Julian Barbour - The End of Time - The Next Revolution in Physics

Second, quantum mechanics as presently formulated needs an external framework. Indeed, the most basic observables, those for position, momentum and angular momentum, all correspond to the ‘lent’ properties mentioned in the discussion of the EPR paradox. They could not exist without the framework of absolute space, and Mach’s principle suggests strongly it is determined by the instantaneous configurations of the universe. Time, moreover, plays an essential role in quantum mechanics yet stands quite outside the description of the quantum state. But we saw in Chapter 6 that time is really just a shorthand for the position of everything in the universe, so the configurations of the universe can be expected to play an essential and direct role in a quantum description of the universe. I cannot see how we can hope to understand the external framework of current quantum theory unless we put them into the foundations of quantum cosmology. This is what leads me to the dualistic picture of Platonia, the collection of all possible configurations of the universe, and the completely different wave function, conceived of as ‘mist’ over Platonia. In the language of Everett’s theory, this introduces a preferred basis. In answer to the question ‘what is real?’, I answer ‘configurations’. My book is the attempt to show that they explain both time and the quantum – as different sides of the same coin.

The Many-Worlds Interpretation(p. 221) Everett’s original Ph.D. thesis, his published paper and the papers of DeWitt (and some other people) relating to the many-worlds idea can be found in the book by DeWitt and Graham (1973).

In this chapter I am going to go into a little detail about how wave mechanics works. This means looking at two equations Schrödinger discovered in 1926 which, Dirac remarked, explained all of chemistry and most of physics. You will need to absorb enough to understand the bearing of the first part of the book on the structure of quantum cosmology. That is the goal; I hope you will find it is worth the effort. I believe it will show us how creation works. No theory can ever explain why anything is – that is the supreme mystery. But theory may be able to tell us why one thing rather than another is created and experienced. What is more, I believe that in every instant we experience creation directly. Creation did not happen in a Big Bang. Creation is here and now, and we can understand the rules that govern it. Schrödinger thought he had found the secret of the quantum prescriptions. Properly understood, what he found were the rules of creation.

Let us get down to business. We shall be considering how the wave function ψ changes. In quantum mechanics, this is all that does change. Forget any idea about the particles themselves moving. The space Q of possible configurations, or structures, is given once and for all: it is a timeless configuration space. The instantaneous position of the system is one point of its Q. Evolution in classical Newtonian mechanics is like a bright spot moving, as time passes, over the landscape of Q. I have argued that this is the wrong way to think about time. There is neither a passing time nor a moving spot, just a timeless path through the landscape, the track taken by the moving spot in the fiction in which there is time.

In quantum mechanics with time, which we are considering now, there is no track at all. Instead, Q is covered by the mists I have been using to illustrate the notion of wave functions and the probabilities associated with them. The red and green mists evolve in a tightly interlocked fashion, while the blue mist, calculated from the other two, describes the change of the probability. All that happens as time passes is that the patterns of mist change. The mists come and go, changing constantly over a landscape that itself never changes.

One of the equations that Schrödinger found governs this process. If ψ is known everywhere in Q at a certain time, you know what ψ will be slightly later. From this new value, you can go on another small step in time, and another, and so on arbitrarily far into the future. The role played here by the red and green mists, the two primary components of ψ, is quite interesting: the way the red mist varies in space determines the rate of change of the green mist in time, and vice versa. The two components play a kind of tennis. This equation is sometimes called the time-dependent Schrödinger equation because time features in it. This is not in fact the first equation that Schrödinger discovered.

The first one he found is now usually called the stationary or time-independent Schrödinger equation. This determines what happens in certain special cases in which the two components of ψ, the red and green ‘mists’, oscillate regularly, the increase of one matching the decrease of the other. This has the consequence, as we have already seen for a momentum eigenstate, that the blue mist (the probability density) has a frozen value – it is independent of time (though its value generally changes over Q). Such a state is called a stationary state . This explains the name given to the second equation – its solutions are stationary states. The standard view is that the time-dependent equation is the fundamental equation of quantum mechanics; the stationary equation is seen as a special case derived from it. This corresponds to an overall scheme in which some state of ψ is created at some time and then evolves until a measurement is made.

There are intriguing hints that in the quantum mechanics of the universe the roles of these two equations are reversed. The stationary equation (or something like it) may be the fundamental equation, from which the time-dependent equation is derived only as an approximation. We think it is fundamental because we have been fooled by circumstances that make it valid for the description of the phenomena we find around us. However, these phenomena deceive us greatly when it comes to the overall story of the universe. In particular, they lead us to believe time exists when it does not.

That this is likely to be so follows from an important property of the two Schrödinger equations. For any quantum system, we can use the time-independent equation to find all the stationary states it can have. Each of these states corresponds to a definite energy, and in each of them the red and green mists oscillate with the same fixed frequency while the blue mist remains constant. These solutions are also solutions of the time-dependent equation, though they are special, being stationary. I have mentioned linearity in quantum mechanics. Here, linearity means that two or more solutions of the time-dependent Schrödinger equation can be simply added together to give another solution. If the special stationary solutions are added, something significant results. In each solution, considered separately, the red and green mists oscillate at a fixed frequency while the blue mist remains constant. However, when we add two such solutions with different frequencies, they interfere: the added intensities of the red and green mists no longer oscillate regularly. More significantly, the blue mist varies in time.

Now this is very characteristic – indeed, it is the essence of quantum evolution. All solutions of the time-dependent equation can be found by adding stationary solutions with different frequencies. Each stationary solution on its own has regular oscillations of its red and green mists, but a constant – in fact static – distribution of its blue mist. But as soon as stationary states with different energies, and hence frequencies, are added together, irregular oscillations commence – in particular in the blue mist, the touchstone of true change. All true change in quantum mechanics comes from interference between stationary states with different energies. In a system described by a stationary state, no change takes place .