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Fundamental non-unitarity

  1. May 26, 2014 #1

    marcus

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    Bei Lok Hu at the University of Maryland has a review article on the "fundamental decoherence" research topic, which devotes detailed attention to the treatment by Gambini and Pullin.

    I think it's an interesting topic for several reasons, so I'll give some links. Here's the paper by B.L. Hu et al.:
    http://inspirehep.net/record/781938 (Intrinsic and Fundamental Decoherence: Issues and Problems)
    http://inspirehep.net/author/profile/B.L.Hu.1 (profile of Bei Lok Hu)

    Here are G&P's papers on this topic:

    http://inspirehep.net/record/645205 47 cites (A Relational solution to the problem of time in quantum mechanics and quantum gravity induces a fundamental mechanism for quantum decoherence)
    http://inspirehep.net/record/653376 38 cites (Realistic clocks, universal decoherence and the black hole information paradox)
    http://inspirehep.net/record/674573 12 cites (Fundamental decoherence in quantum gravity)
    http://inspirehep.net/record/712912 38 cites (Fundamental decoherence from quantum gravity: A Pedagogical review)
    http://inspirehep.net/record/735013 25 cites (Relational physics with real rods and clocks and the measurement problem of quantum mechanics)

    I think the gist of it is that in GR natural processes occur at different rates all over the place. There is no official/ideal time, so the best one can do is correlate the other observables to some choice of *real clock*. The definition of unitary evolution is only as good as the clock.

    It seems that Eugene Wigner came up with a theoretical limit on the precision-lifespan of a real clock (how accurate for how long a time a clock could be made without it foiling you by turning into a black hole). And Gambini Pullin adapted Wigner's limit on real clocks to find a theoretical limit on the lifespan of unitarity.

    You will have to refer to their "Realistic Clocks" paper, http://arxiv.org/abs/hep-th/0406260, because I can't reproduce their argument in detail, but the upshot seems to be that if one focuses on black hole evaporation the unitarity of evaporation dies out on a timescale comparable to the lifespan of the black hole itself.
     
    Last edited: May 26, 2014
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  3. May 26, 2014 #2

    marcus

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    ==quote from conclusions of G&P http://arxiv.org/pdf/hep-th/0406260.pdf ==
    Summarizing, we have shown that unitarity in quantum mechanics only holds when describing the theory in terms of a perfect idealized clocks. If one uses realistic clocks loss of unitarity is introduced. We have estimated a minimum level of loss of unitarity based on constructing the most accurate clocks possible. The loss of unitarity is universal, affecting all physical phenomena. We have shown that although the effect is very small, it may be important enough to avoid the black hole information puzzle.
    ==endquote==

    A moment's reflection reveals the importance of their result. The only operationally meaningful unitarity is unitarity based on a realistic clock. There is a limit to the precision and stability of a realistic clock and therefore unitarity has limits *in principle*. There are limits *in principle* to the timescales over which unitarity is applicable and can be expected.

    Gambini and Pullin find, in the interesting case of black hole evaporation, that the term limit for unitarity is comparable to the estimated evaporation time of the black hole.
     
    Last edited: May 26, 2014
  4. May 31, 2014 #3

    marcus

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    Assuming unitarity in the BH evaporation process leads to various bizarre conclusions.
    For a taste of the current brew skip to shortly after minute 60 of Smerlak's 29 May PIRSA talk.
    http://pirsa.org/14050027/
    Last gasp ‎of a black hole: why unitary evaporation must be non-monotonic
    Speaker(s): Matteo Smerlak
    Abstract: I will describe the relationship between radiated energy and entanglement entropy of massless fields at future null infinity (the "Page curve") in two-dimensional models of black hole evaporation. ‎I will use this connection to derive a general feature of any unitary-preserving evaporation scenario: the Bondi mass of the hole must be non-monotonic. Time permitting, I will comment on time scales in such scenarios.
    Date: 29/05/2014

    the talk is based on two recent papers with Eugenio Bianchi:
    http://arxiv.org/abs/1405.5235
    http://arxiv.org/abs/1404.0602

    I think what we are seeing is more like the "last gasp of unitarity" (where misapplied over time scales comparable to the age of the universe.)

    Between minute 60 and 66 he is simply disposing of the the Hawking (semiclassical) evaporation scenario by showing that it leads to singularities/inconsistency (so "one should stop believing in it")
    Then around minute 67 he introduces other scenarios (which still assume unitarity).
    The first of these is the "Hayward proposal". At minute 73:50 he says that the Hayward scenario is "ruled out" because purification of the final state occurs in a burst, violating energy conservation. The authors have a theorem that purification must be slow. Unfortunately after about minute 70 he was running out of time, and there were a lot of questions from the audience--he didn't have time to prove or even discuss the theorem that purification must be slow (which was included in the initial plan of the talk).
     
    Last edited: May 31, 2014
  5. May 31, 2014 #4

    marcus

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    Some quotes:
    "When one introduces realistic clocks, quantum mechanics ceases to be unitary and a fundamental mechanism of decoherence of quantum states arises. We estimate the rate of universal loss of unitarity using optimal realistic clocks. In particular we observe that the rate is rapid enough to eliminate the black hole information puzzle: all information is lost through the fundamental decoherence before the black hole can evaporate."
    (http://arxiv.org/abs/hep-th/0406260)

    "...general relativity is a generally covariant theory where one needs to describe the evolution in a relational way. One ends up describing how certain objects change when other objects, taken as clocks, change. At the quantum level this relational description will compare the outcomes of measurements of quantum objects."
    (http://arxiv.org/abs/gr-qc/0603090)

    "...as ordinarily formulated, quantum mechanics involves an idealization. That is, the use of a perfect classical clock to measure times. Such a device clearly does not exist in nature, since all measuring devices are subject to some level of quantum fluctuations. The equations of quantum mechanics, when cast in terms of the variable that is really measured by a clock in the laboratory, differ from the traditional Schroedinger description. Although this is an idea that arises naturally in ordinary quantum mechanics, it is of paramount importance when one is discussing quantum gravity. This is due to the fact that general relativity is a generally covariant theory where one needs to describe the evolution in a relational way..."
    (http://arxiv.org/abs/quant-ph/0608243)

    To paraphrase, in the real world duration and distance are defined operationally by physical measurements which is to say by events. In reality there is no such thing as abstract idealized "time". There are various natural processes, whose rates, when compared, depend on their relative motion and their position in the gravitational terrain. Defining an observer's proper time ultimately depends on the choice of some physical process to be the observer's clock.
    Unitarity is only as good as the physical clock, and cannot be taken as absolute. It is potentially misleading to assume unitarity on timescales comparable to the present age of expansion.
     
    Last edited: May 31, 2014
  6. Jun 1, 2014 #5
    I understand 'unitarity' to be a fashionable way of expressing the long-held assumption that mass/energy is conserved, absolutely and always. For example, when a fundamental particle -- say an isolated electron -- is described by a quantum wave function whose absolute square value, integrated over all space, sums to unity, one is expressing a belief that the electron always exists somewhere. By extension, an ordinary carbon atom should also persist permanently, validating the advertisement 'A Diamond is Forever'. Is this now being revealed as just a romantic commercial snare?
     
  7. Jun 1, 2014 #6

    marcus

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    :biggrin: nice example,
    for convenience I'll put some PF links about distinction between pure states (unit vectors in the H-space, can be superpositions) and mixed states. Unitarity preserves pureness.
    https://www.physicsforums.com/showthread.php?t=595708
    https://www.physicsforums.com/showthread.php?t=242158
    https://www.physicsforums.com/showthread.php?t=535917

    A unitary transf. is one that takes pure states to pure states. Unit vectors to unit vectors (hence the name). So the pure/mixed distinction is important here. If system is in a pure state and you know the state, then you know all there is to know (e.g. regarding the spin or polarization) about the system. This does not mean you can predict the outcome of every possible measurement. You just know all the rules of nature allow you to know. E.g. if it has been determined that the spin is "up" in a certain orientation, along a certain axis, and you know that, then you still can't predict what it will be along ANOTHER axis. But you know everything about the spin that nature allows you to know.
    In a mixed state there is missing information---something you could in principle know but don't. Like in a two-slit setup where it actually bumped something while passing thru one of the slits, but you didn't notice which. So for you it is in a state 1/2 R plus 1/2 L.

    There's also this very strenuous technical treatment of the pure/mixed business. I list but don't recommend.
    http://pages.uoregon.edu/svanenk/solutions/Mixed_states.pdf

    Mixed states can have probabilities summing to one, but they aren't unit vectors in the Hilbert space. A unitary transformation takes unit vectors to unit vectors. Conceptually, it preserves completeness of the info. You don't lose info about the system when a unitary is applied, the way you would if it evolved into a mixed state. (a mixture of pure states where you didn't know which one.)

    Loss of unitarity means loss of predictability. I think. Have to go, back later today.
     
    Last edited: Jun 1, 2014
  8. Jun 2, 2014 #7

    marcus

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    I should have said "then you know all you can know about the system" (rather than "all there is to know").
    I want to re-iterate part of the previous post:
    "You just know [as much as] the rules of nature allow you to know. E.g. if it has been determined that the spin is "up" in a certain orientation, along a certain axis, and you know that, then you still can't predict what it will be along ANOTHER axis. But you know everything about the spin that nature allows you to know."

    I think unitarity amounts to more than "a fashionable way of expressing the long-held assumption that mass/energy is conserved,…"

    It seems to be more about information (predictability of future, reconstructibility of past) than about conservation of energy etc. I'll get back to this in the morning when hopefully I can be more coherent :^D
     
    Last edited: Jun 2, 2014
  9. Jun 3, 2014 #8
    Thanks for those links, Marcus. I now see that unitarity is more than just simplistically conserving
    mass/energy, as I had assumed. Instead the conserved feature associated with unitarity seems to
    be probability, as explained in Merzbacher’s discussion of The Optical Theorem (his Quantum
    Mechanics, 2nd ed., p.505). Since quantum mechanics is all about the probablistic evolution of
    ‘systems’ that have attributes like names, mass/energy, charge and angular
    momentum. I suppose that it is the continuity of these attributes, including the continued existence and
    identity of the system itself, say during a scattering event, that demands conservation of an
    appropriate probability, a.k.a. unitarity?
     
  10. Jun 3, 2014 #9

    marcus

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    Hi, Paulibus! I got distracted by other things yesterday and forgot to get back to this thread. Thanks for replying and continuing the conversation despite my negligence. Looking back, I think what I said in #6 and #7 is basically OK. Likewise what you say in #8, although there are two ways of looking at it.

    I think you can look at unitarity as preserving information or as preserving probability. BTW if it is not too off-topic you know the Heisenberg vs Schrödinger difference in perspective on QM? With Schrö there is a wave function (say for a single particle) that evolves in time. With Heisen it is the OPERATORS that evolve in time! Think of the STATE as just a fixed vector and the operators are matrices which can evolve, and to get NUMBERS you apply a matrix to the fixed state vector and take inner product. <ψ, Aψ>

    So in Heisen the "observables" evolve and the the "state"is just a fixed vector in the Hilbertspace that serves as a device to get actual numbers from the evolving operators that represent "observables".

    And then there is an algebraic trick where you represent the evolution of observable matrix A by bracketing it with a unitary matrix U
    Anew = U* Aold U

    And that AMOUNTS TO THE SAME THING as applying the unitary transformation U to the STATE ψ
    TO GET A NEW STATE. But basically the Heisen approach ignores "wave functions" and focuses on matrix mechanics----with states being unit vectors in an abstract inner product vector space. It is "relational" in the sense that it focuses on observations---the matrices correspond to possible measurements or interactions or detections...
    ===================
    For the moment just forget I said that about the Heisenberg perspective. Let's focus on the Schrö picture with just one particle. The wave function is defined on space and is an amplitude to find the particle at that spot and it is square integrable and it is a UNIT VECTOR in an obvious inner product vector space (of functions) and amplitude times its conjugate can be seen as a probability which integrates to ONE. And a unitary transformation on the vector space of wave functions takes unit vectors to unit vectors so by definition it preserves this property of adding up to one.
    So one can say "unitary means it preserves probability". It's nice and intuitive.
    ===================
    But maybe it is the wrong way to look at QM. The wave function is only defined on space or space time when you have a SINGLE particle, and with many particles it is defined on a multidimensional phase space. And as soon as there are interactions the number of particles might change and the dimension of the phase space keeps changing! So mathematically it's an unworkable mess
    and what they actually do is use the HEISENBERG approach to establish a quantum FIELD THEORY in which you have MATRIX-VALUED FUNCTIONS DEFINED ON spacetime.

    With the Schrö approach you have to give up defining functions on ordinary 4D Minkowski spacetime, because your functions are too simple valued, they are just complex amplitude valued. It doesn't work except in the very simple case. But with the Heisen approach you get to keep 4D Minkowski spacetime and all the operator-valued fields are defined on the same spacetime.

    So then the question comes back: what does "unitary" mean, in this field theory context, where we no longer have a "wave function" telling simply the amplitude of a particle to be at some particular place at some moment.

    I think now unitary means more something like preserving information, or preserving coherence, predictability. It is not as clear what the intuitive meaning is. I'll try to suggest it by the most elementary possible (in fact embarrassingly simple) example

    Suppose |R> and |L> are two unit vectors in a complex inner product vector space. And suppose their inner product is zero (they are orthogonal). I'll abbreviate sqrt 0.5 by .707
    Then .707|R> +.707|L> is a unit vector, it has norm = 1
    but .5|R> +.5|L> is not a unit vector, it has norm = 1/2
    That second one corresponds to where the thing bumped something as it was going thru a slit so it went thru one or the other but you did not notice which. So with probability 1/2 it went thru R and with prob 1/2 it went thru L. But you don't know which! You lost some information. The thing decohered. I'll try to present a better example later, maybe involving spin up and spin down, or an "EPR" type Alice and Bobber. The PF threads I linked to earlier provide some helpful discussion.
     
  11. Jun 3, 2014 #10

    marcus

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    Paulibus, I want to recall the essential passages from Gambini Porto Pullin papers I quoted earlier and emphasize their relevance to the Planck star model black hole and to the LQG bounce. If the information engulfed by a black hole has finite shelf life then, in the Planck star context that tells us something about the LQG bounce. And in particular it tells us that it produces a blank slate structure-wise, so that we should not be surprised by the low geometric entropy at the start of cosmological expansion.
    The logic is based on comparing two different time-scales. A distant outside observer (à la Gambini et al) sees information-bearing structure fall into the black hole and then after a very long time that info has expired and does not come out during evaporation. Translating that to the Planck star model, the gamma ray burst (GRB) at the end of the bounce is a total blank. (From the outsider's viewpoint the bounce has been time-dilated so that it takes a long time).

    But consider how it looks from the viewpoint of an observer rides in with the collapse and witnesses the bounce and the ensuing GRB. For him it all happens very quickly. And yet all the structure in what fell in is wiped out.
    Something special was going on. It probably has to do with the fact that during LQG bounce quantum effects make gravity repel instead of attract.

    That understanding of what occurs during black hole bounce carries over to the case of the cosmological bounce. It should not surprise us that the geometry which appears there is even and blank---no pockmarks and wrinkles---doesnt have a lot of structure written into it from the prior contracting phase.
     
    Last edited: Jun 3, 2014
  12. Jun 6, 2014 #11
    Thanks for your thought-provoking posts 9 and 10. From the point of view of someone trying to
    understand where fundamental physics is going I found them stimulating. Some comments:

    I agree and think that the simplest interpretation of 'unitarity' is that it's a preserver of information
    collected under the umbrella of 'identity'; whether the system or object under consideration is
    described mathematically by a wave function or a by an S(scattering)-matrix, it's identity remains
    in focus as it evolves with time.
     
  13. Jun 7, 2014 #12

    marcus

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    Fundamental decay of unitarity (and the "blank rebound problem")

    Paulibus, I'm trying to delineate what I've heard called the "blank rebound problem" or words to that effect, which seems to come up in Loop gravity and cosmology if you make the "Gambinian" assumption that precise timing decays over time. Assuming such, unitarity itself decays over long spans of time. We are talking spans of time that are over a million-fold the present age of the universe. The only meaning unitarity can have is operational (e.g. clocks and rods) and it the concept can be indistinguishable from perfection over measurable spans of time or even conceivable spansm and yet be found wanting on inconceivably long scales like the lifespan of an astrophysical black hole. So I just refer back to those Gambini et al quotes and say "lets assume they're right".

    Then talking about astrophysical objects, the final gamma burst of a Planck star is blank, all info about what fell in to form it has expired. This is from the outsider's viewpoint. But what about an hypothetical observer who falls in and participates in the bounce.

    For him the bounce happens quickly. He has to see everything turn blank in the blink of an eye. His own memory of the past has to be wiped!

    Because when the observer comes flying out in the star's final gamma burst he hast to agree with the outsider that the burst is nondescript virgin energy with no microstate memories. So there must be something special about the Loop black hole bounce. It does a blank rebound.

    How can that happen? That, I guess, is the "blank rebound problem".
     
  14. Jun 7, 2014 #13

    marcus

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    Wait. Maybe "in the blink of an eye" is meaningless. Maybe all clocks malfunction above a certain density.
    Gambini et al, when calculating the theoretical limit on the endurance of unitarity, found that the best clock was a black hole (for long-lasting stability and precision). the observer would cause it to "ring" and keep time with its frequency. But above a certain frequency Loop quantum corrections make gravity repel---so a black hole clock would burp itself out of existence. What clock could function competently at Loop bounce density?

    Can one even say how long the bounce takes, from the participant's viewpoint?

    Unitarity needs the passage of time, measured by some clock, in order to be defined. what if clocks cease to function at some point in the bounce? when the density exceeds a certain threshold?

    I likely must discard the idea of a participant observer and stick with Gambini Porto Pullin's idea of an outside observer---one outside the black hole who can witness the collapse that forms it and then measure how long it takes to evaporate using a separate timepiece.

    I keep having to go back to the quotes (e.g. in post #10) from the Gambini et al papers.

    What I want to do is learn something about the Loop black hole bounce, which will then carry over to the cosmological bounce and explain why one should expect expansion to begin with a "blank slate" low entropy state.

    Can one even say how long the Loop cosmological bounce takes, from the participant's viewpoint?

    And what other viewpoint could there be, since the cosmos is all-inclusive?

    For someone in the past, looking forward, how long does the actual bounce take, since he can never see expansion start. How long does it take for someone in the expanding phase, looking back, since there an hiatus blocks his view of the contracting phase? Maybe time-evolution around that point is ill-defined, essentially because of a shortage of clocks. :-)
     
    Last edited: Jun 7, 2014
  15. Jun 9, 2014 #14

    marcus

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    This looks like it might be relevant. I noticed it does not cite the earlier papers of Gambini et al.
    http://arxiv.org/abs/1403.6457
    Purity is not eternal at the Planck scale
    Michele Arzano
    (Submitted on 25 Mar 2014)
    Theories with Planck-scale deformed symmetries exhibit quantum time evolution in which purity of the density matrix is not preserved. In particular we show that the non-trivial structure of momentum space of these models is reflected in a deformed action of translation generators on operators. Such action in the case of time translation generators leads to a Lindblad-like evolution equation for density matrices when expanded at leading order in the Planckian deformation parameter. This evolution equation is covariant under the deformed realization of Lorentz symmetries characterizing these models.
    6 pages.
     
  16. Jun 9, 2014 #15
    Having read all your posts, I as a layman gather that this research is somewhat of a breakthrough. Again from the point of view of a layman, how ground-breaking is this?

    Is unitarity effectively threatened as a fundamental premise of physics or is this a limited, specific quirk of QM that has little/no implication for the wider scope of physics as a whole.

    For example, I understand that the conservation of energy in GR is problematic on cosmological scales but that does'nt infirm the theory in itself. Is this the same kind of observation that we now have with unitarity?


    IH
     
  17. Jun 9, 2014 #16

    marcus

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    I think that is an intelligent comparison (with conservation of energy in GR) and I'll try thinking of it that way.

    It's good to maintain a sense of proportion, and I think that is primarily what you are looking for. I appreciate having another interested person to talk with about these issues even though I cannot say they represent breakthrough or groundbreaking physics.

    We can't say for sure that the 2004-2006 Gambini Pullin result is right (though it well could be),
    moreover it is only one of several proposed solutions to the BH evaporation paradox
    (all of which so far seem to lead to further paradoxical or bizarre consequences!)
    and even if it is right there would only be a very mild effect on the general principle of unitarity
    (because they conclude that unitarity spoils very slowly, over billions of years, at the rate the best clocks get out of synch).

    G&P are respected influential physicists and so far as I know no one has pointed out any flaw in their argument. So it makes sense, I think, to take seriously the possibility that they could be right (unitarity expires over very long durations, as time itself loses definition) and see where that might lead us.

    For all practical purposes, unitarity is perfect and permanent (on a human and particle-physics timescale) but nevertheless I would like to share with you why I find it interesting to follow out the consequences of Gambini Pullin reasoning.

    The interesting paradox, for me, comes when I combine the LQG black hole bounce picture with Gambini Pullin.
    E.g. google "Planck star" and you get the paper of Rovelli Vidotto which points out that "a black hole is a shortcut to the distant future". At very high density, according to its LQG quantization, gravity repels and causes a bounce. But natural processes are slowed, deep down in a gravity well. so the bounce (and eventual gamma ray burst ,GRB, explosion) takes a long time to happen, seen from outside.

    if it makes sense to think of a conventional observer, with a clock, riding in on the collapse and participating in the bounce and riding out in the GRB explosion then what that observer sees should obey unitarity

    He takes a shortcut to the distant future and for him (unless time measurement for him is somehow interrupted) the whole process from collapse to final explosion takes a short time---so the process should be indistinguishable from ideal unitary.

    but for the outside observer at some distance from the black hole collapse, who waits for the final explosion, the process takes many billions of years and should be recognizably non-unitary.

    That is unless something novel happens right at the point of the LQG bounce which resolves the contradiction. I have been wondering what that could be, and asking if it could have anything to do with what ARZANO wrote about earlier this year. He seems to be suggesting that when information is compressed to very small scale then these very small scale processes can suffer a degradation of unitarity. I just came across his paper yesterday, and need to spend a little time looking at it, or get someone else more expert to explain it to me. (I am NOT expert, more the interested bystander who happens to be fascinated with QG and cosmology.)
     
    Last edited: Jun 9, 2014
  18. Jun 9, 2014 #17

    marcus

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    I found a 2010 video talk by Arzano which can serve as an introduction to some of what is discussed in the March 2014 paper:
    Here is the 2010 online video:
    http://pirsa.org/10050017/ [Broken]
    Fun from none: deformed Fock space and hidden entanglement
    Michele Arzano
    Attempts to go beyond the framework of local quantum field theory include scenarios in which the action of external symmetries on the quantum fields Hilbert space is deformed. A common feature of these models is that the quantum group symmetry of their Hilbert spaces induces additional structure in the multiparticle states which in turns reflects a non-trivial momentum-dependent statistics. In certain particular models which might be relevant for quantum gravity the richer structure of the deformed Fock space allows for the possibility of entanglement between the field modes and certain ''planckian'' degrees of freedom invisible to an observer that cannot probe the Planck scale.
    19/05/2010
     
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  19. Jun 10, 2014 #18

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    I have a bunch of questions about non-unitarity, although I may not be qualified to understand the answers.

    First, the statement about the lack of ideal clocks implying a breakdown of unitarity seems strange to me. Science always has to deal with non-ideal measurements, but we don't usually feel the need to rewrite the laws of physics to take into account our flawed measurement devices. Perhaps the distinction is that in other branches of physics, we can understand our measuring devices as giving an approximation to some ideal, while in quantum gravity, there is no way to even define what the ideal would be?

    Second, in ordinary single-particle quantum mechanics, unitarity is connected with the laws of probability; we want the integrated probability density to equal 1, and unitarity insures that if this is true at one time, it will always be true. If we allow non-unitary evolution, then I'm not sure what that means for the probabilistic interpretation of quantum mechanics.
     
  20. Jun 10, 2014 #19

    marcus

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    Hi Steven, thanks for the comment! I think that's a clear way to put it and goes to the heart of the matter. It's actually true for 1915 GR as well. You've probably heard about the "problem of time" in GR. There is no way to define the "ideal time" in GR until you have solved the equation and fixed on a specific solution--and then you get different proper times for different observers.

    There is a nice brief discussion of this for non-specialist near the beginning of Carlo Rovelli's 2006 essay "unfinished revolution" that served as the first chapter of a book on the various approaches to quantum gravity. If you are curious I think googling "rovelli revolution" would get it.

    So you already have this no-ideal time in the general rel theory and, as the essay explains, the problem is even more serious when you go towards quantum GR, because in quantum theory continuous trajectories do not exist, reality is stripped down to intermittent interactions. Sporadic events. Discrete "observables".

    Well, I have trouble with the single particle Schrödinger "wave function" picture of quantum mechanics. because for one thing when you have many particles you no longer have a wave function defined on space time and if particles are interacting and being created/annihilated the wave function picture gets a bit confusing. So the picture of QM I have is operators defined on a Hilbertspace, that is basically a complex vector space with inner product. I lean towards the Heisenberg version of QM.

    You know what UNIT VECTORS are---the analog of the unit circle in the complex plane---vectors with norm one. For me a unitary operator is one that maps unit vectors to unit vectors.

    Since unit vectors in QM are called "pure states" a unitary operator is one that preserves purity.

    A probabilistic combination of pure states, like for instance 1/2 |R> + 1/2 |L>, would not necessarily have norm one. If |R> and |L> are orthogonal, it would have norm 1/2, instead of norm 1. You can check by multiplying out the inner product (but this could all be familiar to you).
    So this kind of probabilistic combination is a called a MIXED STATE.

    So as I see it you can have transformations which DO preserve probability but which are NOT unitary because they take a pure state ( norm 1) to a mixed state (e.g. norm 1/2).

    You may already know all this stuff but I want to be as clear and basic as I can, about it, because other people might be reading the thread.

    My excuse for taking this point of view is that the prevailing quantum theory, that is the basis of the Standard Model of matter, is quantum field theory (QFT) which is very much a Heisenberg-type formalism. It consists of OPERATORS distributed over the 4D spacetime of special rel.
    "Operator-valued distributions" At any point of spacetime you have a bunch of operators .

    I can think of them as observables corresponding at least in part to measurements or particle detections which I might or might not make around that point.

    And all these operators are transformations on the one Hilbertspace, of "states". A state represents comprehensive information about the system as a whole (all its particles etc), and the chosen state is what allows us to make each and every observable-type operator give us a real number, if we think of making an observation with that operator. There is just ONE STATE for all that huge welter of operators distributed all over the spacetime. You can think of operators evolving over time, while the state remains the same (barring new info input). That's the essential feature of the Heisenberg picture as opposed to the Schrödinger one.

    Basically just explaining a point of view (which I subscribe to) about what unitary means:
    it means "preserves pure states" and gradual loss of unitarity means that a pure state can devolve down into a probabilistic mixture. I hope you can see how it looks from my viewpoint.
     
  21. Jun 10, 2014 #20

    marcus

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    Steven, the "unfinished" character of quantum theory comes from, more than anything else, the fact that QFT is SPECIAL relativistic rather than GENERAL. It is defined on the 4d ("Minkowski") spacetime of special relativity, with its Lorentz or Poincaré group symmetry. And it does not know anything about GR.

    The gradual expiration of unitarity is a GENERAL relativity effect. I don't see how you can even describe it within the mathematical framework of (special) relativistic QFT. Maybe other people can, but I can't. I suspect that what Gambini and Pullin found in their 2004-2006 papers about this could very well be right and yet
    1. on ordinary particle and human timescales it doesn't matter because expiration is so slow as to be undetectable
    2. people won't be able to mathematically EXPRESS the gradual loss of unitarity until they actually have arrived at a general relativistic quantum field theory.

    Since we have turned a page I had better bring forward the quotes from those papers by Gambini Porto Pullin.
     
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