Cuadrivectores Diego Saa

8/16/2019 Cuadrivectores Diego Saa

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Fourvector AlgebraDiego Sa´ a 1

(1) Escuela Polit́ecnica Nacional. Quito – Ecuador.e-mail: [email protected]

Abstract

The algebra of fourvectors is described. The fourvectors are more appropriate than the

Hamilton quaternions for its use in Physics and the sciences in general. The fourvectors

embrace the 3D vectors in a natural form. It is shown the excellent ability to perform

rotations with the use of fourvectors, as well as their use in relativity for producingLorentz boosts, which are understood as simple rotations.

PACS : 02.10.Vr

Key words: fourvectors, division algebra, 3D-rotations, 4D-rotations

Contents

1 Introduction 21.1 General . . . . . . . . . . . . 21.2 The Hamilton quaternions . . 51.3 The Pauli quaternions . . . . 61.4 Dirac matrices . . . . . . . . . 6

2 The fourvectors 72.1 Discussion . . . . . . . . . . . 82.2 Complex fourvectors . . . . . 9

3 Fourvector algebra 103.1 Product with matrices . . . . 123.2 The norm . . . . . . . . . . . 133.3 Identity fourvector . . . . . . 133.4 Multiplicative inverse . . . . . 13

3.5 Scalar multiplication . . . . . 14

3.6 Unit fourvector . . . . . . . . 143.7 Fourvector division . . . . . . 143.8 Right factor of a fourvector . 143.9 Left factor of a fourvector . . 153.10 Solution of quadratic fourvec-

tor polynomials . . . . . . . . 16

4 Fourvector rotation 174.1 Composition of rotations . . . 204.2 Reflections . . . . . . . . . . . 21

5 Conclusions 22

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http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1http://arxiv.org/abs/0711.3220v1


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1 Introduction

1.1 General

In this paper it is suggested the use of fourvectors with the purpose of replacing the3D vectors and the quaternions. Becausethe fourvectors contain the three dimensionalvectors and can be considered a formalizationof them.The discovery of the quaternions is attributedto the Irish mathematician William Rowan

Hamilton in 1843 and they have been used forthe study of several areas of Physics, such asmechanics, electromagnetism, rotations andrelativity [26], [20], [6], [2], [23], [13]. JamesClerk Maxwell used the quaternion calculusin his Treatise on Electricity and Magnetism,published in 1873 [22]. An extensive bibliog-raphy of more than one thousand referencesabout Quaternions in mathematical physics has been compiled by Gsponer and Hurni[14].

The modern vectors were discovered bythe Americans Gibbs and Heaviside between1888 and 1894. Their work may be consid-ered a sort of combination of quaternions andideas developed around 1840 by the GermanHermann Grassman. The notation was pri-marily borrowed from quaternions but thegeometric interpretation was borrowed fromGrassman’s system.

By the end of the nineteenth century

the mathematicians and physicists were hav-ing difficulty in applying the quaternions toPhysics.

Ryan J. Wisnesky [32] explains that “Thedifficulty was a purely pragmatic one, which

Heaviside was expressing when he wrote that

there is much more thinking to be done [to set up quaternion equations ]. In principle, mosteverything done with the new system of vec-tors could be done with quaternions, but theoperations required to make the quaternionsbehave like vectors added difficulty to usingthem and provided little benefit to the physi-cist.”

“Gibbs was acutely aware that quater-nionic methods contained the most importantpieces of his vector methods.” [32]

After a heated debate, “by 1894 the debatehad largely been settled in favor of modernvectors” [32].

Alexander MacFarlane was one of thedebaters and seems to have been one of thefew in realizing what the real problem withthe quaternions was. “MacFarlane’s attitudewas intermediate - between the position of the defenders of the Gibbs–Heaviside systemand that of the quaternionists. He supported

the use of the complete quaternionic productof two vectors, but he accepted that the scalar part of this product should have a

positive sign . According to MacFarlane theequation j k = i was a convention thatshould be interpreted in a geometrical way,but he did not accept that it implied the negative sign of the scalar product ”. [25](The emphases are mine).He incorrectly attributed the problem to asecondary and superficial matter of repre-

sentation of symbols, instead of blaming tothe more profound definition of the quater-nion product. “MacFarlane credited thecontroversy concerning the sign of the scalarproduct to the conceptual mixture done by

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Hamilton and Tait. He made clear that the

negative sign came from the use of the same symbol to represent both a quadrantal versor

and a unitary vector . His view was thatdifferent symbols should be used to representthose different entities.” [25] (The emphasisis mine).

At the beginning of the twenty century,Physics in general, and relativity theoryin particular, was lacking an appropriatemathematical formalism to represent the

new physical quantities that were beingdiscovered. But, despite the fact that allphysical variables such as space-time points,velocities, potentials, currents, etc., were rec-ognized that must be represented with fourvalues, the quaternions were not being usedto represent and manipulate them. It wasnecessary to develop some new mathematicaldevices to manipulate such variables. Be-sides vectors, other systems such as tensors,

spinors, matrices and geometric algebra weredeveloped or used to handle the physicalvariables.

During the twenty century we have wit-nessed further efforts to overcome the diffi-culties remaining, with the development of other algebras, which recast several of theideas of Grassman, Hamilton and Clifford ina slightly different framework. An example inthis direction is Hestenes’ Geometric Algebra

in three dimensions and Space Time Algebrain four dimensions. [16], [17], [21], [8] [18]

The commutativity of the product wasabandoned in all the previous quaternionsand in some algebras, such as the one of Clif-

ford. According to Gaston Casanova [4] “It

was the English Clifford who carried out thedecisive path of abandoning all the commu-tativity for the vectors but conserving theirassociativity.” “This algebra absorbs theHamilton quaternions, the Girard’s complexquaternions, the cross product and the com-plex numbers, the hyperbolic numbers andthe dual numbers.” [4]. Also the Hestenes’“geometric product” conserves associativity[18]. In this sense, the associativity of theproduct is finally abandoned in the fourvector

algebra proposed in the present paper. Thismeans that the fourvectors do not constitutea Clifford Algebra [2] or a Geometric Alge-bra [1]. This is a collateral effect of the pro-posed algebra, and constitutes a hint aboutthe form the fourvectors should handle, forexample, a sequence of rotations; besides, thecomplex numbers are not handled as in theHamilton quaternions, where the real numberis put in the scalar part and the imaginary in

the vector part, but a whole complex numbercan be put in each component, so it is pos-sible to have up to four complex numbers ineach fourvector. But, what is more impor-tant, it is known that in quantum mechanics,observables do not form an associative alge-bra, so this could be the natural algebra forPhysics.

The proposed algebra could have beenalready developed, around 1900, underthe name of hyperbolic quaternion , which

is a mathematical concept introduced byAlexander MacFarlane of Chatham, Ontario.The idea was dismissed for its failure toconform to associativity of multiplication,but it has a legacy in Minkowski space and

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as an extension of “split-complex numbers”.

Like the quaternions, it is a vector spaceover the real numbers of dimension 4. Thereis only the mention to such quaternions butno accessible references to confirm if thosequaternions satisfy the same algebraic rulesgiven in the following for the fourvectors.However our intent is to convince the readerthat the presented here is one of the mostimportant mathematical tools for Physics.

The fourvector representation is without a

doubt a more unified theory in comparison tothe classical vector or matrix representations.

The use of fourvectors allows discerningconstants, variables and relations, previouslyunknown to Physics, which are needed tocomplete and make coherent the theory.

The vectors have lost some ground infavor of the Hamilton quaternions due to

the lack of an appropriate 4D-algebra. Forexample, Douglas Sweetser, who has workedextensively in the application of Hamiltonquaternions to many possible physical areas,in general with very little success, sustainsthese opinions: “Today, quaternions are of interest to historians of mathematics. Vectoranalysis performs the daily mathematicalroutine that could also be done with quater-nions. I personally think that there may be4D roads in physics that can be efficiently

traveled only by quaternions.” [29]

In fact those 4D roads should be traveledonly by properly handled fourvectors. It hasbeen an old dream to express the laws of

Physics using quaternions. But this attempt

has been plagued with recurring pitfalls forreasons until now unknown to both physi-cists and mathematicians. The quaternionshave not been making problem solving easieror simplifying the equations. Very often theHamilton quaternions require an extreme ha-bility to guess when and where a quaternionneeds to be conjugated, in order to obtainsome particular result.

I believe that this has been due to aninternal problem in the mathematical struc-

ture of the Hamilton quaternions, which Iwill try to reveal in the present paper. Thecorrection of such problem constitutes a newnon-commutative, non-associative normedalgebraic structure with which it is possibleto work with fourvectors in an improvedway relative to the Hamilton, Pauli or Diracquaternions, geometric algebra, space–timealgebra and other formalisms.

In the present paper, in particular, thepresent author exhibits the application of thefourvectors to 3D and 4D rotations, whichrequires a reformulation of the Hamiltonianmathematics.

The powerful Mathematica R packageincludes a standard algebra package for themanipulation of the Hamilton quaternions.I have borrowed from that package thesymbol, as double asterisk, to represent the

fourvector product. It is easy to modify thecited package to handle the fourvectors, aswell as to permit not only their numeric butalso symbolic and complex manipulation.Though I have still not been able to figure

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out a simple way to reproduce the trigono-

metric and exponential functions for thefourvectors (if at all possible), which thatpackage allows to compute for the Hamiltonquaternions.

In the following three subsections, a cur-sory revision is made of the Hamilton, Pauliand Dirac quaternions, for an easy compari-son with fourvectors. In section 2 the fourvec-tors are presented. Section 3 can be skimmedby the mathematician, since it is mostly clas-

sic algebra. Finally, in section 4 the formulaeneeded to perform rotations and reflectionswith fourvectors is presented.

1.2 The Hamilton quaternions

Quaternions are “four-dimensional numbers”of the form [31]:

A = a + i ax + j ay + k az, (1)

B = b + i bx + j by + k bz

where the basis elements 1, i, j, k satisfy therelations:

i2 = j2 = k2 = ij k = −1 (2)and also:

i j = − j i = k, (3) j k =

−k j = i, (4)

k i = −i k = j. (5)Here 1 is the usual real unit; its product

with i, j or k leaves them unchanged.Thus, since the products of the basis elements

are non-commutative, we have in general

A**B = B**A, where the double aster-isk represents quaternion multiplication. Un-der these conditions, quaternion multiplica-tion is associative, so that (A**B)**C =A**(B**C) for any three quaternions A, B,C.

The sum of two quaternions is

A + B =(a + b) + i(ax + bx)+

j(ay + by) + k(az + bz), (6)

and, using relations (2) and (3)-(5), the prod-uct is given by:

A ∗ ∗ B = (ab − axbx − ayby − azbz)+i (abx + axb + aybz − azby)+

j (aby − axbz + ayb + azbx)+k (abz + axby − aybx + azb).

(7)

The notation of three-dimensional vec-tor analysis furnish a useful shorthand forquaternion operations. Regarding i, j, k asunit vectors in a Cartesian coordinate system,we interpret the quaternion A as comprisingthe scalar part a and the vector parta = i ax + j ay + k az. Then we write in thesimplified form A = (a , a). With this nota-tion, the sum (6) and the product (7) maymore compactly be expressed as:

A + B =(a + b, a + b) (8)

A ∗ ∗ B =(a b − a · b, a b + b a + a × b)(9)

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where the usual rules for vector sum and

dot and cross products are being invoked.According to the mathematicians, theHamilton quaternions are mathematicalstructures which combine properties of com-plex numbers and vectors.

1.3 The Pauli quaternions

The Hamilton multiplication rules differ fromthe Pauli matrices rules only by the explicitappearance of the fourth basis element.

The basis elements of the Pauli quaternionspace are denoted by s1, s2, s3, s4.

They obey the following multiplicationrules, comparable to (2)-(5):

s21 = s

2

2 = s2

3 = −s24 = −1s1 s2 = −s2 s1 = s3s3 s1 = −s1 s3 = s2s2 s3 = −s3 s2 = s1s4 sk = sk s4 = sk, (k = 1, 2, 3, 4)

(10)

These rules are satisfied, in particular, bythe Pauli spin matrices (only the first threebear this name, because σ4 serves to form theidentity matrix) [18], [20], [3] [10]:

σ1 =

0 11 0

, σ2 =

0 −i i 0

σ3 =

1 00 −1

, σ4 =

i 00 i

(11)

where “1” i n (10) represents the identitymatrix, i the imaginary unit and si = − i σi,

for i ∈ {1, 2, 3, 4}.

The Pauli quaternions evidence one differ-ence with respect to the classical Hamiltonquaternions, being the need of matrices,which in some cases have imaginary units i .

The sum of two Pauli quaternions is of the same form as the given for the Hamil-ton quaternions and its product, using (10),becomes:

A ∗ ∗ B =s4(ab − axbx − ayby − azbz)+s1(abx + axb + aybz − azby)+s2(aby − axbz + ayb + azbx)+s3(abz + axby − aybx + azb).

(12)

In a compact form, the product for thePauli quaternions has exactly the same formas the Hamilton quaternions and, therefore,have the same problems as these:

A∗∗ B = (a b − a·b, a b + b a+ a×b) (13)

1.4 Dirac matrices

The Dirac matrices must satisfy the Klein-Gordon equation, the following relationsshould be satisfied by the Dirac matrices [33]:

αiα j + α jαi = 2δ ij ,

αiβ + βαi = 0, (i = 1, 2, 3)

α2i = β 2 = I

(14)

where I represents a N × N unit matrix.

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For 2 × 2 matrices, only three anti-commuting matrices exist (the Pauli matri-ces). Thus the smallest dimension allowed forthe Dirac matrices is N = 4. If one matrixis diagonal, the others can not be diagonal orthey would commute with the diagonal ma-trix. We can write a representation that ishermitian (a matrix is hermitian if it is equalto the conjugate of its transpose), traceless(trace equal zero), and has eigenvalues of ±1:

αi =

0 σiσi 0

, (i = 1, 2, 3) (15)

and

β =

I 00 −I

(16)

where σi are the 2 × 2 Pauli matrices andI is the 2 × 2 unit matrix.

Finally, we are ready to define the Dirac’s

gamma matrices out of αi and β :

γ 0 ≡ β, γ i ≡ β αi (i = 1, 2, 3) (17)

These matrices satisfy the relations:

(γ 0)2 = 1, (γ i)2 = −1, (18)

and all four matrices anticommute amongthemselves. These relations are compara-ble to the Hamilton basis (2)-(5) and Pauli

basis (10), except for the exchange of sec-ondary importance in the signature of thegamma matrices, (+, −, −, −), to the sig-nature satisfied by the Hamilton and Paulibases: (−, +, +, +).

2 The fourvectors

The fourvectors are four-dimensional num-bers of the form

A = e at + i ax + j ay + k az (19)

or, assuming that the order of the basis el-ements is the indicated, those basis elementscan be suppressed and included implicitly ina notation similar to a vector or 4D point:

A = (at, ax, ay, az) (20)

Where the elements of the fourvector canbe any integer, real, imaginary or complexnumbers.

The four basis elements e, i, j, k satisfy therelations:

e2 = i2 = j2 = k2 = e = e i j k (21)

The following rules are satisfied by the basiselements:

e i = −i e = i,e j = − j e = j,e k = −k e = k,i j = − j i = k,

j k = −k j = i,k i = −i k = j.

(22)

The group of relations (22) gives an im-

portant operational mechanism to reduce anycombination of two or more indices to at mostone.

The “e i j k” bases characterize thefourvector product as not commutative but,

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3. More compact (and faster) than matri-

ces. For computation with rotations,fourvectors offer the advantage of requir-ing only 4 numbers of storage, comparedwith 9 numbers for orthogonal matri-ces [24]. Composition of rotations re-quires 16 multiplications and 12 addi-tions in fourvector representation, but 27multiplications and 18 additions in ma-trix representation...The fourvector rep-resentation is more immune to accumu-lated computational error. [24].

4. Every fourvector formula is a propo-sition in spherical (sometimes degrad-ing to plane) trigonometry, and has thefull advantage of the symmetry of themethod [30].

5. Unit fourvectors can represent a rotationin 4D space.

6. Fourvectors are important because of

their “all-attitude” capability and nu-merical advantages in simulation andcontrol [28].

Quaternions have been often used in com-puter graphics (and associated geometricanalysis) to represent rotations and orienta-tions of objects in 3D space. This choresshould be now undertaken by the fourvec-tors, which are more natural, and more com-pact than other representations such as ma-

trices, and operations on them such as com-position can be computed more efficiently.Fourvectors, as the previous quaternions,will see uses in control theory, signal pro-cessing, attitude control, physics, and or-

bital mechanics, mainly for representing rota-

tions/orientations in three dimensions. Thespacecraft attitude-control systems should becommanded in terms of fourvectors, whichshould also used to telemeter their currentattitude. The rationale is that combiningmany fourvectors transformations is more nu-merically stable than combining many matrixtransformations.

2.2 Complex fourvectors

The only difference with respect to theordinary fourvectors is that the elements arenot purely real but complex numbers.The collection of all complex fourvectorsforms a vector space of four complexdimensions or eight real dimensions. Com-bined with the operations of addition andmultiplication, this collection forms a non-commutative and non-associative algebra.There is no difficulty in obtaining the mul-tiplicative inverse of a complex fourvector,when it exists, within the fourvector algebrasuggested below. However, there are complexfourvectors different from zero whose normis zero. Therefore the complex fourvectorsdo not constitute a division algebra.

It is important to realize that the relationsneeded by the Klein-Gordon equation (14),are directly satisfied by the purely realfourvectors, whereas the relations needed

by the Dirac equation (18), are satisfiedby the fourvectors constituted of imaginarycomponents in the vector part.

This seems to suggest that there are two

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different kinds of physical entities, although

closely related, which need respectively thereal and the imaginary representations. Thisinsight appears potentially useful for Physics.

3 Fourvector algebra

The sum of two fourvectors is anotherfourvector where each component has thesum of the corresponding argument compo-

nents:

A + B =e(at + bt) + i(ax + bx)+

j(ay + by) + k(az + bz) (25)

The difference of two fourvectors is definedsimilarly:

A − B =e(at − bt) + i(ax − bx)+ j(ay − by) + k(az − bz). (26)

The conjugate of a fourvector changes thesigns of the vector part:

A = eat − iax − jay − kaz (27)From this definition it is obvious that

the result of summing a fourvector withits conjugate is a fourvector with onlythe scalar component different from zero.Divided by two, isolates the scalar compo-nent of a fourvector and serves to define

the operator named the anticommutator (similar to the scalar Hamilton’s operatorS ): (A + A)/2 = S A. Similarly, the resultof subtracting the conjugate of a fourvectorfrom itself is a pure fourvector (that is, one

whose scalar component is equal to zero).

Divided by two serves to define the com-mutator (similar to the vector Hamilton’soperator V ): (A − A)/2 = V A

The complex conjugate or hermitian conju-gate of a fourvector changes the signs of theimaginary parts. Given the complex fourvec-tor:

A =e(at + ibt) + i(ax + ibx)+

j(ay + iby) + k(az + iby)

(10.15)

(28)

Then its complex conjugate is:

A∗ =e(at − ibt) + i(ax − ibx)+ j(ay − iby) + k(az − iby)

(29)

Using relations (21) and (22), the fourvec-tor product is given by:

A ∗ ∗ B =e(atbt + axbx + ayby + azbz)+

i (atbx − axbt + aybz − azby)+ j (atby − axbz − aybt + azbx)+k(atbz + axby − aybx − azbt).

(30)

Using the notation of three-dimensionalvector analysis we obtain a shorthand for theproduct. Regarding i, j, k as unit vectors ina Cartesian coordinate system, we interpretthe fourvector A as comprising the scalar a and the vector part a = i ax + j ay + k az.

Then we write it in the simplified form A =(a , a). With this notation, the product (30)is expressed in the compact form:

A∗∗ B = (a b + a ·b, a b−a b + a ×b) (31)

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The following properties for the product

are easily established:

1. If the scalar terms of both argumentfourvectors of the product are zero thenthe resulting fourvector contains theclassical scalar and vector products inits respective components.

2. The product is non-commutative. So, ingeneral, there exist P and Q such thatP**Q = Q**P.

3. Fourvector multiplication is non-associative so, in general, P**(Q**R)= (P**Q)**RNote that this is different from theHamilton quaternions and the so-calledClifford Algebras, see for example [1].

4. The product of a fourvector by itself pro-duces a result different from zero only in

the first or “scalar” component, which isidentified as the norm of the fourvector.In this sense this constitutes the classicaldot product of vector calculus:

A∗∗A = (a2t + a2x +a2y +a2z, 0, 0, 0) (32)

Note that this expression is substantiallydifferent with respect to the Hamiltonquaternions, in which the square of aquaternion is given by

A2 = (a2t − v · v, 2 atv), (33)

where v represents the three vectorterms of the quaternion. Not only the

scalar component has terms with the

sign changed, but appears a non-zeroterm in the vector part of the quaternion.This has been a source of difficulty toapply Hamilton quaternions in Physics,which is overcome by the fourvectors.

5. The multiplicative inverse of a fourvectoris simply the same fourvector divided byits norm.

6. Properties of the product and conju-

gates: P ∗ ∗ Q = Q ∗ ∗ P (34)P ∗ ∗ (Q ∗ ∗ R) = R ∗ ∗ (Q ∗ ∗ P) (35)

P ∗ ∗(P ∗ ∗Q) = P ∗ ∗(P ∗ ∗Q)= |P| ∗ Q (36)

(P ∗ ∗Q) ∗ ∗P = (P ∗ ∗Q) ∗ ∗P= |P| ∗ Q (37)

With an operator notation: The prod-

uct of two fourvectors is equal to theconjugate of the same product in reverseorder: A**B = Conjugate[B**A]

7. For the case that “r” is a rotor (afourvector with |r| = 1) then:

r ∗ ∗(r ∗ ∗Q) = Q = r ∗ ∗(r ∗ ∗Q)(r

∗ ∗Q)

∗ ∗r = Q = (r

∗ ∗Q)

∗ ∗r

((((Q ∗ ∗r) ∗ ∗r) ∗ ∗r) ∗ ∗r) = Q(otherwise, if |r| is not equal to 1, theproducts of this numeral are equal to Qor Q multiplied by |r|.)

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8. The fourvectors do not contain the

complex numbers, as is usually demon-strated for the Hamilton quaternions.The product of the fourvectors: (a, b, 0,0) and (c, d, 0, 0) is(ac+bd, ad-bc, 0, 0); also, the productof the fourvectors (a, i b, 0, 0) and (c, i d, 0, 0) is (ac-bd, i ad-bc, 0, 0), whereasthe product as complex numbers shouldbe: (ac-bd, i ad+bc, 0, 0)

9. Given the fourvectors A and B, thecommutator:

[A, B] = 1

2(A ∗ ∗B − B ∗ ∗A) (38)

= (0, a b − ab + a × b) (39)

gives a fourvector with zero scalar andwith the vector part equal to the vec-tor part of the fourvector product A**B.

For the curious, this commutator satis-fies the properties of antisymmetry andlinearity. The Jacobi identity is satisfiedonly for pure fourvectors.

10. Given two fourvectors, A and B, theanticommutator:

= 1

2

(A

∗ ∗B + B

∗ ∗A) (40)

= (a b + a · b, 0, 0, 0) (41)

gives a fourvector with the scalar equalto the scalar of the fourvector product

A**B and with the vector part equal to

zero.

11. Product is left distributive over sum:

a**(b + c) = a**b + a**c

12. Product is right distributive over sum:

(a + b)**c = a**c + b**c

13. The product of three “pure” fourvectors(defined as those whose scalar compo-

nent is zero) can be expressed with thefollowing vector products:

a ∗ ∗(b ∗ ∗ c) =(a · (b × c), a × (b × c) − a ∗ (b · c) )

Where “·” and “×” are the standardvector dot and cross products and “*”represents the product of the scalar(b · c) by the vector a. The scalar com-ponent of the result can be recognizedas the volume of the parallelepipedhaving edges a, b and c. Consequently,if the three vectors a, b and c are inthe same plane (or parallel to the sameplane) then the scalar component of theresulting fourvector product is zero.

14. The following identity is also satisfied:(a**b) ** (a**b) = (a**a) ** (b**b)

3.1 Product with matrices

Given two fourvectors, p and q:p=(p0,p1,p2,p3),

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q=(q0,q1,q2,q3),

their product can be obtained asR=p**qThe same product can be obtained multiply-ing the following matrix P by the (four)vectorq:

R =

p0 p1 p2 p3− p1 p0 − p3 p2− p2 p3 p0 − p1− p3 − p2 p1 p0

q 0q 1q 2q 3

The P matrix has the propertyP · PT = PT · P = I, where T representsthe transpose and I is the identity matrix.(More precisely the diagonal elements havethe form: p0

2+p12+p2

2+p32, which are equal

to 1 only if p is a unit fourvector; else, inthe diagonal is obtained the norm of the pfourvector).

3.2 The norm

The norm of a fourvector is defined by

|(at, ax, ay, az)| = a2t + a2x + a2y + a2z (42)

It can be computed as the scalar componentof the product of the fourvector by itself.The norm satisfies the properties

|A| = |A| (43)

|P ∗ ∗Q| = |Q ∗ ∗P| = |P| ∗ |Q| (44)

The last property allows to conclude thefollowing form of the four-squares theorem :

(a20 + a21 + a22 + a23)(b20 + b21 + b22 + b23) =

(a0 b0 + a1 b1 + a2 b2 + a3 b3)2+

(a0 b1 − a1 b0 + a2 b3 − a3 b2)2+(a0 b2 − a1 b3 − a2 b0 + a3 b1)2+(a0 b3 + a1 b2 − a2 b1 − a3 b0)2

(45)

3.3 Identity fourvector

The identity fourvector , let us denote with 1,has the scalar part equal to 1 and the vectorpart equal to zero: 1 = (1, 0, 0, 0).It has the following properties, where “p” isany fourvector:

1**p = p

p**1 = pAs you can see, this is the left identity. It ispossible to find the right identity of a fourvec-tor but it is a little more complex. See the

section “Right factor of a fourvector”.

3.4 Multiplicative inverse

The multiplicative inverse or simply inverse of a fourvector P is denoted by P−1.

The inverse of a fourvector P is the samefourvector divided by its norm:

P−1 = P/

|P

| (46)

The inverse operation satisfies the proper-ties:

P**P−1=1P−1**P=1

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(P−1)−1 = P

(P ** Q)−1

= P−1

** Q−1

P−1 = (P)−1

Commutativity of products includinginverses:P ** (P−1 ** Q) = P−1 ** (P ** Q)(P−1 ** Q) ** P = (P ** Q) ** P−1

Q = P**(P−1 ** Q) = P−1**(P ** Q)

Q = (P−1 ** Q)**P = (P ** Q)**P−1

3.5 Scalar multiplication

Scalar multiplication If c is a scalar, or ascalar fourvector, and q=(a , v) a fourvector,then c q = (c, 0) ∗ ∗q = (c, 0) ∗ ∗(a , v) =(ca + 0 · v, c v − a 0 + 0 × v)Simplifying:c (a , v) = (ca , c v)

3.6 Unit fourvector

A unit fourvector has the norm equal to 1.It is obtained dividing the original fourvectorby its magnitude or absolute value, that isthe square root of the norm. The product of two unit fourvectors is a unit fourvector. Aunit fourvector can be represented with theuse of trigonometric functions

ŵ = (± cos(α) ± û sin(α))

where û is in general a 3D vector of unitlength.

The product of two unit fourvectors :Assume the unit fourvectors a and b:

a = (cos(α), sin(α), 0, 0)

b = (cos(β ), sin(β ), 0, 0)

Its product isa**b = (cos(β − α), sin(β − α), 0, 0)so, if a = b, then the resulting fourvector

is the identity fourvector .

The inverse of a unit fourvector is the sameunit fourvector. This is because the productof the fourvector by itself produces the iden-tity fourvector, or the norm, equal to 1, in

the scalar component.

3.7 Fourvector division

The fourvector division is performed bymultiplying the “numerator fourvector”, P,by the “denominator fourvector”, Q, dividedby its norm (or rather multiplying P by theinverse of Q):

P ∗ ∗Q/|Q| = P ∗ ∗Q−1

If P and Q are parallel “vectors” (purefourvectors or with scalar part equal to zero),then the division produces, in the scalar part,the proportion between both vectors. For ex-ample:P=(1, 2, 3, 4)Q=(3, 6, 9, 12)

P ∗ ∗Q/|Q| = (1/3, 0, 0, 0)

3.8 Right factor of a fourvector

Let us try to solve the following equation for“X”

A == B ** X

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Let us assume that A and B are constant

fourvectors and X is an unknown fourvector

A=(a0,a1,a2,a3)B=(b0,b1,b2,b3)

X=(x0,x1,x2,x3)

Where the components can be integer, ra-tional, real or complex.

The solution for X can be obtained withthe expression

X = B −1

∗ ∗A (47)

For example, let us try to determine whatthe value of X should be in order to satisfythe equation A == B ** X, when

A=(7,1,-3,5)

B=(1,3+i 5,2,-1)

According to equation (47) the solution is

X = (−310

− i 25

, 910

− i 45

, 1225

− i 5350

, 925

− i 2150

)

Replacing this solution into the productB ** X it can be verified that it reproducesthe A fourvector.

3.9 Left factor of a fourvector

In a similar form, let us try to solve the fol-

lowing equation, where the unknown “Y” isnow a left factor of the constant fourvector B:

A == Y ** B

Y is obtained with the expression

Y = Hprod[A, B−1] (48)

Where “Hprod” represents the product forthe classical Hamilton quaternions or Grass-man product.

For example, for the same fourvectors Aand B from the previous example, let usapply the equation (48) and find

Y = ( 425

−i 150

, 3950

−i 2925

,

−19

25 + i 11

50,

−11

50 + i 21

25)

Replacing this solution into the productY ** B it can be verified that it reproducesthe A fourvector.

Note that the left and right factors of somefourvector such as B are different, althoughboth factors have the same norm and satisfythe equality:

X ∗ ∗ Y−1 == X−1 ∗ ∗ Y

Both products, B ** X and Y ** B areequal to A.

Formula (48) can be used to determine thesingle rotation fourvector that produces thesame effect as a rotor. However, the resultsobtained are, in general, more complex thana classical rotor. Nevertheless, if we needthe fourvector L, which produces the samerotation of the fourvector p as the rotorfourvector r, that is:

L ∗ ∗ p = Rotate[p, r] = r ∗ ∗ (p ∗ ∗ r)

applying equation (48) we solve for L:

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L = Hprod[(p ∗ ∗ r) ∗ ∗ r], p−1]

Where “Hprod” is the Grassman product.

3.10 Solution of quadraticfourvector polynomials

There can be an infinite number of solutionsfor a quadratic fourvector polynomial. Con-sider the quadratic equation

q2 == 1.

Then, the fourvector q = (cos(x),sin(x),0,0), where x is any real number isa collection of solutions for this equationbecause the norm of q is 1:

q ** q == 1

So the above choice for q satisfies the

equation q2 == 1 for all real values of x.

When there is a solution of a quadraticequation, it can be computed as in thefollowing.Assume a quadratic polynomial of the form:

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

where:

q=(q0,q1,q2,q3) is an unknown fourvectorand

j=(0,-1,1,0) and k=(-1,0,0,1) are constant

fourvectors

From here, the four equations, obtainedequating the four components, are:

1+q02- q1+ q12+ q2+ q22+ q32==0,-q0+q3==0,q0+q3==0,-1-q1-q2==0.

This system of equations has two solutionsfor the four components of the fourvector:

q1 = (0, −1 − i /

√ 2, i /

√ 2, 0) (49)

q2 = (0, 1 + i /√

2, −i /√

2, 0) (50)

Replacing q1 (or q2) by its value, in thefollowing expression, which is the left handside of the given equation, q**q+q** j, the

value returned is: (-1,0,0,1),

Which is the value of the right hand side.

For a comparable quadratic equation, butnow affecting the “ j” fourvector to the leftof “q” instead of to the right,

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

The two solutions are:

q1 = (0, −i /√ 2, 1/2(−2 + i √ 2), 0) (51)q2 = (0, i /

√ 2, 1/2(−2 − i

√ 2), 0) (52)

Which, replacing in q ** q + j ** q

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produce the value: (-1,0,0,1)

Note that the fourvectors form a division algebra since they have a (left) multiplicativeidentity element 1=0 and every non-zeroelement a has a multiplicative inverse (i.e.an element x with a x = x a = 1).

4 Fourvector rotation

Mathematically, a rotation is a linear trans-formation that leaves the norm invariant.These are called orthogonal transformations.

There are several methods to representrotation, besides quaternions, includingEuler angles, orthonormal matrices, Paulispin matrices, Cayley-Klein parameters, andextended Rodrigues parameters.

From the several ways to represent theattitude of a rigid body one of the mostpopular is a set of three Euler angles.Some sets of Euler angles are so widelyused that they have names, such as theroll , pitch , and yaw of an airplane. Themain disadvantages of Euler angles arethat certain functions of Euler angles haveambiguity or singularity for certain angles.This produces, for example, the so-called“gimbal lock”. Also, they are less accurate

than unit quaternions when used to integrateincremental changes in attitude over time [7].

The handling of rotations by means of quaternions has constituted the technical

foundation of modern inertial guidance sys-

tems in the aerospace industry for the orien-tation or “attitude” of satellites and aircrafts.This task is to be shown in the following thatshould be carried out by fourvectors.

Many graphics applications that need tocarry out or interpolate the rotations of objects in computer animations have alsoused quaternions because they avoid thedifficulties incurred when Euler angles areused. The form to replace by fourvectors isnot performed in the present paper, although

it should be perfectly possible.

The use of matrices is neither intuitive forthe localization of the axis of rotation norefficient for computation. But one of themost important disadvantages is the asso-ciativity of both the matrix and Hamilton’squaternion products where, for example,A · (B · C) is equal to (A · B) · C. In fact itis rather well known that the composition of

rotations, when either matrices or Hamiltonquaternions are used, is associative. Thismeans that these mathematical tools producethe same result no matter what the groupingof a sequence of two or more rotations. Thisposes a serious technical problem to theengineers who need to distinguish betweentwo sequences of the same rotations. Toillustrate with an example, let us assumethat you are piloting an airplane with a localframe of reference whose origin is attached

at the center of the airplane. Assume that,at a certain instant, the “x” axis, which ispointing toward the front of the plane, ishorizontal according to an observer in theearth, let us say directed toward the North

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pole, the “y” axis points to the right wing,

that is pointing to the East, and the “z” axispoints toward the earth’s center. If underthese conditions you maneuver to produce a90◦ roll (a quarter circle clockwise rotationabout the local x axis) followed by a 90◦

yaw (rotation “to the right” around thelocal z axis) your plane should be fallingperpendicularly to the earth; but if youinterchange the order of these rotations,that is first the yaw and then the roll,this should put your plane in a horizontal

heading toward the East, although with theright wing pointing toward the earth’s center.

To rotate a vector u around an Euler vectorn through an angle θ, one has to apply thefollowing equation (see Fig. 1 and references[24], [28]):

v = ucosθ + n(n · u)(1 − cosθ) + (n × u)sinθ(53)

Figure 1: Graphic of a rotation

According to Silberstein, [26]: “It has beenremarked by Cayley, as early as in 1854, that

the rotations in a four-dimensional space may

be effected by means of a pair of quaternionsapplied, one as a prefactor and the other asa postfactor, to the quaternion u whose com-ponents are the four coordinates of a space-point, say v = a u b”. This phrase appliesdirectly to fourvectors if “quaternion(s)” isreplaced by “fourvector(s)”.In the case of pure rotation, a and b mustbe either unit-fourvectors or the norm of theirproduct must be 1: |a| ∗ |b| = 1.

It follows, from the rule:

|a ∗ ∗ b| = |a| ∗ |b|, (54)the multiplication of the fourvector being ro-tated by unitary fourvectors a and b, effectsan orthogonal transformation.This form can be simplified so instead of twodifferent unitary fourvectors is selected onlyone, let us name it r.

A possible fourvector r that produces therotation of any fourvector V about a certain

axis “n”, through an angle θ, has the form([24], [5], [12] [15], [19], [9]):

r = (cos(θ

2), nx sin(

θ

2), ny sin(

θ

2), nz sin(

θ

2))

(55)

or simply

r = (cos(θ

2), n sin(

θ

2)) (56)

The rotation is carried out with the following

product:V’ = r ∗ ∗ (V ∗ ∗ r) (57)

The rotation operand needs to be conjugated.This is different with respect to the rotation

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using Hamilton quaternions [26], where the

inverse or the conjugate of the second rotoris needed.Fig. 2 shows the vector V rotated aroundthe unit vector k, through an angle θ, withrotor r. The rotation operator is linear. It

Figure 2: Example rotation

can be proved that the rotation of either theproduct or the sum of two fourvectors s and twith the rotor r, is the same as, respectively,the product or the sum of the rotations of each fourvector:

Rotate[s ∗ ∗ t, r] ≡ Rotate[s, r] ∗ ∗ Rotate[t, r](58)

Rotate[s + t, r] ≡ Rotate[s, r] + Rotate[t, r](59)

Let us define the following example rotationfourvectors, or rotors, whose norm is the unit:

qi ∗ ∗ qi = 1, and cause rotations about thex axis:

q1 = (±cos(α/2), ±sin(α/2), 0, 0) (60)q2 = (±cosh(α/2), ±i sinh(α/2), 0, 0) (61)

q3 = (

γ + 1

2 , i

γ − 1

2 , 0, 0) (62)

where γ is the Lorentz contraction factor.

The rotation fourvector q3 was obtained bytransforming the following fourvector

(± γ , ± i β γ , 0, 0)in such a way that it produces a rotation

of half the hyperbolic angle, as with q2.

Let us multiply any one of the previousfourvectors by the following one that repre-sents a differential of interval:

ds = (c dt, i dx, i dy, i dz)

The products are of the form:

Rotate[ds, qi] = qi ∗ ∗ (ds ∗ ∗ qi) (63)

where qi is anyone of the above list of unitfourvectors.Then, the norm of the result is the square of the differential of interval:

ds2 = c2 dt2 - dx2 - dy2 - dz2

This means that any one of these trans-

formations (rotations) preserves the intervalinvariant.

But first the rotation of a real fourvectora = (at, ax, ay, az) with the rotor q1 producesthe classical formulas for rotation of a vector

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about the x axis:

Rotate[a, q1] = (at, ax,

ay cos(α) − az sin(α),az cos(α) + ay sin(α))

(64)

The opposite rotation can be done usingas rotor the inverse of q1, which changesthe sign of the angle α. For example, if werotate the last result with the conjugate orthe inverse of q1 then the original fourvectora is recovered.

The rotation of the differential of intervalwith q2 gives the complex fourvector:

Rotate[ds, q2] = (cdt, i dx,

dz sinh(α) + i dy cosh(α),

−dy sinh(α) + i dz cosh(α))(65)

The rotations with q3 produce Lorentzboosts. Let us apply to ds:

Rotate[ds, q3] = (cdt, i dx,

i γ (dy − i β dz ),i γ (dz + i β dy))

(66)

If the fourvector to be rotated is theprevious a and the rotor fourvector is

r = (cos(α/2), 0, 0, sin(α/2))

Then the double rotation:

r ∗ ∗(r ∗ ∗(a ∗ ∗r) ∗ ∗r)

Is equal to a single rotation through thedouble angle 2α. The result is:

(at, ax cos(2α) − ay sin(2α),ay cos(2α) + ax sin(2α), az)

(67)

4.1 Composition of rotations

The rotation through an angle α followedby another rotation through an angle β isequivalent to a single rotation through anangle α + β :

Assume, for example that the rotations areproduced by application of the following ro-tors:

roth1 = (cosh(α/2),isinh(α/2), 0, 0) (68)

roth2 = (cosh(β/2),isinh(β/2), 0, 0) (69)

Let us apply these rotors over the fourvec-tor M=(a,b,c,d) with the operation:

M1 = Rotate[M,roth1] (70)

followed by the following rotation:

M2 = Rotate[M1,roth2] (71)

We obtain the following result:

(a,ib,iccosh(α + β ) + d sinh(α + β ),

idcosh(α + β ) − c sinh(α + β )) (72)

Which is identical to the rotation producedby directly applying over M the rotor:

roth =

cosh(α + β

2 ),isinh(

α + β

2 ), 0, 0

(73)

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4.2 Reflections

Let us show how the reflection in a planewith unit normal a can be done (see nextfigure)

Figure 3: Reflection

the normal vector a, which defines thedirection of the plane, generates a reflectionof the vector x if we rotate x around thevector a through an angle of π radians andthen change every sign of x (otherwise weend up with an arrow pointing at the samepoint as x). This rotation is done by fixingat π radians the angle α of rotation of arotor of the form q1 of section 4, i.e.

q[Cos[α/2],a Sin[α/2]].

Consequently, the cosine term disappearsand the sine term becomes equal to 1, withwhich we are left with the vector a alone, asrotor.

The vector x, after reflection, is:x’ = −a ∗ ∗(x ∗ ∗a)To simplify this expression, let us note that

the rotation through π radians clockwise isthe same as the rotation through π radians

counterclockwise, so we could include conju-

gations of the vector a. Also, the conjugationof the vector x can be canceled with the neg-ative sign, so the final equation is:

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 thevector x’ that describes the direction of emer-gence of a ray of light initially propagating

with direction x and reflecting off a sequenceof plane surfaces with unit normals a1; a2;. . . ;an:

x’ =an**(...(a2**((a1**(x**a1))**a2)...an)

Figure 4: Reflections

Hestenes [15] shows other applications

for rotations and reflections, for examplein crystals. Note that in his “geometricalgebra” Hestenes needs the negative signthat we don’t need.

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5 Conclusions

The mathematical structure of quaternionshas always been considered as more appro-priate than the simple vectors to representthe real physical variables. Nevertheless,the quaternions were dismissed for thedifficulties and complications produced bytheir quaternion product. With the newproduct, suggested in the present paper forfourvectors, all those difficulties disappear.Of course there must be a delicate balance

between the correct mathematical tools andthe real physical objects being studied andhandled. One has to also be aware thatmathematics clearly affects the ontology of physics [11].

The fourvector algebra proposed in thepresent paper seems to be the correctmathematical tool to study the fundamentalphysical variables and their describing equa-tions.

This new mathematical structure is anextension of the classical vectors. Its sim-plicity contributes to the possibility of moreextended and fruitful use in all branches of science.

The applications of the Hamilton quater-nions for rotations in three dimensionshave been the more extended in current

Physics. Such uses, as well as reflections,are still permitted by the fourvectors. Theapplications to Lorentz boosts had problemswith the old quaternions, so this area opensup for the scientists.

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