{ } According to the mathematicians, the The Pauli quaternions evidence one differ- Hamilton quaternions are mathematical ence with respect to the classical Hamilton structures which combine properties of com- quaternions, being the need of matrices, plex numbers and vectors. which in some cases have imaginary units i.

1.3 The Pauli quaternions The sum of two Pauli quaternions is of the same form as the given for the Hamil- The Hamilton multiplication rules differ from ton quaternions and its product, using (10), the rules only by the explicit becomes: appearance of the fourth basis element.

A B =s4(ab axbx ayby azbz)+ The basis elements of the Pauli quaternion ∗∗ − − − s1(abx + axb + aybz azby)+ space are denoted by s1, s2, s3, s4. − s2(aby axbz + ayb + azbx)+ − They obey the following multiplication s3(abz + axby aybx + azb). − rules, comparable to (2)-(5): (12) 2 2 2 2 In a compact form, the product for the s1 = s2 = s3 = s4 = 1 − − Pauli quaternions has exactly the same form s1 s2 = s2 s1 = s3 − as the Hamilton quaternions and, therefore, s3 s1 = s1 s3 = s2 (10) − have the same problems as these: s2 s3 = s3 s2 = s1 − s4 sk = sk s4 = sk, (k =1, 2, 3, 4) A B =(a b a b, a b + b a+ a b) (13) ∗∗ − · × These rules are satisfied, in particular, by the Pauli spin matrices (only the first three 1.4 Dirac matrices bear this name, because σ4 serves to form the The Dirac matrices must satisfy the Klein- identity matrix) [18], [20], [3] [10]: Gordon equation, the following relations should be satisfied by the Dirac matrices [33]: 0 1 0 i σ1 = , σ2 = − 1 0 i 0 α α + α α =2δ , (11) i j j i ij 1 0 i 0 α β + βα =0, (i =1, 2, 3) (14) σ3 = , σ4 = i i 0 1 0 i 2 2 − αi = β = I where “1” in (10) represents the identity where I represents a N N unit matrix. × matrix, i the imaginary unit and si = i σi, − 6 For 2 2 matrices, only three anti- 2 The fourvectors commuting× matrices exist (the Pauli matri- ces). Thus the smallest dimension allowed for The fourvectors are four-dimensional num- the Dirac matrices is N = 4. If one matrix bers of the form is diagonal, the others can not be diagonal or they would commute with the diagonal ma- A = e at + i ax + j ay + k az (19) trix. We can write a representation that is or, assuming that the order of the basis el- hermitian (a matrix is hermitian if it is equal ements is the indicated, those basis elements to the conjugate of its transpose), traceless can be suppressed and included implicitly in (trace equal zero), and has eigenvalues of 1: ± a notation similar to a vector or 4D point:

A =(at, ax, ay, az) (20) 0 σi αi = , (i =1, 2, 3) (15) Where the elements of the fourvector can σi 0  be any , real, imaginary or complex and numbers. I 0 β = (16) 0 I The four basis elements e, i, j, k satisfy the − relations: where σi are the 2 2 Pauli matrices and × 2 2 2 2 I is the 2 2 unit matrix. e = i = j = k = e = eijk (21) × Finally, we are ready to define the Dirac’s The following rules are satisfied by the basis elements: gamma matrices out of αi and β:

0 i γ β, γ βαi (i =1, 2, 3) (17) e i = i e = i, ≡ ≡ − e j = j e = j, These matrices satisfy the relations: − e k = k e = k, − (22) (γ0)2 =1, (γi)2 = 1, (18) i j = j i = k, − − j k = k j = i, − and all four matrices anticommute among k i = i k = j. themselves. These relations are compara- − ble to the Hamilton basis (2)-(5) and Pauli The of relations (22) gives an im- basis (10), except for the exchange of sec- portant operational mechanism to reduce any ondary importance in the signature of the combination of two or more indices to at most gamma matrices, (+, , , ), to the sig- one. − − − nature satisfied by the Hamilton and Pauli The “eijk” bases characterize the bases: ( , +, +, +). fourvector product as not commutative but, − 7 what is more important and different with 2.1 Discussion respect to the previous Hamilton and Pauli Tait and Gibbs discussed about the rela- quaternions as well as to the Clifford Algebra (see [4], p. 5 axiom 3), the product is not tive merits of the “vector product” against always associative. For example consider the “quaternion product”. Tait appeared to gain a slight advantage by pointing out that the product of the following four symbols “iejk”, in the order given; if, with the use quaternion products are associative, whereas the cross product is not. Tait used the follow- of (22), we first reduce “i e” to i then ing example to reveal the supposed deficiency “ i j” to k and finally “ k k” to− e we obtain− one− result; but, if− we first− reduce of the vectors [32]: the two middle basis elements “e j” to j, i (j j)= 0 =(i j) j = i (23) then “i j” to k and then “k k” to e, we × × 6 × × − get the same result but with the sign changed. Note that the four-vector product proposed in the previous section, although not associa- tive, or precisely because of that, gives the If we put these rules into a multiplication correct result for the problem at hand: table they look in the following way: i (j j) i e i × × → × → − (24) ** e i j k (i j) j k j i e e i j k × × → × → − i –i e k –j The fourvectors have extensive applica- j –j –k e i tions in electrodynamics and relativity. The k –k j –i e present author believes that the use of the fourvectors, with the proposed algebra, can replace advantageously the matrices, vec- Let us assume that each basis unit is tors and tensors in representation. Some of affixed its proper number (as components of the advantages proposed for the Hamilton a tensor): quaternions, Geometric Algebra and Space- (e, i, j, k) (q0, q1, q2, q3)= q, Time Algebra, which are also extended to the then the→ fourvector multiplication satisfies fourvectors, are: the following compact relation, where the symbol χ has some similitude with the 1. Fourvectors can express rotation as a ro- Levi-Civita symbol. Refer to [27] for the tation angle about a rotation axis. This meaning of such symbol. is a more natural way to perceive rota- tion than Euler angles [5].

j qiqk = δikq0 + χikqj i, j, k = 0, 1, 2, 3 2. Non singular representation (compared with Euler angles, for example)

8 3. More compact (and faster) than matri- bital mechanics, mainly for representing rota- ces. For computation with rotations, tions/orientations in three dimensions. The fourvectors offer the advantage of requir- spacecraft attitude-control systems should be ing only 4 numbers of storage, compared commanded in terms of fourvectors, which with 9 numbers for orthogonal matri- should also used to telemeter their current ces [24]. Composition of rotations re- attitude. The rationale is that combining quires 16 multiplications and 12 addi- many fourvectors transformations is more nu- tions in fourvector representation, but 27 merically stable than combining many matrix multiplications and 18 additions in ma- transformations. trix representation...The fourvector rep- resentation is more immune to accumu- 2.2 Complex fourvectors lated computational error. [24]. The only difference with respect to the 4. Every fourvector formula is a propo- ordinary fourvectors is that the elements are sition in spherical (sometimes degrad- not purely real but complex numbers. ing to plane) trigonometry, and has the The collection of all complex fourvectors full advantage of the symmetry of the forms a vector space of four complex method [30]. dimensions or eight real dimensions. Com- bined with the operations of addition and 5. Unit fourvectors can represent a rotation multiplication, this collection forms a non- in 4D space. commutative and non-associative algebra. 6. Fourvectors are important because of There is no difficulty in obtaining the mul- their “all-attitude” capability and nu- tiplicative inverse of a complex fourvector, merical advantages in simulation and when it exists, within the fourvector algebra control [28]. suggested below. However, there are complex fourvectors different from zero whose norm Quaternions have been often used in com- is zero. Therefore the complex fourvectors puter graphics (and associated geometric do not constitute a . analysis) to represent rotations and orienta- tions of objects in 3D space. This chores It is important to realize that the relations should be now undertaken by the fourvec- needed by the Klein-Gordon equation (14), tors, which are more natural, and more com- are directly satisfied by the purely real pact than other representations such as ma- fourvectors, whereas the relations needed trices, and operations on them such as com- by the Dirac equation (18), are satisfied position can be computed more efficiently. by the fourvectors constituted of imaginary Fourvectors, as the previous quaternions, components in the vector part. will see uses in control theory, signal pro- cessing, attitude control, physics, and or- This seems to suggest that there are two

9 different kinds of physical entities, although whose scalar component is equal to zero). closely related, which need respectively the Divided by two serves to define the com- real and the imaginary representations. This mutator (similar to the vector Hamilton’s insight appears potentially useful for Physics. operator V ): (A A)/2= V A − The complex conjugate or hermitian conju- gate of a fourvector changes the signs of the 3 Fourvector algebra imaginary parts. Given the complex fourvec- tor: The sum of two fourvectors is another fourvector where each component has the A =e(at + ibt)+ i(ax + ibx)+ sum of the corresponding argument compo- j(ay + iby)+ k(az + iby) (28) nents: (10.15)

Then its complex conjugate is: A + B =e(at + bt)+ i(ax + bx)+ (25) ∗ j(ay + by)+ k(az + bz) A =e(at ibt)+ i(ax ibx)+ − − (29) j(ay iby)+ k(az iby) The difference of two fourvectors is defined − − similarly: Using relations (21) and (22), the fourvec- tor product is given by: A B =e(at bt)+ i(ax bx)+ − − − (26) j(ay by)+ k(az bz). A B =e(atbt + axbx + ayby + azbz)+ − − ∗∗ i (atbx axbt + aybz azby)+ The conjugate of a fourvector changes the − − signs of the vector part: j (atby axbz aybt + azbx)+ − − k(atbz + axby aybx azbt). − − A = eat iax jay kaz (27) (30) − − − From this definition it is obvious that Using the notation of three-dimensional the result of summing a fourvector with we obtain a shorthand for the its conjugate is a fourvector with only product. Regarding i, j, k as unit vectors in the scalar component different from zero. a Cartesian coordinate system, we interpret Divided by two, isolates the scalar compo- the fourvector A as comprising the scalar a nent of a fourvector and serves to define and the vector part a = i ax + j ay + k az. the operator named the anticommutator Then we write it in the simplified form A = (similar to the scalar Hamilton’s operator (a, a). With this notation, the product (30) S): (A + A)/2 = SA. Similarly, the result is expressed in the compact form: of subtracting the conjugate of a fourvector from itself is a pure fourvector (that is, one A B =(a b + a b, a b a b + a b) (31) ∗∗ · − × 10 The following properties for the product scalar component has terms with the are easily established: sign changed, but appears a non-zero term in the vector part of the quaternion. This has been a source of difficulty to 1. If the scalar terms of both argument apply Hamilton quaternions in Physics, fourvectors of the product are zero then which is overcome by the fourvectors. the resulting fourvector contains the classical scalar and vector products in 5. The multiplicative inverse of a fourvector its respective components. is simply the same fourvector divided by its norm.

2. The product is non-commutative. So, in 6. Properties of the product and conju- general, there exist P and Q such that gates: P**Q = Q**P. P Q = Q P (34) 6 ∗∗ ∗∗ 3. Fourvector multiplication is non- P (Q R)= R (Q P) (35) associative so, in general, P**(Q**R) ∗∗ ∗∗ ∗∗ ∗∗ = (P**Q)**R P (P Q)= P (P Q) 6 ∗∗ ∗∗ ∗∗ ∗∗ (36) Note that this is different from the = P Q Hamilton quaternions and the so-called | |∗ (P Q) P =(P Q) P Clifford Algebras, see for example [1]. ∗∗ ∗∗ ∗∗ ∗∗ (37) = P Q 4. The product of a fourvector by itself pro- | |∗ duces a result different from zero only in With an operator notation: The prod- the first or “scalar” component, which is uct of two fourvectors is equal to the identified as the norm of the fourvector. conjugate of the same product in reverse In this sense this constitutes the classical order: A**B = Conjugate[B**A] of vector calculus:

A A =(a2 +a2 +a2 +a2, 0, 0, 0) (32) ∗∗ t x y z 7. For the case that “r” is a rotor (a fourvector with r = 1) then: Note that this expression is substantially | | different with respect to the Hamilton r (r Q)= Q = r (r Q) quaternions, in which the square of a ∗∗ ∗∗ ∗∗ ∗∗ quaternion is given by (r Q) r = Q =(r Q) r ∗∗ ∗∗ ∗∗ ∗∗ 2 2 ((((Q r) r) r) r)= Q A =(a v v, 2 atv), (33) t − · ∗∗ ∗∗ ∗∗ ∗∗ (otherwise, if r is not equal to 1, the | | where v represents the three vector products of this numeral are equal to Q terms of the quaternion. Not only the or Q multiplied by r .) | | 11 8. The fourvectors do not contain the A**B and with the vector part equal to complex numbers, as is usually demon- zero. strated for the Hamilton quaternions. The product of the fourvectors: (a, b, 0, 0) and (c, d, 0, 0) is 11. Product is left distributive over sum: (ac+bd, ad-bc, 0, 0); also, the product a**(b + c) = a**b + a**c of the fourvectors (a, i b, 0, 0) and (c, i d, 0, 0) is (ac-bd, i ad-bc, 0, 0), whereas 12. Product is right distributive over sum: the product as complex numbers should (a + b)**c = a**c + b**c be: (ac-bd, i ad+bc, 0, 0) 13. The product of three “pure” fourvectors (defined as those whose scalar compo- 9. Given the fourvectors A and B, the nent is zero) can be expressed with the commutator: following vector products:

1 a (b c)= [A, B]= (A B B A) (38) ∗∗ ∗∗ 2 ∗∗ − ∗∗ (a (b c), a (b c) a (b c) ) · × × × − ∗ · = (0, ab ab + a b) (39) − × Where “ ” and “ ” are the standard · × gives a fourvector with zero scalar and vector dot and cross products and “*” with the vector part equal to the vec- represents the product of the scalar tor part of the fourvector product A**B. (b c) by the vector a. The scalar com- · ponent of the result can be recognized For the curious, this commutator satis- as the volume of the parallelepiped fies the properties of antisymmetry and having edges a, b and c. Consequently, linearity. The Jacobi identity is satisfied if the three vectors a, b and c are in only for pure fourvectors. the same plane (or parallel to the same plane) then the scalar component of the 10. Given two fourvectors, A and B, the resulting fourvector product is zero. anticommutator:

14. The following identity is also satisfied: 1 = (A B + B A) (40) (a**b) ** (a**b)=(a**a) ** (b**b) 2 ∗∗ ∗∗ =(a b + a b, 0, 0, 0) (41) · 3.1 Product with matrices gives a fourvector with the scalar equal Given two fourvectors, p and q: to the scalar of the fourvector product p=(p0,p1,p2,p3),

12 q=(q0,q1,q2,q3), their product can be obtained as 2 2 2 2 2 2 2 2 (a0 + a1 + a2 + a3)(b0 + b1 + b2 + b3)= R=p**q 2 The same product can be obtained multiply- (a0 b0 + a1 b1 + a2 b2 + a3 b3) + 2 ing the following matrix P by the (four)vector (a0 b1 a1 b0 + a2 b3 a3 b2) + − − 2 q: (a0 b2 a1 b3 a2 b0 + a3 b1) + − − 2 (a0 b3 + a1 b2 a2 b1 a3 b0) p0 p1 p2 p3 q0 − − (45)  p1 p0 p3 p2   q1  R = − − p2 p3 p0 p1 q2  − −     p3 p2 p1 p0   q3  3.3 Identity fourvector  − −    The P matrix has the property The identity fourvector, let us denote with 1, P PT = PT P = I, where T represents has the scalar part equal to 1 and the vector · · the transpose and I is the identity matrix. part equal to zero: 1 = (1, 0, 0, 0). (More precisely the diagonal elements have It has the following properties, where “p” is 2 2 2 2 the form: p0 +p1 +p2 +p3 , which are equal any fourvector: to 1 only if p is a unit fourvector; else, in 1**p = p the diagonal is obtained the norm of the p fourvector). p**1 = p As you can see, this is the left identity. It is possible to find the right identity of a fourvec- 3.2 The norm tor but it is a little more complex. See the The norm of a fourvector is defined by section “Right factor of a fourvector”.

2 2 2 2 (at, ax, ay, az) = a + a + a + a (42) | | t x y z 3.4 Multiplicative inverse It can be computed as the scalar component The multiplicative inverse or simply inverse −1 of the product of the fourvector by itself. of a fourvector P is denoted by P . The norm satisfies the properties The inverse of a fourvector P is the same A = A (43) fourvector divided by its norm: | | | | − P 1 = P/ P (46) | | P Q = Q P = P Q (44) The inverse operation satisfies the proper- | ∗∗ | | ∗∗ | | |∗| | ties: The last property allows to conclude the P**P−1=1 following form of the four-squares theorem: P−1**P=1

13 (P−1)−1 = P a = (cos(α), sin(α), 0, 0) (P ** Q)−1 = P−1 ** Q−1 b = (cos(β), sin(β), 0, 0) − P 1 =(P)−1 Commutativity of products including Its product is a**b = (cos(β α), sin(β α), 0, 0) inverses: − − P ** (P−1 ** Q) = P−1 ** (P ** Q) so, if a = b, then the resulting fourvector (P−1 ** Q) ** P = (P ** Q) ** P−1 is the identity fourvector. − Q = P**(P−1 ** Q) = P 1**(P ** Q) − −1 The inverse of a unit fourvector is the same Q = (P 1 ** Q)**P = (P ** Q)**P unit fourvector. This is because the product of the fourvector by itself produces the iden- tity fourvector, or the norm, equal to 1, in 3.5 Scalar multiplication the scalar component. Scalar multiplication If c is a scalar, or a scalar fourvector, and q=(a, v) a fourvector, 3.7 Fourvector division then c q = (c, 0) q = (c, 0) (a, v) = ∗ ∗ ∗ ∗ The fourvector division is performed by (ca + 0 v, cv a0 + 0 v) · − × multiplying the “numerator fourvector”, P, Simplifying: by the “denominator fourvector”, Q, divided c (a, v)=(ca, c v) by its norm (or rather multiplying P by the inverse of Q):

3.6 Unit fourvector − P Q/ Q = P Q 1 A unit fourvector has the norm equal to 1. ∗∗ | | ∗∗ It is obtained dividing the original fourvector If P and Q are parallel “vectors” (pure by its magnitude or absolute value, that is fourvectors or with scalar part equal to zero), the square root of the norm. The product of then the division produces, in the scalar part, two unit fourvectors is a unit fourvector. A the proportion between both vectors. For ex- unit fourvector can be represented with the ample: use of trigonometric functions P=(1, 2, 3, 4) Q=(3, 6, 9, 12) ˆw = ( cos(α) ˆu sin(α)) P Q/ Q = (1/3, 0, 0, 0) ± ± ∗∗ | | where ˆu is in general a 3D vector of unit 3.8 Right factor of a fourvector length. Let us try to solve the following equation for The product of two unit fourvectors: “X” Assume the unit fourvectors a and b: A == B ** X

14 Let us assume that A and B are constant Y is obtained with the expression fourvectors and X is an unknown fourvector − Y = Hprod[A, B 1] (48) A=(a0,a1,a2,a3) B=(b0,b1,b2,b3) Where “Hprod” represents the product for X=(x0,x1,x2,x3) the classical Hamilton quaternions or Grass- man product. Where the components can be integer, ra- tional, real or complex. For example, for the same fourvectors A and B from the previous example, let us The solution for X can be obtained with apply the equation (48) and find the expression 4 1 39 29 19 11 11 21 −1 Y =( i , i , + i , + i ) X = B A (47) 25 − 50 50 − 25 − 25 50 − 50 25 ∗∗ For example, let us try to determine what Replacing this solution into the product the value of X should be in order to satisfy Y ** B it can be verified that it reproduces the equation A == B ** X, when the A fourvector. Note that the left and right factors of some fourvector such as B are different, although A=(7,1,-3,5) both factors have the same norm and satisfy the equality: B=(1,3+i 5,2,-1) − − X Y 1 == X 1 Y According to equation (47) the solution is ∗∗ ∗∗ Both products, B ** X and Y ** B are X =( −3 i 2 , 9 i 4 , 12 i 53 , 9 i 21 ) 10 − 5 10 − 5 25 − 50 25 − 50 equal to A. Replacing this solution into the product Formula (48) can be used to determine the B ** X it can be verified that it reproduces single rotation fourvector that produces the the A fourvector. same effect as a rotor. However, the results obtained are, in general, more complex than a classical rotor. Nevertheless, if we need 3.9 Left factor of a fourvector the fourvector L, which produces the same rotation of the fourvector p as the rotor In a similar form, let us try to solve the fol- fourvector r, that is: lowing equation, where the unknown “Y” is now a left factor of the constant fourvector B: L p = Rotate[p, r]= r (p r) ∗∗ ∗∗ ∗∗ A == Y ** B applying equation (48) we solve for L:

15 L = Hprod[(p r) r], p−1] fourvectors ∗∗ ∗∗ From here, the four equations, obtained Where “Hprod” is the Grassman product. equating the four components, are:

1+q02- q1+ q12+ q2+ q22+ q32==0, -q0+q3==0, 3.10 Solution of quadratic q0+q3==0, fourvector polynomials -1-q1-q2==0.

There can be an infinite number of solutions This system of equations has two solutions for a quadratic fourvector polynomial. Con- for the four components of the fourvector: sider the quadratic equation

q2 == 1.

q1 = (0, 1 i/√2, i/√2, 0) (49) Then, the fourvector q = (cos(x), − − q2 = (0, 1+ i/√2, i/√2, 0) (50) sin(x),0,0), where x is any is − a collection of solutions for this equation because the norm of q is 1:

Replacing q1 (or q2) by its value, in the q ** q == 1 following expression, which is the left hand side of the given equation, q**q+q**j, the So the above choice for q satisfies the value returned is: (-1,0,0,1), equation q2 == 1 for all real values of x. Which is the value of the right hand side. When there is a solution of a quadratic equation, it can be computed as in the For a comparable quadratic equation, but following. now affecting the “j” fourvector to the left Assume a quadratic polynomial of the form: of “q” instead of to the right,

q**q+q**j==k q**q + j**q==k

where: The two solutions are:

q=(q0,q1,q2,q3) is an unknown fourvector q1 = (0, i/√2, 1/2( 2+ i√2), 0) (51) − − and q2 = (0, i/√2, 1/2( 2 i√2), 0) (52) − − j=(0,-1,1,0) and k=(-1,0,0,1) are constant Which, replacing in q ** q + j ** q

16 produce the value: (-1,0,0,1) foundation of modern inertial guidance sys- tems in the aerospace industry for the orien- Note that the fourvectors form a division tation or “attitude” of satellites and aircrafts. algebra since they have a (left) multiplicative This task is to be shown in the following that identity element 1=0 and every non-zero should be carried out by fourvectors. element a has a multiplicative6 inverse (i.e. Many graphics applications that need to an element x with a x = x a = 1). carry out or interpolate the rotations of objects in computer animations have also used quaternions because they avoid the difficulties incurred when Euler angles are 4 Fourvector rotation used. The form to replace by fourvectors is not performed in the present paper, although Mathematically, a rotation is a linear trans- it should be perfectly possible. formation that leaves the norm invariant. These are called orthogonal transformations. The use of matrices is neither intuitive for the localization of the axis of rotation nor There are several methods to represent efficient for computation. But one of the rotation, besides quaternions, including most important disadvantages is the asso- Euler angles, orthonormal matrices, Pauli ciativity of both the matrix and Hamilton’s spin matrices, Cayley-Klein parameters, and quaternion products where, for example, extended Rodrigues parameters. A (B C) is equal to (A B) C. In fact it is rather· · well known that the· composition· of From the several ways to represent the rotations, when either matrices or Hamilton attitude of a rigid body one of the most quaternions are used, is associative. This popular is a set of three Euler angles. means that these mathematical tools produce Some sets of Euler angles are so widely the same result no matter what the grouping used that they have names, such as the of a sequence of two or more rotations. This roll, pitch, and yaw of an airplane. The poses a serious technical problem to the main disadvantages of Euler angles are engineers who need to distinguish between that certain functions of Euler angles have two sequences of the same rotations. To ambiguity or singularity for certain angles. illustrate with an example, let us assume This produces, for example, the so-called that you are piloting an airplane with a local “gimbal lock”. Also, they are less accurate frame of reference whose origin is attached than unit quaternions when used to integrate at the center of the airplane. Assume that, incremental changes in attitude over time [7]. at a certain instant, the “x” axis, which is pointing toward the front of the plane, is The handling of rotations by means of horizontal according to an observer in the quaternions has constituted the technical earth, let us say directed toward the North

17 pole, the “y” axis points to the right wing, the rotations in a four-dimensional space may that is pointing to the East, and the “z” axis be effected by means of a pair of quaternions points toward the earth’s center. If under applied, one as a prefactor and the other as these conditions you maneuver to produce a a postfactor, to the quaternion u whose com- 90◦ roll (a quarter circle clockwise rotation ponents are the four coordinates of a space- about the local x axis) followed by a 90◦ point, say v = a u b”. This phrase applies yaw (rotation “to the right” around the directly to fourvectors if “quaternion(s)” is local z axis) your plane should be falling replaced by “fourvector(s)”. perpendicularly to the earth; but if you In the case of pure rotation, a and b must interchange the order of these rotations, be either unit-fourvectors or the norm of their that is first the yaw and then the roll, product must be 1: a b = 1. this should put your plane in a horizontal It follows, from the| |∗| rule: | heading toward the East, although with the a b = a b , (54) right wing pointing toward the earth’s center. | ∗∗ | | |∗| | the multiplication of the fourvector being ro- To rotate a vector u around an Euler vector tated by unitary fourvectors a and b, effects n through an angle θ, one has to apply the an orthogonal transformation. following equation (see Fig. 1 and references This form can be simplified so instead of two [24], [28]): different unitary fourvectors is selected only one, let us name it r. v = ucosθ + n(n u)(1 cosθ)+(n u)sinθ · − × A possible fourvector r that produces the (53) rotation of any fourvector V about a certain axis “n”, through an angle θ, has the form ([24], [5], [12] [15], [19], [9]): θ θ θ θ r =(cos( ), n sin( ), n sin( ), n sin( )) 2 x 2 y 2 z 2 (55) or simply θ θ r =(cos( ), n sin( )) (56) 2 2 The rotation is carried out with the following product: Figure 1: Graphic of a rotation V’ = r (V r) (57) ∗∗ ∗∗ According to Silberstein, [26]: “It has been The rotation operand needs to be conjugated. remarked by Cayley, as early as in 1854, that This is different with respect to the rotation

18 using Hamilton quaternions [26], where the qi qi = 1, and cause rotations about the inverse or the conjugate of the second rotor x axis:∗∗ is needed. Fig. 2 shows the vector V rotated around the unit vector k, through an angle θ, with q1 =( cos(α/2), sin(α/2), 0, 0) (60) rotor r. The rotation operator is linear. It ± ± q2 =( cosh(α/2), i sinh(α/2), 0, 0) (61) ± ± γ +1 γ 1 q3 =( , i − , 0, 0) (62) r 2 r 2

where γ is the Lorentz contraction factor.

The rotation fourvector q3 was obtained by transforming the following fourvector ( γ, i β γ, 0, 0) ± ± in such a way that it produces a rotation of half the , as with q2. Let us multiply any one of the previous fourvectors by the following one that repre- sents a differential of interval: ds = (c dt, i dx, i dy, i dz)

Figure 2: Example rotation The products are of the form:

Rotate[ds, q ]= q (ds q ) (63) can be proved that the rotation of either the i i ∗∗ ∗∗ i product or the sum of two fourvectors s and t with the rotor r, is the same as, respectively, where qi is anyone of the above list of unit the product or the sum of the rotations of fourvectors. each fourvector: Then, the norm of the result is the square of the differential of interval: 2 2 2 2 2 2 Rotate[s t, r] Rotate[s, r] Rotate[t, r] ds = c dt - dx - dy - dz ∗∗ ≡ ∗∗ (58) This means that any one of these trans- Rotate[s + t, r] Rotate[s, r]+ Rotate[t, r] formations (rotations) preserves the interval ≡ (59) invariant. But first the rotation of a real fourvector Let us define the following example rotation a =(at, ax, ay, az) with the rotor q1 produces fourvectors, or rotors, whose norm is the unit: the classical formulas for rotation of a vector

19 about the x axis: (at, ax cos(2α) ay sin(2α), Rotate[a, q1]=(at, ax, − (67) ay cos(2α)+ ax sin(2α), az) ay cos(α) az sin(α), (64) − az cos(α)+ ay sin(α)) 4.1 Composition of rotations The opposite rotation can be done using The rotation through an angle α followed as rotor the inverse of q1, which changes by another rotation through an angle β is the sign of the angle α. For example, if we equivalent to a single rotation through an rotate the last result with the conjugate or angle α + β: the inverse of q1 then the original fourvector a is recovered. Assume, for example that the rotations are produced by application of the following ro- The rotation of the differential of interval tors: with q2 gives the complex fourvector: roth1=(cosh(α/2), i sinh(α/2), 0, 0) (68) Rotate[ds, q2]=(c dt, i dx, roth2=(cosh(β/2), i sinh(β/2), 0, 0) (69) dz sinh(α)+ i dy cosh(α), (65) dy sinh(α)+ i dz cosh(α)) − Let us apply these rotors over the fourvec- tor M=(a,b,c,d) with the operation: The rotations with q3 produce Lorentz boosts. Let us apply to ds: M1 = Rotate[M, roth1] (70) i Rotate[ds, q3]=(cdt, dx, followed by the following rotation: i γ (dy i β dz), (66) − i γ (dz + i β dy)) M2 = Rotate[M1, roth2] (71) If the fourvector to be rotated is the We obtain the following result: previous a and the rotor fourvector is (a,ib,iccosh(α + β)+ d sinh(α + β), r = (cos(α/2), 0, 0, sin(α/2)) (72) idcosh(α + β) c sinh(α + β)) − Then the double rotation: Which is identical to the rotation produced by directly applying over M the rotor: r (r (a r) r) ∗∗ ∗∗ ∗∗ ∗∗ α + β α + β roth = cosh( ), i sinh( ), 0, 0 Is equal to a single rotation through the 2 2 double angle 2α. The result is: (73)

20 4.2 Reflections counterclockwise, so we could include conju- gations of the vector a. Also, the conjugation Let us show how the reflection in a plane of the vector x can be canceled with the neg- with unit normal a can be done (see next ative sign, so the final equation is: figure) x’ = a**(x**a).

As was suggested at the end of the previ- ous section, the rotations can be composed. So that, by multiple application of this re- flection formula it is possible to compute the vector x’ that describes the direction of emer- gence of a ray of light initially propagating with direction x and reflecting off a sequence of plane surfaces with unit normals a1; a2; Figure 3: Reflection ...;an: x’ =an**(...(a2**((a1**(x**a1))**a2)...an)

the normal vector a, which defines the direction of the plane, generates a reflection of the vector x if we rotate x around the vector a through an angle of π radians and then change every sign of x (otherwise we end up with an arrow pointing at the same point as x). This rotation is done by fixing at π radians the angle α of rotation of a rotor of the form q1 of section 4, i.e.

q[Cos[α/2],a Sin[α/2]].

Consequently, the cosine term disappears Figure 4: Reflections and the sine term becomes equal to 1, with which we are left with the vector a alone, as rotor. Hestenes [15] shows other applications The vector x, after reflection, is: for rotations and reflections, for example x’ = a (x a) in crystals. Note that in his “geometric − ∗∗ ∗∗ To simplify this expression, let us note that algebra” Hestenes needs the negative sign the rotation through π radians clockwise is that we don’t need. the same as the rotation through π radians

21 5 Conclusions

The mathematical structure of quaternions has always been considered as more appro- References priate than the simple vectors to represent the real physical variables. Nevertheless, [1] G. Arag´on, J.L. Arag´on and M.A. the quaternions were dismissed for the Rodr´ıguez. Clifford algebras and geomet- difficulties and complications produced by ric algebra. Advances in Applied Clifford their quaternion product. With the new Algebras 7 No. 2, 91-102. 1997. product, suggested in the present paper for [2] John C. Baez. The . fourvectors, all those difficulties disappear. At Cornell University arXiv: Of course there must be a delicate balance math.RA/0105155 v4. May 16, 2001. between the correct mathematical tools and the real physical objects being studied and [3] R. G. Beil and K. L. Ketner. Peirce, handled. One has to also be aware that Clifford, and Quantum Theory. Inter- mathematics clearly affects the ontology of national Journal of Theoretical Physics, physics [11]. Vol. 42, No. 9. September 2003.

The fourvector algebra proposed in the [4] Gaston Casanova. L’Alg`ebre de Clifford present paper seems to be the correct et ses applications. Advances in Ap- mathematical tool to study the fundamental plied Clifford Algebras. Universidad Na- physical variables and their describing equa- cional Aut´onoma de M´exico. Vol 12. tions. May, 2002. [5] Erik Dam, Martin Koch and Martin Lill- This new mathematical structure is an holm. Quaternions, Interpolation and extension of the classical vectors. Its sim- Animation. Technical Report DIKU- plicity contributes to the possibility of more TR-98/5. Department of Computer Sci- extended and fruitful use in all branches of ence. University of Copenhagen. Den- science. mark. July 17, 1998.

The applications of the Hamilton quater- [6] Stefano De Leo. Quaternions and special nions for rotations in three dimensions relativity. American Institute of Physics. have been the more extended in current J. Math. Phys. 37 (6). June 1996. Physics. Such uses, as well as reflections, are still permitted by the fourvectors. The [7] James Diebel. Representing Attitude: applications to Lorentz boosts had problems Euler Angles, Unit Quaternions, and with the old quaternions, so this area opens Rotation Vectors. Stanford University. up for the scientists. 20 October 2006.

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