bi.dk" ax.n ed E-mail addresses: tal eyl ber tof w biv y" of um ed that y states of UGH@n , tal scales lo cated at 1 y 4=3+1 the n it.edu" and \R hnology ysics, ue,  ec en teresting that the exp erimen vn , Denmark ugh eyl equation: It is observ e to fundamen eyl-equation form a basis. It is also [email protected] dimensionized as the matrices in the W ta. y the W usetts Av h Abstract e in 3+1 dimensions? 2 Weyl n . So just in this dimension (in fact, also in a trivial bi.dk", \R View metadata,citationandsimilarpapersatcore.ac.uk d y to o ccur in an as large as p ossible basin of allo ax.n e liv Svend Erik R bridge, MA 02139, U.S.A. usetts Institute of T tally true space-time dimensionalit ter for Theoretical Ph e from The Niels Bohr Institute (1/1 - 1/7 1993). h biv The Niels Bohr Institute, Holger Bech Nielsen v ej 17, 2100 Kb enha 77 Massac

HEP-TH-9407011 terpret these features to signal a sp ecial \form stabilit t matrices Cam Cen ery high energies) it b ecomes in On lea  etoin ydow Massac \HBECH@n Blegdamsv Wh ould lik = 1) do the sigma-matrices of the W eyl spinor for all nonzero momen eyl equation in the phenomenologically true dimension of space-time. In an ew d eyl - rather than Dirac - particles (relativ W Noticing that really the fermions of the Standard Mo del are b est though haracteristic for this dimension that there is no degeneracy of helicit of linearly indep enden equation equals the dimension case c the W the W attempt of making this stabilit precisely in the exp erimen space-time dimension is singled out b as W some presumably v provided byCERNDocumentServer brought toyouby CORE

mo di cations we discuss whether it is p ossible to de ne what we could p ossibly

mean by \stability of Natural laws". 2

1 Intro duction

Since ancient times [1] it has b een a challenging question whywehave just

three space dimensions. In the light of sp ecial relativity one may formulate

an even slightly extended question: Whydowehave just 3+1 dimensions,

one time and three space dimensions?

It is the purp ose in this contribution to elab orate on the idea [4] that

the numb er of dimensions 3+1 is connected with prop erties of the Weyl

equation, the latter b eing esp ecially \stable" in these dimensions (stability

under p erturbations which do not keep Lorentz invariance, say, a priori).

In this connection we shall however also challenge the concept of \stability

of Natural laws" (so much studied by one of us [4], [5]). This (second) part

of our article is a preliminary e ort where we try to seek some \principal

b oundaries" for the set of ideas whichwe call the \random dynamics pro ject".

Before we drift into a detailed discussion of these issues, i.e. our own

p oints of view, whichthus aims at connecting the space-time dimensionality

with certain prop erties of the Weyl equation, wewould like to mention some

other ideas on how to arrive at the space-time dimensionality 3+1. Ideas

which to some extent are comp eting (cf. nal remarks, sec. 4.1.) with our

own p oint of view taken here.

Let us rst remark, that one of the purp oses of this article is to emphasize

the scienti c value of reading information from phenomenology, i.e. from

theories whichhave b een successfully confronted with exp eriments. Such

inspiration from phenomenology supplement more wild and sp eculative con-

1

structions (theories of \everything"? ) which, in case they do not a priori

op erate in 3+1 dimensions, clearly have to o er some explanation for why

our space-time is 3+1 dimensional.

Phenomenologically inspired attempts of understanding 3+1 dimensions

have indeed app eared, cf. e.g. P. Ehrenfest [2], G.J. Whitrow [3] and the b o ok

by J.D. Barrow & F.J. Tipler [1]. The arguments presented in this context

seems dominantly to b e based on observations involving the Coulombor

Newton p otential, connected to considerations dealing with electromagnetic

and gravitational interactions, resp ectively. Assuming a Laplace equation

for the (electromagnetic or gravitational) eld, it has a p ower related to the

dimension of the space. Thus, it can easily b e calculated that the Laplace

equation in d indep endentvariables, 4 (x ; :::x ) = 0, will corresp ond to

s 1 d

s

an inverse (d 1)th p ower law of force (i.e. an inverse (d 2)th p ower lawof

s s

the p otential) in a Euclidian space of d dimensions. Moreover, this p ower in

s

the p otential is related to the stability of atoms or planetary systems. If the

numerical value of the p ower is larger than just the unity corresp onding to

1

We claim, however, that in a certain elementary sense of the word \everything" it

is not (theoretically) ever p ossible to arrive at a theory of everything since an \in nite

regress" argumentvery fast sets in - which stops, only, if language itself (which parts of

it?) unambiguously could select a unique construction of a \T.O.E.". Not even then,

however, do es our search for a \truth b ehind" stop, since we are then, obviously, forced

into a philosophy of the (mysterious) p ower of our language. H.B.Nielsen and S.E.Rugh,

ongoing discussions. See also S.E. Rugh [9]. 2

four dimensions - the atom or planetary system gets unstable against falling

into the singularityat r= 0 (or it go es o to in nity, cf. P. Ehrenfest [2]).

This instability concerns an unstable state in the terminology of subsection

3.1 b elow. The argument b eing the main study of this article turns out to

b e based on stability in a di erent sense: Stability against variations in the

dynamics (rather than the state).

Other - but more sp eculative - attempts have b een made to explain that

there should b e 3+1 dimensions: The following list is, we admit, highly

selective and random - in uenced by talks and pap ers whichwe accidently

have come across. G.W. Gibb ons [11] try to get 3+1 dimensions out of

a membrane-mo del in a higher dimensional setting; K. Maeda [12 ] study

how to get 3+1 dimensional universes as attractor universes in a space of

higher dimensional cosmologies and a group of physicists and astrophysicists

in Poland, M. Biesiada, M. Demianski, M. Heller, M. Szydlowski and J.

Szczesny [13], pursue the viewp oint of dimensional reduction - i.e. arriving

at our 3+1 dimensional universe from higher dimensional cosmologies.

In Cop enhagen, pregeometric mo dels have recently b een investigated by

e.g. F. Antonsen [14 ]. He arrived at the spatial dimension d =42=17 

r m ef f

2:5 from a pregeometric toy-mo del for \quantum gravity" based on ran-

dom graphs, but according to Antonsen this value is probably to o low, since

the mo del was only tested with simplexes of dimension 0 4. E. Alvarez,

J. Cisp edes and E. Verdaguer [15] also arrives at dimensionalities around

d  1:5 2:5 in pregeometric toy-mo dels assigned with a (quantum) metric

describ ed by random matrices.

D. Ho chb erg and J.T. Wheeler [16] contemplate whether the dimension

may app ear from a variational principle. Also Je Greensite [17] uses a

variational principle (roughly making the signature of the metric a dynamical

variable) in order to get esp ecially the splitting 3+1 into space and time.

Of course, any development sensitive to dimension and working in the

exp erimental dimension 3+1 may b e considered an explanation of this di-

mension, b ecause an alternative dimension might not b e compatible with

the same theory.For example, twistor theory (cf. R. Penrose [18 ]) is sucha

theory, suggesting the dimension. In fact, it do es it in a way exceptionally

close to the \explanation" presented in the present article, since b oth are

based on Weyl-spinors.

How sup erstrings arrive at 3+1 dimensions ?

Sup erstring theory [19] has b een much studied for a decade as a candidate for

a fundamental theory. Let us therefore nally discuss - more lengthy- how

the sup erstring, by construction living in 10 or 26 dimensions, may arriveat

our 3+1 dimensional spacetime.

The space-time dimension 3 + 1 (on our scales) is in sup erstring theory

singled out by several, to some extent, indep endent considerations:

(1) An argument for the dimensionality d  3 + 1 (cf. R. Brandenb erger

&C.Vafa [20]): If one assumes some initial b eginning - a big bang, say- 3

with a lot of sup erstrings wound around various dimensions of space (think

for simplicity of a torus) there should statistically b e strings of opp osite

orientations normally enclosing the same dimensions, and they would \as

time passes" comp ensate each other and disapp ear provided they manage

to hit each other and switch their top ology, so that such comp ensation is

made p ossible. As long as there are less than 3+1 (= 4) large (i.e. extended)

dimensions there is a very high chance that strings will hit and shift top ology

b ecause it is very dicult for strings in 3 or less space-dimensions to avoid

hitting each other. It turns out that it is basically the numb er of large

dimensions, that counts for whether \hits" take place easily or not.

As the dimension of the large directions reaches 3+1 the pro cess of un-

folding slows drastically down and one may imagine this b eing the reason for

there just b eing 4 dimensions. We remark, that this argument is truly very

\stringy" since it makes the dimension of space-time b ecome just twice that

of the time track of the string (i.e. 2 + 2=4=3+1).

(2) A compacti cation argument (\dimensional reduction"): A Calabi-

Yau space has exactly 6 dimensions, and so the 10 = 9 + 1 dimensions needed

for the sup erstring to exist (b ecause of the requirement of cancellation of

anomalies, or, say, to get the needed zero-p oint mass squared - contribution

as needed) enforces that using 6 of the 10 dimensions in the compacti cation

leaves just 4 = 3 + 1 for the \large" dimensions But, why should we select a

Calabi-Yau space as the compactifying space? It is motivated by assuming

that N = 1 sup ersymmetry should survive far b elow the compacti cation

scale of energy, i.e. by the requirement of not breaking this sup ersymmetry.

While the argument using Calabi-Yau space and the 10 dimensionalityof

the sup erstring can hardly stand alone without other supp ort for the string

theory, several of the other arguments are suciently primitive to b e trusted

by itself, since based on phenomenologically supp orted ideas. Note, for in-

stance, that argument (1) for string theory ab ovewas also to some extent

phenomenological.

Let us now, for a moment, b elieve in the compacti cation mechanism,

reducing the dimensionality from 10 to 4=3+1. Then, it b ecomes a central

issue of clari cation at which scales this compacti cation takes place. It is

challenging for the sup erstring that it has b een suggested byAntoniadis [21 ]

that already at a few TeV one should exp ect to see the additional dimensions.

If true, this has - most likely - severe cosmological implications. One of us

[21 ] would like to make the arguments of Antoniadis completely rigorous, so

that strings can not escap e this prediction (mo dulo very general assumptions

ab out string-constructions).

It is interesting, though, that recent attempts tend to formulate sup er-

string theories from the b eginning in four spacetime dimensions, rather than

existing in 10 or 26 dimensions, of which all but the four extended dimensions

of our spacetime somehow b ecome compacti ed. (Cf, e.g., the summary talk

by Steven Weinb erg [22] and references therein).

But in that case the sup erstring has evidently a somewhat less impressive

capacityofpower as regards an explanation of whywe live in 3+1 dimensions. 4

1.1 Getting inspiration from phenomenology

Instead of relating the dimension numb er \3+1" to highly sp eculative con-

structions, which could b e plain wrong, such as strings, wewould rather try

to identify structures in the well known and established laws of Nature which

p ointtowards that the numb er \3+1" is distinguished.

You may consider this pro ject a sub-pro ject of a program consisting in

2

\near reading" of the empirically based structure we knowtoday, i.e. the

formal structure and the parameters of the Standard Mo del (describing the

electroweak and strong \spin 1" gauge-interactions) and Einsteins theory for

the gravitational \spin 2" interactions.

Among the information we do not understand to day (and whichwe neces-

sarily have to b e \inspired" from) the following issues were esp ecially stressed

by Steven Weinb erg [22] at the recent XXVI Conference on High Energy

Physics in Austin (August 1992):

 Why do the parameters of the Standard Mo del take the observed values ?

 Why are there three generations of quarks and leptons ?

 Why is the Standard Mo del group S (U  U ) ' U (1) SU (2) SU (3) ?

2 3

And here wewould like to add:

 Whydowe live in 3+1 dimensions ?

In fact, wehave found a sp ecial prop erty in the Weyl equation which p ervades

so strongly the Standard Mo del of the electroweak and strong forces and the

3

known material constituents. This sp ecial prop erty (whichwe enco de by

the concept of \stability" and whichwe shall discuss in this contribution) is

a prop erty which distinguishes 4=3+1 and 1=0+1 as standing out relative

to all other space-time dimensionalities. The numb er of linearly indep endent

matrices of the typ e that app ear in the Weyl equation is exactly equal to the

2

The desp eration in this pro ject is that all this information can b e immensely condensed

down (to a page or so) so wehave actually only very little structure to \read" (and to

b e inspired from): The truly observed sector of the Standard Mo del has only of the

order of 19 real parameters, some of which are hardly determinable in practice, even

using baryon numb er generation physics: 12 quark and lepton masses of which the three

neutrino masses (in the simplest Standard Mo del) are understo o d to b e just zero, three

nestructure constants, four parameters in the Kobayshi-Maskawa-Cabibb o matrix and

the  -angles.

In this program of \near reading" information from the Standard Mo del wehave also

tried to read o remarkable features of the Standard Mo del gauge group S (U  U )

2 3

compared to other groups with up to 12 generators (12 gauge b osons), say, and wehave

found that the standard mo del group is remarkable by its \skewness" (remarkably few

automorphisms), cf. H.B. Nielsen & N. Brene [7].

3

Note, that even relativetothe W mass scales most of the quarks and leptons are light

particles, and certainly they are all \massless" relative to the Planck scale. Since also right

and left comp onents couple di erently to the weak gauge b osons, a description in terms

of Weyl particles is strongly suggested, cf. also intro ductory discussion in H.B. Nielsen &

S.E. Rugh [8]. 5

dimensionality d of space-time if d = 4 (or, in fact, if d = 1).

Are the elementary matter constituents, we knowtoday (i.e. the fermions)

not Dirac particles rather than Weyl particles? Well, the Weyl spinors - in-

tro duced by Hermann Weyl - were originally rejected as candidates for our

constituents of matter b ecause they were incompatible with parity conser-

vation. However, seeking fundamental physics (at the Planck scale, say) we

2

ignore the small masses of a few hundred GeV =c of the quarks and the

weak gauge b osons, rendering the connection b etween right and left handed

comp onents completely negligible. Thus we consider all the material con-

stituents (quarks and leptons) we knowtodayasWeyl particles rather than

Dirac particles!

So to read some sp ecial feature of dimension 3+1 connecting to the Weyl

equation means to read it o from all known material elds of whichwe are

build up! (cf., also, the more detailed intro ductory discussion in H.B. Nielsen

& S.E. Rugh (1992) [8]).

The outline for our contribution is as follows: In the following section

(section 2) we remind the reader ab out the Dirac and Weyl equations in

an arbitrary numb er of dimensions. Esp ecially,we calculate the number of

comp onents of the Weyl spinor and the crucial observation of the equalityof

dimension and the numb er of elements in a basis for the matrices in the Weyl

equation is made. In section 3 we then seek to unravel the message to b e

learned from this observation: It is connected with \a stability" under p er-

turbing the fundamental \dynamics". We also discuss the general limitations

for p ostulating such a stability. In section 4 we resume and put forward the

concluding remarks and dreams, and it is discussed if the observation could

just b e an accident?

2 The Dirac and Weyl equations in d dimen-

sions

In order to note the sp eciality of the 4=3+1 dimensional Weyl equation we

shall see that in the general d dimensional case the numb er of comp onents

n of the spinor of what is reasonably called the Weyl equation

Weyl



 D =0 (2.1)



is

(

d=21

2 for d even,

(2.2) n =

Weyl

(d1)=2

2 for d o dd.

In fact, the Weyl equation is at rst glance only de ned for even dimensions

and, thus, wehave to agree up on, what we will call a Weyl equation in o dd

dimensions. We outline shortly how to build up a Weyl equation in even

and in o dd dimensions. 6

2.1 Construction of the Weyl-equati on (in even and

o dd dimensions) from the Dirac-equation (in even

dimension)

The Weyl-equation in arbitrary dimensions is most easily constructed from

the Dirac equation, whichwe shall in the next section construct in even di-

mensions via an inductive construction. It has only half as many comp onents

d+1 5

and is obtained by\ -pro jection" (think here of the -pro jection in the

4 dimensional case) meaning that we rst de ne

d+1 0 1 2 d1 d

=


(2.3)

mo dulo an optional extra phase factor i so as to b e hermitean, then cho ose -

d+1

matrix representation so that this b ecomes diagonal (Weyl-representation),

and then thirdly write only that part of the Dirac equation



(i D m) =0; (2.4)



which concerns those comp onents of the Dirac eld which corresp ond to one

5

of the eigenvalues of ,say +1. Wenow drop the mass-term. Since the only



remaining term in the Dirac-equation is thereby the \kinetic" one i D =



d+1

0 and the gamma-matrices only have matrix elements connecting ( = 1)-

d+1

comp onents to ( = 1)-comp onents the Weyl equation can b e written

using only matrices with half as many comp onents as the corresp onding Dirac

equation from which it is obtained. Really we nd the Weyl matrices as o -

diagonal blo cks in the Dirac matrices in the Weyl representation. That is to

 



say the Weyl-gamma-matrices whichwe call  (or  and  ) if wewantto

+

5

distinguish if we pro jected on the equal to minus or plus 1 comp onents)

are given by

5 5

1 1+

 

 

 = ;  (2.5) =

+

2 2

with successive removal of the unnecessary entries in the matrices, or b etter

by

!



0 

 +

: (2.6) =



0 

2.2 Inductive construction of the matrices (by addi-

tion of two dimensions at a time)

If we are in d time dimensions and d space dimensions we denote by rel-

t s

ativistic invariance the invariance of the equations (to b e constructed right

now) under the (generalized Lorentz) group O (d ;d )  O(d ;d d ).

t s t t

Generalizing the construction by Dirac, we seek a construction in d =

1 2 d

d + d dimensions, whichinvolves a set of matrices, ; ; ::::; shap ed

t s

to ob ey a sp eci c anti-commutator algebra



 

(2.7) f ; g =2g

(d) 7



(where g denotes the metric tensor, taken just to b e a series of 1's and

(d)

1's along the diagonal).

If equation (2.7) is satis ed for the matrices, the Dirac equation imply

a corresp onding Klein-Gordon equation

   2  

(i D + m)(i D m) =(g D D m [ ; ]eF ) = 0 (2.8)

    

(with magnetic moment term in it, and where we used [D ;D ]= 2eF )

  

and that is what is wanted to have the usual relativistic disp ersion relation

2 2

p m =0 :

It is not dicult to show the existence of such matrices by induction in

steps of two in the dimensions, starting, e.g., from two dimensions, to get

the even ones.

Start of the induction in two dimensions

In two space-time dimensions one may use two of the three Pauli matrices

as gamma-matrices, e.g.

0 1

=  and = i (2.9)

x y

in the Minkowski-space case and simply

0 1

=  and =  (2.10)

x y

in the euklidian(ized) two dimensional space. These are the Dirac-matrices

and they have n = 2 comp onents.

Dirac

Step from d 2 to d dimensions.

Having the Dirac equation in d 2 dimensions (with d even) we can in an

inductive construction make the Dirac-matrices for two dimensions higher by

the following construction:

Supp ose that we already have constructed the gamma-matrices for d 2

dimensions and denoted them with a tilde in order to distinguish them from

the gamma-matrices for d dimensions whichwe construct successively. They

ob ey the anti-commutation algebra



 

f ~ ; ~ g =2g (2.11)

(d2)

with

;  =1;2;3; :::; d 3;d 2: (2.12)

Then we de ne the d-dimensional gamma-matrices for the rst d 2values

of  by:

!



0 ~



= (2.13)



~ 0 8

and the two new gamma-matricesby:

!

1 0

d1

= sign
;

0 1

(2.14)

!

0 i1

d

= sign
:

i1 0

The signs sign (which, without loss of generality, need only to takevalues

1ori) are to b e chosen so as to arrange the wanted signature for the anti-

4

commutator algebra



 

f ; g =2g (2.15)

(d)

which is easily checked with the presented ansatz, cf. (2.13) and (2.14).

Numb er of comp onents of the Weyl-and Dirac spinors

Thus we see that the numb er of comp onents for a Dirac equation for even

5

dimension d of space-time is

d=2

n =2 for d even (2.16)

Dirac

and thus since the Weyl equation has just half as many comp onents it has:

d=21

n = n =2 for d even (2.17)

Weyl

Note, that in d = 4 dimensions the Weyl spinor gets two comp onents (the

neutrino), while in 10 dimensions, say, the Weyl spinor gets 16 comp onents (a

complicated neutrino, indeed) so we see that the numb er of comp onents grows

very fast, b eing an exp onential function of the dimension d. I.e. whereas a

\spin 1/2" particle has 2 spin-states (if it is a massive Dirac spinor) as we are

used to in four space-time dimensions, a \spin 1/2" particle in 10 dimensions

has 16 spin-states (but 32 Dirac-comp onents).

4

In a systematic exp osition of this construction one would go into Cli ord algebras and

all that.

5

This numb er of comp onents is only a minimal numb er in the sense that the Weyl

spinors could have additional degrees of freedom (as they indeed have, for instance \color").

When wehave-until now - ignored such additional degrees of freedom, this is, mathemat-

ically sp eaking, that wehave build up the irreducible representations of the algebra (2.7).

The question ab out additional degrees of freedom of the Weyl spinor turns out to b e a

ma jor \killing attempt" of the entire approach pursued here, cf. the section \An almost

true theorem ..." (subsection 3.6). 9

2.3 The o dd dimensional cases

(d1)+1

In o dd dimensions one may simply include the -matrix as the matrix

number d and use the d 1even dimensional Dirac equation, whichwehave

already constructed. That is to say,we take the Dirac equation one unit of

dimension lower than the o dd-dimensional equation wewant to construct.

Then wehave all the -matrices already except for the last one - the dth one.

For the latter we then use the construction of taking the pro duct of all the

5

rst d 1 matrices. Since it is well known that sucha\ "-typ e pro duct

will anti-commute with all the other gamma-matrices if constructed for even

dimension, and since it can by an optional factor i have the sign of its square

as wanted, it obviously will satisfy all the needed prop erties for joining the

set of gamma-matrices to get a set for the o dd dimension d.

The equation, with gamma-matrices, obtained in this way for the o dd di-

mension d is in twoways b est considered the o dd-dimensionalWeyl equation:

6

1. It violates parity , just like the even-dimensional Weyl equation,

5 d+1

2. It cannot b e further reduced by\ "-pro jection (really ),

since its \gamma- ve" would b ecome the unit matrix.

However, this \Weyl equation" in o dd dimensions deviate from the Weyl

equation prop erties by allowing a mass term, since there is only a trivial

d+1

= 1 that can b e constructed. There is no \chirality" which protects

against generation of mass m 6= 0 for the Weyl spinor eld .

One might then de ne an equation with the double numb er of comp o-

nents and consider that the o dd dimensional Dirac equation. By comp osing

a couple of o dd dimensional Weyl equations, b eing mirror images of each

other one easily obtains a \Dirac equation" in o dd dimensions whichisboth

symmetric under parity transformations and can b e reduced back to its Weyl-

comp onents.

With the suggested notation we easily nd the numb er of comp onents for

the o dd dimensional case:

d+1

(d1)=2

2

n =2
2 =2 for d odd (2.18)

Dirac

for the Dirac equation (the doubled one) and for the Weyl equation

d1

(d1)=2

2

for d odd: (2.19) n =2 =2

Weyl

2.4 An observation



With n comp onents the numb er of linearly indep endent  typ e matri-

Weyl

2 2

ces that can b e formed is n . Out of these n p ossibilities the Weyl

Weyl Weyl

equation makes use of d. Our crucial observation is that in the phenomeno-

2

logically true case d =4=3+1ithapp ens that all the n p ossible

Weyl

matrices are just used onceeach since in fact

2

n = d: (2.20)

Weyl

6

In o dd space-time and thus even space dimension the p oint-re ection is just a rotation

and when we talk ab out paritywe mean a re ection in a mirror (true mirror symmetry). 10

Fig.1. This gure shows the graphical way of solving equation (2.21) by

presenting as functions of the dimension d a semilogarithmic plot of :

1) the identity function d (the curved curve), 2) right hand sides of (2.21)

interpolatedtoreal numbers separately for even and odd d (the two straight

lines). Note, that there is also coincidence for d =1which is however a

completely empty and trivial theory. The one for d  0:33 is of course not a

true solution.

This equation written as an equation for the dimension d of space-time

b ecomes

(

d=21 2 d2

(2 ) = 2 for d even,

d = (2.21)

(d1)=2 2 d1

(2 ) = 2 for d odd:

As can b e easily seen from the gure the only acceptable solutions to this

equation are

(

1

d = (2.22)

4

b ecause the solutions app earing at rst also d = 2 (if 2 had b een o dd) and

an irrational solution around d  0:3 (had it b een even) are not acceptable. 11

2

= d as a message ab out 3 The equation n

Weyl

robustness and stability of the Weyl equa-

tion in 3+1 dimensions

2

What could p ossibly b e the reason for this coincidence of numb ers n and

Weyl

dimension d ? (If it is not an accident, cf. the nal remarks).



It means that the sigma-matrices  make up a basis for all the p ossible

matrices that could p ossibly multiply the Weyl spinor in d dimensions, if

7

d = 4 (or in d = 1, whichisavery trivial case indeed ). In particular, this

means that



 D



may b e considered as the most general linear op erator that can act on , pro-

vided we could consider the D 's completely general expansion co ecients.



Now, in fact, D is the covariant derivative D = @ ieA . So is this

   

really the most general op erator ? Yes, it is indeed the most general form

of the rst two terms in a Taylor expansion of any reasonable well b ehaved

di erential op erator

2

@ (x; p =0) 1 @ (x; p =0)

(x; p)  (x; p =0)+ p + p p + :::: (3.1)

  

@p 2 @p @p

  

The most general equation, whichwe supp ose to b e homogenous and linear

in the Weyl-spinor eld (see discussion later), is thus

(x; p) (x)=0 ; p=p =i@ (3.2)

 

where now denotes an 2  2 matrix op erator, where wehave considered

~

only op erators which map Weyl-spinor waves (x)= (x;t)toWeyl-

spinor waves (as in the Weyl equation).

Being a 2  2 matrix op erator, can b e expanded in terms of the basis

 2

matrices  (in the case where the equation (2.20), i.e. n = d, is ful lled

Weyl



we can use the set of the matrices  as a basis). Let denote the basis



co ecients, i.e. de ne



(x; p)= (x; p) (3.3)



WeTaylor expand:

@ (x; p =0)



(x; p) = (x; p =0)+ p + ::: (3.4)

  

@p



 

(x)p +::: (3.5) eA (x)+V = V

 

 

7

Considering this one dimension a time dimension, there is a zero dimensional space

or, in other words, only one p oint. The Weyl equation has in this case the following trivial

form, D =(@ ieA ) = 0, where A can b e gauged away, rendering the single-

0 t 0 0

comp onentWeyl spinor completely constant. Wethus have a Hilb ert space with only one

dimension, and thus there is only one quantum state for the Weyl spinor, leaving no ro om

for developments in that \universe". 12

In the last equation (which is merely notation) wehave de ned

@

1



V (x)= (x; p =0) ; eA (x)=V (x; p =0): (3.6)

  

 

@p





By identifying A (x) with an electromagnetic eld and V (x) with a





vierb ein (in a gravitational theory) wehavethus made the interpretation of

the lowest order terms in the Taylor expansion of the general 2  2 matrix

op erator equation (3.2) as b eing nothing but the usual Weyl equation for our

Weyl-spinor (coupled to an electromagnetic eld A and to a gravitational





eld in the usual manner by the vierb ein eld V ), see also discussion in



section 3.5.

We can summarize that Nature has organized just such a dimension d of

space-time as to make the Weyl equation op erator (with couplings to some

external electromagnetic and gravitational elds unavoidably app earing in

the same go) the most general op erator in the Taylor expanded limit of small

momenta p = p (small derivatives @  ).

 x

Whydowe need all these Taylor expansions? Well, that we arrive at the

Weyl equation via Taylor expansions (keeping only the lowest order terms)

is a very natural thing, b earing in mind that we are living in the \infrared

limit" compared to some presumed fundamental scale (e.g., the Planck scale,

say) in the sense that wehave-even with our b est accelerators to day - access

to very small 4-momenta p only.

In fact, to use sucha Taylor expansion (at exp erimentally accessible ener-

gies) is completely analogous to Taylor expanding away higher order curva-

ture terms in the gravitational action thereby arriving at the Einstein-Hilb ert

action in the \low energy limit". And just like higher order curvature terms

will blow up in the ultraviolet in the action for the gravitational eld we could

exp ect that the Weyl equation will b e mo di ed in the ultraviolet (for higher

 



momenta) and blow up terms quadratic in the momenta like  @ @ 

 



(here  denotes some vector eld, which do not necessarily, in fact b etter



not, have to b e the A eld) etc. etc

The ab ove means, esp ecially, that we can make a reinterpretation of any

little additivechange

 

 D =0 ! ( D +W ) =0 (3.7)

 



in the op erator  D acting on as a change in the D 's which again is just

 

a shift in the \electromagnetic eld" A and/or the \gravitational" vierb ein





V .



Here W = W (x; p) could b e any22 matrix op erator, whichhavea

well b ehaved Taylor expansion in sucha way that higher order terms than

those which are linear in 4-momenta p may b e neglected when p is small.

Notice that the typ e of mo di cation terms W (x; p)we here consider

are not at all Lorentz invariant a priori. That the Weyl equation keeps its

form and thus Lorentz invariance is only achieved at the cost of allowing the



(and A ) to b e mo di ed, but it is still remarkable. vierb ein V



 13

Wewould thus liketointerpret the dimension coincidence as a signal of

a certain kind of \form stability", whichwe are going to discuss further in a

moment.

In 4 dimensions there is only one helicity state (no degeneracy) of

the Weyl spinor:

An alternativeway, to observe the \stability" of the Weyl spinor in exactly 4

dimensions is to notice that a Weyl particle (antiparticle not included !) for a

given momentum has just the following number of Weyl particle p olarization

states

(

d=22

n

2 for d even,

Weyl

= (3.8)

(d3)=2

2 for d odd:

2

which means that it is just for d =3ord= 4 that there is just no degeneracy

of particle states and just one state p er momentum! Several states for one

momentum signals an instability under mo di cations W (x; p), b ecause in

general such a mo di cation will cause level repulsion (of the eigenvalues), as

8

is well known from p erturbations of levels in quantum mechanics. We can

(at least) say that d = 4 is the highest dimension without degeneracy!

3.1 Is it p ossible to de ne \structural stability" for

Natural Laws?

Let us for a moment set aside the Weyl equation (we return to that) and

digress into a discussion of howwe p ossibly can assign a meaning to the notion

of \stability of Natural laws" (and, in particular, the stipulated stabilityof

the Weyl equation).

It is remarkable that there are over fty generally accepted de nitions of

stability (whichwe shall not go into here, cf, e.g., Szeb ehely (1984) or E.

Atlee Jackson, p.41 [25] ). These de nitions deal, however, with the stability

of a certain state, i.e. a sp eci c con guration of the degrees of freedom of a

xed dynamical system, a xed Hamiltonian, say.

The Weyl equation is a law of Nature (or a dynamics), not a \state".So

let us distinguish the concept of stability of a \state" under p erturbations in

the space of p ossible states (with the Natural laws xed) from that of stabil-

ity of the laws (the dynamics) under p erturbations in the space of p ossible

laws of Nature (p ossible Hamiltonians, say):

8

E.g. for a 6-dimensional Weyl equation, which according to the observation ab ove, has

degeneracies of states, this degeneracy would disapp ear under (almost) any mo di cation,

and hence the mo di ed (p erturb ed) equation could not b e interpreted as a 6-dimensional

Weyl equation. 14

I. Stability of states (keeping the dynamics, i.e. the Natural laws,

xed)

Taking into account the ubiquitous bath of small p erturbations and \noise"

which surrounds and \attacks" every ob ject (in some sp eci c \state"), it

necessarily has to p ossess some degree of stability (along the directions of

the most likely p erturbations which \bath" the ob ject) in order to exist in

that \state" for a longer interval of time.

This is the typ e of stability that distinguishes a needle standing vertically

on its tip as unstable versus a stable ball, say, lying in a (little) depression.

To make the needle balance for a long interval of time requires enormously

accurate ne tuning, although it is in principle p ossible.

I I. Stability of a dynamics (a set of equations)

Rather than dealing with the stability of a \state" wewould like to fo cus

on the \stability of a dynamics". In this case, the questions whichwe will

address are whether certain features of the dynamical development are stable

under mo di cations of the Hamiltonian or action, say.

I I(a) \Structural stability" of a dynamical system

In particular, the concept of \structural stability" ( rst put forward byA.

Andronov and L.S. Pontryagin more than ftyyears ago) of a dynamical

system is de ned in the following way:

_

A given physical mo del, based on the system  = F (; c) is

"structurally stable" if a slightchange in F ,

n

F ! F +  F ; jj F ( )jj < 82R

do es not result in "essential" changes in the solutions of this sys-

_

tem, whereby is meant that the phase p ortraits of  = F ( ) and

_

 = F ( ) +  F are topological ly orbitally equivalent. In other

words, there exist homeomorphisms that transform the phase

space tra jectories of the vector eld F to those of the p erturb ed

one F +  F (Cf., e.g., E. Atlee Jackson [25] Vol.I, p. 102, p.380.

See, also, e.g. A.A. Coley and R.K. Tavakol [26 ])

I I(b) \Stability of Natural laws" (in the \random dynamics" spirit)

Removing the fo cus from \orbital top ological equivalence" wewould attempt

to consider as the essential feature (the stability of which is to b e de ned) 15

9

some e ectivelaws or regularities appearing in some limit .Wemaysay that

wehave stability in the spirit of \random dynamics" [5] if such e ectivelaws

or regularities are unchanged under p erturbations (\small" mo di cations) of

the (fundamental) dynamics of the system.

By analogy: Just like a \state" of a given Hamiltonian, say,haveto

p ossess a certain degree of stability in order to exist for a longer span of time

(otherwise the p erturbations have to b e netuned enormously), we sp eculate:

Could it b e that the \Natural laws" also have to p ossess some degree of

stability ? And howmay one pursue this idea ?

10

If it is so that the Natural laws dep end substantially on netuned

implementations of structure at the Planck scale, say, then there is very

little \ro om" for \mis-implementations" and \small errors", as regards the

structure (the Natural laws) at this fundamental scale.

The b elief that this is not the case, i.e. the b elief that the Natural laws

indeed will show up to exhibit some degree of stability in order to \exist" -

that is the \random dynamics spirit"! (and the viewp ointhave b een investi-

gated and implemented in the context of many di erent \toy-mo dels". Cf.,

e.g., C.D. Froggatt & H.B. Nielsen [5] and references therein.

However, the \random dynamics" p oint of view have inherently some

small diculties and obstacles whichwe shall now seek to discuss. (Section

3.2-3.5).

Note, also, that structural stability in the strict sense of I I(a) demands

complete equivalence b etween phase p ortraits whereas \the random dynam-

ics" spirit stability only aspire to arrive at the same \form" in some \limit",

where the same \form" means that you can interpret it as following the same

law(s). Even though I I(b) is weaker requirement of stability than I I(a) in

the sense of only caring for a limit, it is of course much stronger if we take

it, I I(b), to mean stability of the Natural laws under \completely general"

p erturbations (whatever that should mean ?, cf. discussions b elow).

9

Dominantly,we shall have in mind this limit to b e an infrared limit, i.e. physics at our

\human scales" where all the particle and interaction constituents have small momenta

and energies compared to some \presumed" fundamental scale of the \world machinery"

_

(this world machinery corresp onds to the system  = F ( ) of I I.(a) ab ove).

But also other limits could b e thought of: For instance the isospin symmetry of strong

interactions app ears in the \limit" of b oth the up and the down quarks happ ening to b e

.Wehave, also, a prop osal for understand- very light compared to the scale given by

MS

ing the linearity and to some extend hermiticity of the Schrodinger equation as features

app earing in the limit of waiting very long, i.e. very late times. See, e.g., C.D. Froggatt

& H.B. Nielsen [5]

10 15

Down to scales of  10 meter or so, the Natural laws apparently do not change

signi cantly at di erent space-time p oints, since we seemingly have go o d conservation

of energy and momentum.Itmay b e a bit hard even to imagine how there could b e a

breaking of translation symmetry at even smaller scales - without causing problems by

b eing observable. But wemay, either, e.g. imagine the coupling to long waves to b e only

tiny[5] or some sort of quantum uid [24]. 16

3.2 Howwehaveweakened the concept of stabilityin

order for the Weyl equation to b e stable?

Note esp ecially, that in order to claim the stability of the Weyl equation we

restricted the class of p erturbations W (mo di cations) of that equation

considerably: We outline b elow in which sense this basin of p erturbations

was restricted and comment for each item howwe imagine one could - to

some extent - relax the restrictions made.

1. The equation is restricted to remain linear and homogeneous in

the Weyl spinor eld

2 3

One may argue that additional terms like , ,...would b e suppressed if the

eld itself is considered small. However there is a problem how to suppress

the zeroth order term - i.e. the term not dep ending on - in the limit of

weak , unless one somehow argues that attention can b e restricted to the

homogeneous part of the equation, the inhomogeneous solution b eing just

\background".

2. The numb er of degrees of freedom (of the Weyl spinor) remained

unchanged

We mo di ed the Weyl equation for the Weyl spinor (with two degrees of

freedom) with p erturbations W restricted to b e of the op erator-form of 2  2

matrices.

\Elementary constituents" (elementary particles) often turns out to b e

\comp osite" when they are lo oked up on at higher energy scales (cf. the

\quantum ladder"). I.e. they semi-always turn up to have additional degrees

of freedom (\frozen in" in the infrared limit) which gradually \wake up"

towards the ultraviolet. That is, the e ectivenumb er of degrees of freedom

of some ob ject dep ends on the energy by whichwe \lo ok" at it.

In particular, one could imagine a \random dynamics" pro ject where the

numb er of comp onents of what b ecomes the Weyl-spinor eld itself (in the

infrared) did dep endent on the energy-scales.

Indeed, the toy-mo del for a \world machinery" describ ed in detail in

section 3 of H.B. Nielsen & S.E. Rugh [8] is to b e considered as one p ossible

attempt of relaxation on exactly this p oint. It is in fact argued that - genericly

- the Fermi surface would consist mainly of p oints of the disp ersion relation

with at most two degenerate states, b ecause the p oints with more than two

11

cannot separate lled and un lled states for top ological reasons .Very near

the Fermi surface there should b e at most two relevant comp onents, i.e.

e ectively n=2.

We will return to the issue ab out additional degrees of freedom for the

Weyl-spinor in section 3.6 where we are able to o er \An almost true theo-

rem ab out the numb er of gauge b osons and fermion comp onents". However,

this theorem is not ful lled for the left handed quarks (Weyl spinors) - and

there is, in fact, some instability of the left handed Weyl spinors (in the

11

This remark stems from old unpublished work in collab oration with Sudhir Chadha. 17

restricted sense wehave talked ab out here) when we takeinto account the

gauge degrees of freedom of the Weyl spinor. But it is remarkable that all

the other Weyl spinors, i.e. the right handed quarks and the leptons, are

stable (in our sense) - provided you do not allow the basin of p erturbations

to allow mixing of one irreducible representation with another one.

3. The basin of equations are chosen to b e selected from the class

of di erential equations.

This restriction could b e relaxed in manyways. One often implementphysics

on a ne lattice (for instance with di erence equations) in the ultraviolet,

whichwould lead to di erential equations only as \e ective" equations in the

infrared. But that would really only work if the \lattice" used has enough

structure to b e approximately a manifold from a long distance p oint of view.

Even in the section 3 of H.B. Nielsen & S.E. Rugh [8] where we attempt to

make a sp eculative mo del which relaxes many of the prop erties of the Weyl

equation we only relax the equation-op erator to b e an analytic one in x's

(or q 's) and p's or equivalently di erential op erators p = i@ =@ x. Since

analytic functions are at least approximated by p olynomials it means that,

approximately,we did only allow p olynomials in the di erential op erators

and thus essentially kept the requirement of there b eing a di erential op er-

ator in the equation. We approximately kept a di erential equation there.

Most imp ortant is presumably that wekeep a kind of \lo cality" bykeeping a

di erential equation. To attempt to relax this assumption might of course b e

interesting, but very likely this can only b e done by asserting (in the same

go) that in almost any structure you can invent an approximate top ology

and thus pretend to see some lo cality.

4. The space-time dimensionality d is kept xed under the p ertur-

bation W of the Weyl equation

Even if you allow a higher dimension of space(time) there would b e no 

matrices to go with the extra co ordinates and momenta and one could show

that there would b e no motion in the extra directions of the mo di ed Weyl

particle. There would, however, b e no obvious reason why a right handed

top quark and a right handed charm quark, say (i.e. fermions which b elong

to di erent irreducible representations), should select the same directions in

which not to move. The story of \no motion" in more than 3 space direc-

tions is also only true as long as the numb er of comp onents of the Weyl

spinor is xed to 2. However, we (at least) cite suggestive arguments in H.B.

Nielsen & S.E. Rugh [8]) for that genericly there will e ectively only b e two

comp onents (see also [5, 10 ]).

In particular - provided no mechanism prevents di erenttyp es of Weyl

particles from selecting di erent \directions of no motion" - human observers,

say, comp osed of Weyl particles of these di erent sorts, would b e split up

in subparts (moving in di erent directions) each of which consists of Weyl

particles of a given typ e in the sense discussed ab ove! Also, there would b e

access to more than 3+1 dimensions using several typ es of Weyl particles in 18

this way.

When a higher dimensional Weyl spinor with d  4 can move \fast"

in more than three spatial co ordinate directions this is a prop erty that is

strongly unstable (even restricting to a very small piece around a Fermi sur-

face).

We remark, that every time we attempt to relax one of the restrictions

1,2,3 and 4 mentioned ab ove, wevery fast end up with a class of mo dels

which is in nitely much bigger than the restricted basin of the de ned p er-

turbations.

3.3 Preliminary discussion of some \principial b ound-

aries" for which \basin of p erturbations" we can

formulate?

Wewould like to state the \stability" of the Weyl equation as impressiveas

p ossible. I.e. the Weyl equation would b e stable under \all" p erturbations -

conceivable as well as inconceivable. This is, clearly, not p ossible to claim.

We cannot claim that the Weyl equation comes out from everything (an

12

elephant, say) - even in some limit? Nevertheless this is the \spirit in the

random dynamics"!

Besides those restrictions of the basin just mentioned ab ove (item 1,2,3

and 4) there mayeven b e restrictions of a kind whichwehumans may not b e

fully aware - and whichmay represent principal b oundaries wemay therefore

neither b e able to transcend nor fully explore, even in the distant future (and

therefore we cannot easily display such b oundaries here).

If one should go so far as to even consider logic a part of physics, and

thus make a \random physics" mo del to b e also one which relaxes on the

principles of our Aristotelian logic, it would get very hard indeed to formulate

the basin or class of mo dels of this typ e, let alone to prove something ab out

that class.

The b oundaries for what can b e conceived or not are, surely,very hard to

lo cate. As a simple illustration of a p ossible p erturbation whichyou would

hardly imagine with \old fashioned group theory" is the existence of \quan-

tum groups" (e.g. SU (2) ) conceived of as a continuous deformation of a

q

group. If one should suggest a p erturbation of the Weyl equation whichwas

\analogous" to such continuous deformation of groups (\quantum groups")

wewould havea very hard job to do! For instance, p erturbation of the equa-

tion into some \relation" among symb ols whichwas not even an equation.

However, it is - in our p oint of view - an interesting pro ject to identify,

and try to formulate in print, such fundamental b oundaries and limitations

12

If we take, say, the limit of extremely low temp erature the resulting \deeply frozen"

elephant mighthave left only a very tiny fraction of its originally active degrees of freedom

and it is not so easy to exclude totally that the remaining vibrations in the trunk and the

still active spins could not, somehow, b e interpret cleverly as Weyl equation(s) and/or the

Standard Mo del ? 19

for what can - ever - b e thought and accomplished. For example, it is well

known that the german philosopher Immanuel Kant and the danish physicist

13

Niels Bohr were - seriously - contemplating such issues.

Niels Bohr, in particular, emphasized the imp ortant role played by our

\daily language" which presents a \b oundary" in the sense that it is hard

to transcend - since we ultimately have to communicate all our ideas and

knowledge in \daily language".

We might add that any attempt to make \great theories" at a very funda-

mental level esp ecially has to b e accompanied with some interpretation back to

the daily language. Any such mo del is thus necessarily complicated by b eing

provided with such an unavoidable system of interpretation assumptions. Ir-

resp ectiveofhow \uni ed" and how simple some ambitious \T.O.E." would

b e it cannot b e without some assumptions providing the translation from its

own formalism into that of daily language. This fact sets a lower limit for

its simplicity:

We cannot transcend that limit, but could we reach it?

You maysay that the \random dynamics" pro ject precisely intends to reach

it. The goal (in the strongest form of \random dynamics") is to argue that

whatever the basic theory is - almost - we can by appropriate interpretation,

i.e. by adding the appropriate \translation assumptions" into daily language,

make it b ecome a go o d and well functioning \T.O.E." But then it b ecomes

also tempting to really make use of this \interpretation assumptions" to the

uttermost.

3.4 So what could (should) \random dynamics" hop e

to arrive at?

Rather than claiming - as ideal \random dynamics" (a very strong form of

14

random dynamics indeed [4]) - that al l (or almost all ) mo dels at the funda-

mental level lead to the well known phenomenology (i.e. the Weyl equation,

general relativity, the standard mo del with its particular \skew" gauge group

etc.) we should seek a large class of models that will lead to the same infrared

phenomenology. The \random dynamics" pro ject intend to lo cate b oundaries

\@ B " for this (huge?) class of mo dels and the pro ject (however, at the level

of \Natural laws") resembles somewhat the pro ject of nding \universality

classes" in phase transitions or other dynamical b ehaviors.

It is a natural contemplation, that if these b oundaries are not to o narrow,

it leaves (logically) the p ossibilityofhaving \chaos in the fundamental laws"

13

Cf, e.g., I. Kant \Prolegomena to Any Future Metaphysics" (a shorter, more easily

readable version of \Critique of PureReason") [27]. Niels Bohr has discussed our \Con-

ditions for description of Nature" at very many o ccasions, cf. e.g. [28].

14

Both the concept \all" and \almost" are not de ned a priori. (Presumably, they could

only b e sp eci ed in a way that would ultimately turn out to b e rather arbitrary). 20

15

- i.e. the \fundamental structure" may b e selected randomly - if it is just

selected from the class of mo dels restricted by these b oundaries.

Thus, in some sense, there is a no need for netuning of the fundamental

structure, except that it has to lie within the b oundaries.

Nevertheless, we note, that these b oundaries \@ B " (if these could b e

made precise enough to b e stated formally,say,by mathematical formulas)

in a certain sense themselves have status as \Natural Laws" - and so on, ad

in nitum. Therefore, this way, one do es not circumvent the concept of some

\Natural laws" to b e implemented.

It would b e interesting if the the \random dynamics" pro ject could lead

to de nitive conclusions as regards the class of fundamental mo dels which

would lead to the Standard Mo del in the infrared limit. It may turn out to

b e of small (zero) measure (in which case the fundamental structure haveto

b e netuned enormously) or - which is more in the random dynamics spirit

-itmay turn out that the mo dels not leading to the standard mo del would

have small (zero) measure.

\Standard philosophy", as materialized in the last decades, for instance,

in the search of one unique \sup erstring" is inclined towards the rst view-

p oint, while many arguments for the second viewp ointhave b een collected

in C.D. Froggatt & H.B. Nielsen [5].

Apparently, it is easier to show \instability" than \stability", b ecause the

latter requires stabilitytowards all conceivable (and inconceivable, as well)

p erturbations, while the rst requires only the unambiguous demonstration

of instability in some directions of the conceivable p erturbations.

If it turned out that the random dynamics pro ject failed in the sense that

it could p ointtowards some regularities that has an instability - and thus

these regularities have to b e netuned - this conclusion would in our opinion

beavery interesting one, and p ointtowards some essential feature of the

entire \world machinery" of whichwehumans participate (so shortly).

In any case it is thus justi ed to investigate if a netuning or not is needed.

Working the way that one for every discovered law of Nature (which is based

on empirical facts) rememb ers to investigate its stability could b e conceived

of as using \random dynamics " as a scienti c method.

3.5 Turning to the Weyl equation again

In order to allow ourselves to call the Weyl equation stable, wehave- in

addition to restricting the basin of p erturbations (as discussed ab ove)-to

diminish the ambition of stabilityby restricting the features considered es-

sential and therefore required to b e unchanged under the mo di cations (the

15

And the structure maythus very well di er - in a random way - in various sub-parts

of our \world machinery". This idea is (in spirit) in some contrast to the idea that there

is only one unique and universal p ervasive \sup erstring", say, (a \T.O.E.") which-at

some fundamental scale - is ubiquitously implemented \everywhere" - and implemented

so \precisely" that it do es not di er, not even by the tiniest \small error", in any sub-part

of our \world machinery". 21

p erturbations).

Thus, when we claimed stability of the Weyl equation, under the consid-

ered class of p erturbations (3.7), we only achieved this b ecause we claimed

that it remained the Weyl equation in spite of the fact that it actually was a

completely new equation in the sense that the functional forms of the A (x)





and the V (x) had changed. It is only when we maintain that these func-



tional forms are not essential for it b eing the Weyl equation that the Weyl

equation can b e called stable, as we did.

This deemphasis (needed for the stability) of the imp ortance of the func-

tional forms of the x-dep endent co ecients A (electromagnetic four p oten-





tial) and V (vierb ein), is achieved, or strengthened at least, by declaring



that they are \dynamical". By declaring them \dynamical" we mean that

they dep end on some b osonic degrees of freedom, say, so that they b elong to

initial conditions rather than to the laws of Nature. They should b e more

like the p osition or the momentum of a particle or the value of a eld than

likesay the mass or the charge of a particle or some coupling constant. Very



luckily for such a p ostulate of A and V b eing \dynamical" is the fact that





we happ en to know phenomenologically already elds precisely having the



typeofinteraction with the Weyl particles which A and V have! These







elds are for A the electric 4-p otential and for V the vierb ein, whichis





easily identi ed with a gravitational eld. In this sense wemaysay that we

need the electromagnetic and the gravitational elds in order to pretend the

\stability" of the Weyl-equation! So instead of considering this need for not

considering the functional form essential a weakness of the idea of claiming

the Weyl equation exceptionally stable in just 4 dimensions, it turns out to

just t nicely provided wehave electromagnetic and gravitational elds (as

dynamical elds). Rather than having a trouble for our claim we got a p ost-

16

diction of a couple of well established forces in nature!

Is the two dimensional Weyl equation really two dimensional?

Note, that if one imp oses the stability requirement (of the restricted kind,

wehave discussed here), then all higher dimensions than 4 have b een com-

pletely ruled out, they are not stable! Therefore, if \our creator" had chosen

6 dimensions say, he had to \ netune" the Weyl equation, for example in

order to avoid terms like some di erential op erator or some other quantity

multiplying a matrix not b elonging to the  matrices of the 6 dimensional

Weyl equation!

16

In the \toy mo del" whichwe put forward in section 3 of H.B. Nielsen & S.E. Rugh

[8] we construct (p ostulate) a very general and complicated quantum mechanical system

(whichwe call the \b osonic" part of the full mo del) interacting in a random way with the

\fermionic world machineries" which represented the Weyl particles. Since all the e ect of



the \b osonic part" on the Weyl particles is represented just by A and V it means that





this \b osonic part" provides precisely electromagnetic and gravitational elds. The mo del,

however, has a few \small" troubles: The interaction of the vierb ein and electromagnetic

elds with themselves are a priori non-lo cal, we still have the dimensions in which motion

do es not take place and, moreover, one needs very strong input assumptions ab out the

state of the \world machinery" (the latter may, though, in principle b e o.k.). 22

If we liketocharacterize, however, the dimension 4 by the discussed sta-

bility of the Weyl equation we meet the problem that the d = 2 dimensional

Weyl equation is equally stable - or even more stable in a sense - than the

four dimensional one. For d = 2 there are namely n = 1 comp onent only

Weyl

2

and hence only 1 = 1 linearly indep endent matrices that can o ccur in the

equation.

From the ab ove formulas for the numb er of comp onents it is easily seen

that the Weyl equation in d = 2 dimensions has only n = n = 1 comp o-

Weyl

2

nent and thus wehave in this case d>n and the Weyl equation must

Weyl

even use the same \ -matrix" more than once. This means that it is very

stable, but if one have to use the same  -matrix twice, one can with go o d

reason ask if this equation - the two dimensional Weyl equation - is truly two-

dimensional! Wewould say that it is really not two dimensional, b ecause the

Weyl particle in two dimensional Minkowskian theory,say, can only movein

one direction and always with the sp eed of light. It really only makes use of

+ 0 1

one light-cone variable, e.g. x = x + x .Thus, it is rather to b e considered

only 1-dimensional! If you accept this p oint of view a \truly d-dimensional

Weyl equation which is stable in the sense describ ed ab ovemust have equal-

2

itybetween the numb er of linearly indep endent  or matrices n and the

2

numb er of dimensions d, i.e. n = d for the Weyl case.

Weyl

3.6 An almost true theorem ab out the numb er of gauge

b osons and the numb er of di erentWeyl particles

So far, wehave considered only one single Weyl particle, but we oughtto

consider (at least) the states related by gauge transformations as various

comp onents of a single equation. Taking into account that there can b e

several gauge comp onents, the numb er of comp onents of the Weyl spinor

17

increases . Itiswell known, that the Weyl spinor, indeed, has suchinternal

degrees of freedom (in Nature) - although this has b een neglected in the

previous discussion - as the reader mayhave noticed. For instance we know

to day that the Weyl spinor has color and other gauge degrees of freedom.

With such gauge elds interacting with the Weyl particle the Weyl equation

still takes formally the form (2.1) but now the covariant derivativeis not

simply

1

D = (x; p)=@ ieA (x) (3.9)

  

i

(ignoring the vierb ein by taking it to unit matrix), but rather, say

a

1 

a

D = (x; p)=@ ie A (x) (3.10)

 



i 2

a



's make up a basis in the representation r (for the Weyl particle where the

2

in question) for the representation of the Lie algebra of the Yang Mills gauge

group.

17

This observation (obstacle) is due to a question by Prof. Dragon at the conference 23

a

I.e. there is one  matrix for each dimension of the Lie algebra, and that

a

again means one  matrix for each \color state" of the gauge particle (each



gauge particle we could also say). In the Weyl-equation op erator  D thus



o ccurs the term

a



 a

 A : (3.11)



2

in the Yang-Mills case and thus the Weyl eld really must have n
d

weyl r

comp onents, where d denotes the dimension of the representation r under

r

which the Weyl particle transform under gauge transformations. This repre-

sentation dimension d is

r

!

numb er of di erent

d =

r

Weyl particles

while, as we rememb er,

!

numb er of comp onents

: n =

Weyl

of the Weyl spinor

The numb er of matrices that are needed to form a basis is the square of

2

the total numb er of comp onents, meaning (n
d ) . The condition for

Weyl r

stability of the Weyl equation (now coupled to \color" Yang-Mills degrees

of freedom) - if we de ne it in an analogous manner to the equation (2.20)

ab ove - is therefore given by the following:

\Almost true" theorem

! !

2 2

numb er of comp onents numb er of di erent

 =

of the Weyl spinor Weyl particles

! !

space-time number of

 (3.12)

dimension gauge b osons

a

Here wehave used that the numb er of di erent A (x)-comp onents is



! !

number of space-time

d d =  ;

g

gauge b osons dimension

This can b e seen easily, since there is an a-value for eachofthed dimensions

g

of the gauge group.

If we takeinto account that (at least empirically) wehave already the

equation (2.20), our main observation implies

! !

2

numb er of comp onents space-time

= : (3.13)

of the Weyl spinor dimension

and combining this equation with the more general stability condition (\the

almost true theorem") we get : 24

! !

2

numb er of di erent number of

= (3.14)

Weyl particles gauge b osons

Note, that this splitting of equation (3.12) into (3.13) and (3.14) can b e

derived from the fact that otherwise the matrices o ccurring in (3.11) would

not b e linearly indep endent.

So this is a prediction from the extension of the observed stability of the

Weyl equation. Is it true empirically?

Had it not b een for left handed quarks, this would have b een an entirely

true prediction provided, though very strongly, that we (as seen by Prof.

Dragon) only considers one single irreducible representation at a time. That

is to say: For each irreducible representation of Weyl particles - such as the

right handed strange quarks, say-we count the numb er of gauge particles

coupling non-trivially to this irreducible representation and use that as \the

numb er of gauge particles". As the numb er of di erentWeyl particles we

use the numb er of di erentWeyl particles in the irreducible representation

in question.

Then, indeed, the equation (3.14) is ful l led for al l the irreducible repre-

sentations - except for the left handed quarks!

For example, for the left handed p ositron only one linear combination

of gauge b osons couples - that of the weak hyp ercharge - and there is only

one left handed p ositron in the irreducible representation, so our equation is

ful lled with 1 = 1. Actually, in this case our previous discussion worked (in

whichwe neglected that there is more than one comp onent).

Another example is, say, one of the left handed antiquarks: There are 3

di erentWeyl particles in the irreducible representation, and to them couple

8 gluons and the weak hyp ercharge gauge b oson (a certain linear combination

0 2

of and Z ). That is 8 + 1 = 3 .

Only for an irreducible representation of left handed quark we are o : 12

2

gauge b osons but 6 Weyl particles. That do es not t since 12 6=6 .

In fact it is so that for all other irreducible representations than the left

handed quarks one nds a true representation of a U (N ) factor group of the

standard mo del group SM G = S(U (2)  U (3)). Here e.g. N = 1 for the left

handed p ositron, N = 2 for the left handed leptons , and N = 3 for b oth

typ es of left handed antiquark representations. In such cases one nds

!

numb er of di erent

d = = N (3.15)

r

Weyl particles

and

!

number of

2

d = = N (3.16)

g

gauge b osons

so wewould have the equality in just 4 dimensions (3 + 1). If we make the

very imp ortant assumption of lo oking at only one irreducible representation

of the gauge group at a time we nd that indeed wehave the stability in the 25

standard mo del for all of them except for the left handed quark representa-

tions.

So had it not b een for left handed quarks we could have claimed that the

stability of the Weyl equation as used by Nature (i.e. the Standard Mo del)

not only supp orts the dimensionality of space-time but also the typ e of gauge

group representations found in the standard mo del. Unfortunately for the

achievements of this principle of stabilitywehave an exception by the left

handed quarks, or in other words: it would predict that there should not

have b een left handed quarks, but otherwise it would haveworked well for

each irreducible representation separately.

This latter lack of b eauty means that the stability principle only is o.k.

provided there is some rule forbidding that di erent irreducible representa-

tion of fermions can get mixed in the attempts to mo dify the Weyl equa-

tion(s).

The fact, that instability o ccurs when mixings are allowed (here mixings

between irreducible representations) is also what wewould see if we thought

of the Dirac equation instead of the Weyl equation. That would namely mean

allowing mixing of left and right handed comp onents, and again no stability.

In conclusion, it app ears that \our creator" has made the Weyl equations

as stable as p ossible! This has b een achieved b oth bycho osing the dimension

to b e 4 = 3 + 1 (and not, for instance, 8) and bycho osing the representa-

tions and the gauge group so that in (at least) most cases each irreducible

representation taken separately is stable.

The exception of the left handed quarks it is, in fact, rather unavoidable

if there have to b e gauge anomaly cancellations and we still want mass pro-

tected particles. So, very likely, the choice of the Standard Mo del is - in some

way of measuring it - the most \stable" p ossibility.

So stabilitymayeven b e a principle b ehind the representation choice,

to o! Not only (as we already saw) b ehind the dimension and the existence

of gauge and gravitational elds.

It must b e admitted, however, that there is the little problem with the

working of this principle now where wehave included several comp onents

and non-Ab elian gauge elds: We also get the vierb ein and thereby the

gravitational eld into a nontrivial representation of the gauge group. That

particular feature is not so go o d from the empirical p oint of view! 26

4 Concluding remarks and dreams

In the present contribution wehave made an empirical observation (sec. 2.4)

connected to the Weyl equation, suggested an interpretation in terms of \form

stability" of the Weyl equation (sec.3), and - in a search of expanding the

basin of allowed p erturbations (mo di cations) of the Weyl equation (under

whichwewould like to claim its stability) - wehave also challenged the

concept of \stability of Natural laws". The latter part (sec. 3.2-3.5) is a

kind of preliminary attempt to lo cate principal b oundaries for the set of

ideas whichwe call the \random dynamics" pro ject, describ ed, e.g. in C.D.

Froggatt & H.B. Nielsen [5].

As concerns our empirical observation, wehave seen that the dimension

d = 4 ob ey the equation (2.21),

(

d=21 2 d2

(2 ) = 2 for d even,

d = ;

(d1)=2 2 d1

(2 ) = 2 for d odd:

which - via the step of the sigma-matrices forming a basis (sec.3.1) - leads to

the suggestion that just in the phenomenologically true dimension d =4 is

the Weyl equation esp ecially stable. This is so to say the message from \our

creator" brought via the dimension. In order to implement this stabilityit

was at rst noticed that at low energy (and momentum)Taylor expansions

had to b e used and - secondly - that we needed b oth gravitation and elec-

tromagnetism. Wemay put it the way that in order to make the idea of the

Weyl equation b eing stable, as suggested, one predict in retrosp ect these two

forces of nature. Really we could use the non-Ab elian Yang Mills elds, as

found in the Standard Mo del, instead of simple Ab elian electro dynamics but

there are a couple of severe problems at this p oint:

1. One must b e satis ed with lo oking only for stability under p ertur-

bations in which elds in the same irreducible representation are allowed

to mix, i.e. wemust require linearity and homogeneity in each irreducible

representation of Weyl fermions separately. (In the sp eculative mo del put

up in section 3 of H.B. Nielsen & S.E. Rugh [8] one might hop e that mild

continuity requirements will turn out to b e enough for implementing sucha

rule. But, a priori, it complicates matters to need such a separation of the

di erent irreducible representations).

2. Even with the restriction to \no mixing" of di erent irreducible repre-

sentations the left handed quarks do not t into the scheme. The problem is

that the entire standard mo del group has only dimension 12 while the num-

b er of left handed quark Weyl elds is 6 times the numb er of comp onents for

a single Weyl two-comp onenet eld. This means that the numb er of allowed

2

matrices is enhanced by a factor 6 = 36 due to the color and weak isospin

degrees of freedom. The number of Yang-Mills eld comp onents is, however,

corresp ondingly only enhanced by 12, and that is less than 36. In order

that the \form stability" of the Weyl equation shall work for the left handed

quarks it would b e appropriate with a U (6)-gauge theory rather than, just, 27

the Standard Mo del. Such a mo del would, however, have gauge anomalies

and not have a consistent gauge symmetry.

We presume, that a strongly related expression for the sp ecial \form

stability" of the Weyl equation is the fact that just in the dimensions d =3

and d = 4 is the numb er of helicity states of a Weyl particle (antiparticle not

included) precisely one, so that there is no degeneracy.

In b oth ways of lo oking at our result there is a need for a separate dis-

18

cussion of the two dimensional (d = 2) case since this is, at rst, even more

stable than the exp erimentally observed case (d = 4). However, one may

claim that this case is not truly two-dimensional since it do es not (truly)

make use of b oth dimensions (cf. section 3.5).

Taking (in spite of the \small"troubles) the sp ecial stability of the 4-

dimensional Weyl equation as a trace in the search for fundamental principles

in physics wehave then lo oked into howmuchwemayexpand the basin of

allowed p erturbations (mo di cations) and still claim the Weyl equation to

remain \form stable". In a sense, such a search is, precisely, what is done in

H.B. Nielsen & S.E. Rugh [8] in whichwehave put up a very general \toy-

mo del" for a pregeometry (a \world machinery"). The latter typ e of mo del

may in fact b e thought of as representing a very general typ e of linear homo-

geneous di erential equation conceivable as describing a rather general quan-

tum mechanical \machinery" having partly a classical analogue and ob eying

the smo othness conditions asso ciated with the smo othness prop erties of this

classical analogue. It must, however, b e admitted that e.g. Fermi statistics

and the identical particle prop erty is rather put in than derived, so far.

Even if there are several troubles (as wehave describ ed, sec. 3.2-3.6)

in interpreting the Weyl equation as totally stable against all imaginable

p erturbations, wewould still claim that the choice of just 4 dimensions,

indeed, p oints to some sort of stability as a guiding principle.

\Our creator" has also chosen the representations in the standard mo del

so that the Weyl equations b ecome as stable as p ossible ! (at least very

stable compared to what could have happ ened). When He namely use - as

is the case except for the left handed quarks - de ning representations of an

2

gauge SU (N ) group having also a nontrivial Ab elian charge there are N

2

elds eachwithd=4Lorentz comp onents and also 4 times N linearly

indep endent matrices. Only for the left handed quarks for whichwe get 6

times 2 comp onents there are 144 linearly indep endent matrices but only 12

times 4 gauge p otentials to adjust (i.e. 48 6= 144).

It turns out that there is a con ict b etween making the Weyl equation

stable and keeping mass-protection (i.e. unless we satisfy the no anomaly

condition trivially). For instance, if one simply p ostulated some gauge elds

in a group SU(6) extending the group SU (2)  SU (3)  SU (6) providing all

the (sp ecial) unitary transformations of an irreducible representation corre-

18

The \no degeneracy" principle leaves also d = 3 as a comp etitor to the exp erimental

dimension d = 4, but if we assume that we only have access to low energies and thus only

shall see mass-protected particles we could on the that ground exclude the o dd space-time

dimensions anyway. 28

sp onding to a set of left handed quarks this eld would have anomalies in

the gauge symmetry. I.e. the gauge symmetry could not b e uphold at the

quantum level unless further particles were added to the mo del to comp en-

sate the anomaly. But then, of course, the added eld might b e in danger of

lacking stability again.

Hence, a claim that there should b e enough gauge elds to realize the

\form stability" prop osed seems not to b e fully realized in Nature - although

Nature comes close - but it is also not realizable with mass protected fermions:

One simply cannot cancel the gauge anomalies and at the same time have

all the fermions in the de ning representations unless wecho ose a trivial

cancellation which will sp oil the mass-protection. With so many de ning

representations as are used in the Standard Mo del one might then think that

19

the Standard Mo del do es its b est p ossible to b e maximally \stable"!

4.1 Could our observation b e an accident?

Which \signals" in phenomenology carry much information and which \sig-

nals" carry only small amounts of information ?

The purp ose of reading out \signals" from phenomenology is of course that

this is the waywemay really learn from Nature rather than only from our own

sp eculations. It is, however, only to b e trusted when the numb er explained

is suciently complicated that it is not to o easy to nd an explanation for it.

There must b e a fair amount information (whichhave previously not b een

understo o d) that nds an explanation, otherwise the explanation will not b e

convincing.

Moreover, there can only be one explanation of a given phenomenon -

such as the dimensionality of space-time considered here. If there were two

fundamental (indep endent) explanations of a given interesting numb er (such

as 4 = 3 + 1 dimensions) then it would b e a strange mathematical accident

that these two explanations would give the same result.

In that sense fundamentally di erent attempts of arriving at the number

4 (for instance, the alternative attempts [11] - [21 ] mentioned in the list of

references) cannot b e true - provided, for instance, that our observation is

20

not an \accident".

19

In fact, what favours the stability is that the representation is so \small" as p ossible

compared to the \size" of the gauge group or rather the part that is not represented

trivially. So maximal stability of the Weyl equation would roughly corresp ond to smallest

p ossible representations of the fermions. Now it happ ens that S.Chadha and one of us

p ointed out that the Standard Mo del representations are characterized as the \smallest"

p ossible ones with mass-protection and no gauge-anomalies (we thank J. Sidenius for

nding that a single alternative solution is excluded by the requirement of no mixed

anomaly). So it is from that work really suggested that the Standard Mo del is as stable

as it would have b een p ossible to construct it without lo osing mass-protection and/or the

anomaly cancellations.

20

That our observation is not an accidentwould b e strongly supp orted if we could

obtain from the Weyl equation - in a related way - not only the dimension 4 but also its

splitting into 3+1. The Weyl equation supplemented with the - rather mild and reasonable

- assumption that by appropriate multiplication with a matrix the Weyl-equation op erator 29

Note, that the dimensionality of space-time is known with several digits

precision,

d =4=4:00::::

But almost all these digits are \0"s, and therefore they do not carry so much

information - mo dulo that most underlying theories, by construction, have

an integer numb er of dimensions.

It would b e easier to get reliable information from the electromagnetic

21

nestructure constant, say,

2

e

= =1=(137:0359895  0:0000061)

hc

which is suchan \irregular" numb er (measured to a precision of 8 digits

or so) that it probably carries a huge amount of information to supp ort or

destroy a theory which aims at predicting this constant(we remark that no

22

theory exist to day which claims to arriveat with 8 digits precision ).

Compare, also, e.g. with the Standard Mo del group S (U  U ). If we

2 3

want to predict this group, relative to other p ossible groups with up to 12

generators (gauge b osons), wehave of the order of 240 groups [7]. We con-

clude that to predict the true gauge group is comparable to predicting a

numb er with two signi cant digits.

If we, by accident, nd several fundamentally di erent explanations for

the observed dimensionality d = 4 of our space-time then it would presum-

ably indicate that the numb er \4" is to o small and to o simple (there is

not enco ded information enough in this numb er) for reading out interesting

structural information from it. In the light of that several explanations -

which seems reasonable and fundamental, e.g. the stability of (classical or

quantum) systems governed by the Coulomb or Newtons law - co exist, we

fear that the numb er \4" is to o simple to b e readable in this sense.



 D b ecomes Hermitean can actually only b e realized provided the splitting is into an



odd numb er of times (and an o dd numb er of space-dimensions; we think of the even d

case). A relatively easy series of algebraic manipulations shows this fact. For the troubles

(as far as the Dirac or Weyl equations are concerned) caused in \euklidianization" due to

the di erence b etween \4" and \3+1", see e.g. S. Coleman [23].

21

Cf. ,e,g., \Review of Particle Properties",Phys. Lett. B 204 (April 1988).

22

Actually it happ ens that one of us would claim to have quite an impressive \expla-

nation" of all the three nestructure constants in the Standard Mo del by relating them

to the (or a) \multi-critical point" (where several phases of a - lattice - gauge theory

meet) But these ts have rather only one signi cant digit, although that seems to us al-

ready suggesting that there is some truth b ehind. Cf. D.L. Bennett & H.B. Nielsen, in

preparation. 30

Acknowledgements

We b oth have pleasure in thanking for travel-supp ort the EEC-grant CS1-

D430-C which has b een of signi cance for developing the ideas of the present

article for instance in Rome and Heraklion and in Berlin, where the confer-

ence was held.

S.E.R. would also like to thank Prof. John Negele and Prof. Ken John-

son for the warm hospitality extended to him at the Center for Theoretical

Physics, where parts of this work was written up in some nal stages.

Supp ort from the Danish Natural Science Research Council (Grant No.

11-8705-1 ) is gratefully acknowledged.

We thank Hans Frisk for discussions and imp ortant input at a certain

stage of the development of this work, in particular, discussion ab out the

\mo de conversion surfaces", a central concept for the toy-mo del of a \world

machinery" describ ed in a previous work [8] on these issues.

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[27] I. Kant \Prolegomena to Any Future Metaphysics" (1783) (The Paul

Carus Translation extensively revised by J.W. Ellington is available from

Hackett Publ. Co, 1977). This little b o ok a ords the reader a compact

(and more accessible) overall view of \Critique of PureReason" (1781).

[28] N. Bohr \Atomic Theory and the Description of Nature" (Cambridge,

1934), \Atomic Physics and Human Know ledge" (New York, N.Y., 1958),

\Essays 1958-1962 on Atomic Physics and Human Know ledge" (New

York, N.Y., 1963). 33