Hilbert space in Everettian QM

[Posy McPostface]

Is it assumed that Hilbert space is an infinite manifold that the non-collapsing wave function occupies in Everettian QM?

Thank you.

 

[Posy McPostface]

Sorry, I think I phrased the question wrong. I was concerned with whether Hilbert space is infinite dimensional in Everettian QM? Don't know if that is different than the OP question, but at least more clear.

 

[Posy McPostface]

Not to self-bump, but if the assumption that the wavefunction occupies a state space in infinite dimensional Hilbert space, then doesn't that smell of metaphysics?

 

[Demystifier]

Wave function lives in an infinite dimensional Hilbert space in any QM, not just Everettian QM. The question (somewhat metaphysical if you like) is whether this wave function is ontic or epistemic.

 

[Gigaz]

The Hilbert space can have finite or infinite number of dimensions. I think most physicists would concede that any real Quantum system is probably limited to a finite (though excessively large) number of states because of the finite size of the universe and Planck limits.

 

[Posy McPostface]

So, my next question is that if Hilbert space is infinite dimensional in QM, does that contribute to quantum indeterminacy? Is it pertinent to quantum indeterminacy or just a confounding factor?

 

[Nugatory]

So, my next question is that if Hilbert space is infinite dimensional in QM, does that contribute to quantum indeterminacy?

The indeterminacy is inherent in the Born rule, which works similarly no matter how many or few dimensions are in the Hilbert space.

 

[cosmik debris]

because of the finite size of the universe

Can you cite the source for this?

Cheers

 

[bhobba]

The Hilbert space can have finite or infinite number of dimensions. I think most physicists would concede that any real Quantum system is probably limited to a finite (though excessively large) number of states because of the finite size of the universe and Planck limits.

Indeed.

That's one reason I like the Rigged Hilbert Space formulation.

From the start you recognize the states you work with are not physically releasable - its just introduced for mathematical convenience.

Thanks
Bill

 

[Posy McPostface]

That's one reason I like the Rigged Hilbert Space formulation.

From the start you recognize the states you work with are not physically releasable - its just introduced for mathematical convenience.

Doesn't that imply metaphysics or some variety of Platonism for or given the existence of the wavefunction in infinite dimensional Hilbert space?

 

[Nugatory]

Doesn't that imply metaphysics or some variety of Platonism for or given the existence of the wavefunction in infinite dimensional Hilbert space?

No. It's just math, and math routinely works with things that aren't physically realizable.

 

[bhobba]

Doesn't that imply metaphysics or some variety of Platonism for or given the existence of the wavefunction in infinite dimensional Hilbert space?

Physics is a mathematical model. In applied math in general, not just physics, some mathematical assumptions that often have zero impact on anything actually measurable are routinely made. This is just another case. As an example in modeling a hammer strike you often use a Dirac Delta function. Of course the actual strike isn't infinitely high and of zero duration - it's just very large and of such a short duration its fine for most purposes to model it that way.

Same in QM. You can think of the output of some observation as some kind of digital readout. It can be a very long string of numbers - but not actually infinite. So what you do is a mathematical trick - you take all the row vectors of finite length. These are the physically realizable outcomes. You then take what's called the dual and that's the space you work in in QM. The original space is called a test space. By doing that you can apply the methods of the calculus etc and chose whatever subset of that dual is most convenient. Actually its a bit trickier than that - you chose a subset with nice mathematical properties of that big dual as your test space and take the dual of that - the choice is purely done for technical mathematical convenience.

It is often said in physics its best to avoid words like reality etc in discussing physics. If pushed to it I define reality as what our theories describe. The above shows its more subtle than that glib remark - but the whole issue is a bit of a morass best avoided - philosophers argue about it all the time - the problem is they get nowhere but physics actually makes progress.

Thanks
Bill

 

[MrRobotoToo]

The Everett interpretation doesn't make any a priori assumptions about the dimensionality of the Hilbert space. The number of dimensions will depend on the specific model one is using to describe the universe. For example, if one supposes that we live in a de Sitter universe (and that the big bang is therefore just the most recent low entropy fluctuation), then chances are that the associated Hilbert space is finite dimensional, since a de Sitter universe has a finite maximum entropy (proportional to the surface area of the cosmic horizon).

 

[PhysicsExplorer]

My friend, who is a physicist, had this to say to me not long ago, when I confronted him on Hilbert space ~

''If the representation is to be unitary, it is infinite-dimensional by the compactness of Lorentz group. These representations exist. In this case sigma matrices Sigma_munu defined as their commutators would be hermitian matrices and the formula would make sense as quantum expectation value. It would represent the curvature tensor as quantum operator. The problem with infinite-D representations of Lorentz group is that they are not encountered in particle physics, where representations are finite-D (spinors, vectors, tensors) and these representations do not allow Hilbert spzce interpretation. Poincare invariance saves the situation: one can have instead of finite-D representations of Lorentz group representations of entire Poincare group by quantum fields associated with finite-D representations of Lorentz group and everything is OK. But the problem with GRT is of course that the Poincare invariance is lost.''and he further spoke about the subject''The infinite-D representation follows only the condition that Lorentz transformation are represented unitarily. This guarantees the curvature tensor is Hermitian operator. It is essential that one has only Lorentz invariance. If one has Poincare invariance, finite-D representations which indeed correspond to physical particles, are enough. And the problem is that GRT does not allow Poincare invariance except as approximate symmetry.''Which does have some interesting information in it.

 

[tom.stoer]

Doesn't that imply metaphysics or some variety of Platonism for or given the existence of the wavefunction in infinite dimensional Hilbert space?

It doesn't imply Platonism, but Everett's idea makes sense or is the result of a platonist view of physics.

For an instrumentalist there is no reason to worry about collapse or no collapse, what a collapse means or what the wave function "really describes"; all she cares is if the wave function can be used to calculate possible outcomes, probabilities and expectation values for measurements. For a Platonist - who understands the mathematical formalism as something that encodes reality behind mere observations, i.e. something that really exists out there, for somebody who thinks beyond empirism - the collapse contradicts the unitary time evolution and cannot be real in the same sense as this time evolution. Therefore she will avoid the collapse at all cost, including the acceptance of the reality of the branch structure of the wave function predicted by unitary quantum dynamics, especially due to decoherence.

Many proponents of the Everett interpretation seem to be Platonists; some do explain this in detail, e.g. Deutsch.

 

[vanhees71]

My friend, who is a physicist, had this to say to me not long ago, when I confronted him on Hilbert space ~

''If the representation is to be unitary, it is infinite-dimensional by the compactness of Lorentz group.[...]''

He meant "non-compactness" of the Lorentz group (I'd rather say Poincare group since the spatial translations also play a crucial role).

 

[Posy McPostface]

It doesn't imply Platonism, but Everett's idea makes sense or is the result of a platonist view of physics.

For an instrumentalist there is no reason to worry about collapse or no collapse, what a collapse means or what the wave function "really describes"; all she cares is if the wave function can be used to calculate possible outcomes, probabilities and expectation values for measurements. For a Platonist - who understands the mathematical formalism as something that encodes reality behind mere observations, i.e. something that really exists out there, for somebody who thinks beyond empirism - the collapse contradicts the unitary time evolution and cannot be real in the same sense as this time evolution. Therefore she will avoid the collapse at all cost, including the acceptance of the reality of the branch structure of the wave function predicted by unitary quantum dynamics, especially due to decoherence.

Many proponents of the Everett interpretation seem to be Platonists; some do explain this in detail, e.g. Deutsch.

I'm a big fan of Max Tegmark and Deutsch. I see the instrumentality of using mathematics in physics as, as close a proof of Platonism as one can get.

If we take something like the Church-Turing-Deutsch principle, then there's little way to prove it to be true ad hoc. Then, there are some hard limits that contribute to such a situation, like Godel's incompleteness theorem, which is just a personal opinion although I'm interested in what you think about the CTD principle and what kind of relation it may or may not have with Godel's incompleteness theorems.

 

[Posy McPostface]

Many proponents of the Everett interpretation seem to be Platonists; some do explain this in detail, e.g. Deutsch.

Could you please provide links to the reference you are alluding to in your post. Thank you.

 

[Posy McPostface]

Well, Ii went straight to the source and e-mailed David Deutsch and sent a Facebook message (actually, I think I'll just e-mail him too) to Max Tegmark regarding Godel and the CTD principle, and if the CTD principle is a measure of the truth of Platonism in the real world, and if there are any ways to prove that principle.

If anyone is interested I can share their responses here (with their consent).

Thanks.