SciPost Submission Page
Classical mechanics as the high-entropy limit of quantum mechanics
by Gabriele Carcassi, Manuele Landini, Christine A. Aidala
Submission summary
| Authors (as registered SciPost users): | Christine Aidala |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202412_00006v1 (pdf) |
| Date submitted: | 2024-12-03 15:10 |
| Submitted by: | Aidala, Christine |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We show that classical mechanics can be recovered as the high-entropy limit of quantum mechanics. That is, the high entropy masks quantum effects, and mixed states of high enough entropy can be approximated with classical distributions. The mathematical limit ℏ→0 can be reinterpreted as setting the zero entropy of pure states to −∞, in the same way that non-relativistic mechanics can be recovered mathematically with c→∞. Physically, these limits are more appropriately defined as S≫0 and v≪c. Both limits can then be understood as approximations independently of what circumstances allow those approximations to be valid. Consequently, the limit presented is independent of possible underlying mechanisms and of what interpretation is chosen for both quantum states and entropy.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1. The authors argue that the limit of high entropies reduces quantum mechanics to classical mechanics. This is a new viewpoint on the correspondence principle, which represents a pillar of our understanding of quantum theory.
2. The authors provide illustrative and accessible examples that support their main point.
3. The authors define and investigate stretching maps, which are interesting mathematical objects that may find applications elsewhere.
4. The manuscript is well written and accessible to a broad audience.
Weaknesses
1. The main goal of the manuscript is in my opinion not properly supported by the content.
1.1. Section 2 does not properly support the main goal. In Sec. 2.1., the authors provide examples illustrating the main point. This is a strong start for Section 2. In Sec. 2.2, the authors provide another example, Gaussian states and their uncertainty, complementing Sec. 2.1. Section 2.3 is the first that claims some generality. However, it largely focuses on finite-dimensional systems, where as the authors state, their results do not apply. To my understanding, the main message of Sec. 2.3 is that when entropy is maximized, the system behaves classically. But this is simply the statement that a maximally mixed state behaves classical, which is not a novel statement.
1.2 Section 3 does not properly support the main goal. Section 3 provides yet another two examples. These are however not novel. Even on Wikipedia (page of Rayleigh–Jeans law) it is stated that Planck's law reduces to Rayleigh- Jeans at high temperatures. The second example is also well known, that at high temperatures vacuum fluctuations become irrelevant compared to thermal fluctuations in the Wigner function.
1.3 Section 4 does not properly support the main goal. Section 4 is devoted to finding and investigating stretching maps, which increase entropy, and show how states become classical under these stretching maps. While these maps are interesting, they are to my understanding not sufficient to derive the main point. The authors write: "All of this is done under general conditions, and
is independent of the mechanism that performs the entropy increase." To my understanding, the stretching map singles out a particular mechanism of increasing entropy. It is unclear to me how the stretching map can show that there does not exist another way of increasing entropy that maintains some quantum properties.
2. The authors focus on the classical behavior of states in the high entropy regime but they spend very little time on discussing how the dynamics of quantum system reduces to classical dynamics. They do address this in Sec. 4, in particular with Eq. (54), where they argue that the more a state is stretched, the less important its derivatives becomes. This is in my opinion a very handwaving argument, as one could envision that quantum dynamics induces fast variations on an otherwise broad Wigner function which grow over time.
3. The arguments of the authors only hold for continuous variable systems. The authors argue that a finite-dimensional system has a finite maximal entropy and thus the large entropy limit does not apply. However, also finite-dimensional systems behave classical at high entropies (as mentioned by the authors in Sec. 2.3), where coherence can be neglected and a qubit or a quantum spin simply behaves like a classical two-level system.
4. Overall, the results of the manuscript are not well connected to established correspondence principles, which were an important contribution towards our understanding of quantum theory.
Report
In my opinion, the present manuscript does not "present a breakthrough on a previously-identified and long-standing research stumbling block". I can therefore not recommend publication. Below, I provide a list of requested changes that may help the authors to improve their manuscript. I believe that with these changes, the manuscript may be suitable for SciPost Physics Core.
As discussed above, many of the aspects discussed by the authors are already well-known in the literature and they are not being put on a firmer footing by the present manuscript. The main novelty is the approach using the stretching map but to my understanding this is not sufficient for a proof that the high-entropy regime always behaves classically.
Requested changes
1. The conclusions that can be drawn from the stretching map need to be clarified, in particular with respect to the generality of the statement: high entropies result in classical behavior. To my understanding, the only conclusion that can be rigorously drawn is: stretching results in classical behavior.
2. The authors should include a discussion on existing correspondence principles and contrast those to their results.
3. The authors use the inequality S>>0. This inequality is to my understanding not adequate. Any finite number is "much bigger" than zero. I suggest the authors try to find a more adequate inequality that reflects the high-entropy limit.
4. Much of Sec. 2.3. considers a two-level system. Since the main results of the manuscript does not apply to two-level systems, the authors should clarify what the role of the two-level system for understanding the main messages of the manuscript is.
5. In Sec. 2.3., the kets (quantum states) are not rendered properly.
Recommendation
Reject