What is a Pure State and Mixed State?

[koantum]

the notion of "present time" is only a construction of our subjective experience

I hesitate to use the word "construction". Like qualia, the experiential now defies objectification, true enough. But how, in what sense, does experience construct? I'd use the word for our theoretical activities; we construct theories with maths as our chief or only tool.

I don't mind seeing quantum theory as "just a tool to calculate probabilities of outcomes". I think it is the minimalistic version (the one that should adhered to when first being introduced to the theory). But I fail to see the refusal to try to make more sense of it.

I don’t refuse to do this. Quite the contrary. I'll return to this in a separate thread.
Next, I said if the world is an incompletely differentiated spatiotemporal whole, then determinism is out of the window, to which you replied

I don't think so ; and, from the beginning, determinism has no issue here. By coincidence, the Schroedinger equation is deterministic, but that's no issue.

Huh? Determinism is very much an issue. You believe in an ontology that evolves unitarily and therefore deterministically. I don't. Of course, omitting the term evolution and speaking of, say, a "unitarily structured spatiotemporal whole" leaves this issue untouched. Is this what you meant?
Next, I said that any theory postulating deterministic evolution presupposes the existence of a completely differentiated spacetime manifold. If the latter is an exploded myth, so is the former. Your response to this:

I think you attach too much importance to the split between "state" and "evolution". It is a practical way of talking about the overall structure

I agree, but this is beside the point. You can't have a deterministic evolution and a reality that is incompletely differentiated with respect to space and time. You can't have the cake and eat it too.

But hey, I'm already very happy to talk with someone who SEES the issue of many times ; mostly I get the silly reaction that there are not many times, because t is the value of current time or something of the kind

I know exactly what you mean.

you SOMETIMES have indications of the existence of the other branches.

If I am not mistaken, the world splitting of Everett's original MWI is as irreversible as the collapses of collapse theories (and therefore has the same measurement problem). If you allow re-interference, you aren’t really an Everettic; you are an existentialist à la Zurek, whose "existential interpretation" makes the consequences of "taking unitary evolution seriously" very clear. He arrives at a double relativity of "existence." One, existence is relative to branches: there is one for each branch. Two, the existence of a branch is relative rather than absolute: there can be more or less of it. The less a branch is capable of re-interference with other branches, the more it exists. This is the kind of "philosophy" that makes most physicists abhor philosophy.

"To tell you the truth, I think most of my colleagues are terrified of talking to philosophers - like being caught coming out of a pornographic cinema." (Max Tegmark, University of Pennsylvania)​

My assessment of Zurek's interpretation can be found at http://in.arxiv.org/abs/quant-ph/0401179" or in the International Journal of Quantum Information 2(2), 201-220, 2004.

mathematical structures are the only things of which we can make sense, apart from our qualia

Come on! You have never read a book, seen a movie, listened to a piece of music that made a lot of sense? If you did, then please show me how you reduce it to maths and qualia.

"Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover." - http://plato.stanford.edu/entries/russell/"​

As to MWI, I consider that it is the most evident interpretation of the *current mathematical formulation* of quantum theory

Rather, the most simple-minded interpretation.

What is, in my eyes, an aberation, however, is to consider ontologies which GO AGAINST the mathematical structure of quantum theory. The projection postulate does such a thing.

There I agree. If quantum mechanics only correlates measurement outcomes, you don’t need collapsible wave functions.

if you leave out projection (and HENCE place yourself in an MWI context)

That's a non sequitur.

 

[koantum]

In quantum ontology, wave functions apply to situations in an indefinite future. That is, they correspond to experiments that have not yet been performed.

Careful! The Born rule is time-symmetric in that it allows us to assign posterior probabilities (probabilities of possible outcomes of earlier measurements on the basis of later outcomes) as well as prior probabilities (probabilities of possible outcomes of later measurements on the basis of earlier outcomes). Quantum mechanics even allows us to assign probabilities that are time-symmetric in the sense that they are assigned on the basis of later as well as earlier outcomes. For this you need to use the ABL rule (after Aharonov, Bergmann, and Lebowitz) instead of the Born rule. Take a look at my paper on the time-symmetry of quantum mechanics (http://in.arxiv.org/abs/quant-ph/0006116" , American Journal of Physics 69, 864-873, August 2001). This symmetry of the formalism is spoilt by every interpretation that gives more importance to quantum states evolving from past to future than to quantum states evolving from future to past.

 

[CarlB]

Careful! The Born rule is time-symmetric in that it allows us to assign posterior probabilities (probabilities of possible outcomes of earlier measurements on the basis of later outcomes) as well as prior probabilities (probabilities of possible outcomes of later measurements on the basis of earlier outcomes).

My point is that there is an obvious ontology for splitting the domain of relativity and quantum mechanics, one that is compatible with common sense and provides a natural arrow of time. I don't mean to suggest that this is the unique interpretation compatible with quantum mechanics. What I'm saying here is that if one restricts the domains of these two theories, the incompatibility between them disappears. I don't mean to say that either theory is particularly fond of having its domain restricted, just that the two theories are not, in themselves, incompatible with common sense.

Oh, there are issues when you try to split spacetime into two domains, past and future, this way because the split itself defines a preferred reference frame, but the assumption of no preferred reference frame is also in violation of common sense. Adding an (apparently) undetectable preferred reference frame to special relativity changes absolutely (pun) no predictions of the theory and therefore does the theory no great damage. except in the eyes of those who prefer purity in their physics to common sense. To me, relativity is a guide to indicate when a physical prediction is incompatible with known observations, not something that rules out a preferred reference frame. I think that Lorentz symmetry is an accidental symmetry, not a true part of nature, and I expect to live long enough to see it experimentally disproved.

By the way, I love your writing and point of view on QM, and have downloaded all 20 of your arxiv papers. I look forward with great joy to reading them at my leisure. If there is something I'm missing elsewhere, some articles that didn't make it to arxiv, please do point me.

Carl

 

[koantum]

Adding an (apparently) undetectable preferred reference frame to special relativity changes absolutely (pun) no predictions of the theory and therefore does the theory no great damage.

But what do you gain by this?

If there is something I'm missing elsewhere, some articles that didn't make it to arxiv, please do point me.

You find a complete list of my papers at http://thisquantumworld.com/papers.htm" . Those you don't find in the arxiv are rather more philosophical and less commonsensical, I'm afraid.

 

[koantum]

This is just to tell you that I have started https://www.physicsforums.com/showpost.php?p=956010&postcount=1".

 

[koantum]

could you please expand a bit more the idea of the world as an "incompletely differentiated spatiotemporal whole"?

I'll get to it in https://www.physicsforums.com/showpost.php?p=956010&postcount=1"what you're looking for.

 

[vanesch]

I hesitate to use the word "construction". Like qualia, the experiential now defies objectification, true enough. But how, in what sense, does experience construct? I'd use the word for our theoretical activities; we construct theories with maths as our chief or only tool.

I agree, the word "constructed" was poorly chosen ; sometimes I quickly type a response in between playing with the kid, making dinner and talking to my wife

Huh? Determinism is very much an issue. You believe in an ontology that evolves unitarily and therefore deterministically. I don't. Of course, omitting the term evolution and speaking of, say, a "unitarily structured spatiotemporal whole" leaves this issue untouched. Is this what you meant?

Yes. And for us, "temporaly oriented beings", it is convenient to split this in "state" and "evolution", but indeed, I mean the entire structure. As such, there's no difference between, say, the Heisenberg and the Schroedinger view.

I agree, but this is beside the point. You can't have a deterministic evolution and a reality that is incompletely differentiated with respect to space and time. You can't have the cake and eat it too.

I don't know. Imagine a sphere in 3 dimensions, and consider this sphere as an "incompletely differentiated object with respect to the xy and the z axis". I can write down a deterministic evolution equation that "evolves" each xy slice of the sphere in its next slice, along the z-axis, up to one ambiguity, which is, whether we're in the lower or the upper half of the sphere.

If I am not mistaken, the world splitting of Everett's original MWI is as irreversible as the collapses of collapse theories (and therefore has the same measurement problem). If you allow re-interference, you aren’t really an Everettic; you are an existentialist à la Zurek, whose "existential interpretation" makes the consequences of "taking unitary evolution seriously" very clear. He arrives at a double relativity of "existence." One, existence is relative to branches: there is one for each branch. Two, the existence of a branch is relative rather than absolute: there can be more or less of it. The less a branch is capable of re-interference with other branches, the more it exists. This is the kind of "philosophy" that makes most physicists abhor philosophy.

I put all these variants under the MWI denomination - I didn't even hear about this particular |school". For me, MWI is when you give some ontological status to the mathematics of the unitary part of quantum theory, and you refuse to introduce collapse as a physical phenomenon - which leads to the "existence of all possible outcomes" and then you need to fiddle around with conscious observation in one way or another to say that you only "experience one" of these outcomes. There's a multitude of variations on the concept, the differentiation of which I do not find very illuminating.

As to your original point of "irreversibility" of branching: I'd say that a priori, it is just as "reversible" (the fusion of two branches into one) as is the reversibility in classical physics:
If say,
|bob1>|u> evolves into |bob0>|w>
and
|bob2>|v> evolves into |bob0>|y>

then the two "bob" branches
a |bob1>|u> + b |bob2>|v>

evolve into
|bob0> (a |w> + b |y>)

Of course, now, bob0 doesn't remember from which branch "he came", so nothing unusual !

But, just as in the case of classical physics, it is not because this is in principle possible that we happen to live "that part of the evolution" as to witness it - in exactly the same way as irreversibility in classical physics".

"To tell you the truth, I think most of my colleagues are terrified of talking to philosophers - like being caught coming out of a pornographic cinema." (Max Tegmark, University of Pennsylvania)​

I think it has more to do with a kind of "macho" culture together with a lack of education of physicists. I've been there too, you know. Got straightened up by my wife, who's a classicist.

"Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover." - http://plato.stanford.edu/entries/russell/"​

Yes. That's about what I meant. We can only abstractly (once we do away with all intuition, qualia,...) talk about mathematical constructions. As an ontology is an abstract construction which we invent to explain our subjective experiences, I don't see what it can be outside of this.

cheers,
Patrick.

 

[CarlB]

But what do you gain by this?

As I mentioned, one needs some sort of split in order to distinguish between a region of spacetime over which QM has domain (i.e. the future), while still keeping the other to be covered by the usual intuition of point particles. One also gains the ability to consider theories that violate Lorentz symmetry.

Thanks for the extra articles, which I am sure I will enjoy as I did the Bhagavad Gita so many years ago.

Carl

 

[koantum]

Hi Patrick (aka vanesch),
I said: "You can't have a deterministic evolution and a reality that is incompletely differentiated with respect to space and time." Your response was "I don't know." Haven't thought of it, eh? Well, I intend to be more explicit in the "promises to keep" tread, where you'll get a second chance.

For me, MWI is when you give some ontological status to the mathematics of the unitary part of quantum theory, and you refuse to introduce collapse as a physical phenomenon - which leads to the "existence of all possible outcomes" and then you need to fiddle around with conscious observation in one way or another to say that you only "experience one" of these outcomes. There's a multitude of variations on the concept, the differentiation of which I do not find very illuminating.

In other words, you live in the faith that all problems arising from this conception will in the end be solved one way or the other. What you do not find very illuminating is attempts to solve these problems or to expose the absurdity of that conception by ferreting out its absurd consequences. I call this playing the ostrich.

As to your original point of "irreversibility" of branching: I'd say that a priori, it is just as "reversible" (the fusion of two branches into one) as is the reversibility in classical physics

This is what compels Zurek to deny that each branch exists in an absolute, unqualified sense.

Russell: Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.
You: Yes. That's about what I meant. We can only abstractly (once we do away with all intuition, qualia,...) talk about mathematical constructions.

Why do you want to do away with all intuition, qualia,…? If we admit only quantities then we can only talk about quantities. That's rather boring, I'd say. What does you wife think? Do you belong to the old school according to which qualities are nothing but quantities? I, for one, belong to a school to which quantities are nothing but means of manifesting or realizing qualities. I do not want to reduce everything to physics. I want to understand physics in the context of a whole that includes much more than physics.

an ontology is an abstract construction which we invent to explain our subjective experiences

This is a very narrow (and I'd say rather silly) definition of ontology. You need to explain happiness mathematically?

The very best,
Ulrich

 

[vanesch]

Why do you want to do away with all intuition, qualia,…?

I think you misunderstood me: there is, on one side, the world of qualia, of your subjective experiences. It's the one we know that exists. And then, on the other hand, we tend to set up a mental construction of which we postulate the existence, in order to organize our qualia: that's the ontology. So of course your qualia cannot be involved in the construction of your ontology ! Your ontology needs to be their *explanation*. As this is an abstract mental construction, I do not see what it can be else but a mathematical object, given that it must be an abstract mental construction. I don't know of any OTHER abstract mental constructions.

It is this dichotomy between constructed ontology and subjective experience which makes me reject your idea of "taking measurements as starting positions" if you do not identify "measurements" with "subjective experiences".

If we admit only quantities then we can only talk about quantities. That's rather boring, I'd say. What does you wife think? Do you belong to the old school according to which qualities are nothing but quantities? I, for one, belong to a school to which quantities are nothing but means of manifesting or realizing qualities. I do not want to reduce everything to physics. I want to understand physics in the context of a whole that includes much more than physics.

Ah, that's a fundamental difference then. I think that everything, except for our subjective experiences, is physics, and that even our subjective experiences are *derivable* from physics, although they cannot be part of it of course. That's why I'm a reductionist in heart and bones.
I know not everybody shares this idea with me, but I take it as the *definition* of physics.

This is a very narrow (and I'd say rather silly) definition of ontology. You need to explain happiness mathematically?

Well, I'd say that happiness is first of all a subjective experience, and hence does not belong to an ontological description ; but that we'd rather POSTULATE an ontology from which we can eventually derive our "experience of happiness", given an appropriate rule which will be outside of the ontology proper. So, there will be an ontological ORIGIN of my or your happiness, which will find its origin probably in some neurological state. The very fact that this neurological state will be related to the subjective experience of "happiness" will, IMO, always be something that remains outside of an ontological description itself: I don't think we will ever have a *theory* that will allow us to deduce which matter states are related to "an experience of happiness". But we might find out, by experiment, that specific matter states of the human brain lead to subjects who declare "being happy".

But in any case, to be able to tell exactly what these states are, you need to symbolise these brainstates abstractly, and I don't see how you can do this without making it into a mathematical object. That was the idea. Now, "happiness" is probably a very complicated concept. Let's start with "seeing a red light flash".

 

[Jimmy Snyder]

What is a pure state and a mixed state?

A pure state is one that can be represented by a vector in a Hilbert space. A mixed state is one that cannot: it must be represented by a statistical mixture of pure states.

It seems to me that this thread has gotten far afield of the question that got it started. I browsed through it for a better answer but found none. Perhaps I just missed it.

Contrary to the answer given, pure states and mixed states are both represented by vectors in Hilbert space. A given state is called pure if it is represented by an eigenvector of a given operator. Otherwise it is a mixed state. However, a pure state for one operator may not be a pure state for some other operator. For instance, a pure state for the momentum operator will be a mixed state for the position operator.

In short, a state cannot be said to be pure or mixed except as it relates to some operator. With relation to a given operator, it is a pure state if it is represented by an eigenvector of that operator.

There is a slight ambiguity in this answer in that for degenerate eigenvalues (eigenvalues for which there is more than one eigenvector), pure and mixed states are not so neatly packaged. The way around this is to find a complete set of commuting operators ('complete' meaning that it disambiguates all degeneracy) and define a pure state to be one that is represented by a vector that is an eigenvector of each of the commuting operators.

 

[Doc Al]

Sorry, jimmysnyder, but you are confusing "eigenvector" with "pure state"; they are not the same. Certainly an eigenstate is itself a pure state, but just because a state is not an eigenvector of a particular operator does not mean it's not a pure state.

A "spin up in the z direction" state is a pure state regardless of the eigenbasis used to represent it. True, it can be represented as a linear combination of up/down spin states in the x direction, but that is still a single vector in Hilbert space (and still a pure state).

I stand by my first answer in this thread.

It seems to me that this thread has gotten far afield of the question that got it started.

I certainly agree with that!

 

[Jimmy Snyder]

A "spin up in the z direction" state is a pure state regardless of the eigenbasis used to represent it.

In this matter we agree except for semantics. I would say "pure state of the z component of angular momentum", and you would simply say "pure state". That is, I would mention the name of the operator wrt which it is pure.

True, it can be represented as a linear combination of up/down spin states in the x direction, but that is still a single vector in Hilbert space (and still a pure state).

In this matter we agree except for semantics. I would say "mixed state of the x component of angular momentum" (without contradicting or changing my statement above), and you would say "pure state" (without contradicting or changing your statement above).

I stand by my first answer in this thread.

Here we must disagree (about your first answer, not about where you stand), for in your first answer you said "A mixed state is one that cannot be represented as a vector in a Hilbert space" and I say that a mixed state can be so represented. For instance, take two different pure states of an operator, each one represented by a different vector. A mixed state can be made of these two pure states, represented by the (possibly weighted and) normalized sum of those two vectors. That sum is itself a vector in the same Hilbert space. QED.

 

[vanesch]

I certainly agree with that!

Yes, sorry, I'm partly responsible for that. For my excuse, a point can be made that links the discussion to the OP: that is that for those that give ontological state to the quantum state vector, there is a fundamental conceptual difference between a "pure state" and "a mixture", while for those that see quantum theory as an algorithm, formalism, technique, whatever for calculating probabilities of outcomes, there's no fundamental difference.

 

[vanesch]

Here we must disagree (about your first answer, not about where you stand), for in your first answer you said "A mixed state is one that cannot be represented as a vector in a Hilbert space" and I say that a mixed state can be so represented. For instance, take two different pure states of an operator, each one represented by a different vector. A mixed state can be made of these two pure states, represented by the (possibly weighted and) normalized sum of those two vectors. That sum is itself a vector in the same Hilbert space. QED.

Sorry, but Doc Al is right. It is a matter of terminology, but I don't think there's any ambiguity here. What you are describing is not a "mixture" but a "superposition".

 

[Jimmy Snyder]

What you are describing is not a "mixture" but a "superposition".

In that case, I stand down. Given that a mixed state cannot be represented as a vector in a Hilbert space, is there some other way that it can be represented?

 

[selfAdjoint]

In that case, I stand down. Given that a mixed state cannot be represented as a vector in a Hilbert space, is there some other way that it can be represented?

In the tensor product of two Hilbert spaces.

 

[vanesch]

In the tensor product of two Hilbert spaces.

Never thought of it that way!
I'd have said a density operator. How do you get to a tensor product ?
You mean, H is isomorphic to the dual of H, and then we have a basis like |u><v|, so any linear combination of such a thing is ok.
Isn't this a bit too large ? I mean, a linear combination of |u><u| style elements, yes, but general |u><v| ?

 

[koantum]

What's happened to this thread? I thought that pure and mixed thing was settled long ago. Every quantum state is first and foremost a density operator W. This satisfies a number of conditions that guarantee that the probabilities we get out of it with the help of the trace rule are real, not less than 0, not greater than 1, etc. In addition it satisfies either WW=W or WW<W. In the first case it's a 1 dimensional projector |w><w|, we work with |w>, and we call it a pure state. In the other case we cannot work instead with a vector, and we call it mixed.

 

[koantum]

In the tensor product of two Hilbert spaces.

Who ever heard of such nonsense?

 

[koantum]

there is, on one side, the world of qualia, of your subjective experiences. It's the one we know that exists. And then, on the other hand, we tend to set up a mental construction of which we postulate the existence, in order to organize our qualia: that's the ontology.

This has two possible readings: Qualia are the stuff of reality and the quantum-mechanical correlation laws just structure this stuff. Or else, mathematical reality somehow (God knows how) produces qualia.

So of course your qualia cannot be involved in the construction of your ontology ! Your ontology needs to be their *explanation*. As this is an abstract mental construction, I do not see what it can be else but a mathematical object, given that it must be an abstract mental construction. I don't know of any OTHER abstract mental constructions.

How about taking a course in philosophy? Has it ever occurred to you that reality may not be an abstract mental construction? Do you really expect an abstract mental construction to explain your qualia?

It is this dichotomy between constructed ontology and subjective experience which makes me reject your idea of "taking measurements as starting positions" if you do not identify "measurements" with "subjective experiences".

Do you really think that the human cognitive distinction between conceptions and perceptions makes a sound foundation for ontology? I want to understand physics without having to drag in conscious observers or subjective experiences. This is simple a newer version of the old "God in the gaps".

I think that everything, except for our subjective experiences, is physics, and that even our subjective experiences are *derivable* from physics, although they cannot be part of it of course.

That's why I'm a reductionist in heart and bones.

Me too, except that I prefer to reduce quantities to qualities. (As said, according to me, qualities are nothing but means to realize or manifest qualities.)

I know not everybody shares this idea with me, but I take it as the *definition* of physics.

That's inadmissible. It's not up to you to define physics. You may of course have your private philosophy or religion.

Well, I'd say that happiness is first of all a subjective experience, and hence does not belong to an ontological description ; but that we'd rather POSTULATE an ontology from which we can eventually derive our "experience of happiness", given an appropriate rule which will be outside of the ontology proper.

So, there will be an ontological ORIGIN of my or your happiness, which will find its origin probably in some neurological state.

How about the ontological ORIGIN of my or your happiness being a self-existent Happiness that creates neurological states to realize itself variously?

 

[Jimmy Snyder]

In the tensor product of two Hilbert spaces.

Thanks to Doc Al, vanesch, selfAdjoint, and others and to physicsforums for putting me straight. This forum is an invaluable resource.

 

[CarlB]

Every quantum state is first and foremost a density operator W.

Yes.

In addition it satisfies either WW=W or WW<W. In the first case it's a 1 dimensional projector |w><w|, we work with |w>, and we call it a pure state.

For spin-1/2 particles, it is in going from W to |w> that the U(1) gauge freedom is introduced. If one generalizes the density matrix formalism by the Schwinger measurement algebra, one can allow a single W to represent various distinct spin-1/2 particles in a manner similar to how a double spinor represents an electron or positron. My guess is that if one computes the density matrix for that larger W, one can eliminate the general gauge freedom. However in doing this, one ends up having to assume preons.

Carl

 

[koantum]

For spin-1/2 particles, it is in going from W to |w> that the U(1) gauge freedom is introduced. If one generalizes the density matrix formalism by the Schwinger measurement algebra, one can allow a single W to represent various distinct spin-1/2 particles in a manner similar to how a double spinor represents an electron or positron. My guess is that if one computes the density matrix for that larger W, one can eliminate the general gauge freedom. However in doing this, one ends up having to assume preons.

With http://en.wikipedia.org/wiki/Preon" we are going beyond the standard model (nothing wrong with that) but as my interest is the philosophical side of physics, it would be premature (for me) to deal with these unconfirmed (and partly disconfirmed) areas.

 

[vanesch]

I'll not respond to the rest of the post as this will lead to polemic which is of no use to nobody. But I'll make an exception for this one:

How about taking a course in philosophy? Has it ever occurred to you that reality may not be an abstract mental construction?

Given the unfalsifiability of solipsism, I don't see what else an ontology can be but a *hypothesis*. Now what's a hypothesis but an abstract mental construction ?

 

[koantum]

I'll not respond to the rest of the post as this will lead to polemic which is of no use to nobody. But I'll make an exception for this one: Given the unfalsifiability of solipsism, I don't see what else an ontology can be but a *hypothesis*. Now what's a hypothesis but an abstract mental construction ?

I agree, and I also make one exception: I find it more fun to live in the real world rather than in an abstract mental construction.

 

[koantum]

one needs some sort of split in order to distinguish between a region of spacetime over which QM has domain (i.e. the future), while still keeping the other to be covered by the usual intuition of point particles. One also gains the ability to consider theories that violate Lorentz symmetry.

If you recall my posts # 56 and # 62 in this thread, I am convinced (along with greats like Einstein, compared to whom I am less than nobody, that's understood) that the experiential now and therefore the concept of an objective split between an open future and a "fixed and settled" past is an illegitimate projection into the objective world of physics of our self-experience as agents in a successively experienced world.
To my mind, the domain of quantum mechanics is the spatiotemporal whole, in which it correlates measurement outcomes.
And why remain stuck with the "usual intuition" of point particles? Take a look at my page http://thisquantumworld.com/form.htm" , which lists several reasons for thinking of a structureless particle as a formless entity rather than as possessing a pointlike form. One of them is that there is no way of explaining the origin of this pointlike form, whereas with formless "ultimate constituents" we can fully comprehend the realization or coming into being of form – cosmic morphogenesis, if you like – for then all existing forms resolve themselves into fuzzy relative positions between formless entities. (Heisenberg once said something to this effect: if you want to explain the features of this world, you cannot postulate entities already in possession of these features.)

 

[Hurkyl]

I agree, and I also make one exception: I find it more fun to live in the real world rather than in an abstract mental construction.

Of course, your perceptino of the real world is an abstract mental construction, but hey!

 

[koantum]

Of course, your perceptino…

A new species of particle?

…of the real world is an abstract mental construction, but hey!

Isn't there a difference between perception and conception? But my objection ought to be seen in its context, the claim that abstract mental constructions have to be mathematical constructions. The great metaphysical systems of the past are imposing abstract mental constructions but certainly not mathematical.

 

[Bertrand]

Hi,

Being new to this forum, I'm not sure this is the right place to ask my question. I had no time to read all the posts of this very interesting debate, and maybe this subject was already addressed somewhere...

My question is :

B. D'Espagnat states that the reduced density matrix of a subsystem A of a composite system "A+B", obtained through the partial trace operation on B, doesn't necessarily represent a mixture, but what he calls an "improper mixture".

in the following article :

http://arxiv.org/PS_cache/quant-ph/pdf/0109/0109146.pdf

the author, K.A. Kirkpatrick shows that D'Espagnat is wrong somewhere in his reasoning, and that, therefore, this reduced matrix may indeed be considered as representing a "true mixture".

Kirkpatrick's argument is based on indistinguishability, but I don't really get it.

Could someone explain it ? My last question is, finally, who is right and who is wrong ?

Thanks for help,

Bertrand