Quantum Mechanics without Hilbert Space

[unusualname]

Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.
http://www.bbk.ac.uk/tpru/BasilHiley/Algebraic Quantum Mechanic 5.pdf

...These results have only intensified my curiosity as to why most if not all of the results
can be obtained without seemingly the need to resort to Hilbert space. This goes against
the prevailing orthodoxy that appears to insist that quantum mechanics cannot be done
except in the context of a Hilbert space. Yet there have been other voices raised against
the necessity of Hilbert space. Von Neumann himself wrote to Birkoff (1966) writing "I
would like to make a confession which may seem immoral: I do not believe absolutely in
Hilbert space any more." (A detailed discussion of why von Neumann made this
comment can be found in Rédei 1996).
But there are more important reasons why an algebraic approach has advantages. As
Dirac (1965) has stressed, when algebraic methods are used for systems with an infinite
number of degrees of freedom...
...We have shown how an approach to quantum mechanics can be built from the algebraic
structure of the Clifford algebra and the discrete Weyl algebra (or the generalised
Clifford algebra). These algebras can be treated by the same techniques that do not
require Hilbert space yet enable us to calculating mean values required in quantum
mechanics...------------
Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles.
http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4031v1.pdf.

yeah, I think they were worried about the "completeness" part of Hlibert Space not being a fundamental requirement (to describe reality), they'll probably be proved right soon.

 

[Chopin]

I guess I'm curious why folks seem to be so set on finding a way to do QM without a Hilbert space in the first place. Is there something distasteful about using it? Even if it is possible to do, whatever other mechanism you come up with is going to have to reproduce many of the features of a Hilbert space, so it's not like you can get around understanding how it works. The reason we use it is that the main feature of quantum mechanics--superposition--is handled very naturally by the linear algebra of a Hilbert space.

So if your motivation in this is that you saw some examples that dealt with state vectors, linear operators, or bra-ket notation and decided that you didn't want to learn all of that stuff, then you're going to be disappointed. There aren't any shortcuts on this one--you've got to learn how all that works if you want to have any kind of grasp on quantum mechanics. Understanding why a Hilbert space is so useful for describing the features of QM is a necessary step before you're going to have any hope of finding another way of doing it.

 

[Varon]

I guess I'm curious why folks seem to be so set on finding a way to do QM without a Hilbert space in the first place. Is there something distasteful about using it? Even if it is possible to do, whatever other mechanism you come up with is going to have to reproduce many of the features of a Hilbert space, so it's not like you can get around understanding how it works. The reason we use it is that the main feature of quantum mechanics--superposition--is handled very naturally by the linear algebra of a Hilbert space.

So if your motivation in this is that you saw some examples that dealt with state vectors, linear operators, or bra-ket notation and decided that you didn't want to learn all of that stuff, then you're going to be disappointed. There aren't any shortcuts on this one--you've got to learn how all that works if you want to have any kind of grasp on quantum mechanics. Understanding why a Hilbert space is so useful for describing the features of QM is a necessary step before you're going to have any hope of finding another way of doing it.

What if one learns QM by starting with Hilbert Space and then latter the plain Fourier based Wave function like depicted in the site:
http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html

I'm more familiar with Hilbert space having started understanding QM comtemplating on the double slit experiment and Schrodinger Cat in superposition.. so I know about pure state, mixed state, decoherence.

I won't go into full details of QM as I'm not a physics student. I just need a basic mathematical grasp to understand the different interpretations as the reason we novices get into QM is because of the weirdness in it.. lol... so Hilbert space is sufficient for us.. right?

 

[Varon]

I mean I just want to use vector space. Most course in QM started with the particle in a box and hamiltonian and energy and all that and it takes a long time. Since I just want to focus on Schroedinger cat.. I don't need to use energy.. but just position.. so I wonder if I can just focus on Hilbert space or vector space and later go to Hamiltonian of energy of electrons in atoms.. which I don't really need or not urgent right now..

 

[Varon]

But I have read the best novice book on particle physics. Deep Down Things. It summarizes what is the Schroedinger Equation:

"Thus, the Schroedinger Equation is just the wave-mechanical statement that the sum of the kinetic and potential energies at any given point is just equal to the total energy - the Schroedinger equation is simply the quantum-mechanical version of the notion of energy conservation."

Is the "energy" bias is because particle physics is the application? I can't find the thread where I read this (if anyone know the thread title where this argument is given that the "energy' is due to the some kind of bias, pls. let me know). The book never mentions about Hilbert Space or about the cat or double slit. It goes on to talking about Quantum Field theory and Gauge theory. So it seems the Schroedinger equation has two major usages. In particle physics to determine the kinetic energy, potential, etc. of particles. and in general case, to determine superpositions like in double slit. But maybe one can give a general statement of what is the Schrodinger equation aside from merely telling it is "simply the quantum-mechanical version of the notion of energy conservation"

 

[Fredrik]

The most explicit connection between the Schrödinger equation and energy conservation is that in wave mechanics (the quantum theory of a single spin-0 particle in Galilean spacetime), "momentum" and "energy" are represented by the operators

-i\nabla,\quad i\frac{d}{dt}

If you just take the classical relation E=p²/2m, and make the substitutions

p\rightarrow-i\nabla,\quad E\rightarrow i\frac{\partial}{\partial t}

what you get is the Schrödinger equation (without a potential). (I believe that this is what your book is referring to). I don't like to emphasize this too much, because this substitution trick doesn't work in special relativistic QM, not even for single particle theories. (You get a useful equation, but the function it acts on can't be interpreted as a wavefunction).

A more sophisticated approach (which works with both non-relativistic and special relativistic QM) is to start by noting that there must exist a time evolution operation U(t) (for each t) that takes a state vector to the state vector that represents the state of the system time t later, and it must satisfy U(t+s)=U(t)U(s). It's possible to show that this condition implies that there exists a self-adjoint operator H such that U(t)=e^{-iHt}. This H is called the Hamiltonian. (This should be taken as the definition of H). The above implies U'(t)=-iHU(t), or equivalently,

i\frac{dU(t)}{dt}=HU(t)

This is another version of the Schrödinger equation. If you define \psi(x,t)=U(t)\psi(x,0) (this is only done in non-relativistic QM), and have the operators in the equation above act on \psi(x,0), you get the more familiar version

i\frac{\partial\psi(x,t)}{\partial t}=H\psi(x,t)

You asked for a general statement of what the Schrödinger equation is. I would just say that it's the equation that tells us how state vectors change with time. So in addition to being the QM counterpart of E=p²/2m, it's also the QM counterpart of F=ma. (Newton's 2nd is what tells us how states change with time in classical mechanics).

 

[Varon]

The most explicit connection between the Schrödinger equation and energy conservation is that in wave mechanics (the quantum theory of a single spin-0 particle in Galilean spacetime), "momentum" and "energy" are represented by the operators

-i\nabla,\quad i\frac{d}{dt}

If you just take the classical relation E=p²/2m, and make the substitutions

p\rightarrow-i\nabla,\quad E\rightarrow i\frac{\partial}{\partial t}

what you get is the Schrödinger equation (without a potential). (I believe that this is what your book is referring to). I don't like to emphasize this too much, because this substitution trick doesn't work in special relativistic QM, not even for single particle theories. (You get a useful equation, but the function it acts on can't be interpreted as a wavefunction).

A more sophisticated approach (which works with both non-relativistic and special relativistic QM) is to start by noting that there must exist a time evolution operation U(t) (for each t) that takes a state vector to the state vector that represents the state of the system time t later, and it must satisfy U(t+s)=U(t)U(s). It's possible to show that this condition implies that there exists a self-adjoint operator H such that U(t)=e^{-iHt}. This H is called the Hamiltonian. (This should be taken as the definition of H). The above implies U'(t)=-iHU(t), or equivalently,

i\frac{dU(t)}{dt}=HU(t)

This is another version of the Schrödinger equation. If you define \psi(x,t)=U(t)\psi(x,0) (this is only done in non-relativistic QM), and have the operators in the equation above act on \psi(x,0), you get the more familiar version

i\frac{\partial\psi(x,t)}{\partial t}=H\psi(x,t)

You asked for a general statement of what the Schrödinger equation is. I would just say that it's the equation that tells us how state vectors change with time. So in addition to being the QM counterpart of E=p²/2m, it's also the QM counterpart of F=ma. (Newton's 2nd is what tells us how states change with time in classical mechanics).

You mentioned "state vector". So your formulation is in Hilbert Space format? But almost all introductory QM doesn't use any Hilbert Space. See:

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html

Without state vector. What is the equivalent of your formulation in the same mathematical language as the web site? Many thanks.

 

[Chopin]

You mentioned "state vector". So your formulation is in Hilbert Space format? But almost all introductory QM doesn't use any Hilbert Space. See:

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html

Without state vector. What is the equivalent of your formulation in the same mathematical language as the web site? Many thanks.

\Psi is a state vector. It's a state in a Hilbert space, they're just going to lengths to avoid calling it that. When we specify the state of a system with a wavefunction, what we're saying is that the system is in a superposition of position states--specifically, that the probability that the particle is at x_0 is |\Psi(x_0)|^2, the probability that it's at x_1 is |\Psi(x_1)|^2, etc.

We can cast this into bra-ket notation (the language of the Hilbert space) by denoting |x\rangle as the state with position x. If we do so, then our particle's state is simply \Psi(x_0)|x_0\rangle + \Psi(x_1)|x_1\rangle + .... Or, put more simply, \int{\Psi(x) |x\rangle dx}. So in this case, wavefunctions and Hilbert spaces are exactly equivalent.

If you're looking at a text that doesn't talk about Hilbert spaces, they're just hiding the concept inside other terms. For instance, the site you linked to talks about building up a wavefunction out of multiple basis states, called \Psi_1, \Psi_2, \Psi_3, .... These are exactly the same thing as state vectors--you could just as easily call them |a_1\rangle, |a_2\rangle, |a_3\rangle, .... The math is the same.

Hilbert spaces are just more general than the wavefunction--they allow you to talk about discrete-valued operators like spin and polarization, they let you talk about situations with multiple species of particle, and they let you talk about situations where the number of particles changes (for instance, pair production.) The simple wavefunction can do none of these things. If you move on to advanced quantum mechanics or quantum field theory, you will find no mention of the wavefunction anymore--it's all bras and kets. For instance, in quantum field theory, the equation for determining how an initial state transforms into a final state is given by:

\langle\phi_{final}|T(e^{i\int{H(t) dt}})|\phi_{initial}\rangle

There's no \Psi--only bras and kets. So if you really want to understand quantum mechanics at anything past a basic level, you're going to have to learn how to do this stuff. The sites you mentioned are doing basic enough stuff that they don't need to get into it, but if you understand all of that, then learning this should be your next step.

 

[Fredrik]

I wouldn't say that the introductory texts aren't using Hilbert spaces. They kind of are. What they're doing is very close to introducing a specific Hilbert space, instead of stating the general definition of a Hilbert space and leaving the details of what Hilbert space we're dealing with unspecified. I'm saying "very close to", because they're leaving out one important detail, which will be mentioned below.

They're considering functions from \mathbb R^3 into \mathbb C that are square-integrable, i.e. if f is such a function, then

\int |f(x)|^2 d^3x

is a real number. The set of such functions is given the structure of a vector space by the "obvious" definitions of addition and scalar multiplication:

(f+g)(x)=f(x)+g(x)

(af)(x)=a(f(x))

So these functions are vectors, in the sense "members of a vector space". We can define a semi-inner product and a semi-norm by

\langle f,g\rangle =\int f(x)^*g(x) d^3x

\|f\|^2=\langle f,f\rangle

The former fails to satisfy the definition of an inner product, and the latter fails to satisfy the definition of a norm, because this condition isn't satisfied:

\langle f,f\rangle=0\ \Rightarrow f=0

For example, if we define f(x)=1 when x=(x_1,x_2,x_3) is a triple of rational numbers, and f(x)=0 otherwise, then \|f\|=0.

The detail I had in mind when I said that there's one thing the introductory books are leaving out is the trick that gives us a Hilbert space:

We define an equivalence relation on this set of functions, by saying that f and g are equivalent if and only if the set where they take different values has Lebesgue measure 0. (Roughly speaking, this means that it has a well-defined size, and that that size is 0). In particular, since the set of numbers with rational coordinates has Lebesgue measure 0, the function that has the value 1 on that set and 0 everywhere else, belongs to the same equivalence class as the constant function that has the value 0 everywhere.

The set of such equivalence classes is denoted by L^2(\mathbb R^3). Let's denote the equivalence class that f belongs to by [f], and define

a[f]+b[g]=[af+bg]

\langle [f],[g]\rangle=\langle f,g\rangle

\|[f]\|=\|f\|

Now we have a space with an actual inner product, and an actual norm, and it happens to be one of the simplest examples of an infinite-dimensional Hilbert space.

So the space that the introductory books are working with is a semi-inner product space that can be turned into a Hilbert space quite easily. The reason why they can work with this semi-inner product space instead of the Hilbert space is a) that many results that hold for Hilbert spaces hold for semi-inner product spaces too (e.g. the Cauchy-Bunyakovsky-Schwarz inequality), and b) that they're not going to do things rigorously anyway. These things make the differences between these two spaces pretty much irrelevant.

 

[Varon]

Thanks for the valuable input that many introductory quantum text without Hilbert space doesn't even explain the connection.

Anyway. There is an issue of primary importance. Many said that the state vector and collapse is just our knowledge of the system. But in the double slit, the detector indeed collapse the wave function even if you don't do calculations. So collapse exists independently from your knowledge. What do you say about this?

 

[Fredrik]

Many said that the state vector and collapse is just our knowledge of the system. But in the double slit, the detector indeed collapse the wave function even if you don't do calculations. So collapse exists independently from your knowledge. What do you say about this?

I have never liked the claim that a wavefunction represents the observer's knowledge of the system. What does that even mean? I think that this is just a less accurate way of saying that a wavefunction is something mathematical that's used to calculate probabilities of possible results of experiments. If that's what it's supposed to mean, then why not just say it that way? There really is no need to dumb it down to a statement the meaning of which is unclear.

This could mean that people who use that phrase really mean something else, but I don't see how it can mean something different unless it comes with a definition of the term "knowledge".

Some of my earlier posts about the topics you brought up ("collapse" and "observers"): 1, 2, 3.

 

[Varon]

I have never liked the claim that a wavefunction represents the observer's knowledge of the system. What does that even mean? I think that this is just a less accurate way of saying that a wavefunction is something mathematical that's used to calculate probabilities of possible results of experiments. If that's what it's supposed to mean, then why not just say it that way? There really is no need to dumb it down to a statement the meaning of which is unclear.

This could mean that people who use that phrase really mean something else, but I don't see how it can mean something different unless it comes with a definition of the term "knowledge".

Some of my earlier posts about the topics you brought up ("collapse" and "observers"): 1, 2, 3.

So if they meant a wavefunction is something mathematical that's used to calculate probabilities of possible results of experiments. So does the electron have classical trajectory to either left or right slit and we just don't know what it is? Or doesn't it have trajectory? Or in Copenhagen, do they just try to suppress this fact by pretending the electron and slits are not really there physically?

What bothers me is decoherence. Since collapse (as most commonly believed) is supposed to be only our knowledge of the system during observation. But we never observe decoherence.. so in your view.. is decoherence just purely mathematical? I want to have a physical picture of decoherence in Copenhagen. Can I imaginine that when I walk in the street.. billions of particles in the street are physically entangling with my body. Can I imagine waves interfering with my body waves (and it actually occurring). But since the wave function is supposed to be just knowledge of the observer and not ontological. Then in Copenhagen view. There isn't any actual waves interfering in decoherence but just a math trick? But could it be merely math when my body atoms really entangled with the environment. This bothers me for many months and probably years now. Hope we can settle this. What do you think? Let's just focus on the Copenhagen for now as I just want to understand how this is understood by mainstream physicists (not what we'd like it to be and how much some of us dislike it but how it is understood by others. Because it is by knowing how it is commonly understood that we can know what is really wrong. Let's also avoid Many worlds for this discussion as Many worlds is simply an easy way out and if this world is only one world. Many worlds is escaping from reality).

 

[Fredrik]

So if they meant a wavefunction is something mathematical that's used to calculate probabilities of possible results of experiments. So does the electron have classical trajectory to either left or right slit and we just don't know what it is? Or doesn't it have trajectory?

I would say that it's the latter, but for the last few weeks, I've been really baffled by the fact that Ballentine's 1970 article on the statistical interpretation suggests that particles with wavefunctions that aren't sharply peaked can have definite but unknown positions. I'm going to have to take the time to read that article one of these days, to find out if I have misunderstood something fundamental. (This suggestion seems completely wrong to me, so I really want to know).

What bothers me is decoherence. Since collapse (as most commonly believed) is supposed to be only our knowledge of the system during observation. But we never observe decoherence.. so in your view.. is decoherence just purely mathematical?

In a double-slit experiment with C70 molecules moving through air, the interactions between the molecules and the air destroy more of the interference pattern the higher the air pressure is. This is decoherence in action, so it's definitely not purely mathematical.

I want to have a physical picture of decoherence in Copenhagen. Can I imaginine that when I walk in the street.. billions of particles in the street are physically entangling with my body. Can I imagine waves interfering with my body waves (and it actually occurring).

Yes. The surrounding air is going to have a much larger effect on you than on a single C70 molecule.

But since the wave function is supposed to be just knowledge of the observer and not ontological. Then in Copenhagen view. There isn't any actual waves interfering in decoherence but just a math trick? But could it be merely math when my body atoms really entangled with the environment. This bothers me for many months and probably years now. Hope we can settle this. What do you think?

This isn't something we can settle. It's conceivable that QM assigns probabilities to possible results of experiments with fantastic accuracy without giving us a detailed description of what's "actually happening" to the system at all times. It's also conceivable that the reason why those probability assignments are so accurate is that there is a description of what's "actually happening" somewhere in QM, that we just don't understand.

 

[Varon]

I would say that it's the latter, but for the last few weeks, I've been really baffled by the fact that Ballentine's 1970 article on the statistical interpretation suggests that particles with wavefunctions that aren't sharply peaked can have definite but unknown positions. I'm going to have to take the time to read that article one of these days, to find out if I have misunderstood something fundamental. (This suggestion seems completely wrong to me, so I really want to know).


In a double-slit experiment with C70 molecules moving through air, the interactions between the molecules and the air destroy more of the interference pattern the higher the air pressure is. This is decoherence in action, so it's definitely not purely mathematical.



Yes. The surrounding air is going to have a much larger effect on you than on a single C70 molecule.


This isn't something we can settle. It's conceivable that QM assigns probabilities to possible results of experiments with fantastic accuracy without giving us a detailed description of what's "actually happening" to the system at all times. It's also conceivable that the reason why those probability assignments are so accurate is that there is a description of what's "actually happening" somewhere in QM, that we just don't understand.

Using all your knowledge of Hilbert space and QM. Please let me know if the following scenerio is possible. Got it from Peter Ryser article:

"Everett considers the many worlds as real, in an ontological sense. However, it is not necessary to adopt this assumption. Following Squires (1991) I will consider the many worlds as possibilities or, as Popper (1977) described it, as ‘propensities’. I will assume that a single universal mind experiences only one of the many possible realities. In terms of the Copenhagen Interpretation this would mean: A single universal mind collapses the universal wave-function. In this picture there is no local wave-function collapse and no artificial distinction between classical and quantum systems. There is only the universal wave function and a universal mind that moves along one of the many branches of this function. I will also assume that the universal mind can, to a certain degree, ‘choose’ which branch is realized."

~~~~~~~~~~~~~~~~~~~

How much should you alter QM mathematics and conceptual foundations to make this scenerio possible where the other Everett branches are not real worlds but just possibilities. And if it was not chosen by the single universal mind, the branches cease to exist. Only the branch chosen becomes real. This scenerio differs from standard Many Worlds where all the worlds are real. What kind of alteration must you do to the mathematics of QM to make this only one branch becoming real possible? You may say this is weird.. well.. standard Collapse theory is just as weird, and Many worlds all real is equally weird (or strange) too. So we must not reject any theory on the basis of weirdness. Let's be open to all possibilities.

 

[Fredrik]

Using all your knowledge of Hilbert space and QM. Please let me know if the following scenerio is possible. Got it from Peter Ryser article:

"Everett considers the many worlds as real, in an ontological sense. However, it is not necessary to adopt this assumption. Following Squires (1991) I will consider the many worlds as possibilities or, as Popper (1977) described it, as ‘propensities’. I will assume that a single universal mind experiences only one of the many possible realities. In terms of the Copenhagen Interpretation this would mean: A single universal mind collapses the universal wave-function. In this picture there is no local wave-function collapse and no artificial distinction between classical and quantum systems. There is only the universal wave function and a universal mind that moves along one of the many branches of this function. I will also assume that the universal mind can, to a certain degree, ‘choose’ which branch is realized."

Is this "single universal mind" named YHWH by any chance? This is a science forum, not a religion forum.

If it's observing at all times, then no system would be isolated from its environment and QM would fail completely. There wouldn't be any superpositions at all.

What kind of alteration must you do to the mathematics of QM to make this only one branch becoming real possible?

I don't see any other way than to just add what you just said as an additional assumption on top of QM.

You may say this is weird.. well.. standard Collapse theory is just as weird, and Many worlds all real is equally weird (or strange) too. So we must not reject any theory on the basis of weirdness. Let's be open to all possibilities.

The problem isn't that it's weird. It's that the assumption is completely unjustified and doesn't change any of the theory's predictions. It's like adding an invisible blue giraffe that doesn't interact with matter to the axioms of special relativity.

I don't know if there is such a thing as "standard collapse theory". I assume that this would describe how "collapse" is a physical process. I'm not familiar with anything like that.

 

[Fredrik]

No. According to Ryser. "Individual minds can only influence the indeterminacy that
has its origin in their brains while the indeterminacy of the environment belongs to the realm of the universal mind." Hence the realm of superpositions don't belong to the universal mind but to individual mind (but with zero probability of effects occurring outside the brain lest we can control superpositions).

This sounds like crackpot stuff. Do you have a reference to a peer-reviewed physics journal? If he hasn't been able to publish, it doesn't seem worthy of any deeper analysis.

We need a radical idea.

I don't think we do. We may just have to lower our expectations about what a good theory can tell us.

Any idea that's good enough to improve on the current situation would be a new theory, not an interpretation of QM.

Copenhagen is already getting outdated as you agreed.

Did I agree to that? Maybe you just confused me with Fra (the guy who signs his posts /Fredrik). My view on "the Copenhagen interpretation" is that the term is useless, because there's no standard definition of the term. (There isn't even a standard view of what an interpretation is). And I think that the idea that QM is just a set of rules that assigns probabilities to possible results of experiments is a "Copenhagenish" interpretation. It includes most of the ideas that people tend to slap the Copenhagen label on. The main detail that's left out is the idea that a wavefunction is a complete description of all the features of the system, but I don't know if Niels Bohr really held that view.

 

[Fredrik]

I still don't think that the article is worth the time it would take to read it.

 

[Varon]

I still don't think that the article is worth the time it would take to read it.

Ok let's just ignore it then. In Copenhagen. There is collapse. In Many worlds there is none. By putting Collapse back to Many worlds. It's redundant. So I guess "Copenhagen Many Worlds' interpretation is thus refuted. So at the end of the day. There is a million Fredriks after all. I hope none of my billion other copies have shot Obama because I sometimes dreamt of it and uncomfortable thinking my other copy has done it.

 

[Varon]

I would say that it's the latter, but for the last few weeks, I've been really baffled by the fact that Ballentine's 1970 article on the statistical interpretation suggests that particles with wavefunctions that aren't sharply peaked can have definite but unknown positions. I'm going to have to take the time to read that article one of these days, to find out if I have misunderstood something fundamental. (This suggestion seems completely wrong to me, so I really want to know).

What are you baffled with this. It's written in 1970 so maybe outdated already and refuted?

http://www.kevinaylward.co.uk/qm/ballentine_ensemble_interpretation_1970.pdf

Bohr proposed that in the absence of measurement to determine its position, the electron has no position. It probably exists as ghostly mist in Hilbert Space with no definite basis. In the ensemble interpretation (which I presume is identical with the statistical interpretation). It's like the electrons are brownian motion of gas? But why did they have wave characteristic. Can't we even refute such differences using experiments?

Anyway. Here's how someone refutes it as wiki:

http://en.wikipedia.org/wiki/Ensemble_interpretation

"Criticism

Arnold Neumaier finds fault with the applicability of the ensemble interpretation to small systems.


"Among the traditional interpretations, the statistical interpretation discussed by Ballentine in Rev. Mod. Phys. 42, 358-381 (1970) is the least demanding (assumes less than the Copenhagen interpretation and the Many Worlds interpretation) and the most consistent one. It explains almost everything, and only has the disadvantage that it explicitly excludes the applicability of QM to single systems or very small ensembles (such as the few solar neutrinos or top quarks actually detected so far), and does not bridge the gulf between the classical domain (for the description of detectors) and the quantum domain (for the description of the microscopic system)". (spelling amended) [5]"

In a double-slit experiment with C70 molecules moving through air, the interactions between the molecules and the air destroy more of the interference pattern the higher the air pressure is. This is decoherence in action, so it's definitely not purely mathematical.



Yes. The surrounding air is going to have a much larger effect on you than on a single C70 molecule.


This isn't something we can settle. It's conceivable that QM assigns probabilities to possible results of experiments with fantastic accuracy without giving us a detailed description of what's "actually happening" to the system at all times. It's also conceivable that the reason why those probability assignments are so accurate is that there is a description of what's "actually happening" somewhere in QM, that we just don't understand.

 

[yoda jedi]

I have never liked the claim that a wavefunction represents the observer's knowledge of the system. What does that even mean?

that the reality is more than what we see.


.

 

[strangerep]

[Ballentine, Statistical Interpretation was] written in 1970 so maybe outdated already and refuted?

No.

BTW, it's not a good look for anyone to pass assessments (positive or negative)
on material they haven't studied properly. :-)

 

[Varon]

No.

BTW, it's not a good look for anyone to pass assessments (positive or negative)
on material they haven't studied properly. :-)

That is why there is a question mark.

So in the Statistical Interpretation, the electron has position and trajectory at all times? How does this differ to Bohmian Mechanics. How come the latter has to propose a separate real wave function and quantum potential to push the particle while in the Statistical Interpretation )SI), these two extra ingredients are not necessary? Hope experts in the SI can share how Bohmian is identical to SI and how SI differs to de Broglie/Bohm Mechanics. Thanks.

 

[strangerep]

Hope experts in the SI can share how Bohmian is identical to SI and how SI differs to de Broglie/Bohm Mechanics.

They are not at all identical -- as anyone who has actually studied them would know.
But there's no need for me (or anyone else) to write a tutorial on SI here,
since Ballentine has already written a good paper and a good textbook.

But I'm happy to discuss specific points in either his paper or the textbook,
with anyone who has conscientiously studied them.

 

[Fredrik]

But I'm happy to discuss specific points in either his paper or the textbook,
with anyone who has conscientiously studied them.

I started to read the article a couple of days ago. I have only read a few pages a day, so I haven't made it to the end yet. There's definitely a lot of good stuff in there. For example, the discussion of the uncertainty relations is the best I've seen. But there are a few specific points that I disagree with. If I still think he's wrong about those things when I get to the end of the article, I will start a thread about it.

 

[Varon]

They are not at all identical -- as anyone who has actually studied them would know.
But there's no need for me (or anyone else) to write a tutorial on SI here,
since Ballentine has already written a good paper and a good textbook.

But I'm happy to discuss specific points in either his paper or the textbook,
with anyone who has conscientiously studied them.

I have read the paper although not 100% because I don't understand all the math. We who delve in Many worlds and Copenhagen or even Bohmian would love any interpretation that has promise of a resolution of the measurement problem with least assumptions. So the Statistical Interpretation would be promising if it were true. But it didn't appear to be. Although Bohmian can describe individual system where the particle is push by pilot wave or quantum potential by a real wave function located in some configuration space. Statistical interpretation is only valid for ensemble of similarly prepared experiments. Now here is where its weakness lies according to this site:

http://implications-of-quantum-physics.com/qp24_ensemble-interpretation.html

"24. The Ensemble or Statistical Interpretation.

Summary

The ensemble or statistical interpretation is unsatisfactory because it is vague and does not take advantage of all we know about quantum physics.

There are interpretations (championed by Einstein) in which it is assumed that quantum physics gives only statistical information. It is assumed that there is a collection, or ensemble, of copies of the physical system and our perceived world corresponds to only one of them. The wave function then gives statistical information about which one of these copies corresponds to our actual world.

But such interpretations do not say what the actual world is ‘made of.’
And they do not explain why the copies change in time in a way that is consistent with the changes in the wave function. That is, the dynamics of the actual copies of the physical world are not specified. In my opinion, these schemes are not well-formulated enough to say whether or not they constitute a valid interpretation."

Also note in http://en.wikipedia.org/wiki/Bell_test_experiments that bell test experiments only started in 1972 (2 years after the article was written) so it didn't take into account that Bell's Thorem is violated categorically especially in light of Alain Aspect more rigorous experiment. So Statistical Interpretation is not designed to totally explain the correlation of Bell's theorem. It only mentions at the end of the 1970 Ballentine paper that it mentioned that it departed from the formalism of quantum theory but there was no subsequent updated work that would make it explain the correlation of Bell's Theorem.

I also read elsewhere about Einstein similar idea of Statistical interpretation which he presented at the 1927 Solvay Congress but he and many didn't push thru with it because it couldn't describe individual system or even a single atom. That is.. it couldn't explain a single atom electronic behavior hence Einstein didn't completely support it.

So it is more likely that the Statistical Interpretation is not representative of reality at all. Or it is very incomplete. Or if it could be model of reality, Neumaier approach may extend where it left off... that is if Neumaier was right. But Neuameir stated that the 430 atom buckyball simply vanish after it reach the detector or become smeared as wave... this doesn't seem to make sense.

 

[StevieTNZ]

This sounds like crackpot stuff. Do you have a reference to a peer-reviewed physics journal? If he hasn't been able to publish, it doesn't seem worthy of any deeper analysis.

Try http://www.neuroquantology.com/journal/index.php/nq/index - it says its peer-reviewed, and its also a journal on physics.

 

[Fredrik]

So the Statistical Interpretation would be promising if it were true. But it didn't appear to be.

What makes you say that? The statistical interpretation is really just quantum mechanics with no additional assumptions. (I've read enough of the article now to see that he doesn't actually assume that every particle has a well-defined position and momentum at all times. He just claims that this wouldn't contradict QM. Maybe he's right, maybe he's wrong. Either way, it doesn't affect the statistical interpretation, since that claim isn't a part of its definition). So how could the statistical interpretation be wrong? Is your point that QM is wrong?

Now here is where its weakness lies according to this site:

You should probably select your sources more carefully. It's actually against the forum rules to post links to questionable material. This guy rejects just about everything except the idea that a mind isn't physical. (It gets funny when he rejects particles because of "no evidence", but doesn't require evidence for his claim about minds).

But such interpretations do not say what the actual world is ‘made of.’

So? What makes him think that there's a version of QM that can tell us that?

Also note in http://en.wikipedia.org/wiki/Bell_test_experiments that bell test experiments only started in 1972 (2 years after the article was written) so it didn't take into account that Bell's Thorem is violated categorically especially in light of Alain Aspect more rigorous experiment.

He was certainly aware that QM violates Bell inequalities. The experiments only confirmed that reality does too. So the experiments weren't that relevant.

So it is more likely that the Statistical Interpretation is not representative of reality at all. Or it is very incomplete.

Right, but if that's true, then the same can be said about QM itself (since the statistical interpretation is just QM without additional assumptions), so that problem can't be solved by another interpretation.

 

[strangerep]

I have read the [Ballentine] paper although not 100% because I don't understand all the math. [...]

Fredrick has already responded appropriately to most of your post.
I'll just say two things...

(1) Try to acquire more understanding of the math so you can understand such
papers completely. Worry a bit less about intepretation until you've done so.

(2) Regarding the following quote:

Also note in http://en.wikipedia.org/wiki/Bell_test_experiments that bell test experiments only started in 1972 (2 years after the article was written) so it didn't take into account that Bell's Thorem is violated categorically especially in light of Alain Aspect more rigorous experiment. So Statistical Interpretation is not designed to totally explain the correlation of Bell's theorem. It only mentions at the end of the 1970 Ballentine paper that it mentioned that it departed from the formalism of quantum theory but there was no subsequent updated work that would make it explain the correlation of Bell's Theorem.

One need only read Ballentine's much more recent textbook to see that this is rubbish.

 

[Fra]

I didn't follow the cascade of threads about interpretations lately, but here is one comment to Varon, connecting loosely to some previously threades before the cascade.

To speak for myself, as long as we talk about the small subsystem abstraction, and complex observer - ie the cases where the statistical ensemble can in fact the realized in some principal sense, then then ensemble view is pretty close to my view. I think it's good.

But IMHO, the obvious problem is that the, the general case (ie. think cosmological models, or intrinsic measurement theory where the observer is EMBEDDED IN the much more complex system it's trying to observe), then this abstraction fails.

What I seek is the generalization, where the statistical enterpretation corresponds to one limiting case.

In cosmological models, you switch from "statistics" of similarly prepared system, to "statistics" of many measurements perhaps of the same system... but what is lacking is a coherent abstracting on this that makes sense in the general case - in particular are we switching from descpritive views to a decition theoretic view where we do not have acces to the limiting cases of "perfectly known ensembles" to beeing forced to evaluate the action based upo incomplete but rational "counting of evidence"; without reference to ensembles (because they only exists in the limiting sense, and the limit isn't at hand in cosmologilca or inside views).

My motivation is taht we need to understand this general case to solve the open issues of unification/theory scaling and QG.

/Fredrik

 

[Varon]

What makes you say that? The statistical interpretation is really just quantum mechanics with no additional assumptions. (I've read enough of the article now to see that he doesn't actually assume that every particle has a well-defined position and momentum at all times. He just claims that this wouldn't contradict QM. Maybe he's right, maybe he's wrong. Either way, it doesn't affect the statistical interpretation, since that claim isn't a part of its definition). So how could the statistical interpretation be wrong? Is your point that QM is wrong?


You should probably select your sources more carefully. It's actually against the forum rules to post links to questionable material. This guy rejects just about everything except the idea that a mind isn't physical. (It gets funny when he rejects particles because of "no evidence", but doesn't require evidence for his claim about minds).

Oh. I just saw that single web page. I didn't see the main index or other parts so didn't know he is a crackhead.

So? What makes him think that there's a version of QM that can tell us that?


He was certainly aware that QM violates Bell inequalities. The experiments only confirmed that reality does too. So the experiments weren't that relevant.


Right, but if that's true, then the same can be said about QM itself (since the statistical interpretation is just QM without additional assumptions), so that problem can't be solved by another interpretation.

I thought in the Solvay congress in 1927. Einstein already saw the weakness in the Statistical Interpretation (I was reading a history of the debate). Maybe Ballentine's formulation is an update. Anyway. In a single 430 atom-buckyball at a time double slit experiment.. how does the single buckyball move from the emitter to the detector? Did the Statistical Interpretation answer this or doesn't it know? If it doesn't know. Then it is not an explanation. It just smears it into ensemble of 'don't knows', isn't it.

 

[Varon]

Here's Gary Bowman take on the Statistical Interpretation in his book "Essential Quantum Mechanics":

"Most texts adopt—implicitly or explicitly—the so-called Copenhagen interpretation
of quantum mechanics. Like other, conflicting interpretations, Copenhagen is an attempt to tell us what’s really “going on”—to inform us of the quantum world beyond the formalism itself.

By contrast, the statistical interpretation arguably is not an interpretation, but a broad framework that describes how quantum mechanics works in actual practice. You may worry that by learning quantum mechanics from this perspective, you’ll be at a disadvantage. On the contrary, the statistical interpretation provides an understanding one can have confidence in: because it’s “just” a framework, it remains compatible with other approaches, such as Copenhagen, while avoiding many conceptual puzzles
that arise within them.

But what is the statistical interpretation? In a sense, it simply amounts to the following edict: take seriously what quantum mechanics does tell us, and don’t take seriously what quantum mechanics doesn’t tell us."

~~~~~~~~~~~~~~~~~~~~~

I wonder if there are many variants of the Statistical Interpretations (like Many worlds or Copenhagen which have many variants) or there is only one SI which came from Ballentine. Can anyone confirm?

So Statistical Interpretation is simply a framework. Not really an interpretation. It is the pragmatist framework. This means it is compatible with all interpretations and if you are serious about knowing what made QM tick at the very heart, you still have to entertain other interpretations after accepting the Statistical I. Framework. Agree??

 

[strangerep]

[...] take seriously what quantum mechanics does tell us, and don’t take seriously what quantum mechanics doesn’t tell us. [...]

Your post was reasonable up to this point.

But then...

[...] if you are serious about knowing what made QM tick at the very heart, you still have to entertain other interpretations after accepting the Statistical I. Framework.

No, the opposite is true. You don't need to waste any brain energy at all on fictional stories that have become grafted onto the exterior of QM theory.

 

[Fredrik]

I thought in the Solvay congress in 1927. Einstein already saw the weakness in the Statistical Interpretation (I was reading a history of the debate). Maybe Ballentine's formulation is an update.

I don't think so. As far as I know, he only rejected the unnecessary and unjustified assumption that a wavefunction describes all features of a single system. Ballentine takes that rejection as the definition of the statistical interpretation.

In a single 430 atom-buckyball at a time double slit experiment.. how does the single buckyball move from the emitter to the detector? Did the Statistical Interpretation answer this or doesn't it know? If it doesn't know. Then it is not an explanation. It just smears it into ensemble of 'don't knows', isn't it.

That's right. (Close enough anyway). But why would you think that QM actually contains an explanation of that sort? This isn't implied by anything that's known for sure.

I agree with Strangerep's comments in the post above this one.

 

[Varon]

What makes you say that? The statistical interpretation is really just quantum mechanics with no additional assumptions. (I've read enough of the article now to see that he doesn't actually assume that every particle has a well-defined position and momentum at all times. He just claims that this wouldn't contradict QM. Maybe he's right, maybe he's wrong. Either way, it doesn't affect the statistical interpretation, since that claim isn't a part of its definition). So how could the statistical interpretation be wrong? Is your point that QM is wrong?

I found out Ballentine specifically claimed that a particle has a well defined position and momentum at all times! Here's the relevant quotes. Don't you agree with him? Why?

Page 8.

"This statement is often supported by one or both of the following arguments:

(i) A measurement of q causes an unpredictable and uncontrollable disturbance of p, and vice versa. [This was first proposed by Heisenberg (1927) and is widely repeated in textbooks].

(ii) The position and momentum of a particle do not even exist with simultaneously and pefectly well defined (though perhaps unknown) values (Bohm, 1951, p.100)"
<snip>
"Argument (ii) is easily seen to be unjustified"
<snip>
"Using de Broglie's relation between momentum and wavelength, p = h / wavelength, it is then asserted that a particle cannot have definite values of both position and momentum at any instant. But this conclusion rests on the almost literal identication of the particle with the wave packet (or what amounts to the same thing, the assumption that the wave function provides an exhaustive description of the properties of the particle)."
<snip>
"A consistent application of the Statistical Interpretation yields the correct conclusion that the division of the wavepacket yields the relative probabilities for transmission and reflection of particles. But there is no justification for assertion (ii)"

You should probably select your sources more carefully. It's actually against the forum rules to post links to questionable material. This guy rejects just about everything except the idea that a mind isn't physical. (It gets funny when he rejects particles because of "no evidence", but doesn't require evidence for his claim about minds).


So? What makes him think that there's a version of QM that can tell us that?


He was certainly aware that QM violates Bell inequalities. The experiments only confirmed that reality does too. So the experiments weren't that relevant.


Right, but if that's true, then the same can be said about QM itself (since the statistical interpretation is just QM without additional assumptions), so that problem can't be solved by another interpretation.

 

[Varon]

Ballentine paper assumes all particles have positions at all times. This means in Bell's Theorem. He indeed believed that the particles were connected with superluminal link? In Ballentine 1989 textbook which I studied, he mentioned:

Are the experiments conclusive?
If we accept the theoretical arguments that quantum mechanics is incompatible with locality, the next question is whether the experiments are adequate for ruling out locality. We have already seen that, strictly speaking, they are not, because of inefficiencies of the detectors and other instrumental problems. However, the fact that those photon pairs that are detected are correlated in the manner predicted by quantum theory is certainly strong evidence for the correctness of those predictions. Although it is possible to devise local models that would obey Bell’s inequality for ideal detectors, but which agree with quantum theory for the imperfect instruments presently available, such models seem rather contrived. This is especially true in view of the fact that the effect of the various systematic errors that experimentalists have studied is to reduce the coincidence detection rate. But quantum theory predicts a
coincidence rate that is greater than is permitted by Bell’s inequality.

Question. Anything wrong by assuming entangled particles exist at all times even 100 billion light years away and since Bell's Theorem is violated, they really are connected with superluminal link? This is the consequence of Ballentine's Statistical Interpretation.

Bohr arguments was the particles attributes like position didn't exist before measurements, so there was no non-local link because the particles wasn't there at all.

 

[Fredrik]

I found out Ballentine specifically claimed that a particle has a well defined position and momentum at all times!

In section 5, titled "Joint probability distributions", he says

...quantum theory is not inconsistent with the supposition that a particle has at any instant both a definite position and a definite momentum, although there is a widespread folklore to the contrary. Of course we are not compelled either to accept or reject this supposition, but it is of interest to explore it on a tentative basis.​

So he isn't really claiming that particles have well-defined positions and momenta at all times. He's just saying that they might have well-defined positions and momenta at all times. I'm not convinced that this is true. However, if he's wrong, it doesn't have any implications for the statistical interpretation, since this assumption isn't part of its definition.

 

[Varon]

In section 5, titled "Joint probability distributions", he says

...quantum theory is not inconsistent with the supposition that a particle has at any instant both a definite position and a definite momentum, although there is a widespread folklore to the contrary. Of course we are not compelled either to accept or reject this supposition, but it is of interest to explore it on a tentative basis.​

So he isn't really claiming that particles have well-defined positions and momenta at all times. He's just saying that they might have well-defined positions and momenta at all times. I'm not convinced that this is true. However, if he's wrong, it doesn't have any implications for the statistical interpretation, since this assumption isn't part of its definition.

maybe not the statistical interpretation but he mentioned that when using the capital Statistical Interpretation, it's his string of ideas. So maybe we must distinguish them in the future by using smaller or capital letters when mentioning this particular interpretation to denote general or Ballentine version.

 

[Fredrik]

I don't think it would be helpful to have two different definitions of "statistical interpretation" and "Statistical Interpretation".

I read section 1.3 again. It bothers me a lot that he says

In contrast, the Statistical Interpretation considers a particle to always be at some position in space, each position being realized with relative frequency |\psi(\vec r)|^2 in an ensemble of similarly prepared experiments.​

This bothered me just as much the first time I read it, but when I got to the part (section 5) where he says that it's not necessary to assume that particles always have well-defined positions and momenta, I sort of forgot how strong the statement in section 1.3 is.

What bothers me the most is that section 1.3 is supposed to be the one that defines the statistical interpretation. This means that the statistical interpretation does include the assumption that particles always have well-defined positions. I find this very strange. The main idea is to reject an unnecessary assumption, and then he goes and makes another unnecessary assumption!?

Edit: On the other hand, in section 1.2, he says

Although there are many shades of interpretation (Bunge, 1956), we wish to distinguish only two:

(I) The Statistical Interpretation, according to which a pure state (and hence also a general state) provides a description of certain similarly prepared systems, but need not provide a complete description of an individual system.

(II) Interpretations which assert that a pure state provides a complete and exhaustive description of an individual system (e.g. an electron).
​

So he's sort of contradicting himself. Section 1.3 says that the statistical interpretation includes the assumption that particles have well-defined positions at all times. Section 5 says that there's no need to assume that. And yet, section 1.2 talks about "the" statistical interpretation, as if there's only one. It would have made more sense to talk about "statistical interpretations".

 

[strangerep]

I read [Ballentine, SI of QM paper] section 1.3 again. It bothers me a lot that he says

In contrast, the Statistical Interpretation considers a particle to always be at some position in space, each position being realized with relative frequency |\psi(\vec r)|^2 in an ensemble of similarly prepared experiments.​

[...]

What bothers me the most is that section 1.3 is supposed to be the one that defines the statistical interpretation. This means that the statistical interpretation does include the assumption that particles always have well-defined positions.

No it doesn't. Ballentine said "some position", not "well-defined position".

Edit: On the other hand, in section 1.2, he says

Although there are many shades of interpretation (Bunge, 1956), we wish to distinguish only two:

(I) The Statistical Interpretation, according to which a pure state (and hence also a general state) provides a description of certain similarly prepared systems, but need not provide a complete description of an individual system.

(II) Interpretations which assert that a pure state provides a complete and exhaustive description of an individual system (e.g. an electron).
​

So he's sort of contradicting himself. Section 1.3 says that the statistical interpretation includes the assumption that particles have well-defined positions at all times.

He's not contradicting himself. That quote from sect 1.2 is about sharply distinguishing different classes of interpretations of QM. And as I said above, sect 1.3 doesn't say what you said it does.

 

[Jaisonsc]

Im May, 7, Chopin sad: In addition to using a Hilbert state to describe a particle at a position, you can use it to describe a particle with a certain charge, or a certain momentum, or even something more abstract like which slit a particle goes through in the double-slit experiment.

Somebody know's how is possible to do the idea that I put in italics types?
Some article ou so on...
Thanks!

 

[IRobot]

Could we say that in the path integral formalism we don't need a Hilbert space, at least to write down the functional integral, not to calculate with it.