Quantum Fluctuations

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41 Responses to Quantum Fluctuations

  1. Patrice Ayme says:

    SC: Let’s conjure some science up in here. Science is good for the soul.

    Why is science good for the soul? Because the human soul is centered on finding truth. Science is truth, thus science is human.

    SC: …what are “quantum fluctuations,” anyway? Talk about quantum fluctuations can be vague. There are really 3 different types of fluctuations: Boltzmann, Vacuum, & Measurement. Boltzmann Fluctuations are basically classical: random motions of things lead to unlikely events, even in equilibrium.

    Richard Feynman, at the end of his chapter on entropy in the Feynman Lectures on Physics, ponders how to get an arrow of time in a universe governed by time-symmetric underlying laws.

    “So far as we know, all the fundamental laws of physics, such as Newton’s equations, are reversible. Then were does irreversibility come from? It comes from order going to disorder, but we do not understand this until we know the origin of the order. Why is it that the situations we find ourselves in every day are always out of equilibrium?”

    Is that really true? Are equations time-symmetric? Not really. First, equations don’t stand alone. Differential equations depend upon initial conditions. Obviously, even if the equations are time-symmetric, the initial conditions are not: the final state cannot be exchanged with the initial state.

    Quantum Physics make this observation even more important. The generic Quantum set-up depends upon a geometric space S in which the equation(s) of motion will evolve. Take for example the 2-slit: the space one considers generally, S, is the space AFTER the 2-slit. The one before the 2-slit, C, (for coherence) is generally ignored. S is ordered by Quantum interference.

    The full situation is made of: (C, S & Quantum interference). It’s not symmetric. The Quantum depends upon the space (it could be a so-called “phase space”) in which it deploys. That makes it time-assymmetric. An example: the Casimir Effect.

  2. Sean and co-authors,

    My “mileage” is of course zero, but this paper makes for an enjoyable read (and the tweeter feed is even better!) I mean, the write-up is as if it has been written to be understood. Is it because graduate students were involved? Or is it because the paper is based on some plain conversations / talks that occurred *before* beginning writing it? I guess the second. In any case, it’s an easy-to-read paper, and in that sense *good*.

    BTW, is there any QFT *text*-book that would be as easily understandable to an engineer (like me) as this paper is? (I first thought of asking you for “the Griffiths of QFT,” but then reformulated the question). Thanks in advance for any recommendations.



  3. Magnema says:

    @Patrice: Actually, from a differential equations standpoint, the final equation CAN be exchanged with the initial condition – there is no distinction (at least in ODEs; PDEs get a bit more technical, but in terms of fundamentally-physical PDEs, the same ideas broadly apply). The mathematical theorems are not direction-dependent. Differential equations don’t really depend on initial conditions; they depend on SOME additional constraint, but this could equally validly be ICs, FCs, or BCs (for ODEs; PDEs, again, are more complicated).

    As for the quantum, if you watch/read his time course, he talks about it; essentially, if you both “confine” the initial conditions and the final conditions (by setting up the IC and only counting certain measured FCs), then you get time-symmetric “middle conditions.” In other words: if you ask time-symmetric questions, you get time-symmetric answers – and this principle holds in all of fundamental physics (up to a few asterisks of CPT, which are of no significance to the core question).

    What you say is “ignored” about Quantum Mechanics (in the “before states” vs. “after states”) is merely ignored in most popularizations. In practice, the initial state and the final state are considered on at least near-equal ground (“near” for a few practical reasons, which are contingent on the apparent arrow of time and in no way on the fundamental physics being experimented on).

  4. Sean Carroll says:

    Ajit– There is a book titled “Quantum Field Theory for the Gifted Amateur,” but their idea of an “amateur” is pretty optimistic.

  5. phayes says:

    “Worth pointing out that my discussion of quantum fluctuations betrays my Everettian (many-worlds) sympathies.”

    Well that is betraying Everettian sympathies but not in a good way. It’s been pointed out before in comments here that the confusion of uncertainty and fluctuation isn’t even specific to QM / noncommutative probability theory, let alone a mistake only Everettians (can) avoid.

  6. Daniele says:


    I’ve read From Eternity To Here with great pleasure.

    I hope this is a good medium to ask a question I’ve been meaning to ask for a while.

    The argument for Boltzmann brains always puzzled me. Brains and thinking things are made of heavy atoms and complex molecules. A Boltzmann brain would be the most likely explanation of our existence only if we were made of particular arrangements of simple particles. However, our brains are made of heavy elements that require high temperatures and pressures to make.

    The easy way for these elements to be created is in the furnace of stars. My understanding is that they could also pop into existence thanks to quantum fluctuations which have to randomly overcome huge energy barriers. My question is, which one is more likely?

    Is it more likely for a few trillion heavy atoms to randomly fluctuate into existence? Or is it more likely for 10^85 particles to arrange themselves in a low entropy state and then kick off the long process that leads to us?

    Fundamentally my main point is about the fact that the Boltzmann brain argument doesn’t take into account the fact that the atoms in the brain are hard to make from fundamental particles.

    I hope my question is clear.

    Thank you,

  7. Sean Carroll says:

    Daniele– The answer is pretty simple. Starting from thermal equilibrium, it is enormously more likely that a few trillion heavy atoms randomly fluctuate into existence than having all the particles in the universe arrange themselves in a low-entropy state. Quantitatively, the former probability is something like one part in 10^(10^28), while the latter is one part in 10^(10^120). It’s not even close.

  8. Daniele says:

    Sean, thanks for the answer. I guess my point is that while it’s orders and orders of magnitude fewer particles, each particle has to be arranged in a way that is very very unlikely to happen randomly.

    Just for the sake of argument, say that an Iron atom only has one in 10^100 chances of randomly fluctuating into existence, then the orders of magnitude get close.

    I don’t possess the mathematical tools to compute the probability above. But my argument is that fluctuating an atom of iron into existence requires crossing energy barriers that perhaps are very hard to cross via random fluctuations. This fact may skew the probabilities significantly.

    Thanks again.

  9. Steady State Theorist says:

    A neutron has a mass of 0.9 geV/C^2 and the vacuum energy is 6.24 GeV/m^3. That would mean that for a particle to fluctuate into existence would require a significant fraction of the energy to converge in a space the size of a neutron . I’m pretty sure that is way more likely around a black hole and might happen around one of the relativistic jets. Which is what I will pretend this paper argues. If black holes create neutrons and anti-neutrons, because we don’t want to violate conservation laws, then we would expect there to be a lot more hydrogen near black holes because a neutron decays into a proton and electron. Would that mean that stars in the center of galaxies near supermassive black holes contain more hydrogen than we expect ?

  10. Wow, Sean! One look at it (Lancaster and Blundell’s, QFT), and I know already that I am not gifted. But the book seems to have been written with such fantastic level of essentialization that I am damn sure it just won’t be possible for me to calmly keep it aside either. (I feel like it would have been better had I never come to know about its existence! Ignorance *is* bliss…) … Thanks, anyway!

  11. Patrice Ayme says:

    [Partly trying to answer Magnema.]

    SC: “Nothing actually “fluctuates” in vacuum fluctuations! The system can be perfectly static. Just that quantum states are more spread out.”

    Indeed. Quantum states are, intrinsically, more spread out. Why?

    One has to go back to the basics. What is Quantum Physics about? Some, mostly the “Copenhagen Interpretation” followers, claim Quantum Physics is a subset of functional analysis (mathematician Von Neumann was the founder of this system of thought, and famously claimed that De Broglie and Bohmian mechanics were impossible… Von Neumann had made a logical mistake). The Quantum-as-functional analysis school is actually dominant. It had great successes in the past. It allows to view Quantum Physics as “Non Commutative Geometry”. However, contrarily to repute, it’s not the most fundamental view.

    Where does Quantum-as-functional-analysis come from? A Quantum system is made of a (“configuration”) space S and an equation E (which is a PDE). Out of S and E is created a Hilbert Space with a basis, the “eigenstates”.

    In practice, the eigenstates are fundamental waves. They can be clearly seen, with the mind’s eye, in the case of the Casimir Effect with two metallic plates: there is a maximal size for the electromagnetic wavelengths between the plates (as they have to zero out where they touch the metal). The notion of wave is more general than the notion of eigenstate (Dirac pushed, successfully, the notion of wave so far that it created space, and QFT has done more of the same).

    Historically, De Broglie suggested in 1923 (several publications to the French Academy of Science) that to each particle was associated a (relativistic) wave. De Broglie’s reasons were looked at by Einstein, who was impressed. The De Broglie’s wave appears on page 111 of De Broglie’s 1924 thesis, which has 118 pages (and contains, among other things, the Schrodinger wave equation, and, of course, the uncertainty principle, something obvious when one tries to localize waves with waves!)

    Consider a space S made available to (classical) Boltzmann particles: S is progressively invaded by particles occupying ever more states. When the same space S is made available as part of a Quantum System, the situation is strikingly different. As Sean points out, the situation is immediately static, it provides an order (as Bohm insisted it did).

    What’s a difference with a classical system? The classical system evolves, from a given order, to one, more disordered. The Quantum system does not evolve through increasing disorder. Instead, the space S, once accessed, becomes not so much an initial condition, but a global order The afore mentioned Hilbert Space with its eigenstates). So the Quantum System is static in some sense (from standing Quantum Waves, it sorts of vibrates through time).

    So Quantum Systems have an intrinsic time-assymmetry (at least when dealing with cavities). When there are no cavities, entanglement causes assymmetry: once an interaction has happened, until observation, there is entanglement. Before interaction, there was no entanglement. Two classical billiards balls are not entangled either before or after they interact, so the interaction by collision is fully time reversible.

    Entanglement is also something waves exhibit, once they have interacted and not before, which classical particles are deprived of.

  12. Magnema says:

    @Patrice: Entanglement is (in principle) just as reversible as any other law of physics. Your assumption that things are not entangled before interaction is simply not necessarily true, generically, unless you specify that the two particles have never, through any chain of effects looking backwards in time, interacted (and even then, it is questionable). The key here is that being “entangled” is the generic case – being disentangled is actually a very special condition, which we usually specify on our problems more out of (a) a desire to simplify them or (b) simple epistemological uncertainty, in the usual statistical-mechanical sense.

    By analogy: if you have some expression awy+bwz+cxy+dxz (for a,b,c,d numbers and w,x,y,z variables), then being “disentangled” is equivalent to this expression being able to be written (ew+fx)(gy+hz) for e,f,g,h some numbers. Since the latter has nontrivial restrictions, the solution space is a lower-dimensional submanifold of the original space (in particular, “disentangled states” between two particles form a three-dimensional submanifold out of a four-dimensional state). This is why disentangled systems become entangled – because being disentangled is a highly nongeneric state which, unless explicitly preserved by the Hamiltonian, will immediately dissociate.

    In this case, generically speaking, quantum particles ARE entangled from the beginning, unless you put some additional restrictions on initial conditions, that things aren’t entangled – and even if you do so as a cosmological criterion (which I don’t think is unreasonable), for most practical experiments, everything has been interacting for long enough that it is all entangled anyway.

    As for your wave discussion: The wave equation (and the S.E., more to the point of quantum mechanics) is completely time-reversible, so you can’t get irreversibility out of that, even if you restrict to a cavity – including the Casimir effect. This is ignoring the fact that waves only get you so far in QM – in particular, waves cannot account for finite-dimensional system, and particularly not angular momentum, unlike the linear algebra approach. (I would also disagree that waves are more fundamental than eigenstates, seeing as “waves” in the sense of sine waves are simply eigenstates of a particular linear operator PDE, and even general manifold derivatives involve linear algebra… but that’s beside the point, at the end of the day, of the arguments of irreversibility.)

  13. phayes says:

    @Patrice Ayme

    “What is Quantum Physics about? Some, mostly the “Copenhagen Interpretation” followers, claim Quantum Physics is a subset of functional analysis”

    Not so. The claim is that quantum physics is the application of (noncommutative) probability theory¹ to physics. As (Copenhagenist) Ray Streater says, “It took some time before it was understood that quantum theory is a generalisation of probability, rather than a modification of the laws of mechanics.”²

    1) Does physics need probability theory? Yes.³
    2) Does it have noncommuting observables? Yes.³
    3) P̶r̶o̶f̶i̶t̶!!! Quantum theory.

    “The notion of wave is more general than the notion of eigenstate ”

    Depending on context / definition of eigenstate it might be. It certainly isn’t more general than the notion of state and wave function states are not enough for physics (cf. Chris Isham’s Lectures on QT: “there are many examples of quantum-mechanical systems whose states cannot be represented as wave functions”).

    “The classical system evolves, from a given order, to one, more disordered. The Quantum system does not evolve through increasing disorder. ”

    Neither the quantum entropy ( https://en.wikipedia.org/wiki/Von_Neumann_entropy ) nor its classical counterpart (Gibbs entropy) increase: http://www.ucl.ac.uk/~ucesjph/reality/entropy/text.html

    ¹ https://terrytao.wordpress.com/2010/02/10/245a-notes-5-free-probability/
    ² http://arxiv.org/abs/math-ph/0002049
    ³ http://www.nobelprize.org/nobel_prizes/physics/laureates/1954/born-lecture.html

  14. BobC says:

    What? No Boltzman Brains?

    And here I was hoping a quantum fluctuation between my ears would give me an upgrade.

  15. Haruki Chou says:

    Patrice Ayme says: “..how to get an arrow of time in a universe governed by time-symmetric underlying laws.”
    But don’t we already know that the underlying laws are not time-symmetric?
    (CPT is a symmetry, but T alone is not)

  16. Haruki Chou says:

    Daniele says: ..about Boltzmann brains..

    I believe brains are more probable than Boltzmann brain.
    Any big primordial cloud (of H and He) is any of a multitude of configurations will coalesce under gravity to form stars and planets where natural selection will generate brains.

    Fluctuating a Boltzmann brain requires a quite precise fluctuation, while any coalescing mass will generate stars and planets and life. I think the latter to be overwhelmingly more probable than the former.

  17. Josh says:

    If quantum fluctuations don’t really “fluctuate,” how can we abide the old mantra that these fluctuations explain the initial expansion (and inhomgenities) of the early universe?

    It would seem only a “measurement fluctuation” could account for this right? Do we think something “measured” the whole of the universe then?

    Maybe I’m confused though…?

  18. Magnema says:

    @Haruki: The distinction between T-symmetry and CPT-symmetry is not really of fundamental importance to the ideas that underlie statistical mechanics. You know this because inverting CP would not change the second law of thermodynamics (at least, it wouldn’t intuitively).

    As for Boltzmann brains, it’s possible that you are correct that they are nongeneric, because the theory of entropy with gravity is not completely understood, but your reasoning (as given) has holes. In particular, when you take a “big cloud,” you are asserting that it will then collapse – which means that it was not in thermodynamic equilibrium in the first place, so you haven’t let your system sit for long enough to consider the kinds of arguments that lead to Boltzmann brains.

  19. Patrice Ayme says:

    Trying to answer Magnema:
    Magnema says:”Entanglement is (in principle) just as reversible as any other law of physics. ”

    Patrice Ayme: Entanglement gets un-entangled (it’s called decoherence) NON-LOCALLY. Occasionally Free Will will even be involved (say choosing the direction of polarization measurement, or a magnetic field in a Stern-Gerlach device, etc.). Without evoking Free Will, decoherence will involve a large macroscopic object, thus a large space, like a gravitational field, but with a life of its own, so to speak, thus irreversible. As Bohr would say, the Quantum System cannot be isolated when “measured” (or made to decohere).

    Non-Locality does not look reversible to me: it involves non topologically trivial “inner” geometry (see below).

    Magnema: “Your assumption that things are not entangled before interaction is simply not necessarily true, generically, unless you specify that the two particles have never, through any chain of effects looking backwards in time, interacted (and even then, it is questionable).”

    PA: The entanglement is caused by an interaction. In general. Always.

    Could the particles have been entangled before, without suspecting it? This is the subject of present research, but many in the know do not doubt what will be found. The question has been asked in the case of Einstein-Podolski-Rosen (type) experiments (“EPR”). These involve making a measurement on an element of a pair of “particles” A and B which are entangled, yet “separated” by what I would call a “CLASSICAL DISTANCE”.

    Starting with Alain Aspect (who got the Wolf Prize in Physics for it), it has been shown that spin measurement can be made in flight, where the direction of the polarization is changed haphazardly in flight, and the measurement on A “immediately” changes B (“immediately” meaning out of the light cone of A).

    The suggestion has been made that A and B could be actually within the same much larger, much older light cone, and thus causally related in the classical relativistic sense. An esteemed team of American experimenters will thus run a version of the Aspect experiment, where the polarization direction are obtained by signals from quasars so distant from each other that they cannot plausibly be causally related.

    It seems extremely unlikely that the experiment will reveal that, after all all correlations observed and predicted (by the Quantum axiomatics) were the fruit on long set hidden causality.


    As far as I know, (Quantum) entanglement arises only from interaction. Thus picking two particles haphazardly, they should NOT have, generically, a common entanglement (albeit we don’t know what they are entangled with, some could end entangled with each other).

    Because of its non-local character, Quantum entanglement is impossible to reverse. Why? It has a non-trivial geometry and topology. It’s not the (simple) one of relativistic spacetime, it is the geometry of what happened before: not the sum of all histories, just the (geometrical and topological) nature of a particular history. Once again, not the geometry of spacetime, but a subset with a non trivial topology of the geometry of much higher dimensional manifold in which spacetime is embedded (see Nash embedding theorem).

    Now for waves. I remember Dirac, initially an engineer, saying he was looking for the simplest relativistic wave (simpler that a solution of the Klein-Gordon equation, a second order PDE). it struck him that that the simplest wave had to be a first order PDE (waves on a string like sines answer to second order PDEs; other waves, such as the KdV solitons, answer to nonlinear SE PDEs: waves don’t have to be just trigonometric functions!).

    Angular momentum in its simplest form arises from spinor geometry, not linear algebra (it was discovered by Elie Cartan before World War One, in a purely geometrical setting). It’s a square root of the Riemannian metric.


    Dirac rediscovered it by postulating the simplest relativistic wave, a solution to the simplest, thus first order, Partial Differential Equation giving what can be called a wave. It’s equivalent to taking the square root of the relativistic WAVE operator (the usual Laplacian).


    However, Michael Atiyah said:

    “No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors.”

    May I suggest the same is true of the geometry of entanglement?… With the caveat that it is irreversible?

  20. Kurt Stocklmeir says:

    I think arrow of time changes direction all the time for small amounts of time and this influences vacuum flucs but my vacuum flucs are different than what people think Kurt Stocklmeir

  21. Patrice Ayme says:

    Trying to answer phayes. Phayes says:
    @Patrice Ayme
    “What is Quantum Physics about? Some, mostly the “Copenhagen Interpretation” followers, claim Quantum Physics is a subset of functional analysis”

    Not so. The claim is that quantum physics is the application of (noncommutative) probability theory¹ to physics. As (Copenhagenist) Ray Streater says, “It took some time before it was understood that quantum theory is a generalisation of probability, rather than a modification of the laws of mechanics.”²

    Well, this is your opinion and that of Mr. Streater. Actually interpretations of the Copenhagen Interpretation vary considerably. You can consider:


    As engineers are trying to make functional Quantum computers and Quantum simulators, they have to figure out exactly what causes “decoherence” or “wave function collapse”. “Interpreting” Quantum Physics has become an experimental subject.

    Quantum Physics has even been interpreted as a change of logic (with no less than Birkhoff and Von Neumann as the original authors):


    As you can see therein, the subject is not closed. It involves now supersymmetry, supergeometry and thus non-commutative mathematics.

    Waves: first, there is no general definition of “wave”. That’s, precisely the force of the concept (whereas there is a strict definition of eigenstate, that’s a weakness). If a Quantum state has not (yet!) be demonstrated to be caused by waves, that does not mean it’s not. It just mean we don’t know. To be sure why something does not happen, one has to be sure why it does not happen. In this particular case, one would have to know more than what the Copenhagen Interpretation says. “Copenhagist” who claim to know more than Copenhagen, are not really Copenhagists. That’s rather ironical.

  22. Robin says:

    I thought that in EQM everything was, in effect, static.

  23. Robin says:

    With Boltzmann brains, it depends entirely on what you think a brain does. If you think that what a brain does is computing, then a Boltzmann brain does not have to be a similar to the atomic structure of a brain, merely a computer that is running an equivalent algorithm. It is highly unlikely that evolution has provided us with the most efficient route to running this algorithm.

    In addition a universal computer is a very simple thing (minus program) it can be described in about 32 bits. If just about anything can fluctuate into existence, surely universal computers will outnumber universes by several orders of magnitude!

  24. Robin says:

    In fact it seems to me that the resources necessary to process the information I am processing at any particular moment in time are comparatively trivial and that makes it vanishingly improbable that I am living in a real universe.

  25. Robin says:

    Further, it seems that a universe is an incredibly complex way of making a computer that can observe itself. So this universe fluctuated into existence from something that had no simpler ways of making a computer that can observe itself?