I've never understood why proof that the universe is "holographic" (= 2+1 dimensions of information projected as 3+1) does not fall out of the Schrödinger’s/Maxwell’s field equations.
After all, the equations, by their very nature as equations, constrain the dimensionality of possible universes (field configurations) by one, from 4 down to 3. The fourth is always derivable from the other three (e.g., X-Y-Z intial conditions at T=0 define X-Y-Z-T fields for all T).
To believe that the universe contains four dimensions of information (i.e., is not a hologram), would imply that the field equations do not universally hold. So what this experiment is actually testing is the truth of QED, which implies holography.
Does anyone know why this is not so? (I tried asking a while ago on Physics StackExchange and only got flippant responses.)
(As an analogy for CS types: consider the game of Life. It is 3-dimensional (2 space + 1 time), but constrained by the Life equation. So it cannot contain three dimensions full of arbitrary information; only a two-dimensional slice can be arbitrarily instantiated. The analogy is not perfect, as Life is neither reversible nor fully observable from any 2D slice, but it is close.)
I may be off base here, but I think the distinction is between number of degrees of freedom (what you describe) and density of bits of information. Bekenstein's result showed that the information in a black hole is proportional to its surface area, not its volume. This seems unintuitive in the sense that if you, e.g., stuff more ram chips into a box, its information content seems to increase as a function of volume; the disparity lies in the fact that bits in ram are not as densely packed as in a black hole.
I think there is no distinction to be made. Consider encoding information in a 4-cube (hypercube) constrained by Maxwell's equations. (Let's make the space discrete so our heads don't explode.) You will find that you can only encode an amount of information proportional to the cube of the 4-cube's edge length, not the 4th power, because you are constrained by Maxwell's equations. If you further insist on keeping the information unchanged with respect to time, you'll be restricted to an amount proportional to the square of the 4-cube's edge length.
Given that according to quantum mechanics, empty space continually experiences small, random fluctuations[1], I think you can argue that in the normal space, information does on average expand proportionally to the volume encompassed, even if the proportionality constant is rather low (edit: this applies to space-time volume too since the fluctuations are random, distinct on each timeslice).
Indeed, I think quantum mechanics would be the only system that guarantees this. I suppose a purely classical system that was entirely reversible couldn't contain information in the time dimension because each moment would imply every other moment. But claiming that any system determined by classical differential equations is "holographic" might be a stretch.
First of all, a spatial dimension is a degree of freedom per particle (two if you count velocity).
Second of all, physics equations often represent many constraints. For example, `p2 = p1 + v t` is actually three constraints. Each particle has these sorts of constraints, so all of them cut down the degrees of freedom in the system.
So it's not 4-1=3. In Newtonian mechanics it's 1 (time) + 3n (positions of particles) + 3n (velocities of particles) - 3n (velocities determined by forces) - 3n (positions determined by velocities) = 1+3n+3n-3n-3n = 1. Which makes sense, because otherwise you wouldn't get one solution per time step.
EDIT: jamii seems to understand that we are working with very different definitions of dimensionality.
My (and I suspect most readers') understanding of dimensionality is as follows. If I have a graph with x and y axes with six RGB-colored points on it, that does not make it a 30-dimensional graph. It's two dimensional because there are two independent variables in the 5-tuple relation that graph is representing.
Number of particles and number of dependent attributes do not affect dimensionality.
Correct. There's spatial dimensions, where you can put the points, and degrees of freedom, the number of numbers you specify when setting up a system. The word dimension can refer to either definition (and others), depending on the context.
What I was trying to communicate to the gp is that they were mixing the two concepts. Constraints reduce the number of degrees of freedom, not the number of spatial dimensions, so the intuition that you get the holographic principle by adding a constraint to 3d space simply doesn't type-check.
No, they really do reduce the number of spatial dimensions. Take EM fields. You have a 6-tuple relation, x×y×z×t×e×m. Maxwell's equations aside, this relation is four-dimensional, because two of its dimensions (e and m) are dependent on the other four (x, y, z, and t), making the relation a function: x×y×z×t→e×m. Meaning, for any values of x, y, z, and t, they are related to exactly one value of both e and m in the relation. Without any constraints, e and m may be freely chosen for each value of x, y, z, and t in the relation.
Now, if I assume Maxwell's equations, I am free to rewrite this relation as dependent on only three variables; let's say x, y, and z, but it really doesn't matter: x×y×z→t×e×m. To further refine the example, if I took t=0 for all x, y, and z, I'd have initial conditions at t=0 (very standard).
Given Maxwell's equations however, I can derive exactly one relation of the form x×y×z×t→e×m from my x×y×z→t×e×m initial conditions. Hence P(x×y×z×t→e×m) and P(x×y×z→t×e×m) are in bijection, |P(x×y×z×t→e×m)| and |P(x×y×z→t×e×m)| have exactly the same cardinality, and therefore the same spatial dimension, three.
I believe Strilanc is talking about dimensions in phase space rather than in physical space. If you plot all the possible configurations of those points on a graph, that graph would have 30 dimensions. Whether or not that is a valid point is beyond me.
Your point is very similar (equivalent IMHO) to Julian Barbour's case in The End of Time, that you can express the laws of physics without reference to time, since time can be defined with respect to a special distance metric between universe states (the Machian distinguished simplifier).
In fact, I think the analogy between Game of Life's governing rule and the Schrodinger equation is very clarifying!
After all, the equations, by their very nature as equations, constrain the dimensionality of possible universes (field configurations) by one, from 4 down to 3. The fourth is always derivable from the other three (e.g., X-Y-Z intial conditions at T=0 define X-Y-Z-T fields for all T).
To believe that the universe contains four dimensions of information (i.e., is not a hologram), would imply that the field equations do not universally hold. So what this experiment is actually testing is the truth of QED, which implies holography.
Does anyone know why this is not so? (I tried asking a while ago on Physics StackExchange and only got flippant responses.)
(As an analogy for CS types: consider the game of Life. It is 3-dimensional (2 space + 1 time), but constrained by the Life equation. So it cannot contain three dimensions full of arbitrary information; only a two-dimensional slice can be arbitrarily instantiated. The analogy is not perfect, as Life is neither reversible nor fully observable from any 2D slice, but it is close.)