Quaternions in Eletrodynamis André Waser * Issued: 9.07.000 Last modifiation: 04.07.001 At the advent of MAXWELL s eletrodynamis the quaternion notation was often used, but today this is replaed in all text books with the vetor notation. If the founders of eletrodynami would have used the quaternion notation onsequently with their most unique property namely the fourdimensionality they would have disovered relativity muh before VOIGT, LORENTZ and EINSTEIN. A short desription of eletrodynami with quaternions is given. As a result a new set of MAXWELL s equations is proposed, whih transform in today s equations when the LORENTZ gauge is applied. In addition an appliation of this new quaternion notation to quantum mehanis and other disiplines is presented. Introdution One of the most emotional disputes in the late nineteenth-entury eletrodynamis was about the mathematial notation to use with eletrodynami equations [1]. [1]. The today s vetor notation was not fully developed at that time and many physiist one of them was James Clerk MAXWELL are onvined to use the quaternion notation. The quaternion was invented in 1843 by Sir William Rowan HAMILTON [5]. Peter Guthrie TAIT [11] was the most outstanding promoter of quaternions. On the other side Oliver HEAVISIDE [6] and Josiah Willard GIBBS [1] both deided independently that they ould use a part of the quaternion system better than the entire system, why they proeeded further with that, what is today alled the vetor notation. Generally the vetor notation used in pre-einstein eletrodynamis uses threedimensional vetors. The quaternion on the other hand is a four-dimensional number. To make the quaternion usable for the three-dimensional eletrodynami of MAXWELL, HAMILTON and TAIT indiated the salar part by prefixing an S to the quaternion and the vetor part by prefixing a V. This notation was also used by MAXWELL in his Treatise [8], where he published twenty quaternion equation with this notation. But with applying this prefixes the whole benefit of quaternions is not used. Therefore MAXWELL has never done alulations with quaternions but only presented the final equations in a quaternion form. It was then merely a alulation with vetors and salars as today pratied. * André Waser, Birhli 35, CH-8840 Einsiedeln; andre.waser@aw-verlag.h opyright (000) by AW-Verlag; www.aw-verlag.h Page 1
HAMILTON s Quaternions A general quaternion has both a salar (real) and a vetor (imaginary) part. In the example below a is the salar part and ib + j + kd is the vetor part. Q = a + bi + j + dk (1) Where a, b,, and d are real numbers and i, j, k are so-alled HAMILTON ian unit vetors with the magnitude of -1. They satisfy the equations and i = j = k = ijk = 1 () ij = k jk = i ki = j ij = ji jk = kj ki = ik A nie explanation about the rotation apabilities of the HAMILTON ian units in a three-dimensional ARGAND diagram was published by GOUGH [3]. A quaternion is a hyperomplex number. The quaternion radii (or magnitude) in four-dimensional spae is defined similar as for ordinary omplex numbers as: Q x + x + x + x (3) 0 1 3 as also shown by Walker [1]. By introduing a onjugate quaternion number the quaternion magnitude is also Q* = a - bi - j - dk (4) * Q QQ = x + x + x + x. (5) 0 1 3 The four-dimensional quaternion is very suitable to represent an event in fourdimensional vetor spae : 4 = ( x + x + x + x ) x + x i+ x j+ x k X i j k X. (6) 0 1 3 0 1 3 Expansion of Quaternions with Imaginary Numbers An expansion of quaternions to eight-dimensional numbers an be ahieved when the real variables a, b, and d now are replaed with imaginary numbers. A omplex quaternion is then: Q = (a + ia) + (b + ib)i + ( + ic)j + (d + id)k (7) Beause the HAMILTON ian unit vetors i, j, k are still valid beside the imaginary (GAUSS ian) unit i this new introdued quaternion is different from the known otionions (known form LIE algebra). A omplex quaternion is a number whih represents two interseting four-dimensional spaes. The omplex quaternion Q an be splitted into two sub entities: ( X0 ix0) ( x1 ix1) i ( x ix) j ( x3 ix3) k ( ix x i x j x k) ( X ix i ix j ix k ) Q = + + + + + + + = + + + + + + + = Q+ Q 0 1 3 0 1 3 The first term (upper line in above equation) is very suitable for a ompat notation of eletrodynamis if the seond term (lower line) is always set to zero as some investigations have shown. With this the omplex quaternion redues again to a fourdimensional number with the speialty, that no real number exists anymore. It will be shown, that this omplex quaternion is also useful for other appliations. The unit vetors in three-dimensional spae are printed in bold letters with i, j, k, the HAMILTON ian units i, j, k in itali letters and the imaginary unit i in normal letters without serifs. Page opyright (000) by AW-Verlag; www.aw-verlag.h
Definitions The following definitions applies always to the omplex quaternion Q. Definition 1: A omplex quaternion is splitted into two parts, whereas the seond part is always set to zero aording to: ( X0 ix0) ( x1 ix1) i ( x ix) j ( x3 ix3) k ( ix x i x j x k 0 1 3 ) ( X ix i ix j ix k 0 1 3 ) ( ix x i x j x k) Q = + + + + + + + = + + + + + + + + + + =Q 0 1 3 Definition : The quaternion Nabla operator is: i = + i+ j+ k X t x x x (9) 1 3 Definition 3: The quaternion LAPLACE (or D ALEMBERT ) operator is: 1 1 = = = i (10) t x x x t Definition 4: The total derivative of time is: 1 3 (8) d = V (11) The salar part is analogue to the well-known equation d = + v t Definition 5: The amount (radii) of a quaternion is: Q x + x + x + x (1) 0 1 3 Quaternions in four-dimensional Spae In 191 the first attempt to use quaternions for relativity was published by SILVER- STEIN [10]. He used two quaternion operator to formulate relativity. But from the presentation above about the flat four-dimensional spae an analogue representation of the urved four-dimensional MINKOWSKI spae an be given with one single quaternion with: 4 = ( it + x + x + x ) it + x i+ x j+ x k X i j k X. (13) The magnitude (radii) aording to (3) leads to: 1 3 1 3 X = = xx + xx + xx t X 1 1 3 3 (14) This magnitude is different to the definition 5, but is aording to the speial theory of relativity invariant to transformations between inertial systems. This applies also for the differential dx. A division with i results in another invariant: 1 1 v dx = ( dx + dx + dx 1 3) = 1 = dτ i This is the time dilatation known form relativity. For the differential of an event we an further write: 1 3 (15) dx = i + idx + jdx + kdx. (16) opyright (000) by AW-Verlag; www.aw-verlag.h Page 3
so that we get the known four-dimensional veloity U: dx i iv + j v + k v U = = + dτ v v 1 1 1 3 The derivation of the time dilatation follows from the magnitude of an event vetor, whereas the derivation of the four-dimensional veloity is applied to the vetor itself. This is not onsistent as Yong-Gwan YI [15] already has found. The onept of time dilatation and of the four-dimensional veloity (17) above do exlude eah other. Without the onept of urved spae the four-dimensional veloity an also be formulated in an other way: (17) dx V = = i + iv + j v + k v (18) 1 3 This effetive (absolute) veloity of a body in flat spae should now be used in the further explanations. Quaternions in Eletrodynamis Analogue to the veloity we define the quaternion potentials with: The quaternion urrent is then: So we get with the potential: A ϕ A i + ia + j A + k A (19) 1 3 J iρ+ i J + j J + k J (0) 1 3 1 A A A 1 3 i A1 i A A 3 = ϕ + + + + + ϕ + t x1 x x3 t x1 x x3 i i A i ϕ A1 A 3 i A3 i ϕ A A 1 + + + j+ + + k t x x x t x x x 3 1 3 1 Then with the substitutions we get the ompat formula A E = ϕ+, B = A (1)a, b t 1 ϕ -i -i -i A = + i A + E 1+ B1 i+ E+ B j + E3 + B3 k t () Aording to definition 1 the real salar term is zero together with the imaginary vetor part. From this results the known LORENTZ ondition (gauge): with 1 ϕ 0 + i A = (3) t B = A and E = 0 With definition 1 the LORENTZ ondition is stritly valid for E = 0, but otherwise it an be freely hosen. Aording to the definition of the potentials (1) it is possible to hoose an arbitrary definition beause of the divergene of the vetor potential A, but for E = 0 the LORENTZ ondition is ultimate. Then for the urrent density we an derive: Page 4 opyright (000) by AW-Verlag; www.aw-verlag.h
J J J J 1 3 i J J J 1 3 = ρ + + + + + i ρ + i t x1 x x3 t x1 x x3 i J ρ J1 J 3 i J3 ρ J J 1 + + i + j+ + i + k t x x x t x x x 3 1 3 1 Here we an also introdue the substitutions: J = ρ+ whereas follows the ompat formula opyright (000) by AW-Verlag; www.aw-verlag.h Page 5 (4) G, C = J (5)a, b t ρ -i -i -i J = + i J + G1+ C1 i+ G + C j+ G3 + C3 k t (6) Again aording to definition 1 the real salar and the imaginary vetor part is set to zero. From this follows the known ontinuity equation (onservation of eletrial harge): with ρ + i J = 0 (7) t C = J and G = 0 New for this formulation is, that the ontinuity ondition similar to the LORENTZ ondition is only stritly valid for G = 0, but elsewhere it an be hosen freely beause of C = J. The LORENTZ Fore If we define the quaternion momentum as i P W+ ip + j p + k p (8) 1 3 then we get for the quaternion momentum of an eletrial harge q in a potential field: P = qa (9) From this follows immediately the total energy W q of the harge q q Wq as well as the three-dimensional momentum as The, if we define the four-vetor fore with q = ϕ q (30) p = qa (31) i F P+ i F + jf + kf (3) 1 3 then we get for the fore of the potential field on the free harge q: From this we get for the vetor part (writing the prefix V.): da F = q = q q ( V ) A (33) 1 ϕ v Fq = V. Pq = q ( E+ v B) v + ia + i B E t (34) Aording to definition 1 the first and seond term orresponds to a real physial fore. The first term desribes the known LORENTZ fore on a free harge. Fq = q( E+ v B ) (35)
The seond term orresponds the LORENTZ ondition (3) and is not neessarily zero. The third term in (34) is aording to definition 1 equal to zero. From this we get the know relation: The salar part of (33) is = v B E (36) i 1 ϕ Pq = S. Pq = q vb i + ve i i + ia t (37) Again with definition 1 the first and seond term is physially existent, whereas we find for the third term: vb= i 0 (38) If the path of a free moving harge follows exatly the field lines of B there will be no fore exerted on the harge. The seond term depends again on the LORENTZ ondition (3). The first term represents the energy intake or release (power) P q = dw q / of the harge moving in an eletri field: The Field Transformation dwq P = = qve i (39) q Equation (36) an be inserted in (35) to write the LORENTZ equation (35) without the magneti indution field B: If the power P q = 0 then it is v E = 0 and from (40) follows: v Fq = q E+ ( v E ) (40) v F = q 1 q E (41) This formula has also been published by MEYL [9], but it is, as shown above, only valid for the ondition P q = 0 (uniform motion). MAXWELL s Equations If the quaternion LAPLACE operator is applied to the potentials, we get with (1) and some algebra for the salar part (using the prefix S.): and for the vetor part i 1 ϕ S. A = ib+ + + ie ia t t (4) 1 E 1 ϕ i B V. A = B + + t ia t E t (43) Similar equations has been presented by HONIG [7]. From the imaginary part we get with definition 1 diretly AMPERE s law: B + E = 0 (44) t Page 6 opyright (000) by AW-Verlag; www.aw-verlag.h
Now we hose A =µ J (45) then with the imaginary salar part follows with (4) the expanded COULOMB law 1 ϕ ρ ie+ + = t ia t (46) ε When foring the Lorentz ondition (3) this transforms into GAUSS s law: ε ie= id =ρ (47) Doing the same with equation (43) delivers the expanded FARADAY law 1 E 1 ϕ B + + =µ t ia t J (48) what again transforms into a known MAXWELL equation when foring the Lorentz ondition (3): And finally by setting the real salar part to zero we get 1 E B =µ J (49) t i B = 0 (50) The expanded MAXWELL equations without foring the Lorentz ondition are (valid for linear media): 1 E 1 ϕ B =µ J + t ia t (48) B E + = 0 (44) t ρ 1 ϕ ie= + t ia ε t (46) i B = 0 (50) With the lassial and mostly used LORENTZ ondition the expanded MAXWELL equations presented above transforms into the ordinary known set of MAXWELL s equations. Now we take a short look at the quaternion urrent density. With the Quaternion veloity(18) in flat spae this an be expanded to: J ρ V = iρ+ i J + j J + k J (51) 1 3 This means, a harge density ρ is moving through four dimensional spae. This orresponds exatly to the onept of an eletri urrent. opyright (000) by AW-Verlag; www.aw-verlag.h Page 7
Kinematis analogue to Eletrodynamis The momentum of a veloity field U on a mass m is in quaternion notation P = mu (5) m The analogue writing with (9) is lear to see. It is possible to write the Equations of Eletrodynamis similar to those of fluid mehanis beause the eletri potential field follows the same momentum laws as the veloity fields with masses. The eletri four potential field orresponds to the veloity field and the harge q orresponds to the mass m. In the definition of the harge momentum (9) an external field is the ause whereas normally in the definition for the mass momentum the veloity of the mass (relative to an observer) is taken. Therefore the momentum laws for mass and harge are not defined exatly in the same way. In this setion this is now hanged. The fourdimensional veloity of a mass m is then transformed into an external veloity filed: U i + iu + j U + ku = i + iu + ju + ku = i +U (53) 1 3 1 3 From this we get wit the definition of the momentum (8) the total energy of a mass m Wm as well as the three-dimensional momentum m = m (54) p = mu (55) In four-dimensional spae eah body moves along a world line, as also defined in MINKOWSKI spae. The definition of an absolute veloity U doesn t make sense in physis, but merely the definition of a relative veloity u between two bodies. This relative veloity follows form the differene of the four-dimensional veloity of two bodies: ( ) ( ) ( ) U = V W = i v w + j v w + k v w = iu + ju + ku =u (56) 1 1 3 3 1 3 It is only possible to measure the momentum with a relative veloity u, that means, at least two bodies are neessary to determine momentum. Therefore the momentum is also defined as: p = mu (57) m Interestingly the total energy of a mass is not derived from a relative movement between two masses but it is defined for a single mass. Therefore (54) is not valid anymore. For the fores between masses a similar set of formulas an be written as used for eletrodynamis, whih desribes fores between harges. The equivalent ansatz to (1) is: Fm U G = = φ and T= U (58)a,b m t with F m : Fore on a mass m [N] = [kg m / s ] G: Kinemassi fore field [m / s ] U: Three-dimensional veloity of mass m [m / s] φ: Gravity potential [m / s ] T: Rotary or osillation field [1 / s] Page 8 opyright (000) by AW-Verlag; www.aw-verlag.h
Then the kinemassi fore field an be deomposed in two known parts: G = φ and U G = G T t with G G : Gravity field [m / s ] G T : Inertia field [m / s ] Then with (3) it follows for the fore on a mass m with self veloity V: (59)a,b From this we get for the vetor part: du F = m = m m ( V ) U (60) 1 φ V Fm = V.Fm = m ( G+ V T) V + iu + i T G t (61) The first term orresponds to a gravitational LORENTZ fore Fm = m( G+ V T ) (6) The seond term is analogue to the LORENTZ ondition and the third term shows the relation: The salar part of (60) is: = V T G (63) i 1 φ Pm = S. Fm = m vg i i + is + vt i t (64) The first term orresponds to energy onsumption or release (power) P m = dw/ of a mass, whih is moving in a gravitational field: P m dw = = mvg i (65) As with the eletri fore also the gravitation fore an be summarized when (63) is inserted in (6) If the power P m = 0 then is v G = 0, and from (66) follows v Fm = m G+ ( v G ) (66) v F = m 1 m G (67) The fores on a mass m an be formulated analogue to eletrodynamis, as shown above. But if one wished to desribe fores between masses as effets of fores between harges only, it is not allowed to introdue the new potentials φ and U, whih are independent of ϕ and A. It should be shown in an other paper [1] that this ould probably be possible. opyright (000) by AW-Verlag; www.aw-verlag.h Page 9
Kinematis aording to Newton In opposite to the desription above in NEWTON s mehanis the veloity or an aeleration of a mass is always onsidered relative to an other mass. From there we have the basi equation: or in quaternion notation: F = ma (68) N dv F = m = m N ( V ) V (69) Thus we get immediately for P = iw/ + ip 1 + jp + kp 3 and V = i + iv 1 + jv + kv 3 : W = m and = m p v (70) Before we take a loser look to (69) the quaternion Nabla operator is now applied to the veloity, as previous done: i v V = i v+ v+ (71) t then we get with definition 1 for the salar part the known ontinuity equation i v = 0 (7) whih is only stritly valid for v/ t = 0 (imaginary vetor part equal zero). Then the vetor part of (69) gilt nun: v v v FN = V.F N = m + v( iv ) v ( v) i v + (73) t t Again the real part represents a physial real fore. This fore orresponds to the inertia fore of an aelerated mass, whih points in opposite diretion than the aeleration. The inertia fore (real vetor part) does orrespond with the known formula of fluid mehanis of the aeleration field: v v a = + v ( v ). (74) t Then by setting the imaginary term equal zero we get a new relation: v v v (75) t = Again the strong analogy to the equation (36) of eletrodynamis an be seen. Now, the salar part of (69) is: v v PN = S. FN = m vi( v) + i iv + i (76) t With definition 1 we an see for the real part, that v must always be perpendiular to v to guarantee that this term will always be zero. Page 10 opyright (000) by AW-Verlag; www.aw-verlag.h
Quantum mehanis Relativisti wave equation IN 1937 CONWAY [] has shown a possible notation for the relativisti wave equation, if the Hamilton ian units are used as pre- and post fators in his formulas. In this setion we derive the relativisti wave equation with the herein desribed quaternion notation. We start with the momentum law: and wit the definition of the total energy of a mass m P = mv (77) E P (78) so that with that and with definition 5 the EINSTEIN formula an be derived for µ = 1..3: 4 E = m + p µ (79) µ Until this point we have used a salar for the energy, whih for example takes a onstant value for a resting body. But quantum physis (i.e. experiments) have shown, that this is not quite orret but that energy is merely on osillating phenomenon and therefore must satisfy a wave equation. Therefore equation (79) an be seen as a stati average equation of a olletion of many energy osillators. But for a single partile the osillatory behavior of its energy an learly be observed. Equation (79) is already in a quadrati form as this is the ase also for the differential operators of a wave equation. To find the wave equation behind the energy equation, we use the established substitutions for the differentials as known in quantum mehanis: ħ E i t and p µ ħ i x µ, (80) where ħ is PLANCK s onstant divided by π. Now this differentials must be applied to a new unknown funtion. This is nothing else than the (dimensionless) wave funtion Y. This wave funtion is again a quaternion: Y= iψ +ψ i+ψ j+ψ k (81) 0 1 3 Then we get the Dira relativisti wave equation by inserting (81) in (79): 1 m t ħ Y - Y + Y = 0 (8) Partile without external potential fields DIRAC has solved the energy equation (79) E m p µ µ =± + (83) by taking the square root with the introdution of 4x4 matrixes. On the other side equation (78) offers an other possibility without taking a square root, when diretly taking the quaternion momentum. But the sign of the quaternion momentum should not hange. This an be ahieved with a modifiation of equation (78): E ±P then it is E = E =± P (84) opyright (000) by AW-Verlag; www.aw-verlag.h Page 11
The two possible sign for the energy state in (83) and (84) has motivated DIRAC to postulate the existene of anti-partiles espeially the positron. By inserting of the substitutions (80) into (84) an other DIRAC equation a be obtained: Y Y Y 1 Y ħ i+ j+ k+ my = 0, (85) x1 x x3 t On a first glane this equation differs form the original DIRAC equation beause no matrixes are used. But the HAMILTON ian units an also be written as matrixes (see appendix A). The by multipliation of (85) with this HAMILTON ian matrixes it follows the equation system: ħ ψ0 ħ ψ1 ψ ψ 3 mψ 0 0 + + = t i x1 x x3 ħ ψ 1 ψ0 ψ3 ψ mψ 1 + ħ i + = 0 t x x x 1 3, (86) ħ ψ ψ ψ ψ 3 0 1 mψ + ħ + i = 0 t x x x 1 3 ħ ψ 3 ψ ψ ψ 1 0 mψ ħ i 0 3 = t x1 x x3 This system ontains four equations for a partile without external fields as originally proposed by DIRAC. Now, in an analogue way the equation for a partile within external fields an be desribed. Partile within external potential field The momentum on a harged partile hanges with an external potential field. If the momentum of an external field on a harged partile q is defined as then it follows for its energy q P = qa (87) q E P = q A (88) q and So the total energy is now E q = qa (89) ( q ) E = ± P A (90) Again the extended DIRAC equation follows with the substitution of energy and momentum to: 1 ħ qa i qa j qa k 1 + + 3 + Y ( m+ qϕ ) Y = 0 (91) x1 x x3 t Again quaternions an be used instead of matrixes. Page 1 opyright (000) by AW-Verlag; www.aw-verlag.h
Summary With the introdution of the omplex quaternion aording to definitions 1 to 4 all basi equations of eletrodynamis an be formulated in a very ompat way. Furthermore there result some new suggestions about an extension of MAXWELL s equations, whih are dependent on the use of the LORENTZ ondition. Of ourse this result an not over that there are not satisfatory explanations available at the moment, why exatly this type of a four-dimensional quaternion notation an be applied to eletrodynamis suessfully. Beneath eletrodynamis the quaternion is also very suitable for appliation in other disiplines in physis, as for example in quantum mehanis or in kinematis. The used struture of the omplex quaternion is interesting beause it does not ontain one single reel number anymore. There is more work neessary to bring more light into the geometrial meaning of this notation and of definition 1. Aknowledgement I gratefully thank Koen J. VAN VLAENDEREN for his orretions onerning some sign errors as well as to the definition of the quaternion Laplae operator. Further findings will be published in a separate paper. Referenes [1] BORK Alfred M., Vetors Versus Quaternions The Letters of Nature, Amerian Journal of Physis 34 (1966) 0-11 [] CONWAY A. W., Quaternion Treatment of the Relativisti Wave Equation, Proeedings of the Royal Soiety London, Series A 16 (15 September 1937) 145-154 [3] EINSTEIN Albert, Zur Elektrodynamik bewegter Körper, Annalen der Physik und Chemie 17 (30. Juni 1905) 891-91 [4] GOUGH W., Quaternions and spherial harmonis, European Journal of Physis 5 (1984) 163-171 [5] HAMILTON William Rowan, On a new Speies of Imaginary Quantities onneted with a theory of Quaternions, Proeedings of the Royal Irish Aademy (13 November 1843) 44-434 [6] HEAVISIDE Oliver, On the Fores, Stresses and Fluxes of Energy in the Eletromagneti Field, Philosophial Transations of the Royal Soiety 183A (189) 43 [7] HONIG William M., Quaternioni Eletromagneti Wave Equation and a Dual Charge- Filled Spae, Lettere al Nuovo Cimento, Ser. 19 No.4 (8 Maggio 1977) 137-140 [8] MAXWELL James Clerk, A Treatise on Eletriity & Magnetism, (1893) Dover Publiations, New York ISBN 0-486-60636-8 (Vol. 1) & 0-486-60637-6 (Vol. ) [9] Meyl Konstantin, Elektromagnetishe Umweltverträglihkeit, Indel Verlag, Villingen- Shwenningen Teil 1 ISBN 3-98054-8-3 (1996) 105 [10] SILBERSTEIN L., Quaternioni Form of Relativity, Philosphial Magazine, Ser. 6 3 (191) 790-809 [11] TAIT Peter Guthrie, An elementary Treatise on Quaternions, Oxford University Press 1 st Edition (1875) [1] WALKER Marshall J., Quaternions as 4-Vetors, Amerian Journal of Physis 4 (1956) 515-5 [13] WASER André, Gravity as an effet of a variable light speed?, AW-Verlag, www.awverlag.h (in publiation) [14] WILSON E. B., Vetor Analysis of Josiah Willard Gibbs The History of a Great Mind, Charles Sribner s Sons New York (1901) [15] YI Yong-Gwan, On the Nature of Relativisti Phenomena, Apeiron 6 Nr.3-4 (July-Ot. 1999) opyright (000) by AW-Verlag; www.aw-verlag.h Page 13
Appendix A Aording to Arthur CAYLEY it is possible to represent omplex numbers as a matrix: Example: a a+ ib= b with 0 1 i = 1 0 0 1 0 1 1 0 i = ii = = = 1 1 01 0 0 1 The HAMILTON ian units of a quaternion build together with the numbers 1 and 1 a non ABEL ian group of eight order. Its first four positive elements are: i 0 i = 0 i 0 1 j = 1 0 0 i k = i 0 1 0 1 = 0 1 If we now replae the imaginary unit i with the orresponding matrix (9), then it follows: (9) (93) (94) 0-1 0 0 1 0 0 0 i = 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 k = 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 j = 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 = 0 0 1 0 0 0 0 1 (95) The square of this matrixes always returns the value 1 as requested by the definition of the HAMILTON ian unit vetors (). Page 14 opyright (000) by AW-Verlag; www.aw-verlag.h