Quantised Spacetime

[mtvessel]

Jun 2018
Reality is Not What it Seems - Carlo Rovelli – Publisher, Date
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Given his nationality, it is perhaps not surprising that Professor Rovelli is something of a renaissance man. Although this is primarily a book about physics, he also discusses ancient Greek philosophers - Aristotle, Plato and particularly Democritus - quotes poetry by Lucretius and Shakespeare, and uses Dante's plan of Paradise as an example of a 3 sphere. As he puts it, "our culture is foolish to keep science and poetry separated; they are two tools to open our eyes to the complexity of and beauty of the world." I strongly suspect that the richness of his mental toolkit is one of the reasons why he is one of the leading experts on Loop Quantum Gravity (LQG) and can explain it so clearly.

For regular popular science readers, the first half of the book is mostly a refresher on quantum weirdness and relativity (Rovelli's description of the latter is good, but I must plug the wonderful spacetime globe invented by the guy behind Minute Physics, which demonstrates Lorentz transformations of timelines in a beautifully physical way). Rovelli makes an interesting point about quantum theory, which is that it is about relationships rather than atoms. If you take the mathematics literally, then quantum particles only exist when they are interacting - as he puts it, an electron is a "combination of leaps from one interaction to another." When an electron is on its own, it effectively does not exist; its properties become undefined. It pops into existence when it interacts with something else. The property values it acquires at that point - such as the orbital that the electron appears in - are on a spectrum of limited values, and both the range of the spectrum and their relative likelihoods at the next interaction are determined by Dirac's equations. This is very different from Einstein's equations which describe a continuous spacetime, infinitely divisible. Relativity is analogue, quantum mechanics is digital. Both theories give rise to predictions that have been proved accurate to a mindboggling degree of precision, but only by ignoring the other.

This distinction has existed for almost one hundred years and is possibly the most embarrassing discrepancy in physics. It can be partially reconciled; quantum field theory combines special relativity and quantum mechanics by treating particles as excited states of quantum fields, and is the basis of the highly successful Standard Model. However, it cannot explain the existence of gravity, which is a fairly major omission.

It is interesting to take a step back and think about how one could in principle go about reconciling these two models. One approach is to say that both Bohr and Einstein were right (or wrong) - relativity and quantum mechanics are approximations to a greater, underlying concept from which both can be derived. As I understand it, this is the approach of string theory. The other way is to pick a side and assume that the principles of relativity or quantum mechanics will produce a formulation that will explain the other. This is what quantum gravity tries to do. As its name implies, it assumes that Einstein was wrong and Bohr was right. Spacetime, it says, is a quantum field - it is grainy, probabilistic and manifests through interactions. Space and time consist of little packets (we know their size - the Planck length of approximately 10-35 m) that interact in a flickering dance. Since the packets are not interacting "in" anything - you can conceive of them as all piled on top of each other - the only information about them that needs to be considered is which packets are connected to which. This can be modelled as a network or graph of nodes with lines between them. The graph is known as a spin network, because the quantum mechanical property of spin as a half-integer quantum value is involved in labelling the lines (where this comes from and what it means for the calculations I'm not quite sure - Rovelli gets rather hand-wavey about this). Each spin network describes a single quantum state of space, and the combination of all possible spin networks in various degrees of probability describes the effects of space on particles, just as the combinations of states in quantum field theory describes the interactions of particles.

So where does the curviness of general relativity come from? One important feature of the spin network is that it is cyclic - you can step from node to neighbouring node and eventually return to where you came from. By considering the direction of an imaginary arrow as you do this, it is possible to calculate an average curvature across all closed loops in the spin network and - lo and behold! - the curvature of space appears.

You may have noticed that time does not get a mention in this formulation. It too derives from the spin network, but in a rather interesting way. Imagine taking a three-dimensional spin graph and moving it, so that the nodes trace out lines and the lines form planes. Also, let nodes coalesce or split apart to form a different spin network. The resulting soap-bubble-like structure - a spinfoam - represents the history of an evolving spin network. You can then apply Feynman's Sum over Histories idea across all possible spin networks that have a particular boundary to get an integral that represents spacetime.

This approach is so elegant and parsimonious, using the principles of quantum mechanics to model spacetime with no additional concepts required, that I am surprised that so much research effort has been expended on its rivals. Presumably some of the maths doesn't work out quite as cleanly as Rovelli suggests. And, of course, there is the problem of how you test it. But, as Rovelli points out, LQG solves a number of the issues plaguing physical theories where the very large and the very small meet. The infinite energy densities - singularities - that you get in general relativity-based models of the initial state of the Big Bang and of black holes disappear if there is a minimum size that spacetime can be (LQG actually suggests a big bounce rather than a big bang - the same quantum repulsion effect that stops electrons from falling into the nucleus of atoms would cause a contracting universe to rebound explosively). LQG also provides a possible explanation for the "heat" of black holes. And there are observations that could be made that would demonstrate that gravity is quantised. Small perturbations in the distribution of the cosmic background radiation might show that the universe had a bouncy rather than an explosive beginning. Investigation of the expected gravitational wave background by experiments such as LISA may also give hints of its quantum origins. And finally, LQG predicts that old black holes will explode due to the same quantum bounce effect proposed for the big bang. If we see such an explosion - and they have been suggested as one possible cause of Fast Radio Bursts - it would be strong support for the theory.

Now of course this beautiful model could be slain by ugly facts, or simply by further investigation into the underlying mathematics that shows it to be internally inconsistent. But given that supersymmetric particles have failed to appear at the LHC, covariant quantum fields (that is, quantum fields that don't pre-suppose a spacetime matrix) must surely be the front runner for a theory of everything now. And Professor Rovelli, living embodiment of the falsity of that other great divide - between art and science - is a powerful advocate for it.

Comments

[ingaborg]

What a great review. Thank you.