What is a Pure State and Mixed State?

[vanesch]

You will get into trouble if you take wave functions for real, for then a cat can be both alive and dead, which is pure nonsense.

I have no fundamental problems with saying that a cat is both alive and dead, if I also have the explanation of why I only see one of both. I mean: postulating something extra which is not observable is not "nonsense", it is neutral. If I tell you that at exactly the place where you are, there's a fire-spitting dragon, made of stuff which with you cannot interact, and which spits fire with which you cannot interact, then that is a totally neutral statement: it is not a priori true, or it is not a priori false. It should not be classified as "obviously not true" because you cannot know.
Now, if the only aim of the introduction of this dragon, is, well, to be able to say that there is such a dragon, then you can rightly object to this extraneous statement, which serves no other purpose than to talk about non-interacting dragons, because of Occam's razor. However, if the introduction of this dragon makes go away a lot of OTHER problems, then I don't see what's wrong with it. After all, you don't know, and there's no way to know, if there are not many of these dragons around ! There's no way for you to state that they are not there.

To come back to the cat: if I can resolve the "formal ontology", "locality" and "measurement/interaction dichotomy" problems, by saying that there is an unobserved dead cat when I see a live cat, then I find that a totally acceptable statement. This is the difference with the dragon: the dragon came "out of the blue". The cat was suggested by the formalism. In both cases, we're talking about something that is "real but unobservable". But the first one gets (rightly) cut away because of Occam's razor (the dragon doesn't serve any purpose) ; in the second case, it serves the purpose of simplifying the concept.

Again, if you don't like cats which are live and dead, be my guest. But it is not nonsensical to say so - especially if it can solve other issues.

You won't arrive at this nonsense of you understand that the quantum formalism does nothing but correlate measurement outcomes, whose existence it presupposes. To make your approach consistent with the existence of measurement outcomes, you need many worlds. I want to understand this one world. The plural of "world" is (for me) a contradiction in terms.

Because you know as well as I do that this is just a colloquial way of talking. The "worlds" are nothing else but the alternative subjective perceptions of the observer, which only experiences one.
There's not much difference with "time" in relativity: there, yesterday and tomorrow "exist" as much as "today" exists. But you have the impression that only "today" exists, while yesterday "doesn't exist anymore" and tomorrow "has not yet come into existence". Nevertheless, the whole 4-manifold "exists" and all time slices have equivalent ontology. Why do you only experience one slice ? Are these also "parallel worlds" ? Is a "copy of you" experiencing yesterday, while you are experiencing "today", and another copy of you is experiencing "tomorrow" ? It is not so different.

Absolutely not. I haven’t yet told you what my fundamental ontological entities are. To be able to conceive of them, you need to accept the quantum formalism as being fundamentally a probability algorithm.

I'd like to hear that. And I wonder how you are going to do this, without introducing them as a mathematical object!

Apropos of Kolmogorov. There are two misconceptions about quantum-mechanical probabilities:

• that they are subjective rather than objective,
• that they are absolute rather than conditional. "Every probability is a conditional probability" - Hans Primas, "Time–Entanglement Between Mind and Matter" in Mind and Matter 1 , 81–119, 2003.

What could it mean, "objective probabilities" except for the fact that the alternatives "exist", and not that "only one exists, but we don't know which one" ?
And of course all probabilities are conditional ! They are conditional on the initial state you care to specify.

If you take the wave function (rather than the propagator) as the primary object, you will take the probabilities it defines in an absolute sense, as depending on nothing but the wave function. If you take the propagator as the primary object, then it is obvious that the wave function is only a tool for calculating conditional probabilities – probabilities that are determined by the outcomes of actual measurements and the time of the measurement to the possible outcomes of which they are assigned.

Of course. I agree with what you write here: what is of course real is not the "wavefunction at a certain moment", but the entire unitary structure over time. You can even go to the Heisenberg picture if you want to, that doesn't change the idea. "taking the wavefunction seriously" does not mean that one should attach a specific meaning to psi(t) for a given value of t (especially in a relativistic setting). The wavefunction is nothing else but something like a "spacelike slice" of this unitary structure, in a similar way as space is a spacelike slice of minkowski space.
The propagators are another way to look upon this structure, this time more along the timelike axis. It is as if we were going to have a discussion to what's real: spacelike slices of Minkowski space, or world lines of particles. That's a hollow discussion. This is like arguing over the meaning of phase space, and how this meaning gets altered under canonical transformations. It's the entire structure of course, and not one specific "coordinate description" which is real.

Besides, you know the enormous advantage of the propagator formalism over the wave function formalism – its explicit relativistic invariance. With the wave function formalism you schlep with you the useless burden of a preferred reference frame, which of course is as unobservable as your evolving wave function.

I fully agree here. The two things go hand in hand, and are in fact different aspects of the unitary structure over hilbert space introduced by the time evolution operator.

Wrong. There is a crucial difference between macroscopic objects and all the rest. But (once again) to be able to understand it, you need to accept the quantum formalism as being fundamentally a probability algorithm.

Again, this is what I refuse to do: make any distinction between an electron and an apple, in principle. A universal theory must treat them in the same way. Sorry, it's my religion I'm convinced that from the moment you do this, you introduce too much intuitive naive realism into the theory.

 

[Steve Esser]

koantum and vanesch: Thank you for the valuable debate. (Don't stop!)

 

[CarlB]

Again, this is what I refuse to do: make any distinction between an electron and an apple, in principle. A universal theory must treat them in the same way. Sorry, it's my religion I'm convinced that from the moment you do this, you introduce too much intuitive naive realism into the theory.

This is another way of saying that you believe that quantum mechanics is correct and complete and you will therefore refuse to acknowledge any evidence to the contrary. You should realize that you are going against not only Einstein in this belief, but also against all those physicsists busily attempting to unite gravity and QM.

If the MWI interpretation were correct, then the quantum gravity problem should already have been solved, along with all the other mysteries of physics so well described by Smolin in "A Crisis in Fundamental Physics".

http://www.nyas.org/publications/Upd...sp?UpdateID=41

Instead, the situation after MWI is identical to the situation before MWI. It brought no new information to the table, no new predictions, nothing of any use. The same could be said of the various string theories, as well as the things that are dear to my own heart, Bohmian mechanics and David Hestenes' geometric algebra.

When Einstein discovered relativity, a flood of applications soon followed. When various people put QM together, again a flood of applications followed. By contrast, all the various modern attempts to refound QM and or relativity have been completely barren. That's another word for useless. All these theories have provided is badly justified new possibilities for religious devotion.

When a new foundation for QM and or relativity appears, you will know it by its utility.

Carl

 

[koantum]

Smolin,"A Crisis in Fundamental Physics".
http://www.nyas.org/publications/Upd...sp?UpdateID=41

That link didn't work for me. http://www.nyas.org/snc/update.asp?updateID=41&page=1#" did.

 

[koantum]

I have no fundamental problems with saying that a cat is both alive and dead, if I also have the explanation of why I only see one of both.

Unfortunately I don’t have that explanation. Fortunately I don’t need it. As far as I am concerned, the two cat states are possibilities. Whereas there is only one actual world, there can of course be many possible ones.

postulating something extra which is not observable is not "nonsense", it is neutral.

There I agree. I play the same game, but the extra I postulate is not an ontology isomorphic to a probability algorithm (that is, a mathematical tool for calculating probabilities of possible measurement outcomes on the basis of actual ones).

If I tell you that at exactly the place where you are, there's a fire-spitting dragon, made of stuff which with you cannot interact…

Then what about Popper's definition of a scientific theory? To be scientific, it has to be falsifiable.

However, if the introduction of this dragon makes go away a lot of OTHER problems, then I don't see what's wrong with it.

That's a BIG if. What if it creates a lot of OTHER problems?

The cat was suggested by the formalism.

Not by the formalism but by what I call the "evolutionary paradigm": the belief that physics is neatly divisible into kinematics (concerned with the description of systems at anyone time, their "states") and dynamics (concerned with the evolution of states from earlier to later times).
This hangover from classical times keeps alive the belief in the existence of evolving, instantaneous states that are descriptive of physical reality. And it leaves us with no alternative to seizing upon the wave function as that evolving, instantaneous ontological state. Hence the mother of all pseudo-problems: why two modes of evolution rather than one? It's a pseudo-problem because the real number of modes of evolution is zero.

The "worlds" are nothing else but the alternative subjective perceptions of the observer, which only experiences one.

So we had better call it the "many minds interpretation" (David Z. Albert, Quantum Mechanics and Experience, Harvard UP 1992). According to Albert, everyone schleps with them a non-denumerable infinity of minds. Schizophrenia with a vengeance!

There's not much difference with "time" in relativity: there, yesterday and tomorrow "exist" as much as "today" exists. But you have the impression that only "today" exists, while yesterday "doesn't exist anymore" and tomorrow "has not yet come into existence". Nevertheless, the whole 4-manifold "exists" and all time slices have equivalent ontology. Why do you only experience one slice ? Are these also "parallel worlds" ? Is a "copy of you" experiencing yesterday, while you are experiencing "today", and another copy of you is experiencing "tomorrow" ? It is not so different.

There is a significant difference. I'll discuss it in a separate post.

I'd like to hear that. And I wonder how you are going to do this, without introducing them as a mathematical object!

This deserves a separate thread.

What could it mean, "objective probabilities" except for the fact that the alternatives "exist", and not that "only one exists, but we don't know which one" ?

What it means is that the hydrogen atom (and everything made of atoms) is "fluffed out" by the objective fuzziness of relative positions and momenta (not excluding those between macroscopic objects) rather than by our ignorance of the exact values of these observables. We use objective probabilities to describe objective fuzziness and subjective probabilities to describe subjective ignorance.

And of course all probabilities are conditional ! They are conditional on the initial state you care to specify.

This illustrates the nefariousness of the evolutionary paradigm. There is no initial wave function. The wave function is only a tool that helps in the calculation of probabilities of later outcomes given earlier outcomes, just as the electromagnetic field is only a tool that helps calculating (later) effects on the behavior of charges caused by the (earlier) behavior of other charges.

what is of course real is not the "wave function at a certain moment", but the entire unitary structure over time.

That starts a completely new ballgame!

"taking the wave function seriously" does not mean that one should attach a specific meaning to psi(t) for a given value of t (especially in a relativistic setting). The wave function is nothing else but something like a "spacelike slice" of this unitary structure, in a similar way as space is a spacelike slice of minkowski space.

A spacelike slice? May I remind you of your post in which you agreed with me that the wave function evolves in the configuration space of the universe? (Pardon me, I meant to say: that the wave function exists in the configuration spacetime of the universe?)

the time evolution operator

What's that? If what exists is the spatiotemporal whole, how can anything evolve?

 

[koantum]

The "worlds" are nothing else but the alternative subjective perceptions of the observer, which only experiences one. There's not much difference with "time" in relativity: there, yesterday and tomorrow "exist" as much as "today" exists. But you have the impression that only "today" exists, while yesterday "doesn't exist anymore" and tomorrow "has not yet come into existence". Nevertheless, the whole 4-manifold "exists" and all time slices have equivalent ontology. Why do you only experience one slice ? Are these also "parallel worlds" ? Is a "copy of you" experiencing yesterday, while you are experiencing "today", and another copy of you is experiencing "tomorrow" ? It is not so different.

You are right in that the experiential now has no counterpart in the physical world. There simply is no objective way to characterize the present, and since the past and the future are defined relative to the present, they too cannot be defined in physical terms. The temporal modes past, present, and future can be characterized only by how they relate to us as conscious subjects: through memory, through the present-tense immediacy of qualia, or through anticipation. The proper view of physical reality therefore is not only what the philosopher http://philosophy.fas.nyu.edu/object/thomasnagel" has called "the view from nowhen" (the physical world does not contain a preferred time corresponding to the particular moment from which or at which I experience it).
To philosophers, the perplexities and absurdities entailed by the notion of an advancing objective present or a flowing objective time are well-known. To physicists, the unreality of a temporally unextended yet persistent and continually changing present was brought home by the discovery of the relativity of simultaneity. For any two events A,B there exist two reference frames F_A and F_B and a third event C such that C is simultaneous with A in F_A and simultaneous with B in F_B. This "simultaneity by proxy" compels us to conceive of all parts of the spatiotemporal whole as coexistent and as equally real.
It has been argued that quantum mechanics is just barely consistent with relativity. An inconsistency is perceived by wave function enthusiast who (unlike you) feel the need to reintroduce an absolute simultaneity. Here is what http://www.columbia.edu/cu/philosophy/Faculty/_facultypages/davidAlbert.html"'s "peaceful coexistence" of the two.
Case in point: If I assume that quantum mechanics is nothing but a tool for calculating the probabilities of possible outcomes given actual outcomes, then I can demonstrate that the spatiotemporal differentiation of reality doesn’t go "all the way down." Nothing in the world corresponds to a precise position or a sharp instant of time. We should therefore conceive of the world first and foremost as a whole. Our theoretical model of it should not be built on spacetime points or temporally unextended "nows" and spatial points, but instead should be built "from the top down," by a differentiation that Nature has not carried as far as conceivable.
Quantum mechanics thus agrees with special relativity in that it too is inconsistent with presentism: the view that only the continually changing present is real. Special relativity because of the relativity of simultaneity, and quantum mechanics because it implies that reality is not built up from infinitely thin successive presents but instead is an incompletely differentiated spatiotemporal whole.
Now comes the punch line: if the world is an incompletely differentiated spatiotemporal whole, then determinism is out of the window. Any theory postulating deterministic evolution presupposes the existence of a completely differentiated spacetime manifold. If that is a fiction, so is determinism.
I almost forgot to point out the difference between many worlds and many times. I am aware of the many times but not of the many worlds. Of course I am not simultaneously aware of the many times – that would be a contradiction in terms. But I have subjective evidence of the many times and no subjective evidence whatever of the many worlds. The problem of the apparent impossibility of objectifying the Technicolor reality of the experiential present (which deeply troubled Einstein) is therefore genuine. Your invocation of a multitude of coexistent worlds of which we are unaware, on the other hand, is a gratuitous solution of a pseudo-problem.

 

[antfm]

Hi Ulrich (koantum),
Thank you very much to you (and rest of debatants) for this beautiful discusion. Sorry for the interruption, but could you please expand a bit more the idea of the world as an "incompletely differentiated spatiotemporal whole"?
I think I grasp the idea, but doesn't it imply somehow a framework over which to establish the differentiation?
Am I missing something? Thanks again, and sorry if this is too elementary.
Antonio

 

[vanesch]

This is another way of saying that you believe that quantum mechanics is correct and complete and you will therefore refuse to acknowledge any evidence to the contrary.

On the contrary! The belief that the only sensible *hypothesis* (and, as I outlined several times, it will always remain a hypothesis because of the non-falsifiability of solipsism) of an ontology is a mathematical structure (because mathematical structures are the only things of which we can make sense, apart from our qualia) has nothing to do with "the belief that quantum mechanics is correct and complete". You are not allowed to use qualia as the defining properties of an ontology - almost by definition of an ontology ; so what remains are mathematical structures. Only, which one ? I'd say that you have to deal with the structures you have, at the moment, of the theories that are seen as "most fundamental" today. This can change, tomorrow.

What I'm saying is, that IF YOU WANT TO GIVE A SENSE to quantum theory, then you have to take (one of its) mathematical representations seriously. And if you want to give a sense to general relativity, then you have to take its mathematical representation seriously. And if you want to give a sense to classical physics, then you have to take its mathematical formalism seriously. In each case, you have a different *hypothesis* of ontology - simply because you have a different mathematical formalism.
If I do general relativity, then "what is out there" is a 4-dim manifold. And if I do quantum theory, then "what is out there" is the unitary structure of the theory (hilbert space + evolution operator). If I do classical physics, then "what is out there" is the phase space.

Now, it is of course rather disturbing to have to CHANGE ontology, according to with which theory you're working. But if you understand that a certain theory's mathematical formalism is an approximation of an underlying theory, then of course you can keep the underlying formalism - with its ontology - to interpret the approximative theory: the mathematical objets of the approximative theory correspond to approximations of the mathematical objects of the underlying theory. This is what happens, for instance, when going from general relativity to Newtonian physics: the 4-manifold (which is then taken to be "real") is to be viewed as sliced up according to a certain time coordinate, and this coordinate is then seen as "the time axis" of Newtonian physics, while the spacelike slices are seen as seen as "space" in Newtonian physics. We now see that what were considered two distinct ontological objects (time, and space), are in fact a specific slicing up of the "encompassing" ontological object (4-manifold).

As to MWI, I consider that it is the most evident interpretation of the *current mathematical formulation* of quantum theory as we know it - so as long as you think that this formalism "stands" I think that it is the clear ontology that goes with it, simply because it's what the mathematical structure says. And - I repeated this often - the day that this formalism needs a change, then this ontology will change also. After all, ontology is nothing but a hypothesis: there doesn't need to be a real world out there in the first place.
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. It violates unitarity, it violates Lorentz invariance, and it isn't even specified WHEN it should be applied - you're left to your intuition to do so.
But again, the day that one changes the formalism, one changes also the ontology - so I see no point of your claim that I believe quantum theory is ultimately correct. I even think that as long as the challenge with GR is open, that all bets are off. Which doesn't mean, that we know WHAT will give and what will remain. It might very well be that the superposition principle remains - in one form or another. It might be that in a future theory, we might understand what the projection postulate actually meant, because it is then naturally incorporated in the structure of the theory. Or not. But it doesn't make any sense to speculate over that if you don't know which way it will go, and if you don't have a reasonable mathematical formalism that does so.

You should realize that you are going against not only Einstein in this belief, but also against all those physicsists busily attempting to unite gravity and QM.

First of all, except for its historical value, I don't think it matters much what Einstein thought. After all, what I say probably also goes against what thought Saint Augustin. So what ?

On the other hand, I think you're mistaking when you think that I go against what people are doing on the GR-QM front, on the contrary. Many of these schemes SIMPLY DON'T MAKE SENSE if you consider projection. Most, if not all, "official" approaches stick to strict unitarity. Even Hawking admitted that probably, the evaporation of a black hole is a strictly unitary process. The entire issue of conservation of information (read: unitarity, which is OPPOSITE to projection) in a quantum process such as the formation and evaporation of a black hole only makes sense if you leave out projection (and HENCE place yourself in an MWI context).
So the entire question is: will it be possible to conserve the superposition principle, and unitary dynamics, while unifying with gravity, or does one need a fundamental change of this entire machinery. In fact, (Penrose style thinking) gravity is the last hope of killing off unitarity (and hence opening the possibility of introducing the projection postulate as a dynamical phenomenon). Even Penrose, who is certainly not in favor of the MWI view, thinks that the MWI view is the _only sensible interpretation_ if quantum theory turns out to remain strictly unitary in the presence of gravity. Hence his hope that gravity will introduce a deviation from unitarity.

 

[vanesch]

The temporal modes past, present, and future can be characterized only by how they relate to us as conscious subjects: through memory, through the present-tense immediacy of qualia, or through anticipation. The proper view of physical reality therefore is not only what the philosopher http://philosophy.fas.nyu.edu/object/thomasnagel" has called "the view from nowhen" (the physical world does not contain a preferred time corresponding to the particular moment from which or at which I experience it).

Exactly. Well said. So you agree with me that the notion of "present time" is only a construction of our subjective experience, as you state so well: "temporal modes past, present, and future can be characterized only by how they relate to us as conscious subjects". Couldn't agree more.

To physicists, the unreality of a temporally unextended yet persistent and continually changing present was brought home by the discovery of the relativity of simultaneity.

Very good. It's what I've been trying to say already quite a while.

For any two events A,B there exist two reference frames F_A and F_B and a third event C such that C is simultaneous with A in F_A and simultaneous with B in F_B. This "simultaneity by proxy" compels us to conceive of all parts of the spatiotemporal whole as coexistent and as equally real.

Case in point: If I assume that quantum mechanics is nothing but a tool for calculating the probabilities of possible outcomes given actual outcomes, then I can demonstrate that the spatiotemporal differentiation of reality doesn’t go "all the way down."

I agree with that. But I think that you can say more about it. Again, you don't HAVE to, but you can. Again, 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.

Nothing in the world corresponds to a precise position or a sharp instant of time. We should therefore conceive of the world first and foremost as a whole. Our theoretical model of it should not be built on spacetime points or temporally unextended "nows" and spatial points, but instead should be built "from the top down," by a differentiation that Nature has not carried as far as conceivable.

Sure. That's what I call "the mathematical structure" that should correspond to the proposed ontology. I never said that it had to split into something that is "at an instant in time" or so.

Quantum mechanics thus agrees with special relativity in that it too is inconsistent with presentism: the view that only the continually changing present is real. Special relativity because of the relativity of simultaneity, and quantum mechanics because it implies that reality is not built up from infinitely thin successive presents but instead is an incompletely differentiated spatiotemporal whole.

Yes, I am with you here. That's why it is not simply the wavefunction, but the entire unitary structure which is that famous mathematical structure I like to refer to.

Now comes the punch line: if the world is an incompletely differentiated spatiotemporal whole, then determinism is out of the window.

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. I think you want to say that "dynamics" is out of the window. But I don't think that that is correct: after all, "dynamics" is nothing else but symmetries of the mathematical structure of nature "in the direction of what we experience as time", while what's usually called kinematics is more related to the symmetries of the mathematical structure "perpendicular" to what we call time.

Any theory postulating deterministic evolution presupposes the existence of a completely differentiated spacetime manifold. If that is a fiction, so is determinism.

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, just as "spacelike surface" and "world line" are practical ways of talking about spacetime.

I almost forgot to point out the difference between many worlds and many times. I am aware of the many times but not of the many worlds. Of course I am not simultaneously aware of the many times – that would be a contradiction in terms. But I have subjective evidence of the many times and no subjective evidence whatever of the many worlds. The problem of the apparent impossibility of objectifying the Technicolor reality of the experiential present (which deeply troubled Einstein) is therefore genuine. Your invocation of a multitude of coexistent worlds of which we are unaware, on the other hand, is a gratuitous solution of a pseudo-problem.

Well, I think you are seeing too much distinctions: I think that the "many times" and the "many worlds" are much more analogous concepts. 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 think both ideas are very similar in the following sense:
in the same way you can say that "you are only aware of "now" and other times simply don't exist ; history is simply an algorithm to calculate the present world, but the past doesn't exist", you can say that, of the many branches, only the one I live in, exists. But, in the same way as history has a clear influence on what is there today, you SOMETIMES have indications of the existence of the other branches. I gave in a parallel thread the example of the EPR style experiment: for me, this is the empirical verification that these branches exist. As I said there, if we wouldn't have had the entire discussion about local/non-local, and Einstein and Bell, I think that the EPR experiment would have been the best demonstration of the empirical evidence of parallel branches: namely of the quantum interference between two very macroscopic systems (namely Alice and Bob and their notebooks) which shows up in the EPR correlations.
Using locality and the spacelike separation, you allow for the phases of the two branches not to decohere, and when you bring them together again, you get an interference pattern ; in the same way as you get such a pattern when a lightbeam splits and comes together.

 

[CarlB]

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

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. It violates unitarity, it violates Lorentz invariance, ...

Yes, these are true. Bohmian mechanics, an alternate ontology to the MWI that has been promoted by some damned smart guys, violates unitarity and, indeed, violates Lorentz invariance.

... and it isn't even specified WHEN it should be applied - you're left to your intuition to do so.

This is not at all true. This gets to the heart of the "many times" viewpoint, I think.

In the quantum ontology, wave functions apply to situations in an indefinite future. That is, they correspond to experiments that have not yet been performed. To collapse a wave function requires that one wait for the experiment to be done, which is a passage of time that is not modeled in either QM or relativity. I agree that the transition between the future and past (as opposed to the evolution of wave functions as the time coordinate in spacetime is changed either positively or negatively, or the evolution of particle positions evolves in a classical theory) is not modeled in any of these theories. But it is the existence of this gulf, the difference between the past and future, that makes relativity compatible with quantum mechanics. In short, they do not apply to the same region of OUR experience, as opposed to spacetime. If anything, the only indication the difference between these theories suggest is that the ontology of spacetime which is in the future of a given observer is different from the ontology of the spacetime which is in his past.

So the entire question is: will it be possible to conserve the superposition principle, and unitary dynamics, while unifying with gravity, or does one need a fundamental change of this entire machinery. In fact, (Penrose style thinking) gravity is the last hope of killing off unitarity (and hence opening the possibility of introducing the projection postulate as a dynamical phenomenon). Even Penrose, who is certainly not in favor of the MWI view, thinks that the MWI view is the _only sensible interpretation_ if quantum theory turns out to remain strictly unitary in the presence of gravity. Hence his hope that gravity will introduce a deviation from unitarity.

To unite the wave and particle of a given experiment in the sense of the mathematical ontology, one must put the particle description into a form which is the same. One does this by using a few simple tricks from Bohmian mechanics. These tricks allow one to write a version of Schroedinger's equation where the Heisenberg uncertainty principle is violated.

A student really should spend an hour or two trying to set up an initial condition for the Schroedinger wave equation that violates the HUP. One eventually discovers that the reason that it is impossible is because the amplitude and the phase are related. To separate them out, one must rewrite the Schroedinger equation in a way that splits the information that gives the probability density from the information that gives, in the Bohmian interpretation, the velocity field for the possible particle tracks. Having split these, one can add an extra parameter that changes the wave function from one that obeys the HUP to one that allows particles to be restricted to a Bohmian trajectory.

I think this is a lot more natural than the MWI, which requires great straining of intuition, and it preserves the (approximate) correctness of the ontology of QM and relativity.

By the way, I've just released my first great hope to get allowed onto Arxiv.org. It gives a prediction for the masses of the neutrinos from the Koide mass formula for the charged leptons:
http://brannenworks.com/MASSES.pdf

Carl

 

[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

 

[vanesch]

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

I only skimmed through the article, but I think Kirkpatrick is somehow wrong when attacking Hughes' argument (with which I think I'm familiar) on top of p 2, because I think this argument is correct.
There's a difference between having independent statistics for the systems S and M, given by the two reduced density matrices, and the correlated statistics that will result when applied to the pure state. However, this correlation will not show, of course, if all measurements are of the form A_S x 1 or 1 x B_M (in other words, when we do not measure correlations between S and M, but do independent measurements on S alone, or on M alone), at least, in the case when the systems S and M are entangled (that means, that the pure state is not a product state). If the systems S and M are not entangled, then the individual reduced density matrices will also give rise to individual pure states. The only case in which the reduced density matrix is not pure, while the "master" state is pure, is when the system is entangled.

So the point of Hughes (which I think is correct), is that he physical state of a global system (S + M) in a pure state is not correctly described by only the mixtures given by the reduced density matrices. These only give the correct result for measurements who do not try to establish correlations between the systems S and M. Only the pure state gives the right correlations, while the local density matrices assume statistical independence and hence erroneous results for these correlations.Kirkpatrick might be right, however, that in absense of explicit projection, there's no such thing as a proper mixture if we didn't start with one. But that's known: it is the entire issue of the measurement problem: how to produce a genuine mixture, by unitary evolution, from a pure state (which doesn't work).

Simple example:

|psi> = a |+> |-> + b |->|+>

This gives, for the first system, a reduced density matrix as mixture:
a^2 |+><+| + b^2 |-><-|

and for the second system:
b^2 |+><+| + a^2 |-><-|

But an overall operator measuring the state:
a |+>|-> + b|->|+> ==> outcome 1
all other orthogonal states ==> outcome 0

would give expectation value = 1 when applied to the pure state,
and an expectation value below 1 when applied assuming statistical
independence.

 

[selfAdjoint]

From the paper

"Consider a composite system in the pure state \rho^{S \oplus M}, of which the component states are the mixed states \rho^S and \rho^M. For the sake of
argument, assume that \rho^S = a1 | u1 &gt;&lt; u1 | +<br /> a2 | u2 &gt;&lt; u2 |, while \rho^M = b1 | v1 &gt;&lt; v1 | +<br /> b2 | v2 &gt;&lt; v2 |, with a1 \neq a2 and b1 \ne b2, so there are no problems of degeneracy. Then, according
to the ignorance interpretation of \rho^S and \rho^M,
system S is really in one of the pure states | u1 > or | u2 >, and system M is really in one of the pure states | v1 > or | v2 >. . . . But this would mean that the composite system is really in one of the four states | u_j v_k &gt;, with probabilities a_jb_k respectively — in other words, that the composite system is in a mixed state. Since this contradicts our original assumption, the ignorance interpretation simply will not do."
This argument is so clearly stated by Hughes that its error stands out: the claim that “the composite system is in a mixed state” is not supportable — nothing external to S ⊕M distinguishes those states from one another. We must add the state vectors (not the projectors): |\rho^{S \oplus M}&gt; = \sum_{jk} \psi_{jk} | u_j v_k &gt; — a pure state.

So the indistinguishable states are the supposed components of the mixed states |u_jv_k&gt;, and they are indistinguishable because their difference cannot be observed.

 

[CarlB]

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.

Einstein's relativity, and classical mechanics in general, has to do with the rules governing the "fixed and settled" past. Quantum waves have to do with the open future, with the relation between these being given by the probability postulate. At least that is my version of things.

I realize that Einstein was not a fan of this sort of thing, but then again, he wasn't much of a fan of quantum mechanics in general. As long as physics rejects an objective split between the past and the future, it should be no great surprise that the laws of physics do not possesses an explicit arrow of time. The act of measurement give an explicit arrow of time.

Yes, the act of measurement can be reversed (at least in thought), and the laws of physics will work just fine, but that hardly proves anything. It doesn't make sense to apply probabilistic laws to an experiment that has already been made and whose result is known, so applying the laws of physics backwards is not a very useful thing to do. It is the action of measurement that indicates the arrow of time of course this arrow can be pointed in whatever direction one wishes to assume. But in our physical world, there can be no doubt which direction the arrow points.

Carl

 

[vanesch]

From the paper

<quotation>

So the indistinguishable states are the supposed components of the mixed states |u_jv_k&gt;, and they are indistinguishable because their difference cannot be observed.

No, that's the point. It is sufficient to use a measurement on the combined system which has eigenvalue 1 for exactly this pure state, and 0 for all the others, and this is a measurement that can distinguish between the pure state and the mixture. It is only if you lock yourself up in the basis of the original measurement that you can't tell the difference between a mixture and a pure state.

This measurement is of course a correlation measurement, and not a measurement on one of the systems alone.

 

[Bertrand]

Hello Vanesh,

Thank you for this very clear explanation !

I have now another one

You say this :

"Kirkpatrick might be right, however, that in absense of explicit projection, there's no such thing as a proper mixture if we didn't start with one. But that's known: it is the entire issue of the measurement problem: how to produce a genuine mixture, by unitary evolution, from a pure state (which doesn't work)."

In this sentence, I undersatnd that for the moment it is not proved that one may obtain a projection operator from unitary evolution.

However, recently I read an article from Zurek (Decoherence and the Transition from Quantum to Classical-Revisited) who states (page 4) :

" Decoherence leads to the environment-induced superselection (einselection) that justifies the existence of the preferred pointer states. It enables one to draw an effective border between the quantum and the classical in straightforward terms, which do not appeal to the "collapse of the wavepacket" or any other such deus ex machina"

In this sentence, I understand that Zurek claims he has solved this problem of constructing a projection operator from unitary operators. Is it really what he claims ?

Maybe this is not the right place to ask this question. I guess this topic must have already been addressed somewhere else in the forum.

Thanks again for your explanations,

Bertrand

 

[vanesch]

Hello Vanesh,

Thank you for this very clear explanation !

I have now another one

You say this :

"Kirkpatrick might be right, however, that in absense of explicit projection, there's no such thing as a proper mixture if we didn't start with one. But that's known: it is the entire issue of the measurement problem: how to produce a genuine mixture, by unitary evolution, from a pure state (which doesn't work)."

In this sentence, I undersatnd that for the moment it is not proved that one may obtain a projection operator from unitary evolution.

Worse, it is simply demonstrated that this cannot be the case! It takes 5 lines or so to show that a unitary operator can never be a projector. von Neumann knew that already, hence his distinction between "process 1" and "process 2". (process 2 was the unitary evolution, and process 1 was the projection)

In the language of hilbert states, process 1 is a projection (writing out the state in a specific measurement basis (eigenbasis of the measurement operator), and statistically picking out one component). In the language of density operators, process 1 is writing out the density operator in a specific basis (the measurement basis) and putting the non-diagonal elements to 0. This changes a pure state into a mixture.

Now, a density operator, under process 2 (unitary evolution) changes as follows: rho(t) = U(t) rho(0) U+(t)
and the condition for a pure state is: rho^2 = rho

Now, from this, follows simply that, if rho is a pure state at t = 0, then rho remains a pure state under process 2:

rho(t)^2 = rho(t) rho(t) = U(t) rho(0) U+(t) U(t) rho(0) U+(t)
= U(t) rho(0)^2 U+(t) = U(t) rho(0) U+(t) = rho(t)

So, once a pure state, and unitary evolution, always a pure state.

EDIT: I would like to add, that taking the reduced density matrices by partial trace, and putting these reduced density matrices again into an overall density matrix (which makes the measurements on the two systems statistically independent) is half way between process 1 and process 2: it puts the elements in the original overall density matrix to 0, which deal with correlations between the two systems, but doesn't put all non-diagonal elements to 0. But in any case, this cannot be obtained through unitary evolution.

However, recently I read an article from Zurek (Decoherence and the Transition from Quantum to Classical-Revisited) who states (page 4) :

" Decoherence leads to the environment-induced superselection (einselection) that justifies the existence of the preferred pointer states. It enables one to draw an effective border between the quantum and the classical in straightforward terms, which do not appeal to the "collapse of the wavepacket" or any other such deus ex machina"

In this sentence, I understand that Zurek claims he has solved this problem of constructing a projection operator from unitary operators. Is it really what he claims ?

Well, no. Decoherence doesn't solve the measurement problem. However, it can solve the preferred basis problem, and as thus, suggest a set of associated projection operators. The suggested preferred basis is then simply a basis that remains robust under time evolution (that is, each robust term by itself evolves while remaining within its "superselection subspace").

In fact, in the book "decoherence and the appearance of a classical world" by Joos, Zeh et al, it is clearly stated that decoherence does not solve entirely the measurement problem but only illuminates a specific aspect of it. Zeh himself wrote that (in chapter 2) and explained carefully what decoherence does, and what it doesn't. I know some claim that it does, but this is an erroneous claim.
However, *if* you have a way to tackle the measurement problem (by accepting one or other interpretational scheme), then decoherence shines some light on otherwise strange phenomena. The most natural setting to consider decoherence is an Everett-style interpretation of course, because of the fact that the apparatus and environment are provided with quantum states.

 

[Bertrand]

Hello, Vanesh !

Your answer has the merit to be clear ! I had difficulties to find out, from the articles I read, whether or not Zurek's decoherence theory was able or not to explain completely the measurement process. Some people I discussed with told me with certitude that yes indeed it did, however, I remained very doubtfull, some points remaining very unclear for me, especially, how does the theory explain why a given "pointer state" is "really selected" among all the possible states.

So far as I understand it, Everett's theory gives the same degree of reality to all the potential results of a measurement, and therefore requires that somehow the universe splits each time a measurement is performed. If this is really the correct interpretation, well, maybe the theory is interesting for some mathématical aspects, but for the rest, it seems crazy.

Concerning measurement, an other aspect that must be taken into account by a theory supposed to explain the detail of the measurement process, without making use of the projection principle contained in the Copenhague Interpretation, is the time irreversibility.

My understanding of his theory, is that Zurek obtains this time- irreversibility through a mecanism akin to the second principle of thermodynamics, that is, the principle of the increase of the entropy.

He seems to say that the measuring process is irreversible, because the "information" flows into the degrees of freedom of the environment, and the time reversal of this process is highly unprobable.

Do I correctly get his point, and in case of a positive answer, what do you think of this argument ?

Bertrand

 

[vanesch]

Your answer has the merit to be clear ! I had difficulties to find out, from the articles I read, whether or not Zurek's decoherence theory was able or not to explain completely the measurement process. Some people I discussed with told me with certitude that yes indeed it did, however, I remained very doubtfull, some points remaining very unclear for me, especially, how does the theory explain why a given "pointer state" is "really selected" among all the possible states.

Well, if you place yourself in an Everett view, then decoherence does describe the measurement *process* of course, but you're still confronted with the fact that all branches are present in the wavefunction.
Now, in an Everett-like view, you then just say that an observer is not a physical object, but a physical state, so if an object occurs in several entangled states, then that corresponds to so many subjective worlds (with different probabilities to be experienced).

So the "resolution" of the measurement *problem* comes from the interpretation, but at least the *process* is described using decoherence. what decoherence brings in (once one has a scheme to resolve the measurement process), is two things: first of all, the absence of quantum interference between branches. This is important: even if there are parallel branches, in order for us to live our subjective life in one of them, it is important that most of the time, we do not get disturbed by "neighbouring branches". In other words, decoherence helps us understand why the "neighbouring branches" become unobservable.
The second point decoherence brings us, is: why things seem to have definite positions (in a branch). This comes from the structure of the interaction hamiltonian between a system and the environment, which is highly position-sensitive. This means that only coarse-grained states with approximately given positions to objects can be robust against time evolution.

So far as I understand it, Everett's theory gives the same degree of reality to all the potential results of a measurement, and therefore requires that somehow the universe splits each time a measurement is performed. If this is really the correct interpretation, well, maybe the theory is interesting for some mathématical aspects, but for the rest, it seems crazy.

I'm in fact an Everettian. Not that I think that this is ultimately the correct world description (I'm ignorant of that), but I think it is the correct view on the formalism of quantum mechanics as it stands now, simply because it is the only view in which there is no arbitrary distinction between "measurement" and "evolution", and because it is the only view that allows for an analysis of the measurement process itself. Moreover, the Everett view is the only ontological view that is entirely compatible with the standard QM predictions, and locality. All other views which give an ontological interpretation need some form of non-locality.

All other interpretations are in fact cutting away "a piece of quantum mechanics", and do not want to face what this theory tells us. It would be a bit like using the Lorentz transformations, but saying that time dilatation doesn't apply to human beings or so.


Now, I've exposed so many times my views on this, here, that I'm not going to reiterate them (do a search on my name and MWI or something if you're interested). However, there's one common misconception of the Everett view I want to correct: the *universe* doesn't branch, it is only the *observer* that branches.

Concerning measurement, an other aspect that must be taken into account by a theory supposed to explain the detail of the measurement process, without making use of the projection principle contained in the Copenhague Interpretation, is the time irreversibility.

My understanding of his theory, is that Zurek obtains this time- irreversibility through a mecanism akin to the second principle of thermodynamics, that is, the principle of the increase of the entropy.

He seems to say that the measuring process is irreversible, because the "information" flows into the degrees of freedom of the environment, and the time reversal of this process is highly unprobable.

Do I correctly get his point, and in case of a positive answer, what do you think of this argument ?

I think so, yes. This is the way irreversibility can be reconciled with time-reversible microdynamics: you start from a very unprobable initial state, and then it evolves most probably into more probable states. This is "irreversibility". It's only statistical, but if the probabilities are small enough to go backward (which they are), you will not observe it.

 

[Bertrand]

"Now, I've exposed so many times my views on this, here, that I'm not going to reiterate them (do a search on my name and MWI or something if you're interested)"

OK Vanesh, I'll do this when I come back in 3 weeks.

Thank you again for all your explanations,

Bertrand