How to think about Quantum Mechanics—Part 2: Vacuum fluctuations

[Other parts in this series: 1,2,3,4,5,6.]

Although it is possible to use the term “vacuum fluctuations” in a consistent manner, referring to well-defined phenomena, people are usually way too sloppy. Most physicists never think clearly about quantum measurements, so the term is widely misunderstood and should be avoided if possible. Maybe the most dangerous result of this is the confident, unexplained use of this term by experienced physicists talking to students; it has the awful effect of giving these student the impression that their inevitable confusion is normal and not indicative of deep misunderstanding“Professor, where do the wiggles in the cosmic microwave background come from?” “Quantum fluctuations”. “Oh, um…OK.” (Yudkowsky has usefully called this a “curiosity-stopper”, although I’m sure there’s another term for this used by philosophers of science.) a  .

Here is everything you need to know:

  1. A measurement is specified by a basis, not by an observable. (If you demand to think in terms of observables, just replace “measurement basis” with “eigenbasis of the measured observable” in everything that follows.)
  2. Real-life processes amplify microscopic phenomena to macroscopic scales all the time, thereby effectively performing a quantum measurement. (This includes inducing the implied wave-function collapse). These do not need to involve a physicist in a lab, but the basis being measured must be an orthogonal one.W. H. Zurek, Phys. Rev. A 76, 052110 (2007). [arXiv:quant-ph/0703160] b  
  3. “Quantum fluctuations” are when any measurement (whether involving a human or not) is made in a basis which doesn’t commute with the initial state of the system.
  4. A “vacuum fluctuation” is when the ground state of a system is measured in a basis that does not include the ground state; it’s merely a special case of a quantum fluctuation.

More explicitly: Amplification requires (1) a system that can take be in one of many states, and (2) an amplifier that records that state in multiple macroscopic slots. Let \mathcal{S} be the system taking multiple states \vert S_i \rangle, and let the amplifier be \mathcal{A} = \mathcal{A}^{(1)} \otimes \cdots \otimes \mathcal{A}^{(N)}, where each \mathcal{A}^{(n)} is one component of the amplifier taking possible states \vert A_i \rangle. Then amplification is a special type of physical evolution that obeys

(1)   \begin{align*} \vert S_i \rangle  \vert A_0 \rangle  \cdots \vert A_0 \rangle \to \vert S_i \rangle  \vert A_i \rangle \cdots \vert A_i \rangle  \end{align*}

That means the evolution for an arbitrary state \vert \psi \rangle = \sum_i c_i \vert S_i \rangle is

(2)   \begin{align*} \vert \psi \rangle \vert A_0 \rangle  \cdots \vert A_0 \rangle \to \sum_i c_i \vert S_i \rangle \vert A_i \rangle  \cdots  \vert A_i \rangle  \end{align*}

The fact that unitary evolution preserves the inner product means that the \vert S_i \rangle must be orthogonal for any choice of non-overlapping \vert A_i \rangle.

If \vert \psi \rangle is the ground state for some quantum system, and the \vert A_i \rangle are macroscopically distinguishable states, then this is a (single) vacuum fluctuation. An observer that looks at the state of a component of the amplifier will find it in one of the states \vert A_i \rangle with probability p_i = \vert c_i \vert^2. Importantly, the above evolution need not involve any measuring equipment of the observer, and in fact could have taken place billions of years before humans existed (e.g. shortly after the big bang).

If the system naturally relaxes to the ground state \vert \psi \rangle after some time, after which the amplification process is repeated, then we will get another fluctuation. More generally, the amplification process and the relaxation are happening concurrently, with a continuous generation of entropy (i.e. a truly quantum random number generator).

You may have heard that the vacuum in QFT is ‘constantly fluctuating’ or that it’s a ‘sea of particle-antiparticle pairs momentarily popping into existence before annihilating’. Well. That’s true in exactly the same sense that it’s true that an electron in the ground state of the hydrogen atom is ‘constantly whirling around the atom’ or is ‘disappearing and reappearing on opposite sides of the nucleus’. Which is to say, these are intuitive physical descriptions of certain basis vectors (e.g., position eigenstates) which have nonzero overlap with the ground state (or vacuum), and which have a nonzero probability of being observed if the corresponding measurement were made, but which do not sensibly describe ground state before it is measured. Indeed, we know it’s not sensible, when a quantum system is in some given state, to pretend that it’s in one of a number of eigenstates (not including the current state) of some observable, and that a measurement simply reveals which. Moreover, the ground state (or vacuum) is a stationary state. Nothing is happening.This post was prompted and partially inspired by the paper on Boltzmann brain fluctuations by Kim Boddy, Sean Carroll, and Jason Pollack, who come to similar conclusions. I’m not sure I can agree with everything they write, but their description of a quantum fluctuation in their Section II is the clearest one I know in the literature. I encourage you to check it out: K. Boddy, S. Carroll, and J. Pollack, “De Sitter Space Without Quantum Fluctuations”. [arXiv:1405.0298] c   You can of course tell a story about unobservable events or dynamics that supplement the quantum state, and thereby instantiate hidden variables, but we all know these are necessarily nonlocal and correspondingly righteously shunned.

[Originally titled “Bad ways to think about Quantum Mechanics…”, but that turned out to be easily misinterpreted. Largely re-written on 2016-11-12.]

Footnotes

(↵ returns to text)

  1. “Professor, where do the wiggles in the cosmic microwave background come from?” “Quantum fluctuations”. “Oh, um…OK.” (Yudkowsky has usefully called this a “curiosity-stopper“, although I’m sure there’s another term for this used by philosophers of science.)
  2. W. H. Zurek, Phys. Rev. A 76, 052110 (2007). [arXiv:quant-ph/0703160]
  3. This post was prompted and partially inspired by the paper on Boltzmann brain fluctuations by Kim Boddy, Sean Carroll, and Jason Pollack, who come to similar conclusions. I’m not sure I can agree with everything they write, but their description of a quantum fluctuation in their Section II is the clearest one I know in the literature. I encourage you to check it out: K. Boddy, S. Carroll, and J. Pollack, “De Sitter Space Without Quantum Fluctuations”. [arXiv:1405.0298]
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10 Comments

  1. Theo Nieuwenhuizen

    Dear Jess, trusting you’re young enough to learn:

    Your statements about “everything you need to know” for quantum measurements
    do not represent what solutions of models for measurements teach us. May I call your attention to our recent overview: Understanding quantum measurement from the solution of dynamical models, Phys. Rep. 525 (2013) 1-166,
    Armen E. Allahverdyan, Roger Balian, Theo M. Nieuwenhuizen.

    • You’ve misunderstood the point of this post. I am not saying this is everything you need to know about quantum measurement. (If that were true, I’d get a new job.) I’m saying this is everything you need to know to understand what the term “vacuum fluctuation” means given a (however incomplete) understanding of quantum measurement.

  2. Dear Jess,

    Isn’t it more helpful to stress how quantum fluctuations differ from “normal” fluctuations? I am not sure adding to the already cumbersome jargon of quantum mechanics (e.g. by saying that we need to think of measurements in term of basis and not in term of observables period) is that insightful for students (or for myself). And by the way, what is it that you are really trying to clarify? What is the right definition of a quantum fluctuation or how not to confuse the students?

    • > Isn’t it more helpful to stress how quantum fluctuations differ from “normal” fluctuations?

      I’d guess probably not, because in order to compare them you need a good understanding of what “classical” fluctuation looks like in complete quantum description of a system, and that’s much harder.

      This is like the problem of trying to define what people mean by a “true” quantum random number generator versus a “classical” random number generator. The former just means we have a complete description of the physical process that connects a quantum amplification event to our readings (e.g. quantum random number generators that operate by directing photons to half-silvered mirrors and amplifiying, using an avalanche diode, the outcome of whether a photon is strikes a sensor behind the mirror). The later is a much murkier concept, and whose best definition seems to be that we *don’t* have such a clear description. In the case of cryptography, this is important because we can be sure of the randomness whose source we can trace back to a quantum event, but we can’t be sure of the randomness of something that is operating in a way we can’t control, since it might (at least in principle) have subtle correlations.

      > I am not sure adding to the already cumbersome jargon of quantum mechanics (e.g. by saying that we need to think of measurements in term of basis and not in term of observables period) is that insightful for students (or for myself).

      Ahh! Good, because that means I have something to write about in a forthcoming post. But do note that this all works with “basis” –> “eigenbasis of the amplified observable”, as I say in my footnote.

      > And by the way, what is it that you are really trying to clarify? What is the right definition of a quantum fluctuation or how not to confuse the students?

      Well, my intention was to point folks to the Boddy-Carroll-Pollock paper if my short description wasn’t enough. But I’ve now added to the post, so please let me know if you think it’s still not helpful!

      • I’d guess probably not, because in order to compare them you need a good
        understanding of what “classical” fluctuation looks like in complete
        quantum description of a system, and that’s much harder.

        > I agree with you that this can be a tricky task indeed. However, in the context of the teacher-student case that you specify: ” Professor, where do the wiggles in the cosmic microwave background come from?” “Quantum fluctuations”.”. The point seems to be more to stress the “nature” or origin of the fluctuations. Here, putting the adjective “Quantum” implies (this is an implicature linguistically speaking) that the mentioned fluctuations are neither “Thermal” nor of any other classical kind we know of. I am not sure that the technical details you give, although very instructive, will explain anything more than the teacher you mention.

        Ahh! Good, because that means I have something to write about in a
        forthcoming post. But do note that this all works with “basis” –>
        “eigenbasis of the amplified observable”, as I say in my footnote.

        > I look forward to reading your forthcoming post then :). Regarding the eigenbasis of the amplifier stuff, how does this view differ from the one proposed by Allahverdyan et al. (pointed by Theo Nieuwenhuizen) where they have been dealing with this problem for almost ten years but looking specifically at the density matrix instead (which constitutes the most general description of a quantum state –in virtue of Gleason’s theorem–)?

        If is the ground state for some quantum system, and the are macroscopically distinguishable states, then this is a (single) vacuum fluctuation

        > I really appreciate the effort you made in describing in slightly more details what it was about…..I must be stupid but I don’t understand at all what is “the (single) vacuum fluctuation” you talk about in your text. This must be a writing or reading problem but do you mean that if you were to observe $|A_i rangle$* for one component of the amplifier and if its probability isnot equal to one then it is a vacuum (quantum) fluctuation?

        * My apologies I don’t know if I can use Latex on this page

        • > I am not sure that the technical details you give, although very instructive, will explain anything more than the teacher you mention.

          The equations I gave are the bare minimum amount of mathematical details one would need to understand what a vacuum fluctuation is. If someone couldn’t reproduce that, then I’d say they simply didn’t know what the term meant.

          > Regarding the eigenbasis of the amplifier stuff, how does this view differ from the one proposed by Allahverdyan et al. (pointed by Theo Nieuwenhuizen) where they have been dealing with this problem for almost ten years but looking specifically at the density matrix instead

          The point of this post is not to propose a new view, it’s to explain what people mean when they say “vacuum fluctuation” — insofar as the terms is used in a way that has any coherent meaning — especially with regards to events that do not involve laboratory measurements, such vacuum fluctuations in the CMB.

          Based on the abstract, Allahverdyan et al. appear to be following the standard practice of analyzing a quantum amplification process by following the complete evolution of the global quantum state, including the environment. The result of these types of analyses, which first appeared many decades ago, is that the wavefunction branches into a sum of orthogonal terms corresponding to different macroscopic outcomes, only one of which is seen by the experimenter. This post is very compatible with that picture.

          > I don’t understand at all what is “the (single) vacuum fluctuation” you talk about in your text. This must be a writing or reading problem but do you mean that if you were to observe $|A_i rangle$* for one component of the amplifier and if its probability isnot equal to one then it is a vacuum (quantum) fluctuation?

          Well, observing a macroscopic object whose current state is the result of amplifying the state of a quantum system in some basis (so long as that basis does not include its initial state) would make it a “quantum fluctuation”. (This is true whether the amplifying system is an experimental apparatus designed by a human, or a natural process.) “Vacuum fluctuations” are just a special subset of quantum fluctuations for which the initial state of the amplified system is its ground state.

          > My apologies I don’t know if I can use Latex on this page

          Unfortunately, the comments are handled by Disqus and they do not support LaTeX. I would like to fix this at some point.

          • > The point of this post is not to propose a new view, it’s to explain what people mean when they say “vacuum fluctuation” — insofar as the terms is used in a way that hasany coherent meaning — especially with regards to events that do not involve laboratory measurements, such vacuum fluctuations in the CMB.

            Yes, I see what you mean. But strangely enough, people don’t seem too bothered about quantum fluctuations when it comes to explaining the stability of the hydrogen atom for example. I mean, if I believe in the existence of the hydrogen atom independently of me observing it with some experiment, I can’t understand how it has a definite ground state without relying on quantum fluctuations. In this sense I don’t see what is so special about the CMB.

            > Well, observing a macroscopic object whose current state is the result of amplifying the state of a quantum system in some basis (so long as that basis does not include its initial state) would make it a “quantum fluctuation”.

            Yes right. I just didn’t get this meaning from the actual sentence. It must have to do with me not being a native English speaker.

            > Lastly, to come back to the tweet by Sean Carroll that says that there is not consensus on what means a quantum fluctuation. I wonder if part of the confusion doesn’t come from the fact that people are trying to characterize it at different levels. One could discriminate at least two levels: an operational one and a theoretical one where the former would refer to my original query about how do you actually discriminate a “quantum” fluctuation from fluctuations well captured by classical probability theory and the later would actually define it from a particular framework, namely what we call quantum mechanics (and this, you have explained it quite brilliantly in this thread).
            What are your thoughts on that point? Do you think that everybody agrees that the problem is to non ambiguously define quantum fluctuations within the theoretical framework called quantum mechanics (in any case, I reckon it is a very important thing to do anyway)?

          • > I mean, if I believe in the existence of the hydrogen atom independently of me observing it with some experiment, I can’t understand how it has a definite ground state without relying on quantum fluctuations.

            The existence of its ground state is a mathematical fact about its Hamiltonian. (Take the Hamiltonian, diagonalize it, find the eigenvector corresponding to the lowest eigenvalue.) In what way does this require quantum fluctuations as I defined above? Or do you mean some alternative notion of a fluctuation?

            > One could discriminate at least two levels: an operational one and a theoretical one where the former would refer to my original query about how do you actually discriminate a “quantum” fluctuation from fluctuations well captured by classical probability theory and the later would actually define it from a particular framework, namely what we call quantum mechanics…What are your thoughts on that point?

            Yes, I strongly agree that the difference between an operational view and a theoretical view leads different people to use apparently contradictory language to describe the same phenomenon. (Maybe the most common manifestation of this, which I have seen Serge Haroche himself complain about, is conflict on whether measurement processes reveal a pre-existing uncertainty in a system or induce that uncertainty through the measurement process itself. There is a sense in which both are true, but of course there is no contradiction.)

            The operational point of view is fine as long as one sticks strictly to statements about macroscopic variables, and indeed it is a healthy agnostic way of thinking about things, especially for folks who are searching for theories that would go beyond quantum mechanics. The problem is that there’s isn’t a way to talk operationally about whether the source of apparent randomness is classical or quantum, because the source isn’t macroscopic. As far as I can tell, all one can do is define classical fluctuations to be those for which a classical description appears to suffice, and call everything else quantum. (I welcome a better operational definition from someone who takes this point of view.) But this leads to an unstable definition because it relies on how carefully the phenomenon is tested. In principle, one can always trace back classical fluctuations to some quantum amplification process. And you can’t prove it’s quantum unless it violates a Bell inequality.

            Often, people who think they are reasoning operationally (i.e. just referring to macroscopic variables) are really reasoning classically about microscopic variables, and then call things “quantum” if they run into a problem.

            > Do you think that everybody agrees that the problem is to non ambiguously define quantum fluctuations within the theoretical framework called quantum mechanics…?

            Not only do I think there is no agreement between different people, I think that the same very smart person will often use two contradictory ways of thinking in the same breath. For many reasons, some psychological and some historical, this seems to be a bigger problem in quantum mechanics than any other field of physics. (However, a close runner up is the confusing talk about “when” things have fallen into a back hole, and what happens to them afterwards.)

            The worst part is that people who really do have a clear understanding will use sloppy language because it’s easier, and this will mislead other people who don’t have a clear understanding into thinking that none exists. Listening to two confused people argue about quantum mechanics can lead even smart observers to conclude that there is no fact of the matter, and that all this talk about quantum measurement problem is hopeless nonsense.

          • > The existence of its ground state is a mathematical fact about its Hamiltonian. (Take the Hamiltonian, diagonalize it, find the eigenvector corresponding to the lowest eigenvalue.) In what way does this require quantum fluctuations as I defined above? Or do you mean some alternative notion of a fluctuation?

            We seem to have some difficulties to communicate (I know the mathematical explanation, I just think it is not explanatory enough and could be more insightful in light of your post). The question that I want to raise is why is it that the eigenvalues of the hamiltonian are bounded from below? As far as I am concerned, this is deeply connected to the fact that position and momentum operators do not commute or equivalently due to the Heisenberg inequalities. In fact, although retrieving the whole energy spectrum requires indeed to diagonalize the hamiltonian, the existence of a lower bound doesn’t ( http://scitation.aip.org/content/aapt/journal/ajp/58/6/10.1119/1.16448?ver=pdfcov ). The same should hold for the lowest energy state of say the harmonic oscillator. If now, what you say is that the definition of quantum fluctuations you emphasize doesn’t yield, as a particular case, the zero point fluctuations of a simple harmonic oscillator, I am not sure I find it so insightful anymore (I think they do anyway because even you prepare your system in an eigenstate of the hamiltonian, if the amplifier has to do with the position of the particle then it will never be collinear with a single |x> state.). In fact, the absence of quantum fluctuations would then be that there doesn’t exist a pair of states (initial state, amplifier state) for which the probability of the amplifier state to be the initial state is neither zero nor one. In this case, one retrieve the usual classical logic for which we know that the simplest mathematical structure to define a probability measure is the usual classical probability.

            In some respect that would make sense because a state |psi> corresponds more or less to what the actual state of the system “is”. While, in classical physics, the “being” part is believed to be either true or false, as a consequence, a particle is either at |x> or it is not, period (for ANY |x>).

            In this very naive interpretation, quantum fluctuations are simply an additional layer of uncertainty about some things that we thought had definite truth values; while they don’t in general.

            This type of hand waving interpretation can be made more tangible I believe by looking at path integral formulations of quantum mechanics which provide a complementary picture on what “are” quantum fluctuations without relying heavily on operators and their eigenvectors/eigenvalues.

  3. (Sorry for the delay in allowing this message to appear; it got trapped in the spam filter for no good reason.)

    > would it be fair to think of a basis as a reference frame?

    Yep, a basis of a Hilbert space shares a lot in common with a reference frame. They are both used for constructing a coordinate system i.e. as a way for mapping pure numbers to physical states.

    Many of the deepest symmetries of physics are expressed by the equivalence of writing the laws of physics in different reference frames. In quantum mechanics, a shift in reference frame is usually equivalent to a shift of basis, and moreover there are additional symmetries expressed as shifts of basis (such as internal symmetries in quantum field theory) that do not correspondend to shifts inreference frames.

    One difference between bases and reference frames is that the former doesn’t have an interpretation in terms of observers. Reference frames with spatial offsets or relative velocities can be described as corresponding to observers located at different places traveling at different speeds, but in general a change of basis doesn’t have a simple interpretation.

    > And quantum fluctuations as things we have to admit into our descriptions of reality when we try to patch together multiple points of view?

    This is probably a bad way to think of things. Quantum fluctuations arise when the basis in which a quantum system is measured (aka the eigenbasis of the measured observable) doesn’t commute with the state of the system. As you probably know, commutation is a mathematical conditions meaning that the relevant variables can take simultaneous exact values. For instance, position and momentum don’t commute for a point particle in the same direction, resulting in an uncertainty principle, but there isn’t an uncertainty principle preventing one from giving the position and momentum of the same particle in two orthogonal directions (e.g. Y-position and Z-momentum) or for the position of one particle and the momentum of another one. This is because these pairs of variables commute.

    Now, the choice of reference frames is usually a matter of calculational convenience; you can use any frame you like as long as you’re consistent. However, the basis in which a measurement takes place is an objective property of the physical measurement (or amplification) process. One can take apart a Stern-Gerlach experiment and determine which spin basis is measured.

    Likewise, initial quantum state of a system is an objective property of the system. (There was actually a recent important paper on this, although the only formal result was to rule out an interpretation of quantum mechanics that no one took seriously anyways.) Therefore, the lack of commutation, which produces the fluctuation, is not just a change of perspective by an observer patching together a point of view. It is a real-life objective fact, agreed upon by all observers.

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