# Does the wave function/density state actually exist?

I have been reading with interest the debates here on whether the wave function/density state actually collapses or not, or whether it is subjective Bayesian or objective with actual complex numbered values for each component. I have been struck, though, by the implicit assumption made by all camps that something like the wave function/density state actually exists, whether subjective or objective. What if it does not? That would render all of these debates moot.

The wf/ds is a highly abstract theoretical inference of what is measured experimentally. All one measures are correlations between definite measured outcomes. Pragmatically, it has been found such correlations can pretty much only be calculated accurately within the Dirac/von Neumann two stage framework where one first postulates some abstract wf/ds evolving unitarily alternating with a collapse during measurement to the eigenspaces of some 'measured observable' according to the Born rule, whatever that really means. This makes no ontological assumptions per se; it is just what anyone needs to go through to calculate the measured correlations.

Many worlds people argue wf/ds is objective with no collapse and all the branches co-exist. Copenhagenists argue for collapse. But what if the wf/ds doesn't exist? Then they are both wrong and missing the point. Other than the fact the only time wf/ds shows up is in abstract symbolic calculations, why its existence be assumed?

Here, the situation differs from classical probability distributions. The probabilities are still linear for the density state, but not the wave function, but negative probabilities appear and that makes all the difference. In classical Bayesian updating, there is some leeway in when the updating happens because the different updates evolve independently and the original distribution always evolves as a nonnegative weighted sum of the individual independent contributions. In the quantum density state case, destructive interference due to oscillations between positive and negative 'probabilities' exist and the different outcomes can no longer be said to be in any way independent. Decoherence does not really explain it away because the suppression of interference is not exact, takes time, and is potentially reversible in principle. What if the density state does not exist?

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In your last paragraph, you say "Decoherence does not really explain it away". What is "it"? – Mitchell Porter Nov 17 '11 at 9:26
– anna v Nov 18 '11 at 4:53

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Can you please ask the question in a way that doesn't mention existence? What if I don't exist. So what. What do I care whether I exist. You need to make it clear what it means for something not to exist. Also, you can measure the wavefunction for a collection of identically prepared systems (or the density matrix for that matter), and so there are circumstances where it is certainly a measurable object. – Ron Maimon Nov 18 '11 at 5:45

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To answer politically: "That depends on what your definition of 'is' is" – Lagerbaer Dec 1 '11 at 23:11

## 8 Answers

You ask "Does the wave function/density state actually exist?" but this is a question that can't be answered. Quantum Mechanics is a mathematical model that gives an excellent description of the real world. QM is based upon the assumption that the wf/ds is a real object, but whether QM is a "real" description of the world is a question we need to leave to the philosophers.

In a comment to another question, Is there a mechanism for time symmetry breaking?, someone mentioned this paper. Although I'm not sure it revolutionises our understanding of the wavefunction it makes an interesting read in the context of this question.

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There is only one universe with only one quantum state. Density states are statistical descriptions and do not and cannot possibly apply to only one universe. Only with an ensemble of many many copies of the same quantum state can we associate a density state according to the limiting ratios of frequencies of the measured values of observables. No ensemble, no density state and no wavefunction either.

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Even if you believe this, you can believe that there is one universe with a state we don't know, and our ignorance leads to a probability distribution. If someone gave you a statistical description of cosmology using a density matrix, would you reject it our of hand? I think this is more of a philsophical statement, that there is "really" a pure state underneath, but what positivist meaning can you give to this statement, when a state which is rotated $\epsilon$ with respect to this pure state will be indistinguishable from it almost certainly, and we observe only one universe? – Ron Maimon Nov 19 '11 at 7:08

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You're completely ignoring the fact that density states arise from partial traces as well as from statistical descriptions; partial traces are meaningful even if there is just one universe. – Peter Shor Dec 1 '11 at 20:51

Let me quote Zurek here. There is no information without representation. If the wave function or density state exists as some form of actual information, it has to be represented by matter in some physical way. It has to, in the colorful terminology of a very interesting character, be made of potatoes. Clearly, there is no such material representation within our universe, but it could be turtles all the way down and materially represented on some higher hypostatis.

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There is a hidden assumption made here by most people including Pusey et al. The assumption is strict causality in time. Give me any particular instant in time, and realism combined with this assumption states that there is some complete set of information we can specify about the state at that time such that the probabilities for future outcomes can be determined to the best possible in principle based solely and uniquely upon the complete set of information at the given instant. This overlooks retrocausal interpretations where the actual observed outcome depends upon a transaction of alternations between forward causal influences and retrocausal influences. In such interpretations, the wavefunction need not exist.

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The answer all boils down to which interpretation you adopt. Obviously, the many worlds interpretation deals with an existing wave function. The consistent histories interpretation also requires an existing real wave function because one of its requirements is consistent families and what the permissible consistent families are is very sensitive to the actual values of the components of the physical density state. The opinion of the Copenhagen interpretation is the wave function is only a computational tool for getting the probabilities of measured outcomes.

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Theoretical Physics does not actually use the concept of 'existence'. That word does not appear in the usual axioms of QM and does not appear (much) in normal Physics textbooks either (I just checked Sommerfeld's Mechanics: as is typical, it only uses the word for mathematical existence, in an informal way: there exists a solution to the equation, etc.). How on earth would one define it, anyway? (Mathematical Logic doesn't use our normal intuitive concept of existence, either, as one can see by the fact that what might seem to be something like that, the existential quantifier, is in fact avoidable by using the universal quantifier and negation instead.)

Physics is, on the other hand, full of such statements as 'If the system is in the quantum state psi_o at time t=0, then....'

I think this certainly supports Mr. Rennie's point. I myself have noticed the same thing about the word 'event', I don't recall seeing it even once in any Mechanics textbook I have looked at...

Then the real difference between Bayesians and more 'classic' Quantum theorists such as Dirac and Wigner is that the latter simply say 'The set of quantum states of a system are the set of rays in a Hilbert Space' and go on to say many statements such as 'If the system is in the quantum state $v$ at time $t=0$ then ...' But Bayesians are forced to say 'The quantum state of a system encodes all our knowledge about the system' and they cannot avoid introducing subjective concepts like 'my' or 'knowledge'.

This is not a difference as to the probabilistic or statistical interpretation of the wave function, both a 'classic' QM-er and a Bayesian can both say that the interpretation of the wave function is that the modulus squared of its values are the probabilities that ....etc. And a Bayesian could say the wave function or quantum state is 'real', depending on their philosphic notions of reality and existence, which like Mr. Rennie said, are a separate issue from Physics. That is why I stated the real difference is between formally introducing subjective concepts like 'knowledge' or 'observer' or not introducing them. Dirac carefully avoids using either word: he says 'result of a measurement process' just as if no one was watching or cared.

A symptom of the difference between a Bayesian and a classic QMer is that the former expand the old idea of quantum state to include density matrices. The classic axioms make a sharp distinction: quantum state is a primitive concept and its connection with the probabilities of the results of measurement processes are given by axioms. Then the mixed states and density matrices are define in terms of these, and the rules for calculating probabilities of results of measurements applied to mixed states are derived as theorems. For a logically careful classic QMer, all quantum states are pure and all systems are closed.

If you ask me, it is Bayesians who are trying to make the concept of quantum state palatable to our everyday intuition by dragging in subjectivism and knowledge issues. Dirac simply said one could only develop an intuition about quantum concepts by using them... To me, it seems that it is Bayesians who are trying to interpret a quantum concept and include that interpretation in the axiom or formal system, whereas Dirac wanted it to stay uninterpreted.

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Your statement about mathematical logic is incorrect. The negation of a universal quantifier being an existential quantifier is essentially the equivalence of the statements "There is a crow of some color other than black" and "It is not true that all crows are black," which is perfectly valid using the standard definition of existence. – Peter Shor  Dec 10 '11 at 19:26
Possibly you are being confused by the fact that in mathematical logic, if there are no crows, then all crows are black (or any color you please, for that matter). – Peter Shor  Dec 10 '11 at 23:26
Existence is not a predicate.  If $x$ is a proper name, mathematical logic cannot say « $x$ exists ».  The fact that one can use quantifiers and we gave one of them the same name as « existence » does not prove that it covers the same concept as we have in ordinary language or physics when we say `exists'.  I am not sure what you mean by « the standard definition » of existence.  If existence is not one of the primitive undefined concepts, but is defined, then its meaning will vary with differing interpretations of the undefined concepts.  (That is what underlies the Skolem paradox.) – joseph f. johnson Dec 15 '11 at 1:21

I think you're misinterpreting the debates a bit. The wave function isn't real. It's a bookkeeping device. QM has these things called observables. Nothing else is posited to be real. The wave function is not an observable. If you go look at the development of QM in Schwinger's book, you see that the wave function shows up purely as a mathematical intermediate to make a lot of calculations straightforward.

Copenhagen posits that there is an act, "measurement", which forces a particle into a pure state. Many worlds folks say that there is an act, "measurement", which splits the universe into multiple paths. Neither of them claim any reality for the wave function. There are interpretations, such as the Bohm-de Broglie pilot wave, which imply reality for the wave function. They're worth knowing about because they have been a useful intuitive tool for folks like John Bell.

The two things that are really worth reading about foundations of quantum mechanics at this point are van Kampen's 'Ten Theorems on Quantum Mechanical Measurements' (let me know if you can't find a copy), and Griffiths's book 'Consistent Quantum Theory' (available online).

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-1: This is not true. You can measure the wavefunction on a collection of identically prepared system, for example, a bunch of hydrogen atoms. Schwinger does not make the assumption that the wavefunction is an artifact, he just gives an unusual set of axioms on observations to reproduce quantum mechanics. – Ron Maimon Nov 18 '11 at 5:43
I would agree with that it is a bookkeeping device.It is similar to discussions on the barycenter of the solar system in weather related blogs; like asking :" is the barycenter real". It is a confusion of language levels, which is how paradoxes arise. The barycenter, (as the wavefunction), is a real mathematical point; it is calculable continuously with newtonian mechanics. That is its reality. In the reality framework of the barycenter plowing through the sun and creating effects it is nonsense. Real is "for a certain value of reality", i.e. the metalevel of the observation. – anna v Dec 10 '11 at 5:27

I want you to think computationally. Obviously, nature can and does compute quantum mechanics. It remains to be asked how nature does it. The Copenhagenists will smugly tell you it does not matter how nature does it or what happens in between. Inputs go in, and then a miracle happens and we are supposed to stay hush hush about it and not peek inside, and outputs come out. Aren't you the least bit curious what happens in between? Do you want to listen to the Copenhagenists telling you nothing happens in between? Then, by implication, nature computes by magic and somehow get it right.

If the Church-Turing thesis is right, nature needs to compute with some scratch space and "registers" and "RAM"s. Otherwise, nature is some hypercomputational machine. The contents of the scratch space which happen in between are what Bell termed "beables". What are the minimal information needed for a beable? Some would say the wave function. Others the density state. Others the path integral. If it is not possible to do with any less, the beable has to be real, or do you still insist upon objecting and fighting? The beable has to be real.

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Have you ever heard of "nomological Bohmian mechanics"? The wavefunction is not treated as a thing, instead it is absorbed into the equation of motion of the classical configuration. That equation of motion can be reduced to local classical forces and a nonlocal quantum force whose specific form depends on the specific wavefunction that you started with. The point is that you do not need an exponentially large beable in order to get quantum dynamics. – Mitchell Porter Nov 19 '11 at 6:42

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@Mitchell Porter: If you write a simulation of Bohmian mechanics, you still need a ton of RAM to store the wavefunction data, and there is no reduction in the amount of necessary data from the particle positions. The particle positions are just extra baggage. From a computation perspective, Bohm is worse than quantum mechanics, and the words you use like "nomological" don't make any difference to the computational heft of the wavefunction. – Ron Maimon Nov 19 '11 at 7:12
Ron, in nomological Bohmian mechanics you can get rid of the wavefunction entirely. You can go backwards from the Hamilton-Jacobi picture and just think of the net force which you calculate as the gradient of the phase, and then that force breaks into two parts as I described. Quantum dynamics then arises from the nonlocal part in the equations of motion for the classical beables. – Mitchell Porter Nov 21 '11 at 0:03

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@Mitchell:you can repeat, you are still wrong. Just to specify the initial wavefunction, you need a function on 3N dimensions, which requires 10 to the 3N values even for only positions on a 10 cubed grid. The idea that the force only needs to be computed at the particle positions is seductive, but wrong in principle, because the wavefunction diffused around based on its own wave equation. You can calculate the wavefunction by summing over all paths, but there are exponentially many paths. – Ron Maimon Nov 21 '11 at 18:18

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There is absolutely no reason to believe that the universe is being computed on some computer living in some hyperverse. In fact, if you take this view (as you seem to), you are left with the question of what is doing the computation that runs this hyperverse. Also, the absence of observations of "bugs" in the laws of physics argues strongly against this possibility. – Peter Shor Dec 1 '11 at 20:52
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