Free will

Eisenstein g_3 modular form

As a physicist, I get to think about free will just like the rest of ’em.  I was recently prompted to set my thoughts to writing on the talk page of the Wikipedia article free will theorem. I think I can string together a few more pieces, and clarify how it actually “all works”. Caution: the rest of this article is about physics and math. So good luck with that, if you are not widely read.

The free will theorem starts with three postulates: an upper bound to the speed of information; the importance of spinors, and quantum entanglement. Not a bad foundation. Lets see how it plays out.

In some theories of entanglement, the resolution of wave function collapse happens via propagation of the phases of the wave-function into the past, see, for example, the two-state vector formalism. Here, it is not information that is travelling backwards in time, but rather the geometric phase (aka the holonomy). There is a U(1)-connection relating the phase of the quantum-mechanical particle (taken to be the phase of the spinor on a spin manifold); that phase has to be consistent across time (and not just space, as it is in the Aharonov–Bohm effect); to describe this consistency, one has to use the holonomy of the connection. If the phase in the future, after wave function collapse, is going to be consistent with the past, then you have to propagate it “all the way around”, “into the past” as well as “into the future” — to close the loop — the holonomic loop (aka the Wilson loop). That’s what makes the two state-vector formalism work. Roughly speaking, it is not “information” in the sense of “classical bits”, that propagates into the past, its the qubits. Anyway, that’s the general idea, as I understand it. The article on Aharonov–Bohm effect has a tortured, painful explanation of the space-like only version of this; I’ve seen far more elegant explanations elsewhere. Aim your search-engine at “U(1) holonomy” for details. Maybe throw in “Dirac string” into the search.

Anyway, that is my understanding of wave-function collapse. There’s no speed-limit in this. Basically, you can think of the past as being “not yet fully frozen” or “not yet fully determined” until the future forces those holonomic loops to close; when they finally freeze up, that wave-front of “freezing up” is what necessarily propagates at the speed of light. (I suppose if you are bold, you can claim that the U(1) of the quantum mechanical phase its exactly the U(1) of electromagnetism; this would explain why its the “speed of light” and not some other speed that is involved. I don’t know if one can be that bold, or not, but it sure seems reasonable.) The loops are what is carrying the “classical bits” of information, the classical bit corresponding to the question “is this loop closed yet, or not?” which has a clear yes/no answer.

The elegance of this is that it replaces a fairly nebulous concept of “causality” with something very concrete and algebraic: the holonomy, and Wilson loops, more generally, with which one can do explicit calculations: it is the cornerstone of algebraic topology. You can do calculations with loops, things like spectral sequences or more generally the Postnikov tower. You cannot do comparable calculations with “causality”. You can’t stick “causality” and “determinism” into some equation and turn the crank. It doesn’t work.

The other problem with the naive concept of the “speed of information” is with what happens at the event horizon at a black hole. My (faulty and naive) understanding is that, again, this is where the holonomy plays a key role; the holonomy in a certain sense “escapes” the black-hole information paradox. The holonomic loops are free to thread through the event horizon; that is because they are not “physical particles” and have no “speed” and thus no “speed limit”. Whatever is entangled inside the horizon must still be phase coherent across the horizon with whatever is going on outside. The evaporation is what “tunnels” the phase from the inside to the outside. Thus, it is not “information” that is being radiated away during during evaporation, it is the end-points of the holonomic loops; when these finally close, the “information” that they are closed is now outside of the BH. For evaporation (Hawking radiation), the end-points happen to be entangled spinors. They carry no information by themselves, the information is “created” when the wave-functions that embody them are a part of collapse. That is where “information” comes from. It is also “why” it looks like information “lives on” the event horizon; the information is a count of the not-yet-closed loops that are waiting for closure. This is consistent with the replica trick (from spin glasses) that is used to resolve the ER=EPR suggestion. The ER’s are the places through which the Wilson loops thread through.

In more abstract terms, information is a cobordism, or rather, the content of what is required to specify a specific cobordant arrangement. From what I can tell,  its got something to do with spectral triples, but I don’t entirely get it. The spectral triples describe the operators needed for the operator product expansion across  the event horizon boundary. Or something like that. I dunno.

So, the above is a sketch that offers up the mathematical details for why “causality” and “determinism” are faulty concepts. John Baez explained one aspect of this elegantly  where he argued for getting rid of category “Set” and replacing it by category “nBord” and category “Hilb”. (See “Quantum Quandaries: A Category-Theoretic Perspective” in “The Structural
Foundations of Quantum Gravity” (2006)) It gets rid of the stupidities with set theory and functions, which are the same stupidities of “causality” and “determinism”, that everyone gets so hung up about, and replaces them with e.g. the infinity category (or the infinity groupoid as that’s more appropriate.)

So where does “free will” come from? Roger Penrose suggests a path. Let me suggest a revised model. Modern physics uses the Standard Model to describe particle interactions.  For this discussion, let’s fall back to a simpler description, which can be used in generic settings (including in curved space-time): this is the resonant interaction. In this case, the conservation of energy becomes a Diophantine equation. Now, Hilbert’s tenth problem asks for the enumeration of such solutions, and it is now known that this is algorithmically impossible — there is no computer program that can achieve this. What does this mean? Well, “determinism” or “causality” is that thing which results when you use digital algorithms; such systems have no “free will”.

Put it this way: whatever free will is, it is certainly not a deterministic coupling. When you say “1+1=2”, it is what it is and there can be no other way. Any set of equations that couple together a bunch of different things “determine” those things. For example, mathematical proofs are “determined” by their premises; they proceed in a purely mechanical way unto their inevitable conclusion. When a certain path is not Turing computable, one gains a certain freedom, as it were; one is not forced to march down that path. So, for example,   when pondering the conservation of energy in the resonant interaction, the resonance condition is “fixed”: one must have a balance of energy. Re-interpreting this as Diophantine equations, we equally see that they are “fixed”, determinate. Balancing these against Hilbert’s tenth problem, we’ve got a rub: A certain set of deterministic, unbreakable equations have solutions that are not recursively enumerable. It is, as it were, that these equations can “make a choice”: they can say “I choose to be solved like so, or like so.”  There is no algorithm forcing their hand.  From the point of view of the Conway-Kochen free will theorem, an electron can choose to do this, or to do that. I’m being crazier here: a Diophantine equation can “choose” to express itself this way or that way.

This is how one builds the bridge from undecidability (Turing incompleteness) to “free will” in physics. To be crystal clear: the outcome of the interaction between physical (quantum mechanical) particles (in a curved space-time background) requires a decision problem to be solved, that cannot be solved using algorithms/Turing machines. Ergo “free will”.

(Footnote: It is not clear whether or not geometric finite automata (GFA), for example, the quantum finite automata (QFA) can evade these non-computability results. That is, can a QFA or GFA ever be an oracle? That is an interesting question in itself.)

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