The Clifford algebra of space-time applied to field theories,part 1Citation for published version (APA):Vroegindeweij, P. G. (1987). The Clifford algebra of space-time applied to field theories, part 1. (EUT-Report;Vol. 87-WSK-03). Eindhoven: Technische Universiteit Eindhoven.

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TECHNISCHE UNIVERSITEIT EINDHOVEN

NEDERLAND

EINDHOVEN UNIVERSITY OF TECHNOLOGY

THE NETHERLANDS

FACULTEIT WISKUNDE EN

INFORMATICA

DEPARTMENT OF MATHETMATICS AND

COMPUTING SCIENCE

The Clifford algebra of Space-time

applied to Field Theories

Part I

by

P.G. Vroegindeweij

AMS Subject Classifications: 15A66, 81Exx, 83A05, 81Gxx

EUT Report 87-WSK-03

ISSN 0167-9708

Coden: TEUEDE

Eindhoven, June 1987

- 0 -

Contents

Preface

1. The Clifford algebra of 3-space

2. The Clifford algebra of space-time

3. The Clifford algebra of n-space

4. Differential operators in Clifford algebras

5a. The partial differential equation of b. Electrodynamics

6a. The partial differential equation D~e5

b. Dirac fields

References

]

m¢

1

3

12

24

32

40

44

45

53

63

- 1 -

Preface

'rhe origin of this report is somewhat remarkable.

Around Christmas 1983 I read the then bestseller "Wholeness and the impli­

cate order", written by David Bohm [BO].

Especially the algebraic point of view of Bohm about space-time, as explain­

ed in this book, highly fascinated me.

Meanwhile, one of Bohms cooperators, Basil Hiley, had published a paper (to­

gether with F.A.M. Frescura) entitled "The implicate order, algebras and the

spinor [FH].

In this paper the authors advocate the use of Clifford algebras to describe

physical phenomena, in contrast to the usual description using vector spaces.

They eventually make contact with the "pregeometry" of Wheeler.

Looking for literature about applications of Clifford algebras I met the vast

and versatile work of David Hestenes. Besides his first book [Hl] he had

written a large number of papers about the subject under consideration ([H2J­

[H22]l, some of them together with other authors [GH], [HG], [HL].

Meanwhile he has published two new books in the same topic [HS], [H18]. One

of the greatest problems for me was to overcome the very divergent levels of

comprehensibility of Hestenes' papers.

The approved means to conquer this problem was to make a survey of results

which were scattered in a lot of papers and three books. The underlying re­

port presents such a survey. It should be accessible to a broad circle of

readers.

During the winter 1984-1985 I attended the lectures in Tensor calculus and

Differential Geometry of J. de Graaf at the Eindhoven University of Techno­

logy. That grew out to a very pleasant and fruitful cooperation, it gave

rise to a number of improvements in this report and last but not least it

led to Theorems 4.1 and 4.2 in section 4 of this report.

Of course the author is very indebted for this stimulating support. He also

gratefully acknowledges the willingness of S.J.L. v. Eijndhoven to read a

former version of the manuscript. It also led to a number of clarifications

and the correction of some embarrassing errors. Finally he would like to

thank Mrs. Marese Wolfs for her excellent typing of the manuscript.

- 2 -

Next a few remarks about the contents of the report.

William Kingdon Clifford (1845-1879) was the founder of the ideas as exposed

here. Had he not died young, the algebra bearing his name might well have

replaced Gibbs' vector algebra (including the cross product a x b) as a fun­

dament for mathematical descriptions in physics. An interesting survey of

Cliffords life and work can be found in [CC].

This report has been organized in the following way:

Sections 1 and 2 give an introduction to Clifford algebras of 3-space and

space-time, readable for pedestrians. Section 3 generalizes the matter of

sections 1 and 2 in a coordinate free way.

Section 4 deals with differential operators starting with basis dependent

contemplations and ending with the general and coordinate free relation

a == d - O.

Sections 5 and 6 give a survey of the two most obvious applications, the

Maxwell theory of the electromagnetic field and the Dirac field theory of

the electron and the photon.

The mathematics has been gathered in sections Sa and 6a while the physical

and more interpretative remarks can be found in sections 5b and 6b.

I hope to return to the subject in part II of this report, dealing with

Yang-Mills fields, strong and electroweak fields, Glashow-Weinberg-Salam

fields and, perhaps, an alternative for the Higgs-mechanism.

Eindhoven, june 1987

P.G. Vroegindeweij

- 3 -

1. The Clifford algebra of 3-space

Let E denote Euclidean 3-space endowed with a righthanded orthonormal basis

{~1,e:2'£3}' We want to define a multiplication of vectors x in E, satisfying

the rule x2 :; II xII 2 .

Using our basis {E1 '£2'£3} this requirement can be expressed by

This valids for

2 2 £ = E: = 1 2 3

£1 £2 + £2 £1 0

E:2 £3 + E:3 £2 0

£3£1 + £1£3 = 0

Or shortly

£k £ Q, + £Q,£k 2ok Q, , k I Q, 1,2,3

Thus we extended our 3-space to a 23-dimensional (associative) algebra with

basis

This algebra is usually named the real Pauli algebra P.

It is easy to check that

and that

k 1,2,3 •

The quantity £1£2£3' called pseudoscalar, has similar algebraic properties

as the imaginary unit i E t. For that reason we often write shortly

i

Note that C1£2€3 depends only on the orientation of the chosen orthonormal

basis. For left bases one finds t: <.: c = -i. Compare the expression 123

det(c 1£2t: 3 ) ±1 for right and left bases.

- 4 -

Clifford has introduced the algebra that bears his name in a similar way as

ske tched above. 2

Grassman followed the same lines, but his condition was (x1

€1 +x2

€2 +x]E3) 0

and also Dirac did by factorization of the expression

We call €lE2' E3 and E3El unit bivectors. They satisfy the same algebraic

1 H 'lt t' 'k dO' ,2 k 2 02 1 ru es as am1 ons qua ern10nS], an N, V1Z.] = N = - and

jk = -kj L

A vector can be described by a line, a real scalar and a direction.

Similarly a bivector can be described by a plane, a real scalar and a direc­

tion.

J

Now let a

A simple and straightforward computation yields:

ab

and similarly

ba

E2

alb

1 + a

2b

2 + a

3b

3 + a

1 b

i

E3 El €1€2

a2

a3

b2

b3

J

be vectors in E.

We recognize the symmetric part of ab as the inner product a·b and we write

:;, (ab + bal

The remaining and antisymmetric part a A b of ab is the Hodge dual * of

Gibbs' cross product a x b and one can write:

- 5 -

lEi £2 £3

a A b ~(ab - ba) *(a x b) i(a x b) i a1

a2

b1

b2

b3

where i E1E2 E: 3 ·

Thus we splitted the product ab in two parts a'b and a A b, satisfying the

fundamental rule of Clifford

ab = a • b + a A b

In particular we have:

For k ~ ~: Ek • E~ 2

E:k = £k • £k

0, hence EkE~ 0, whence

1.

Any element A E P can be written as

A

or as

and also as

A

where i 1,2,3.

Further note that the wellknown Pauli matrices

[0 -i]; a = [1 0] i 0 3 0-1

with i ~ satisfy the same algebraic rules as our basis vectors £l,E2

,L3

"

Hence our 8-dimensional Pauli algebra is isomorphic to the algebra ~(2) ,

consisting of all 2 x 2-matrices over the field «. Moreover the expression

A

reveals that the Pauli algebra is also isomorphic to the algebra of quater­

nions with coefficients in the field ~.

- 6 -

Note in passing that in contradistinction to the algebra lH of real quater­

nions our algebra is not a division algebra.

Indeed the inverse element

does not exist if

o , a,B,y,o E It •

Remark. In Quantum mechanical textbooks (e.g. A Messiah, Quantum Mechanics,

page 546) one often meets the identity

(0 • a) «J • b) = (a b)I2

+ icr • (a x b)

where (J • a means al

a1

+ 02

a2

+ (J3a3 and so on.

As can immediately be seen this rule is merely a guise of Cliffords rule

ab = a • b + a A b.

We also often write

A = A + A + A + A 1 123

where AO means the scalar part of A, A1 the vectorial part, A2 the bivec~o­

rial part and A3 the trivectorial part, also called pseudoscalar part.

A is called a multivector.

A is called even if Al = A3 o. The subalgebra of even multivectors is isomorphic to the field of real qua­

ternions lH.

Next we introduce a number of automorphic maps of P.

For A E P we define:

* * A ~ A where A AO - A + A2 - A 1 3

A t+ A t

where At AO + Al - A2 A3

A tr A where A AO - A1 - A2 + A3

Note that A

- 7

* One recognizes the restriction of the map A + A to 3-space as space re-

flection, also often called parity. Using the relation ab

for the vectors a and b we conclude immediately:

* (ab) * * t t t "". ~~ a b , ( ab ) = b a , ab =. ba .

For multi vectors A and B the same rules valid viz.

* * * t (AB) = A B , (AB) BA

a • b + a A b

The proofs of the latter three results are not difficult but somewhat leng­

t.hy (compare section 3 for generalizations to the Clifford algebra of eucli­

dean n-space) .

* Observe that A~ A is an involutory map (called inversion or main involu-

* * * * * tion) because one has (AB) = A B and (A) = A.

It can be considered as a generalization of complex conjugation.

A ~ At is the so-called

by the properties (AB)t

main antiautomorphism or reversion, characterized

BtAt and (At)t = A. t Moreover, in matrix representation, A even corresponds to the hermitean

conjugate of the matrix A.

Using At we can introduce the norm or length II All of a multivector A in the

following way:

II All = "At A) 0 .

As is easy to check (AtA)O 2

-A3

are 2:: O.

222 o because AO' AI' -A2 and

Finally, A ~ A is the map of A onto its conjugate if P is considered as the

complex quaternion algebra.

Indeed A a - Bj - yk o~ corresponds to A

The Hodge star operator *, as ego introduced in [AMR] yields for the algebra

of 3-space with positive definite metric:

1 *e:

2 *E

3 *£ =:

123 E: E: E:

2 3 e: E

1

1 2 E E

k where E = Ek

, k =: 1,2,3.

- 8 -

1 2 *(E E: )

3 E:

231 *(E:E) £

2 E

123 *(e: £ E ) "" 1

Note that in this special case **

cyclic onto their duals.

1 and that * changes the quantities

Both the Hodge * operator and the operator i map multivectors onto their

duals but there is also a difference because on the one hand ** = 1 and on

the other hand i 2 -1. Therefore it is useful to compare both operators.

One finds for i:

i1 = i

i 2

E

iE: 2 3

iE 3 1

E

Whence we find:

*A

*A == Ati

Note that indeed:

**A

1 C

· ( 1 2) 3 1 E E -E

· ( 2 3) 1 1 E E -E:

· ( 3 1) 1 E E

· ( 1 2 3) 1 E E E -1 .

i(Ao + Al

iAt = Ati

-iAi =: A .

A minimal of an algebra is an ideal that contains only itself and the

zero ideal as ideals. Minimal ideals are generated by primitive idempotents

([BJ, page 60).

- 9 -

Appealing to the ~(2) representation of our algebra one immediately sees

that - up to isomorphy - there are two independent primitive idempotents,

viz. ~(l + 83

) and £ ~(l - 83),

and £ generate the minimal left l ) ideals 1+ and I .

In the next table we present orthonormal bases for 1+ and I and their ma­

trix representations,

I

c = 4

Note that ~bll:2 and ~cll:2 are idempotent and that b2

,b3 ,c2,c

3 are nilpotent.

The matrix representations of the elements of 1+ and I are given by (a 0) b 0

and (~ ~) where a,b,c,d E ~. Next we make some remarks about the orthogonal group of 3-space E. Let b be

any vector in E with IIbli = 1 and let x be any vector.

Consider the map Sb: x ~ x' defined by

x' (x) -bxb .

Sb is the reflection in the plane, perpendicular to b.

(b.x) b x

x-(b.x)b

, '~ (x) x - 2(b.x)b = x bxb - xbb == -bxb •

1) Obviously the same is true for right ideals, but they do not playa role

in this paper.

- 10 -

Next consider the composition R of two reflections Sa and Sb' where

II all = Ilbll 1

x" = R(x) = S Sb(x) = -S (bxb) = abxba . a a

As known, R is a rotation with axis a x b and angle twice the angle ~ be­

tween a and b.

We can write:

ab = a • b + a A b = cos ~ + u sin a = e~u

where u denotes the unit bivector

a A b lIa A bll

One has u 2 -1.

Now write Rx ~u -~u

we can e xe

au Note especially that the one symbol e contains exactly the three charac-

teristics of a rotation, viz.

1. The axis of rotation perpendicular to the plane in which the rotation

takes place. This axis is determined by u.

2. The angle of rotation 2a.

3. The orienta"tion of the rotation determined by the orientation of the bi-

vector u.

(Hore details can be found in [HiS], chapter 5.)

Even multivectors in the Pauli algebra P are called spinors. For a spinor

A one can write:

where p

~ ~J' A = A + A = P e o 2

A1 - A2 is a positive scalar, ~ is a scalar given by o 2

cos a

and j is the unit bivector A2/11 A211 with j2 -1.

Yet we find AA = AA p.

- 11 -

The matrix representation of a spinor

r a + is 6 + \

r 1J! 1 -~2l iYJ aI2 + S0 10 2 + Y020 3 + 6° 3° 1 iy is l1J!2 1J!1 t- 6 + a -

As we will see in the sequel a spinor can also be represented by the matrix

[1J!1 0] E 1+ or

1J!2 0

Compare these representations with [::]. known as the Pauli spinor.

-- 12 -

2. The Clifford algebra of space-time

Following the same lines as in section 1, we now deal with the Clifford al­

qt~bra of real Minkowski space-time M with orthonormal basis {eO

,e1

,e2

,e3

}.

M is endowed with an indefinite quadratic form II xII 2 which on the given basis

can be expressed as

Writing x = xOeO

+ x1

e 1 + x2

e2

+ x3

e3

the requirement II xII 2

the relations

1

-1 , k 1,2,3

o , k -F 9., and k,9., 0,1,2,3 .

Summarizing:

diag. (1,-1,-1,-1) .

2 x now yields

We write the (Lorentz invariant) expression eO

e1e

2e

3 as e

S' i.e.

whence

and k 0,1,2,3 .

The 24-dimensional algebra, thus introduced, mostly is named real Space-Time

Algebra ~ or real Dirac Algebra.

Remark. opposite to the complex Dirac algebra, used in relativistic quantum

mechanics, and with dimension 32, our real space-time algebra has only dimen­

sion 16.

The quantity e5

a role similar to i

portant difference is that e5

and eO

,e1,e

2,

[1'£2'£3 are commutative.

E1E2E3 in section 1, but an im­

anticommute, while i and

As in section 1 we can introduce the inner and outer products a • band

a A b as

- 13 -

a • b !.:! (ab + ba) and

a A b ~(ab ba)

There is no equivalent here for Gibbs' cross product a x b (the dual of a

bivector is not a vector in this case but a bivector again) .

We want to embed the Pauli algebra P of section 1 in the space-time algebra

STA just introduced.

To that end we define the monomorphic map'~: P ~ STA in the following way:

(n(E ) = E = e e 'V k k k 0 ' k 1,2,3 •

Obviously ~ depends on the chosen bases in 3-space as well as in space-time.

Especially notice that vectors of 3-space are mapped onto bivectors in space­

time and that

Observe also that in this algebraic set-up 3-space is not isomorphic to a

subspace of space-time but rather that the Clifford algebra of space is iso­

morphic to a subalgebra of space-time algebra.

Bemark. At first sight it seems to be more natural to embed 3-space algebra

into STA by the map

k 1,2,3

but this turns out to be undesirable. Indeed for timelike vectors we find a

negative separation in that case Le. IIxll2 < 0 for timelike vectors. More-

over in that case we would have

-eO

e 5

This relation suffers from the undesirable property that Wei) depends on

the choice of eO'

We can build up a chain of (bases dependent) monomorphic maps in the follow­

ing way:

- 14

In this monomorphic chain the image of any algebra is the even subalgebra

of the next one.

We return to this chain in a more general setting in the next section.

Compare also the similar chain in Cayleys theory of octaves where eventual­

ly associativity is dropped. Here we dropped the claim that P and STA are

division algebras.

As already out in section 1, P contains non-trivial ideals indeed.

Any element A of STA can be written as

A

and consequently as

A ~(B) + ~(C)eO

where Band C belong to 3-space algebra P.

It is also possible to represent eO

,e1,e

2,e

3 and hence

YO'Yl'Y2'Y3'Y5'

We mostly use the 4 x 4-matrices

k 1,2,3

whence one finds

with k 1,2,3 and i E It.

3

as matrices

As we shall see in the next section it is also possible to represent the

elements of STA as 2 x 2-matrices over the field of quaternions lB. In this

IH (2) representation eO

,e1

, ,e3

respectively correspond to

1 0 0 j 0 k 0 ~

[ l ' 1 [ 1 [ 1 o -1 • [1 O' k O· , 0 •

(Note that lB(2) is a non-commutative algebra over a non-commutative field.)

- is -

Just like in 3-space algebra we can write an element of STA as

A = AO + Ai + A2 + A3 + A4 .

We introduce the mapping A I+A by

-A = A

0 - A

1 + A2 - A 3

+ A4 = -e Ae 5 5

In textbooks the restriction to M of this map is called space-time inversion

pt.

Further we introduce the automorphic maps of STA:

A = AO + Ai A2 - A3 + A4

* A eOAe O

At eOAeO

Note that A = (A *) t.

* - * The restrictions to M of the maps A 1+ A I resp. A 1+ (A) correspond to

space reflection p and time reflection t.

- -We next define A is even iff A = A, and A is odd iff A -A, whence A is

The even part A of A and the odd part Aodd of A are given by even

A = ~(A - eSAe S) even

Obvious computations show that the symbols ~, * and t with respect to STA

are chosen such that they correspond to the same symbols in P by way of the

monomorphic map (p defined above. The deviations in question are caused by

the choice of <.p.

- * t A, A, A and A have similar commutation properties as their equivalents in

P. Compare again section 1.

- 16 -

The Hodge star operator * (compare e.g. [AMR]) yields for STA:

1 2 3 e e

o 123 *e = e e e

102 3 *e e e e

3 0 I 2 *e e e e

o I *(e e )

o 2 *(e e )

3 2 e e

1 3 e e

o 3 2 1 *(e e ) :: e e

1 2 *(e e )

o 3 e e

2 3 0 I *(e e) e e

3 1 *(e e )

023 *(e e e )

013 *(e e e )

012 *(e e e )

I 2 3 *(e e e )

o 2 e e

1 e

2 e

3 e

0 = e

o 123 *(e e e e ) = -1

where (as usually) is written eO :: k eO and e

k+l In this special case we find ** = (-1) where k

for scalars, vectors, bivectors and so on.

5 Writing e o e e one finds:

5 0 I 2 3 Ie = e e e e

o 5 e e

1 5 e e

2 5 e e

e

123 e e e

o 2 3 e e e

o I 3 -e e e

012 e e e

(eOe 1)

025 (e e ) e

2 3 e e

1 3 -e e

( 0 3) 5 e e e 1 2

e e

5 Note that e

o 1 2 3 e e e e

( 1 2 3) e e e

( 0 2 3) 5 e e e e

o (e e

( 0 1 2) e e e

( 2 3) 5 e e e

(e

3

o -e

1 -e

2 e

3 -e

o 1 -e e

o 2 e e

o 3 -e e

-1.

1,2,3.

0,1,2,3,4 respectively

- 17 -

It follows easily that

*A = *(AO + A1 + A2 + A3 +

S (AO + A1 - A2 - A3 + A4)e

(AO - A1 - A2 + A3 + A4)

*A =

We find furthermore:

**A

**A -A

in accordance with ** (_l)k+l.

~ S Ae

Evidently the minimal ideals in STA are closely related to the ideals in P,

as treated in section 1. Using the IH(2) representation of our 16-dimensio­

nal STA it is obvious that STA contains (again up to isomorphy) also two

independent minimal left ideals 1+ and I •

Remark that the complex Dirac algebra (with dimension 32) and isomorphic to

~(4) contains four independent minimal ideals.

The generators of 1+ and I in STA are e.g. ~(1 + e3e

O) and ~(1 - e 3e O).

Orthonormal bases of 1+ and I can be given as follows:

1+ I

----_1 (1 d 1

+ e3

eO

) 1 e3e

O) d = -(1 -

12 9 12 d2 e 1d 1 d

10 e

1dg

d3

e2

d1 d

ll e

2dg

d4 e 3d

1 d 12 e3dg

dS

e1

e2

d1

d13

=0 e1e

2dg

d6

e2

e3

d1 d

14 e

2e

3dg

d7 e

3e

1 d

1 d1S

e3e

1dg

dS

e1e

2e

3d

1 d16

e1e

2e

3dg

- 18

Note that ~dl and ~d912 are idempotents (even primitive idempotents) and

that d4 , d6 , d7

, d8

, d12

, d14

, d15

and d16

are nilpotents, however no proper

nilpotents (e.g. -~l2e3d4 is even an idempotent).

We now turn to some theorems about the orthogonal group, acting on space­

time, called Lorentz group L. Our main goal is to present a global view on

this group within STA.

We suppose that the reader is acquainted with the property that every Lorentz

transformation, modulo the parity operator p, the time reversal t and their

product pt is a so-calledrestricteaLorentz transformation L! (determinant

== +1 and pointing into the future). t

More precisely we have the group relation L+ - L/V4

where V4 {e,p,t,pt},

the Kleinian four group.

For the Lorentz group, acting on M, obviously t == -p and for the matrix re-

presentation of 9.- Lt we have t tt is the transposed matrix E p == t pt, where + of t-

Further it is possible to decompose an element of t

L+ -in a unique way- in

an element u of SO(3) and a so-called hyperbolic screw h.

u 1 u t and pup-1 == u (SO(3) is the centralizer of p) are well-known pro­

perties as well. -1 -1 t 1

Moreover php hand h = h and finally h == uzu where z is a pure

Lorentz transformation, known from physical textbooks.

Using the fact that SL(2,£) is the double covering of L! one finds in the

same way the polar decomposition of an element of SL{2,~) in a factor that

belongs to SU{2,~) and a factor that belongs to the group of positive defi­

nite Hermitean 2 x 2-matrices (again unique). For more details the reader is

referred .to [v].

We want to describe all these decompositions in terms of space-time algebra

STA.

Let x be a vector in M and let R be an element of STA with properties

-RR 1 and R = R •

As proved in [H1], page 47 the transformation L, given by

Lx RxR X E M

is a restricted Lorentz transformation and conversely.

- 19 -

As remarked previously, we have:

* space reflection (parity) p: x ~ X

time reversal

composed

-* t: x ~ x

-pt : x ~ X •

Theorem 2.1. If RR 1 and R R then R B ±e where B is a bivector in STA.

Remark. We consider the proof of this theorem in detail because argumentary

as used in that proof very often plays a role in the next sections.

Proof. For every bivector B in STA we find

B2 "" B • B + B A B .

We distinguish bivectors in complex bivectors with B A B ~ 0 and simple

bivectors with B A B = O.

Simple bivectors are called timelike if B -B > 0, lightlike if B • B "" 0 and

spacelike if B • B < O.

If we have the condition R R (i.e. R even) we can write

We distinguish the (rather trivial) case R; 2

case R2 ~ O.

o and the (more complicated)

. ) 2 0 ~ R2 = •

'rhe condition RR = 1 yields

(RO + R2+ R4) (RO

- R2 + R4

) 1

hence (R2

0) : 2

which yields after splitting:

and 2ROR4 o .

o would imply R: 1 < 0 whence R4 ±1.

- 20 -

1 gives R

RO == -1 gives R -1 + R 2

2 ii) R2 f 0 whence

1 .

It is possible to decompose the bivector R2 in a simple timelike part ~lbl

and a simple space like part a2

b2

where b1

and b2

are commutative and hence

orthogonal, i.e.

R2 = ~lbl + a 2b 2

with b~ > 0, b~ < 0, b1b

2 = b

2b

l, b

i . b;) = O.

We can adjust the magnitudes of b1

and b2

and hence the coefficients a1

and

a 2 " Substituting R2 = a1b

1 + a

2b

2 one finds

and

Choose a1,a2 such that a

Ia

2 RO.

That gives:

222 2 (a

1 - b

2) (a

2 - b

1) 1

R4 == b 1b 2

R == (a1

+ b2

) (a2

+ b1

)

Choose a1

such that 2

b2

1. a - = 1 2

Thence 2 _ b 2

1 , b2

< 0 a1 2 2

2 2 1, b

2 ~2 b

i 1 > 0

1

.

Take unit bivectors ul

and u2

along bi

and b2

, then one finds:

with 1

2 u

2 = -1 •

1) b • b 1 2

Introduce scalars tl and t2 by

This yields

R

and the proof is complete.

- 21 -

B e

Notice that u1

and u2

and therefore B1 and B2 are commutative.

Theorem 2.2a. If UU 1 and U -U * U one can write U

and spacelike.

aU 2 Remark. Compare particularly the expression e with u

for 3-dimensional rotations.

B1 Proof. We borrow from theorem 2.1 that U ~ ±e . Hence

* U

* *

B1 e with B1 simple

-1 in section 1

The condition U = u therefore implies that B1 ~ B1 and that means that B1

is simple and spacelike. (Notice that iBI is timelike.) In the minus sign

case we find

B' 1

e

where j is a bivector with j2 -1.

Note that bivectors in P correspond to simple spacelike bivectors in STA

and that vectors in P correspond to simple and timelike bivectors in STA,

but of course not conversely.

o

o

If HH :; 1 and H H Ht one can write H B2

±e with B2 simple

and timelike.

- 22 -

Proof. As in the proof of theorem 2.2a we have again, starting from H

* -B + 2 _e

* Hence H Ht yields B2 which means that B2 is simple and timelike.

Note that in contradistinction to the situation in theorem 2.2a it is here B2

impossible to hide the minus sign into e .

Theorem 2.3. :!T!!h~e.....£~~..2!~~~~~~

If RR

and H

* 1 and R R then R = HU where U = U t B = H = ±e 2, B2 simple and timelike.

Proof. We call A RRt and find respectively:

i) AA

ii) A

1

A t

A

B1 e B1 simple and space like

Appealing to theorem 2.2b, we can write A = RRt

Now choose H !:i * A and U = HR and the proof is complete.

[1

[]

Remark 1. Contrary to the proof of theorem 2.1 in this case B1 and B2 do not

commute.

Remark 2. We can also write R a ib h db- 1 . l-k ±e e were a an are S1mp e t1me 1 e non-

bivectors.

Summarizing we have found that every Lorentz transformation can be written

as (B any bivector) and decomposed into the factors (if present)

* 1. x f-r X (pari ty)

-* 2. X '* x (time reversal)

a -a x i+ e xe 3. (hyperbolic screw)

ib -ib x '* e xe 4. (pure rotation)

where a and bare timelike bivectors.

An even multi vector A in STA is called a

23 -

Theorem 2.4. Let A be a spinor then A C e where C is also a spinor.

Proof. We split A in parts

A AO + A2 + A4

whence

A AO - A2 + A4

and

(AO + 2 2 Se

S AA A4) A2 a + be

S pe

1 -~Se5 -~ Next introduce R by R ~ peA. This yields RR 1 and appealing to

theorem 2.1 we find that R = ±eB and thence

A + ~ is/2 B -p c e

wh(:!re C is any spinor.

Remark. Notice that (in contrast to spinors in the Pauli Algebra P) it is

possible that A ~ 0 but P = O.

For example A = 1 + e3

eO

~ 0, but AA = p2 0, hence p O.

In the sequel the expressions

k 0,1,2,3

will play an important role (in particular the cases k

spinor in STA.

0,3) where ~ is a

~

First note that wekW is a vector, k = 0,1,2,3. Indeed, in general a multi-

a vector iff v = v and v = -v and as can ~ --

Wekw wekw and ~ekW = -~ek$·

[I

vector v in space-time algebra is

be immediately seen we have that

Moreover we conclude that wek~

formation and p is a scalar.

B -B pe e e = pLe where L is a Lorentz trans-k k

- 24 -

3. The Clifford

In this section we generalize a number of notions of section 1 (observe the

different notation for in section 2). First of all we shall de-

fine our notions independent of a particular1) chosen basis and secondl;­

we now deal with dimension n in stead of the particular choice n = 3.

Let V denote a linear space over a field ]F. T is the (associative) tensor

algebra of V, denoted by

T = I #v . p

From now on we confine ourselves to cases where

1. ]F is commutative, mostly]F IR

2. V has finite dimension n

3. V is endowed with a non-degenerate symmetric 2-tensor (0,.)

4. On V a (positive) oriented volume is assigned in accordance to (',').

Next we consider the ideal 1 c T generated by all expressions of the forrn

x ® x - (x,x).l where x E V.

Thence the Clifford algebra of V is defined by

C = T/I .

Notice that for any vector x E: V we have that x2 = (x,x) = IIxll2 E IR. For

more details the reader is referred to [wJ, volume 2 , page 41.

We can decompose every element A in C in terms Ak which are homogeneous of

k, i.e.

n

A I Ak k=O

where AO denotes the scalar part of A, A1 the vectorial part, A2 the bivec­

torial part and so on. A is called pseudoscalar part. n

Given

A and B

1 ) except for the table of matrix representations.

- 25 -

we shall write

n AB I AkB~.

k,9.,=O

One can prove the very fundamental rule ([HS], page 10):

r (3. 1) AkB£ = I (AkB9.,) I k-£ 1 +2m

m=O

where r = l:!(k + £) - l:! 1 k - £ I·

This rule shows which grades of AkB£ actually occur.

Next define the inner product Ak • B9., of Ak and B9., as the first term in the

righthand expression of 3.1, i.e. the Ik - £I-grade element

if k > 0, 9., > 0

and

if k£ = 0 .

Similarly we define the outer product Ak A B£ of Ak and B9., as the last term

of 3. 1, viz.:

~ • B9., and Ak A B9., satisfy the following commutation rules:

• B 9., • A

k k $ 9.,

Regarding V as a part of C and applying all these rules and definitions to

a,b E V (Le. k 9., = 1) one easily finds

ab = a • b + a A b

where a • b = ~(ab + ba) I the symmetrical part of ab and a A b ~(ab

the anti symmetrical part of abo

Note that ab + ba (a + b)2 _ a 2 2 b E IR. Compare also the results of

sections 1 and 2.

Similarly one can derive e.g. the relations

ba) I

- 26 -

k - (-1) ~a)

and so on.

A more complete list of rules as the ones, presented above can be found in

J. We only have mentioned the rules that we need in this paper.

* The conjugate Ak of ~ is defined by

We next define the inversion or main involution as the mapping that takes

* into Ak ·

One has

* n

(_l)k~ A I . k=O

* * A is called even iff A A and A is odd iff A

Let Aev be the even part of A and Aod the odd

tivectors A and B can be written as:

-A •

• The product of the mul-

A B + A B d + A dB + A dB d • ev ev ev 0 0 ev 0 0

Using 3. 1 we find

and

(AB) ev

(AB) od

Thence follows immediately

* Property 1. (AB) * * A B •

+ A B od ev

The next operation, called reversion, or main anti automorphism is introduced

by the Hermitean conjugate A: of Ak in the following way:

and thence

- 27 -

As to be suggested by its name, we have

t t t Property 2. (AS) = SA.

We do not give the proof in detail of property 2. The most straightforward

way is to introduce an orthonormal basis {el, ... ,e } in V. n

Then one finds that

e. e. lk lk-l

and so on.

e. e. l2 II

e. e. lk_l lk

We can combine both operations to a third one, viz.

A (A*)t (At)*

One finds immediately

'-v'

Property 3. AS BA.

The scalar product A * B of A and B is defined by

A * B

with property A * B B * A.

We shall say that A is orthogonal to B iff A * B = O.

Now it is possible to introduce the square length II AU 2

of a multivector A

as

II All 2

Note that in case of a positive definite metric one has:

II All '=' 0 iff A = 0

The following properties are easy to check:

o if k f:. I/,

ifk II, > 0

- 28 -

A * B

The next property necessary and sufficient conditions for a multi-

vector x to be a vector.

Let, I denote the (only orientation dependent) unit pseudoscalar n

I n

• •• A e n

A multivector x is a vector if and only if I A X n

Proof. Let x be a vector, then I x I A X + I --- n n n whence I x I . x . n n

.

o and I x n

x . Obviously

Conversely, let I . x = I x and I A X 0, this yields n n n

I x I A X + I • x • n n n

On the other hand, applying 3.1 to Ak = In and B.I'. x, we find

In - ~I + 2 = n + land n > l •

Hence .I'. 1, i.e. x is a vector.

I • x. n

I A x n

we borrow the introduction of the Hodge star operator * from [AMRJ.

It is a dual operator (it maps k-grades onto (n - k)-grades).

It can be introduced in the following way;

0

o

Let w be a given k-form (i.e. grade kl, u a chosen n-form (orientation) and

e any form.

The dual * w of W can be defined by

e A * W = (O,wiu .

Remarks.

1. Evidently * depends on the chosen orientation and the given inner product.

2. * is independent of the chosen basis.

3. ** is a scalar operator. One can derive that

k (n-k) + n-t ** (-1) 2

where k is the grade and t p - q the sign of the inner product and

p + q n.

4.

Sa. * u

5b. * 1

e 1\ * ex

n-t

(-1) 2

u .

- 29 -

Introducing an orthonormal basis {el, ••• ,en

} in V

II xII 2 (x2

1 + ... + x2

) p+ p+q

and

1 1 s k s P

-1 p + 1 k s n

k f R, •

V n

V we can write: p,q

p + q n

The associated Clifford algebras C of V can be represented by matrices p,q p,q

over one of the fields IR, ~ or IH if p - q # 4m - 1 and by pairs of such

matrices if P q 4m + 1 (whence n is odd). The number of rows and columns

of the intended matrices are powers of 2.

We write the matrices as IF (2r

) where IF = IR, <t or IH and the pairs as

2lF (2 r ). Note that in all cases C has dimension 2n. pq

In the next table the reader can find the basic tools and necessary rules to

present the matrix representations of all Clifford algebras. In the table

the matrices are presented for n ~ 9.

Especially C30

and C13

are indicated in the table because they play a very

dominant role in the sequel.

All details can be found in [ABS] and in [p], page 250.

- 30 -

Table

Bb.::'l(, 't-ools Rulc-s ~~~~~-

- !Ji Ci0

2!Ji COl! l. C C (symmetry)

p+Lq q+l ~¥

COl = lR

2 " Cp+1,q+l ® lR (2) C

10 COl = TIl 2"

lR (; 3. C ® m (16) (periOolClty:'

2 p,q C

03 Ih

C20

Cll

CO2

(04 = ll! (2) m (2) m (2) ll!

- C{4) Pauli

c;] C21

C12

C03

= Ii( (8) 2m (2) 2iH II: (2) [(2)

r (8) 2 2 2 Dirac -07 x +y +20

C40

C31

C22 ~

C04

'~C'b = IR 116) m (2) lR (4) lR (4) m (2) m (2)

222 2 t -x -y -20

C41

C32

C23

C14

2m (2) [(4) 2m (4) 11:(4) (2) (:(4)

C42

C33

C24

C06

m (4) m (4) m (8) m (Sl m (4) m (4) m(s)

C52 C43 C34

C25 C16 eC,)

«8) 2m (4) IC (S) 2lR (8) 1C(8) 2m (4) 11:(8) 2

m (8)

Cso CE,:/ C53 c44 C35 C26 C08

m(6) lH (8) m (8) m (16) m (16) mrS) mrs) m (16) lR (16)

C54 C45 C36 C27 C18

(16) [(16) 2m (8) [(16) 2m; 16) e(16) (8) 1[:(16) 2m; 16) It ( 1 E.

Symmetry

- 31 -

Again from [pJ we borrow a simple rule to obtain the even subalgebra cO pq

from the Clifford algebra C , viz. pq

cO '" C and cO C p,q+l pq p+l,q q,p

Examples:

1. 0

cOl cOO lR

2. 0 '" CO2 = COl C

3. 0 c30

cO2

m

4. 0 1[:(2) c13

C12

p

5. 0 '" 0 ~ '" m (2) '" c14 = c 41 = C

13 = = STA

The table gives moreover that cl4

'" 2m (2) and c41

- 1[:(4).

Compare especially the monomorphic chain (section 2):

We close this section with the introduction of a spinor in the Clifford al­

gebra C . As in the sections 1 and 2 we mean by a spinor an element of the n

even subalgebra cO of C • n n

As proved in several textbooks the unitary spinors (length 1) constitute

the group, called spin V and this group double covers the rotation group of

lRn

V.

Compare the example at the end of section 2 where the spinor A # 0 but 2

p AA O.

- 32 -

4. Differential Clifford

He:; s:':,ac thJ..s section 'Nith some elementdry and basis dependent"_ oE':finitior,s.

Let {C1'€2' } denote an orthonormal righthanded basis in E.

We introduce the vectorial differential operator V by

Applying v to scalar fields ~ and to vector fields a one easily finds

v~ grad ~

Va II • a + II 1\ a v . a + i(V x a)

= diva + i rot a .

Note that ak

2 The scalar operator II represents the Laplace operator.

One has:

v • V

Turning to Minkowski space-time with orthonormal basis {eO

,e1,e

2,e

3} we next

define:

a eOaO - e 1a1 - e 2d2 - e 3a3

01230 Introducing the dual basis {e ,e ,e ,e } with e k

we can write more generally a = e aka

k eO and e -e

k, k := 1,2,3

Hestenes [ uses the symbol k o as 0 := e ak

instead of a but this seems

confusing, for in most textbooks 0 denotes the d'Alembert operator i.e. k 2

o = (e ak

) .

In recent physical literature one often meets the symbol P (d-slash) as

'. = yk"k where yO and yk (k 1 2 3) t . t t' p 0 := YO = -Yk =" are ma rLX represen a Lons

for eO

,e1

,

we mention the following simple rules for the vector operator d:

1.

where Al denotes a differentiable vector field in STA.

2. k 0,1,2,3 and x

a := e • d k k

- 33 -

In section 2 we introduced the monomorphic map ~: P + STA defined by

Considering V as a vectorlike operator we shall also write:

w(,'1) = V

Whence we find:

Eld l + E2 d2 + E3 d3

e1e

03

1 + e

2e

032 + e

3e

03

3

The field theories we want to describe in the next sections need the use of

differential operators. Just above we introduced this operators as

V = E:l d l + E:2

d2

+ E3 d3 and 3 = e0

30

- e1

d1 - e2

32

- e3

33

,

Obviously it is more convenient to introduce upper indices and to write k k

V = E 3k

and 3 = e dk

•

From an algebraic point of view these operators behave like vectors. Both 123

operators are expressed in the chosen orthonormal bases {c ,E: ,E } and o 123

{e ,e ,e ,e } but the laws of physics we want to investigate cannot depend

on the frame of reference that we chose for the purpose of our description.

Therefore we look for possibilities to introduce differential operators in

a coordinate free way.

Inevitably we then arrive at the exterior derivative d of Cartan and the

codifferential o. Our first aim is to investigate their behaviour in the

Clifford algebras of 3-space and space-time. We start with a brief summary

of the properties of d and o. Details can be found in [AMRJ,

For convenience we make the following identifications:

in 3-space respectively

in space-time.

Let

1 e

2 e dz

- 34 -

e

be a k-form, then the exterior derivative de is defined by

de

Remarks:

1. d maps k-forms onto (k + i)-forms.

jk A ••• A du

2. The definition of d does not use any inner product or orientation.

3. It can be proved that the definition of d is independent of the chosen

basis.

4. d2

== O. £

5. d(a A S) == da A S + (-1) a A dS where a is any £-form and S any m-form.

The codifferential 0 is defined by

n-t n(k+i)+i~

(-1) *d*

and has the following properties:

1. 6 maps k-forms onto (k -i)-forms.

2. 62

== 0 (** is a scalar operator).

Example 1. 3-space with (dual) orthonormal basis {81

,82

,83

}.

Let A be any multivector field of the Pauli algebra P.

We write (as earlier)

A == AO + Al + A2 + A3

k k £ 1 2 3 a + a

k8 + a

k£8 8 + Se: e: e:

We find for the exterior derivative d:

1. dAO d1

ae: 1

d2

ae: 2

+ d3ae: 3

+

2. dAi

(d2

a3

-2 3 3 1

d3

a2

)e: e: + (d3

a1

dl

a3

)e: e:

1 2 + (d 1 a 2 - d

2a

1)e: e:

- 35

3.

4. dA3

= 0 •

Further we can derive: (0 = (_l)k*d* in this special case)

1.

2.

3.

4.

o

-dlul - d2U 2 - d3a 3

1 ( d 2 a 12 - d 3 a 31 ) e: + ( d,3 a 23

3 + (d

1U

31 - d

2a

23)e:

2 3 3 1 1 2 -(dlSe e: + d2ae e + d38e e )

In this example we write (as earlier)

and find:

1.

2.

3.

4.

5.

A = AO + Al + A2 + A3 + A4

dA1

k k t k ~ mOl 2 3 a + uke + uk~e e + ak~me e e + Be e e e

o 3 1 2 + (d

Oa

3 - d

3a

O)e e + (d

1a 2 - d2u 1)e e +

2 3 3 1 d

3a 2 )e e + (d

la 3 - d

3u1}e e

013 (d

Ou l3 - dl a

03 + d

3u

Ol)e e e +

012 + (d

Oa

12 - d

1a

02 + d

2u

01)e e e +

023 + (d

Oa23 - 3

2u

03 + 3

3u

02)e e e +

1 2 3 + (31u23 - 3

2a

13 + d

3a

12)e e e

o .

- 36 -

Finally we find for 0 *d*

1. OAO 0

2. OA1 dOaO - d1a 1 - 320. 2 - °

30. 3

3. OA2

-(31

0.01 + 320.02 + 3

3(

03)e 0

-(dO

a01

+ 32

0. 12 + 33( 13)e 1

-<dO

a02 - °10. 12 + °3a.23 )e 2

- (dOa.0'3 - 31

0.13

- 32

(23

)e 3

0A3 (32

et012

+ o 1 1 2 3

3et

013)e e - <00a.012 - 3

3(

123)e e 4.

+ (aIa012 -

o 2 03et023)e e - (aoct

023 -

2 3 31(123)e e

1 3 - (° 10.013 + 3

20.

123) - (3

00.013 + 3

2(

012)e e

5. OA4

I 2 3 dOSe e e

023 + 0ISe e e +

031 32Se e e

o 1 + d3Se e

k k Applying v ~ E dk and 3 = e dk

to multivectors in P and in STA, one finds:

In

\l • A 0 -OA 0

0 v " AO dAO

V • A1 -OA 1 v " A1 dA

l

v · A2 -OA v " A2 dA2 2

V · A3 -OA V " A3 dA3 . 3

Whence summarizing: v . A -OA and v " A dA and therefore

\l d - 0

Compare aAk = a " ~ + a • Ak and aA = a " A + a • A in section 3.

The first righthand expression has grade k +1 and the second one has grade k-l.

Similarly we have for v:

dA - oA k k

- 37 -

where \j " Ak d~ has grade k + 1 aqd 'V • Ak = -OAk has grade k -1.

We left to the reader to check that for space-time we find in the same way:

aA 3 A A + a • A = dA - oA

3 A A = dA and a • A = -oA

Summarizing our main conclusion is that both in space and in space-time we

have the fundamental rule

As known d and 0 are coordinate free and hence a is.

Now we are justified to define for space and space-time the operator a as

a = d - 0 also independent of any frame of reference.

a d - 0 is known as the Hodge-de Rham operator.

-(do + od) is a scalar operator known as

Laplace-Beltrami operator.

In section 5 (electrodynamics) and in section 6 (Dirac fields) we make use

of the results obtained here.

In [CC], page 346, Hestenes constructs more generally a one-to-one mapping

of the Clifford algebra STA onto the algebra of differential forms with

respect to space-time. This representation of STA by differential forms is

called the Kahler algebra.

We return to this algebra of Kahler in section 6 where we point out the

Dirac-Kahler equation.

Closing this section we propose an alternative for Leibniz' rule for diffe­

rential operators working on Clifford algebra valued functions. We start

with some notations:

k For of = e dkF we write

k

k shortly a = e d

k and likewise 3 • F

be written as a • k

(w,3)F = (w,e ldkF

e • ok' as (w, a) =

k d A F = e A dkF as

k (w,e )3k

(ordinary directional derivative).

k a A = e A Ok and

- 38 -

The relations dA d'A+dAA -oA + dA, obtained above, now can be

written as

d • = -8, d A

Consequently one has:

d ==

and in 3-space

rot k x d e k

d and a

k -e • d k

d - 0 .

Definition. Let A be a multivector field. The directional derivative (A • a)

is defined by

(A • a) F k

~(e A

Example. If A is a vector field w then one finds (w • a)F as introduced above.

Theorem 4.1. 1) (Leibniz' rule for (A • a».Let F and G be multivector fields,

then

Proof.

In this

* (A • a) (FG) «A • a)F)G + (AF • a)G - A «F' a)G)

k (A • 3) (FG) = ~{e A * k k * k A e ) (dkF)G + ~(e A - A e )F(akG)

k *k *k **k] «A' d)F)G + ~e AdkG + [-~{A e F - Fe) - ~A F e dkG

«A' a)F)G + (AF • a)G - A*{F • a)G •

we used the fact that in any algebra

~t(FG) dF G dG dt + F dt

'I'heorem 4.2. 1) (Leibniz' rule for a). Let F and G be multivector fields then

d(FG) = (3F)G + F*{aG) + 2(F • a)G .

l)Due to J. de Graaf (oral communication)

II

- 39 -

Proof.

((IF)G + 2(F • a)G + F*«lG) •

Remarks.

1. If G is a constant multivector field then (l(FG) ((IF) G.

2. If F is a constant multivector field then (l(FG) ~ F(aG) in general.

3. a(aF)

In case of a positive inner product and an orthonormal base (ek ) this be­

comes

4. Similarly one finds

k (a • a) F = (e • '\) aF

and

(d A (l)F = 0 •

Now we are justified to write

a • (l + 0 a • a •

(compare a • a for vectors a.)

o

- 40 -

Sa. The partial differential equation of = J

In this section we analyse the simplest coordinate free differential equa­

tion that can be written down in space-time algebra.

Let F be a bivector field and J a vector field in space-time.

We consider the equation

~=J.

Splitting yields:

o • F + 0 A F J

whence

d • F J (vectorial part)

oAF o (trivectorial part)

Remark. Using the relation 3 = d - 0 one finds

dF = 0 and of = -J .

Compare [AMRJ, page 500, where has been stated dF = 0 and of = J. This dif­

ference in sign is caused by the chosen metric in [AMR] (-,+,+,+) in stead

of our choice (+, ,-,-).

Now we try to express F as the gradient of a vector potential A. viz.

F aA d • A + a A A

Since F is a bivector field this is equivalent to F

the condition1

) a • A = 0 is satisfied.

a A A provided that

The vector A is not uniquely determined by F = aA. Consider A1 A + oX 1 with a2

Xl = 0 then evidently

This freedom in the choice of A is called gauge freedom.

Here we meet the first and rather trivial of an (Abelian) gauge group.

Compare A + a(X 1 + X2

) and of course

holds,X = 0 is the unit element and -X is the inverse element

of X.

1) The so-called Lorentz condition, compare section 5b.

- 41 -

Given aF = J and F aA we find

d • J + 3 A J and thence

3-J=0.

F

which is sometimes written in the matrix representation:

v -+

11 0 -E -E -E + 1 2 3

El a B3 -B F 11V 2

E2 -B 3

a Bl

E3 B2 -B 1

0

We shall also write F

E

and

B

Appealing to the results of section 2 we find with respect to F:

G

whence by substituting -B for E and E for B in FllV :

r 0 Bl B2 B3

0 E3 -E -B1 2

-B2 -E

3 0 El

-B 3 E2 -E

1 0

The equation dF J, described in STA can also be interpreted as an equation

in time dependent multivector fields in 3-space algebra P.

In fact it means that we pull back the equation, given in space-time, into

equations in 3-space.

- 42 -

In section 2 we introduced the monomorphism~: P ~ STA by

Therefore in suggestive 3-space notation we get:

E + iB

From section 2 we quote the decomposition

For convenience we shall from now on drop the caps and shall write V instead

of V and i instead of i = e5

, unless confusion is likely.

Yet we write eO] = p - J where

J JOe O + J

1e

1 + J 2e

2 + J3

e3

p JO

J J1

C1

+ J2

C2

+ J3

C3

.

Starting from dF = J, or equivalently eOdF

(dO + V) (E + iB) = p - J or

dOE + VE + idOB + iVB = P - J , i.e.

dOE + VoE + iV x E + idOB + iV 0 B - V x B = P - J .

Splitting in scalar, vectorial, bivectorial, trivectorial and pseudoscalar

parts we get:

(5.1) V 0 E P or div E = P

(5.2) aoE V x B -J or dE - - rot B dt

= -J

(5.3) V x E + dOB 0 dB

or rot E dt

(5.4) V 0 E 0 or div B 0

It is also possible to solve E and B in terms of A and F. (Again we then

pull back from space-time to space.)

F 2

dA or equivalently E+e5

B = deOA. We write

- 43 -

~ AO

A = A1El + A2E2 + A3 E3

whence eOA == q, - A.

Now E + eSB == de~A yields:

E + iB (dO - V) (<fJ - A) Le.

E + iB = aO~ + VA - v<fJ - dOA

or equivalently

E + iB

Considering dOq, + V • A == 0 (d • A == 0) one finds after splitting:

(5.S)

(5.6)

B v x A

E == -v<fJ dA at

The expression F2 has only scalar and pseudoscalar parts:

F • F + F A F •

Using coordinates and F ~FJlVe e one finds Jl v

F . F _~F]J\.lF E2 _ B2 )J\.l

F A F _Fa8FJlV E i == i(E . B) a i3]l 'V

where E 0 is the antisymmetric Levi-Civita tensor. al-']Jv

Finally one can consider the scalar expression

L = -~F • F - J • A

better known if written in components

The results of this section are due to M. Riesz [RJ.

- 44 -

Sb. Electrodynamics

Perhaps there are some readers who in the mathematical descriptions in sec­

tion Sa recognized some physics. For that reason we make in this section

some interpretative remarks.

The scalar field ¢ = AO is known as the potential and the 3-space vector

field A as the vector potential.

Similarly we have p = jo is the charge and J is the three-current.

The components FVV of F are usually written as FVV

The electric and magnetic components E and B of F = E + eSB are too familiar

to reguire comment and the same is true for the equations 5.1 - 5.6 of sec­

tion Sa.

a • A

a J~ Jl

o or a AV = 0 is called the Lorentz condition and a • J = 0 or Jl

o is known as the continuity equation or the conservation of charge.

L is known as the Lagrangean density of the electromagnetic field and the

present coupling of J and A plays a fundamental role in Dirac fields (sec­

tion 6b), Yang-Mills fields (section 7b) and Glashow-Weinberg-Salam fields

(section 9b).

- 45 -

6a. The partial differential equation D¢e 5 = m¢

Let A be a vector field and F a bivector field, satisfying F

in grades

F = <lA

yields (as in section 5)

a • A = 0 and F = a A A .

Consider now the transformations $ + $ and A + A, given by

(6.0)

qA qA - aa.

<lA. Splitting

where q is a constant and a. is a scalar field, satisfying a2a. O.

Yet we introduce the differential operator V

Finally considers the vectors

Theorem 6. 1 •

~

a) ] ] c) F == F

b) 13 = S d) V~

Proof.

j ~ -a.eSe 3e O a.e5e3eO~

a) ljIeO$ =: we eOe W

- ';:: -ae5

e3

eO a.e5e3eO~

b) 13 'Pe 31j1 = ljIe e 3e W

e l1V by Jl

,..., ] $eOW

ljIe3'iP' = S

.

-a.eSe 3eO Observe that e

5e 3e

O and hence e commute with e

3 and eO but not with

e1

and e 2

c) F d A A o A (A - 1 act) q

oAA=F.

o A A - 1. a A aa. q

d)

whence

v ~ J.l

d ( J.l

d IjI J.l

Next introduce the expression

(6.1) L

Theorem 6.2. L L.

Proof. We can write

- 46 -

• F

• F •

and one immediately sees that the claim is a direct consequence of theorem

o

6.1 Cd. 0

Remark In section 6b we shall prove that starting from (6.1) it is

to derive the equations:

(6.2)

or

or (6.2' )

i) dF = J (compare section 5)

ii) 31j1eS

e3

eO

+ qAIjI

OljleSe 3e O = mljle O

Oljle = -mljie 5 3

Remark. As is immediately clear equation (6.2) is invariant under the trans­

formation 6.0.

- 47 -

Theorem 6.3. a • J = O.

Proof. One can write (6.2) as

(6.2")

Lemma.

(6.3) a • J

Proof of the lemma.

a • J = (oJ) o

(ell(o,,1jJleof)o + (e ll1jJe 0 $) = ,.. ° \l 0

(e\1(o\l1jJ)eO*)O + «a\l~eofe\1)o =

e \1 ( a \11)1) eO

$) 0 + e \1 ( a i) e O$') 0 =

substituting (6.2") in (6.3) one finds:

-2q(eSA1)Ie3$)0 + 2m(1)Iele2Wlo =

~ i8 ~ -2qp(esARe3R)0 + 2mp(e Re 1RRe2R)o.

Now observe that ek = RekR, k = 0,1,2,3 is an orthonormal basis of space­

time. Therefore:

substituting A = Alle~ one finds easily (esARe3R)o

the proof of theorem 6.3.

Starting from the relation

we yet derive:

o and that completes

D

r

- 48 -

Because we have:

v V l/! ]J \)

Whence we find:

Le. (V V - V V )l/! = -qed A - d

vA,,)$e

Se

3e

O ]J \) \) ]J ]J V ~

[V]J,VvJI/! = -qF]J\)~eSe3eO •

We now return to the equation

(6.2)

and multiply both sides on the right with the factor ~(1 + e O)(1 + e3

eO)'

One finds:

~d$e5e3(1 + eO (1 + e 3e O) + qtj;~ (1 + eO

)(1 + e3

eO

)

ml/!~eO (1 + eO

)(1 + e3

eO

) or

a~l/! (1 + eO) (1 + e3

eO

) e2

e1

+ qAl/!~(l + eO) (1 + e3

eO

)

ml/!~(1 + eO) (1 + e 3e O) .

We call ~I/! (1 + ~ and multiply on the right by the factor

• This yields

(6.4) a~e5 + qA~ = m~ or

Introducing

we can write shortly

(6.5)

with the condition ¢e3

eO

¢.

Remark 1. Note that $, satisfying ~e3eO

as pointed out in section 2.

$ belongs to the ideal 1+ in STA,

- 49 -

Remark 2. Equation 6.5 seems to be more symmetric than its equivalent

V~e5 = -~me3 but observe that ~ is a spinor (even multi vector) while

~ = ~~(1 + eO) (1 + e3e

O) is a much more complicated expression.

One can prove the equivalence of

(6.6)

and

(6.7)

{d~eSe3eO + qA~ = m~eO

$ even

Starting from 6.6 we showed above that one finds 6.7.

Conversely we can write 6.7 as

{d¢e3e Oe S + qA¢ = m~

cjle3

eO

= cjl •

Substitution of ¢ $1 + $2e O with Wi and $2 even, gives:

Splitting in odd and even parts yields:

and a~le2el + qA~l = ml/J2eO

a~2eOe2e1 + qA~2eO = m$l •

Multiply the second one on the right by eO and conclude

31/11 e 5e 3eO + qA~l

3W2e5e3eO + qAl/J2

Add both equations and name $1 + $2 = ~ (1/1 even).

Hence

m~eo ' i/I even •

0, i.e. cjl

- 50 -

Thus we have derived (6.6) from (6.7).

Observe that the inverse of the relation ~

presented by

(of course accompanied by ~e3eO = ~).

Finally we want to transform the following relations -associated with w­into equivalent relations associated with ~

a) J .: WeO

$'

~

b) s = We 3W

c) W !/Ie -cte

5e

3e

O

qA = qA - act

d)

e)

We find successively

a' )

b' )

C '}

d ')

e' )

J jl

s JJ

L

t (~ eOeJJ~)O

t -(¢ eOefleS¢eS)O

-ae <Pe 5

For the proofs note in particular that

~ ~

and ¢ ~ ~(1 - e

3e

O) (1 + e

O)$

¢t ~(1 + e3

eO

) (1 + eo)w t .

- 51 -

Proof of a'.

(~

t ~ ($ (1 + eO) (1 + e 3e O)( 1 + e 3e O) (1 + eO) 1jJ e Oe

l1) 0

t ; ~(1jJ(1 + eO) (1 + e 3e O) (1 + e O)1jJ e Oe

l1)O

t t (1jJ(1 + e

O)$ e

Oe

l1}O ; (tP1jJ e

Oe

l1}O

(the omitted factor is odd and hence has zero scalar part)

Proof of b'.

(after omitting the odd part)

-= (tPe 1/Je) ~ e • 1jJe 1jJ = s 3 11 0 11 3 11

Proof of c I. _ -ae

5e

3e

O ~ = ~ljJ (1 + eO) (1 + e

3e

O) = ~1jJe (1 + eO) (1 + e

3e

O)

l:;1jJ (1 +

~1jJ(1 +

J[(l + e3eO)cos a

-ae5

) (1 + e3

eO

)e

e 5sin a(! + e 3eO)]

'-ae <pe 5

Proof of d'. Multiplication on the right of VtP

~(1 + eO) (1 + e 3e O) yields:

- S2 -

Now multiply by e3e

O on the right, then

Proof of e'.

t ~d1jJ (1 + eO) (1 + e 3e

O) eS~ (1 + e

3e

O) (1 + eO) IJ! eO) 0

t + (qA - m)~lfJ(l + e

O)(l + e

3e

O)(1 + eO)tjJ eO)O ==

~

(~a\j!eS (1 - eO) (1 + e3

eO

) (1 + eO) \j!) 0

+ (qA - m)~1jJ(1 + eO) (1 + e3

eO

) (1 + eo)~)o

(Cll/leS

e3

(1 + eo);; + (qA - m)l/IO + eo)~)o =

(aljJe5e3~ + qAlJ!eo~ - ml/l~) 0

«ClljJeS

e3

eO

+ qAljJ - m1jJeo)eo~)o [l

- 53 -

6b. Dirac fields

In this section we want to connect the mathematical descriptions of section

6a with Quantum Electro Dynamics (Q.E.D.).

For that reason we start with a brief summary of Q.E.D. as one can find in

e.g. [JPJ.

The state of a classical physical system can be described by a set of com­

plex-valued functions cp l' ... ,4> k' called fields.

The Lagrangean density

is a scalar function on these fields and their first partial derivatives.

The evolution of the system, i.e. the time dependence of the fields is de­

termined by the field equations

(6.8)

4>k and L describe the physical system.

In case of Q.E.D. a system of electrons, positrons and photons is described

by the Lagrangean density

where F = a A - a A ~ ~v ~ v v v'

{y~}, ~ = 0,1,2,3 are the 4

t '¥ YO'

x 4 Dirac matrices characterized by

y~yv + YVy~ 2g~V where g~V diag{l,-l,-l,-l}. The electron has charge

q = -e.

Putting <Pk ~ in 6.8 one finds the Dirac equation

(6.9)

Remark. As mentioned in section 1 Dirac obtained this equation by factoriza­

tion of

( O~ 1~ 2~ 3~ )2 Y 0 0 + Y °1 + Y °2 + Y °3

- 54 -

If we put 1>k A in 6.8 we get: 11

dL d

dL 0 ~ \> d(d A )

11 \) ]1

where

and

whence

Together with its dual one recognizes the Maxwell equations, here written

in tensor form (compare again section 5).

For more details, see [JPJ.

Besides the current J11 = ~y]1w one can introduce the spin current

The so-called local gauge transformation can be defined by

-ia(x) , e 1jI

qA = qA 11 11

d a . 11

The gauge invariant derivative is

v = a - iqA II Jl ).I

with property [V ,V ] II \)

-iqF (compare section 6a). ]1\>

As known from literature one has:

-J J

).I' S F F

11\)' L L

Jl \1 11\1 and

1) - e-ia(x)V I/J ).IW .

11

The easiest way to make contact with the developed mathematics in section

6a is the following:

Write

- 55 -

r 1/1 1 a1 + i8

1

'¥ = 1/1 2 a

2 + i82

Wl~l + 1/12~2 + 1/13V3 + W4V4 W3 a3

+ iS3

W4J a4

+ i84J

where ~1 = [~ 1 and so on.

D J . Substitution of the (evident) relations

yields:

The quantity in parenthesis appears exactly as the matrix representation of

an even multi vector (spinor) of the real Dirac algebra.

We write

(6.10)

where M(~) means the matrix representation of the spinor ~ in STA.

As is immediately clear from (6.10) there is a I-1-correspondence between ~

and W.

From the Dirac equation

(6.9)

one finds

Although ~1 does not have an inverse, we can write

(6.11) D

56 -

because the relations

show that the matrix representation of even multivectors, operating on ~1'

generate a complete basis for Dirac spinors.

Note that the same is true for odd multivectors, because YO~l ~1'

(6.11) is the matrix representation of

i. e.

(6.6) {a~e5e3eO + qA~ = m~eO

tjJ even .

Evidently it is possible, starting from 6.6 to obtain 6.9 and therefore the

equivalence of 6.6 and 6.9 (and hence of 6.7) is proved.

Remarks.

1. From (6.10) one can derive

~1 -l/J 2 l/J 3 l/J 4 l/J2 l/J 1 l/!4 -l/J

M(1j!) 3

= l/!3 tjJ4 l/J 1 -l/J

2

W4 -l/! 3 W2 W1

whence we find

Tr(M(l/!»

2. Note that e 2 ,e 1 and eO in (6.2) are vectors and that Y2'Yl and YO in (6.9)

are matrices.

3. (6.9) contains a factor i E ~. In (6.2) the equivalent factor is

e2

e1

, a unit bivector in real space-time.

Yet we show the equivalence of the following expressions:

a) J l/!eOl/! a ') J '¥y'l' ~ ~

b) s = 1).Ie 1).1 b ') s -i i¥Y~Y5'1' s ~

- -aeSe 3eO -ia c) l/! l/Je c ') 'I' e 'I'

- 57 -

d) d ') V = a - iqA 11 IJ IJ

e)

The proofs need the following observations:

H) J , sand L are real scalars. 1J IJ

Hi) 1J11Ji = l:i(1 + YO) (1 - iy2y1) = diag(1,O,O,O).

iv) l:iTrM(lj!) = (lj!)O'

Proof of

qiy '¥ 11

~

\Tr{yl1

lj!(1 + YO) (1 - iY2Yl)YOlj!) ==

~ ~

l:iTr(yll

lj!(1 + YO)lj!) == (yl1

lj!(l + YO)lj!)O ==

~ ~ ~

(ylJlj!lj!)O + (Ylllj!YOlj!)O 0 + (YllWYOlj!)O = Jll

• lj!YOlj!

Proof of b) •

-i~y Y '¥ = _i'¥ty Y Y '¥ = 11 5 0115

-l:iTr(Y1JY5$(1 + YO}Y2Y1YOlj!} =

-\ Tr(YllY5$Y 1YO$) - l:i Tr(YllY5lj!YOY2YlYO$)

~ ~

\ Tr(Yu

lj!y 3lj!) - 0 = (Y1Jlj!y3lj!}o

For convenience we identify $ and M(lj!).

o

o

- 58 -

Proof of c •

-in e ~ = (cos a - i sin a)~~l

-ae2e

1 and that corresponds to ~e •

Proof of

which corresponds to

Proof of e .

·~(iy~V~ - m)'!' = ny~()fl~ - q\iiy~Afl'!' mn = Ll + L2 - L3

Ll = i'!'y~()~~ = i Tr ~tYOY~()fl'!' =

i Tr(fl;wtYoY~dpWfll) = i Tr(yfldpWfllfliwtyo)

~i(Tr yfldp$(l + YO) (1 -' iY2Yl>~tyO) ~ p t fl ~

(y dfl~(l + YO)Y2Yl~ YO'O = (y {dpW)Y2Y1YO~)O «dW)y2Y1YO~)O

L2 = q\iiyPAflW = q Tr(~tYoyflAfl~> =

~ ~ t t fl qTr(,!,,!,l yoY Afll = qTr(Wfl 1fl 1W yoy Afl>

~q Tr(~{l + YO) (1 - iY2Yl)WtyoyflAfl)O

= q(W~tYoyflAfl)O = q(A$Yo$)o •

o

[l

- 59 -

t t t m Tr(~~l~l~ yO) = ~ Tr(~(l + YO) (1 - iY2Yl)~ YO)

t ~

m(~(l + YO) (1 - iY2Yl)~ YO'O = m(~~)O

Summarizing:

({VI/Jy 2 Y 1

t (~ YO{V1/IY2Y1

which is the matrix representation of

In the next table the expressions for 1,s, 1/1, V1/I and L in the three repre­

sentations associated with 6.6, 6.7 and 6.9 are compared. Especially note

the similarity of II and III in the table where e 2 = -1 and i 2 = -1 but 5

is a pseudoscalar in the Dirac algebra and i is a complex number without

geometrical meaning.

o

Table

I (6.6)

Dirac equation ,)lj;eS

e3e

O + qA1j; m1j;e

O

<Ii even

current J =: WeOlP

~

spin current s = 1jJe 3w

- -aeS

e 3eO gauge transformation 1); = l./Je

gauge invariant derivative Vw 3lji - qA1/Ie Se 3eO

L t - m1/leO})O Lagrangean density = (1/1 eO (01);eSe3eO

II (6.7)

a~eS + qA¢ = mcp

¢e3e O = ¢

t J = {cjl eOejl<p}O

U

t s = -(¢ eOe).leS¢eS}O ).l

¢ == ¢e -ae

S

V¢ == 3<p - qAifie s

L t - mrp» 0 == (<p eO Wifie S

III (6.9)

,gn + qA'¥ == m'¥

1ft If J = U YOYu

-i'¥\OY]JYSIf s = ]J

.",.~ •.. --~,~-- .. - .•...

~ -ia 'l' == 'l'e

V If' == d If' - qAlf'i ]J 11

L == 'f'\o(y11Vl - m),±,

0'1 o

- 61 -

Final remarks.

1. It is possible to give a similar interpretation for the vectors ~el~ a~d

~e2~ as is given for :he current J ~eO~ and the spin current s we3~'

It turns out that we1~ and ~e2~ are related to nuclear spin. See [H3].

2. e 2e 1 is one of the three generators of the group SU(2} whereas eS

gene­

rates the group U(1) .

Note that J ~eOW (contrary to s = ~e3~) is not only invariant under the -ae

2e

1 -8e

1e

O -ye 3e 2

transformation W ~e but also under W we and ~ = we

corresponding to the remaining generators of SU(2).

Evidently J is invariant under the U(l) group, i.e. the transformation oe

S ~ ~e

Summarizing we find for J the invariance group SU(2) x U(1), known from

Glashow-Weinberg-Salam theory. We return ample to this group in section 9.

3. It is possible, starting from 6.6, to derive expressions for the Pauli

e~uation and the Schroedinger equation. See [GH] and [HGJ.

4. The conservation of probability usually expressed by a~J ~

o corresponds

to our theorem 6.3.

5. In section 4 we dealt with the operator a = d - 0 in the Pauli algebra

and the Dirac algebra. It is possible to extend this operator to curved

space-time as is shown in a number of papers in [eeJ.

The Dirac equation 6.9 turns out to be the so-called Dirac-Kahler equa­

tion (d - 0 - m)$ = 0 for pseudo Riemannian manifolds, but in our situa­

tion considered for the special case of flat Minkowski space-time.

6. In part II of this report we shall deal with the perfect symmetrical

equation

where $ is any element of the real Dirac algebra.

This equation describes the pair of leptons (e,v) in the following way:

where 1 )

CPe "" <Pee3e O (compare (6.7)

and 1 )

CPv -cp ve 3e O

1) Here again we recognize the ideals 1+ and I as treated in section 2.

- 62 -

Unfortunately this description suffers from the unrealistic assumption

m m = O. e \!

This problem has been solved in the Glashow-Weinberg-Salam theory by symme-

try breaking, but that yields an uninvitated quest -the so-called Higgs

boson- and up to now the existence of this boson is not experimentally ve­

rified (it is planned for 1988 with the aid of the L.E.P. collider PETRA).

We shall try to contribute to a solution of all these problems in part II

of this report.

- 63 -

References

[ABS] Atiyah, M.F., Batt, R., Shapiro, A., Clifford modules. Topology 3,

3-38, 1964.

[AMR] Abraham, R., Marsden, J.E., Ratiu, T., Manifolds, Tensor analysis and

applications. Addison Wesley Publ. Camp., inc., London, 1983.

[B] Boerner, H., Representations of groups. North Hall. Publ. Comp., Amster­

dam 1970.

[Bo] Bohm, D., Wholeness and the implicate order. Routledge and kegan Paul

Ltd, London, 1981 (paperback).

[CC] Chisholm, J.S.R. and Common, A.K., eds. Clifford Algebras and their ap­

plications in physics. Proc. of the NATO and SERC workshop 15-27

sept. 1985. Reidel, Dordrecht, 1986.

[FH] Frescura, F.A.M. and Hiley, B.J., The implicate order, algebras and

spinors. Foundations of physics lQ, 7-31, 1980.

[GH] Gurtler, R., Hestenes, D., Consistency in the formulation of the Dirac,

Pauli and Schrodinger theories. Journal of Math. Phys. !t, 573-584,

1975.

[H1] Hestenes, D., Space-time algebra. Gordon and Breach science publishers

Inc., New York, 1966.

[H2] Hestenes, D., Real Spinor fields. J. of Math. Ph. ~, 798-80S, 1967.

[H3] Hestenes, D., Spin and isospin. Journal of Mathematical Phys. ~, S09-S12,

1967.

[H4] Hestenes, D., Spinor fields as distortions of space-time. J. M. Ph. ~,

1046-1050, 1967.

[H5] Hestenes, D., Multivector calculus. Journal of Math. Anal. and Appl. ~,

313-325, 1968.

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