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An objective collapse model without state dependent stochasticity
by Lotte Mertens, Matthijs Wesseling, Jasper van Wezel
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Lotte Mertens · Jasper van Wezel |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202301_00035v2 (pdf) |
| Date accepted: | 2023-03-27 |
| Date submitted: | 2023-02-28 11:15 |
| Submitted by: | van Wezel, Jasper |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approaches: | Theoretical, Computational |
Abstract
The impossibility of describing measurement in quantum mechanics while using a quantum mechanical model for the measurement machine, remains one of its central problems. Objective collapse theories attempt to resolve this problem by proposing alterations to Schrodinger's equation. Here, we present a minimal model for an objective collapse theory that, in contrast to previous proposals, does not employ state dependent stochastic terms in its construction. It is an explicit proof of principle that it is possible for Born's rule to emerge from a stochastic evolution in which no properties of the stochastic process depend on the state being evolved. We propose the presented model as a basis from which more realistic objective collapse theories can be constructed.
Author comments upon resubmission
Errors in user-supplied markup (flagged; corrections coming soon)
REFEREE 1:
We again thank the referee for their support of our work. We are happy to adopt their suggestions.
We implemented the following corrections to the manuscript, as proposed by the referee:
1.) First page, second paragraph, last sentence: "large molecule" corrected to "a large molecule".
2.) Second page, second column, third full paragraph: Out of place citation and period fixed.
3.) Second page, last sentence: extra comma fixed.
4.) Third page, left column: equation at the bottom is no longer split.
5.) Third page, last paragraph: avoided the repeated phrase "with vanishingly small $\epsilon$" in back-to-back sentences.
6.) Fourth page, second column, first full paragraph: missing period and funky citations fixed.
7.) Appendix, directly below the equality in (A.4): "fo" changed to "for".
We thank the referee for pointing out these typos.
Then, the referee writes:
'8) My last comment is that in the first sentence of section II the authors assert, "all objective collapse theories propose an alteration or correction to Schrodinger’s equation". They then outline their theory in a manner that resembles the equation (1). However, in section V the authors state, "We, therefore, do not argue that the current model presents a realistic objective collapse theory". Which seems contradictory to the sentence in section II. Clarification is required here.'
--> Reply:
We thank the referee for pointing out the potential confusion between these sentences. What we intended to convey, is that even if the proposed dynamics constitutes a valid objective collapse model, it cannot be a realistic theory of nature. We adjusted this sentence to make it more clear.
We again thank the referee for their constructive remarks and for taking the time to consider our work.
--------------------------------
REFEREE 2:
The referee writes:
'Generally, I find the present manuscript very well written and largely understandable even to a non-expert such as myself. It avoids the mistakes of many other such manuscripts which often use very specialist jargon. I enjoyed reading the manuscript. And I finished reading it. With a sense of having understood its main points. I largely only have a few technical comments. Should these be taken into account by the authors, I would enthusiastically recommend publication. I have also had read of the rebuttal provided by the authors following the original round of reviewer comments.'
--> Reply:
We thank the referee for supporting publication of our work, for their postive remarks, and for the time and effort taken in considering the manuscript. We address their comments point by point below.
1) First page, first paragraph. Why do the authors state that macroscopic realm of objects is comprised of about 10 to the 18 atoms or more? I would have normally expected to see a number such as 10 to the 23. If this is not a typo, then maybe the 10 to the 18th should be explained.
--> Reply:
The power in 10^18 is intentional, and we added a sentence in the revised manuscript explaining its origin.
For objects containing more than about 10^18 atoms, there have been direct experiments confirming their dynamics follow the laws of classical physics. Similarly, quantum dynamics has been directly tested for objects for objects with less than about 10^6 atoms. In between these limits, however, there is no direct experimental data on the dynamics encountered in typical quantum measurement protocols. It is therefore usual to classify the region of objects containing between 10^6 and 10^18 atoms as the ``mesoscopic'' regime. We explain this in the revised manuscript, and add appropriate citations.
2) Page 1. Left-hand column, second paragraph. I think it would be helpful to support some of the discussion of interpretations of quantum mechanics by citing a few relevant books, and not just the single review article as is presently done. As such a book, I can for example suggest: Jonathan Allay, Quantum reality: Theory and philosophy, second edition, as published by Taylor and Francis.
--> Reply:
We add this citation and thank the referee for suggesting it.
3. First page. Second paragraph of the right tend column. When speaking about the “existing objective collapse theories … with a non-linear state dependent factor”, relevant citations should be provided beyond the simple citation of the review article.
--> Reply:
We agree with the referee and add appropriate citations.
4) Page 1. Right and column. The authors state that “origin of the coupling remains unexplained.” Perhaps the use of the word “unexplained” explains some of the antagonism of reviewer 2. Perhaps a reformulation can help. For example using “unclear”, “controversial” or “mysterious” instead of “unexplained”.
--> Reply:
We thank the referee for pointing out the poor choice of wording, and adjust it in the revised manuscript.
5) Page 1 right and column last paragraph. The authors state that their model distinguishes from “existing collapsed models … without … a state dependent probability distribution function.” Perhaps here one could add the comment regarding “fine tuning” discussed by reviewer 2 and in the rebuttal. Only a small comment needs to be made here. And a discussion similar to the one in the rebuttal could be added later in the discussion section.
--> Reply:
We thank the referee for the suggestion and include a sentence in the introduction as well as a new paragraph in section IV of the manuscript to comment on the existence of a relation between parameters (the "fine-tuning") of the model.
6. Page 2 bottom of the left-hand column. “The first part of this process.... the latter part.” I'm not quite sure what the “first” and the “latter” part refer to. This should be clarified.
--> Reply:
We clarify this in the revised text.
7) Page 2, right hand column, third paragraph. “... all of the predictions of quantum mechanics are recovered.” Is it true to say “all” the predictions are recovered? A “time-dependent norm”, as used in the following sentence, is for example not a prediction of quantum mechanics. Perhaps it is simply best to remove “all of” Or to reformulate more precisely.
--> Reply:
We thank the referee for pointing out this sentence is not sufficiently clear.
As the norm of a quantum state can never be measured, quantum mechanics in fact makes no prediction about its value and no predictions of physical observables are altered by having a time-dependent norm. We clarify this sentence to more clearly convey the intention that no predictions of physical observables depend on the value of the (unobservable) norm.
8) Page 5. Right and column top paragraph. I presume the discussion of the microscopic region corresponds to N epsilon ->0 as in the equation 4. It would be good to use the mathematical expression here.
--> Reply:
We agree with the referee, and add the "N" in the revised manuscript.
9) Page 5 section B first paragraph, centre of. “pi in orange “ should use the symbol for Pi.
--> Reply:
We thank the referee for finding this typo and fix it in the revised manuscript.
10) Page 6 just before the start of section C. That $cos^2 (phi0/2)$ Is indeed the desired Born rule becomes much clearer if equation (6) was cited again here.
--> We thank the referee for this suggestion and will implement it.
11) Figure 2. The legend. In the Figure 4 be not equals 1.0. Is near unreadably faint. I suggest to simply use black for that label.
--> Reply:
We chose this colour because it matches the colour of the line it corresponds to. We appreciate the suggestion of the referee, but prefer to keep it the original colouring.
12) Page 7. Right Hand column. The authors assume that “the infinitesimal symmetry breaking field is beyond the control of any experiment.” Doesn't that, however, introduce a stochastic coupling for the state of the system? One could imagine an experimenter who does control the state well enough and for then it would be a coupling to the state with some stochasticity. I feel that this assumption might very well be the weak point of the authors approach here. At least it should be discussed.
--> Reply:
Just as in the standard theory of spontaneous symmetry breaking, the field required to break the unitarity of time evolution scales as 1/N. Applying this scaling to macroscopic systems, the field required to violate unitarity is not just very small, but fundamentally out of reach of any conceivable experiment. It is similar to the classic illustration of standard spontaneous symmetry breaking, in which the clapping of a butterfly's wings on the other side of the world creates far more than the required perturbation on our side to break the translational symmetry of any human-sized object. Controlling, shielding, or even measuring these types of infinitesimal perturbations is not just practically out of reach, but simply impossible.
13) Appendix equation A1. The symbol G used here as a general time evolution operator clashes with G introduced an equation 1 where it is the collapse-generating operator. I suggest using an operator such as F.
--> Reply:
We thank the referee for pointing this out, and adjust it in the revised manuscript.
14) Appendix just after equation A5. Why do the overall phase theta and normalisation n not appear in the equations for phi-dot and theta-dot (sorry, no greek). There should be a physics-based explanation.
--> Reply:
The absence of these quantities in the evolution of the physically relevant phi and theta angles is a direct indication of the fact that the overall phase and normalisation are can physical, measurable quantities. We include a sentence explaining this in the Appendix.
15) When reading the rebuttal to reviewer 2 I find the authors’ explanations largely convincing. However, I also find that the issue of non-signalling and the fine tuning may very well be something that a reader might find puzzling. Hence, I would like to suggest including the discussion of non-signalling and fine tuning like what is done in the rebuttal also in the main text of the manuscript.
--> Reply:
We thank the referee for the suggestion, and include a paragraph and sentence on the relation between parameter values, as well as a sentence about non-signalling.
We again thank the referee for their supportive comments and careful consideration of our work.
REFEREE 1:
We again thank the referee for their support of our work. We are happy to adopt their suggestions.
We implemented the following corrections to the manuscript, as proposed by the referee:
1.) First page, second paragraph, last sentence: "large molecule" corrected to "a large molecule".
2.) Second page, second column, third full paragraph: Out of place citation and period fixed.
3.) Second page, last sentence: extra comma fixed.
4.) Third page, left column: equation at the bottom is no longer split.
5.) Third page, last paragraph: avoided the repeated phrase "with vanishingly small $\epsilon$" in back-to-back sentences.
6.) Fourth page, second column, first full paragraph: missing period and funky citations fixed.
7.) Appendix, directly below the equality in (A.4): "fo" changed to "for".
We thank the referee for pointing out these typos.
Then, the referee writes:
'8) My last comment is that in the first sentence of section II the authors assert, "all objective collapse theories propose an alteration or correction to Schrodinger’s equation". They then outline their theory in a manner that resembles the equation (1). However, in section V the authors state, "We, therefore, do not argue that the current model presents a realistic objective collapse theory". Which seems contradictory to the sentence in section II. Clarification is required here.'
--> Reply:
We thank the referee for pointing out the potential confusion between these sentences. What we intended to convey, is that even if the proposed dynamics constitutes a valid objective collapse model, it cannot be a realistic theory of nature. We adjusted this sentence to make it more clear.
We again thank the referee for their constructive remarks and for taking the time to consider our work.
--------------------------------
REFEREE 2:
The referee writes:
'Generally, I find the present manuscript very well written and largely understandable even to a non-expert such as myself. It avoids the mistakes of many other such manuscripts which often use very specialist jargon. I enjoyed reading the manuscript. And I finished reading it. With a sense of having understood its main points. I largely only have a few technical comments. Should these be taken into account by the authors, I would enthusiastically recommend publication. I have also had read of the rebuttal provided by the authors following the original round of reviewer comments.'
--> Reply:
We thank the referee for supporting publication of our work, for their postive remarks, and for the time and effort taken in considering the manuscript. We address their comments point by point below.
1) First page, first paragraph. Why do the authors state that macroscopic realm of objects is comprised of about 10 to the 18 atoms or more? I would have normally expected to see a number such as 10 to the 23. If this is not a typo, then maybe the 10 to the 18th should be explained.
--> Reply:
The power in 10^18 is intentional, and we added a sentence in the revised manuscript explaining its origin.
For objects containing more than about 10^18 atoms, there have been direct experiments confirming their dynamics follow the laws of classical physics. Similarly, quantum dynamics has been directly tested for objects for objects with less than about 10^6 atoms. In between these limits, however, there is no direct experimental data on the dynamics encountered in typical quantum measurement protocols. It is therefore usual to classify the region of objects containing between 10^6 and 10^18 atoms as the ``mesoscopic'' regime. We explain this in the revised manuscript, and add appropriate citations.
2) Page 1. Left-hand column, second paragraph. I think it would be helpful to support some of the discussion of interpretations of quantum mechanics by citing a few relevant books, and not just the single review article as is presently done. As such a book, I can for example suggest: Jonathan Allay, Quantum reality: Theory and philosophy, second edition, as published by Taylor and Francis.
--> Reply:
We add this citation and thank the referee for suggesting it.
3. First page. Second paragraph of the right tend column. When speaking about the “existing objective collapse theories … with a non-linear state dependent factor”, relevant citations should be provided beyond the simple citation of the review article.
--> Reply:
We agree with the referee and add appropriate citations.
4) Page 1. Right and column. The authors state that “origin of the coupling remains unexplained.” Perhaps the use of the word “unexplained” explains some of the antagonism of reviewer 2. Perhaps a reformulation can help. For example using “unclear”, “controversial” or “mysterious” instead of “unexplained”.
--> Reply:
We thank the referee for pointing out the poor choice of wording, and adjust it in the revised manuscript.
5) Page 1 right and column last paragraph. The authors state that their model distinguishes from “existing collapsed models … without … a state dependent probability distribution function.” Perhaps here one could add the comment regarding “fine tuning” discussed by reviewer 2 and in the rebuttal. Only a small comment needs to be made here. And a discussion similar to the one in the rebuttal could be added later in the discussion section.
--> Reply:
We thank the referee for the suggestion and include a sentence in the introduction as well as a new paragraph in section IV of the manuscript to comment on the existence of a relation between parameters (the "fine-tuning") of the model.
6. Page 2 bottom of the left-hand column. “The first part of this process.... the latter part.” I'm not quite sure what the “first” and the “latter” part refer to. This should be clarified.
--> Reply:
We clarify this in the revised text.
7) Page 2, right hand column, third paragraph. “... all of the predictions of quantum mechanics are recovered.” Is it true to say “all” the predictions are recovered? A “time-dependent norm”, as used in the following sentence, is for example not a prediction of quantum mechanics. Perhaps it is simply best to remove “all of” Or to reformulate more precisely.
--> Reply:
We thank the referee for pointing out this sentence is not sufficiently clear.
As the norm of a quantum state can never be measured, quantum mechanics in fact makes no prediction about its value and no predictions of physical observables are altered by having a time-dependent norm. We clarify this sentence to more clearly convey the intention that no predictions of physical observables depend on the value of the (unobservable) norm.
8) Page 5. Right and column top paragraph. I presume the discussion of the microscopic region corresponds to N epsilon ->0 as in the equation 4. It would be good to use the mathematical expression here.
--> Reply:
We agree with the referee, and add the "N" in the revised manuscript.
9) Page 5 section B first paragraph, centre of. “pi in orange “ should use the symbol for Pi.
--> Reply:
We thank the referee for finding this typo and fix it in the revised manuscript.
10) Page 6 just before the start of section C. That $cos^2 (phi0/2)$ Is indeed the desired Born rule becomes much clearer if equation (6) was cited again here.
--> We thank the referee for this suggestion and will implement it.
11) Figure 2. The legend. In the Figure 4 be not equals 1.0. Is near unreadably faint. I suggest to simply use black for that label.
--> Reply:
We chose this colour because it matches the colour of the line it corresponds to. We appreciate the suggestion of the referee, but prefer to keep it the original colouring.
12) Page 7. Right Hand column. The authors assume that “the infinitesimal symmetry breaking field is beyond the control of any experiment.” Doesn't that, however, introduce a stochastic coupling for the state of the system? One could imagine an experimenter who does control the state well enough and for then it would be a coupling to the state with some stochasticity. I feel that this assumption might very well be the weak point of the authors approach here. At least it should be discussed.
--> Reply:
Just as in the standard theory of spontaneous symmetry breaking, the field required to break the unitarity of time evolution scales as 1/N. Applying this scaling to macroscopic systems, the field required to violate unitarity is not just very small, but fundamentally out of reach of any conceivable experiment. It is similar to the classic illustration of standard spontaneous symmetry breaking, in which the clapping of a butterfly's wings on the other side of the world creates far more than the required perturbation on our side to break the translational symmetry of any human-sized object. Controlling, shielding, or even measuring these types of infinitesimal perturbations is not just practically out of reach, but simply impossible.
13) Appendix equation A1. The symbol G used here as a general time evolution operator clashes with G introduced an equation 1 where it is the collapse-generating operator. I suggest using an operator such as F.
--> Reply:
We thank the referee for pointing this out, and adjust it in the revised manuscript.
14) Appendix just after equation A5. Why do the overall phase theta and normalisation n not appear in the equations for phi-dot and theta-dot (sorry, no greek). There should be a physics-based explanation.
--> Reply:
The absence of these quantities in the evolution of the physically relevant phi and theta angles is a direct indication of the fact that the overall phase and normalisation are can physical, measurable quantities. We include a sentence explaining this in the Appendix.
15) When reading the rebuttal to reviewer 2 I find the authors’ explanations largely convincing. However, I also find that the issue of non-signalling and the fine tuning may very well be something that a reader might find puzzling. Hence, I would like to suggest including the discussion of non-signalling and fine tuning like what is done in the rebuttal also in the main text of the manuscript.
--> Reply:
We thank the referee for the suggestion, and include a paragraph and sentence on the relation between parameter values, as well as a sentence about non-signalling.
We again thank the referee for their supportive comments and careful consideration of our work.
Published as SciPost Phys. 14, 114 (2023)
Reports on this Submission
Report 2 by Eric Aspling on 2023-3-2 (Invited Report)
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I'm pleased with the changes made. I recommend publication.
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I am satisfied by the replies that the authors have made. I am not keen on searching through the manuscript and trust that the editor has checked that the necessary changes have indeed been made. If this is the case, then I would be happy to support publication of the manuscript in its revised form.