Subject: Bringing the Dirac Coincidences up to date
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Date: Wed, 5 May 2004 13:07:40 -0500

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)

where

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)

where

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.

Conclusion

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.

References

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