I do not think so. I can give you an example: consider the travelling salesman problem for 10000 cities. If P not equal NP, then finding the smallest route that passes through all cities will probably take more than the age of the universe. However I'd say that such a route is very very real, it exists, however we cannot find it (compute it) in a reasonable amount of time.

Such an example poses a logical circularity: Let us suppose that there exists a smallest route passing through N cities. Then, since such a route (i.e., a solution to the particular NP-complete problem) really exists, it would definitely correspond to an element of physical reality.

But then again, why do we need to solve the problem if we already know the answer?

It is not absolutely accurate.

 The wave functions of the Universe are the exact solutions to the Wheeler-DeWitt (WDW) second-order functional differential equation – the quantum gravitational analogue of the Schrödinger equation (SE). So, similar to the quantum-mechanical wave functions (i.e., the exact solutions to the SE), the exact solutions to the WDW equation contain the maximum information about the Universe, that is, all of the information about the geometry of the Universe and its matter content, which obviously includes all planets, stars, galaxies, the contents of intergalactic space, the smallest subatomic particles, and observers with their measurement apparatuses.

In Roland Omnes nice book 'Understanding Quantum Mechanics'(1999) we read (p. 237):

'People who believe in a wave function of the universe have given a curious twist to the second assumtion of realism: ... What mathematics can lead some people to dream of is strange ....'

To give a specific equation fullfilled by the 'wave function of the universe' does not help. What you write about maximum information is exactly of the kind  ' What mathematics can lead some people to dream of is strange ....' A world in which you are missing and we would count one polar bear more than we have in our world, would be consistent with the Schrödinger equation in the same silly sense in which any world can satisfy a equation.

You know, in physics, unlike literature or poetry, emotional words and expressions – even if they are of high quality, great beauty, sincerity or intensity, or profound insight – do not make much difference.

It does not matter how amusingly clever in perception and expression you can laugh at the wave functions of the Universe. It does not matter whether or not you consider this construct pompous. What is really matter is only this: Can you prove or at least show within the frame of known theories and formal logic that those wave functions of the Universe physically meaningless?

@Arkady, if you believe in Goedel's proofs about incompleteness of any logical system, then assuming a global wavefunction of the universe (say |PSI>), we will always have undecidable questions, i.e. projections onto |PSI>

Arkady,

givent that nobody so far was able to make reasonable use of wavefunctions of computers, automobiles, airplanes, cities, ... is it reasonable to take the concept for granted in the case of the universe? How can you expect help from formal logic in this case?

@Vlad like your answer.

I think computability is an interesting question--let's leave aside whether Wheeler-DeWitt is "right".

The idea that the universe is a (the only?) closed quantum system described by unitary evolution is appealing and seems to make some sense--apart from the fact that quantum gravity is not something we have under control!

Your question is really "if we can not compute the wave function, is it an element of reality?". That may be relevant for small isolated systems (if they exist!). If you do allow quantum computers, the answer is obviously yes!  Almost by definition, the information contained in the universe requires a universe to simulate (compute) it--so yours seems a circular argument!

That is Vlad's projection statement: no-one can extract information about the whole universe, because they are part of the universe. As soon as you divide the universe between an observer and the remaining universe, we get all kinds of interesting questions, to which the answer would be cute. The fact that we can't observe the whole universe makes things even more tricky.

Okay people, in order to continue a discussion we need to throw some obvious things out of the way.

First, as much as physics is not poetry, it is not philosophy either. This implies that questions like “Can the Universe be measured?” or “Can someone – being a part of the Universe – extract the information about the Universe?” should be answered strictly within physical science, not philosophy. And whatever hot debates they may stir in philosophy, in physics these questions are answered in the affirmative.

Second, nor physics is mathematical theory. This means that even theoretical physics is not built as a formal language (on a set of well-formed statements) and therefore Gödel’s original incompleteness theorems (based on formal language theory) are not applicable to physics. More correctly speaking, their application would extract the trivial and well-known result – physics as a science is not complete (and it would never be). That is, there exist statements in physics that are true but that cannot be proved to be true within physical theory. For example, confining the velocity v of a physical object to the speed of light in vacuum c cannot be proved to be true within relativistic theory. 

Third, skepticism in science is useful but it should know its limitation and not become nihilism. Sure, one can doubt that the Universe is a closed system, but doing so, he should present analysis of faults and problems in the existed theoretical scheme, which is based on the assumption of a closed system or leads to it. Merely saying that we really do not know what is going on with the Universe does not help us to know more about what is going on with the Universe.

It is a little more complicated than that, Charles. It is an argument based on conservative heuristic ideas that proceed from the extrapolation of established and empirically confirmed concepts beyond their present range of application.

For example, Isaac Newton applied principles of macroscopic physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data. Consequently, he created celestial mechanics – a mechanical model of the Universe. Later on, in the 20th century, applying the superposition principle of quantum mechanics (established and empirically confirmed for systems made up of a small number of microscopic constituent particles) to the Universe as a whole system, Feynman, Hartle, Hawking and some others came out with the construct of quantum gravity.

Thus, it is a way that physics usually evolves. The provocative question is, can we put the finger on something within physical theory or mathematics that makes such a way impossible?

But ask yourself why? Why we can stretch one physical principle (physical universalism) but cannot do the same with another (the superposition principle)?

 No, you’re mistaken if you think it is simple. Actually, the opposite is true: this puzzle has withstood the mighty efforts of Einstein, Bohr, Feynman, John Bell, and many others.

The quantum superposition principle results from the property of linearity of the solutions to the Schrödinger equation; but this equation contains nothing prohibiting its application to any objects including macroscopic ones, and even the whole Universe.

In other words, the Schrödinger equation has no factor, which can ‘switch’ it off in situations where microscopic systems grow into macroscopic ones. The lack of such a ‘switch’ gave birth to various foundational problems and paradoxes in quantum mechanics including the Schrodinger’s cat paradox and the measurement problem.

Indeed, if quantum theory did not have any ‘switch’ at all, then Schrödinger’s equation would be scalable on the system’s constituent particle number N. This means that the dynamics of a macroscopic system composed of N microscopic parts (whose behavior is described by the laws of quantum mechanics) would depend, through the corresponding many-body Schrödinger equation, on a macroscopic quantum Hamiltonian.

Even so microscopic parts of the macroscopic system can actually be in a superposition state, it is not a priori clear what physical properties should be associated with the macroscopic system described by a generic quantum superposition resulted from the macroscopic (i.e., many-body) Schrödinger equation.

Aha, I got it. You are promoting your own formulation of quantum mechanics (QM). I am familiar with your Arxiv paper, I came across it some time ago. Nevertheless, I read it again. 

I might – but prefer not to – classify your formulation trying to (more-or-less successfully) assign it to one or a few known formulations/interpretations of nonrelativistic QM. I do not want to do this because as well as any other QM formulation yours uses the Schrödinger equation as “a way of calculating the probability for the outcome of an experiment”. However, given the NP-hardness of this equation, this makes your interpretation of QM infeasible, i.e., practically useless, for anything else that significantly differs from a microscopic system.

And that is the essence of the problem.

I will read your relativistic treatment. Meantime, let me tell you what I think about various alternative interpretations of quantum mechanics.

The problem is not to come out with an interpretation (a formulation), which on the one hand, would allow quantum theory to be a complete theory seamlessly encompassing all systems in Nature (including observers with quantum states of their own) but on the other hand, would leave the widely accepted mathematical structure of quantum theory essentially intact. The real problem is to do this with the number of assumptions as little as possible.

Indeed, according to Occam’s razor principle, the interpretation of quantum mechanics with the fewest assumptions is the best one. In other words, among competing interpretations, the one with the fewest assumptions should be selected.

And therein lies the dilemma. The traditional Copenhagen interpretation, with its axiomatic Born rule for computing empirical outcome probabilities and its notion of wave-function collapse for establishing the persistence of measurement outcomes, has the fewest assumptions ever. And this explains why according to surveys, the Copenhagen interpretation is still the most popular interpretation today.

Unfortunately, your interpretation has too many assumptions to be chosen over the Copenhagen interpretation.

But explaining “what the maths actually means” connotes interpreting since usually “interpretation” is understood as a way to establish some relations between the mathematical formalism of quantum theory (‘math’) and elements of physical reality.

Look, Charles, I do not criticize your formulation, and neither do I advocate the Copenhagen interpretation. What I am trying to say here is that the Copenhagen interpretation has managed to establish those relations (between ‘math’ and physical reality) in the most efficient (‘economical’) way using the fewest assumptions possible.

Does it mean that this interpretation of quantum mechanics is rightest one? If the concept of simplicity or elegance means something in science, then yes, we have to agree that the Copenhagen interpretation should be selected.

Dear Charles, 

I think your interpretation of quantum mechanics (QM) differs from the Copenhagen (‘standard’) interpretation in at least two fundamental points.

First, unlike the standard QM interpretation where the states of a quantum mechanical system are vectors (kets) in a complex separable Hilbert space, the observables are Hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections, in your interpretation of QM Hilbert space is considered “as a formal language which allows us to mathematically describe the behavior of matter in a universe in which position exists only as a relative quantity”, and so kets are interpreted “as formal conditional clauses, rather than as propositions”. 

Second, your interpretation does not follow the attempts of the standard interpretation to treat the macroscopic world (of observers and classical apparatuses) and the quantum world separately, i.e., as being governed by the different sets of laws. Quite the opposite, your interpretation of QM is based on the von Neumann’s idea, of applying quantum mechanics not only to microscopic quantum systems but also to the entire measurement process and its macroscopic apparatuses. 

Again, I do not criticize your approach and I am not going to seek its internal flaws. But admit: even if your interpretation were flawless, it would be overloaded with far more extra assumptions completely absent in the Copenhagen interpretation.

Arkady, the Copenhagen interpretation is, regarding simplicity, and even more elegance, a mess. 

It contains a subdivision into quantum and classical part, which is completely nonsensical conceptually, because fundamentally the laws of nature are the same for small and big things. It contains a subdivision between "normal" Schrödinger evolution and "measurement", where it is clear, that measurement is, of course, a quite normal physical interaction. 

It was sufficient for all practical purposes - for the particular situation of experimental science. Accepting that its philosophy does not really make sense, it has been reduced to the minimial "shut up and calculate" interpretation. 

First understand what the wave function stands for, then you immediately see that it is impossible for the universe to own a wave function.

See http://www.e-physics.eu/MathematicalModelOfReality.pdf

Dear Ilja,

As I said before, the depth of someone’s feeling towards a particular theoretical concept does not make his arguments for or against the given concept more logical or more convincing. You may call the Copenhagen interpretation of quantum mechanics “a mess”, “senseless” or whatever you like but it does not make the majority of physicists – practicing ones and theoreticians alike – agree to abandon this interpretation for good. To bring about an argument of logic, not an argument of emotion, that would demand the rejection of the Copenhagen interpretation with necessity you need something more than just sentiments.

On more serious (less emotional) note, the Copenhagen based interpretations constitute a consistent framework for the description of the physical world by means of the quantum mechanical formalism.

As to the assumption that the dynamics of the macroscopic apparatus also depends through the Schrödinger equation on a quantum Hamiltonian and thus the measurement process takes place between the two quantum systems, it is just an assumption, regardless how extremely reasonable it may seem.

Dear Charles,

I do not use my "own set of assumptions and call it standard Copenhagen interpretation". In the standard Copenhagen position it is postulated that the world is governed by different laws: quantum mechanics for the microscopic world, and classical physics for the macroscopic, directly accessible, world. Moreover, even though the microscopic system can actually be in a superposition state, in its interaction with a macroscopic apparatus, perhaps the observer herself, it will necessarily “collapse" to an eigenstate corresponding to the observed physical quantity. The boundary between microscopic and macroscopic regime is not specified. 

In the future, I recommend you to check your facts more thoroughly before accuse someone over the Internet.

Dear Hans, 

Could you be more specific and elaborate please what exactly you said in your post? Do you assert that I do not understand what the wave function stands for? Or you meant something else?

Arkady, if you want less emotional arguments, no problems, see http://arxiv.org/abs/0909.0175 and don't forget that is only something I consider as new. All the old problems, starting with the measurement problem (which is, in de Broglie-Bohm interpretation, not a problem at all), and all what you can find in Bell's "speakable and unspeakable" (ok, this also contains such words like "unprofessional" ;-) ) should not be forgotten too. 

Dear Ilja,

I am aware of your argument that there is no way “to save the Copenhagen interpretation: Once it is recognized (because of the non-uniqueness problem) that the classical part contains nontrivial physical information, and as long as this classical part remains vague, there seems no way to find out the true symmetry group of a quantum theory in this interpretation.”

 Effectively, this may be expressed in one sentence: Within the Copenhagen interpretation, microscopic-macroscopic interactions cannot be described in the pure terms of quantum theory. For the record, I agree with that.

 With all that being said, I do not want to open here a discussion on the merits and demerits of the Copenhagen interpretations and its alternatives since I do not believe that Research Gate is an appropriate place for this. The issue is too much serious and too much academic to decide it here by up- or down-voting.

What the wave function stands for is not clear. It is generally accepted that it is a probability amplitude distribution and that its squared modulus defines a probability density distribution, where each function value defines the probability of detecting the owner of the wave at or near the location that is defined by the parameter of the wave function. However, even if this is agreed, then still what is described can be all sorts of things. For example Let the owner be a point-like object and let the density distribution correspond to a discrete location distribution (a swarm) that describes where the owner was, is or will be during a small period of time. 

This means that the wave function reflects the fine grain behavior of the owner during a probably very short period. This picture would fit an elementary particle, but not a complete universe.

It also shows that the wave function hides very much of the fine grain structure and fine grain behavior of the things that it describes.

Dear Hans,

What the wave function stands for strongly depends on whom you ask this question. Some would answer you that it represents information in the mind of the observer, i.e. a measure of our knowledge of reality. But others would insist that the wave function really exists, i.e., it must have an objective, physical existence.

Withal no one would argue about what the wave function describes: It describes the quantum state of a system and its evolution over time. And this makes it possible to assign the wave function to the whole Universe.

For example, Hartle and Hawking in their celebrated paper “Wave function of the Universe” describe the quantum state of a spatially closed universe by a wave function, which is a functional on the geometries of compact three-manifolds and on the values of the matter fields on these manifolds. This wave function obeys the Wheeler-DeWitt second-order functional differential equation.

Dear Charles,

Please read my posts carefully: I never used the term “standard Copenhagen”. In fact, I described the position of the Copenhagen (‘standard’) interpretation. So, if you would be so obliging, do not put your words in my mouth (and then argue with them).

I do not know what you call ‘orthodox’ interpretation since you never gave the definition of it. As to the Copenhagen interpretation (which many theorists call the ‘standard’ one), it explicitly emphasizes the necessity of classical concepts for the description of the quantum phenomena. For what it’s worth, it is still the most popular interpretation of quantum mechanics.

Dear Charles,

Again, please read my posts carefully: I repeatedly said I do not criticize your approach and I do not promote the Copenhagen interpretation.

I think our personal communication (which, by the way, did never contribute even a bit to the main question of this thread) long ago passed the point of being cool, polite, and – more importantly – useful for others. So, let us stop it right now.

If nevertheless you feel like you want to continue you can reach me by email.

Lee Smolin in “Elegance and Enigma: The Quantum Interviews”: "[...] To put it differently, the only interpretations of quantum mechanics that make sense to me are those that treat quantum mechanics as a theory of the information that observers in one subsystem of the universe can have about another subsystem. This makes it seem likely that quantum mechanics is an approximation of another theory, which might apply to the whole universe and not just to subsystems of it. [...]"

Charles,

The 1936 paper of von Neumann and Birkhoff defined a skeleton relational structure (which they called quantum logic) is present in separable Hilbert spaces as a set of closed subspaces. When this is explored, then this quickly leads to a set of discrete locations that can be described by a wave function. The set is a spatial reflection of the fine grain behavior of a point-like object that is the owner of this wave function,

See: http://www.e-physics.eu/MathematicalModelOfReality.pdf

Charles, You better read the paper first and then take conclusions.

In my model, the wave function of an elementary particle is constantly regenerated at a very high frequency. In the model, the wave function describes a location swarm. That location swarm is the spatial reflection of the fine grain dynamics of a point-like object.

When considered as a wave package, the wave function would disperse during the movement of its owner!

On the other hand this descriptor of a discrete set is a continuous function that in addition has a Fourier transform. This fact means that the swarm owns a displacement generator and as a consequence in first approximation the swarm can be considered to move as one unit.

All these consequences are due to the fact that at every movement step of the owner the wave function is renewed.

The location swarm is recreated during a fixed duration period and is constantly reflecting the dynamic properties of its owner. As a consequence a hopping path exists that walks along the elements of the swarm. If the swarm does not move, then this path is a closed string.

Dear Joris,

That particular idea of Lee Smolin regarding interpretations of quantum mechanics consists of at least two conceptual elements. The first element is that quantum mechanics is a theory of the information, and the second one is that quantum mechanics is an approximation of another theory.

With all due respect to Prof. Smolin (and I do have great respect for his vision of quantum theory), I can accept such an idea only with some reservations concerning both of these elements.

First, while nowdays it is quite common to characterize quantum theory in terms of information-theoretic principles – see for example Bub’s approach [Stud. Hist. Phil. Mod. Phys. 35 B, 241–266 (2004); Found. Phys. 35(4), 541–560 (2005)] that treats quantum theory as a “principle” theory about (quantum) information rather than a “constructive” theory about the dynamics of quantum systems – such an approach cannot be considered flawless. As Hagar and Hemmo demonstrate [Found. Phys. (2006)] if the mathematical formalism of quantum mechanics remains intact then there is no escape route from solving the measurement problem by constructive theories, and so quantum mechanics on the information-theoretic approach is incomplete.

Second, the idea that beneath quantum mechanics, there may be a deterministic theory with (local) information loss is not a new one; see for example Gerard ’t Hooft paper “Determinism Beneath Quantum Mechanics”, [arXiv:quant-ph/0212095v1 16 Dec 2002]. As ’t Hooft notes, this may lead to a sufficiently complex vacuum state, and to an apparent non-locality in the relation between the deterministic (“ontological”) states and the quantum states, of the kind needed to explain away the Bell inequalities.

However, no empirical hint for such a drastic modification of quantum theory exists so far. Moreover, a modification of quantum mechanics such as this would involve some radical changes in our views on Nature. For example, many or all of the familiar symmetries of Nature, such as translation, rotation, Lorentz and isospin symmetry, would have to be symmetries that relate beables to changeables. This means that the ‘ontological’ theory behind quantum mechanics would not have these symmetries in a conventional form.

Dear Hans,

I read your essay with great attention and as reading it, I could not help but wondering what was the aim of your paper? While I completely agree with some of your revelations (e.g., physicists tend to deny completely or largely deduced models), I did not get what was your motivation to produce this paper? Do you consider that current interpretations of quantum mechanics are deficient and so you come forward with your own one? Or do you have another reason in mind? Please share.

Dear Arkady, thank you for sharing valuable comments on 2 concepts contained in Lee Smolin's citation. Besides this, the citation also indicates that quantum theory deals with subsystems, suggesting that the wavefunction for the universe lies out of scope of well established quantum theory as we presently know it ...

Charles, "I also know how to prove that there can be no determinism underlying quantum mechanics, and am not much impressed by people who have disregarded the proof."

And every else will not be much impressed by your proofs, given the counterexample - de Broglie-Bohm theory - is well-known and sufficiently simple. One adds a simple single equation and quantum theory is made deterministic.

Dear Joris,

That part of Smolin’s citation, which mentions that quantum mechanics deals with subsystems, refers to the decoherence program.

Decoherence has been claimed to provide an explanation for the quantum-to-classical transition by appealing to the ubiquitous immersion of virtually all physical systems in their environment (“environmental monitoring”). However, one issue is looming big as a foundation of the whole decoherence program: It is the question of what are the “systems” which play such a crucial role in all the discussions of the emergent classicality. As a matter of fact, a compelling explanation of what are the systems—how to define them given, say, the overall Hamiltonian in some suitably large Hilbert space—is still absent in the decoherence program.

By definition, the universe as a whole is a closed system, and therefore there are no “unobserved degrees of freedom” of an external environment, which would allow for the application of decoherence to determine the space of quasiclassical observables of the universe in its entirety.

However, this is not the only problem in quantum gravity. Other problems are the problem of time, the information loss paradox (or black hole information paradox), just to name a few.

Dear Ilja,

In passing, let me note that despite the fact that the variety of interpretations of quantum mechanics has acted to divide the physics community into camps (for example, one might be a “Bohmian” or an “Everettian” or in the “I shut up and calculate” camp), there is virtually no travel between camps, but only campaigning for new recruits.

Arkady,

The statement the universe as a whole is a closed system, and therefore there are no “unobserved degrees of freedom” of an external environment, bothers me. We know that the observable space only comprises our past light cone, and that the cosmic expansion continuously reveals new space which were outside yesterday's horizon.

Thus there indeed is an external environment with unobserved degrees of freedom.

It is then questionable what is meant by the universe as a whole.

Decoherence suffers from some defects. But the well established quantum theory as we presently know it, without the need of invoking decoherence, refers to a subsystem as measured by an external observer. The observer compares his measurement results with his knowledge about the system (described by wavefunction or density operator) taking into account the measurement setup (described by measurement operators) using Born’s rule. From this follows that we cannot describe the universe which per definition cannot be measured by an external observer. The universe seems to be out of scope of well established "ordinary" quantum theory.

Arkady, what do you mean by "no travel"? Simply no communication between the different camps? There is some, I have participated in this communication by reacting against criticism of dBB/Nelson coming from other camps ( http://arxiv.org/abs/0904.0764 Overlaps in pilot wave field theories, http://arxiv.org/abs/1101.5774  An answer to the Wallstrom objection against Nelsonian stochastics) and by proposing new arguments against these other interpretations ( http://arxiv.org/abs/0909.0175  A symmetry problem in the Copenhagen interpretation, http://arxiv.org/abs/0903.4657 Pure quantum interpretations are not viable, http://arxiv.org/abs/0901.3262  Why the Hamilton operator alone is not enough).

No people switching from one camp to the other? I disagree too, because all those who support dBB today have been coming from somewhere else, nobody learns dBB in its first course. No development of new interpretations?  Wrong too, I have been in the dBB camp for some time, coming from the "shut up and calculate" camp before, but have developed now my own interpretation, see http://arxiv.org/abs/1103.3506  

Which is, in some sense, a combination of the dBB interpretation with the Bayesian one (with some influence coming from Fuchs). 

What happens to two entangled states outside our light cone when one of them comes into sight because of the expanding horizon?

Charles, no, there is no confusion, except on your side. What is necessary to prove is that 

d_t |psi(x,t)|^2 = nabla ( |psi(x,t)|^2 v(x,t))  

where v(x,t) is defined by the guiding equation. This is a consequence of the Schrödinger equation, which is also a partial differential equation which is defined at every point x,t.  

Charles, whatever v means in your version of quantum theory is irrelevant.  In dBB theory v is defined by the guiding equation, and the equivalence proof uses the v as defined by the guiding equation and uses the Schrödinger equation for this purpose, as it is, as a partial differential equation.  

So, if you have invented some notion of "wave velocities", fine, be happy with it, it does not matter, because all what is necessary and used is the Schrödinger equation itself.  There is no obligation at all that the v of Bohmian mechanics has to be named "velocity" or so in some quantum text book or so.  Forget about naming something "velocity", all we need is to prove that 

d_t |psi(x,t)|^2 = nabla ( |psi(x,t)|^2 d_i(Im ln psi(x,t)))

follows from the Schrödinger equation. 

Charles, there is no need to look for these notions, I know them, fine that you already know that there is more than one meaning of "velocity" (group and phase), may be in some future you will also learn that the in dBB theory there is yet another notion, and learn not to mingle them. 

Then, the next step would be to understand how measurements are described in dBB theory, to understand the simple theorem that the results of quantum theory are recovered, but that, for example, a momentum measurement in dBB does not measure m v of the dBB velocity. 

Dear Thomas,

Thank you for taking part in this (still active) discussion.

Unfortunately I cannot share your believe in physical structures of wave functions of the Universe. Here I rather hold the position of Quantum Bayesianism (QBism) asserting that a quantum state of any system does not represent an element of physical reality but an agent's personal probability assignments, reflecting his subjective degrees of belief about the future content of his experience. According to QBism, a wave function is not physical and thus cannot possess an energy (as well as any other physical characteristic).

Dear Matt,

I've just noticed the problem you posted in the last November: “What happens to two entangled states outside our light cone when one of them comes into sight because of the expanding horizon?”

I found this problem interesting. The first that came to my mind is that such a phenomenon can be understood by recalling that in quantum field theory the vacuum is populated by virtual pairs of particles and antiparticles. So, the expanding horizon can promote one member of a pair into a real outgoing particle (generating a radiation similar to Hawking’s one). On the other hand, the expanding Universe expands into itself (and not into some empty space populated by virtual particle pairs). Thus, we should conclude that the question is not correct. To be exact, even without a complete theory of quantum gravity we can say that there is nothing beyond the expanding horizon.

Charles, may be for you "that means it is not actually a particle velocity at all", but who cares about what positivists think about what is really a particle velocity? 

I name velocity the first time derivative of the position. You may name velocity some quantum measurement connected with momentum or so, whatever you like. Once there is no trajectory in positivistic interpretations of quantum theory, you have to use something else, but this is not my problem. 

Arkady,

I meant the horizon of a Schwarzschild black hole, not the finite horizon of the part of the Universe observable to us.

Dear Matt,

Sorry for misreading your question. Now I am totally confused what this question was about. Did you ask about the last epoch of a black hole evolution when the black hole is evaporating and its horizon is shrinking (and thus expanding for the observer placed at infinity)?

If so, then the remote observer would register an increasing Hawking’s radiation followed by the explosion of the remnant of the black hole. But then again, since such predictions were made using static black hole background and you question concerns the dynamic process, it is difficult to be absolutely sure in the correction of the above picture.

Charles, the initial condition of a particle itself is insufficient, one needs, of course, also the initial value for its wave function. Then, if you measure something, you also need the particular measurement interaction and the initial values for the measurement device, that means, its wave function and position.

Then, the momentum of a quantum particle is measured in the same way as in quantum theory.  Thus, you create an interaction using an interaction Hamiltonian which measures p. The initial value will be some delta(q_p)psi(q_sys), with psi=sum_p a_p e^ipx, the final result will be some sum_p a_p delta(q_p-p)e^ipx, all this together with trajectories q_p(t) of the pointer of the measurement device and q_sys(t) of the system itself. 

Then the new effective wave function for the particle is obtained by putting q_p(t)=r into the full wave function and normalization, which gives e^irq.  Thus, you see that the result also depends on the particular way of the momentum measurement and the initial value of the measurement device itself. 

If everything was initially (system and measurement device) in quantum equilibrium, we can also compute the probability of the value of r as being a_r.  Only in this case, the QM prediction is recovered. 

Arkady,

I was referring to the stable situation when the black hole is accreting matter from the outside of its horizon. Among that matter there may be quantum-mechanically entangled particle pairs. When one of the pair crosses the horizon there cannot be any entanglement anymore, so the other particle becomes real and free.

It is the same situation when one observes one entangled particle. The observation destroys the entanglement so the other particle becomes real and free. 

Charles, as you can see reading my description, the result of momentum measurement is predefined but depends on a lot of things - the particular way of measurement, the initial value of the wave function of the measurement device and of above positions of the system as well as the measurment device.  So, it is not defined by the position and the wave function of the particle alone. Given only the data you mention (position and wave function of the system) the result of momentum measurement is indeed not defined in a deterministic way.   You also have to specify the initial data (inclusive position of the measurement device) to compute a deterministic prediction.  Without these additional data no determinism.

And QM predicions are recovered only for quantum equilibrium - for the system itself as for the measurement device - that means, in a state where we do not know the positions, nor of the system itself, nor of the measurement device. 

If Garret Birkhoff and John von Neumann in 1936 discovered a skeleton relational structure that forms the foundation of a model of physical reality, then this structure can be interpreted as part of a recipe for modular construction. This skeleton relational structure is present as the set of the closed subspaces of a separable Hilbert space. It means that every discrete object in universe can be represented by a closed subspace of that Hilbert space and each discrete object is either a module or a modular system.

Closed subspaces of Hilbert spaces are spanned by eigenvectors of normal operators., thus these subspaces correspond to sets of eigenvalues. These eigenvalues can be real numbers, complex numbers or quaternions. Thus discrete objects are represented by sets of quaternions. These sets can constitute coherent location swarms. A coherent location swarm can be described by a continuous location density distribution. The squared modulus of the wave function of an elementary particle corresponds to such a continuous location density distribution.

In this way it is easy to interpret the wave function of an elementary particle as a descriptor of that particle. The descriptor of the whole universe would consist of the descriptor of the superposition of all closed subspaces that represent discrete objects. It describes a huge closed subspace of the Hilbert space. However, it does not describe a coherent location swarm. Instead it describes a set of separate coherent location swarms. The notion of a wave function does not make much sense for that set.

Dear Matts,

Thank you for clarification: Now I can see clearly the problem you are referring. Nevertheless, if I may, I would rephrase your problem as follows: Does a black hole destroy quantum entanglement?

I consider such a problem very similar to the black hole information loss paradox. Indeed, let us take a quantum system comprising a pair of coupled (entangled) particles in a pure state and throw it into a black hole of mass M. Wait until after the hole has evaporated enough to return to its mass previous to throwing anything in. As we have ended up with a mixed (thermal) state and a black hole of mass M, we have found a process that destroys the information about the initial quantum state of the system.

Now let us throw only one of the coupled particles into the black hole. Again, as in the previous example, after the black hole evaporates, the information about the initial quantum state of the system describing particle entanglement will be destroyed and thus the remained particle will be set free.

Charles, of course in quantum equilibrium we have the uncertainty principle.  And once the state is in quantum equilibrium, it remains there.  So we have no way to prepare states with a given wave function together with a position (q, psi(q)).  

This impossibility to prepare such a state does not change the fact that the evolution of this state is deterministic, which can be easily seen because of the equations. 

Charles, the non-normalizable delta function is a degenerated case one does not have to care about. In a meaningful limiting procedure with localized but normalizable wave functions everything works as in the general theorem.  

I have explained here how momentum is measured, the explanation was completely deterministic, but requires also the wave function of the measurement device and its position. So reread that post, or any other introduction into dBB theory, and stop to distribute claims which are false and have been rejected many times. 

Charles, of course "IljaMechanics" is simply standard dBB theory, which you fail or refuse to understand.  The hidden variables are the standard variables of dBB theory, that means, the wave function and the configuration.  Once we want to describe an interaction named "measurement" in Copenhagen-like interpretations, one needs the full wave function and the full configuration of all interacting things to predict the result in a deterministic way.

Of course, the wave function of the apparatus may be a separate one only at the begin, during the interaction (measurement) it becomes entangled with the "measured" system. 

Are we talking about dBB theory or some versions of Copenhagen?  In dBB theory we have, of course, a wave function for everything we want to consider. 

But your refusal to accept that for momentum measurements the "hidden" variable of the measurement device is necessary for a deterministic prediction is interesting. Because this is the well-known property of contextuality of dBB theory. 

Charles, I have read your paper, and have criticized it here, already, and I simply inform you here about elementary errors in your understanding of dBB theory, errors elementary enough that each reasonable introduction into dBB theory would be sufficient to understand them.  I know that the mainstream rejects dBB, but the arguments for this rejection are quite different, than your errorneous misinterpretations. One can understand that a mainstream scientist rejects dBB because it requires a preferred frame, or that it destroys the symmetry between p and q in the quantum formalism because it prefers q, these are arguments worth to be discussed. But you make here elementary false claims, in contradiction with proven equivalence theorems, This is a different category.  

Charles, the predictions of quantum theory are reproduced for all measurements, including momentum measurements.  And, again, quantum predictions are predictions in quantum equilibrium, thus, of course not deterministic, but probabilistic, as in the classical Liouville equation. The theory makes deterministic predictions if wave function and configuration is specified, which is not done in quantum equilibrium. 

I have reproduced standard dBB theory, not "imagined some way of introducing more hidden parameters", because the cofiguration of the apparatus is in dBB necessary for the equations of a "measurement" process. Its you who makes false claims in contradiction with proven equivalence theorems, proven long ago and not questioned by the mainstream. 

It is clear and obvious that dBB theory does not fulfill the conditions of von Neumann's theorem as well as all the other impossibility theorems, because it is contextual and violates Einstein causality. 

Charles, no, it is done for momentum meausurement, I have given the formulas, of course, one needs the cofiguration of the measurement device as well as the full wave function for this too, and, again, this is not speculation but the prescription, the necessary consequence of the dBB equations, and this is well-known since Bohm's paper. Because this extension from position measurement to all other quantum measurements was the main new result of Bohm's paper in comparison with de Broglies original theory. 

Only repetition of already rejected arguments, so it seems time to finish this discussion. Its already clear that you have no counterarguments against the points I have made, but are unwilling to accept anything.  Bye.