The End of Time: The Next Revolution in Physics

Hamilton’s work opens up a way to reconcile contradictory pictures of the world. Quantum mechanics and the Wheeler-DeWitt equation suggest that reality is a static mist that covers Platonia. But all our personal experience and evidence we find throughout the universe speak to us with great insistence of the existence of a past – history – and a fleeting present. The paths that can be followed anywhere in Platonia where the mist does form a regular wave pattern can be seen as histories, present at least as latent possibilities.

I feel sure that the mystery of our deep sense and awareness of history can be unravelled from the timeless mists of Platonia through the latent histories that Hamilton showed can be there. But just how is the connection to be made? In the remainder of this chapter I shall explain Schrödinger’s valiant, illuminating, but unsuccessful attempt to manufacture a unique history out of Hamilton’s many latent histories. Then, in the next chapter, I shall consider the alternative – that all histories are present.

AIRY NOTHING AND A LOCAL HABITATION

When Schrödinger discovered wave mechanics he was well aware of Hamilton’s work, since de Broglie had used the deep and curious connection between wave theory and particle mechanics in his own proposal. De Broglie’s genius was to suggest that Hamilton’s principal function was not just an auxiliary mathematical construct but a real physical wave field that actually guided a particle by forcing it to run perpendicular to the wave crests. Schrödinger sought to exploit Hamilton’s work somewhat differently. His instinct was to interpret the wave function as some real physical thing – say, charge density. Of course, this could not be concentrated at a point, since its behaviour was governed by a wave equation, and waves are by nature spread out. Nevertheless, Schrödinger initially believed that his wave theory would permit relatively concentrated distributions to hold together indefinitely and move like a particle. His work led to the very fruitful notion of wave packets. These can be constructed using the most regular wave patterns of all – plane waves like the example in Figure 45. A plane wave has a direction of propagation and a definite wavelength. All the lines that run perpendicular to the wave crests are then latent, or potential, particle ‘trajectories’.

Because the Schrödinger equation has the vital property of linearity mentioned earlier, we can always add two or more solutions and get another. In particular, we can add plane waves. Although each separate solution is a regular wave throughout space, when the solutions are added the interference between them can create surprising patterns. This makes possible the beautiful construction of Schrödinger’s wave packets (Box 15).

BOX 15 Static Wave Packets

A wave with its latent classical histories perpendicular to the wave crests is shown at the top of Figure 47. Using the linearity, we add an identical wave with crests inclined by 5° to the original wave. The lower part of the computer-generated diagram shows the resulting probability density (blue mist). The superposition of the inclined waves has a dramatic effect. Ridges parallel to the bisector of the angle between them (i.e. nearly perpendicular to the original wave fields) appear, and start to ‘highlight’ the latent histories. In fact, these emergent ridges are the interference fringes that show up in the two-slit experiment (Box 11), in which two nearly plane waves are superimposed at a small angle, and also in Young’s illustration of interference (Figure 22).

Much more dramatic things happen if we add many waves, especially if they all have a crest (are in phase) at the same point. At that point all the waves add constructively, and a ‘spike’ of probability density begins to form. At other points the waves sometimes add constructively, though to a lesser extent, and sometimes destructively. Wave patterns like those shown in Figure 48 are obtained.

Figure 47 If two inclined but otherwise identical plane waves like the one at the top are added, the figure at the bottom is obtained. The ridges run along the direction of the light rays’ in the original plane waves. (The top figure shows the amplitude, the bottom the square of the added waves, since in quantum mechanics that measures the probability density.)

Figure 48 brings to mind a passage in A Midsummer Night’s Dream that has haunted poets for centures:

And, as imagination bodies forth

The forms of things unknown, the poet’s pen

Turns them to shapes, and gives to airy nothing

A local habitation, and a name.