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mfb
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A proper Hamiltonian is a full description of the physics of the system - at least in theory.
So... as well as location and spatial geometry, it would describe the dynamics of the electroweak forces, color, mass and gravitation?mfb said:A proper Hamiltonian is a full description of the physics of the system - at least in theory.
Is there an layman level explanation as to how the Hamiltonian projects future states? My curiosity applies particularly to whether the degree of state reduction becomes less "refined" as the system moves forward in time. Demystifier suggested earlier (post #5) that decoherence is not irreversible... at least not in simple systems.mfb said:It allows you to find the state at any time in the future or the past if you know the state at a specific point in time.
Excellent. Thank you. Let me roll this around in my head for a bit.mfb said:The Hamiltonian allows to calculate the time-evolution of a state. In nonrelativistic quantum mechanics, this is just the Schroedinger equation, for example. Note that no collapses or similar processes happen here. For systems of sufficient complexity, the time-evolution often leads to states that can be split into multiple pieces with practically no interaction between those pieces. That is decoherence. It is irreversible, as Demystifier said in post #5. An interaction that looks like a measurement can be reversible, but then it does not lead to decoherence.
You can now assume that (a) all but one piece magically disappear and the remaining piece gets a larger amplitude, (b) those pieces just stay independent and keep evolving according to the laws of quantum mechanics, (c) your initial state was not correct or didn't describe everything, (d) ... a few other things.
(a) leads to collapse-like interpretations, (b) to many worlds, (c) to de-Broglie-Bohm, and so on. Your description of the state after a while depends on the interpretation you choose. But all those things are interpretations, not measurement results, they are not necessary for making predictions.
I'm trying to imagine how the Hamiltonian changes if we expand the system being considered away from the locality of the dust particle to include the photon emitting events (assuming that "a few" photons are required) that might be countless light years away. Does the uncertainty in the possible path of the localizing photon have any bearing on anything?mfb said:The Hamiltonian allows to calculate the time-evolution of a state. In nonrelativistic quantum mechanics, this is just the Schroedinger equation, for example. Note that no collapses or similar processes happen here. For systems of sufficient complexity, the time-evolution often leads to states that can be split into multiple pieces with practically no interaction between those pieces. That is decoherence. It is irreversible, as Demystifier said in post #5. An interaction that looks like a measurement can be reversible, but then it does not lead to decoherence.
Feeble Wonk said:I've got to assume that the lack of response to my last question means that I, once again, phrased the idea in terms that are absurd and/or meaningless. So, I'd like to ask it in a different way.
You've patiently explained that the evolution of the quantum state of the system describing the dust particle evolves over time, and decoherence localizes the particle "after" interaction with one or more photons.
I believe it was further explained that in any system of sufficient complexity, the interactions are frequent enough that the decoherence is irreversible (such that the mixture of potential states will remain decohered forever as the separate Hamiltonians continue to evolve along different lines).
All of this makes sense to me. However, my confusion remains in regard to the assertion that the "potential" interaction with the photon(s) constitutes a measurement/observation. And this relates to my question regarding the "expanded" quantum system being considered.
When we consider the system to include not only the (potential) dust particle, but also the (potential) photon emitting event(s), it seems that the (potential) decohered states would be incalculably increased as a result. So yes, IF we stipulate that a dust particle in a specific location in space time interacts with one or more photons that have been emitted from any number of possible sources, then I suppose the location of the dust particle would be defined in one of those (incalculably various) potential states.
But how are we to say that such a logical limitation (of quantum state reduction) has "really" been observed/measured simply because the mathematics define the decohered state reduction IF that were the case?
Does that question make any more sense?
I suppose you could just replace the words "logical limitation" with "realized state reduction", "collapse", or something like that.cube137 said:Your choice of words are kinda vague. Please rephrase the sentences such a way Bhohha, atyy or others can understand.. like what you mean logical limitation has been observed..
Feeble Wonk said:I suppose you could just replace the words "logical limitation" with "realized state reduction", "collapse", or something like that.
I fully accept that the mathematics of decoherence limit the quantum states that CAN occur in such a way that macroscopic superposition will not be observed WHEN an observation is made. I have no qualms about that. Yet, particularly when we are considering a large system (as I've described earlier), it seems to me that the vast array of potential states would remain part of the Hamiltonian expression describing the "potential" locality of the "potential" dust particle.
So, yes, IF we stipulate that one or more photons from any of the countless potential photon emitting events interact with a dust particle at a given location, then I suppose you could say that the mathematics of decoherence would define the physical locality of the dust particle within the Hilbert described by that expression. However, that seems (to me anyway) to be an arbitrary assumption in the absence of a "realized" observation of the photon(s)/dust particle interaction, if we are considering the larger system including the potential photon emission events.
I would appreciate a response as well. I might anticipate a statement that the photon/particle interaction IS the "observation" that produces the decoherence, but that feels very unsatisfying to me. That argument seems like a basic logical "if/then" statement of post-facto causation. IF the photon/dust particle interaction occurs at this point, THEN there must be a dust particle localized at that point and not in superposition over other potential points.cube137 said:Ping any science advisor. Can you please respond the above as I'm interested in what he is saying. Thank you.
Feeble Wonk said:it seems to me that the vast array of potential states would remain part of the Hamiltonian expression describing the "potential" locality of the "potential" dust particle.
Feeble Wonk said:I suppose you could just replace the words "logical limitation" with "realized state reduction", "collapse", or something like that.
I fully accept that the mathematics of decoherence limit the quantum states that CAN occur in such a way that macroscopic superposition will not be observed WHEN an observation is made. I have no qualms about that. Yet, particularly when we are considering a large system (as I've described earlier), it seems to me that the vast array of potential states would remain part of the Hamiltonian expression describing the "potential" locality of the "potential" dust particle.
So, yes, IF we stipulate that one or more photons from any of the countless potential photon emitting events interact with a dust particle at a given location, then I suppose you could say that the mathematics of decoherence would define the physical locality of the dust particle within the Hilbert space described by that expression. However, that seems (to me anyway) to be an arbitrary assumption in the absence of a "realized" observation of the photon(s)/dust particle interaction, if we are considering the larger system including the potential photon emission events.
cube137 said:Before you spent 4 hours reading the book Bhobba suggested to you and perhaps spending 4 years taking physics course to even understand the book. Can you please explain in standard terms what you mean by the following:
bhobba said:It will take more than 4 hours - probably a few weeks spending an hour or so each day. But the pay-off is immense. I can quite easily explain what's going on with just a bit of technical background like that book provides - otherwise forget it. I remember similar discussions around the difference between a pure and mixed states. It went on and on and on. Post after post, and I am not sure after people were really any the wiser. However if you know the Bra-Ket notation it can be explained in a couple of lines. The same with the basic notion of superposition.
Thanks
Bill
cube137 said:What I meant was he spent 4 years taking course in physics first then spending 4 hours reading it (4 years later)
cube137 said:Like I think maybe what he meant was that while they are not yet collapsed, the eigenstates are all there is..
Feeble Wonk said:I suppose you could just replace the words "logical limitation" with "realized state reduction", "collapse", or something like that.
I fully accept that the mathematics of decoherence limit the quantum states that CAN occur in such a way that macroscopic superposition will not be observed WHEN an observation is made. I have no qualms about that. Yet, particularly when we are considering a large system (as I've described earlier), it seems to me that the vast array of potential states would remain part of the Hamiltonian expression describing the "potential" locality of the "potential" dust particle.
So, yes, IF we stipulate that one or more photons from any of the countless potential photon emitting events interact with a dust particle at a given location, then I suppose you could say that the mathematics of decoherence would define the physical locality of the dust particle within the Hilbert space described by that expression. However, that seems (to me anyway) to be an arbitrary assumption in the absence of a "realized" observation of the photon(s)/dust particle interaction, if we are considering the larger system including the potential photon emission events.
cube137 said:except Bill who I know would suggest for us to read dense textbooks
Yes. I think you've hit it pretty much on the head cube. Thank you.cube137 said:.
I wonder if it is the famous factorization problem or others? I think what Feeble was saying above (contemplating on it many hours) is that photon emission was probabilistic.. so how does it behave when it acts as the environment of the system (dust particle). Right?
While I sincerely appreciate your support cube, I also sympathize with Bill's frustration. He's absolutely right about the difficult position I've put him in. My moniker was not chosen randomly. I'm painfully aware of my mathematical limitations, and how difficult that makes it for him (or others) to explain the mathematical aspects of the formalism to me. I obviously have a very poor grasp of even the appropriate parlance, let alone having the mathematical chops to actually run the numbers.bhobba said:You are venturing into territory that's impossible, utterly impossible, to discuss linguistically. You must do the math.
bhobba said:Here is the genuine explanation.
You have |a> representing the state of the photons, |b> the state of the dust particle. The combined state is u = |a>|b>. Due to interactions between the two described by a Hamiltonian and Schroedinger's equation that state changes. The equation is i∂u/∂t = Hu where H is the Hamiltonian. You solve it to get the state at any time t. Now what we find is this new state is no longer factorisable into the state of the photons and the state of the dust particle. They are entangled. However if we just observe the dust particle we find its in a mixed state of position ie |b> = Σpi |bi><bi| where each |bi><bi| is an eigenstate of position. This means it can be interpreted as having a definite position with probability pi. Note I have used the notation for a state |a> and |a><a| interchangeably. What a state is and what it means is explained in the references - that's one of the things you need to understand. You can't understand this without it.
Feeble Wonk said:If so, should I think of the photon/dust particle interaction event as being the "measurement" that determines the reduced state (differentiates between the potential reduced states) of the entangled system?
bhobba said:Basically - yes.
Feeble Wonk said:When I recognize that the state of the entire expanded system is reduced correspondingly as the dust particle becomes localized, it seems as though the logical ramifications could (should?) expand exponentially throughout large swaths of the universe. Yes?
Is this because the information regarding the photon emission is so minimal (and potentially reversible) relative to the localization of the dust particle?bhobba said:No.
Thanks
Bill
...and yes I'm only asking in general terms. I strongly suspect that this is another one of those things that I'm not going to properly understand without the necessary command of the mathematics. I simply mean, is the reason that the state reduction not more expansive because the photon emission event is reversible in terms of the decoherence, or is it just because so little information is exchanged?bhobba said:No.
Thanks
Bill
Feeble Wonk said:...and yes I'm only asking in general terms. I strongly suspect that this is another one of those things that I'm not going to properly understand without the necessary command of the mathematics. I simply mean, is the reason that the state reduction not more expansive because the photon emission event is reversible in terms of the decoherence, or is it just because so little information is exchanged?
Feeble Wonk said:Fair enough.
cube137 said:Please note that Hobba and Neumaier are Ph.D. in mathematics
cube137 said:Also note Hobba and Neumaier hate the idea of wave functions as representing the objects in actual.
bhobba said:I don't have a Phd in math - I have the equivalent of an Honours degree in applied math and computer science. I have partially completed a masters but had to give it away for various reasons. I am self taught in physics so know exactly what is required to understand this stuff. A few weeks of an hour or so study each night is all that's required to learn enough to understand the technical explanation I gave.
If you don't want to do that you will be in a constant state of confusion going around in circles trying to understand what can't be understood without it. I gave the correct explanation - all that is needed is to understand it.
You may think its a mater of patience. IMHO it isn't. It's that this stuff can't be understood on the terms you want to understand it - it can't be done - wishful thinking that someone with more patience will do it simply will not work. This is not just my view eg have a look at the likes MFB gave my posts.
That's incorrect. But fits in with my observation you will not understand what's really being said if you don't know at least a smattering of the technicalities.
My ignorance ensemble interpretation is anostic to such things.
Thanks
Bill
cube137 said:What is funny is you don't understand Feeble question when he asked:
Decoherence is a quantum phenomenon in which a system becomes entangled with its environment, causing the system to lose its quantum coherence and behave classically. The dust particle is often used as an example to illustrate decoherence because it is a macroscopic object that interacts with its environment, leading to the loss of its quantum properties.
Decoherence causes the dust particle to lose its quantum properties, such as superposition and entanglement, and behave classically. This means that the dust particle will no longer exhibit wave-like behavior and will instead behave like a classical particle, following a definite trajectory and having a well-defined position and momentum.
No, decoherence is an irreversible process. Once a system becomes entangled with its environment, it is nearly impossible to reverse the process and restore its quantum coherence. This is why macroscopic objects, like the dust particle, do not exhibit quantum behavior in our everyday experience.
The environment plays a crucial role in decoherence as it is responsible for the loss of quantum coherence in the dust particle. The larger and more complex the environment, the faster the decoherence process occurs. This is why macroscopic objects, which interact with a large number of particles in their environment, experience decoherence more quickly than microscopic objects.
Decoherence is one of the biggest challenges in quantum computing as it can cause errors and lead to the loss of quantum information. Scientists are working on ways to minimize the effects of decoherence, such as using error-correcting codes and designing quantum systems with longer coherence times. However, until we find a way to completely eliminate decoherence, it will continue to be a major obstacle in the development of practical quantum computers.