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Bill S. Offline OP
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Your 12 observations provide something to think about. Let’s take this in bite-sized chunks.

1. I think I have the general sense of this, but I guess I would need more maths than I have to be able to get to grips with Hilbert space; not to mention “linear operators on the Hilbert space”. Any simplification on the Wiki article would be appreciated.

2. OK with that.

3. So far, so good.

4. I would not be able to explain that to someone else, so I guess I don’t understand it.


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Originally Posted By: Bill S.

1. I think I have the general sense of this, but I guess I would need more maths than I have to be able to get to grips with Hilbert space; not to mention “linear operators on the Hilbert space”. Any simplification on the Wiki article would be appreciated.

Hilbert spaces are well covered in wikipedia it's not that complex most layman tend to think in normal Euclidean space but I know you have familiar with the idea of curved space and you are familiar with latitude and longitude and curvature on earth.

It's the same problem while you sit at your desk and plot a path from London to New York you think in a straight line but you are quite aware that you are actually taking a curved path. So a Hilbert space allows for the shape of space to not be flat it's a generalized description of space it's not implicitly flat and actually it covers a range of shapes space could have that are weird and whacky.

The operators are the sorts of measurements etc and so we have to know certain things work. A few are listed in the wikipedia article such as Pythagorean theorem must hold or you aren't on a Hilbert space. Think on earths surface if we plot a huge triangle so long as you account for the curvature of the earth Pythagorean theory still holds and so we assume that of space.

Linear means exactly that it's linear ... Pythagorean theory holds without having to make some sort of algebraic adjustment for curvature which would be the other option.

Non linear option is defined like so
http://www.encyclopediaofmath.org/index.php/Non-linear_operator

So if you want it in a simplification it is basically saying you can extend our normal geometry laws out into space and they work correctly even if space isn't flat as that is the only observational facts we have so that is what we go with. Now we extend it a bit more than just geometry to motion etc but you get the idea basically it says our patch of space works like space as a whole regardless of it's shape.

See each point spells out the basis on which it is made you can attack this point on the basis "our area of space is special" because we can't not observationally exclude that.

Originally Posted By: Bill S.
4. I would not be able to explain that to someone else, so I guess I don’t understand it.

Four is simply saying that the only thing we see change between the quantum mechanics and classic physics is the decoherence change. We have measured and found no other changes so in the same way we create a boundary between classic states like solid/liquid/gas (boundary condition is atom or molecule movement) it is formalising that classic physics is simply a state of quantum mechanics with a defined boundary due to decoherence.

Our ancestors probably never realized that water vapour, water and ice were all the same thing so science felt it important to document the observational fact, something you probably take for granted these days.

So it is a consistent approach water vapour becomes water and then becomes ice as the molecules slow speed number 4 is simply stating classic physics is the QM world viewed with a loss of coherence. Water vapour, water and ice are all the same thing so the classic world and the QM world are the same thing with a boundary condition.

If someone was to find something else change QM is dead. Remember all of these 12 key ideas are continually under attack scientists are trying to find pilot waves and find other things between QM and the classic physics but these have withstood all tests so far.

Last edited by Orac; 05/31/14 04:03 AM.

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Bill S. Offline OP
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Struggling, but getting there slowly.

5. What are “off-diagonal elements”? I tried to get to grips with off-diagonal long range order, when I was looking at the work of Mazur and Chapline, but got no further than realizing it had something to do with the organization of particles in a quantum state.

6. My understanding of the Hamiltonian is that it quantifies the total energy (kinetic and potential) of the particles in a system, but beyond that it is a mass of mathematical esoterica.

Quote:
everyone should calculate or review at least five order-of-magnitude estimates of the "critical" quantities where the classical limit becomes valid


I believe that everyone should do it, and would probably join in if I knew what that meant. smile

Quote:
in many cases, the quantum-classical transition can actually be observed (at the predicted place)


Could you give an example of such an observation, please.

7. I’ve no idea what the preferred basis vectors of the Hilbert space are, but this bit seems to be saying that in the classical world what we observe is our reality, and it’s not going to just vanish.

I think I get the bit about consistent histories, but the link between Schrödinger’s cat and the Hamiltonian eludes me.

8. I thought I was doing quite well with this one until I got to the “diagonalization of the density matrix in the preferred basis”.
Lost!


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Originally Posted By: Bill S.
5. What are “off-diagonal elements”? I tried to get to grips with off-diagonal long range order, when I was looking at the work of Mazur and Chapline, but got no further than realizing it had something to do with the organization of particles in a quantum state.

Ok above we defined classic physics as a state of QM so now we have to deal between pure and mixed states. Just like in real world we have mixture of ice, water and water vapour the states don't exist in isolation.

I am going to do a separate post in a bit on this because it's interesting lets just see how many bits I need to tidy up first.

For now what I would like you to look at is the example in the following link example Beam A prepared in superposition and Beam B organized 50% spin up and 50% spin down. See if you can follow why the result.

http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/DensityMatrix.htm

Originally Posted By: Bill S.
6. My understanding of the Hamiltonian is that it quantifies the total energy (kinetic and potential) of the particles in a system, but beyond that it is a mass of mathematical esoterica.


That is correct and the mathematics basically reflects observation. It wouldn't be valid to pose an answer gravity is only valid when you get more or less than X amount of matter or the universe is X. The mathematics of gravity observation and subsequent calculation does not indicate a limit and so it is not valid, the same is true of QM.

So it's is saying as QM is involved in energy it clearly goes from the smallest size to the largest size and over all time scales.
Quote:
in many cases, the quantum-classical transition can actually be observed (at the predicted place)

Could you give an example of such an observation, please.


Just google "macro quantum entanglement" there are now thousands of them. All these experiment are large macro objects you would clearly call classical yet you can make them behave in a manner that only QM predicts.

Originally Posted By: Bill S.
7. I’ve no idea what the preferred basis vectors of the Hilbert space are, but this bit seems to be saying that in the classical world what we observe is our reality, and it’s not going to just vanish.

I think I get the bit about consistent histories, but the link between Schrödinger’s cat and the Hamiltonian eludes me.

You almost got it spot on except add in that the world exists with or without you it's just in a quantum superposition. The Schrödinger’s cat reference is needed because of Many World Interpretation and for that I will refer you to this reading

http://plato.stanford.edu/entries/qm-manyworlds/

Originally Posted By: Bill S.
8. I thought I was doing quite well with this one until I got to the “diagonalization of the density matrix in the preferred basis”.
Lost!


Yeah that's back to point 5 and the density matrix and mixed states and we will cross that off in detail next if you like.

If you are happy with that we will try and get you understanding mixed and pure states and the density matrix it's not as hard as you may think.

Last edited by Orac; 06/03/14 04:32 AM.

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Bill S. Offline OP
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Quote:
If you are happy with that


That will need a bit of thought. smile

I'm going to have to spend a bit of time trying to work round the maths in the first link. The second looks like just what I need, especially as I am involved in a discussion about multiverse theories in another forum.

It may be a while before I can get to grips with this, but I'll be back.


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