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I Decoherence and standard formalism

  1. Oct 26, 2016 #1
    In the standard mathematical formalism, the environment were treated classically, this is because observers (being macroscopic recording mechanisms) are treated classically, so the system is isolated. Decoherence is about open system, so how is decoherence compatible with Copenhagen or the standard formalism at all?
    How can you make the standard formalism have an open system-environment too? Or perhaps is it correct to think the standard mathematical formalism is already updated and Copenhagen is already outdated? How do you link the two if you were to give a lecture about this in class (which I'll do)?
     
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  3. Oct 26, 2016 #2

    PeterDonis

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    Please provide a reference for this statement. It doesn't look correct to me.
     
  4. Oct 26, 2016 #3
    I mean, in the Copenhagen interpretation, the environment were treated classically because observers were treated classically. But I'm not interested in interpretations but in the standard mathematical formalism. How does the standard mathematical formalism deal with the environment? In the density matrix approach, the environment is traced out.. but this after there is entanglement between the system and environment. In the Copenhagen, there is no entanglement between the system and environment because the environment is treated classically and only the quantum system is treated in isolated. So I'm confused how many physicists could say the Copenhagen is the standard mathematical formalism when it treated the environment as classical and there is no entanglement between environment and system in the Copenhagen. How do you resolve the two?
     
  5. Oct 26, 2016 #4

    Simon Phoenix

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    That's definitely not correct. One of the motivations for the decoherence treatment was to explain how we can get something that looks like a non-unitary irreversible process from a theory (QM) in which the basic interactions are all unitary (and reversible). It's a similar problem to explaining the second law of thermodynamics within classical physics where all the fundamental processes at an individual level are obeying time-reversible evolutions.

    So the basic question here is that if the laws of QM govern everything then surely they also govern the bits and pieces that make up a measurement device. So where does this discontinuous and irreversible change that we call a 'measurement' come from? The system and the measuring device are all made of things that obey the laws of QM which mean that all of the interactions between the things that make up our system and the things that make up our measuring device are unitary interactions.

    So the very essence of the decoherence treatment is to treat everything quantum mechanically, not classically.

    If, for example, we had a single EM field mode inside a cavity then there's going to be some leakage of that radiation inside the cavity to the outside world. How do we model this? Well, if some radiation is getting out of the cavity then there has to be a coupling of the field mode inside the cavity to the field modes outside the cavity. Field modes in QM are essentially just quantum harmonic oscillators and so we can model this system as a single (cavity) field mode coupled to a number of field modes outside the cavity. Essentially just a system then of coupled harmonic oscillators.

    We can then make some reasonable assumptions about the form of that coupling and assume there are an infinite number of discrete field modes outside the cavity (all distinguished by some frequency). With this we can write down a Hamiltonian for the cavity mode plus outside modes. What we're after is an equation that describes the evolution of the field mode inside the cavity, because that's our system of interest. The field modes outside the cavity are our environment and we're looking for a kind of 'averaged' equation of motion for our cavity mode as a result of all of the myriad interactions with the environmental (outside) modes.

    To proceed further we can take the continuum limit for the environmental mode (a continuous distribution of frequencies), assume some basic initial state for the environmental modes (thermal states, for example) and then do a coarse-grained averaging procedure to end up with a master equation for our cavity field mode. What we then have is something that can model a dissipative process in QM fully quantum mechanically. It's no different in spirit to the treatment of spontaneous emission in a fully quantum way.

    What's interesting is that if we construct such a model then for certain initial environmental conditions the master equation for the cavity mode can be solved exactly and we find that the effect of the environment is to rapidly drive the cavity field mode into a diagonal density operator - which is the density operator we get from performing a measurement in which we remain ignorant of the result.

    For this cavity field mode example if we begin with a cavity field mode prepared in a superposition of two coherent states then there is an exponential decay of the off-diagonal elements of the cavity field density matrix with a decay rate that is proportional to the square of the 'distance' between them - so any macroscopic (big difference between the coherent states) superpositions get driven to mixtures very, very quickly.

    Zeh and Zurek did some magnificent work to look at this kind of thing in more general terms and showed that this decoherence is actually a more general feature of interactions of a 'small' quantum system with a large (but still quantum) environment. The idea being that pretty much any 'sensible' quantum environment is going to have this diagonalizing effect on the system of interest. So this leads to the idea that a measuring device can be modelled as a quantum object that is coupled to a large environment and if we treat it this way then we can get something out of it that kind of looks like a quantum measurement (it has the right density matrix for an ensemble).

    As a way to model dissipative effects in QM - wonderful; as a solution to the 'measurement problem' I'm less convinced but clearly some 'decoherence' process must be happening in any measuring device - which is a suitably 'large' quantum object, of course. Whether that's quite enough to explain all the features of what constitutes a quantum measurement I don't think so - but it's a fantastic step closer to fully solving that puzzle I reckon.
     
  6. Oct 26, 2016 #5
    Appreciated your explanations. What I was saying was that Copenhagen was not compatible with the idea the environment is also quantum. Hence just saying the two are not compatible and whether we should make Copenhagen outdated once and for all.

    By the way. Your first paragraph was not exactly right in that decoherence doesn't automatically means the born rule applied. It just delocalized the phases.. to get one outcome.. you still need the born rule.
     
  7. Oct 26, 2016 #6

    PeterDonis

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    And I asked you for a reference for this statement. Either provide one or stop making this claim.

    The standard theory of decoherence, which Simon Phoenix described, is interpretation neutral. It doesn't take any position on whether collapse happens or not.
     
  8. Oct 26, 2016 #7

    Simon Phoenix

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    Yes I misunderstood your initial post slightly - sorry. The issue is to understand how we can get something that looks like 'collapse' from a theory in which all the interactions are governed by time-reversible laws of evolution - so strictly speaking that's mathematically impossible. What we get is something that, for all practical purposes, looks a lot like collapse. It neatly explains why we don't see 'macroscopic' superpositions in the real world.

    The issue of why we get one particular outcome (an eigenstate) is not wholly explained within the decoherence treatment - which is one of the reasons I'm not convinced it's a full solution to the so-called 'measurement problem' in QM. I would say that it has to be a big part of that solution though.
     
  9. Oct 27, 2016 #8

    bhobba

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    That is false.

    It is silent on the issue. It simply assumes a classical world exists and observations appear in it. It says nothing about what that world is ie if its quantum or not. We now know everything is quantum even the classical world of Copenhagen.

    Thanks
    Bill
     
  10. Oct 27, 2016 #9
    Here:

    https://arxiv.org/pdf/quant-ph/0312059v4.pdf

    "The Copenhagen interpretation additionally postulates
    that classicality is not to be derived from quantum
    mechanics, for example, as the macroscopic limit
    of an underlying quantum structure (as is in some sense
    assumed, but not explicitely derived, in the standard interpretation),
    but instead that it be viewed as an indispensable
    and irreducible element of a complete quantum
    theory—and, in fact, be considered as a concept prior to
    quantum theory. In particular, the Copenhagen interpretation
    assumes the existence of macroscopic measurement
    apparatuses that obey classical physics and that
    are not supposed to be described in quantum mechanical
    terms (in sharp contrast to the von Neumann measurement
    scheme, which rather belongs to the standard
    interpretation); such a classical apparatus is considered
    necessary in order to make quantum-mechanical phenomena
    accessible to us in terms of the “classical” world of
    our experience. This strict dualism between the system
    S, to be described by quantum mechanics, and the apparatus
    A, obeying classical physics, also entails the existence
    of an essentially fixed boundary between S and A,
    which separates the microworld from the macroworld (the
    “Heisenberg cut”). This boundary cannot be moved significantly
    without destroying the observed phenomenon
    (i.e., the full interacting compound SA).


    Especially in the light of the insights gained from decoherence
    it seems impossible to uphold the notion of a
    fixed quantum–classical boundary on a fundamental level
    of the theory."

    How is Copenhagen based on a classical environment compatible with environment based on quantum? I'm running out of word to say in a lecture so hope you can share how they are still related. Maybe can one say Copenhagen has less explanatory power even if the mathematics can be fitted to either?
     
  11. Oct 27, 2016 #10

    bhobba

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    Last edited by a moderator: May 8, 2017
  12. Oct 27, 2016 #11

    bhobba

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    That is false - it makes no such assumption.

    I can give a link that carefully explains the details but unfortunately the source strictly speaking doesn't meet our guidelines.

    The textbook I mentioned does meet our guidelines and explains it all in excruciating detail as well as some of the issues that still remain.

    Thanks
    Bill
     
  13. Oct 27, 2016 #12

    Simon Phoenix

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    Last edited by a moderator: May 8, 2017
  14. Oct 27, 2016 #13

    bhobba

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    It is THE book on the issue - worth every cent IMHO. But it is no easy read - you need at least the background of Griffiths or similar to understand it, but likely with effort get by with Susskind.

    Thanks
    Bill
     
  15. Oct 27, 2016 #14

    Simon Phoenix

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    I think I might just about manage :wink:
     
  16. Oct 27, 2016 #15
    Ey, that paper was written by the same author as the book! The paper was a condensed version of his book. I have the book. I'll find the relevant passages.
     
  17. Oct 27, 2016 #16
    Ok. Here's the very passage from the textbook you suggested to Simon https://www.amazon.com/dp/354035773...e=df0&creative=395093&creativeASIN=3540357734
    or Maximilian Schlosshauer "Decoherence and the Quantum-to-Classical Transition". In page 335. Maximilian quoted

    "The Copenhagen interpretation additionally postulates that classicality is not to be derived from quantum mechanics..."

    So Maximilian is wrong??
     
    Last edited by a moderator: May 8, 2017
  18. Oct 27, 2016 #17

    bhobba

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    On the surface yes - but context is everything eg their are a number of variants of Copenhagen and different contexts on what classically means eg some take classical to mean exactly how macro objects emerged which Copenhagen does not explain.

    That book is my bible on decoherence and I know for a fact that is not his view which is that there are 3 parts to the measurement problem.

    1. The problem of non observance of interference
    2. How the preferred basis emerges.This is why, for example, classical objects nearly always have a definite position
    3. How an improper mixed state becomes a proper one.

    The first 2 is explained by decoherence, the third some interpretations simply assume, while others explain.

    The three taken together imply classicality because they explain how objects for all practical purposes have definite values which is what classical physics is.

    Now I could give you the page he says that but in this type of situation where context can be a problem I would like you to explain to me, in your own words, not a quote, but in your own words, how objects with definite values is not classical. In particular exactly what do you mean by classicality.

    Just as a hint to what might be going on is the so called factorisation problem which IMHO is way over hyped, but if its your worry requires another thread:
    https://arxiv.org/abs/1210.8447

    Thanks
    Bill
     
  19. Oct 27, 2016 #18

    bhobba

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    The above is very very true.

    Now I ask you to think a bit. If there is no fixed boundary, and there isn't, (everything is quantum) whats a classical object to begin with?

    Once you nut it (not a quote but in your own words) out please let us know what a classical object is for your claims to make sense.

    Hint - the divide is a human construct and lies outside the theory - its also related to the factorisation issue. Either way its a very difficult problem but usually has some common characteristics such as exact position etc.

    In modern times we don't worry about it - its simply when decoherence occurs.

    Thanks
    Bill
     
  20. Oct 27, 2016 #19

    bhobba

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    Its not. Its treated via QM and randomises phases so its irreversible.

    Thanks
    Bill
     
  21. Oct 27, 2016 #20

    Demystifier

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