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

Mathematically, we can in fact construct wave packets in which the positions are restricted to a small range, from q to q + Δq , and the momenta to a correspondingly small range, from p to p + Δp. Any attempt to make Δ q smaller necessarily makes Δ p larger, and vice versa. Heisenberg’s great insight – his uncertainty relation – was the physics counterpart of this mathematics. There is always a minimum uncertainty: the product ΔqΔp is always greater than or, at best, equal to Planck’s constant h divided by 4π . If you try to pin down the position, the momentum becomes more uncertain, and vice versa. This is the uncertainty relation. Moreover, a wave packet of minimum dimensions will in general spread: the uncertainty in the position will increase. This is what in quantum mechanics is known as the ‘spreading of wave packets’.

Since Planck’s constant h is so small, an object like a pea or even a grain of sand can effectively have both a definite position and definite momentum, and the spreading of its wave packet takes place extremely slowly. This explains why all the macroscopic objects we see around us can seem to have definite positions. But though the quantum laws allow objects to be localized in space and to have effectively definite velocities, there is no apparent reason in the equations why this should habitually be so. They also allow – encourage, one might even say – a pea’s wave packet to be localized in two or more places at once. Nothing forces ψ to ‘localize’ around a single point. Einstein used to look at the Moon and ask why we do not see two. It is a real problem. Quantum measurements on microscopic systems are actually designed to create situations in which a macroscopic instrument pointer is, according to the equations, in many places at once. Yet we always see it at only one.

We shall come back to this mystery, which is one aspect of another: Hilbert space and transformation theory. If you find this section a bit abstract, don’t worry; it is helpful at least to mention these things. In quantum mechanics, position and momentum (and other observables) play a role rather like coordinates – ‘grid lines’ – on a map. Just as in relativity the coordinates on space-time can be ‘painted’ in different ways, so too in quantum mechanics there are many mathematically equivalent ways of arranging the coordinates. This was one of Dirac’s first great insights, and it led to his transformation theory .

According to this, the state of a quantum system is some definite but abstract thing in an equally abstract Hilbert space. The one state can, so to speak, be looked at from different points of view. A Cubist painting might give you a flavour of the idea. In relativity, different coordinate systems on space-time correspond to different decompositions into space and time. In quantum mechanics, the different coordinate systems, or bases , are equally startling in their physical significance. They determine what will happen if different kinds of measurement, say of position or of momentum, are made on the system by instruments that are external to the system. The state in Hilbert space is an enigmatic gem that presents a different aspect on all the innumerable sides from which it can be examined. As Leibniz would say, it is a city multiplied in perspective. Dirac was entranced, and spoke of the ‘darling transformation theory’. He knew he had seen into the structure of things. What he saw was some real but abstract thing not at all amenable to easy visualization. But the multiplication of viewpoints and the mathematical freedom it furnished delighted him.

In The Principles of Quantum Mechanics , a veritable bible for quantum mechanicians, Dirac says that in classical physics ‘one could form a mental picture in space and time of the whole scheme’ but ‘It has become increasingly evident that Nature works on a different plan. Her fundamental laws do not govern the world as it appears in our mental picture in any very direct way ...’. I have quoted these words because, with all respect to the greatness of his discoveries and the clarity of his thought, Dirac may have gone too far with his dismissal of simple mental pictures. But what kind of mental pictures are we talking about here? Dirac was reacting against Einstein and Schrödinger, who longed to form mental pictures in space and time. Schrödinger, for example, had commented in his second paper on wave mechanics that some people

had questioned whether the things that happen in the atom could be incorporated in the space-time form of thought at all. Philosophically, I would regard a final decision in this sense as the same as complete capitulation. For we cannot actually change the forms of thought, and what we cannot understand within them cannot be understood at all. There are such things – but I do not think atomic structure is one of them.

This appeal to ineluctable forms of thought, an echo of the eighteenth-century German philosopher Immanuel Kant’s belief that space and time are an a priori framework without which we cannot even begin to form a picture of the world, is doubly ironic. Schrödinger was strongly drawn to the holistic notions of eastern mysticism but would not accept them in his own theory, where they seem inescapable. Even more ironically, he himself changed the forms of thought. He created new mental images just as transparent as the space and time to which he and Einstein clung for dear life. That is the topic of the next chapter.

(p. 202) Wheeler and Zurek (1983) have published an excellent collection of original papers on the interpretational problems of quantum mechanics.

The true heart of quantum mechanics and the way to quantum cosmology is the way in which it describes composite systems – that is, systems consisting of several particles. It is an exciting, indeed extraordinary story, though it is seldom well told. When Schrödinger discovered wave mechanics, he said it could be generalized and ‘touches very deeply the true essence [ wahre Wesen ] of the quantum prescriptions’. But it was not just the Bohr quantization prescriptions that came into focus: at stake here are the rules of creation. A bold claim, but one I hope to justify as the book goes on. First, we have to see how Schrödinger opened the door onto a vast new arena.

The central concept of this book is Platonia. It is a relative configuration space. The new arena that Schrödinger introduced is something similar, a configuration space (without the ‘relative’). The notion is easily explained. Each possible relative arrangement of three particles is a triangle and corresponds to a single point in the three-dimensional Triangle Land. But now imagine the three particles located in absolute space. Besides the triangle they form, which is specified by three numbers (the lengths of its sides), we now have to consider the location of its centre of mass in absolute space, which requires three more numbers, and also its orientation in absolute space, which also requires three more numbers. Location in Triangle Land needs three numbers, in absolute space it needs nine. Just as each triangle corresponds to one point in three-dimensional Triangle Land, the triangle and its location in absolute space correspond to one point in a nine-dimensional configuration space. The tetrahedron formed by four particles corresponds to one point in six-dimensional Tetrahedron Land and one point in the corresponding twelve-dimensional configuration space. For any Platonia corresponding to the relative arrangements of a certain number of particles, the matching configuration space has six extra dimensions. Schrödinger called such a space a Q, and I shall follow his example. Such a Q is a ‘hybrid Platonia’, since it contains both absolute and relative elements. This hybrid nature is very significant, as will become apparent.