The last couple days have seen various discussions online generated by a piece at Quanta Magazine with the dubious headline Why the Laws of Physics Are Inevitable and an even worse sub-headline claiming “physicists working on the ‘bootstrap’ have rederived the four known forces” (this is utter nonsense). For some of this discussion, see Sabine Hossenfelder, John Baez and Will Kinney.

One reason this is getting a lot of attention is that the overall quality of reporting on math and physics at the relatively new Quanta Magazine has been very high, a welcome relief from the often highly dubious reporting at many mainstream science media outlets. The lessons of what happens when the information sources society relies on are polluted with ideologically driven nonsense are all around us, so seeing this happen at a place like Quanta is disturbing. If you want to understand where this current piece of nonsense comes from, there is an ideology-driven source you need to be aware of.

A major line of defense of their subject by string theorists has essentially been the claim that, while it may lack any experimental support, string theory is “the only consistent way to combine quantum theory and general relativity”. I’ve often explained what the problem with this is, won’t go on about it again here. Nima Arkani-Hamed is at this point likely the most influential theorist around, for some good reasons. The roots of the problem with the Quanta article lie in taking too seriously the kind of arguments he tends to make in the many talks he gives. He’s trying to make as strong as possible a case for the research program he is pursuing, so unfortunately gives all-too-convincingly a very tendentious take on the scientific issues involved. For more about this, see a posting here about the problems with the recent Quanta article that motivated the latest one.

Debates over generalities about whether the “laws of physics are inevitable” are sterile and I don’t want to engage in them here, but I thought it would be a good idea to explain what the serious ideas are that Arkani-Hamed and others are trying to refer to when they make dubious statements like “there’s just no freedom in the laws of physics”. Here’s an attempt at outlining this story:

**Quantum mechanics and special relativity:**

A mathematically precise implication of putting together fundamental ideas about quantum mechanics and special relativity is that the state space of the theory should carry a unitary linear representation (this is the QM part) of the Poincare group (this is the special relativity part). You also generally assume that the time translation part of the Poincare group action satisfies a “positive energy” condition. To the extent you can identify “elementary particles”, these should correspond to irreducible representations. The irreducible unitary representations of the Poincare group were first understood and classified by Wigner in the late 1930s. My QM textbook has a discussion in chapter 42. If you impose the condition of positive energy and for simplicity consider the case of non-zero mass, you find that the irreducible representations are classified by the mass and spin (which is 0,1/2,1,3/2, etc.). Non-interacting theories are completely determined by the representation theory and exist for all values of the mass and spin.

**Extensions of Poincare and the No-go theorem of Coleman-Mandula**

To get further constraints on a fundamental theory, one obvious idea is to extend the Poincare group to something larger. States then should transform according to unitary representations of this larger group, carrying extra structure. Restricting to the Poincare subgroup, one hopes to get additional constraints on which Poincare representations can occur (they’ll be those that are restrictions of the representations of the larger group). The problem with this is the Coleman-Mandula theorem (1967) which implies that for interacting theories the larger group can only be a product of Poincare times an internal symmetry group. Representations will just be products of the Poincare group representations and representations of the internal group, with space-time symmetries and internal symmetries having nothing to do with each other. This is why the Quanta headline about “rederiving the four known forces” is nonsense: the three non-gravitational forces are determined by internal symmetries, have nothing to do with what the Quanta article is describing, work on space-time symmetries.

One way to avoid the Coleman-Mandula theorem is to work with not Lie algebras but Lie superalgebras. Here you do get a non-trivial extension of the Poincare group and a prediction that Poincare representations should occur in specific supermultiplets. The problem is that there is no evidence for such supermultiplets.

Another possible extension of the Poincare group is the conformal group. Here the problem is that the new symmetry implications are too strong, they rule out the massive Poincare group representations that we know exist. One can work with the conformal group if one sticks to massless particles, and this is what the methods advertised in the Quanta article do.

The idea that our fundamental space-time symmetry group is the conformal group is mathematically an extremely attractive one, with the twistor picture of space-time playing a natural role in this context. I strongly suspect that any future truly unified theory will somehow exploit this. Unfortunately, as far as I know, no one has yet come up with a way of exploiting this symmetry consistent with what we know about elementary particles. Likely a really good new deep idea is missing.

**Quantum field theory**

To get stronger constraints than the ones coming from Poincare symmetry, one needs to decide how one is going to introduce interactions. One way to go is quantum field theory, with a principle of locality of interactions. This gets encoded in a condition of (anti)commutativity of the fields at space-like separations, which then implies various analyticity properties of correlation functions and scattering amplitudes. The analyticity properties can then be used to prove things like the CPT theorem and the spin-statistics theorem, which provide some new constraints.

Given a method of constructing a Poincare invariant quantum field theory, typically done by choosing a set of classical fields and a Lagrangian, one can try and realize the various possible Poincare group representations as interacting theories. What one finds is that, for spins greater than two one runs into various seemingly intractable problems with the construction. One also finds exceptionally beautiful theories in the spin 1/2 and spin 1 cases that exhibit an infinite dimensional group of gauge symmetries. An example of these is the Standard Model. Unfortunately, we know of no principle or symmetry that would provide a constraint that picks out the Standard Model. If we did, we might be tempted to announce that the principle or symmetry is “inevitable” and thus the “laws of physics are inevitable”. We’re not there yet…

**Amplitudes and the S-matrix philosophy**

In the S-matrix philosophy one takes the analyticity properties as fundamental, working with amplitudes, not local quantum fields. The 1960s version of this program (also often called the “bootstrap” program) was based on the hope that certain physically plausible analyticity assumptions would so tightly constrain the theory of strong interactions that it was essentially uniquely determined. This didn’t work out. In his recent introductory lecture for his course at Harvard, Arkani-Hamed explains why. The research program he and others are currently pursuing is in some sense a modernized version of the failed 60s program. The hope is that new structures in amplitudes can be found that will replace the structures one gets from local quantum fields.

Amplitudes based arguments about, for instance, why you don’t see fundamental higher-spin states, and why spin 1/2 particles have forces of the kind given by gauge theory have a long history, see for instance work on massless particles by Weinberg in the mid-sixties and Weinberg-Witten in 1980.

As far as I can tell, the work referred to in the Quanta article gives new amplitudes-based arguments of this kind for massless particles, exploiting conformal symmetry. It’s not clear to me exactly what’s new here as opposed to earlier such arguments, or how strong an argument about real world physics one can make using these new ideas. One thing that is clear though is that the Quanta quote that what has been discovered implies that “There’s just no freedom in the laws of physics” is as much nonsense as the “we rederived the four known forces” business.

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