Infinite information in finite volumes of space time?

Question

It recently occurred to me that uncertainty at the quantum level, unintuitive as it may be to our human scale experiences, may solve a fundamental problem: Wouldn't perfect determinism require infinite information e.g. for every elementary "particle", or more general, space time volume? Because in order to predict the result of a quantum (or, indeed, any) interaction with arbitrary precision, the concerned space time volumes would need to hold arbitrarily or rather infinitely precise information about their state.

Are there fundamental physics reasons why that would be impossible? Isn't there a minimum energy/work necessary to hold or exchange information?

In other words, isn't uncertainty the only way things can be for very fundamental reasons?

Answer 1

Uncertainty can be derived from the fundamental axioms of quantum mechanics without reference to information (see also these lecture notes). But it is important to distinguish between (1) quantum mechanics itself, (2) the physics that "runs on it". What fields and interactions are possible is not solely defined by quantum mechanics, and theoretical physicists invent new ones all the time to see how they play out. The information content of the world depends on both 1 and 2.

In my opinion the best explanation for the link is due to Scott Aaronson, who outlined an argument (here shortened a bit):

1. "Relativity—even just Galilean relativity—demands that, in flat space, the laws of physics must have the same form for all inertial observers"
2. "Anything in the physical world that varies in space—say, a field that encodes different bits of information at different locations—also varies in time, from the perspective of an observer who moves through the field at a constant speed."
3. "Combining 1 and 2, we conclude that anything that can vary in space can also vary in time."
4. The conversion factor is $c$ due to SR.
5. "Anything that varies across time carries energy. Why? Because this is essentially the definition of energy in quantum mechanics! Up to a constant multiple (namely, Planck’s constant), energy is the expected speed of rotation of the global phase of the wavefunction, when you apply your Hamiltonian. "
6. "Combining 3 and 5, any field that varies across space carries energy."
7. "More strongly, combining 4 and 5, if we know how quickly a field varies across space, we can lower-bound how much energy it has to contain."
8. "In general relativity, anything that carries energy couples to the gravitational field. This means that anything that carries energy necessarily has an observable effect: if nothing else, its effect on the warping of spacetime."
9. "Combining 6 and 8, any field that varies across space couples to the gravitational field."
10. "More strongly, combining 7 and 8, if we know how quickly a field varies across space, then we can lower-bound by how much it has to warp spacetime. "
11. "But in GR, spacetime can only be warped by so much before we create a black hole: this is the famous Schwarzschild bound."
12. "Combining 10 and 11, the information contained in a physical field can only vary so quickly across space, before it causes spacetime to collapse to a black hole."

He concludes "Summarizing where we’ve gotten, we could say: any information that’s spatially localized at all, can only be localized so precisely. "

Note that this argument hinges on both quantum mechanics (the energy-time link, which is because they are complementary, and hence also imply the uncertainty tradeoff between them), and the special and general relativity (the kind of physics we got). In other kinds of spacetimes it might not hold (indeed, when you allow exotic fields that prevent black hole formation in extreme cases you can get infinite information). Plus, of course, we don't have a proper quantum gravity theory so there is more than a bit of handwaving in this argument.

Taking "infinite information density is not possible" as an axiom might allow you to remove another axiom somewhere in this construction. It is noteworthy that one can view the uncertainty relation as derived from "the statement that the position and momentum distributions cannot both be arbitrarily narrow" which would indeed follow from such an axiom. But as mentioned above, it can be derived from the other quantum mechanics axioms too. Whether you get an interesting axiomatic system by assuming uncertainty and dropping one of the others or using a different set is a bit beyond me, but would not surprise me. Often several different axiom sets can act as "base vectors" for the same theory, and reformulating it to a more information-oriented basis might be illuminating.