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Physicist248
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Does chaos amplify inherent quantum level randomness/uncertainty to macroscopic level?
You can't, because the quantum evolution equation, Schrodinger's Equation, is linear, and you can't have chaos with linear dynamics.Filip Larsen said:If you have a chaotic system of, say, interacting quantum particles
I understand that the coupling between chaos and quantum effects are not as straight forward as my intuition from the two fields alone tries to tell me, and that good examples of experiments should a clear coupling are indeed difficult to find.PeterDonis said:You can't, because the quantum evolution equation, Schrodinger's Equation, is linear
PeterDonis said:You can't, because the quantum evolution equation, Schrodinger's Equation, is linear, and you can't have chaos with linear dynamics.
Collapse doesn't have any dynamics in basic QM; it's just a mathematical operation you perform after you know the result of a measurement. Discussion of particular interpretations that give more meaning to collapse than that belongs in the interpretations subforum.Jarvis323 said:What about collapse though?
Nothing like this occurs in standard QM. Whether quantum gravity can allow something like this is an open question.Filip Larsen said:I imagine that chaos enabling non-linearity should be possible to "arranged" by suitable choice of geometry and/or driving fields, e.g. to enable topological mixing of quantum particles.
This paper is proposing a speculative quantum gravity framework that goes beyond standard QM and standard GR. Even if such a framework turns out to be right, however, it is still not clear that "chaos amplifying quantum level randomness" would be a valid description of what it says. The chaos this paper is talking about comes from the fact that the Einstein Field Equation is nonlinear.Filip Larsen said:The Phys Letter B paper Can chaos be observed in quantum gravity, while being a bit over my engineering head, do seem to conclude there are plenty of challenges pointing to a general link between quantum effects and chaos.
I don't think that this argument is valid. The Schrödinger equation is linear on configuration space, but chaos happens in normal 3D space. And since it is part of a probabilistic description, it is not even a big surprise that it is linear. The Liouville equations provide an analogous probabilistic description of classical systems (where the probabilistic description just described the uncertainty in the initial conditions), and these are also linear.PeterDonis said:You can't, because the quantum evolution equation, Schrodinger's Equation, is linear, and you can't have chaos with linear dynamics.
A problem here is that in QM the Hamiltonian is supposed to exist and be precise; it is not subject to inference in the foundations(which we need not discuss further here wether it ought to or not). So the unitary evolution is essentially postulated. But check this with experiment one could ask: if the system appears chaotic, does it make sense to presume that the process tomography to infer the hamiltonian (from a chaos) makes sense? If not, does it make sense to use the argument against chaos in the first place?Revisiting the role of chaos in quantum tomographyPeterDonis said:You can't, because the quantum evolution equation, Schrodinger's Equation, is linear, and you can't have chaos with linear dynamics.
One must be careful with such overly general statements. See, for example, Linear Chaos by Erdmann and Manguillot.PeterDonis said:You can't, because the quantum evolution equation, Schrodinger's Equation, is linear, and you can't have chaos with linear dynamics.
Your remark fits well into the context of ergodic theory. (Also see the book mentioned above.)gentzen said:I don't think that this argument is valid. The Schrödinger equation is linear on configuration space, but chaos happens in normal 3D space. And since it is part of a probabilistic description, it is not even a big surprise that it is linear. The Liouville equations provide an analogous probabilistic description of classical systems (where the probabilistic description just described the uncertainty in the initial conditions), and these are also linear.
But if the temporal behavior of a quantum system depends also on collapse, then shouldn't we say the quantum system includes more than just unitary evolution?PeterDonis said:Collapse doesn't have any dynamics in basic QM; it's just a mathematical operation you perform after you know the result of a measurement. Discussion of particular interpretations that give more meaning to collapse than that belongs in the interpretations subforum.
There are interpretations (for example MWI) that don’t include collapse, yet successfully predict that temporal behavior. So it can’t be that the behavior depends on collapse.Jarvis323 said:But if the temporal behavior of a quantum system depends also on collapse…
No, chaos happens in whatever space the dynamics happens in. For classical mechanics, that is normal 3D space. But for QM, or at least for non-relativistic QM, it's configuration space. The point is that chaos can only happen if the dynamics is nonlinear.gentzen said:The Schrödinger equation is linear on configuration space, but chaos happens in normal 3D space.
If you want to substitute a different theory for standard QM, and it makes correct predictions, go ahead and publish it and then we can discuss it here. But until then, this forum is discussing standard QM, not some speculative replacement for it.Fra said:if the system appears chaotic, does it make sense to presume that the process tomography to infer the hamiltonian (from a chaos) makes sense?
QPT is just a practical inference scheme formalized to infer the dynamical laws presuming QM. So it's not a new theory, its just an inverse problem in QM.PeterDonis said:If you want to substitute a different theory for standard QM, and it makes correct predictions, go ahead and publish it and then we can discuss it here. But until then, this forum is discussing standard QM, not some speculative replacement for it.
What is QPT?Fra said:QPT
I don't think anyone here is contending that.PeterDonis said:The point is that chaos can only happen if the dynamics is nonlinear.
Quantum Process Tomography - trying to infer the hamiltonian of a system from a large set of interactions.PeterDonis said:What is QPT?
For a moment I feared that our disagreement is over the meaning of 3D space and configuration space. Luckily, on closer reading of your response I see that this is not the case.PeterDonis said:No, chaos happens in whatever space the dynamics happens in. For classical mechanics, that is normal 3D space. But for QM, or at least for non-relativistic QM, it's configuration space. The point is that chaos can only happen if the dynamics is nonlinear.
Filip Larsen said:I don't think anyone here is contending that.
Do you have a good reference for this?Fra said:Quantum Process Tomography - trying to infer the hamiltonian of a system from a large set of interactions.
Do you have a reference for this?Filip Larsen said:Chaos in such a system is, as far as I know, considered an emergent behavior that does not have to be brought about by the underlying interactions themselves, but more due to the macroscopic motion of the system with both local amplification along the flow (i.e. positive Lyapunov exponent) and topological mixing of the flows in state space.
Only if by "Bohmian mechanics" you mean a different theory from standard QM, not just an interpretation of standard QM. As an interpretation of standard QM, the nonlinearity of the guiding equation is irrelevant since individual particle positions are unknown and are not used to make predictions about the results of experiments; the only role the particle positions play in actual predictions of measurement results (as opposed to the story that is told about measurement results after they are known) is in the assumption that their initial distribution matches the probabilities given by the wave function.gentzen said:Our disagreement seems easiest to discuss in Bohmian mechanics
Do you have a reference for this?Jarvis323 said:chaos is possible for an infinite dimensional linear system.
Which part? That chaos is considered an emergent phenomenon? Or chaos theory in general?PeterDonis said:Do you have a reference for this?
PeterDonis said:Do you have a reference for this?
https://link.springer.com/article/10.1007/s003329900069Linear ChaosThe authors present two results on infinite-dimensional linear dynamical systems with chaoticity. One is about the chaoticity of the backward shift map in the space of infinite sequences on a general Fréchet space. The other is about the chaoticity of a translation map in the space of real continuous functions. The chaos is shown in the senses of both Li-Yorke and Wiggins. Treating dimensions as freedoms, the two results imply that in the case of an infinite number of freedoms, a system may exhibit complexity even when the action is linear. Finally, the authors discuss physical applications of infinite-dimensional linear chaotic dynamical systems.
https://link.springer.com/book/10.1007/978-1-4471-2170-1It is commonly believed that chaos is linked to non-linearity, however many (even quite natural) linear dynamical systems exhibit chaotic behavior. The study of these systems is a young and remarkably active field of research, which has seen many landmark results over the past two decades. Linear dynamics lies at the crossroads of several areas of mathematics including operator theory, complex analysis, ergodic theory and partial differential equations. At the same time its basic ideas can be easily understood by a wide audience.
Written by two renowned specialists, Linear Chaos provides a welcome introduction to this theory. Split into two parts, part I presents a self-contained introduction to the dynamics of linear operators, while part II covers selected, largely independent topics from linear dynamics. More than 350 exercises and many illustrations are included, and each chapter contains a further ‘Sources and Comments’ section.
The only prerequisites are a familiarity with metric spaces, the basic theory of Hilbert and Banach spaces and fundamentals of complex analysis. More advanced tools, only needed occasionally, are provided in two appendices.
A self-contained exposition, this book will be suitable for self-study and will appeal to advanced undergraduate or beginning graduate students. It will also be of use to researchers in other areas of mathematics such as partial differential equations, dynamical systems and ergodic theory.
Yes.Filip Larsen said:Which part? That chaos is considered an emergent phenomenon?
It is almost by definition of what emergence means in general, i.e. in this case the structures in chaotic systems, like strange attractors, which appear only when you have the full system, that is, the emergent structure cannot be identified as to originate from a single part of the system. It is very common to denote structures or patterns formed by complex systems as emergent structures.PeterDonis said:Do you have a reference for this?
Most classical dynamical systems are chaotic. The trajectories of two identical systems prepared in infinitesimally different initial conditions diverge exponentially with time. Quantum systems, instead, exhibit quasi-periodicity due to their discrete spectrum. Nonetheless, the dynamics of quantum systems whose classical counterparts are chaotic are expected to show some features that resemble chaotic motion. Among the many controversial aspects of the quantum-classical boundary, the emergence of chaos remains among the least experimentally verified. Time-resolved observations of quantum chaotic dynamics are particularly rare, and as yet unachieved in a single particle, where the subtle interplay between chaos and quantum measurement could be explored at its deepest levels. We present here a realistic proposal to construct a chaotic driven top from the nuclear spin of a single donor atom in silicon, in the presence of a nuclear quadrupole interaction. This system is exquisitely measurable and controllable, and possesses extremely long intrinsic quantum coherence times, allowing for the observation of subtle dynamical behavior over extended periods. We show that signatures of chaos are expected to arise for experimentally realizable parameters of the system, allowing the study of the relation between quantum decoherence and classical chaos, and the observation of dynamical tunneling.
gentzen said:Our disagreement seems easiest to discuss in Bohmian mechanics:
I don't get to your objection. I even explicitly started by acknowledging that the nonlinearity of the guiding equation is irrelevant:PeterDonis said:Only if by "Bohmian mechanics" you mean a different theory from standard QM, not just an interpretation of standard QM. As an interpretation of standard QM, the nonlinearity of the guiding equation is irrelevant since ...
gentzen said:their guiding equation is nonlinear, and ... But you say that this is not the relevant dynamic, and I even agree with this.
gentzen said:You want to conclude from the linearity of that dynamics (of the wavefunction) that it cannot exhibit chaos, and "I don't think that this argument is valid".
General statements about emergence are not the same as a specific derivation of chaos as emergent from the particular dynamics of QM. The latter is what is relevant to this thread.Filip Larsen said:It is almost by definition of what emergence means in general
And then you contradicted yourself by arguing that it is. If it's irrelevant, then it's irrelevant and can't be used as the basis of any argument that is relevant. If it can be used as the basis of a valid argument for this topic, then it's not irrelevant.gentzen said:I even explicitly started by acknowledging that the nonlinearity of the guiding equation is irrelevant