Wikipedia says there are some axiomatic systems for Quantum Field Theory. I bet it is complex enough to have an undecidable statement (invoking Godel). What would be an example of such a statement? Is it anything you could arrange in the real world? If you did would anything weird happen, or just something you couldn’t have decided until you arranged it in the real world, and then the physical system plays out in an unsurprising manner?
Most known undecidable statements in mathematics are very contrived in form, and self-referential in some way. Though this probably mostly just reflects the difficulty in proving statements undecidable.
Once you get into physics, though, it’s easy to come up with mathematically-undecidable statements. Any statement that starts with “the mass of the _____ particle is equal to…”, for instance, will be mathematically undecidable. You can add an axiom for each such statement, but those axioms then must either have attached error bars, or be of unknown applicability to the real world.
These axioms aren’t the statements of formal (in the technical sense) logic involved in Godel’s theorem. Rather, they’re an attempt to create a mathematical object that provides the behavior we expect and observe in physics. To axiomatize classical quantum mechanics, for example, we could start by saying that a particle is a function in L[SUP]2/SUP for some sufficiently nice space X that satisfies some reasonable smoothness properties, belongs to certain representations of the Poincare group SO(1, 3), etc.
Well, one could interpret undecidable as non-deterministic. And even in Newtonian physics there is non-determinism: one example being Norton’s Dome which, if you place a marble at the top, should start rolling away in some finite time–but when, and in which direction? One can’t say.
You could have a perfectly deterministic system like a simple cellular automation which exhibits formally undecidable behaviour (since you cannot observe it for an infinite length of time to see what it does), so mathematical undecidability is not a substitute for non-determinism in the sense that it may be impossible to know the outcome of a coin flip in advance.
Would an undecidable statement in QFT necessarily involve infinite time or energy or some other physical quantity approaching infinity? If space is infinitely divisible could the statement involve finite quantities in a finite volume (but the “trick” to construct the undecidable statement involves infinite subdivisions of the space)?
When I wrote my comment, I meant “undecidable” in the sense that you cannot decide whether a given Turing machine halts, even though you know everything about its behaviour. This is of course related to the impossibility of checking certain arithmetical statements, since you can’t plug in all the numbers. “Non-deterministic” means that you cannot predict the future state of a system, despite knowing the initial conditions.
QM is obviously(?) non-deterministic; since you are asking about undecidable statements, I suppose one has to start with a mathematical description of such a theory, such that one can write down formal statements (that may or may not be decidable). This would involve something like the Araki–Haag–Kastler axioms or Wightman axioms. I don’t know off the top of my head a simple undecidable statement in those frameworks, but I do not see why it would need to explicitly invoke infinite quantities or limits any more than do undecidable arithmetical statements, or undecidable statements in functional analysis: they involve some logical quantifiers, like “for all integers a, b, c, d, e, f, g, h, i the quantity p(a,b,c,d,e,f,g,h,i) [some explicit polynomial function] never equals zero”, but you just have a statement about integers. Another undecidable statement is “the ideal of compact operators on the infinite-dimensional separable Hilbert space can be written as a sum of two smaller ideals.” So the quantum-mechanical statement would involve some such formula on quantum observables [certain Hermitian operators, as in Post #3].