Can a layman learn GR and QM? REALLY learn?

Half_Man_Half_Wit · February 12, 2011, 7:50am

Completely forgot to mention: Nobel laureate Gerard t’Hooft has a site called ‘Theoretical Physics as a Challenge’, where he goes over what topics to learn, and presents some resources for most of them.

Asympotically_fat · February 12, 2011, 11:49am

Half_Man_Half_Wit:

Perhaps, if you get hung up about Hilbert spaces, let’s do away with them for the moment, and consider the more familiar phase space of classical mechanics, in which, loosely speaking, the points represent configurations of the system under consideration, represented through position and momentum variables. This is the natural home of Hamiltonian mechanics, which is just a (somewhat more mathematically sophisticated) reformulation of Newtonian mechanics, where everything nicely follows classical equations of motion. Those equations of motion are dictated by the Hamiltonian, which is, in the simplest sense, just the sum of potential and kinetic energy (and thus, the total energy) (I’m sorry, by the way, if that’s all old hat to you, but I usually feel it’s better to spell out things in as much detail as feasible than to get tangled up in unnecessary confusions). For a bit more on this, see the wiki.

On this phase space, you can do statistical mechanics – you can, for instance, for an ensemble of particles, ask, ‘what is the probability of finding a certain (classical) particle at a certain point in phase space (i.e. at a certain position with a certain momentum)?’ This probability is given by the Liouville density, which evolves in time according to Liouville’s equation. Up to this point, I hope everything seems sufficiently sensible from a physical point of view. Now here’s where the weirdness enters: I’ve noted above the ‘old quantum condition’, which essentially says that if you’re a quantum particle, there’s some rule that your position and momentum have to obey. In a sense, this amounts to the fact that a quantum particle can’t be ‘localized’ arbitrarily well in phase space; roughly, there’s a minimum area (of dimension [momentumposition] = [energytime] = Js, which is not coincidentally the dimension of Planck’s constant) it must ‘occupy’. This ties to the uncertainty relation.

This of course means a complication for finding out how likely it is that a certain particle exists at a certain point in phase space, as the notion of ‘a certain point’ is no longer meaningful. I’ll have to mumble through the details a bit (so much for spelling it all out), but what you do in order to keep track of this mathematically amounts to a deformation of the algebra of functions on phase space (i.e. functions of where a particle is and how fast it’s going somewhere else, bluntly). This is known as phase-space or Weyl quantization. You end up with a (unique up to isomorphism) structure that is in a sense a generalization of classical (Hamiltonian) mechanics on phase space: if you let Planck’s constant (which is the deformation parameter, coming from the ‘quantum condition’/the ‘discretization’ of phase space) tend to 0, you recover the classical situation completely. In particular, all that really changes is that ordinary multiplication goes over to a (noncommutative, as it must be to do quantum mechanics) star product, (the Poisson bracket is replaced by the Moyal bracket, but I’ve not talked about that, so I only mention it in some vain attempt at an illusion of completeness), and the above-mentioned Liouville density gets replaced by the proper analogue, the somewhat unwieldily named Wigner quasi-probability distribution (sometimes Wigner density function), whose evolution is given by Moyal’s evolution equation. Everything else essentially retains its physical interpretation – allowing for some necessary generalizations, you can do something very much like Hamiltonian (statistical) mechanics on your new, noncommutative, deformed phase space.

And, as it turns out, this (to me, much more intuitive) picture of quantum mechanics as deformed statistical mechanics on phase space is completely equivalent to standard Hilbert space quantum mechanics! All phase space functions can be mapped onto Hilbert space operators using the Weyl map (and back using the Wigner map) (especially, the Hamilton function then yields the Hamilton operator, which perhaps gives some motivation for the idea that the eigenvalues of the latter have something to do with energy). In particular, the Wigner density function gets mapped to the quantum mechanical density matrix, which is a description of the quantum state of a system equivalent to the perhaps more familiar wave function (the line, or ray, in Hilbert space giving the state of the system). It evolves via the von Neumann equation (an equivalent to the Schrödinger equation), which is the Weyl transform of the Moyal evolution equation, which was the deformed version of Liouville’s equation. (Also, the Moyal bracket gets mapped to the commutator of operators in Hilbert space.)

So, you can view the Hilbert space formalism simply as an isomorphic mathematical description of deformed statistical phase-space mechanics. Liouville’s equation is what describes the time evolution of the phase space distribution function, which describes the probability of finding the system in a certain phase space volume – i.e., in the case of a single particle, of the particle having a certain momentum and position. The physical content is that at a certain time t, there is a certain probability that the particle (thought taken from an ensemble of particles) has position x and momentum p. That physical content stays (up to some awkwardness related to the occurrence of ‘negative probabilities’ in the Wigner density function) basically the same with the Moyal equation describing the time evolution of the Wigner function obtained after deforming the phase space algebra. The Hilbert space formalism, in which density matrices evolve according to the von Neumann equation (which is equivalent to wave functions evolving according to the Schrödinger equation), is then just a bit of mathematical re-writing.

That’s very interesting!

That said I think it’s still best to learn QM the old-fashioned way, becuase quanisation schemes are a level above basic QM. Though it’s helpful to know they exist.

Density matrices themselves are of course operators on the Hilbert space that is the state space of the system, representing statistical ensembles of quantum states.

Half_Man_Half_Wit · February 12, 2011, 1:59pm

Glad you think so, I always find myself wondering why it’s not better known. Well, I suppose it’s a bit harder to actually do anything useful with, but it may be useful in ‘anchoring’ a few otherwise perhaps rather abstract concepts.

I agree. Basically, I think it’s completely wrong-headed to try and slap quantumness on pre-existing classical theories – it ought to be the other way round, derive classical theories as the large h limit of quantum ones, but it’s a bit hard to get started that way. Everyone needs conceptual bridges, if only to burn them behind themselves!

Yes. I just hoped to get what we’re talking about when we talk about quantum mechanics at least within shouting distance of what has a more immediate physical content to those not used to* the more abstract elements of the theory. It’s still a far cry, but I tried what I could.
*I think what von Neumann said applies here as well, to a certain extent: “In mathematics you don’t understand things. You just get used to them.”

Asympotically_fat · February 12, 2011, 2:47pm

I have a few questions actually:

So in the simplest terms, it’s correct to say that the quantum state space of a system is a transformation of a deformation of it’s classical phase space?

In this quantum phase space indvidual points represent pure quantum states?

The dimesnionality of the quantum phase space corresponds to the dimensionality of the classical phase space?

The fact that we end up with density matrices comes from the fact we started off with a statistical description of the system?

Half_Man_Half_Wit · February 12, 2011, 4:40pm

I’ll do my best, but it’s been a while since I really looked into the stuff. Lest I mislead you, I remember this paper to be quite useful as a short introduction.

Depends on what you mean by that, I think. The algebra of operators on the Hilbert space of the system is isomorphic to the Moyal algebra (the deformation of the Poisson algebra) on its phase space.

Well, you don’t really have a notion of a precise point in phase space quantum mechanically, as that would entail both a definite momentum and position. The state is coded in the Wigner function, which handles pure as well as mixed ones (just as the density matrix).

I don’t see why it wouldn’t, but I haven’t really thought about it.

In the way I have presented it, yes. But you can also start from Hamilton’s equations and end up pretty well at the same point (as mentioned in the wiki). The fundamental input is the uncertainty principle.

brocks · February 13, 2011, 8:33pm

Thanks again. That was exactly the kind of thing I was looking for when I started this thread.

Jamicat · February 14, 2011, 2:55am

Anyone inclined to learn something, can learn something.

I just read up on all the sciences.
Do the occasional thought experiment with my new info.
Some data correlates, others not so much.
I don’t run any of the math, just the ideas.

Usually, not so long afterward a show pops up about it, with proofs to support what I was thinking.

Kinda fun discovering stuff.
Keeps me outta trouble.

Panurge · February 14, 2011, 9:22am

I definitely think you could make it. One doesn’t need to be gifted to be a good scientist.
For starters, I think you could do worse than starting by watching a few of Yale astrophysicist Charles Bailyn’s lessons in his Introduction to Astrophysics (for GR, that is - no QM IIRC). In the lessons on black holes and relativity he sort of outlines the concepts of GR without going too much into the maths of it all. A very sensible approach for the novice, I think. Try to get a feel for the thought processes involved instead of the mathematics.
The same site has a series of QM lessons by Leonard Susskin from Stanford. I haven’t watched them, but it could be a starting point.