constants and cosmology: the nature and origin of fundamental constants in astrophysics and particle...

C O N S T A N T S A N D C O S M O L O G Y : T H E N A T U R E A N D O R I G I N

O F F U N D A M E N T A L C O N S T A N T S I N A S T R O P H Y S I C S A N D

P A R T I C L E P H Y S I C S

P A U L S. W E S S O N

Space Sciences Laboratory, University of California, Berkeley, CA 94720, U.S.A.

"These our actors, As I foretold you, were all spirits, and Are melted into air, into thin air."

SHAKESPEARE, The Tempest

Abstract. We ask about the nature and origin of the fundamental constants of astrophysics and particle physics, notably the speed of light c, the gravitational constant G, Planck's constant h, and the magnitude of the electron charge e. We consider general relativity and the theories of the electromagnetic, weak and strong interactions that make up the Standard Model; together with the Lagrangians of Einstein, Maxwell, Schr6dinger, Klein-Gordon, Dirac, Proca, and Yang-Mills. Then we look in a more qualitative way at how the equations of physics are set up, their dimensional content, and the removal of constants from them by a suitable choice of units. We conclude with Hoyle and Narlikar, Jeffreys and McCrea that parameters like c, G, and h are merely manmade dimensional conversion constants. They arise because of our subjective view that mass, length, and time are different concepts. These constants can be removed in a manner analogous to the removal of the permittivity of free space t o from electrodynamics, and none are really fundamental. The charge e is different, being the low-energy limit of a function related to properties of the vacuum, but because of this it is not a fundamental constant either. We suggest there are no constants which truly deserve to be called fundamental, and that an aim of physics ought to be to write down laws in which no constants appear.

1. Introduction

The so-ca l led f u n d a m e n t a l cons t an t s are widely regarded as a k ind o f dis t i l la t ion o f

physics . The i r d i m e n s i o n s are re la ted to the fo rms o f phys ica l laws, w h o s e s t ructure can

in m a n y cases be r e c o v e r e d f rom the cons t an t s by d imens iona l analysis. The i r sizes

a l low the phys ica l laws to be wri t ten as equat ions , which can be c o m p a r e d to obse rva -

t ion given s o m e cho ice o f units. Desp i t e their pe rce ived f u n d a m e n t a l na ture , however ,

there is no theory o f the cons t an t s as such. F o r example , there is no general ly accep ted

fo rma l i sm tha t tells us h o w the cons t an t s originate, h o w they re la te to one another , their

re la t ive sizes, or h o w m a n y of t hem are necessa ry to descr ibe physics . This lack o f

b a c k g r o u n d seems o d d for pa rame te r s tha t are widely r ega rded as basic. I t is the a im

o f this art icle to c o m e to a bet ter unde r s t and ing o f the f u n d a m e n t a l cons t an t s by asking

(and p rov id ing ten ta t ive answers to) s o m e f u n d a m e n t a l ques t ions abou t their origin and

nature .

W e will p r e s u m e a work ing-phys ic i s t ' s knowledge o f the cons t an t s conce rned , the

* Permanent address: Department of Physics, University of Waterloo, Waterloo, Ontario N2L3G1, Canada.

Space Science Reviews 59: 365-406, 1992. �9 1992 Kluwer Academic Publishers. Printed in Belgium.

366 P A U L S. W E S S O N

relevant laws, and techniques such as dimensional analysis. This will allow the argu-

ments to be made fairly concise, even though the subect matter is far-ranging and in places ambiguous. There are several reviews available that deal with the nature of the

fundamental constants and related things. Notably there are the books by Wesson

(1978), Petley (1985), and Barrow and Tipler (1986); the conference proceedings edited

by McCrea and Rees (1983); and the articles by Barrow (1981) and Wesson (1984a).

Readers who desire more background information are referred to these sources and their bibliographies.

The constants we will be primarily concerned with are those that figure in astrophysics

and particle physics. It is convenient to collect the main ones here, along with their dimensions and sizes in c.g.s, units:

Speed of light c L T - ~ 3.0 x 10 l~

Gravitational constant G M - 1L3 T - 2 6.7 x 10- 8,

Planck's constant h M L 2 T -1 6.6 X 10 - 2 7 ,

Electron charge (modulus) e M1/ZL3/ZT - 1 4.8 x 10- lO

Here e is measured in electrostatic or Gaussian units. We will use e.s.u, in the bulk of what follows, though S.I. will be found useful in places. The two systems of units are

of course related by 4roe o, where the permittivity of free space is eo = 8.9 x 10 - 12 C 2 m - 3 s 2 kg - 1. In S.I., e = 1.6 x 10- 19 C (Coulombs: see Jackson,

1975, pp. 29, 817; and Griffiths, 1987, p. 9; the permeability of free space/~o is not an

independent constant because c 2 - 1/to#o). The above constants have values that are

determined by fairly well-defined experiments (Petley, 1985). These values depend of course on a choice of units, and a comment about this is in order before we proceed.

The practice of choosing units which make the magnitudes of certain constants equal

one may cause us to suspect that some of them are not truly fundamental but does not

prove it. This because they retain their characteristic dimensions, and because it is impossible to choose units which make an arbitrarily large number of constants equal unity and still agree with the real world. (If we chose units that made e = 1,

h ( - h / 2 r 0 = 1, and c = 1, this would make the fine-structure constant e2/hc equal 1

whereas it is really about ~ . ) In fact, the origins and sizes of the fundamental constants are obscure. These things have to do with the way physics is set up, and in particular

with the equations of physical theory. These will be studied in Section 2 following, which deals with the main theories of gravitation and quantum mechanics and the parameters that enter them. Then in Section 3, a general discussion will be given to try and elucidate the status and meaning of the fundamental constants. Some conclusions will be collected

together in Section 4.

CONSTANTS AND COSMOLOGY 367

2. Theories of Physics and Fundamental Constants

In this section we will study the four known interactions of physics, starting with gravity,

including Newton's theory and Einstein's theory of general relativity (GR). We will then consider electromagnetism, including Maxwell's classical theory and quantum electrodynamics (QED). Next the weak interaction will be treated, including its unification with electromagnetism in the Glashow-Weinberg-Salam (GWS) or electroweak theory. Lastly we will look at the modern theory of the strong or nuclear interaction, quantum chromodynamics (QCD). The last three interactions have similar descriptions, and in recent years have been grouped together in the Standard Model (see, e.g., Griffiths, 1987; and Collins et al., 1989, for introductory and advanced accounts). The force of gravity remains separate, however; and instead of being described by particles, including the graviton of hypothetical quantum theories of gravity, it is still to all practical purposes described by fields specified by Einstein's equations (see, e.g., Lawden, 1982; and Weinberg, 1972, for introductory and advanced accounts). Thus the following discussion

divides naturally into two sub-sections.

2.1. MACROSCOPIC PHYSICS

Gravity is described by 10 dimensionless potentials which are the independent elements in a symmetric 4 x 4 matrix or tensor go" The interval between two nearby points in a 4D spacetime is given by ds 2 = go dxi dxJ, where there is summation over repeated indices. The coordinates are identified as x ~ = x, x 2 = y, x 3 = z, x 4 = ct. Here x, y, z

are space-coordinates and t is time. We see that the role of the speed of light here is

to dimensionally transpose a quantity with dimension T to one with dimension L (see Section 3 for a more detailed discussion of this). This can be done at any point, because go can be reduced to the Minkowski form r/~ = diagonal( + 1 - 1 - 1 - 1) at any point even in the presence of fields, so c and the dimensional transposition are well defined. By contrast, the usual view of c as being an upper limit to the speed of propagation of causal effects is less clear, because when ]gol are not unity in the presence of fields the

definition of the speed of causal propagation is bound up with the choice of coordinates, which in GR are arbitrary. Light that propagates between two points follows a null geodesic with ds 2 = 0. For example, propagation along the x-axis with a diagonal metric wherein gll = g~(t, x) # - 1 and g44 = g44( t, X) =/6 + 1 leads to the relation dx/dt = +_(g44/g~a)~/2c. We can interpret this to mean that the propagation speed is not c in common-or-garden Minkowski-type coordinates, or we can write it c = + (g~{2 dx)/(g]/42 dr) -= + dx'/dt' and make it true by a re-definition of coordinates. The point is that if spacetime is curved, the pedantic view that 3-velocities must be less than c and that this parameter is an upper limit to causal propagation loses its validity, and in fact must be replaced by the more appropriate concept of a causally-connected region bounded by an horizon (see Rindler, 1977, pp. 215, 216, and below). Thus the meaning ofc as a fundamental constant becomes blurred when we go from the no-field flat spacetime of special relativity to the finite-field curved spacetime of general relativity.

368 PAUL S. WESSON

When a particle moves under the influence of gravitational fields in a curved spacetime, it does so along a path that makes the interval s a minimum. This of course defines the concept of geodesic, which symbolically is the path satisfying 6 I ds = 0 (where ds2 = go dx~ d x j as above). Written out, the latter relation yields the 4 components of the geodesic equation:

d u k + ~ u ' u : = O. (1)

ds ~ j

Here u k =- d x k / d s are 4-velocities and the Christoffel symbols ~ are functions involving the first partial derivatives of g,y. In general spacetime is curved, and it is convenient to introduce a derivative that takes account of this, the so-called covariant derivative. This is usually denoted by a semi-colon, and the covariant derivative of a vector V,. is given by Vi; j = ~V,./~?x j - I~ jV k. In the special case where there is no gravitational field and spacetime is flat (g,j = constants, U~j = 0), the particle simply does not accelerate and moves with constant 4-velocity. This is the standard interpretation and easy to understand. But there is more to the concepts of interval and geodesic, and other interpretations are more difficult to understand. Consider for example two particles that exchange a photon, so ds 2 = 0. Instead of the conventional interpretation of this as an effect that propagates between the two particles in 3D, it is valid to say that the two particles have zero separation and are, therefore, coincident in 4D. We normally ignore this interpretation, presumably because it brings up the awkward problem of the particles' distinguishability. In practice, this problem is avoided by assigning the particles more labels (e.g., masses), in addition to their 4 spacetime coordinates. In theory, it is possible to lift the degeneracy inherent in 4D relativity by extending the theory to higher dimensions. This is actually done in the so-called Kaluza-Klein theories, though most of these use a different motivation for extending spacetime and merely add extra space-like dimensions. We see that the idea of a manifold, with an interval which when minimized gives a physical relation like (1), can be extended beyond our usual concepts

of 3D space and time. The geodesic equation can be solved if we know the metric coefficients go, and these

are given in turn by solutions of field equations. These are set up to involve tensors, which since they have the property of being unchanged under a change in the coordinates reflect our belief that physics should not depend on our choice of coordinates and should be covariant. The Riemannian geometry based on ds a = g~ dx i dx j with a symmetric g,j involves only a small number of distinct tensors, and this makes it relatively clear how to set up field equations. In GR, we match two second-rank, divergence-free symmetric tensors. The Einstein tensor Ga is geometrical, and is given by G o. = R~j - Rgo./2, where R U and R are the Ricci tensor and scalar, respectively. The energy-momentum tensor T o. is physical, and is given for a perfect fluid by Ty = (p + pc2)u iu j - pg~, where p and p are the pressure and density, respectively. The match between geometry and physics


is given by writing G o = - ~cTo., where tr is a constant (the minus sign here and various

powers ofc 2 that appear in setting up the field equations are the result only of convention

and are not important). It is possible to add a term to the ones just defined because the metric tensor gu behaves as a constant under covariant differentiation. This term is usually written A g ~ / c 2 and is added to the left-hand side. Thus the final form of the field

equations is

R o. - R~ + A g o. = - tc[(p + DC2)bliltj --pgo]. 2

(2)

In vacuum (p = 0 = p), this contracts to give R = 4A, which when used to eliminate R allows (2) to be written as R• = Ag~j. Now A is known to be small, since it corresponds in the weak-field limit to a force per unit mass A c Z r / 3 whose effects have not been

detected. There are also some reasons for thinking it might be zero. For example, the

anthropic principle implies our existence is predicated on the fact the Universe has zero spatial curvature and zero cosmological constant (Hawking, 1983; Carter, 1983; Barrow, 1983a; Barrow and Tipler, 1986). Or there may be a constraint from particle physics that requires space to be flat over macroscopic distances (Hawking, 1983). Anyway, if A is small, or equivalently if the constant length it involves is very large (A has dimensions L - 2), then the extra term can be neglected. The vacuum field equations

are then just

R~ = 0 . (3)

These equations have been tested astrophysically and are in remarkable agreement with observations. By contrast, the non-vacuum equations (2) have only been tested indirectly and inexactly, notably by comparing cosmological models based on them with observations of the real Universe (see below). However, they make sense insofar as their weak-field, low-velocity limit gives back Newtonian gravity. In this case, it follows from the geodesic equation (1) or the equations of motion that the main term in the metric is g44 - (1 + 20 / c2 ) , where ~p is the Newtonian potential. The main part of the field equations (2) is the 4-4 component, which with R44 ~-~ (1/2)72g44 and T44 "~ pc 2 gives back Poisson's equation V2q5 = 4rcGp of classical theory provided ~ = 87~G/c 4. Here G is Newton's constant of gravity, which in Einstein's theory is seen to be bound up with the speed of light.

Solutions of Einstein's field equations (2) for fluids help us understand better how G and c figure in GR. The Newtonian properties of fluids are actually recovered from

the right-hand side of (2) in the 4 equations T,j;j = 0. These give back the equations of motion in ordinary 3-space and the equation of continuity. There is actually a redun- dancy in GR involved here, because the equations of motion of a particle in the fluid are also contained in the geodesic equation (1). It should also be noted that while the 4-4 component of To; j = 0 gives back the equation of continuity, this is not synonymous with the statement that mass is conserved: mass in GR is actually defined as a function of the fluid properties (p and p for a perfect fluid) and the metric properties (gij and their

370 PAUL S. WESSON

partial derivatives), and the mass contained in any region is not a constant in time if the pressure is finite (see Ellis, 1984; and Wesson, 1986a, for reviews). The pressure in GR acts in rather a peculiar way. This can be appreciated by looking at the general equation of motion, in the form derived by Raychaudhuri, and the continuity or conservation equation:

,

=2(co ~ - G ~ ) + u , c ~ - - - ( 3 p + pc2), ( 4 . 1 )

3k ~c 2 = - (p + pc 2) - - (4.2)

R

Here R is the scale factor of a region of fluid with vorticity co, shear ~, peculiar acceleration ~", and uniform pressure and density (see Ellis, 1984). A dot denotes the total derivative with respect to time. From (4.1) we see that the effective mass depends on the combination (3p + pc 2) in the last term, so for mass to be positive and attractive we need (3p + pc 2) > 0. From (4.2) we see that the rate of change of density depends on the combination (p + pc 2) in the last term, so for matter to be stable in some sense we need (p + pc 2) > 0. However, neither of these inequalities should be taken as sacrosanct. Any quantum interaction that results in an attraction or tension between the particles of the fluid corresponds in the convenient but simplistic perfect-fluid descrip-

tion to p < 0. This may arise naturally, and lead to phenomena such as particle produc- tion in the early universe (see Ellis, 1984; Wesson, 1986a; Itzykson and Zuber, 1980, p. 141; Abbott et al., 1982; Henriksen et al., 1983; McCrea, 1986; Shuryak, 1988; Press

et al., 1989). In fact p is a phenomenological parameter, lacking an underlying theory, and there is no fundamental reason why physical situations involving tensions should not be handled by allowing p to be negative.

The same comment applies to p, and even though it is hard to reconcile with our everyday experience there are situations involving particle physics where it may be required to consider p < 0 (e.g., Bonnor and Cooperstock, 1989; Cooperstock and Rosen, 1989). The fact thatp and pc 2 have the same dimension by virtue of the existence of c usually goes unremarked, but presumably has some deeper meaning. That G couples to the combination (3p + pc 2) and thereby defines the gravitational mass is a con- sequence of how the energy-momentum tensor is defined and how the equations of GR are set up. This kind of gravitational mass, which is active in the sense of being the origin of the gravitational field, is not the only one in GR. There are also the passive gravitational mass and the inertial mass (Bondi, 1957; Bonnor, 1989). These are proportional to each other, and often set equal, by the equivalence principle. They are defined by the combination (p + pc 2) and do not involve G. (This is apparent from (4.2), and may be verified by expanding T~j.j = 0 as done by Bonnor (1989) and others.) Thus to sum up: active gravitational mass has a density (3p + pc 2) and is associated with G, while inertial mass ( = passive gravitational mass) has a density (p + pc 2) and is not associated with G. Clearly mass is not as simple as commonly thought! Neither need mass necessarily

C O N S T A N T S A N D C O S M O L O G Y 371

be positive. It is obvious from (4.1) that p < 0 by itself can lead to a negative active gravitational mass, and result not in attraction but repulsion. In this way particle physics processes, that could have produced regions with effective tension (p < 0) in the early

universe, might have drastically affected the dynamics of the fluid and the process of galaxy formation.

The question of negative mass has been investigated in a very lucid manner by Bonnor (1989), who has given references to other work. He has shown that there is nothing intrinsically impossible about it. However, the validity of the usual laws of mechanics does seem to imply that discrete masses must have the same sign. This of course agrees with our observations. But the situation is less clear, and may admit regions with both signs, for a continuous medium that does not interact directly with matter. Such a medium is what is traditionally called a vacuum. In quantum field theory, the vacuum is now realized to be the seat of significant physical processes, such as vacuum polarization (see below). In gravitational theory, the vacuum is not assigned the same status. Presently, the only effective way to attribute physics to the vacuum is to incorporate the cosmological constant or a parameter like it.

The role of the cosmological constant A in describing the vacuum can be appreciated by looking at the field equations of GR for a perfect fluid with a pressure and density that are both homogeneous and isotropic, namely the Friedmann equations:

3 (~C2 + R2 ) _ a c 2 8 ~ G p = ~ (5.1)

8~rGp 1

C 2 R 2 (kc 2 + k 2 + 2_R'R) + A c 2 . (5.2)

Here the notation is the same as above, with the addition of the curvature constant k = _+ 1 or 0. (Equations (5) with A = 0 give (4.1) with co = a =/ t i = 0 by forming (5.1) + 3 x (5.2), and give (4.2) by forming (5.1)" + (5.1) x (3R/R) + (5.2) x (3R/R).)

An inspection shows we can replace explicit A terms on the right-hand sides of (5.1) and (5.2) by implicit terms in the density and pressure on the left-hand sides. Specifi- cally, we can replace A by the density and pressure of the vacuum:

A c 2 A c 4 p~, = + - - , p ~ - (6)

81tG 8rcG

This ansatz, which is now in widespread use, was first formulated by Zeldovich (1968). It preserves Lorentz invariance, and is not restricted to uniform fluids. Even though there is no direct evidence for the dynamical effects of A, which implies I AI < 10- 56 c m- 2 or [Pv I < 6 x 10- 30 g cm ~ (Coley, 1987), it could be that l A[ was larger during an earlier phase of the Universe (see, e.g., Henriksen et al., 1983). Also,

the fact that A can be sensibly re-interpreted by moving it over to the other side of the field equations is important conceptually. Einstein introduced and abandoned an explicit A, but regarding it as a part of T~j may have been more acceptable to him.


Perhaps even this ploy would only have been interim, however, for Einstein apparently

had the dream of uniting gravity with matter in a way that would transfer the matter

properties to the left-hand side of the field equations (see Salam, 1980, p. 534). We will return to this idea below. For now, we note that while A may be zero, if it is finite it

can be interpreted as a property of the vacuum provided it is combined with G and c.

Then Pv = - Pv is a (maybe unrealistically simple) equation of state for the vacuum. The Friedmann equations (5) for the cosmological fluid can be solved with and

without A, and lead to some useful insights, particularly with regard to c. This constant,

as noted above, is commonly regarded as defining an upper limit to the speed of propagation of causal effects. However, the causally-connected regions of spacetime are

defined rigorously in G R in terms of horizons. In the Friedmann-Robertson-Walker

(FRW) cosmological models, which are solutions of (5), there can be two kinds of

horizon. An event horizon separates those galaxies we can see from those we cannot

e v e r see even as t ~ oe. A particle horizon separates those galaxies we can see from

those we cannot see n o w at t = to(--- 2 x 101~ yr). FRW models exist which have both

kinds of horizon, one but not the other, or neither. A model in the latter category is that of Milne (R oc t, k = - 1, p = p = 0). This model is useful, because it can provide an

initial state for the Universe that may help resolve the well-known horizon problem

posed by the isotropy of the microwave background (Wesson, 1985). But it is not

realistic as a model for the later universe because it is empty. In fact all non-empty

isotropic Big-Bang models in G R have a particle horizon. The distance to this imaginary boundary in space defines the size of the region of the

Universe in electromagnetic causal connection with us now. The proper distance to the horizon can be calculated for various values of the parameters in (5), and for p = A = 0

it has a relatively simple form (Weinberg, 1972, p. 489; Lawden, 1982, p. 196). In terms

of Hubble's parameter now (H o - R o / R o ) and the deceleration parameter now ( % = - _ R ' o R o / R ~ ) , the distance to the particle horizon for each possible value of the

curvature constant k is given by

~ ) dk=+l Ho(2qo- 1) 1/2 cos 1 , % > � 8 9 (7.1)

2c 1 (7.2) d k = o - - 3 C t o , qo - 2 ,

I-Io

= ) ' c cosh-1 1 _ 1 qo < ~ . d k = - i Ho(1 _ 2qo)1/2

Even for the middle case, the Einstein-de Sitter model with flat 3-space sections, the distance to the horizon is n o t c t o. This emphasizes a point alluded to above. The speed of light only has a fundamental meaning in the special-relativity flat 4D case where there is an extended inertial frame. It has a less fundamental meaning in the general-relativity curved 4D case where such frames do not exist (see Rindler, 1977, pp. 215,216). The

CONSTANTS AND COSMOLOGY 3 7 3

preceding discussion may seem overly long to some, but there is a very important point

involved here which is not appreciated by everyone. The speed of light c, which is widely

regarded as the most fundamental o f the constants, is theory-dependent and not as funda-

mental as widely believed. At this stage in our deliberations, it may be helpful to collect here points we shall use

below. In gravitation as described by GR, c is essentially a dimension-transposing constant, the constants G and A (if it exists) often occur combined with c, and by judicious use of these constants the equations can be sensibly re-organized and re-inter-

preted.

2.2. M I C R O S C O P I C P H Y S I C S

Particle physics has evolved along different lines than gravitation. The field equations for the latter may if so desired be obtained from an action or variational principle, first formulated by Hilbert about the same time as Einstein used the more intuitive method of matching tensors. Thus if R is the scalar contraction of the Ricci tensor R 0 and g is the determinant of the metric tensor g•, the only natural choice for a geometrical Lagrange function is R with the corresponding Lagrange density ( - g)~/2R, and the

empty-space field equations R U - Rg~/2 = 0 follow from b ~ R( - g)1/2 d4x = 0. (See Weinberg, 1972; p. 364; Misner et al., 1973; pp. 485,491; Pauli, 1981, pp. 68, 162. The non-empty field equations (2) follow from including the scalar contraction T of Tg combined with the constant ~c defined above.) But the traditional way of obtaining the field equations of GR has been the Einsteinian one of matching tensors, where the latter involve a non-flat g,~. By contrast, the development of particle physics has involved a different philosophy that starts with the definition of an appropriate Lagrangian. The Euler-Lagrange variational equations, which were originally equivalent to the equations of motion of Newtonian mechanics but have a wider application, are then used to obtain field equations. The latter normally involve the flat Minkowskian r/~. (We will use this in the rest of this section, and shift over from the gravity usage i = 1-4 to the particle usage i = 0 - 3.) Some of the more important Lagrangians of particle theory will be looked at below. Here, though, we note that an intermediate position between gravity and particle physics is occupied by Maxwell's theory of electromagnetism.

IfAi is the 4-potential and J~ is the 4-current (covariant and contravariant quantities differ now by at most a sign), Maxwell's equations are contained in the tensor relations

_ ~ ? A k #Ai r ik 4re j k , Fik --- , (8.1) 0X i C ax ~ ~X k

and the identities

a ~ aF~t aF,,. - - + + = 0 , ( 8 . 2 ) 0x t ~?x; ~?x ~

implicit in the definition ofF,. k. However, Maxwell's equations may also be obtained by


substituting the Lagrangian

1 1 5O - Fi~F~.~ - *- J iA i (8.3)

16rc c

in the Euler-Lagrange equations, which give (8.1). Strictly, 5 ~ here is a Lagrangian density and has dimensions energy/volume. Dimensionally, A; = [M1/2L1/2 T - l ] , and below we will use a scalar field which also has q5 = [M1/2L1/2T - 1] and a spinor field

with ~ = [L-3/2]. These conventions presume the c.g.s./e.s.u, system, and are widely employed (see, e.g., Griffiths, 1987, p. 347). The relations (8) may be familiar but are worth recalling because they illustrate that c is the only fundamental constant that figures in classical electromagnetism, at least explicitly. This is actually slightly misleading, because in the S.I. system the permittivity of free space eo would be present, at least implicitly (the absorption of eo is a convention, as discussed in the next section). Insofar

as they both contain two fundamental constants, classical electromagnetism and GR with A = 0 are similar. They both describe fields whose origin at the level of the differential field equations (as opposed to their boundary-value-fixed solutions) do not contain extra dimensional constants. This is significant. It is possible to consider versions of these theories with extra constants, but the latter do not seem to be required. For example, a nonlinear version of classical electromagnetism with a new constant in

the Lagrangian was considered by Born (1934) and Born and Infeld (1934), but it led to departures from Coulomb's law at small distances that have not been detected. The accuracy of Coulomb's law arises of course because electromagnetism is mediated by a spin-1 photon with zero rest mass. Also, straight GR has a small or zero A, and a version with a finite range for gravity proposed by Freund et al. (1969) has not gained observational support. The latter theory involved a finite rest mass for the graviton, which is the presumed spin-2 particle that mediates gravity, but which is massless in GR. Thus the fact classical electromagnetism and gravity have a small number of fundamental constants is connected with the fact they describe forces with infinite range mediated by particles with zero rest mass.

Planck's constant h comes into the field theory of particles when the 3-momentum p and total energy E of a particle are replaced by space and time operators that act on a wave-function ~. Thus the prescriptions p ~ (h/ i)V and E-- , (ih) a/~t applied to the non-relativistic energy equation p2/2m + V = E (where m is the rest mass; V, potential energy; and h = h/2r 0 result in the Schr~Sdinger equation

h 2 6~t[ t - - V2t/t + Vt/t= ih - - (9.1)

2m c3t

The path Lagrangian for this is 5O = T - V in general, which for a particle with charge q moving with a 3-velocity d x / d t ~ c in an electromagnetic field is 5(' = (m/2) (dx/dt ) 2 - (q/c)A i dxi/dt . The path action for this is S = ~2 y dt, where the integral is between two points. The variation bS = 0 gives the equations of motion of the particle between these two points, which in classical theory is a unique path. In


quantum theory, there are non-unique paths, but the sum over paths 2; exp (iS~h) has

the interpretation that the modulus squared is the probability that the particle goes from position 1 to 2. Clearly the phase S/h has to be dimensionless, and this is why h appears in the sum over paths. Instead of including it in the latter thing, however, we could instead use I7 exp (iS) and redefine the Lagrangian to be

( d x ) 2 dx i L.ca = m - q A i - - ( 9 . 2 )

2h \ dt } ch dt

This has been pointed out by Hoyle and Narlikar (1974, p. 102; see also Ramond, 1981, p. 35). They go on to argue that since the second term in (9.2) contains another q implicit in A,, it is the combination qZ/h that is important, and in it h can be absorbed into q2. Also, in the first term in (9.2) it is the combination m/h that is important, and in it h

can be absorbed into m. Thus the Lagrangian reduces back to the form given above, and h disappears. The disposability of h should be kept in mind, but in what follows we will keep it explicit in order to aid understanding.

A similar prescription to that above Pi ~ ih a/Ox ~ applied to the relativistic energy equation E 2 - p Z m 2 = m2c 4 or pip~ = m 2 c 2 for a freely-moving particle (V = 0) results

in the Klein-Gordon equation

1 020 -}- 721;/I = q}- (10.1) C 2 0t 2

Here q~ is a single, scalar field and the Lagrangian is

~Q-Q~ = ~ I ( ~ 0(p')2- (V(r 1Q?)21~ 2 2 "

(10.2)

Equations (10.1) and (10.2) describe a spin-0 particle. Spin-�89 particles were described in another equation formulated by Dirac, who

'factorized' the energy relation p;pi = m2c 2 with the help of four, 4 x 4 matrices 2 i. These latter are related to the metric tensor of Minkowski spacetime by the relation 7;7 j + 7J7 ~ = 2tfl. The Dirac equation is

ihy i - ~ - mct~= O. (11.1) Ox i

Here Ois a bi-spinor field, which can be thought of as a 4-element column matrix (though it is not a 4-vector) in which the upper two elements represent the two possible spin states of an electron while the lower two elements represent the two possible spin states of a positron. The Lagrangian is

S = ihc~7 i 0~ m c 2 ~ 6 . (11.2) c3x i

3 7 6 PAUL S. WESSON

Here ~ is the adjoint spinor defined by ~- - 0* 7 ~ where 0 t is the usual Hermitian or transpose conjugate obtained by transposing 0 from a column to a row matrix and complex conjugating its elements. The Lagrangian (11.2) is for a free particle. It is invariant under the global gauge or phase transformation 0 ~ ei~ 0 (where 0 is any real number), because ~-~ e-~o~ and the exponentials cancel out in the combination ~0. But it is not invariant under the local gauge transformation 0 ~ ei~ 0 which depends on location in spacetime. If the principle of local gauge invariance is desired, it is necessary to replace (11.2) by

= ihc-~,/ gO mc2~O_ q~yeOAi. (11.3) ~x ~

Here A i is a potential which we identify with electromagnetism and which changes under local gauge transformations according to A i ~ A ~ + ~2/~x i where 2(x i) is a scalar function.

While this and the preceding equations may be familiar, they lead to two points that are important for our discussion of the fundamental constants. First, the argument of Hoyle and Narlikar applied above to the primitive Lagrangian (9.2) also applies to the Dirac Lagrangian (11.3), so h can be absorbed in the latter. Second, local gauge invariance as involved in (11.3) actually has little justification from particle physics itself, but may have more justification from cosmology. Thus Griffiths (1987, p. 348) has pointed out that gauge transformations can be carried out independently at space-like separated points that are out of electromagnetic communication with one another. This implies that the principle of gauge invariance could be viewed as a way of requiring gauge-uniformity of the laws of physics in parts of the Universe separated by particle horizons. We can note here that the spatial uniformity of physical laws and the constants they involve can be checked by observing remote objects in the Universe, which while they are all presently within our horizon may themselves have only partly overlapping horizons. Thus Tubbs and Wolfe (1980) have found spatial uniformity to a few parts in 10 4 in the combination ~2gp(m/M) typical of strong-interaction physics (here c~ e is the fine-structure constant of electrodynamics; gp, gyromagnetic ratio of the proton; m, mass of the electron; and M, mass of the proton). Also, time variation is ruled out to

4 the same accuracy for this combination over the last ~ of the history of the Universe. This adds to the considerable body of knowledge which indicates no significant variation with time in any of the fundamental parameters or coupling constants of physics. (See Wesson (1978) for a review, and Reasenberg (1983), Irvine (1983), and Pagel (1983), the last of whom has come to conclusions similar to those of Tubbs and Wolfe.) Another interpretation of gauge invariance follows from noting that a transformation like Q --, e~ corresponds to changing the size, but not the dimensions, of a quantity (Q) by a factor that depends on a dimensionless function of the coordinates (0). If this function depends on coordinates divided by units with the same dimension, then invariance under a change of gauge would appear to be equivalent to invariance under a change of units. The latter concept has been discussed in various guises in field theory,


notably in connection with gravitation. For example, Dicke has argued that it is impossible to establish that two separated rods are of the same length, so physical laws

should be invariant under a coordinate-dependent transformation of units, and he has used such a transformation to put the field equations of the Brans-Dicke scalar-tensor theory into the same form as those of the Einstein theory (Dicke, 1962; Brans and Dicke, 1961 ; see also Canuto and Goldman, 1982, for a discussion of units, scales, and clocks).

The preceding comments suggest that gauge invariance, however it is interpreted, is a unifying theme for particle physics and gravitation. The requirement of local gauge invariance for the Dirac Lagrangian obliges the introduction of the field A t in the last term of (11.3); so if cosmology implies gauge invariance then in some sense cosmology also implies electromagnetism.

Actually the Lagrangian (11.3) should be even further extended by including a 'free' term for the gauge field. In this regard, the transformation A e ~ A t + ~?)~/~3x t leaves F~k -- (~Ak/?x ~ - ~At/Ox k) unchanged, but not a term like A~At. The appropriate term to add to (11.3) is, therefore, ( - 1/16rc)Ukb, k, sO the full Dirac Lagrangian is

= i~c~. / i a ~ _ m c 2 ~ l _ 1 UkF.~ _ q ~ 7 t O A , . gx ~ 16~z

(11.4)

If we define a current density J~ - cq(~Tt0), the last two terms give back Maxwell's Lagrangian (8.3). The Lagrange density (11.4) describes electrons or positrons interacting with an electromagnetic field consisting of massless photons. However, a term like the one we just discarded AtAt may be acceptable in a theory of massive gauge particles. Indeed, a field derived from a vector potential A i associated with a particle of finite rest mass m is described by the Proca equation

- - + - - A k = 0 . (12. t) c3x i

This describes a spin-1 particle such as a massive photon, and can be obtained from the Lagrangian

f _ 1 1 [ m c \ 2 i 16~ gikFik + --~h)8rc A A i " (12.2)

Again we see the combination m/h, so h may be absorbed. If we consider two 4-component Dirac fields, it can be shown that a locally gauge-

invariant Lagrangian can only be obtained if we introduce three vector fields (A 1, A~, A~). These can be thought of as a kind of 3-vector A i. It is also necessary to change the definition of F~. k used above. The 3 components of the new quantity ( F i ~ , F ~ , F 3) can again be though of as a kind of vector, where now Fik -- [~Ak/Ox i - OAJ#x k - (2q/hc) (A; x Ak) ]. Further, the three Pauli matrices (xl, x2, ~ ) can be regarded as a vector ~. Then with dot products between vectors

378 PAUL S. WESSON

defined in the usual way, the Lagrangian is

~z~ = i h c ~ / ~ - m c 2 7 0 - 1 Fik. F; ~ _ (q~? ;~0 ) 'A , . ( 1 3 . 1 ) ~x ~ 16re

Here 0 can be thought of as a column matrix with elements 0! and 02, each of which is a 4-component Dirac spinor. The latter still describe spin-�89 particles of mass m (where

we have assumed both particles to have the same mass for simplicity), and they interact with three gauge fields ,~ ~ A 2 ~3 which by gauge invariance must be massless. The kind ~ i , ~ i ~ ~ i

of gauge invariance obeyed by (13.1) is actually more complex than that involving global and local phase transformations with e ;~ considered above. There 0was a single spinor,

whereas here 0 is a 2-spinor column matrix. This leads us to consider a 2 x 2 matrix which we take to be unitary (U* U = 1). In fact the first two terms in (13.1) are invariant

under the global transformation 0-- ' U0, because 0 ~ ~U* so the combination ~ 0 is

invariant. Just as any complex number of modulus 1 can be written as e i~ with 0 real,

any unitary matrix can be written U = e i~ with H Hermitian (H* = H). Since H is a

2 x 2 matrix it involves 4 real numbers, say 0 and al, a2, a3 which can be regarded as

the components of a 3-vector a. As before, let ~ by the 3-vector whose components are three 2 x 2 Pauli matrices, and let 1 stand for the 2 x 2 unit matrix. Then without loss

of generality we can write H = 01 + ~. a, so U = ei~ i~ " . The first factor here is the old

phase transformation. The second is a 2 x 2 unitary matrix which is special in that the

determinant is actually 1. Thus the transformation 0 ~ e i " " 0 is a global SU(2) transformation. It should be recalled that this global invariance only involves the first two

terms of the Lagrangian (13.1), which resemble the Lagrangian (11.2) of Dirac. The

passage to local invariance along lines similar to those considered above leads to the

other terms in the Lagrangian (13.1) and was made by Yang and Mills.

The full Yang-Mills Lagrangian (13.1) is invariant under local SU(2) gauge transfor-

mations, and leads to field equations that were originally supposed to describe two equal-mass spin-�89 particles interacting with three massless spin-1 (vector) particles. In

this form it is somewhat unrealistic, however, since a set of truly elementary (non- composite) particles with these properties has not been observed. The theory is useful,

though. For example, if we drop the first two terms in (13.1) we obtain a Lagrangian

for the three gauge fields alone which leads to an interesting classical-type field theory

that resembles Maxwell electrodynamics. This correspondence becomes clear if like

before we define currents Jr cq(~7~0), whence, the last two terms in (13.1) give a gauge-field Lagrangian

1 1 j i . S - F 'k ' Fik - - A i . (13.2)

16~t c

This closely resembles the Maxwell Lagrangian (8.3). But of course (13.2) gives rise to a considerably more complicated theory, solutions of which have been reviewed by Actor (1979). Some of these represent magnetic monopoles, which have not been observed. Some represent instantons and merons, which are hypothetical particles that


tunnel between topologically distinct vacuum regions. Tunneling can in principle be important cosmologically. For example, Vilenkin (1982) has suggested that a certain type of instanton tunneling to de Sitter space from nothing can give birth to an infla- tionary universe. However, as noted above it is doubtful if the kinds of particles predicted by pure SU(2) Yang-Mills theory will ever have practical applications. The real importance of this theory is that it showed it was feasible to use a symmetry group involving non-commuting 2 x 2 matrices to constrict a non-Abelian gauge theory. This idea led to more successful theories, notably one for the strong interaction based on SU(3) colour symmetry.

Quantum chromodynamics is described by 3 coloured Dirac spinors that can be denoted I~tred, I//blue , ~]g . . . . and 8 gauge fields given by a kind of 8-vector A i. Each of 0r, Oh, qSg is a 4-component Dirac spinor, and it is convenient to regard them as the elements of a column matrix ~. This describes the colour states of a massive spin-�89 quark. The 8 components of A~ are associated with the 8 Gell-Mann matrices (21 _ s), which are the SU(3) equivalents of the Pauli matrices of SU(2), and describe massless spin-1 gluons. The Lagrangian for QCD can be constructed by adding together 3 Dirac Lagrangians like (11.2) above (one for each colour), insisting on local SU(3) gauge invariance (which brings in the 8 gauge fields), and adding in a free gauge-field term (using Fik as defined above for the original Yang-Mills theory). The complete Lagrangian is

S i~C?~ i ~1 f~lC2~ 1 . . . . . F'k. F~k - (q~?'2O)' A i . (14) ~?x i 16~

This resembles (13.1) above, except that the 3-vector �9 there is replaced by colour charge here (though denoted by the same symbol). Indeed, the electric charge of a quark needs to be a fraction of e in order to account for the common hadrons as quark composites. And particle physics is best described by 6 quarks with different flavours (d, u, s, c, b, t) and different masses m. This means we really need 6 versions of (14.1) with different masses. A gluon does not carry electric charge, but it does carry colour charge (actually, a gluon can carry positive and negative units of colour). This is unlike its analogue the photon in electrodynamics, allowing bound gluon states (glueballs) and making chromodynamics generally quite complicated.

We do not need to go into the intricacies of QCD, especially since good reviews are available (Griffiths, 1987; Collins et al., 1989; Ramond, 1981; Llewellyn Smith, 1982, 1983). But some things are relevant to our discussion of the fundamental constants. A basic thing is of course the existence of electric charges that are a fraction of e and, therefore, dethrone this parameter from its traditional place in physics (charges are still quantized, though: see below). Our perception of e actually needs to be changed for another reason too, and we will see later that in electrodynamics vacuum polarization makes the charge of a particle and the dimensionless coupling 'constant' G increase at small distances. Something similar happens in chromodynamics, involving quark polarization and gluon polarization, though here the competition between these processes


causes the dimensionless strong coupling parameter gs to decrease at small distances. This is the origin of asymptotic freedom, wherein quarks move relatively freely inside a proton (say), but feel a strong restoring force that confines them if they try to move outside. Besides being involved in polarization, the vacuum in QCD leads to other effects (see Shuryak, 1988). We need not be concerned with most of these, but it is worth noting that the m which appears in a Lagrangian like (14.1) is not really a given parameter but is believed to arise form the spontaneous symmetry breaking which exists when a symmetry of the Lagrangian is not shared by the vacuum. Thus a manifestly symmetric Lagrangian with massless gauge-field particles can be rewritten in a less symmetric form by redefining the fields in terms of fluctuations about a particular ground state of the vacuum. This results in the gauge-field particles becoming massive and in the appearance of a massive scalar-field or Higgs particle. In QCD, the quarks are initially taken to be massless, but if they have Yukawa-type couplings to the Higgs

particle then they acquire masses. The Higgs mechanism in QCD, however, is really imported from the theory of the weak interactions, and has been mentioned here to underscore that the masses of the quarks are not really fundamental parameters. (See Griffiths (1987, p. 365) for the Higgs mechanism; and Itzykson and Zuber (1980, p. 642), who note that a classical action with no dimensional constants would normally lead to a scale-invariant theory but that masses break this invariance.) However, while the masses of the quarks are not really fundamental, there is another parameter in the theory we should look at more closely to see if it can be classified as a fundamental

constant. This is the parameter which determines the distance or energy dependence of the

dimensionless running coupling constant. The latter quantity is unique in QCD, since while in principle there could be several such they turn out to be equal in practice due to the Ward identities, which are Green-function constraints derived from symmetries of the Lagrangian (see Ramond, 1981, pp. 351,357, 372). The unique strong coupling

constant gs introduced above can equivalently be expressed via gs = (4n~,) 1/2 in terms of a parameter c~ s analogous to the fine structure constant ~e of electrodynamics. Let us use as variable the change in 4-momentum q of either of two particles undergoing a strong interaction (q is often taken to be a complex variable for algebraic reasons, so q2 < 0 is possible). Then the variability of the coupling can be written e, = es([q21). For QCD with n c colours and nf flavours, the precise dependency is

12~ as = , Iq2l >> (A') 2 . (15)

( l lnc - 2nf) ln[ ]q2l/(A')2 ]

In the Standard Model n c = 3 and nf = 6, but for any similar model it is clear a dimensional parameter A' comes in. This defines the scale at which the coupling can no longer be considered small and perturbation theory breaks down (Ramond, 1981, p. 383; Griffiths, 1987, p. 295). Numerically, A' is believed to be in a range given by 100 MeV < A'c < 500 MeV. Conceptually, A' is a parameter whose origin is obscure. In (15) it has the dimensions of an ordinary momentum, but the appropriate use of


familiar constants can produce quantities with the dimensions of a mass (A'/c), a length (h /A ' ) or a time (h/A'c). Insofar as A' defines a scale, it resembles its namesake the cosmological constant A of GR. However, while the A' of QCD is a constant, it is not

clear to what extent it should be regarded as a fundamental one. The weak interactions are mediated by 3 very massive intermediate vector (spin-l)

bosons, two of which (W -+ ) are electrically charged and one of which (Z ~ is neutral. The original theory of the weak interaction, as it involved beta decay, was due to Fermi. This theory has now been superseded, but it is still common to use the Fermi coupling

constant, given by

_ , , f ~ ( g~ )2_~ 1 x 10-5(GeV) -2 (16) GF

(he) 3 8 \MwCV

Here Mw ~ - 82GeVc -2 is the mass of the charged boson (compared to M z _~ 92 MeV c 2 for the neutral one), and gw ~- 2 is a dimensionless weak coupling

constant. The latter can be re-expressed as c G = g~/4rc ~_ ~ , which is somewhat larger than the ee ~ 1@7 of electrodynamics (see below). We noted above that G decreases at small distances or large energies. The same is predicted to happen for e~, though at a lower rate. Experiments do not yet provide accurate data on this, however (see Griffiths, 1987, pp. 77, 309, and below). Since ew > G at moderate energies, the apparent feebleness of weak interactions compared to electromagnetic ones must be attributed to the massive nature of the W +- and Z ~ compared to the massless nature of the photon.

This was realized by Glashow, who together with Weinberg and Salam proceeded

to investigate the possibility of combining the weak and electromagnetic interactions. The resulting GWS theory proved to be very successful (see the reviews by Weinberg, 1980, 1987; Salam, 1980; and Langacker and Mann, 1989). It is based on the idea of (weak) isospin and hypercharge, taken over from the early form of the theory of strong

1 interactions. In the latter, the proton and neutron were regarded as the 13 = + ~ and 13 = -�89 states of a single particle, where the existence was assumed of a spin-like property called isospin described by a 3-vector like ordinary spin. The electric charge

Q and baryon number A were combined with I 3 in the Gell-Mann/Nishijima formula Q = I 3 + (A + S)/2, where another quantum number S for strangeness came in. The latter was included in a new parameter of convenience, the hypercharge Y = S + A. In the modern form of the theory of weak interactions, the same path is followed, but now applied to the electron and its neutrino. This leads to the weak analog of isospin, with the weak analog of hypercharge given by Q = 13 + Y/2. The symmetry groups relevant to these properties are SU(2) and U(1), respectively, and SU(2) | U(1) unifies them. (Actually, the first group involves left-handed states only, while the second involves both chiralities with a hypercharge that is a mixture of weak and electric charge: see Griffiths (1987, p. 335) and below.) In the GWS theory, the weak isospin currents j couple with strength g~ to a set of 3 vector bosons W, and the weak hypercharge current j r couples with strength g" (say) to a single vector boson B. The terms involved are gwJi" W i + ( g ' / 2 ) J [ Bi, where i = 0, 1, 2, 3 and there is summation over i as before.


However, the SU(2) | U(1) symmetry is spontaneously broken (see above; in the Standard Model the Higgs mechanism is responsible for the masses of the W -+ and Z ~ but the details are controversial and a Higgs particle has not been seen). The two neutral states W 3 and B mix, to produce a massless combination we call the photon and a massive combination we call the Z ~ Symbolically,

A e = B i c o s O w + Wi3 sin0w, B i = - B i s i n O w + We3cosOw, (17)

where A i are the usual components of the electromagnetic potential, B i are the components of the boson field, and 0w is a parameter called the weak mixing angle. Also, the third component of the weak isospin c u r r e n t j 3 mixes with the electric current je to form a weak hypercharge current ji Y = 2j[ - 2j,. 3. But we already know that the electromagnetic coupling in QED involves a t e r m g e j [ A i, where ge = (4~ee) 1/2 is the dimensionless electrodynamic coupling constant. The need to match GWS theory with QED actually requires the various coupling constants to be related by

gw sin 0 w = g'w cos 0 w = ge . (18)

Thus there are only 2 independent parameters (a possible different weak coupling to the

Z ~ turns out to be gz = gel sin Ow COS 0 w and so is not independent). It does not matter which 2 we use, but particle physics tends to employ ge and 0.

The latter cannot be calculated within the Standard Model itself, but experiments give 0w -~ 29 ~ or sin 2 0 w ~- 0.23. This value of 0 is confirmed by the observed masses of the mediators, which are related by M w = M~ cos 0 w and are, therefore, not independent either. What is independent or not is important in deciding just how fundamental are both the parameters and the underlying theory. The GWS or electroweak theory has two surviving dimensionless coupling 'constants' and not one, so to this extent it may be more appropriate to call it an integrated theory rather than a unified theory.

Quantum electrodynamics is something whose basis we have already treated above in the Dirac equation (11.1), which describes spin -1 particles interacting via photons. The details of the interactions of all elementary particles are calculated using Feynman

diagrams and rules. These are now central also in QCD and the GWS theory, but were first developed in the context of QED. It is, therefore, appropriate to review briefly the consequences of Feynman's calculus here, in a general way and in relation to QED. We shall concentrate on the implications of particle interactions for the fundamental constants. These are considerably affected by the fact that while we observe real particles moving along lines external to some region where they interact, the interaction process may involve other particles interior to that region which are unobservable or virtual. For such virtual particles, there is associated with each internal line of interaction a propagator 1/(q 2 - m2c2), where q is the 4-momentum given by q2 = qiqe and m is the mass

of the particle. If the latter were real, its mass would be constrained by the special relativistic energy equation E 2 - p2c2 = m2c 4 orp 2 = pipi = mZc 2, noted above as the basis for the Klein-Gordon equation (10.1). But for a virtual particle q2 ~ m % 2 , which means that the propagator just introduced is finite. Similarly, a photon has associated with it a propagator 1/q 2, with q2 ~ 0, which is finite.


A process involving virtual particles that has serious implications for coupling con-

stants and other parameters is when objects interact via a line that contains a loop,

representing the exchange of a particle that temporarily splits into a pair. An example of such a loop, which we will return to below when we restrict the discussion to QED, is a photon that splits into a virtual electron-positron pair and then goes back to a photon. In general, Feynman loop diagrams lead to divergences in the equations describing the interaction that have to be removed or hidden by renormalization. Thus the amplitude for a process involving a loop depends on an integral, composed of propagators and other factors, which is divergent for large values ofq (the 4-momentum of one of the particles such as an electron or positron in the loop: see Griffiths, 1987, pp. 208, 246). This problem can be avoided by introducing a hypothetical large but finite cutoff mass M. The integral can then be done, and results in a finite part independent of M and a part that diverges as the hypothetical cutoff is removed by letting M--, oo. This may not sound like much of an improvement; but it is, because the divergent M-dependent terms appear now as additions to the coupling constant and the masses. Thus we can write g = gb + bg and m = m b + ~m, where 3g and bm are infinite for M-~ oo. To compensate for these latter, we assume that the 'bare' parameters gb and m b also contain infinities. (We will return to the subject of bare masses and their meaning in Section 3 below.) The point is that the observed parameters g and m are finite. There are also contributions to gb and in b that come from the finite, M-independent loop terms. However, these depend on the ingoing and outgoing momenta of the real particles involved in the interaction. This means that the effective coupling constant and masses of particles involved in an interaction depend on the energies concerned. This is the origin of the running coupling constants that have been referred to in various places above. The effect on the masses is not great at normal energies, but the fact it exists implies particle masses are not fundamental and that what we normally mean by the mass of an elementary particle is merely the low-energy end of a variable function.

A similar effect to what has just been discussed for the mass of a particle also occurs in QED for its charge. A positive charge (say) surrounded by virtual electrons and positrons tend to attract the former and repel the latter, resulting in a screening effect that reduces the effective charge. This vacuum polarization is similar to what happens in a dielectric medium. Its existence implies that particle charges, like masses, are not fundamental. It is traditional in QED to absorb the finite correction to the charge of a particle caused by vacuum polarization into the coupling constant ge (Griffiths, 1987, pp. 249, 293). In terms of the electromagnetic fine structure constant ee = g~/41r, this means ~e = ~e(lq2l) - The precise dependency is

0 ~e ~e = , Iq21 ~ ( m c ) 2 . ( 1 9 )

1 - (c~~ in [I q2 i/(mc)2]

This is the QED analogue of the QCD expression (15). In the QED case the low-energy or ordinary mass of the electron m takes the place of the QCD parameter A' in scaling q, which in (19) is the change in 4-momentum of either of two interacting particles. This

384 PAUL S. WESSON

precise definition of the scaling parameter is possible in QED whereas it is not in QCD, because ee decreases with decreasing energy or increasing distance, so its low-energy value eo is experimentally determined and is just the old fine structure constant e. (This is c~ = e2/hc ,-~ ~ in the e.s.u, system as used in this section, but h and c can be absorbed so e is essentially the square of the electron charge.) An equivalent interpretation of (19) is that for large momenta or energies, one particle gets nearer to the other and sees more

of its charge because the screening caused by vacuum polarization is less, so the effective charge increases with decreasing distance. Whichever way we view things, we see that the fine structure constant and the electron charge are not really constant.

At this stage, it may be helpful to summarize our investigations of how the fundamental constants figure in particle physics, and collect points we shall use below. We have looked at the relevant equations and their associated Lagrangians, including those of Maxwell (8), Schrodinger (9), Klein-Gordon (10), Dirac (11), Proca (12), and Yang-Mills (13). And we have examined in more detail the physics of the main theories, including QCD, GWS theory and QED. It has become clear that h can usually be absorbed into parameters like the mass m and charge e of the electron; that particle masses and charges in general are not absolute constants but have effective values dependent on the energy; and that the coupling constants e~, ew, c~s of the electrodynamic, weak, and strong interactions are in fact dimensionless functions of the energy.

3. The Status and Meaning of the Fundamental Constants

The preceding section was somewhat technical in tone, but the present one will be more

philosophical. This is unavoidable, because it is apparent from what we have already

learned that recent developments, especially in particle physics, have made the meaning of a fundamental constant obscure. In this section we will try to clarify things, by giving a discussion of points that come up in the Standard Model, gravitation and physics in general. The theme running through the rather diverse material we will look at is whether

there are any truly fundamental constants. The need to consider this is illustrated by what was written above about the energy

and distance dependence of the coupling 'constants' for the electromangetic, weak, and

strong interactions (ge, gw, g~ or c~ e, ew, c~s). However, there are other parameters in the Standard Model based on U(1) | SU(2) | SU(3) that need to be considered, notably the masses of the basic particles. Should not these be regarded as fundamental constants? There is no clear-cut answer to this, because the observed masses of particles involve renormalization from divergent and therefore undefined bare masses (see Section 2.2). For most particles renormalization is carried out in such a way as to arrive at the observed masses. These have specific and well-determined values, but their origin is clouded by our lack of understanding of renormalization. For quarks, the latter process does not lead to the 'observed' or effective masses, but rather to a new or intermediate set of finite bare masses. The difference between these two sets of masses is partially understood in terms of the processes of QCD. However, there is no explanation of the value of the finite bare quark masses within the Standard Model.


People prefer to call them parameters rather than fundamental constants. Many believe that the bare quark masses and also the lepton masses, which are treated as input parameters in the Standard Model, may have their origin explained in terms of some more general model (see, e.g., Griffiths, 1987, p. 122; Collins et al., 1989, p. 163). In the modern version of the Standard Model, there are 6 bare quark masses and 6 lepton masses. And if we add in other quantities like the couplings and related things the total is about 20 parameters (Ellis, 1983). One might hope to reduce this number by going to a grand unified theory, but there is no guarantee of this.

However, the fact that with increasing energy ge increases while gw and gs decrease strongly suggests they come together at a unifying energy. The latter is not known, but is commonly believed to be of order 1015 GeV (Weinberg, 1983; Llewellyn Smith, 1983; Ellis, 1983; Kibble, 1983; Griffiths, 1987, p. 77; Collins etal., 1989, p. 159). This is large, but nowhere near as large as the energy corresponding to the Planck mass (see below) which is the scale at which quantum gravity effects are supposed to come in and is of order 1019 GeV. However, the unifying group for U(1), SU(2), and SU(3) is not known. A minimum example is SU(5), but here the need for only one coupling is balanced by the need for other quantities such as boson masses, and there are still in all about 20 parameters (Ellis, 1983). Other unifying groups are of course possible and indeed likely. It is worth noting here that while there are numerous masses in particle physics, the same is not true of electric charges (even with fractionally-charged quarks as noted above and quasi-particles as reviewed by Khurana, 1990). In fact, most unified theories lead to charge quantization of some kind because they embed the U(1) of electromagnetism in a simple group, which implies that charges are related by Clebsch-Gordan coefficients (Ellis, 1983, p. 283). The masses are not so restricted, and in order to understand why there are so many and perhaps reduce them to a smaller number of more fundamental masses, it may be necessary to find a unification scheme for particle physics and gravity.

The most discussed option for this is probably supersymmetry. This is a grand unified theory in which bosons and fermions are treated on the same footing and assigned 1, 2 , . . . , N supersymmetry spinorial charges. The case of N = 8 supergravity with gravitons and gravitinos (spins 2, 3, respectively) has been much discussed as a possible candidate to unify all the interactions at the Planck mass within a theory with only one or perhaps even zero free parameters (Ellis, 1983; see also Hawking, 1983; Hawking and Ro6ek, 1981; Cremmer and Julia, 1979; Collins et al., 1989, p. 221). But supergravity in this and other forms has both conceptual and calculational problems, and this conjecture is not proven. It may be that a unification of particle physics and gravity can only be made in a theory with more than 4 spacetime dimensions. In these and other theories the Newtonian gravitational parameter G need not be a fundamental constant but a parameter related to a unification scale, presumably of the Planck order 1019 GeV (Weinberg, 1983). A related idea is that G may be calculable in terms of fundamental particle masses in a quantum gravity theory of which Einstein's GR is a kind of long-wavelength low-energy limit (Adler, 1983). However, while such ideas are interesting, they leave unanswered basic questions, such as why observed particle masses are

386 PAUL S. WESSON

so much less than the Planck mass and why the cosmological constant is so small (Adler, 1983, p. 276; Weinberg, 1983, p. 250; Collins et al., 1989, p. 420). Generically, grand unified theories assume that symmetry breaking and dynamical processes are somehow responsible for the plethora of masses in particle physics. And the unspoken assumption seems to be that most masses have to do with the particle physics part of the problem rather than the gravitational part. This may sound plausible, but before accepting it we should inquire how far gravitational theory can be applied to particles.

General relativity with charged matter as source can be used to obtain a model for particles like the electron, and there is a fair literature on this. Cohen and Cohen (1969) modelled an electron as a static, spherically-symmetric distribution of mass and charge. They showed that the values of these parameters as measured by a remote observer implies that the charge must be in a shell with radius half the classical size of the electron, which is e2 /mc 2 ~ - 3 x 10-13 cm. Gnfidig e ta l . (1978) reverted to a special relativity

model of the electron in order to study its stability. They modelled it as a charged, conducting spherical bubble with surface tension, and suggested its instability to defor- mations could be solved by including spin. L6pez (1984) did this, and constructed a model in general relativity based on the Kerr-Newman metric. This is normally viewed

as the metric of a rotating, charged black hole, but can also be used as the vacuum metric exterior to another appropriate source. The source in the model of L6pez was taken to be an empty charged, rotating shell with surface tension, where the latter replaced the old Poincar6 stresses that were previously used to stabilize the electron against its own electrostatic repulsion. The shape of this GR model of the electron was found to be an ellipsoid of revolution, with a minor axis equal to the classical electron radius and an equatorial velocity of rotation just under the speed of light. The model was found to be well-behaved and causal, with no closed time-like lines. The charge, mass, and angular momentum could all be matched to those of a real electron.

It is remarkable that this can be done at all using straight GR. But a possible objection to the electron models of both Cohen and Cohen and L6pez is that the sources they

involve are somewhat unrealistic and are not unique. With respect to this, it can be noted that Gron (1985a) interpreted the former model as representing a region of vacuum polarization bounded by a charged spherical shell, and the latter model as representing charged bubbles with negative energy density rotating along a spherical domain wall. The former idea sounds reasonable in view of what was written in Section 2.2 above, but the latter sounds contrived. However, negative energy density or mass may not be unacceptable in itself, because we saw in Section 2.1 that a large stress or negative pressure can result in a negative effective mass. To be specific, if the elements of the energy-momentum tensor are such that the active gravitational or effective mass density satisfies # - (T 4 - r l - T22 - r 3 ) / c 2 < 0, then the active gravitational or effective mass

M - ~ # ( - g)l/2 dV is negative. (See Equation (4.1) above, Gron (1985a) and Bonnor and Cooperstock (1989). Here we reintroduce the usage i = 1-4, so g is the 4D determinant of the metric and dV-- dx 1 dx2 dx3.) If M < 0, then there exists gravitational repulsion. Physically, the cause of this repulsion does not have to be a negative pressure or tension, though it does arise in this way in several contexts such as domain


walls (Ipser and Sikivie, 1984). Another cause is the negative electric field energy associated with a charged particle. To see this we can write T/as a sum of a matter part M/ and an electromagnetic part E i, so t~ = [ M 4 + E 4 - ( M 2 + E 2 ) ] / c 2 where

= 1, 2, 3. For an electron or similar particle, the corresponding active gravitational mass is M = m - e2/c2r, where m and e are the inertial mass and charge of the electron as they appear in the Reissner-Nordstr~Sm metric

as2 = (1 2Gin G e 2 ) c 2 _ d r 2 _ _ _ + dt 2 c2r c 4 r 2 J (1 - 2 G m / c Z r + Ge2 /c4r 2)

- r 2 (d02 + sin 2 0 dcp2). (20)

Here the spherical polar coordinates are r, 0, q~ and since the effective mass M = m - e2/c2r depends on r it can be negative close in. This was noticed by several people (De la Cruz and Israel, 1967; Boulware, 1973; Cohen and Gautreau, 1979).

Subsequently similar investigations were made of the case with rotation, for which the metric is the Kerr-Newman one referred to above, but there remains some disagreement about possible sources for this metric (Israel, 1970; Hamity, 1976; L6pez, 1981, 1983; Cohen and De Felice, 1984; Gron, 1985b). But irrespective of the assumed nature of the source at the centre of the metric, there exists as before gravitational repulsion. However, this now depends not only on the charge but also on the angular momentum

per unit mass (a) of the central source (Israel, 1970). Not much work has been done on this, presumably because of the complexity of the metric. But recently the effective mass of the simpler, uncharged Kerr metric was investigated (Kulkarni et al. , 1988), and found to be M = m - a 2 / G r near the horizon.

Also recently, there has been renewed interest in the Reissner-Nordstr~m and Schwarzschild metrics for charged and uncharged non-rotating black holes. It appears

that part of the gravitational field energy resides inside the horizon when there is charge, but that it all lies outside the horizon when there is no charge (Dadhich and Chellathurai, 1986; Lynden-Bell and Katz, 1985). However, the concept of gravitational potential energy is a slippery one in GR, and it is not clear whether these results agree with other work (Witten, 1981; Horowitz and Strominger, 1983). What with ambiguities in definitions and uncertainties in the energy-momentum tensor of the source, it is really not surprising we do not yet have a definitive GR model of the electron. However, progress has and is being made. For example, it used to be believed that the electron had a size comparable to its classical radius of order 10- 13 cm mentioned above, whereas now it is known from scattering experiments that its size is at most of order 10 - 16 cm, a limit which can be used to say something about its constitution (Bonnor and Cooperstock, 1989; Cooperstock and Rosen, 1989). Let us assume that the electron is a charged sphere, whose interior is described by ordinary matter with electromagnetic contributions to T/ ( = M I + E / ) , and whose exterior is described by the empty Reissner-NordstrOm metric (20). Let the 2 regions be joined at r = r o = 10- 16 cm. Then it will probably come as little surprise to learn that while the ordinary observed inertial mass m of the electron is positive, its internal active gravitational mass M = m - e2/c2ro

388 PAUL S. WESSON

is negative. This is because, as noted above, the active gravitational mass density # = [ M 4 + E 4 - ( M ~ + E 2 ) ] / c 2 is negative. But it may come as a surprise to learn that

the low limit on r o, which causes this, also causes the ordinary rest mass density p = M 4 / c 2 for some values of r < r o to be negative (Bonnor and Cooperstock, 1989). This conclusion is a general one and comes from the GR junction conditions. If it is

unpalatable, it could in principle be avoided by adding in other components to T[ or going to a more complicated geometry. In this regard, the Kerr-Newman solution of the Einstein-Maxwell equations mentioned above should perhaps be re-examined, with a view to finding a more realistic source than the one of Ldpez (1984). After all, the Kerr-Newman solution has spin, charge, and magnetic moment, and a gyromagnetic ratio equal to that of a real electron (Bonnor and Cooperstock, 1989). This provides encouragement that a realistic model of the electron based on GR can be found.

It is actually very reasonable from an objective standpoint that Einstein's equations could be applicable to particle physics. After all, the equations G• = - ~ T o. possess great generality and are fully covariant under coordinate transformations. (The latter is not true of much work in particle theory, which uses r/~ instead ofg,j and so is invariant under Lorentz transformations in flat spacetime but not covariant under coordinate transformations in curved spacetime.) The comments of the preceding paragraph indicate that particle physics might be described by equations like those of GR provided we knew how to choose T,j appropriately. However, this is only half the potential story. In the above we noted that in general an electromagnetic part could be included in T~j, and we saw an example of a vacuum solution for a charged point mass in (20). This way of applying the equations of GR is perfectly valid, and leads to results involving the gravitational effect of the energy associated with electromagnetism. That is, we are still talking about a theory of gravity rather than a hypothetical curved-space theory of electromagnetism itself. But is this restriction in our way of regarding Einstein's equations necessary? If we had simply been given the equations G~ = - ~T~j by an extraterrestrial scientist, who assured us of their validity but gave us no other information, we would have no reason for assuming they only applied to gravity. We normally assume they apply to gravity because Einstein formulated them in this context, and because following this the constant ~c can be identified as ~ = 8~G/c 4. But there is nothing in the equations themselves that tells us they could not be applied in other contexts, provided the appropriate identification of the right-hand side is made. Indeed, without a right-hand side the appropriate equations are simply Rij = 0 as in (3) above, and these are completely 'anonymous'. They could describe a theory of any interaction, including those of particle physics.

Another way in which Einstein's equations are more general than in their conventional usage is in terms of the number of dimensions. A long time ago, Kaluza (1921) and Klein (1926) suggested extending GR from 4D to 5D. And recently the idea of extra dimensions has undergone a renaissance, notably in terms of 5D gravity and 10D and 11D theories of particle physics (see Freedman and Van Nieuwenhuizen, 1985; Gribbin and Wesson, 1988 ; and Collins et al . , 1989, for reviews; the last also has a summary of string theory). Most of these theories take the extra dimensions to be space-like, but rolled up


or 'compactified' so as to have a very small size. In this manner it is possible to account

for certain properties of particles and also explain why we cannot actually see the extra

dimensions. The choice of these latter to be space-like is in a way understandable, since we are most familiar with the 3 dimensions of ordinary space. But while nice theories

can be built on this premise, it is not obligatory. In fact, a little thought leads to the realization that one could, if there was justification for it, add in a new dimension describing any appropriate physical quantity (see below for an example). This may sound unusual. But even in 4D, covariance implies we can define new coordinates which are mixtures of old space and time ones; and insofar as the nature of a dimension in GR involves a coordiate and a sign associated with the signature, we can in effect have dimensions that are neither space nor time. Also, we are not bound by the usual signature. For example, defining an imaginary time yields a metric with a positive- definite signature ( + + + + ) that can be used to describe quantum tunneling (Coleman, 1977; Gibbons and Hawking, 1977a, b; Vilenkin, 1982). Thus Einstein's equations can describe physics with any number and kind of dimensions.

If, for now, we restrict ourselves to ordinary space and time, it is interesting to ask if there is anything special about the (3 + 1) combination with which we are most familiar. This and related topics have been reviewed by Barrow (1983a), who considers a hypothetical manifold consisting of n space dimensions and i time dimension. Then only for n = 3 can Maxwell's theory be founded on an invariant integral form of the action, so making it conformally invariant. Also n < 4 is necessary for stable, periodic orbits in the one-body problem in Newtonian gravity, and n < 3 in Einsteinian gravity. And n < 4 is probably required for the existence of stable atoms and therefore chemistry and life.

This is a form of the anthropic principle, which was mentioned in Section 2.1 in another context. In the present form it goes back to Whitrow (1955), who speculated that in an ensemble of all possible Universes with all possible dimensionalities, observers can only exist in those with n = 3. Another form of the anthropic principle has it that aspects of the physical world necessary for the evolution and support of observers like us are very sensitive to slight changes in the values of the fundamental constants; and that the latter really are independent of epoch, and that coincidences in the sizes of

dimensionless numbers formed from them are the result of invariable laws and the time that has necessarily elapsed for our own evolution (Dicke, 1961; Carter, 1974, 1983; Wesson, 1978; Carr and Rees, 1979; Rees, 1983; Press and Lightman, 1983; Barrow, 1981, 1983a; B arrow and Tipler, 1986). This kind of reasoning is plausible even though indirect. It agrees with the fact there is no convincing empirical evidence for time- variability of the fundamental constants, especially G.

It is interesting to ask what would be the status of constants like G - and those of particle physics e, h, and c - if the world had n spatial dimensions. In that case, Barrow (1983a) has show there would be a dimensionless combined constant given by G ( 3 - n ) / 2 e n - ~h 2 non-4. He has pointed out that for n = 1, 2, 3, 4 the constants of

electromagnetism (e), quantum theory (h), gravity (G), and relativity (c) are absent, respectively. For n > 4, they are all included in a single dimensionless quantity. For


n = 3, G goes out and leaves the rest as e2/hc, which modulo a bar is of course the fine structure constant. Thus Newtonian gravity (though not G R which involves c) and

particle physics (though without masses) separate automatically in a world with 3 space dimensions and 1 time dimension. If one regards the preceding comments as con-

vincing, one must assume the extra Kaluza-Klein space-time dimensions must be so

small as to have no effects except on the sub-atomic level. This in regard to the present

universe. It is, however, possible such extra dimensions were larger and had salient

effects in the early universe, but compactified as it evolved (see, e.g., Chodos and

Detweiler, 1980). Alternatively, the preceding comments may indicate that extra dimensions are not like the familiar ones of space and time.

Let us now move on and discuss dimensions in general. The word 'dimensions' is of course used in several different but related contexts in physics. Thus we speak of the

dimensions of space or spacetime to mean the number of components in a manifold,

like n or n + 1 in the previous paragraph. And we speak of the dimensions of a quantity to mean the number and kind of physical parameters it involves, like in the fundamental

constants noted in Section 1. These two usages overlap somewhat. For example, in

mechanics we use space and time as coordinates but also mass as an independent parameter, and all physical quantities including constants can be expressed in terms of

3 base dimensions M, L, T. The conceptual similarity between space, time, and mass

is also recognized by dimensional analysis, which is an elementary group-theoretic

technique whose elements in mechanics are M, L, T. (See Wesson (1984a) for a review

of these things. It has been noted by Massey (1971) that we are not restricted to using

just M, L, T in dimensional analysis. We could introduce 3 base lengths to match the 3D nature of space. But this is of no practical use. Presumably this is because space,

at least over moderate distances, is isotropic.) In addition to the base dimensions M, L, T of mechanics, we can also include the base dimension Q of electrodynamics. The

fact that charge can be taken as a base dimension is obvious in S.I. or similar units,

where Coulomb's law for the force F between two charges ql, q2 distant r apart in free

space with permittivity e o is F = qlq2/4~eo r2. Here ~o is analogous to G in the equivalent gravitational law of Newton (see below). But unlike in the latter case, e o is often

absorbed by using e.s.u., in which system Coulomb's law is F = qlq2/r 2 and charge

acquires mechanical dimensions of M1/2L3/ZT - 1. This illustrates an important point:

an appropriate choice of units can lead to the disappearance of a base dimension and the

absorption of a fundamental constant. The number of base dimensions and the number of fundamental constants in use has

tended to increase with time as knowledge of physics has increased. (Obviously Q and e o, or M and G, depend on knowing of the existence of the laws of Coulomb and Newton.) However, it is by no means clear that this trend will continue. For as we noted

above it is possible, at least in principle, to devise unified theories which reduce the number of free parameters while simultaneously making physics more sophisticated. At present, in the S.I. system, there are 7 base dimensions or units in use (Petley, 1983, 1985, pp. 26-29). Of these, 3 are the familiar M, L, T of mechanics. Then electric current is used in place of Q. And the other 3 are temperature, luminous intensity, and amount


of substance (mole). It is interesting to note that just as charge or an equivalent base

dimension could be made redundant when electrodynamics and mechanics were linked

via Coulomb's law, so may temperature as a base dimension be made redundant if

thermodynamics and mechanics can be linked via the laws of black holes (Hawking, 1974; Gibbons and Hawking, 1977a). In what follows, we shall only be concerned with

mechanics and electrodynamics, and for convenience we shall take the base dimension or unit for the latter to be Q. With bases M, L, T, Q we can define the dimensions of

all of the traditional fundamental constants, and it is now time to look more closely at

the origin of these. The constants c, G, e o, and h can be regarded either as parameters that define

asymptotic values for different physical quantities, or as parameters that define dimensional transpositions between difference physical quantities (Wesson, 1984a). The first

interpretation is implicit even in elementary physics. Thus, c is the maximum velocity of a massive particle moving in flat spacetime; G defines the limiting potential for a mass

that does not form a black hole in curved spacetime; 8 o is the empty-space or vacuum limit of the dielectric constant; and h defines a minimum amount of energy (alternatively,

h defines a minimum amount of angular momentum). However, the second inter-

pretation in terms of dimensional transposition is really more basic even though it is less

well known. It has been touched on in Section 2.1 above, but can usefully be discussed in slightly more detail here.

Thus, consider a photon or light front which moves freely away from an origin and

whose distance is r at time t. Then r oct. To get numbers out of this, we write it as an equation r = ct. Here c is a constant, which must have dimensions L T - 1. The same kind

of argument applies, up to a dimensionless factor, for the free motion of any kind of particle. We see that c is essentially a dimension-transposing constant, allowing us to

put r and (c)t on the same footing. Via the Minkowski metric, this leads to special

relativity. Now consider a test particle of mass m I which moves under the gravitational attraction of a much larger mass m 2 and whose acceleration is d2r/dt 2 at separation r.

Then rn 1 d2r/dt 2 oc rnlm2/r 2. To get numbers out of this, we write it as an equation

d2r/dt 2 = Grn2/r 2 (we assume that inertial mass and passive gravitational mass are the same). Here G is a constant, which necessarily has dimensions M-1L3 T-2. That is,

G has to come in as a quantity with these dimensions in order for us to be able to

dimensionally transpose something involving a mass on the right-hand side to something

involving length and time on the left-hand side. We do this because we recognize the existence on the right-hand side of something new (mass), which in terms of our

perceptions we think of as different from the things which appear on the left-hand side (length, time). It is basically this anthropocentric, perception-based judgment that leads

us to introduce a new base M and the associated constant G. (If you doubt that this process is subjective and sense-related, think of observing two billiard balls bounce off each other and hefting a cannon-ball in your hand.) An argument of essentially the same kind applies to charge with base Q and the permittivity of free space eo. But there the acceleration of a test particle of mass rn 1 with charge ql which moves under the

electrostatic attraction of a much larger mass m 2 with charge q2 involves the test particle

392 PAUL S. WESSON

mass and is given by m 1 d2r/dt 2 = qlq2/47Ceo r2. Here e o must have dimensions Q 2M- 1L 3 T 2 which include mass. Planck's constant h only involves the mechanical

bases, because the quantization of energy E in terms of frequency v given by E oc v only involves mechanical bases. So writing E -- hv obliges us to introduce a constant h with dimensions M L 2 T - 1

We have gone over the physical laws which underlie the constants c, G, e0, and h in

some detail. But it is important to realize that the dimensions of these constants are

forced on us by our way of thinking of mass, length, time, and charge as things of

different physical quality. This allows us to answer a question that was sometimes raised

in the early days of dimensional analysis (e.g., Bridgman, 1922), namely whether it is necessary that the equations of physics should be dimensionally homogeneous. Our

answer is that they have to be. Base dimensions of mass, length, time, and charge are primitive concepts, created with the idea we should try to balance them on either side

of an equals sign; and if they should not balance, we make them do so by inventing a

constant with the appropriate compound dimensions. The tautological nature of this procedure explains why there is no underlying theory

of the dimensional constants, and why occasional attempts to formalize them using

group theory (e.g., Bunge, 1971) lead to nothing new physically. It also explains why these constants can be absorbed in physics without loss of generality. We have already

touched on this in several places. Here we note that the disposability of c, G, e o, h has

actually become a commonplace thing in recent times, insofar as units are often chosen which render the magnitudes of these constants unity and leads to their disappearance

from the equations. The rules for carrying this out in a consistent fashion are well known

(see, e.g., Desloge, 1984). Notably, if there are N constants with N bases, and the determinant of the exponents of the constants' dimensions is nonzero so they are

independent, then their magnitudes can be set to unity. For the constants c, G, e0, h with

bases M, L, T, Q it is obvious ~o and Q can be removed this way. (Actually, setting eo = 1

gives Heaviside-Lorentz units, which are not the same as setting 4~e o = 1 for Gaussian

units, but the principle is clearly the same: see Griffiths, 1987, p. 9.) The determinant of the remaining dimensional combinations M ~ L ~ T - 1, M - ~ L 3 T - 2, M ~ L 2 T - ~ is finite,

so the other constants c, G, h can be set to unity. This ability to remove constants with

impunity, coupled with the other arguments given above, leads us to a significant

conclusion. The fundamen ta l constants c, G, ~o, and h are subjective inventions which can

be consistently removed f r o m physics, and are not therefore in any true sense o f the word

fundamental . The constant e is in a somewhat different class. We noted in Section 2.2 and

Equation (19) that c~ e -- e2/hc is energy or distance dependent. With h and c absorbed by a suitable choice of units, (C~e) ~/2 is just e. And since the variability of ee is experi-

mentally verified, there is no doubt that the magnitude of the electron charge is not a o 1@7 which constant. However, c~ e appears to have a low-energy, large-distance limit ee -~

is constant. It is this limit which defines the traditional electron charge, which we will henceforth continue to denote by e and which has magnitude 4.8 x 10- lO e.s.u. (see

Section 1). How are we to interpret this? One way is to assume that ee is not in fact


perfectly constant at the large distances where it has been measured, but would continue

to decrease and tend to zero at infinite distance. This would imply that charge is of purely

local, dynamical origin and that no truly fundamental unit of charge exists. But it is hard

to see how to prove this conjecture. Another way to interpret c~ ~ and e is to assume they really are constants and that they are related to the properties of the vacuum.

In straight GR, we have seen that the vacuum is related to the cosmological constant A. However, inspection shows it is impossible to simply connect e and A by dimensional

analysis. (In conventional units, it is impossible to write A as a combination of e, c, and

h, say.) Instead it is necessary to introduce the mass m , of a hypothetical, perhaps virtual, particle. Then one can write A oc c4m 2/e 4. Unfortunately, this last combination of parameters, for m , = 10- 27 g ~ electron mass (say), is of order 1025 c m - 2, which

is far from the upper limit of A < 10- 56 c m - 2 noted in Section 2.1. Numerical agree-

ment can be obtained if we drop the idea A is related to e and assume instead that A

is related to m , only. A proposal along these lines was made by Zeldovich (1968; see 2 6 4 also 1977), who suggested we write A oc G m , / h . This combination of parameters for

m , = 10 27 g is of order 10 -69 cm -2, so an m , larger than the electron mass would

lead to a A nearer to its upper limit and have significant dynamical effects in cosmology.

Zeldovich justified his suggestion by considering the gravitational binding energy of

virtual particles of mass m , separated by their Compton wavelength 2 = h/m, c. The density of such vacuum energy would be 2 2 p~c oc (Gm, /2 )2-3 , which with the G R

expression (6) of Pv = Ac2/87cG leads to the noted relation between m , and A. This is

a nice idea, but does not help us directly with the question of how to interpret e. In the

absence of anything concrete it would seem most reasonable to regard the charge of the

electron in the same way as its mass, and assume both are parameters which result from

some grand unified theory. It is often assumed that a grand unified theory of gravity and the interactions of

particle physics will ascribe a basic role to the Planck mass or energy:

m e - _~2.2x 10 S g = 1.2 x 1019GeVc - 2 . (21)

This sounds plausible, because this is the only quantity with the dimensions of a mass

that can be constructed from G, h, and c. But we saw above that these latter constants may well not be truly fundamental, but rather parameters to transpose dimensions. So

we must ask ourselves if the Planck mass is real, in the sense that we can expect to find particles of this mass. In this regard, it is suggestive that no evidence has been found

that masses occur as multiples or submultiples of rap, either in particle physics or

astrophysics. (Energies of order 1019 GeV cannot of course be reached by accelerators, but there is no evidence for particles of mass mp in studies of interstellar matter, cosmic rays, or black holes.) This may indicate that the combination of parameters (hc/G) 1/2 is merely a dimensional accident. This view gains credibility when we think about other possible combinations, which on the face of it are just as acceptable but also do not seem to be realized. Thus meg =- (e2/G) 1/2 ~ 1.8 x 10- 6 g is a possible candidate for a charac-

394 PAUL S. WESSON

teristic mass. We do not hear much about the latter for two reasons. First, meg can be obtained from mp by using the fine structure constant ea/hc ~ 1 - ~ , and the size of this means the two masses only differ by an order of magnitude. Second, meg would

presumably be typical of a unified theory of electromagnetism and gravity, which is not an issue of much interest compared to a quantum theory of gravity which is assumed

to have a typical mass rap. Along these lines, it is also reasonable to ask why no

significant role is played in astrophysics and cosmology by the 'gravitational fine structure constant' Gm2/hc ~ 5 x 10-39 ( C a r r and Rees, 1979). Here mp is the mass

of a proton. However, here and elsewhere one can ask why this mass and not some

other. Particle physics involves numerous masses, with no convincing evidence of a unit.

Cosmology is similar, with a range of masses for galaxies and clusters. Thus while there are discrete classes of objects in the world, mass itself is not quantized.

There are also other reasons for questioning the status of the Planck mass (21). For

example, it is difficult to see why the strong, weak, and electromagnetic interactions unify at an energy of the order of 1015 GeV whereas gravity only joins the flock at 1019 GeV.

This is an aspect of what in particle physics is called the hierarchy problem (see Collins

et al., 1989, pp. 177, 247). A related problem is that the non-gravitational interactions already have a unique dimensionless parameter in the low-energy limit of the fine

structure constant, whereas adding gravity with a real Planck mass might result in more

and different dimensionless parameters. This is unattractive to some, who believe a

genuine grand unified theory should have at most one dimensionless constant. In this regard, it should be noted that there is evidence for the existence of a constant related

to the angular momenta J and masses M of rotating astronomical systems (see Wesson, 1983, for a brief examination and list of references; and also Brosche, 1986; Brosche and Tassie, 1989). This evidence is not totally convincing and is open to several

interpretations, but there is certainly some kind of regularity described b y J = p M 2. Here p ~ 10 - 15 g - 1 cm 2 s 1 is a constant with the same dimensions as G/c, which indicates

it is related to gravity and relativity. This is maybe not too surprising. But what may

be significant is that the dimensionless combination fl - G/pc is numerically of the same order of magnitude as the traditional fine structure constant c~ - eZ/hc. Indeed, to within

uncertainties associated with the observations and possible factors of 2r~ or similar,

c~ = ft. The implication of this is that if we add the new constant p to the traditional ones, we obtain 2 groups (e 2, h, c) and (G,p, c) which are in some sense equivalent dimen-

sionally and numerically. But the Planck mass, which depends on mixing parameters from these groups, has now lost its rationale. (To help appreciate that these 2 groups are essentially the same, a change of bases can be carried out. For the first put X - L T - 1 , y _ M L to give for the dimensions of the constants e 2 = X2y , h = XY,

c = X. For the second put X - L T - 1, y _ M - 1L to give G = X 2 y , p = XY, c = X.)

While the existence of the new constant p is not yet established, what we have just noted involves a general moral. This is that the meaning and significance of the constants of physics cannot be guaranteed but depend on the current state of knowledge.

An illustrative example of this is available from history. Apparently, several centuries ago, the local acceleration due to gravity g used to be regarded as a kind of universal


constant (Bridgman, 1922). This was reasonable at the time, since to discernible

accuracy objects fell at the same rate at all places on the Earth's surface. However, g

lost its status when it was realized, following the formulation of Newton's law of gravity, that g depended on the mass and radius of the Earth, and nowadays several branches of geophysics depend on measuring its spatial variations. This example raises the question of whether some of our present so-called fundamental constants might not be

interpreted in terms of the parameters of underlying quasi-stable physical systems. This appears to be impossible, though. In the case of c, we could imagine it was the ratio of an expanding cosmological surface divided by the time since the Big Bang. But this implies a boundary, which lacks both conceptual and observational support. And if we

extend the idea to other constants it rapidly runs into numerical contradictions with

reality. In fact, it is clear from what we have discussed before and the absence of any better

alternative, that we should regard the majority of the fundamental constants as invented parameters to help us transpose dimensions. This applies to c, G, h, and eo- These constants exist because we regard mass, length, time, and charge as different kinds of physical quantities. That is, we invent the base dimensions M, L, T, and Q to formalize our subjective belief that mass, length, time, and charge are different kinds of things; and the constants c, G, h, and eo have dimensions which are composed of these bases, in accordance, respectively, with the laws of particle motion, gravitation, energy quantization, and electrodynamics. Another way of putting this is that these constants exist because we use different units to measure mass, length, time, and charge. It is not an

accident these 4 constants when expressed in terms of the 4 base dimensions are independent ( = do not form a dimensionless combination). This tells us the constants can have their magnitudes set to unity. And insofar as the purpose of equations is to lead to numerical statements about physics, this implies that these constants can be

absorbed. We have remarked before that this is done routinely in electrodynamics where the

4rte o of the S.I. system of units is absorbed in the Gaussian e.s.u, system. It is not done routinely in other areas of physics (though it is used in theoretical work in GR as a labour-saving device), but it certainly could be. Thus, we could complement the Planck mass m e of (21) with the Planck length lp and the Planck time tp, and use these as units instead of the usual gram, centimetre, and second. (This does not contradict what was stated above, since using the Planck mass as a unit is merely a convention, and is not the same as saying real particles of this mass exist.) The correspondences between these units are as follows:

l m p - - = 2 . 2 x 10 5g,

--= ( G h ) 1/2= 1 .6x 10 .33 cm, 1 le \C3//

l tp----\CS// = 5 . 4 X 1 0 - 4 4 S ,

1 g = 4.6 x 104 rap, (22.1)

1 cm = 6.3 x 1032 le, (22.2)

1 s = 1.9 x 1043 l p . (22.3)

396 PAUL S. WESSON

The existence of natural units like mp, lp, tp was realized by Planck following the

introduction by him of the quantum of action in 1899. (See Barrow, 1983b: similar units

were actually suggested by Stoney somewhat earlier; and some people in the definitions

(22) use h rather than h.) The use of units like (22) has never caught on in mechanics. This is a bit of a pity, because their use would make physics more elegant, at least as

regards the theoretical side of the subject. It can easily be confirmed that the constants

c, G, and h with the values noted in Section 1 in c.g.s, units all become equal to unity

in natural units. Therefore, these constants effectively disappear from the equations,

which become more streamlined. This is not merely a procedural trick. All measurements involve comparing one thing with another similar thing of known size, or unit (see

Dicke, 1962; Bekenstein, 1979; Barrow, 1981; Smith, 1983). So a choice of units is

essential, and we may as well use the most natural ones. Put another way, we always

in effect measure dimensionless ratios, and if the units or denominators in such ratios

were chosen to have the sizes given in the first part of (22), we would not need to encumber theoretical physics with c, G, and h. A possible counter-argument to this,

however, is that numbers of inconvenient sizes would be present in certain areas of

experimental or observational physics. This may be true, but does not alter the principle

involved. Whatever system of units we choose, the fact we can if we wish remove c, G, and h shows we should not attach fundamental importance to these parameters.

This view may sound unorthodox. If it is valid, one could reasonably ask why various

national standards laboratories formerly used to spend considerable resources deter-

mining these parameters to better and better accuracy. Looked at from the viewpoint

outlined above, namely as a means of determining a system of natural units, the answer

must be that the aim of the exercise was to fix the units to better and better accuracy.

This is worthwhile, despite what was stated above. For measurements in physics must

be made in terms of units, and a quantity cannot be expressed to an accuracy better than

we know its units. However, to be practical, the activities &nat ional standards labora-

tories were never seriously directed at establishing a set of natural units. Rather, they were directed at maintaining and improving the stability and reproducibility of standards

(where the latter term includes, but is not restricted to, what most people understand

by units). And nowadays the business of standards is very sophisticated, and directed along lines that do not conflict with the view of the constants espoused above.

A review of constants and units, and how they are and could be defined, has been

given by Smith (1983). The second in S.I. units is defined, as 9 192631770 periods of a microwave oscillator running under well-defined conditions and tuned to maximize the transition rate between two hyperfine levels in the ground state of atoms of 13SCs

moving without collisions in a near vacuum (!). This is a fairly sophisticated definition, which is used because the caesium clock has a long-term stability of 1 part in 1014 and an accuracy of reproducibility of 1 part in 1013. These specifications are better than those of any other apparatus, though in principle a water clock would serve the same purpose. So much for a unit of time. The metre was originally defined as the distance between two scratch marks on a bar of metal kept in Paris. But it was redefined in 1960 to be 1650 763.73 wavelengths of one of the orange-red lines in the spectrum of a 8~Kr


lamp running under certain well-defined conditions. This standard, though, was defined

before the invention of the laser with its high degree of stability, and is not so good. A

better definition of the metre can be made as the distance travelled by light in vacuum

in a time of 1/2 997 924.58 (caesium clock) seconds. Thus we see that a unit of length

can be defined either autonomously or in conjunction with the speed of light. The

kilogram started as a lump of metal in Paris, but unlike its compatriot the metre

continued in use in the form of carefully weighed copies. This was because Avogadro's

number, which gives the number of atoms in a mass of material equal to the atomic number in grams, was not known by traditional means to very high precision. However,

it is possible to obtain a better definition of the kilogram in terms of Avogadro's number

derived from the lattice spacing of a pure crystal of a material like 26Si, where the spacing

can be determined by X-ray diffraction and optical interference. Thus, a unit of mass can be defined either primitively or in terms of the mass of a crystal of known size. We

conclude that most accuracy can be achieved by defining a unit of time, and using this

to define a unit of length, and then employing this to obtain a unit of mass. However,

more direct definitions can be made for all of these quantities, and there is no reason

as far as units are concerned why we should not absorb c, G, and h. The view that these constants are not really fundamental has been arrived at by a few

other workers, though its lack of orthodoxy seems to have prevented its general

acceptance. We mentioned in Section 2.2 that Hoyle and Narlikar have pointed out that

h can be absorbed into the charge and mass of the electron. They have also argued that the c 2 in the common relativistic expression (c2t 2 - X 2 - y2 _ 22) should not be there,

because 'there is no more logical reason for using a different time unit than there would

be for measuring x, y, z in different units'. The velocity of light is unity, and its size in

other units is equivalent to the definition 1 s = 299 792 500 m, where the latter number

is manmade (Hoyle and Narlikar, 1974, pp. 97, 98). The connection between c and a choice of units was actually realized quite some time ago by a few discerning workers,

such as Sommerfeld. It has been investigated by Jeffreys, who has noted than in electrodynamics c is the ratio of electrostatic and electromagnetic units for charge

(Jeffreys, 1973, p. 97; see also pp. 87-94). And recently, McCrea has stated an opinion

very similar to the one arrived at above with regard to h, c, and G, which he regards as 'conversion constants and nothing more' (McCrea, 1986, p. 150). Thus the idea that

c, G, and h are not as fundamental as widely believed has been around for some time,

and the bulk of this section has been an in-depth examination of it which tends to uphold its validity.

The constant e is different. Although we measure the magnitude of the electron's

charge in certain units, it makes no sense to set e to unity. As noted in Section 1, if we did this and also chose units that make h and c unity, then the traditional fine-structure constant e - e2/hc would equal 1, whereas empirically it is about 1@7. This is connected with the experimentally confirmed idea that e is the low-energy limit of a function (see Section 2.2). In this regard, the charges of electrons and other particles are like their masses - parameters that result from the dynamics of the theory. The big mystery,

1 mentioned before, is why the low-energy limit is what it is. That is, why e _ 1~7 and not


some other number. We could appeal to the anthropic principle for an answer to this (Carter, 1974, 1983; Carr and Rees, 1979; Rees, 1983; Press and Lightman, 1983; Barrow, 1981, 1983a; Barrow and Tipler, 1986). Or one could assume it is random, perhaps in the basic sense that it has this value in our world but other values in the many other worlds allowed by the Everett interpretation of quantum mechanics (Everett, 1957; De Witt, 1970). However, it would be more satisfying to have some physical

reason for c~ - 1@7, and to be able to calculate this. We know now, of course, that c~ is not exactly equal to T~7, and this allows us to

pass with only brief comment a series of numerological explanations for c~ that were proposed in times past. The most ambitious attempt to give c~ and many other parts of physics a numerological basis was due to Eddington (1949; see also 1939, 1935, 1929). He believed that physics was subjective, in the sense that much of its content, including the sizes of the constants, could be deduced simply from ratiocination, without the need for empirical input. Anybody who troubles to read Eddington's original work on this will actually find some surprisingly compelling reasons in favour of it conceptually. But it was severely ciriticised and is hardly viable numerically. Jeffreys (1948, p. 284; 1973, p. 255) pointed out that from the viewpoint of statistics, the real constants and Eddington's calculated values actually agreed too well, suggesting the latter were artificial. Kilmister and Tupper (1962) studied the statistical part of Eddington's theory in detail, but found it disjointed and unconvincing. More recently, Barrow (1981) re-examined some of Eddington's key arguments, and one gets the impression that a few of them actually contain useful physics if shorn of their numerological trappings. (Barrow also pointed out that not all numerology is bunk: the early days of atomic spectroscopy were dominated by invented numerical rules which worked excellently and were later underpinned by a consistent theory. It should be noted that Eddington (1935, p. 65) was apparently the first person to appreciate what we now call the anthropic principle.) Even more recently, Gross (1989) severely criticized numerological attempts to calculate c~, which is now known to considerable accuracy to be ~ = 0.00729735. There is no doubt he is right that a number like this cannot be expected to arise from simple formulas involving integers and numbers like ~. But several of his comments are merely facetious, and he offers no explanation for why the zero-energy limit of the electromagnetic coupling is finite and has the noted value.

A related problem of even more fundamental nature is why all electrons in the Universe appear to have the same value ofe. This is often overlooked, because we tend to assume particles had their charges, masses, and other properties impressed on them in the early universe and that they have not altered since. But this is not a very good explanation, because it is puzzling why these parameters show such tremendous uniformity while the matter of the Universe itself displays significant non-uniformity (at least up to moderate distances; see Wesson, 1978, for a discussion of the connected issue of why inertia is uniform). We mentioned the spatial uniformity of the constants in relation to cosmological horizons in Section 2, where we also commented that objects separated in 3D may be regarded as contiguous in 4D. This latter point may sound as if it provides an answer to why all electrons are identical. Because if they exchange light


signals then their 4D interval or separation is zero, so they are effectively at the same

point in 4D and in some sense are therefore the same particle. However, this is not a

complete explanation even if it were acceptable otherwise, because an electron and a muon (say) which exchange a photon do not have the same mass, even though they may

be the same in other respects. Another explanation for the identical nature of all electrons is attributed to J. A. Wheeler. He is reported to have speculated that there may

only be one electron in the Universe, which by dint of being able to transverse wormholes

and thereby gain an effective 3D velocity much greater than that of light appears to us

as numerous electrons at different spatially-separated points. However, ideas like the

ones we have been discussing here have more elegance than substance. If we want to

make some calculational progress towards understanding why all electrons have the

same charge, or equivalently why ~ has the same value everywhere, we should look to some kind of influence between spatially-separated particles carried by an unseen substratum. That is, we should look to the physics of the vacuum.

Attempts to do this have started to appear recently, for example by McCrea (1986)

and Puthoff(1989). Both of these authors have assumed there is a real vacuum electro-

magnetic field with a spectrum such that the energy per unit volume in the frequency interval v to v + dv is given by dE v = 8~hv 3 dv/c 3. This spectrum ensures all freely-

moving, inertial particles experience the same field, and corresponds to an average

energy hv/2 per normal mode. The equivalent form in terms of wavelength 2 = c/v is

dEz = 8nhc d2/25. McCrea has considered this formula in relation to the Casimir effect.

This is an observationally verified effect that arises when a capacitor whose plates are

separated by a finite distance effectively excludes wavelengths longer than this from the

vacuum field, causing a reduction of the energy density between the plates and an effective attractive force between them. (This effect is one possible source of negative

pressure in cosmology: see Section2.1 above and Itzykson and Zuber (1980,

pp. 138-141) for an account.) McCrea has concluded that this and other properties of

the vacuum are consistent with the idea that every so-called 'spontaneous' quantum

transition is actually stimulated by a virtual particle, such as a photon or electron,

associated with a vacuum fluctuation. Puthoff has considered the source of vacuum electromagnetic zero-point energy. He has rejected the hypothesis it is set at the birth

of the Universe or has the nature of a boundary condition. Instead, he has concluded

it can be generated by the charged particles that make up ordinary matter, which are jiggled by the same electromagnetic vacuum fluctuations. That is, there exists a kind of

feedback cycle, which however depends on the total radiation field and thereby on the size of the cosmological horizon. In this way, Puthoff has explained one of the

cosmological coincidences that puzzled Eddington and Dirac, and obtained a relation between the electron charge and the age or size of the observable Universe (see Wesson, 1978; and Barrow, 1981, for discussions of the cosmological coincidences). The articles by McCrea and Puthoff just summarized may be the precursors of other and more detailed attempts to explain the values of some of the 'fundamental' constants in terms

of the properties of the vacuum. These latter can have far-reaching consequences, and may lead not only to an understanding of the low-energy limit e of ee but also to an

4 0 0 PAUL S. WESSON

understanding of the QCD parameter A' that appears in c~, (see Section 2.2). The properties of the vacuum may also have consequences for classical cosmology. For example, a cosmological Casimir effect might exist if there are horizon-sized features such as domain walls, and the attraction this would represent could be interpreted in terms of a finite cosmological constant A. Also, the zero-point radiation field might through its interaction with ordinary matter produce other radiation, and if this happened in such a way that it was thermalized it would represent a contribution to the cosmic

microwave background. Now that we are near the end of this discussion, it is clear that the fundamental

constants fall into two classes. (1)c, G, h, ~o, and similar constants are dimension- transposing parameters. They are put into physics by our insistence on ascribing dimensions to things which we intuitively, but subjectively, classify as different in nature. They can with impunity be removed again. (2) e, A', A, and similar constants like the masses of particles are theory-dependent parameters. They appear to arise from the dynamics of a given theory, and probably represent quasi-stable solutions of a set of equations. They might be calculable if we had a unified theory of all of the interactions

of physics. If we consider these two classes, it is apparent we could in principle construct a grand

unified theory with no fundamental constants in its specification. That is, no constants would appear in the field equations; and the only constants that would appear when these were solved would be ones related to integrations, and these would merely reflect boundary conditions. This proposal is attractive, but along with related suggestions like supergravity (see above), has yet to be realized. However, it would be feeble to leave our discussion without at least showing that it is possible to construct such a theory, at least in principle. Therefore, to conclude this section we shall outline a constant-free

theory, as an example that it can be done. Let us start by asking if it is possible to write the equations of motion without

fundamental constants. We do not know exactly what these equations are for the strong and weak interactions, but we do for gravity and electromagnetism, so let us concentrate for the time being on these. Suppose we are told that motion in one certain direction is described in the weak-field classical limit by the equation

d.(xl)_ r' l_• (23.1)

d(x4) 2 ~X 1 L x U ax 1 Lx~lT'J / / / /

This may not look very familiar. But suppose we are told that all the quantities in (23.1) are lengths, that superscripts label different ones, and that we can rewrite them as

follows: x l = r ' x 4 = c t , l m _ Gm l q = ( _ _ G ,]1/2

c 2 , \41~eocgJ q . (23.2)

Suppose also we are informed that subscripts refer to 'central' and 'test' and are labels for particles. Then (23.2) in (23.1) gives

d2r Gmcmt qcqt (23.3) m t - + _ _

dt 2 r 2 4~eo ra


We recognize this immediately as the radial equation of mot ion for a part icle with mass

and charge. This approach can be generalized, but the example jus t given is sufficient

to show that equations of motion can in principle be written down without explicit

fundamenta l constants .

Without such constants to cloud the issue, a little thought will show that there is really

no dist inction between the coordinates x 1, x 4 and the lengths l m, l q in (23.1). This

suggests we should go further and identify the latter as coordinates too. I f this seems

odd, it may be helpful to put the constants back temperari ly. Thus jus t as c implies we

can dimensional ly t ranspose t to a length coordinate x 4 - ct, so does G imply we can

t ranspose the (rest) mass of a particle to a length coordinate x 5 _-- I m = Gm/c 2. Similarly,

e o in S.I. implies we can t ranspose the charge of a part icle to a length

x 6 - I q = (G/47C~oc4)l/Zq. We often absorb the latter constant , as when we use e.s.u.

instead of the S.I. system. But according to what we have d iscussed above, all the

constants in (23.2) are d isposable in this way. The result is a set of coordinates x i, where

x 1 = x, x 2 = y, x 3 -- z, x 4 = t, x 5 = m, x 6 = q . . . to as many as we need to incorpora te

the interact ions of physics. Wha t we are led to therefore is a kind of K a l u z a - K l e i n theory

(see above) in N dimensions.

This theory can be developed in a way parallel to ordinary GR. Thus to ensure

covar iance the geometry can be taken to be given by

d s 2 = g o d x i d x j , g o = g o ( x i ) , i = 1 - N ( > 6 ) . (24)

As in other Ka luza -K le in theories, the addit ion of extra terms to the metric does not

mean much numerically for most systems. Fo r example, a typical laboratory-s ized

object with a mass of a ki logram and a charge of a microcoulomb has x 5 - 7 x 10 - 28 m

and x 6 ~ 9 • 10 24m, which are both tiny compared to X1'2'3~ l m or x 4 = c

(1 s, say) = 3 x 108 m. However , some significant effects could come in for very large

or very small systems. (For the latter, it should be noted that x 5 - l m = h/mc where m

is the rest mass of a particle is also a valid coordinate . It does not mat ter from the

algebraic viewpoint if we use x 5 = m or x 5 = m - 1, because of the f reedom implicit in

(24) to change coordinates . ) The coefficients in (24) will have to be determined as usual

from field equations. These latter, however, have now to be const ructed without funda-

mental constants . This means there is nothing to couple to a To, so for this and other

reasons (see above) let us consider a theory without a T U . Then we do not need to match

the propert ies of Go, and the simplest tensor is R0-. To ensure recovery of s tandard G R

in the 4D vacuum case, let us consider as field equations the N-dimensional analogs of

(3) above, namely

R o = 0 , i = 1 - N(>_6) . (25)

At this point in what so far has been a logical procedure, Equat ions (25) present us with

a problem, namely how to recover the propert ies of matter. However , this problem is

in principle solvable. The extended equations (25) contain extra terms involving parame-

ters like mass and charge on the left-hand sides. I t might be possible to re-interpret these

terms as appropr ia te propert ies of matter, and if desired move them over to the

402 PAUL S. WESSON

right-hand sides to obtain equations of more familiar form. This is analogous to a technique used for the metric in other Kaluza-Klein theories (Dolan and Duff, 1984; Davidson and Owen, 1985). It should also be noted that some of the information normally contained in the 4D T~ and its divergence is now contained in the ND geodesic equation (1). This latter is free of fundamental constants and is still valid. The 1, 2, 3

components of the geodesic equation give the equations of motion of a particle in ordinary space as before (though with extra terms). But the 5, 6 components are now statements about the conservation of mass and charge. Thus, it may be feasible to recover known physics from a constant-free theory of the type outlined here, as well as providing a unification scheme.

The theory sketched above is an extension of a 5D one proposed a few years ago

(Wesson, 1984b). The latter has some nice properties. It has solutions which show natural compactification, solutions where the weak equivalence principle is obtained as a symmetry of the metric rather than as a postulate, and a weak-field solution that explains some astrophysical data not accounted for by 4D GR (Wesson, 1986b). It has been worked on by a number of people, and appears to be observationally viable (see Wesson, 1990, for a status report and list of references). The extended version of this

theory outlined in this paragraph is being investigated by several workers. However, as a potential grand-unified theory it is still very much in the formative stages and there is nothing unique about it. The aim of the foregoing has been simply to show that it is

possible at least in principle to construct a theory that does not involve any of the so-called

fundamental constants.

4. Conclusion

There are basically two classes of 'fundamental' constants. The first includes c, G, h, and eo (in some systems of units). These are dimension-transposing parameters as they appear in the equations of physics and the appropriate Lagrangians (Section 2). But they can be removed, both numerically by a re-definition of units and conceptually by a re-examination of their dimensional bases (Section 3). In fact, these constants only exist because of our primitive and subjective way of assigning labels ( = base dimensions) to things in our environment. Thus, c exists because we view space and time as different things and choose to measure them in different ways. And in macroscopic and microscopic contexts, respectively, G and h exist because we view mass as different from space and time. Likewise, eo exists if we view charge as a different kind of thing from mass, length, and time. However, these parameters are really just conversion constants. This opinion has been mentioned before, notably by Hoyle and Narlikar, Jeffreys and McCrea. What has been done above is to give an in-depth study of this opinion and confirm its validity. We can, if we wish, drop the different base dimensions and treat the relevant quantities on the same footing. (An analogy: instead of calling things apples, oranges and pears, simply call them all fruit.) Then we do not need c, G, and h.

The second class of 'fundamental' constants includes the magnitude of the classical electron charge e and parameters like the masses of particles. These are not really


fundamental either. For example, the charge of a particle depends via vacuum polarization on the energy or distance at which it is measured (Section 2). Only the low-

energy, large-distance limit appears to be unique. But while presently we do not under- 1

stand exactly why this has the size it does (eZ/hc or e 2 -~ ~ ) , in the future we expect to be able to derive it using a unified theory which contains significant vacuum physics. In the same way, we might be able to understand the origin of the parameter A' of quantum chromodynamics, and perhaps even understand why the cosmological constant A of general relativity is so small (Section 3). There are several potential candidates for a grand-unified theory that might allow us to do these things, but in view of what was said above a reasonable criterion for such a theory is that it should be formulated in terms of differential equations in which no constants appear.

Recognizing that fundamental constants do not really exist is a significant achieve- ment for anyone who does physics. In a way, the suspicion that this might be the case

has been around for a long time in our ability to set the magnitudes of certain constants equal to one. But recognizing it explicitly is important, because it alters the way we think about the equations and laws of physics. If the constants are subjective, might not the laws also be subjective in some sense?

Eddington believed that while an objective external world exists, our laws are subjective in the sense that they are constructed to match our own physical and mental modes of perception (Eddington, 1929, 1935, 1939). He used in several places the analogy of a fisherman, who concludes that all fish are above a certian size, without realizing this is only a result of the size of the mesh in the net he uses. This kind of philosophy runs counter to the traditional view of physics, and has been criticized by several people, including Jeffreys (1948, p. 6; 1973, pp. 15, 254). However, similar ideas have been proposed by other respected workers. Thus, Russell voiced several opinions on the matter but seems largely to have believed that physics is merely the application of necessarily-correct equations, which are themselves derived from more basic branches of mathematics such as number theory and logic (see, e.g., Russell, 1965; Jeffreys, 1973). This may be unpalatable to some. But to the author it does not seem totally impossible that, say, an account of elementary particle masses could be based on number theory. (After all, this does not sound any less likely than that the motions of objects can be

based on curved-space geometry.) Also, Hoyle has speculated that everything in the Universe really happens simultaneously, and that we only invent concepts like time to apply some order to this chaos (Hoyle, 1966; Hoyle and Hoyle, 1963, 1971). A related idea is that our relatively simple laws of physics may be regularities which arise in some way from an underlying very complicated and near random state (Nielsen, 1983). The fact of the matter is that it is very difficult to decide if the laws of physics are discovered or invented (Davies, 1988). What has been shown in the present account is that many of what are traditionally called the fundamental constants are not discovered but invented. It is not out of the question that more of physics may turn out to be like this.

404 PAUL S. WESSON

Acknowledgements

T h a n k s for c o m m e n t s go to R. M a n n , T. Steele, and others . This w o r k was suppor t ed

by the Na tu r a l Sc iences and Engineer ing Resea rch Counc i l o f C a n a d a .

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