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The Vacuum Fluctuation Myth

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This Insight Article is a sequel of the Insight Articles ”The Physics of Virtual Particles” and “Misconceptions about Virtual Particles“ which make precise what a virtual particle is and what being real means, document some of the liberties taken in physics textbooks in the use of this concept, and mention the most prominent misuses. A further Insight Article, ”Vacuum Fluctuations in Experimental Practice”, shows at the example of a recent article in the scientific literature how some authors claim the observation of vacuum fluctuations, justified only by superficial, invalid reasoning.

In short, the concept of virtual particles is well-defined and useful when restricted to its use in Feynman diagrams and associated technical discussions. But it is highly misleading when used to argue about vacuum fluctuations, as if these were processes happening in space and time. The latter is a frequent misunderstanding, a myth that has not the slightest basis in particle physics. (The proper meaning of some terms related to the vacuum is explained at the end of ”The Physics of Virtual Particles”.)

The two articles mentioned do not, however, explain how it is possible that this misunderstanding is so widespread, and even serious experts resort to misleading imagery when explaining the subject to the general public. This is answered here at the example of Steve Carlip’s page on Hawkings radiation, where Steve Carlip, a well-known theoretical physicist working on quantum gravity, gave a lucid but completely mythical narrative about how vacuum fluctuations create Hawking radiation. This vacuum fluctuation myth comes from taking pieces of intuition and connecting them with a plausible narrative. The following is a reconstruction of the sort of thoughts that combine to justify the myth in the eyes of those who use this language. (For those interested, a non-mythical description of Hawking radiation is given in my post on Physics Stack Exchange.)

The starting point is the sound knowledge that there are technical notions of vacuum fluctuations (= nonzero vacuum expectation values), virtual particles (=internal lines in a Feynman diagram), and that in bare quantum field theory with a cutoff, the vacuum is a complicated multiparticle state depending on the cutoff – though in a way that it diverges when the cutoff is removed, so that nothing physical remains. Then the question arises: is there anything about it to convey a bit of this to ordinary people? It is highly unsatisfactory not to be able to talk about what one is doing in one’s research….

So one goes for analogies and images. Already calling internal lines ”virtual particles” is a step in this direction. Allow yourself a little more liberty and combine it with Feynman’s classical absorber theory of radiation; after all Feynman also invented the diagrams bearing his name, possibly even inspired by this analogy. The lines defining the virtual particles look like world lines in a classical process, so why not interpret them (in one’s imagination) as the quantum remnants of the classical world lines of Feynman’s earlier (later abandoned) theory? This happy accident makes the story possible. It is not completely accurate but plausible (in the absence of correction of the intuition by mathematical formulas) because both classical particles and virtual particles are represented pictorially by lines, and it is something that ordinary people can imagine. This is the beginning of the myth. An extra reassurance that you are on a good path is that the arrows that physicists draw on their diagrams (to indicate the sign of conserved quantum numbers) happen to match Feynman’s classical idea that antiparticles are just particles moving backward in time.

To bring in more physics one has to be able to interpret complete Feynman diagrams. Tree diagrams are easy but bring in a new aspect. They talk about real and virtual particles. On an electron line containing two vertices, the electron changes its status from being real (external) to being virtual (internal) and back (external) again. We learn from it a new fact – a virtual particle can become real, and conversely. The interpretation as world lines teaches us other things: A single Feynman diagram should in fact be considered just as a tiny snapshot of an extended web containing all particles in the universe; after all, world lines do not begin and end nowhere. Thus ”in reality” (meaning in the simplified virtual reality pained for the general public) all particles should be viewed as virtual until they are observed (where they obviously are real). This matches a version of the Copenhagen interpretation: Unobserved particles have a sort of ghost existence, since properties emerge only when they are subjected to a quantum measurement. You are pleased by this coincidence – it seems to say that there is a coherent story to be told. Also, since most of the lines in Feynman diagram end, you have a layman’s picture for decaying particles: What you see in a bubble chamber is just a Feynman diagram made visible! This is the first serious manifestation of the myth. In spite of lacking any grounding in real physics (being grounded instead in visual analogy), you feel entitled to make this identification – it serves your final goal to make some of the intricacies of microphysics accessible to the general public. No one there will ever ask,how it can be that two virtual particles can bend as in a Feynman diagram with a loop – so that they find each other exactly at the right place and with exactly the right momentum to annihilate. Therefore such impossibilities – that would spoil the goal of giving a simplified picture of what happens – are silently swept under the carpet.

The next thing is to interpret the bare multiparticle state. It is obviously a complex superposition of bare particles. Make the next move to identify bare particles with virtual particles; after all both are unobservable but appear in some version of the formalism. Now you have the picture of the vacuum as teeming with particles. From the form of Feynman diagrams with one or more loops you can read off that in order to make sense of the narrative these particles have to pop in and out of existence. This is the birth of the next item in the myth. That in a superposition nothing dynamical happens is a small nuisance that you happily sacrifice in order to be understandable to your intended audience. After all you can now give an illusion of having conveyed something of the complexities of the naive perturbative approach without having to talk about perturbation theory. In addition, without asking for it, you have found an unexpected visual interpretation of the notion of a vacuum fluctuation: A teeming vacuum where particles constantly pop in and out of existence clearly fluctuates, and each single act of popping may rightfully be regarded as a fluctuation of the vacuum. Another piece of the myth has found its place. Never mind that there is not the slightest way of justifying this analogy on the level of mathematical formulas. What counts is how the picture appeals to the general public, and it is obvious that drastic simplifications are needed to achieve this goal.

Now one needs to worry about the basic principles of physics in all this. After all, one doesn’t want to talk about particles alone but convey some general physics as well. Let us bring in conservation laws. Everyone knows that energy is conserved in Nature. But wait, doesn’t the creation of particles require some energy? Don’t mind, quantum mechanics comes to the rescue. People will have heard of the Heisenberg uncertainty relation, and if they haven’t this is an opportunity to make your audience acquainted with it. It states the intrinsic uncertainty of position and momentum in nonrelativistic mechanics. What does it tell about energy conservation? Nothing at all, but analogy comes to the rescue. In relativistic physics time is the 4th coordinate of position and energy the 4th coordinate of momentum. Thus we don’t make a big blunder if we consider a time-energy uncertainty relation. (Though time is nowhere in mainstream physics an operator observable.) Uncertain energy can be liberally interpreted as a slightly inaccurate conservation law. After all, one can derive from quantum mechanics only that the expectation of the energy operator is conserved. Expectation brings to mind that whatever you measure inaccurately must be measured many time for getting an improved accuracy. Thus only the average energy needs to be conserved. Reinterpret the average (in the service of simplifying the physics to give your audience a coherent story) as an average in time.

Thus you found the solution: Energy can be borrowed for a short period of time if it is returned on the average. The next item of the myth arrived. Now you are quite confident that you’ll be able to get a full and rich story (for laymen only, so all the small blunders made can be excused) and continue to turn it into something you’ll tell in public (or write in a book). You hope that the attentive audience will not ask where the energy is borrowed from, but unfortunately you told the story first a colleague with an unbiased mind and he insisted on that this should be clarified first. You need to look at some more pieces of information to get the next input. Fortunately you soon find it: The zero-point energy of a harmonic oscillator had in the past always been ignored by saying that only energy differences are observable. Maybe it is the bank from which the virtual particles lining up for popping into existence can borrow their energy. And yes – it turns out that the bare quantum field has a huge amount of zero point energy – an infinite amount if you take the physical limit. Clearly this must be the source – and no ordinary person will be interested to question it. Thus the final piece of the myth arrived. You are happy – it will be a really good story conveying a lot of physics while still being understandable to ordinary people.

That there is no physical mechanism for how the borrowing works is a small nuisance that (for the layman) can be ignored – after all, they want a simple story that they can believe, not a technical discussion of all the problems involved – they know that quantum mechanics is full of unresolved problems. At this point your story is already so convincing that you don’t mind that all observable quantities also become infinite in the limit considered, and that when you instead do a proper renormalization (needed to get the high accuracy predictions quantum field theory is famous for) the whole capital of the vacuum energy bank shrinks to zero!

Now the particle philosophy for the laymen is essentially complete. Only a few – to laymen imperceptible – jumps of the imagination were needed in the service of understandability. Like in a cinema, where the pictures jump in discrete steps but provide a sufficient illusion for the audience to see a continuous story. To make sure that the audience, captured by the imaginative illusion, will not take it for physical reality, and to ensure that your status as a respected scientist is preserved, you begin with a caveat (like Steve Carlip did on his page on Hawkings radiation, in the inconspicuous first line after the heading ”An Incomplete Glossary” – long forgotten at the time the reader enters the mythical narrative linked to above): ”Be warned – the explanations here are, for the most part, drastic oversimplifications, and shouldn’t be taken too literally.” But in spite of this you can instead be sure that most of your audience will ignore this sentence said in the first few seconds in favor of the nice mental pictures that you took a whole hour to explain and make intelligible.

When Hawking discovered what was later called Hawking radiation this picture for the general public was already well entrenched. So he only had to figure out how his discovery would fit in – and it fitted well. Instead of talking about gravitational energy (not visible, hence a sort of vacuum) creating a particle-antiparticle pair one partner of which escapes there is only a small step to saying what the educated general public expects. Since the particles are not (yet) observable by the far away observer seeing only the radiation, they must be sold according to the philosophy developed above as virtual particles created (hence vacuum fluctuations in action). Years later, when one of the particles is finally observed by the far away observer, it becomes real as a piece of the observable Hawking radiation.

Thus if you want to summarize to lay people the Hawking effect in a single phrase, what is more natural than to say that ”vacuum fluctuations cause the Hawking radiation” without repeating the warning that this ”shouldn’t be taken too literally”?

 

 

I am Professor for Computational Mathematics at the University of Vienna, Austria, with a strong interest in theoretical physics and its foundations. I maintain the Theoretical Physics FAQ (http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html), a book-size collection of online articles on quantum mechanics and related topics, in the context of which I answer questions on PhysicsOverflow (http://www.physicsoverflow.org/). My online book Classical and Quantum Mechanics via Lie algebras (http://lanl.arxiv.org/abs/0810.1019) presents quantum mechanics in very close analogy to classical mechanics. A revised version is scheduled to appear in print in 2017.
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  1. HyperStrings
    HyperStrings says:

    Analogies don't create truth

    There is a good physics lecture on 'The Dangers of Analogies' and I agree with you, we should be very discerning of analogy. Though, in the lecture/paper he gives rules to how to properly use analogies safely when discussing/teaching physics and admits that sometimes there is no other way to explain something. Just as all Gaileleo had to prove his theory was an analogy. Its a double edged sword, as we must also be aware, we are heading into a time of where experiments will need to be able to represent complex planckian scale effects, so we may very well have to bite the bullet and start to understand analogous experiments. As it is very possible for them to be useful. With that being said, I will try to be more discreet and specific as I can respect your position.

    Quantum fluctuations are everywhere,

    So, in your opinion, what is causing the ''field theoretic effects'' that oscillate the atom, 'up the slope of the bowl' in a dead, zero point BEC well?

    The paper observed the BEC well oscillations, mathematically predicted the subtle nuance of effects of acoustic oscillations that match the 'vibration' of the field to exactly match the 'field theoretic effect' of the oscillations or quantum fluctuations of the atoms in the BEC well. Then showed, precisely, those same exact quantum fluctuations, with their mathematically predicted phonon model. I have tried to put together any various combination of small changes in the atomic structure that would allow the equivalent of such oscillations but I can't find any correlation that would create these specific, exactly replicated, oscillations. So I am in agreement with the paper.

    (and then there are all the complicated renormalization scheme caveats associated with what we mean by this

    Which brings me to another point that, we are using normalization because of planck scale discrepencies, and 'science' is okay with that, but if you try to isolate those discrepencies with a mathematical application of a 'mistake fixing, re-normalization', science is not okay with that? The very process of normalization is in essence, 'blurring the clarity of the image'. Then a proper re-normalization can result with 'sharpening of the image'.

    As I see it, the vibration of the atoms field creates virtual phonon wave oscillations, the atoms wiggle up the bowls slope and the quantum fluctuations are the wiggle of those vibrations.

  2. Haelfix
    Haelfix says:

    A nonzero vacuum expectation value doesn't mean in any sense that the vacuum is fluctuating. Otherwise the ground state of a single harmonic oscillator would also be fluctuating….

    We completely agree then, although i'm now wondering if the origin of the word in textbooks is precisely when discussing simple harmonic oscillators, particles in a box, and other simple nonrelativistic quantum mechanics.   There it  would presumably me a fluctuation relative to a classical zero.

  3. A. Neumaier
    A. Neumaier says:

    if the origin of the word in textbooks is precisely when discussing simple harmonic oscillators, particles in a box, and other simple nonrelativistic quantum mechanics. There it would presumably mean a fluctuation relative to a classical zero.

    I have seen the word used only in the context of (relativistic or nonrelativistic) quantum field theory. It doesn't make sense for a harmonic oscillator or a particle in a box. The quantum mechanical ground state is dynamically completely inert under the quadratic Hamiltonian that defines the oscillator. Nothing fluctuates. There is an uncertainty about the values of observables not commuting with the energy, but this is because it is impossible to measure them more accurately, not because these would fluctuate in time. The traditional interpretations refrain from saying what happens in between measurememt; none of them claims that these observables have all the time exact but fluctuating values.

  4. A. Neumaier
    A. Neumaier says:

    , what is causing the ''field theoretic effects'' that oscillate the atom

    Well, the interaction with the crystal, or if you wish, the field defined by it cause these effects. Switch the interaction or the mean field off and the effect is gone. This proves that these are the responsible agents. Not mystical quantum fluctuations.

  5. RockyMarciano
    RockyMarciano says:

    Mixing nrqm and qft may lead to confusions. To me quantum fluctuations are defined by the fact that the ground state in qm must also obey the Heisenberg principle, that is what in graphic language fluctuates in quantum fluctuations. When going to the quantum field picture there are so many things that change(for one thing position is no longer a operator while what used to be states are operators,etc…) that there is no longer a good mathematical translation of this, and the vacuum state of the field theory doesn't qualify when formally defined. There, no more mystique.

  6. Haelfix
    Haelfix says:

    . The quantum mechanical ground state is dynamically completely inert under the quadratic Hamiltonian that defines the oscillator. Nothing fluctuates.

    Yep, they are clearly stationary states in the nrqm case.

    So, I would say I have heard the word used more when discussing things like barrier penetration in nrqm.  So an author will write something like "classically you will never measure a particle here, but b/c of 'quantum fluctuations' or 'quantum jitters' you will see a tunneling phenonemon on the other side and the nonzero possibility for the detection of a particle".  So there the word would presumably mean some sort of deviation from classical expectations.

    In the context of inflation, the same sort of pedagogical word choice is frequently used informally in the context of a potential term for a scalar field, where you have either tunneling between false and true vacuums, or alternatively where you have oscillatory behavior at the bottom of a well analogous to the phenomenon which leads to the longitudinal mode in the Higgs phenomenon.

  7. A. Neumaier
    A. Neumaier says:

    To me quantum fluctuations are defined by the fact that the ground state in qm must also obey the Heisenberg principle

    The Heisenberg uncertainty relation is not about quantum fluctuations but about the intrinsic uncertainty in measuring noncommuting observables. Nothing fluctuates there.

  8. A. Neumaier
    A. Neumaier says:

    I have heard the word used more when discussing things like barrier penetration in nrqm. So an author will write something like "classically you will never measure a particle here, but b/c of 'quantum fluctuations' or 'quantum jitters' you will see a tunneling phenomenon on the other side and the nonzero possibility for the detection of a particle".

    Though this is somewhat unrelated to the present topic, let me mention that quantum tunneling is a misnomer. It is motion over the barrier and not through the barrier. For the ''tunneling'' probability tends to zero as the barrier gets higher, and is zero when the barrier is infinitely high. No matter how long a tunnel through the barrier would have to be! Thus it is like the motion of a classical particle with a random kinetic energy – it has a small probability of being kicked over the barrier and ending up outside the well it was in originally.

    Again nothing that fluctuates!

  9. RockyMarciano
    RockyMarciano says:

    The Heisenberg uncertainty relation is not about quantum fluctuations but about the intrinsic uncertainty in measuring noncommuting observables. Nothing fluctuates there.

    This is purely semantic but both the insight and thread are about  semantics so why not get it right?. Fluctuation is a word that is synonim both of oscillation and of indeterminacy or uncertainty. All it means in the quantum context is the Heisenberg indeterminacy of the ground state, and what fluctuates(vacillates i.e. it is intrinsically uncertain) is precisely the noncommuting observables. Of course many people by extension thinks about something moving or oscillating, that I guess it is what you understand if you disregard the meaning of fluctuation as vacillation/indeterminacy. Since Heisenberg indeterminacy lies at the heart of the quantum departure from classical physics, quantum fluctuations by extension are also referred by many as this departure from classicality.

    On the other hand if one is strict with the math not even the fields or the waves actually oscillate, since the math always describes a rigid picture, a shortcoming of analysis. But this should show just how ridiculous can blind strictness get.

  10. ftr
    ftr says:

    This is purely semantic but both the insight and thread are about  semantics so why not get it right?. Fluctuation is a word that is synonim both of oscillation and of indeterminacy or uncertainty. All it means in the quantum context is the Heisenberg indeterminacy of the ground state, and what fluctuates(vacillates i.e. it is intrinsically uncertain) is precisely the noncommuting observables. Of course many people by extension thinks about something moving or oscillating, that I guess it is what you understand if you disregard the meaning of fluctuation as vacillation/indeterminacy. Since Heisenberg indeterminacy lies at the heart of the quantum departure from classical physics, quantum fluctuations by extension are also referred by many as this departure from classicality.

    On the other hand if one is strict with the math not even the fields or the waves actually oscillate, since the math always describes a rigid picture, a shortcoming of analysis. But this should show just how ridiculous can blind strictness get.

    But it seems vacuum fluctuation is mentioned more in QED and you are talking about "quantum fluctuations"

     https://en.wikipedia.org/wiki/Quantum_fluctuation

    https://en.wikipedia.org/wiki/QED_vacuum

    https://en.wikipedia.org/wiki/Vacuum_polarization

    so some "fluctuation" is related to vacuum in the vicinity of interactions others to an otherwise empty interstellar vacuum. It sound like many concepts being mixed up.

  11. A. Neumaier
    A. Neumaier says:

    Fluctuation is a word that is synonim both of oscillation and of indeterminacy or uncertainty.

    No. Fluctuation in today's usage always means change, not just being uncertain! All the major dictionaries agree on that:

    http://www.dictionary.com/browse/fluctuation

    1. continual change from one point or condition to another.

    2. wavelike motion; undulation.

    3. Genetics. a body variation due to environmental factors and not inherited.

    http://dictionary.cambridge.org/dictionary/english/fluctuate

    fluctuate: to change, especially continuously and between one level or thing and another

    https://en.oxforddictionaries.com/definition/fluctuation

    An irregular rising and falling in number or amount; a variation

    https://www.merriam-webster.com/dictionary/fluctuate

    1. to shift back and forth uncertainly

     2. to ebb and flow in waves

    https://www.vocabulary.com/dictionary/fluctuation

    The noun fluctuation refers to the deviations along the path from one point to another. We see frequent fluctuationsin the stock market, as prices go up or down, and also in the weather, which is always changing.

    http://www.macmillandictionary.com/dictionary/british/fluctuation

    frequent changes in the amount, value, or level of something

    Even wikipeedia  describes it as a change, though in a completely unscientific manner (not surprisingly, since it also promotes lots of other nonsense about virtual particles):

    ''In quantum physics, a quantum fluctuation (or quantum vacuum fluctuation or vacuum fluctuation) is the temporary change in the amount of energy in a point in space […]  the field's lowest-energy or ground state, often called the vacuum state, is not, as one might expect from that name, a state with no particles, but rather a quantum superposition of particle number eigenstates with 0, 1, 2…etc. particles.''

    The quality of the Wikipedia statement can be assessed from the second sentence quoted, which is absurd. The vacuum state is always the eigenstate of the number operator with exactly zero particles. There is no uncertainty in the number of particles, since it is an eigenstate.

    what fluctuates(vacillates i.e. it is intrinsically uncertain) is precisely the noncommuting observables.

    This is only you private interpretation of the term.  Never before I heard of someone talk about quantum vaccilations! And even that word means not just uncertainty but wafering uncertainty – a process in time!

  12. ftr
    ftr says:

    Arnold

        What do you think this statement is saying then in wiki

    "So what does the spacelike part of the propagator represent? In QFT the vacuum is an active participant, and particle numbers and field values are related by an uncertainty principle; field values are uncertain even for particle number zero. There is a nonzero probability amplitude to find a significant fluctuation in the vacuum value of the field Φ(x) if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region)."

    https://en.wikipedia.org/wiki/Propagator

  13. A. Neumaier
    A. Neumaier says:

    What do you think this statement is saying

    As stated, it is meaningless since ##Phi(x)## is not an operator, hence not an observable. As remarked in your quote, one has to use a smeared version (averaging over a small open region in space-time) to produce an operator. Even with this amendment, the statement is misleading. The ''vacuum value'' is not a commonly used expression. The nearest expression with a formal meaning is the vacuum expectation value, but this is completely determined and hence certain. What is probably meant is that if one could measure the local value of a smeared field in the vacuum state (don't ask how this ever can be done, as the vacuum contains no particles, hence no observer), the result would have a significant uncertainty. This is formally true if one assumes (as is commonly done) that the Born interpretation holds in this (counterfactual) case.

    But in the statement quoted, the mistake already pointed out is made, that uncertainty and fluctuation are equated. This turns an unconspicuous statement that some measurement result has a significant uncertainty (we know this holds for most measurements) into the remarkable and wrong statement that the value fluctuates with the implied, equally wrong consequence that the vacuum is ''active''. This is typical for the exaggerations made when turning banal news into exciting stories for everyone.

  14. A. Neumaier
    A. Neumaier says:

    The passage

    ''In terms of virtual particles, the propagator at spacelike separation can be thought of as a means of calculating the amplitude for creating a virtual particle-antiparticle pair that eventually disappear into the vacuum, or for detecting a virtual pair emerging from the vacuum. In Feynman's language, such creation and annihilation processes are equivalent to a virtual particle wandering backward and forward through time, which can take it outside of the light cone.''

    from the same wikipedia article is also misleading. Creation and annihilation operators only exist for time-like, on-shell momenta; hence the associated creation and annihilation processes all refer to real particles.

  15. RockyMarciano
    RockyMarciano says:

    No. Fluctuation in today's usage always means change, not just being uncertain!

    This is only your private interpretation of the term.  Never before I heard of someone talk about quantum vaccilations! And even that word means not just uncertainty but wafering uncertainty – a process in time!

    The term uncertainty or indeterminacy is obviously related to change, change in measuring expectations, have you heard about statistical fluctuations referred to the uncertainty in measurements? I mean that's QM.

    Frankly, it looks as though you stubbornly need to hold on to your straw man and the mantra "nothing fluctuates" (indeed a quite private interpretation), when everybody knows since Heisenberg's first modern quantum mechanics paper, the concept of conjugate observables fluctuating as described by Fourier transform coefficients and outlined by Born in its probabilistic rule to capture just that fluctuation in the measurement of noncommuting observables, so hardly my own private interpretation.

  16. A. Neumaier
    A. Neumaier says:

    change in measuring expectations

    This is nonsense. In a stationary setting (such as the ground state of a quantum system), expectations are constant, not fluctuating.

    Measuring expectations means making a lot of individual measurements of different realizations of the same system and taking their mean. Each measurement deviates from the mean, and the minimal mean square deviation is quantified by the uncertainty relation.

    The fluctuation is neither in the quantum system nor in the expectation but in the series of measurements. It is due solely to the measurement process. It comes from the fact that each time a different particle is measured.

    But when you measure a field there is only one field so nothing that could fluctuate. Unless the field itself fluctuates – i.e., changes its values rapidly in time like in turbulence. But a free field in the vacuum state is far from turbulent.

  17. RockyMarciano
    RockyMarciano says:

    This is nonsense. In a stationary setting (such as the ground state of a quantum system), expectations are constant, not fluctuating.

    I wasn't using the word expectation technically, as referred to expectation values there. I was referring to what you explain below(so I'm happy that you got my meaning right after all) and this applies irrespective of the states being stationary or not.

    By the way, even if using your biased concept of the word fluctuation referring only to oscillation, are you saying that stationary waves are not oscillating? I guess they are not waves then either.

    Measuring expectations means making a lot of individual measurements of different realizations of the same system and taking their mean. Each measurement deviates from the mean, and the minimal mean square deviation is quantified by the uncertainty relation.

    I see you have finally understood what quantum fluctuations mean, This minimal mean square deviation refers to something changing(in this case the conjugate noncommuting variables), or there would be no minimal square deviation. So see, it was not so difficult to see how it is this uncertainty that using the statistical language of QM is referred by some people as quantum fluctuation

    The fluctuation is neither in the quantum system nor in the expectation but in the series of measurements. It is due solely to the measurement process. It comes from the fact that each time a different particle is measured.

    Sure, that is where the fluctuation called quantum fluctuation in QM is(I'm leaving out the quantum field extension of the term for reasons I explained in a previous post) and yes, that is the process it is due to.

  18. OCR
    OCR says:

    Arnold, maybe it doesn't matter and you probably know, if you post anywhere on Wikipedia, and you are not a registered user, or signed in, your IP address is visible to anybody and everybody ?

    This little critter, called SineBot, will automatically sign a message with ones registered user name, or in your case… your IP address.

    I blacked yours out, but see…

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