I have just been reading Lee Smolin’s recent book Einstein’s Unfinished Revolution.
My sources for what is going on deep in theoretical physics are Carlo Rovelli (whom I met at How the Light Gets In some years ago), Lee Smolin, and for a different angle Rob Wilson, my former colleague who tends to snipe at physicists for misunderstanding group theory. I really enjoyed a couple of Smolin’s earlier books (Three Roads to Quantum Gravity and The Trouble with Physics), but I have found his more recent books less satisfactory.
In the present book, he explains attempts to go beyond quantum mechanics to produce a theory to satisfy someone like himself who, in philosophical terms, is a realist. I must admit that I don’t really understand that term.
In brief, according to Smolin, quantum mechanics has two rules. Rule 1 says that the wave function evolves in a purely deterministic way, as determined by Schrödinger’s equation; Rule 2 says that something quite different happens when we make a measurement on the system: the wave function instantly changes into an eigenfunction of an operator associated with the quantity we are measuring, and the result of the measurement is the corresponding eigenvalue. Now apparently a realist can accept Rule 1 but not Rule 2; and there are problems, since Rule 1 is time-symmetric whereas the Universe appears not to be.
Smolin begins by tantalising us with the promise of a workable, fullly realist proposal towards the end of the book. But most of the book is taken up with various realist alternatives which have been proposed, most notably pilot wave theory by de Broglie and Bohm, and Everett’s Many Worlds theory. Pilot wave theory proposes that both the particle and the wave are real; the wave satisfies Rule 1, and the particle moves on a trajectory determined by the wave (maybe “steepest ascent”). Aside from other drawbacks, this violates Newton’s third law; the wave affects the particle but the particle has no effect on the wave. Strikingly, when two particles collide, they don’t actually collide at all, but pass through one another without noticing; the waves interact according to the rules of quantum mechanics and then carry the particles with them.
According to Everett’s theory, every time an event occurs with more than one possible outcome, the universe splits into many universes, each of which realises one of the outcomes. I am not sure how this claims to be a realist theory, given the amount of science fiction this idea has powered; but it has other drawbacks too, for example all these universes “really” exist, there are no probabilities in the theory.
Smolin discusses the many attempted fixes for these and other theories, and concludes that none of them are really satisfactory.
Eventually we get his own theory, which he claims is based on ideas due to Leibniz. The Universe is made of atoms called “nads” (so-called because they share some of the properties of Leibniz’s monads), which satisfy various relations. It is not made entirely clear whether these are all binary relations; I will assume so for simplicity. So the Universe is a huge network.
Now a feature of such discrete models is that we have the possibility that space will arise as am emergent property: if we look at the network from so far away that we can’t see details of the nads, it should look like a manifold. The problem is that nobody has yet devised a satisfactory model in which the three dimensions of space arise naturally in this way. (We can do it synthetically, by starting with the desired manifold and sprinkling nads from a “nad sprinkler”, a Poisson process; but that is cheating.) So when later Smolin claims that he can derive the Schrödinger equation in one of his models, it seems to me that there is an unexplained gap.
The two principles he takes from Leibniz are, first, that two objects with identical properties are equal, and second, that our world is the best of all possible worlds.
The first principle is fine in set theory: two sets which contain the same elements are equal. But nads, unlike sets, have no internal structure, and are defined solely by their relations with the other nads. So this principle forbids “twin nads”. Smolin extends this by saying that a nad has a “view” of the universe, consisting of its neighbours out to some specified distance in the network, without telling us what this specified distance is. Now two nads are close together if their views are very similar (they can’t be identical, by the first principle). This explains the phenomenon of non-locality: two nads can be close in this sense, and so influence one another, even if they are far apart in the actual universe (like entangled particles in quantum mechanics which have moved apart).
There is a problem here. Two nads are identical if they have identical views of the Universe, in other words, the things they can see are identical. But there is a vicious circle there. Smolin claims that his principle implies that the network has no symmetry, because two nads related by a symmetry would be identical. But I don’t get this.
The second, Panglossian principle asserts that there is a function, called “perfection”, and the law governing the Universe is that it changes so as to maximise “perfection”. He identifies “perfection” with action (or, strictly, its negative), and so comes close to one of the standard formulations of mechanics, the Principle of Least Action.
However, there is a difficulty for those, like me, who think of symmetry as a kind of perfection. I can’t now remember who suggested to me that homogeneous structures such as the countable random graph have maximum symmetry: any two finite subsets with identical internal structure are equivalent under a global symmetry. (The fact that the random graph is homogeneous can be expressed by the Leibnizian statement “The best of all possible worlds is also the most probable”.) So a universe without symmetry seems to me to be not as nice as one with a lot of symmetry. This may just be a personal prejudice; but perhaps, as I argued above, the proof of “no symmetry” is not valid, and the Universe could have symmetry after all.
Never mind; it is fun to speculate, and I did enjoy the book.
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