Dirac’s "Coincidences" Sixty Years On

Robert A. J. Matthews


Abstract: Sixty years ago, the existence of "coincidences" between dimensionless ratios of sub-atomic and cosmological quantities prompted Dirac to develop a cosmological model based on a suspected connection between physics at the quantum and cosmic levels. While his original proposal has not stood the test of time, recent developments linking cosmology to quantum vacuum processes may yet breathe new life into Dirac’s intriguing hypothesis.

This year marks the 60th anniversary of Paul Dirac’s paper A new basis for cosmology, outlining a new cosmological model based on the existence of a set of extraordinary "coincidences" in cosmic and atomic physics (Dirac 1938). In his paper, Dirac noted that, for some unexplained reason, the ratio of the electrostatic to gravitational force between an electron and a proton is roughly equal to the age of the universe divided by an elementary time constant, implying that


hc/Gmpme ~ mp c2/Hh ~ 1041 (1)


where h is Planck’s constant, c the speed of light, G the Newtonian gravitational constant, me and mp the masses of the electron and proton respectively, and H is the cosmic Hubble parameter. On rearrangement (1) leads to the intriguing implication that the expansion rate of the universe, as measured by H, is linked to sub-atomic quantities via a relationship of the form


H ~ Gcme mp 2/h2 (2)


As is well-known, Dirac’s belief that (1) represented some as-yet undiscovered law of nature led him to argue that the strength of gravity, as determined by G, must vary inversely with time, to compensate for the time-variation of the Hubble parameter H.

For all its fascination, Dirac’s original proposal is now known to be untenable: observational data (e.g. Muller et al. 1991) rule out a time variation in G capable of sustaining (1). In addition, the belief that (1) must reflect a permanent law of nature has been undermined by Bob Dicke’s now-famous use of an "anthropic" argument to show that (1) is a natural consequence of astrophysical constraints on the existence of observers capable of discovering such coincidences (Dicke 1961).

Dicke’s argument does not, of course, imply that Dirac’s interpretation of (1) is incorrect, merely that it is not ineluctable. In what follows, I show that developments in cosmology and quantum field theory unknown 60 years ago hint at a new interpretation of (1) that both supports Dirac’s original contention and which may also cast light on a number of long-standing questions in both cosmology and quantum theory.


Inflation and quantum processes

One of the most exciting developments in modern cosmology has been the discovery of the crucial role of quantum processes in the evolution of the universe. Inflation theory, our current best model of the very early universe, suggests that primordial cosmic dynamics were driven by quantum vacuum energy released by the GUT phase transition at t ~ 10-35 s. Furthermore, current observational evidence hints that quantum effects may still dictate the ultimate fate of the universe, via the so-called cosmological constant (for a review, see e.g. Coles 1998).

These developments suggest that the quantum vacuum is the natural place to start any renewed search for links between quantum and cosmic processes of the type advocated by Dirac. A remarkable feature of inflation theory is that it leads directly to the expression of such a link. Specifically, the field equations of General Relativity show that the cosmological scale-factor R(t) for a universe of energy density u(t) and pressure p(t) are related via


d2R(t)/dt2 = - 4p G[u(t) + 3p(t)]R(t)/3c2 (3)


Inflation theory postulates that at the GUT transition the universe was filled with quantum vacuum energy with a Lorentz-invariant equation of state uvac = - pvac. Inserting this into (3) and solving, we find that


R(t) = R(0)exp(Hvac t) (4)



Hvac = (8p Guvac/3c2)1/2 (5)


This standard result in inflation theory (Guth 1981) is remarkable for being the first explicit expression of a link between cosmic dynamics and quantum processes derived from well-founded theory. It is all the more remarkable for being independent of any detailed field-theoretic assumptions: (5) is derived purely from GR and Lorentz invariance. Furthermore, the existence of this quantum-cosmological link is not restricted solely to the primordial universe: it will also influence the long-term dynamics of any universe containing remnant vacuum energy density, urem. In such a highly-evolved universe, the scale-factor R(t) of the post-inflation universe containing matter in addition to vacuum energy satisfies


[dR(t)/dt]2 = 8p Gu(t)[R(t)]2/3c2 (6)

with u(t) = u(t)mat + urem and du(t) = - 3dR(t)[pmat + prem + u(t)mat + urem ]/R(t). With

pmat ~ 0 and prem = - urem = const. as before, we have u(t)mat = A/R(t)3 so that


[dR(t)/dt]2 = 8p G(A/R(t) 3 + urem )[R(t)2]/3c2 (7)


However, if urem > 0, then by (6) dR(t)/dt is positive for all t, so as t ® ¥ the scale-factor evolution asymptotically approaches


R(t) = R(0)exp(Hasym t) (8)


where this time Hasym is the asymptotic value of the Hubble parameter:


Hasym = (8p Gurem/3c2)1/2 (9)


Thus a universe which retains a finite amount of vacuum energy retains its link with quantum processes throughout its history. Apart from an asymptotically brief matter-dominated interregnum, the dynamics of the universe are dominated by the quantum vacuum, and both start and end with the same scale-factor evolution.

This "symmetry" runs deeper than just the similarity of (5) and (9), however. The form of (4) and (8) shows that both the primordial and asymptotic spacetime is described by the deSitter metric, the maximally-symmetric metric of space-time. This mathematically elegant feature of a universe with a finite vacuum energy term (or, equivalently, non-zero cosmological constant) is worth emphasising, given the common view that such additions to the field equations of General Relativity are in some way "ugly".

As noted earlier, recent observational data have led to the idea of remnant vacuum energy being met with increasing equanimity by cosmologists. Even so, it remains the stuff of nightmares for quantum field theorists. This is because if the remnant quantum vacuum energy is not identically zero, it must have some finite value - and so far all attempts to calculate reasonable finite values have resulted in abject failure.

In essence, the problem lies in the fact that the vacuum energy is made up of contributions of zero-point energy from every mode of every quantum field - and there is no obvious energy "cut-off", Emax above which these contributions should stop. The formal result is a divergent phase-space integral, and thus an infinite vacuum energy density - a physical impossibility. Yet attempts to impose plausible energy cut-offs have met with barely more success. To first order the vacuum energy density is


uvac ~ (Emax)4 /(hc)3 J/m3 (10)


One candidate for Emax is the Planck energy, 1019 GeV, above which the classical picture of spacetime breaks down. Inserting this into (10) leads to uvac ~ 10110 J/m3, 119 orders of magnitude higher than the upper limit of uobs ~ 10-9 J/m3 permitted by observation - a result described as "the worst failure of an order-of-magnitude estimate in the history of science" (Weinberg 1993 p 179). Even setting the cut-off at the lowest plausible non-zero value - the energy scale of the most recent phase transition due to quark-hadron confinement - implies Emax ~ 0.15 GeV and thus uvac ~ 1032 J/m3, still over 40 orders of magnitude too high.

Despite these problems of calculating finite vacuum energy densities, let us take our lead from current observations, and consider the implications of the observational evidence of a non-zero remnant quantum vacuum energy. As we shall see, the outcome may yet help quantum theorists out of their impasse.


Vacuum energy and Dirac’s "coincidences"

As we have seen, while the empirical Dirac "coincidences" hint at the existence of a connection between quantum processes and cosmological dynamics, inflation theory makes this link explicit, by relating quantum vacuum energy density to the Hubble parameter via (5) and (9). This naturally leads to the question: could the Dirac relation (2), and the inflation-based expression (9) actually be manifestations of the same link ?

The obvious next step would be simply to equate (2) and (9) and see what emerges. One problem with this, however, is that Dirac’s relation (2) is based on the evolution of the scale-factor R(t) for the present epoch while (9) is an asymptotic result for large times. Furthermore, while the inflation-based expression for H contains just one mass-scale, related to the quantum vacuum phase transition, Dirac’s empirical coincidences contain a hotchpotch of masses. As the terms in (9) are better-motivated theoretically than those in Dirac’s empirical relationship, let us retain just the dimensional content of the latter, subsuming the various masses into a single quantum mass-scale, m; then eliminating H between (2) and (9), we obtain:


uvac ~ Gm6c4/h 4 ~ 10-4(m/GeV)6 J/m3 (11)


This is an intriguing relationship, for it seems to offer a way of calculating the quantum vacuum energy density which has caused field theorist so much grief. Its chances of success hardly seem bright, however, as it is even more sensitively dependent than (10) on its sole free parameter. Yet if we plug in some plausible candidates for the mass-scale appropriate for (11), something remarkable emerges.

To preserve the successes of inflation, (11) must give a figure for the vacuum energy density associated with the GUT transition of at least 1091 J/m3. Inserting the characteristic GUT mass-scale of 1016 GeV into (11), we find of uGUT ~ 1092 J/m3, in line with observational constraints. Similarly, for the electroweak transition at m ~ 300 GeV, (11) leads to uEW ~ 1011 J/m3, again consistent with observation.

Especially interesting, however, is the value of uvac we obtain for the most recent phase transition, resulting from quark-hadron confinement at m ~ 0.15 GeV: (11) leads to uQH ~ 10-9 J/m3. This is startlingly similar to current observational constraints, and leads to the plausible idea that current cosmic dynamics are influenced by the aftermath of this most recent quantum vacuum state transition.

The upshot, then, is that by combining Dirac’s empirical coincidences with standard inflation theory, one seems to arrive at a way of estimating vacuum energy densities that for the first time gives values consistent with observational constraints at all the critical cosmological epochs. This is all the more impressive given that the resulting values span over 100 orders of magnitude, and come from an expression which is very sensitively dependent on just one free parameter.

Despite this, the suspicion remains that these successes are themselves mere "coincidences". We can, however, uncover glimmerings of a solid field-theoretic basis to (11) if we re-write it slightly in terms of the conventional phase-space result (10):

uvac = a (Emax)4 /(hc)3 = (Gm2/L).(1/L3) (12)


a = (m/MPl)2 (13)

where Mpl is the Planck mass, and L = h/mc is the Compton wavelength associated with the mass-scale m. The resulting gravitational self-energy expression for uvac was first arrived at on empirical grounds over 30 years ago by Zel’dovich (1967). His argument is equivalent to dropping the one-loop terms in the usual calculation of vacuum energy, but keeping those with two loops (Weinberg 1988). Unfortunately, to obtain a sensible figure for the current vacuum energy density, Zel’dovich was compelled to make the ad hoc assumption that the relevant mass-scale involved was that of the proton, which lacked theoretical motivation and in any case gave far too high a vacuum energy density. Our new derivation of Zel’dovich’s self-energy term suggests a much better-motivated and plausible mass-scale: that of symmetry-breaking vacuum phase transitions. The result is a figure for the quantum vacuum energy density that is in remarkable agreement with observational constraints.



Although Dirac’s original proposal for a cosmology based on a link between quantum and cosmic physics has not stood the test of time, its underlying motivation has since become a foundation of modern cosmology. As we have seen, inflation theory contains an explicit link between the Hubble parameter and quantum vacuum processes, derived from nothing more controversial than standard General Relativity and Lorentz invariance.

As inflation theory contains no detailed field-theoretic input, however, we cannot expect it to tell us very much about the nature of this link. Yet by exploring the possibility that Dirac’s "coincidence" may be a manifestation of the same link between quantum and cosmic processes, tentative resolutions emerge to a number of long-standing questions in cosmology and field theory:


O By linking Dirac’s coincidences to concepts drawn from inflation theory, we are led to the possibility that the most recent vacuum phase transition, due to quark-hadron confinement, led to a remnant vacuum energy making an appreciable contribution to the total mass-energy of the current universe. Current observational data do indeed point to the existence of such a contribution.


O Zel’dovich’s original proposal for curing the vacuum energy divergence is put on a firmer theoretical basis: instead of the ad hoc use of the mass of the proton, inflation theory shows that the relevant mass-scale is that associated with symmetry-breaking phase transitions.


O The resulting expression for the vacuum energy density leads to an outcome that has so far eluded quantum field theory: sensible, finite results at every key cosmic epoch.


Of course, many questions remain. For example, the success of the self-energy expression raises the question of why the leading term in the phase-space calculation of vacuum energy cancels out. In addition, why does gravitational self-energy play the key role in determining the size of the vacuum energy ?

Whatever the answers, it seems clear that Dirac’s 60 year old "coincidences" may still have a role to play in resolving deep questions about physics on both the quantum and cosmic scales.


Coles P 1998 Nature 393 741

Dicke R H 1961 Nature 192 440

Dirac P A M 1938 Proceedings of the Royal Society A 165 199

Guth A 1981 Phys Rev D 23 347

Muller J et al 1991 Ap J 382 L101

Weinberg S 1988 Rev Mod Phys 61 1

Weinberg S 1993 Dreams of a Final Theory (London: Hutchinson Radius) 179

Zel’dovich, Ya. B. 1967 JETP Lett 6 316


O Robert Matthews is in the Department of Computer Science and Applied

Mathematics at Aston University, Birmingham, England