my remaining physics questions

As I am reconsidering my relationship to academia, I am looking into the kinds of research directions that feel alive for me. A lot of the questions I had when I started my PhD (mostly technical and philosophical questions about QM, QFT, and GR) I have either answered to my satisfaction, or understood that they are buried in some book that I now have the resources and support I need to understand if i wanted to.

But three questions remain particularly alive, and have been for the last 2 years burning more or less in the background of my day-to-day research activities. These are the questions I still care about:

1. Can we fully account for probability as a description of how complex systems engage in decision making?
2. Can Everettian quantum mechanics account for all observed phenomena, without circularity?
3. Can we understand spacetime as arising from some more fundamental quantum structure?

I’d bet that the answer to all three questions is “yes,” but that this is not in any way trivial. The joy in this research programme would in part in articulating what these questions even mean, before explaining how one gets to an answer! These questions are also deeply related, and indeed it’s likely that if the answer to one question is “no”, then the answer to the questions below will also be no. This is because I think the viability of EQM relies on a certain understanding of probability, while the “emergent spacetime” programmes i find most convincing rely on EQM. The questions are also listed in order of decreasing readiness, ie more work has to be done for the later questions.

While each question deserves its own post, let me get a little into each of them, to give you a flavour of what I am thinking. At the end, I’ll also mention a 4th, bonus question.

1. probability as chances, credences, and frequencies

Most physicists—indeed even most people working in the foundations of QM—do not suspect that question of the philosophical foundations of probability are as murky and controversial as the question of the interpretation of quantum mechanics. But this is totally the case, with lots of different poorly differentiated positions. Are probabilities just frequencies? Are they all beliefs or willingness to bet on stuff? No and no.

Nevertheless, i think there is an almost completely fleshed-out story here. So much so that I can put it into a few sentences. The key is to note that in common usage, the word “probability” stands for 3 different things: credences, chances, and frequencies. Each is a set of non-negative quantities summing to one, each obeys the Kolmogorov laws of probability, but they are fundamentally different: they pertain to personal belief, theoretical models, and observations, respectively.

There is a big if true theorem by Savage (1972) that argues that physical systems acting according to some axioms of rationality will behave as if they are maximising the expected utility of their actions, ie picking the action $a$ that maximises

${\displaystyle ⟨U_a⟩\equiv \sum _i\mathrm{C}\mathrm{r}(i\mathrm{\mid }a)u(i,a),}$

where $i$ runs over the possible outcomes of action $a$, $u(i,a)$ is the utility of the outcome $i$ of action $a$, and $(i|a)$ is credence in this outcome. (The theorem does not state what functions $\mathrm{C}\mathrm{r}$ and $u$ should be, just their properties.) With a couple extra axioms, the theorem by Savage can be extended to show that sequential decision making is done as if the agent is doing bayesian reasoning: updating their credences based on observations and in particular, upon repeated exchangeable experiences, their credences tend towards mirroring the observed frequencies. Chances, on the other hand, are features of the models of the world that we use to make decisions: they are the probabilities that our theories spit out. If you fully believe your theory, you’d set your credences to the chances (this is Lewis’ Principal Principle).

The full story is something like this. The axioms kind of apply to us (especially when we are acting as scientists), so Savage’s axioms apply to us and we develop credences. We are smart, so we come up with theories to help set our credences, and we use data to update our credences of both the facts and our theories.

What I would like to know here is what assumptions need to be made (about the world, about rational reasoning etc) to make this story stick.

I believe that the interpretation of probability is the stumbling block to the interpretation of QM (I am not alone, I’m in the company of some of my favourite authors; see for example (Fuchs and Stacey 2019) and (Wallace 2014). This brings us to the next question.

2. could Everettian QM be the end of the story?

Everettian Quantum Mechanics (EQM, aka Unitary QM, aka the Many-Worlds Interpretation) is perhaps the most maligned interpretations of QM, because it features a mind-boggling continuous proliferation of parallel realities. I used to find it ridiculous until for some reason¹ I seriously engaged with the literature and discovered that the multiverse is not a postulate of the theory but is instead meant to be a prediction of the theory, an emergent feature of our world. I found this a strikingly beautiful idea.

One of the main criticisms of EQM is that we cannot make sense of probability within it. My claim is that we cannot make sense of probability in general, and if we could, we might make sense of it in EQM. This is also one of the main theses in (Wallace 2014). It turns out that the assumptions of Savage’s theorem naturally apply to a branching world too (Greaves and Myrvold 2008), so agents making decisions² in a branching world also choose the action that maximises expected utility, and update their credences to observed frequencies in the long run. Here of course $\mathrm{C}\mathrm{r}$ seems to take on a different meaning, but serves the same functional role of weighting the value of the outcomes. So we have a total account of probabilistic reasoning in a branching world.

Unfortunately, I have a suspicion that this story might be circular. It relies on branching, and branching often is argued via decoherence, which is derived via the partial trace, which in turn is motivated by the fact that partial tracing recovers the correct probabilities—Ouch!

But maybe there is a way out. I have seen proposals to show the existence of branching structure from computational complexity. For example, (Taylor and McCulloch 2023) shows that once the system has evolved for a while, the evolution of the system is the same as if it was an incoherent mixture of some states: the measurable observables and their correlations behave as if there was environment induced decoherence. The main point being something like: if there are real patterns in the observables that behave as if there are different worlds, it means that there are different worlds.

I’d like to get more to the bottom of the status of the branching: what assumptions do we need? is it circular? can the world be split in different families of branches?

The potential I see in EQM is that of a completely naturalistic theory with no special split between classical and quantum, which is rich enough to accomodate all our current theories, and a beautiful playground to develop new ideas and categories of the world, and maybe find a new paradigmatic example of emergence (the best current one of course being how temperature emerges from the movement of atoms). And also, yes, a wild revision of our understanding of reality.

Which brings us to the next question.

3. Emergence of spacetime

The spacetime continuum is how we organise the events in the world: we assign to them 4-coordinates (which can be done in countless different compatible ways) and then we have rules that tell us which events can affect other events, which ones are “close” in space and time, how the presence of energy affects these relations and so on.

But does it have to be this way? The “arena” of classical physics (phase space) is much larger than that, and in quantum mechanics the situation is even more weird, as particles don’t have trajectories, and quantum fields cannot take values at spacetime points. At small scales there need not be any any 4d continuum. Additionally, there might be completely different useful ways of organising data without making any reference to spacetime.

Could we have a quantum theory where spacetime emerges as a useful category only in a certain regime? The answer is: conceivably, yes! A lot of the geometrical properties of spacetime are reflected in the entanglement structure of the fields. What if it’s actually the other way around? Maybe the geometrical properties emerge from the entanglement structure of the quantum systems. Many researchers are working on this programme: Sean Carroll (Cao et al. 2017), Achim Kempf video, Eugenio Bianchi (Bianchi and Myers 2014), Daniele Oriti (Oriti 2018), Fay Dowker (Dowker and Butterfield 2021) are but a few names of the people who are leading efforts from different approaches. There is a vast, disorganised literature on all this.

I have actually been working in collaboration with 6 other researchers (Eugenia Colafranceschi, Joakim Flinckman, Guilherme Franzmann, Jan Głowacki, Niels Linnemann, Florian Niedermann). We call ourselves the EmerGe collaboration³, and for a year, we have been slowly but steadily making our way through this literature. We are prototyping a decentralised non-hierarchical way of working, meeting online and in person in lovely work intensives, being patient with one another not overworking each other on this long project, and we might be putting something out “soon”. Looking forward to see where this goes!

My impression is that the highest chance of this working is in EQM, where we don’t have to rely on a “background” classical reality to make sense of what happens in the world. There are just the quantum state or, more fundamentally, the quantum algebra and their correlations, and somehow, somewhere in there, at least for some solutions, we should see regularities organised in spatiotemporal-like structures.

4. Bonus question: what will we learn from deep learning theory?

Besides all these purely (philosophy of) physics questions that have been fascinating me since before I began my PhD, there is a question that rose more recently, namely, are we ready to develop a mathematical theory of deep learning systems?

Mathematical theories invented to understand human technologies has yielded theories that say deep, deep things about the world:

• pulleys, cannons $⟶$ Newtonian mechanics
• steam engines, lightbulbs $⟶$ thermodynamics, stat mech, and eventually QM.
• telegraphs, computers $⟶$ information theory and computer science
• deep learning systems $⟶$???

We may be close to the development of a whole new branch of science! We need to understand how to describe the incredibly complex behaviour of (relatively simple) networks of nodes, of course, not just understand it, but predict it. Already the description part is going to be hard! we need to find new words and concepts to identify patterns of behaviour. Right now, we are using human/animal behaviour words, computer science machine words, physics words, philosophy words, cognitive science words. It’s kind of all we have. But they are not enough! LLMs don’t “think” like us—like not at all—but they do “think”! To say precisely why that sentence is true will require a whole new set of ideas and concepts, and I am all here for it.

I haven’t gone quite as deep here, as I was focussing on physics, but I have now gotten into this great book (Roberts et al. 2021) where they analyse feed forward neural networks using the tools of effective field theory, and i want to get into this thesis (Gavranović 2024).

It’s a bit crazy, but a theory that explains how deep learning works may yield insights about cognition, conceptual space, perception, and how any of these things exist in the first place! We already have a few fascinating ideas, like the manifold hypothesis and our understanding of the encoder/decoder latent space.

More to do

These are the physics questions that I still feel motivated to explore. Of course, this list does not exhaust all the things I wish I understood better. I’d like to get a much deeper understanding of gauge symmetries and constrained hamiltonian systems (along the lines of (Henneaux and Teitelboim 1992)), have a stronger grip on how thermodynamics underpins causation and free will, and the way thermodynamics emerges from statistical mechanics. I hope I’ll get to these at one point, maybe even as part of the other questions.

I am transitioning away from full-time employment in academia, to have more flexibility in my life, and to distance myself from its career-oriented culture. This means I will research these things while giving space to other, less theoretical, adventures. My hope for this research is to dive deep in literature, to have a lot of exciting conversations, write more posts than articles, and work in collaboration with a bunch of people to explore these questions.

Exploring this new way to do research feels just as as exciting as the research questions themselves!

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Bibliography

Remember that sci-hub and libgen exist!

Image credits to OpenAI.

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1. The reason is that i read The Beginning of Infinity, and thought that David Deutsch made an egregious amount of sense in that book so he could not be completely delusional about QM. So I gave EQM a real shot by reading (Wallace 2014).↩︎

2. Of course this opens up the whole theme of what it means to make decisions if physics is deterministic. I think this is answered in detail by the likes of Daniel Dennett and Carlo Rovelli. The short answer is: free will is a property of our coarse-grained description of physical systems, like wetness. One day I’ll like to write a post about this.↩︎

3. Emergent Geometries.↩︎

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