How to talk about interpretations

[Sugdub]

Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0]. These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Now consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states.

This paragraph looks weird. Obviously a “pure state” qualifies a property owned by a single “system”and the “mixed state” qualifies a property of the “ensemble” of such “systems” that the global iterative experiment has involved.
First one postulates the existence of a “system” (why don't you simply say a “particle”? Would it make any difference?) which IS in a “pure state”. Second one postulates that the “pure state” of this “system” can take one amongst a fixed list of exclusive values, in this example “represented by the vectors [0,1] and [1,0]”. Third one postulates that each single run of the iterative experiment involves a new instance of a “system” so that the ensemble of many different systems (each one with its own state value) leads to the observed statistical distribution. These are not facts.
The assignment of two exclusive values of the “pure state” above does not seem to be restricted to a specific “time”. However this might be contradicted by the assumption whereby the [0, 1] value of the “pure state” property of a single “system” gets transformed into the [1,0] after one second. Due to symmetry considerations, it is logical to assume as well that the second pure state value [1,0] is subject to an analogue transformation in parallel and becomes the [0, 1] pure state. Obviously it is also postulated that the continuous transformation of the “pure state” takes place inside the experimental device, during the experiment. This can't be a fact. Moreover, given that the same U operator applies before and after the mid-way time mark, it is also postulated that the continuous transformation is linear: it holds for all values of the time variable and only depends on the time gap.
So in which state is a "system" at a given time? Is it in a pure state, or is it in a "weird state" corresponding to a "complex vector"?
How does one know at which “time” each “system” is in a pure state, at which time it gets “observed”?
And what about the “complex vector” corresponding to the half-way transformation? It is not a “pure state” (the list of such states has been established earlier), but it results from the linear transformation of a pure state, i.e. of a property inherent to a single “system”. Therefore it cannot either be interpreted as representing a statistical distribution (indeed it is not a mixed state). Do you mean that at a given "time" (whatever that means) an ensemble of "systems" are in (two?) different "weird states" and can be "observed? But how do both weird states map onto the pair of "detectors"? And what is the relevance of the Born rule considering that negative values of relative frequencies are meaningless from a phenomenological standpoint. So what role does the “complex vector” play in this “demonstration”? Where is the link with statistical distributions of events?
This paragraph does not justify the need for-, and the role played by-, complex numbers in the QM formalism. Moreover, given the nature of the many postulates and assumptions that have been injected in order to reach a “conclusion”, whatever that means, we are very, very far from a phenomenological / factual approach.

 

[atyy]

Obviously it is also postulated that the continuous transformation of the “pure state” takes place inside the experimental device, during the experiment. This can't be a fact. Moreover, given that the same U operator applies before and after the mid-way time mark, it is also postulated that the continuous transformation is linear: it holds for all values of the time variable and only depends on the time gap.

The continuous transformation does not refer to time evolution in the particular derivation of finite dimensional QM that bhobba was thinking of. Here is the derivation by Hardy that bhobba referred to http://arxiv.org/abs/quant-ph/0101012. An updated version of Hardy's approach with slightly different axioms is in http://arxiv.org/abs/1104.2066. Hardy's approach is "Copenhagen-like" or "operational" or "instrumental" in the sense that it doesn't solve the measurement problem, and assumes that we intuitively know what a "measurement" is. Hardy himself describes his approach as operational in the discussion of the later paper, and concludes with "After all, it is the desire to understand what reality is like that burns deepest in the soul of any true physicist." I can't tell whether he is being sarcastic or not! :D

 

[atyy]

Excellent question! The constraint does not go away in the classical limit. Instead, in this limit you get classical mechanics in the Hamilton-Jacobi form, which also has a velocity constraint. I never thought about it this way before, but one can use it to argue that Hamilton-Jacobi formulation of classical mechanics is more fundamental than other formulations.

That is a really interesting result. I presume this is not in any paper out there, so I hope you are writing this up:)

So could one say that quantum particles (in any interpretation) do not have simultaneous "canonical" position and momentum that reduce to the strictly Hamiltonian position and momentum in the classical limit?

In the Copenhagen interpretation, this would be true, because the "canonical" ($[q,p]=i\hbar$) position and momentum do not exist simultaneously, although they do reduce to the strictly Hamiltonian position and momentum in the classical limit.

In the Bohmian interpretation, this would be true, because the "canonical" ($\dot{q} = \frac{\partial H(p,q)}{\partial p}, \dot{p} = - \frac{\partial H(p,q)}{\partial q}$, with constraint between $q(0)$ and $p(0)$ ) position and momentum do exist simultaneously, but they do not reduce to the strictly Hamiltonian position and momentum in the classical limit.

I understand the statement is true if I use $[q,p]=i\hbar$ as the definition of "canonical" for every interpretation, but I'm just curious how technically correct one can be if one allows the English to remain ambiguous.

 

[bhobba]

This paragraph looks weird. Obviously a “pure state” qualifies a property owned by a single “system”and the “mixed state” qualifies a property of the “ensemble” of such “systems” that the global iterative experiment has involved.

Your reasoning escapes me.

Imagine a large number of coins. If they are all heads up that is a pure state. If they are all tales up that is a pure state. If some are heads and some tales that is a mixed state and a coin picked randomly will be heads or tales.

First one postulates the existence of a “system” (why don't you simply say a “particle”? Would it make any difference?) which IS in a “pure state”.

I am being general. State can be viewed as heads or tales, the face on a die, the outcome of the Stern–Gerlach experiment - its very general. Its a concept from probability theory - pure state - possible outcomes - events etc etc - same thing.

So in which state is a "system" at a given time? Is it in a pure state, or is it in a "weird state" corresponding to a "complex vector"?

That's the question isn't it - if you allow continuous transformations complex numbers are inevitable.

of a property inherent to a single “system”. Therefore it cannot either be interpreted as representing a statistical distribution (indeed it is not a mixed state)..

Like I said at the start why you think that has me beat.

Moreover, given the nature of the many postulates and assumptions that have been injected in order to reach a “conclusion”, whatever that means, we are very, very far from a phenomenological / factual approach.

You are the one injecting the postulates and assumptions beyond what I wrote and that is confusing you.

What I wrote are general concepts from probability theory, nothing to do with detectors, particles etc etc.

It simply is noting a very general property of standard probability theory. The probability of outcomes, events, whatever terminology you wish to use, can be written as a vector. These are called states. If the probability of the outcome is a cert - ie a single entry contains a 1 - that by definition is called a pure state - the rest by definition are called mixed. Obviously pure states are simply the outcomes. The name pure state comes from the fact its the convex hull of the space of such vectors and has special significance in the theory of general probability models which QM and probability theory are examples of. In QM the pure states are also the convex hull of the state space - but that is just by the by

Now consider something (I am being general) with two possible outcomes ie two pure states. We imagine some kind of process (again I am being general) that after one second transforms one pure state to another. Now we ask what happens after half a second. Chug though the math and you get a complex vector. This can't be interpreted in the formalism - and requires a new interpretation. The new interpretation provided by QM is the Born Rule.

The following paper may help:
https://www.math.ucdavis.edu/~greg/intro.pdf
'To summarize, quantum probability is the most natural non-commutative generalization of classical probability. In this author’s opinion, this description does the most to demystify quantum probability and quantum mechanics.'

That's my view as well, although I would choose continuity or entanglement as the departure point - not non-commutativity - but that's just personal preference - many roads lead to Rome. However I don't agree with the papers view that QM is necessarily Baysian in interpretation. The probabilities of QM via the Born rule can be interpreted either way - in fact in foundational discussions I prefer not to take a view simply using the Kolmogorov axioms. But that's just me.

Thanks
Bill

 

[Sugdub]

You are the one injecting the postulates and assumptions beyond what I wrote and that is confusing you.

I'm not the one who injected the idea that QM deals with “systems” owning properties, the value of which evolves with time, each click on a detector corresponding to one system. This is part of your input. But never mind, I did not intend to hurt you and I am not interested into polemics. BTW you should clarify the requirement for a linear operator in the maths part of your deduction ... on which ground would you believe that the state of any quantum "system" must evolve according to a linear operator? There exist plenty of non-linear processes.
My intention was to open a debate about the possibility of a purely factual approach for describing some key QM experiments such as an SG experiment for which everyone knows that a description in terms of particles getting filtered according to their own properties does not work. The outcome or our first exchange is that your conception of what could be such a factual description is totally different from mine. You keep telling us about physical “systems” owning properties.
Conversely, my suggestion is to STOP talking about “systems”, I mean about what might happen inside the experimental device during the journey of each system from the source to a detector. Just describe the experimental setup and its outcome. Describe the evolution of this outcome in response to changes in the setup and then, try to get it formalised. Because ultimately this is what the QM formalism does, nothing else: it provides an operator transforming the outcome of an experiment into the outcome of another experiment, based on what has changed in the experimental setup. I can't believe this has not been done already and this was the purpose of my question to Demystifier: can you propose a reference? Just in case what I'm looking for is not clear enough, I'll give a first try, although my knowledge of SG experiments is very thin.
An SG experiment has the potential to produce a flow of discrete events distributed over a set of “counters” spatially separated in the same plane. For every relative orientation between the “source” apparatus and the counters' plane, the statistical distribution is stable and reproducible. It constitutes a “macroscopic” property of this SG experiment for this specific relative orientation.
Each statistical distribution can be formalised as a unit vector in a manifold which base vectors correspond one-to-one with the “counters”, in such a way that the orientation of the unit vector in the selected base corresponds to this unique distribution: the simplest way is that the coordinates of the unit vector are set as the square roots of the relative frequencies observed (real positive numbers).
A limited list of transformations of the experimental device can be implemented, leading to new SG experiments showing new statistical distributions which can be derived from the initial distribution: adding new filters, changing their relative orientation, adding shutters... For example the addition of a “shutter” on one channel in front of its counter translates into the erasing of the corresponding component of the distribution. The orientation of the unit vector corresponding to this new setup has changed insofar one of its former contributing components has vanished: the corresponding counter gets no longer activated. One of the former components has been “filtered” out.
In the above, there is no attempt to discuss what might happen inside the experimental device: it only deals with facts and their maths formulation.
Thanks.

 

[bhobba]

on which ground would you believe that the state of any quantum "system" must evolve according to a linear operator? There exist plenty of non-linear processes.

The transformation I gave is via a matrix so is linear. But yes in applying it for half a second I am assuming it remains linear.

My intention was to open a debate about the possibility of a purely factual approach for describing some key QM experiments such as an SG experiment for which everyone knows that a description in terms of particles getting filtered according to their own properties does not work. The outcome or our first exchange is that your conception of what could be such a factual description is totally different from mine. You keep telling us about physical “systems” owning properties.

Can you define exactly what you mean by 'factual' approach?

Conversely, my suggestion is to STOP talking about “systems”,

'System' was used simply as a general concept to apply the probability concepts of event etc etc. But if it worries you think in terms of events and the Kolmogorov axioms.

Describe the evolution of this outcome in response to changes in the setup and then, try to get it formalised.

That's exactly what I am doing. As what you are observing changes in time obviously the outcome of observations changes. I am simply considering a change, during one second, from an observation that will give an outcome for sure, to a different outcome for sure. But what would happen if you instead observed it after half second? Then, and yes one makes the assumption the transformation is still linear, you get a complex state. That is what QM tries to get a grip on - what kind of observational outcomes does such a thing represent. The answer is the Born rule.

I am not being specific in my set-up because I am speaking generally, and indeed QM is a general theory.

It also needs to be pointed out I described my way of viewing it:

For me its simply a variant of probability theory that allows continuous transformations between pure states.

If it doesn't fit in with your 'factual' approach (whatever that is) then we have different ways of looking at things. If you want to put forward a view more in line with what you prefer - go ahead - but I am sticking with mine.

If you want to see a careful and exact working out of the continuous idea, where every t is crossed and i dotted, then, as I mentioned previously, see the following:
http://arxiv.org/pdf/quantph/0101012.pdf

In the above, there is no attempt to discuss what might happen inside the experimental device: it only deals with facts and their maths formulation.

I didn't see much detailed math either.

If you want to formulate an actual different approach you are going to have to rigorously derive the formalism like the paper I linked to above does.

It considers the general set-up used in discussions of QM foundations of preparation, transformation, and measurement and, as far as I can see, meets the criteria of what you are after.

The conclusion is, along with a few very reasonable mathematical assumptions, QM is what inevitably results if you want to model continuous changes - the continuity thing is critical.

Thanks
Bill

 

[Sugdub]

The conclusion is, along with a few very reasonable mathematical assumptions, QM is what inevitably results if you want to model continuous changes - the continuity thing is critical.

Indeed continuity is a key factor. But look at your model: the one-to-one matching of each pure state with each of the detectors is unambiguous. If there exists a continuous transformation between one pure state and the second one, then at some point in-between, there will be a discontinuity of that association. Your model of a pure state subject to a continuous transformation into another pure state implies a discontinuity in the correspondence between the evolving state and the detectors. Therefore the maths demonstration you propose, which is correct, cannot be seen as a formalisation of your physical model. In addition, linearity is also a key requirement in the maths demonstration, but your physical model only addresses it through an ad hoc postulate.

I'm not challenging the need for complex numbers in QM, I do challenge the validity of the justification you propose. Thanks.

 

[Sugdub]

The continuous transformation does not refer to time evolution in the particular derivation of finite dimensional QM that bhobba was thinking of

Indeed the so-called evolution of the state along time is a second-raw metaphor on top of a first-raw metaphor assuming that “systems” move at constant speed from the source to the detector. Hence the distance between the source and whichever analyser is assumed to reflect the time during which the state of the system has evolved. Then the so-called evolution of the state along time factually corresponds to an evolution of the observed statistical distribution in response to a change of the distance between the source and the analyser.
With SG experiments, the factual operational parameter illustrating a continuous transformation of the state vector is neither time nor position, it is the relative orientation between the source and the plane of the SG filter. The continuous evolution of the distribution responds to a continuous evolution of this relative orientation. The correspondence is then factually established between a continuous family of distributions and a continuous family of experiments (and not a continuous family of states of a system). Thanks.

 

[Demystifier]

So could one say that quantum particles (in any interpretation) do not have simultaneous "canonical" position and momentum that reduce to the strictly Hamiltonian position and momentum in the classical limit?

I guess one could.

 

[bhobba]

If there exists a continuous transformation between one pure state and the second one, then at some point in-between, there will be a discontinuity of that association.

On the surface that is contradictory - I think you need to expand on it..

Between any two pure states of QM a continuous transformation can always be found.

Thanks
Bill

 

[Sugdub]

On the surface that is contradictory - I think you need to expand on it..

Between any two pure states of QM a continuous transformation can always be found.

Assuming the initial pure state (1,0) corresponds to clicks on channel A of an SG filter and the final state (0,1) corresponds to clicks on channel B of the same filter, a continuous transformation of state (1,0) into state (0,1) through a continuous family of intermediate states formalised into complex vectors implies that at some point the association between intermediate states and the respective channels A and B will change. This change in the correspondence between components of the state vector and the channels enabling their manifestation as "clicks" of the detectors cannot be continuous.

 

[bhobba]

Assuming the initial pure state (1,0) corresponds to clicks on channel A of an SG filter and the final state (0,1) corresponds to clicks on channel B of the same filter, a continuous transformation of state (1,0) into state (0,1) through a continuous family of intermediate states formalised into complex vectors implies that at some point the association between intermediate states and the respective channels A and B will change. This change in the correspondence between components of the state vector and the channels enabling their manifestation as "clicks" of the detectors cannot be continuous.

That's incorrect.

A continuous transformation exists between such states, as it does between any two states.

A little linear algebra easily shows this.

Thanks
Bill

 

[Dale]

OK, this thread has gone off on a tangent.