Ultimate Explanations of the Universe (9 page)

The inflation case has made it clear that if we want to approach an “ultimate explanation” we shall have to think of something more radical than just bloating up the size of the universe. That little “something” which triggers inflation calls for an explanation as well. But the inflationary mechanisms have drawn our attention to the vacuum problem. The quantum vacuum with which contemporary physics is concerned is not the metaphysical nothingness from which we should like to produce everything that exists (and thereby achieve the “ultimate explanation”), but it is the physical state of least admissible energy, and we suspect that it must have played an important part in the emergence of the universe from something more primordial than the states of the universe that modern physics is capable of describing.

The hypothetical “false vacuum” indispensable for the initiation and maintenance of inflation, as we saw in the previous chapter, is not the same as a “real” physical vacuum, which is defined as the global minimum of the potential energy function. In classical physics it is assumed that this minimum potential energy of the physical fields under consideration is equal to zero (as we know, we may select any point on the energy scale as our zero point). In quantum physics we cannot do this, since in accordance with Heisenberg’s principle, an exact determination of the energy level (even if it were the zero level) would mean an infinite uncertainty of the time for the entire process. On combining this fact with other laws from the relativistic quantum theory we obtain a picture of the quantum vacuum as a container in which there is an “eternal storm” of various processes. In the quantum vacuum pairs of particles and anti-particles are continually being generated, only to be annihilated shortly afterwards. The quantum vacuum is not a static nothingness, but an ocean of fluctuating energy.

Could we not use the quantum vacuum to produce the universe? Of course it would not be the creation out of nothing of the universe that theologians speak of, but undoubtedly it would mark a step forward on the road to finding explanations that go further and further.

In the early 1970s Edward Tryon sent an article to the prestigious periodical
Physical Review Letters
on the emergence of the universe out of the quantum vacuum, but the editors rejected it as too speculative.
¹
Tryon revised it and sent it to the no less prestigious
Nature
. We may safely say that his article was highly successful, launching a new path of research in the quest for the origins of the universe.

Tryon’s idea became rather popular. His model did not provide an “ultimate explanation,” as it did not explain the origin of the quantum vacuum the fluctuation of which gave rise to the universe (Tryon spoke of a “pre-existing quantum vacuum”), but the concept of a vacuum seemed near enough to nothingness for the idea to inspire many researchers who took up this line of reasoning. Some of them found the combination of Tryon’s idea with the concept of inflation particularly exciting. Three Belgian researchers, R. Brout, F. Englert, and E. Gunzig, proposed a model in which the matter emerging from quantum fluctuations had a large negative pressure, giving rise to an inflation scenario.
⁴
A number of other researchers pursued the same line of enquiry. Tryon resorted to this idea to underpin his model. Whatever happens in a quantum vacuum is determined by the law of probability; therefore the chances of a small universe emerging from it are much higher than the chances of a big universe emerging, but our universe is very big. But it cannot be ruled out that in the beginning it was very small and that inflation expanded it to a huge size.
⁵

The idea that the negative energy of a gravitational field could cancel out the positive energy contained in the masses is certainly appealing and carries numerous consequences, but we must not turn a blind eye to the problems which it has to reckon with. It is textbook knowledge that in the general theory of relativity there are serious problems with the definition, independently of choice of co-ordinates, of the localised energy of a gravitational field. Hitherto the only case where such a definition has been successfully obtained was for an asymptotically flat space-time, viz. one which admitted the assumption that at “infinity” (that is at a sufficiently remote distance away from the observer) the gravitational field was weak enough to be ignored. Such a situation definitely does not correspond to any of the more realistic cosmological models. In the general case it has been an open issue, but many researchers have been inclined to conclude that the concept of the total energy of the universe is meaningless. If so, then the whole of Tryon’s construction is unfounded.

There is one more snag to the idea: the universe is not just its “material content,” but also its space-time. We may assume that according to Tryon’s model the universe arises out of a pre-existing quantum vacuum and a pre-existing space-time, but the status of the space-time in this model is not clear. The concept of space-time is one of the tools of relativistic physics rather than quantum physics, but Tryon’s concept does not even have the rudiments of a quantum theory of gravitation. Hence it can be no more than a prelude to, or inspiration for more advanced ideas.

The next stage was the attempt to “produce” the universe, along with its space-time, out of “nothing,” on the assumption of the existence of only the laws of physics. The general consensus among theoreticians was that these laws should combine quantum physics with gravitational physics. But since we still do not have a generally approved quantum theory of gravitation, a set of hypothetical assumptions had to be made regarding this issue. The model which became the best-known concept of “the quantum creation of the universe” was Jim Hartle and Steve Hawking’s proposal (1983)
⁶
: a hybrid of two extremely hypothetical models for the quantisation of gravitation – a model based on the concept of the quantum function for the universe, and the integration over paths model. Before we present the Hartle-Hawking model, we have to turn our attention to these two constituent models.

There are several different approaches to ordinary quantum mechanics. For most issues in this branch of physics they are equivalent, only in applications concerning quantum field theory do some of them turn out to be more useful than others. But the fundamental differences between the various approaches only become conspicuous when we attempt to apply them to the quantisation of the gravitational field. This is where the different strategies arise in the search for a quantum theory of gravitation. To achieve this diverse authors have been trying to apply a variety of approaches to ordinary quantum mechanics.

The most familiar, textbook-style formulation of quantum mechanics boils down to a statement that in a certain space known as the configuration space a wave function (usually labelled Ψ) is defined containing all the available information on the quantum object under consideration (e.g. an electron). The wave function must obey the differential equation governing its evolution. In standard quantum mechanics the wave equation is the well-known Schrödinger equation. Usually its solution and the interpretation of the results obtained mark the end of the theoretical part of the problem.

Difficulties crop up as soon as we try to transfer this method to the quantisation of gravitation. Above all the configuration space turns out to be very complicated. The contemporary theory of gravitation, that is the general theory of relativity, is set in a four-dimensional space-time, but space-time is not a quantum object capable of taking part in a game of quantum probabilities. To transform it into such an object it has to be resolved into all possible three-dimensional spaces. This is a complicated procedure, since in attempting such a resolution it is very easy to produce many copies of the same three-dimensional space differing only by having a different mathematical description. A lot of effort was expended before theoretical physicists learned to carry out this procedure in the right way. But this was not the end of the problems relating to the construction of a configuration space. The three-dimensional spaces have to be equipped with all the possible configurations of geometry and physical fields.
⁷
Only when we have a configuration space constructed in this way may we determine the wave function of the universe on it. But this is where the really big conceptual problems start. What does the “wave function of the
universe
” mean?

In the 1920s, when Schrödinger brought the notion of the wave function of the electron into quantum mechanics, he misinterpreted it himself, and quite a long time had to pass before physicists agreed on its probabilistic interpretation. According to this interpretation those properties of the electron for which the wave function has the biggest value have the greatest probability of being achieved. This interpretation has to be transferred in some way to the wave function of the universe. Every three-dimensional space with determined fields represents a possible state of the universe. There is an infinite number of these states. The wave function of the universe is determined over the space of all of these states. Those states for which the wave function has a bigger value have a higher probability of being achieved. The wave function should have the highest values for those states which describe a universe similar to ours – because that is what the real universe is like.
⁸

The wave function for the universe should obey a differential equation similar to Schrödinger’s equation. It is known as the Wheeler-DeWitt equation. Although it has an analogous role to Schrödinger’s equation in quantum mechanics, it differs significantly from the latter. Schrödinger’s equation describes the evolution of the wave function with time, but how can a wave function determined over all the possible states of the universe evolve? All the possible states of the universe do not exist in time. There is nothing with respect to which the wave function of the universe may evolve. Again a lot of time passed before physicists arrived at the right way to tackle that problem. The heart of the matter lies in the Wheeler-DeWitt equation. The wave function of the universe depends on various parameters characterising the possible states of the universe, and the Wheeler-DeWitt equation describes the changes in the wave function for the universe with respect to all those parameters. Time turns out to be a correlation between some of them. So there is no external time (external with respect to the universe) which can be used to determine the changes and rates of change in the universe. Time is an outcome of the internal relationships between the parameters characterising all the possible states of the universe. The Wheeler-De Witt equation plays the role of a co-ordinator, selecting a set of states out of all the possible states which lead to the emergence of “internal time.”

This theoretical scheme is often referred to as the canonical quantisation of the general theory of relativity. Its chief assets are some interesting conceptual analyses which elucidate the nature of the difficulties encountered in diverse attempts to quantise gravitation. It has been developed as an independent research programme, but it constitutes only one of the two models on which Hartle and Hawking based their idea of the quantum creation of the universe. The other is the integration over paths model, which is frequently applied in quantum field theories.

In this approach we are interested not so much in states as in transitions from one state to another. Let’s consider two states
S1
and
S2
of a quantum system; we want to calculate the probability of a transition from state
S1
to state
S2
. To do this we calculate all the possible paths in the configuration space from
S1
to
S2
. Along each of these paths we calculate a certain integral (referred to as the action integral, in other words to each of the paths we assign a certain number, which is the result of the integration. Effectively we obtain a function defined on all the possible paths from
S1
to
S2
. This function is associated with the probability of a transition of the quantum system from
S1
to
S2
.

This method works very well in quantum field theories, but when we try to apply it to the theory of general relativity we are faced with serious problems. This is what Hartle and Hawking attempted. We shall take a closer look at their procedure.

We shall consider a transition from
S1
to
S2
, just as we would for ordinary quantum mechanics, but now we shall be dealing with states of the universe. Each of these states is a three-dimensional space
S
with an appropriate metric tensor γ (which defines the geometry on
S
), and the appropriate physical fields φ. We shall adopt Hartle and Hawking’s assumption that
S
is a closed space (like a three-dimensional sphere). We shall describe the initial state
S1
as the triple (
S
₁
, γ
₁
, φ
₁
), and the final state
S2
as the triple (
S
₂
, γ
₂
, φ
₂
).

The path from
S1
to
S2
is a sequence of “intermediary” states of the universe, in other words a sequence of closed three-dimensional spaces with appropriate fields γ and φ. Of course certain conditions must be fulfilled for smooth transition from one state to the next. The sequence of states traces a “tube” in the space of all possible states. States
S1
and
S2
are the boundary states of the tube. Now we have to consider all such tubes starting at
S1
and finishing at
S2
and calculate the magnitude (referred to as the propagator) which allows for the determination of the probability of a transition from state
S1
of the universe to state
S2
of the universe. The propagator is usually denoted by the symbol
K
(
S
₁
, γ
₁
, φ
₁
;
S
₂
, γ
₂
, φ
₂
).

Unfortunately there are a number of conceptual and technical difficulties which complicate the solution of the problem. One of the most serious is the fact that in the general theory of relativity three-dimensional spaces “at a fixed point in time” have to fit into a four-dimensional space-time. As we know, in space-time the square of the time coordinate in the expression for the space-time metric takes the opposite sign with respect to the spatial coordinates. We say that the geometry of space-time is Lorentzian, not Riemannian (in the latter all the coordinates take the same sign). The difficulty is that in the Lorentzian case the calculations necessary for the computation of the probability of transition from state to state are generally impossible to carry out, for fundamental reasons.

To get round this difficulty Hartle and Hawking used a certain trick which is sometimes resorted to in ordinary quantum mechanics, in situations when the time coordinate
t
appears in equations. They multiplied
t
by the imaginary unit
i
, the square root of minus 1. This procedure made all the coordinates in the space-time metric assume the same sign and turned the Lorentzian space-time into a four-dimensional Riemannian space. In ordinary quantum mechanics a similar procedure is regarded as a trick in calculations, and after the calculations have been carried out the original sign is restored to the time coordinate. Hartle and Hawking assigned a fundamental meaning to this operation. They interpreted it as a mathematical expression of time at the basic level losing its properties of “the flow of transience” and turning into a fourth spatial coordinate.

Their next investment was the assumption that the propagator is the wave function of the universe, in other words

This is where the programme of the canonical quantisation of gravitation meets the integration over paths programme. The wave function is a conceptual component of the former, and the propagator is a conceptual component of the latter. Moreover, Hartle and Hawking postulate that the wave function of the universe obey the Wheeler-DeWitt equation.

Now comes their most important conceptual innovation. Let’s imagine that the initial state is the “empty” state, viz.
S
₁
= Ø. We shall now calculate the wave function

This step allows us to calculate the probability of the universe’s transition from the “empty” state to state
in other words the probability of the universe arising out of nothing. In addition Hartle and Hawking make one more assumption, that Ψ
₀
is the wave function of the universe in its ground state (in ordinary quantum mechanics the ground state is the state in which the system is at its lowest admissible energy). If the probability of a transition from the “empty” state to any other state has a finite, non-zero value, then, according to Hartle and Hawking, we may speak of a quantum creation of the world from nothing.