Will– The point is that the laws of physics tell us what the acceleration is, if we know the position and momentum. (a = F/m, in Newtonian language.) We don’t need to specify it independently.

On momentum, yes that’s right. The rule is simple: there is one dimension of momentum for every one dimension of space, since the particles can move independently in each direction of space.

I’ve been following along pretty well so far, but I had a couple questions come up on a couple points. I hope that they don’t seem like ridiculously easy questions, but I think I might have missed an assumption or two along the line.

1. On p.129 you specifically state that we don’t need to know the acceleration of the ball to predict the next state. But I’m confused as to how you can safely ignore it. Obviously, by knowing the position and momentum, you can predict the next position of the ball. But since acceleration would change velocity and hence momentum, wouldn’t you need to have that to predict the momentum of the ball at the next instant?

2. On p.132 you described the 4-dimensional space of states for a billiard ball. Then you said that a baseball in 3-space would have a 6-dimensional space of states. I don’t know if I’m counting correctly. Is that 3 for the position, 2 for the direction of momentum, and 1 for the magnitude of momentum? Are there always n-1 dimensions for direction of momentum for n-space? I think I am close to understanding this, but I guess what’s confusing me is that the number of directions of momentum seems to be somewhat arbitrary – it seems like it is possible to use a single dimension to describe that direction even in 3-space (even if it is not very practical). Or you could use two dimensions to describe the direction in 2-space by splitting it into x and y components. What am I missing?

I’m really enjoying the book and discussion so far, and recommending it to my friends. Thanks!

I have read a little, about Hubble, the laws of thermodynamics and how they don’t appear to apply to the universe, donut shaped universes etc., and these ideas remind me of something I saw as a young boy.

I live by the sea, and would sit and watch the waves roll in, break, then rapidly spread across the sand as they lost energy (at least rapidly from my perspective). It occurs to me that the universe is just energy, as was the energy in the wave. That sitting on the cusp of the wave we would know nothing of the depths of the ocean or the heights of the sky. We would only know the energy we were part of, and we would also have the perspective that the wave was increasing in velocity as it was affective by other medium, and as it spread and lost energy.

Is this a completely illfounded idea?

As for the particular issue we are discussing, of course we all accept what the basic model is. There is not much room for disagreement once you do take the wavefunction as the model. I certainly do agree with the Bohr-type view, but I disagree with your saying that it involves things going “missing”. As you know, when things are inherently uncertain, in that the (nature dictates) info cannot exist, that is different from when the info only doesn’t exist. We do not cling to classicism when we are at peace with the notion that the uncertainty is okay.

I hope there is a mention of Feynman’s “can we simulate physics with computers” paper in the book. It would make for a great blog entry!

Note also that in QM, and per Sean’s previous comment: such a reversed world is not like a Laplacian clockwork that is just running backwards and will reliably continue to do so for deterministic reasons. The wave functions have to behave differently (I don’t think their backwards evolution is quite the same – consider how wave packets of particles are supposed to expand inherently as different velocities. Am I missing something? In any case, a reversed world should be constantly under the threat (in model theory …) of the game screwing up as probabilities eventually fail to cooperate in the “conspiracy” to pretend to support a t-reversed world. IOW, the screen that now photons are being emitted *from* will not continue to have them absorbed on the little spot that is (in the reversed world) the presumptive “emitter” of those photons! See more at my link on QM issues of decoherence etc. that might relate to that. (http://tinyurl.com/c7dw2v,sorry it didn’t go thru system OK. )

OXO– I don’t know of any place you can buy a signed copy. If you mail a copy to me with a stamped return envelope, I’m happy to sign it and mail it back.

As you point out, with measurement left out of the picture the evolution of the wave function is in fact deterministic. If you subscribe to something akin to the Cophenhagen interpretation then what you are saying is correct — I am stating that the evolution of a joint probability distribution is deterministic.

If you take an Everettian point of view, quantum mechanics really is the whole story and the “uncertainty” and “probability” are put upon you partially by insisting that you can use a classical description — QM tells you about a wave function that is perfectly well described and uncertainty relations only come about by us trying to use classical notations of position and momentum that CM lead us to believe could be simultaneously determined. It is not that QM is “missing” information, it is that CM gave us the wrong impression of what “complete” information would look like.

The Everett picture tells us that interactions tend to destroy coherence and makes the macroscopic world look like a (very) weakly coupled set of (almost) classical systems. If this is in fact the right way to think about it, QM is in fact a deterministic theory, and the probabilities arise in a manner similar to the way they arose in (classical) stat mech — from our course graining and assigning ourselves to only one of the possible (almost) classical worlds.

I disagree that reverisibility is essentially the same as determinism. In Sean’s book with the “Checkerboard” rules he gives examples of deterministic evolution (i.e. with knowledge of the present state, you can predict the future state at an arbitrarily *later* time) but the lack of reversibility means that how you evolved to a particular state is ambiguous. You can predict, but not retrodict. If you knew the initial state (if it exists) then it is perfectly consistent in such a world for evolution to be deterministic but irreversible. However that does not seem to be the type of rules that nature has employed.

As for LAG’s second comment about determinism and quantum mechanics, if we believe that quantum mechanics is the true story (with no collapse) quantum mechanics does not tell us that we cannot fully measure the state of the universe at one time. QM tells us that the state of the universe (or a isolated system) is given by the wavefunction, which we can write down, and this *is* the full information in the state. The issue with “uncertainty relations” is that things that we have been mislead by classical mechanics into thinking as a complete description of the state (e.g. position + momentum for every particle) cannot be well defined — that technically information about different observables is obtained by projecting the *same* state in terms of different bases that do not line up with one another.

So it may be more useful to say that quantum mechanics fundamentally changes what we mean by “state of the system”, rather than what many popular science books try to do under the guise of uncertainty relations. (Heisenberg’s microscope argument is a particularly blatent offender here.)

LAG– It is basically determinism, and QM raises questions there, which I discuss in Chapter 11. QM certain violates determinism at an apparent level, but I’ll argue that at a deeper level it does not. The Schrodinger equation, in particular, is perfectly deterministic.

” Reversibility is a very deep idea; it implies that the state of the universe at any one moment in time is sufficient (along with the laws of physics) to precisely determine the state at any other time, past or future. …”

But, isn’t that ‘determinism’ which i thought Quantum Mechanics forces us to abandon because we cannot *fully* measure the state of the universe at any one moment and furthermore, whatever happens next is *not* precisely determinable since it is quantum probabilistic ?

So isn’t reversibility inconsistent with the ‘facts on the ground’ so to speak ?