What Are Incomplete Collapses and Tails in Objective Collapse Theories?

[ephen wilb]

What does it mean below that in order to keep these theories from violating the principle of the conservation of energy, the mathematics requires that any collapse be incomplete? What is the meaning of complete collapse vs incomplete (and the "tails")?

http://en.wikipedia.org/wiki/Objective_collapse_theory

GRW collapse theories have unique problems. In order to keep these theories from violating the principle of the conservation of energy, the mathematics requires that any collapse be incomplete. Almost all of the wave function is contained at the one measurable (and measured) value, but there are one or more small "tails" where the function should intuitively equal zero but mathematically does not. It is not clear how to interpret these "tails". They might mean that a small bit of matter has collapsed elsewhere than the measurement indicates, that with very low probability an object might "jump" from one collapsed state to another, or something else entirely. All of these options seem counterintuitive.

The original QMSL models had the drawback that they did not allow dealing with systems with several identical particles, as they did not respect the symmetries or antisymmetries involved. This problem was addessed by a revision of the original GRW proposal known as CSL (continuous spontaneous localization) developed by Ghirardi, Pearle Rimini in 1990.

 

[Agrippa]

If you localise the position wave-function, then (by Heisenberg's uncertainty principle) you smear out the momentum wave-function.
For example, if the position wave-function is delta function (i.e. all probability is confined to a single point in space), then momentum is completely uncertain: all possible momentum states are equiprobable.
But it cannot be that all momentum states spontaneously become equiprobable. Think about what this would entail: you go to measure the momentum of the exactly localised particle and infinite momentum is equally likely as the momentum value immediately prior to localisation. This would entail violation of energy conversation not consistent with experiment, for example, gases spontaneously heating up in a manner that is not observed.
So, GRW relaxed the collapse function from a delta-function to a Gaussian. But a Gaussian has non-vanishing tails. So a particle in an equal superposition of here and there will, post collapse, remain in such a superposition, it's just that the superposition will not be equal: "here"'s probability amplitude spontaneously dramatically increases while "there"'s probably amplitude spontaneously dramatically decreases.
The upside is that although energy conservation is violated, it is not violated to a degree that is inconsistent with what we know by experiment.
The downside is that the Gaussian collapse functions leave behind tails: While Schroedinger's cat "collapses" to a state of being alive, the term in the wave function for the dead-cat still has non-zero probability amplitude, and so by symmetry, may be just as real as the high amplitude alive-cat.

 

[ephen wilb]

In objective collapse, what is the thing that causes collapse... is it some kind of field that can cause collapse? Is this field similar to scalar fields like higgs field? Or why can't the collapser be a field? What should it be then?

 

[StevieTNZ]

For the GRW collapse theory, it is the size of the quantum object which determines when it collapses to one state or the other.

 

[ephen wilb]

For the GRW collapse theory, it is the size of the quantum object which determines when it collapses to one state or the other.

For Copenhagen.. Do you see what is wrong if some kind of field exist whose purpose is to collapse wave function? Since in Copenhagen observation collapses wave function.. what would be wrong to think of a field that has ambient observatory ability that can collapse wave function (or collapse into definite outcome in decoherence)? Has this been proposed before.. what is it called?

 

[StevieTNZ]

The Copenhagen Interpretation is where the macroscopic apparatus collapses the wave function of the quantum system. Decoherence doesn't produce a definite outcome state from a superposition -- all decoherence does is entangle the quantum system with the environment.

I don't understand the nature of a field that could collapse the wave function.

I don't think anyone has envisioned a field collapsing the wave function, so no idea what such a model would be called if there is one.

 

[Agrippa]

In objective collapse, what is the thing that causes collapse...

On the original GRW model, nothing causes collapse - collapse happens spontaneously. In particular, every elementary particle is endowed with a 10^-16 probability per second for spontaneous collapse. This entails that an isolated particle collapses spontaneously about every hundred million years. Consequently, for an ordinary macrosystem composed of entangled (non-isolated) parts, collapse happens about every 10^-7 seconds. So it's as if macro-objects are constantly collapsed. So while nothing causes collapse for GRW, collapse rate for a given system correlates with the size of the system.

... is it some kind of field that can cause collapse? Is this field similar to scalar fields like higgs field?

I'm aware of two collapse models that invoke fields as the cause of collapse. The first is Pearle's continuous spontaneous localization (CSL) model on which a classical field interacts with quantized particles to cause collapse. The second is Penrose's Gravity-induced collapse model on which collapse is caused by the tension in the fabric of spacetime created by mass-energy displacement between quantum states in superposition. The greater the difference the sooner the collapse. Neither have much to do with the Higg's field I think.

Or why can't the collapser be a field? What should it be then?

That is also another potential solution - that the observer's consciousness causes collapse. Here you need a precise scientifically motivated definition of consciousness in order to implement the theory. Perhaps you could consider consciousness to be some kind of field. However, modern implementations of this idea define consciousness using the integrated information theory of consciousness and define collapse rate as a function of a system's level of integrated information.

In general, nobody has any idea as to what causes collapse. Indeed, we don't even know if collapse occurs at all. At this stage, we are just throwing around different empirically consistent models in the hope of finding new insights and feasible experimental predictions.

 

[bhobba]

Since in Copenhagen observation collapses wave function.. what would be wrong to think of a field that has ambient observatory ability that can collapse wave function (or collapse into definite outcome in decoherence)? Has this been proposed before.. what is it called?

There is a bit of confusion in some quarters about Copenhagen.

Here is a good explanation of it:
http://motls.blogspot.com.au/2011/05/copenhagen-interpretation-of-quantum.html

Since the state is subjective knowledge collapse doesn't mean anything physically in Copenhagen. Its like throwing a dice - before you throw it each side has a probability of 1/6 of coming up - after it is thrown one side is a dead cert. The probability suddenly collapsed - but since probabilities, in the Bayesian interpretation, is also subjective it matters not.

As far as GRW goes its been a while since I investigated it detail - but the way it works is it introduces a non-linearity into QM that gets amplified during observation resulting in chaotic behaviour which is the explanation for collapse. Strictly speaking its not an interpretation of QM because the non-linearity is in principle distinguishable from standard QM.

Thanks
Bill

 

[ephen wilb]

On the original GRW model, nothing causes collapse - collapse happens spontaneously. In particular, every elementary particle is endowed with a 10^-16 probability per second for spontaneous collapse. This entails that an isolated particle collapses spontaneously about every hundred million years. Consequently, for an ordinary macrosystem composed of entangled (non-isolated) parts, collapse happens about every 10^-7 seconds. So it's as if macro-objects are constantly collapsed. So while nothing causes collapse for GRW, collapse rate for a given system correlates with the size of the system.
I'm aware of two collapse models that invoke fields as the cause of collapse. The first is Pearle's continuous spontaneous localization (CSL) model on which a classical field interacts with quantized particles to cause collapse. The second is Penrose's Gravity-induced collapse model on which collapse is caused by the tension in the fabric of spacetime created by mass-energy displacement between quantum states in superposition. The greater the difference the sooner the collapse. Neither have much to do with the Higg's field I think.
That is also another potential solution - that the observer's consciousness causes collapse. Here you need a precise scientifically motivated definition of consciousness in order to implement the theory. Perhaps you could consider consciousness to be some kind of field. However, modern implementations of this idea define consciousness using the integrated information theory of consciousness and define collapse rate as a function of a system's level of integrated information.

In general, nobody has any idea as to what causes collapse. Indeed, we don't even know if collapse occurs at all. At this stage, we are just throwing around different empirically consistent models in the hope of finding new insights and feasible experimental predictions.

In theories (like Penrose) where they were field that can collapse wave functions.. if you remove the field, then the different potential outcomes form many worlds? Or is it akin to Heisenberg Potentia.. where the wave functions are wave of possibilities instead of Many worlds?

 

[Agrippa]

In theories (like Penrose) where they were field that can collapse wave functions.. if you remove the field, then the different potential outcomes form many worlds? ?

Yes you're on the right track. But in the specific case of Penrose, the field is the gravitational field, and in his theory if you remove that, you remove all of space-time. So I think Penrose avoids this implication.

But the general point is correct. For example, in the highly unlikely event that no GRW spontaneous collapses occurs over an interval of time T, the universe will generate a branching structure, and will be indistinguishable from many-worlds theory, during T. The problem is particularly acute in observer-based collapse theories: prior to the existence of observers, you have many-worlds, then (somehow) the first observer appears (presumably in one branch) and then brings down the whole wave-function.

Or is it akin to Heisenberg Potentia.. where the wave functions are wave of possibilities instead of Many worlds?

No - a key premise of the collapse theories is realism about the wave-function, where all branches are equally real. Indeed, that is the whole point of introducing collapses a.k.a world-killers, in the first place.

 

[Agrippa]

As far as GRW goes its been a while since I investigated it detail - but the way it works is it introduces a non-linearity into QM that gets amplified during observation resulting in chaotic behaviour which is the explanation for collapse.

In what sense does it introduce non-linearity?

GRW let the wave-function evolve via the linear Schrödinger equation, except, at random times, wave-function experiences a jump of the form:
$\psi_t(x_1, x_2, ..., x_n) \rightarrow \frac{L_n(x)\psi_t(x_1, x_2, ..., x_n)}{||\psi_t(x_1, x_2, ..., x_n)||}$
Where $\psi _t(x_1, x_2, ..., x_n)$ is system state vector prior to jump and L_n(x) is a linear operator equal to:
$L_n(x) = \frac{1}{(\pi r^2_c)^{3/4}}e^{-(q_n - x)^2 / 2r^2_c}$
So the operator they introduce is linear. There is also no breakdown in the principle of linear superposition. So I'm curious as to what you mean.

 

[bhobba]

That random jump in non linear - if it was linear it would remain in superposition and not collapse:
http://en.wikipedia.org/wiki/Objective_collapse_theory
'Collapse is found "within" the evolution of the wavefunction, often by modifying the equations to introduce small amounts of non-linearity. A well-known example is the Ghirardi–Rimini–Weber theory[1] (GRW).'

Thanks
Bill

 

[Agrippa]

That random jump in non linear - if it was linear it would remain in superposition and not collapse:
http://en.wikipedia.org/wiki/Objective_collapse_theory
'Collapse is found "within" the evolution of the wavefunction, often by modifying the equations to introduce small amounts of non-linearity. A well-known example is the Ghirardi–Rimini–Weber theory[1] (GRW).'

No, if you read a bit further down in your link, you'll find that the superpositions always remain - GRW hits never destroy them. So that can't be the reason for why the link talks about "introducing small amounts of non-linearity". The link refers to this as "tails". More precisely, the collapse function I wrote above is a Gaussian with non-vanishing "tails". So I still don't see what's non-linear in GRW.

 

[bhobba]

I read it and can't find anything of the sort.

But aside from that there is no way, no way at all, objective collapse can occur without non-linearity. If it was linear it would remain in superposition.

Added Later
I was so surprised at the idea it wasn't nonlinear when obviously it must be I did a bit of a search. Every single article I came across states, specifically, it must be nonlinear - as logic indicates it must be eg:
http://motls.blogspot.com.au/2011/06/ghirardi-rimini-weber-collapsed.html
'Instead, it keeps Schrödinger's equation only and adds some nonlinear "flashes" into the evolution that are meant to squeeze the state vector in the mantinels that the authors consider "appropriate".'

Thanks
Bill

 

[Agrippa]

I read it and can't find anything of the sort.

But aside from that there is no way, no way at all, objective collapse can occur without non-linearity. If it was linear it would remain in superposition.

Thanks
Bill

Under 'Problems and Drawbacks' the link says "the mathematics requires that any collapse be incomplete". But anyway, your source is Wikipedia and is not rigorous. It is obvious that the system remains in a superposition post collapse from the mathematics of the GRW collapse function:

$L_n(x) = \frac{1}{(\pi r^2_c)^{3/4}}e^{-(q_n - x)^2 / 2r^2_c}$

This function takes the form of a Gaussian. Gaussians have non-vanishing tails. It follows that it is mathematically impossible for GRW collapses to destroy superpositions: superpositions remain necessarily.

So your modus tollens doesn't work and I'm still wondering what's non-linear in GRW.

 

[bhobba]

It is obvious that the system remains in a superposition post collapse

That's impossible. It can't remain in superposition and objectively collapse.

Thanks
Bill

 

[bhobba]

So your modus tollens doesn't work and I'm still wondering what's non-linear in GRW.

http://motls.blogspot.com.au/2011/06/ghirardi-rimini-weber-collapsed.html
'Instead, it keeps Schrödinger's equation only and adds some nonlinear "flashes" into the evolution that are meant to squeeze the state vector in the mantinels that the authors consider "appropriate".'l

Get it? Its utterly obvious.

Again Added Later:
After reacquainting myself with some of the detail from the above link, as it says, every 10^15 seconds the wavefunction discontinuously changes. Linear changes are continuous so it can't be linear.

Thanks
Bill

 

[Agrippa]

That's impossible. It can't remain in superposition and objectively collapse.

Sorry but I just proved it. That's what's nice about mathematics: it's true whether or not you believe it.

Get it? Its utterly obvious.

Another unrigorous link with no explanation. Clearly, neither of us know what exactly the non-linearities are.

 

[bhobba]

Sorry but I just proved it.

What's nice about mistakes is the person that makes them often continues in blissful ignorance.

Personally I am not motivated into delving again into the detail of GRW to spot your exact error - suffice to say you are at odds with every article I know about it - as well as simple logic about collapse.

Thanks
Bill

 

[bhobba]

"the mathematics requires that any collapse be incomplete"

Incomplete collapse meaning it remains in superposition - interesting view.

I think on that note I will take leave of this thread and leave it to someone that is more current with the detail.

Thanks
Bill

 

[Agrippa]

What's nice about mistakes is the person that makes them often continues in blissful ignorance.

Personally I am not motivated into delving again into the detail of GRW to spot your exact error - suffice to say you are at odds with every article I know about it - as well as simple logic about collapse.

You misunderstand. I'm not saying there is no non-linearity in GRW. Rather, I'm just asking you what the source of the non-linearity is. I don't understand why you don't just admit that you don't know - that way perhaps someone who does can enlighten both of us.

The claim about the GRW Gaussian collapse function always leaving behind superpositions is uncontroversial and follows straightforwardly from the Gaussian form of the collapse function. The problem is just that you're using 'collapse' to mean reduction to an eigenstate - which is not what GRW mean, they also use 'collapse' to mean a process that merely approximates a reduction to an eigenstate.

 

[bhobba]

Rather, I'm just asking you what the source of the non-linearity is.

I was going to leave this thread but you seem to be under some kind of confusion.

Discontinuous jumps that occur every 10^15 seconds are non linear.

That however is my last comment.

Thanks
Bill

 

[Agrippa]

I was going to leave this thread but you seem to be under some kind of confusion.

Yes, I wouldn't be asking the question if I wasn't confused about it! I thought that was the point of this forum.

Discontinuous jumps that occur every 10^15 seconds are non linear.

So in the GRW master equation, where $\lambda$ represents the Poisson process:
$\frac{d}{dt}\rho(t) = -\frac{i}{\hbar}[H, \rho(t)] - \lambda[1 - e^{-(x - y)^2 / 4r^2_c}]<x|[\rho(t)]|y>$
Your claim is that $\lambda$ is the nonlinear term (which is attached to a linear operator).
It's still a bit unclear what it means for a Possion term to be nonlinear.

I think what's going on is this:
In the QM literature, sometimes you see the phrase 'linearity of the dynamics' being used to refer to the following idea. Let system S be under constraints C so that if it begins in state |A> it will evolve to state |A'> and if it begins in state |B> it will evolve to state |B'>. The linearity of the dynamics then refers to the claim that given the previous stipulation it follows that If S under C begins in state a|A>+b|B> then it will evolve into a|A'>+b|B'>.

When you introduce collapse (discontinuous or otherwise) this principle breaks down. So I think this is what is meant. (This explanation applies more straightforwardly to continuous spontaneous localization models, which introduce nonlinearity without necessarily introducing discontinuity.)

That however is my last comment.

Thanks for trying!

 

[bhobba]

So in the GRW master equation, where $\lambda$ represents the Poisson process:
$\frac{d}{dt}\rho(t) = -\frac{i}{\hbar}[H, \rho(t)] - \lambda[1 - e^{-(x - y)^2 / 4r^2_c}]<x|[\rho(t)]|y>$

I was going to leave this thread but decided to relent because your error is very easy.

I could spell it out but it's probably better for you to nut it out yourself.

You need to look into stochastic modelling and what exactly a random process is, which a poisson process is an example of - in particular what a random variable is. Then apply that to what $\lambda$ does over time - hint the rate of the process it represents is random and not continuous:
http://en.wikipedia.org/wiki/Poisson_process

See the graph of the process.

Thanks
Bill

 

[StevieTNZ]

Sorry but I just proved it. That's what's nice about mathematics: it's true whether or not you believe it.

I guess if there are really 'tails' after supposed collapse, then no collapse of the wave function has occurred.

 

[Agrippa]

I guess if there are really 'tails' after supposed collapse, then no collapse of the wave function has occurred.

At this point it's verbal. Some agree and prefer to use "state reduction" others just speak of "GRW hits". But most still use "collapse". Indeed the class of theories to which GRW belong are usually referred to as "dynamical collapse theories". So "collapse" refers to a sudden and drastic shift in amplitudes; a reduction to an eigenstate is only an example of such a shift.

I think this actually points to quite a severe problem for dynamical collapse theories. The collapses happen in the position basis. Here reduction to eigenstates are clearly out of the question, since position eigenstates are delta-functions which are unphysical. Still, you might think that to avoid the tails GRW could go in for collapse functions with compact support, rather than Gaussians. But it turns out that from the positivity of the Hamiltonian alone one can prove that tails are acquired infinitely fast if they are somehow destroyed even for a moment. That will cause serious conflict with relativity theory. For example in quantum field theory there is a result called the Reeh-Schlieder theorem, which entails that no state with bounded energy is an eigenstate of any observable associated with a bounded spacetime region. So the tails are here to stay, it seems.