In popular physics articles and even some physics classes I've been to, the vacuum of space is described as being constantly full of quantum fluctuations. Supposedly, all sorts of particle-antiparticle pairs at all scales are constantly appearing and disappearing. We end up with a mental image of the vacuum as a roiling, choppy sea with all sorts of things going on, rather than a calm, placid background.

However, the vacuum, being the lowest-energy state of a theory, should be an energy eigenstate—which means it is time-invariant (except for a physically-irrelevant phase factor). So it seems the vacuum really should not be seen as a dynamic entity with all kinds of stuff happening in it, as we're led to believe.

Jess Riedel wrote in a blog post that

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.

So it sounds as if the existence of vacuum fluctuations is contingent on measuring the vacuum—in particular, measuring something that doesn't commute with energy (such as, I guess, the value of a field at a point).

How much truth is there to the idea that vacuum fluctuations are constantly happening everywhere, all the time? Is that really a useful way to think about it, or just a myth that has been propagated by popularizations of physics?

  • I do not know about the truth, but it is a useful concept that explain many observed phenomenon. You are in the right path in the sense that energy is always conserved, but the eigenstates of the Hamiltonian (energy observable) are not the same as (i.e. the Hamiltonian does not commute with) the particle number. For more details: operators.en.wikipedia.org/wiki/Quantum_fluctuation – Wolphram jonny Nov 11 '14 at 0:29
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    I'd like to clarify the statement you've quoted. As I said in the blog post, "real-life processes amplify microscopic phenomena to macroscopic scales all the time, thereby effectively performing a quantum measurement". Measurements-by-humans are not special or privileged. They are just another example of the sort of amplification processes that occur naturally, i.e., when a certain degree of freedom is copied onto other degrees of freedom. Vacuum fluctuation are contingent on such amplification processes, but they are not contingent on humans. – Jess Riedel Oct 18 '15 at 1:46
  • So yes, in this sense vacuum fluctuations are constantly happening. Such fluctuation are contingent on amplification processes, but are not contingent on humans. – Jess Riedel Oct 18 '15 at 1:52
  • No truth at all; see physics.stackexchange.com/a/250814/7924 – Arnold Neumaier Apr 20 '16 at 9:58
  • I read a few days in the topic as a layman. My conclusion that that quantum fluctuations in vacuum really exist and they can be described by virtual particles. The keyword is virtual here, so these particles do not exist in reality, they are just part of a mathematical model to ease calculations, or something like that. en.wikipedia.org/wiki/Virtual_particle – inf3rno Dec 1 '17 at 23:12
up vote 20 down vote accepted

Particles do not constantly appear out of nothing and disappear shortly after that. This is simply a picture that emerged from taking Feynman diagrams literally. Calculating the energy of the ground state of the field, i.e. the vacuum, involves calculating its so-called vacuum expectation value. In perturbation theory, you achieve this by adding up Feynman diagrams. The Feynman diagrams involved in this process contain internal lines, which are often referred to as "virtual particles". This however does not mean that one should view this as an actual picture of reality. See my answer to this question for a discussion of the nature of virtual particles in general.

I think it’s possible to give a beginners guide to what is meant by vacuum fluctuations, but it necessarily involves taking a few liberties so bear that in mind in what follows.

Before we start let’s remind ourselves of the following key point about superpositions. Suppose we have an operator $\hat{n}$ with eigenfunctions $\psi_i$ and we place it in a superposition:

$$ \Psi = a_0\psi_0 + a_1\psi_1 + \, … $$

Then when we do a measurement of the system using our operator $\hat{n}$ the suprposition will collapse and we will find it on one of the eigenstates $\psi_i$. The probability of finding it in that state is $a_i^2$.

Now suppose we do a measurement, then put the system back into the same superposition and do a second measurement, and keep repeating this. Our measurements will return different results depending on which of the eigenstates the superposition collapses into, so it looks as if our system is fluctuating i.e. changing with time. But of course it isn’t - this is just how quantum measurement works, and we’ll see that something similar to this is responsible for the apparent vacuum fluctuations.

Now let’s turn to quantum field theory, and as usual we’ll start with a non-interacting scalar field as that’s the simplest case. When we quantise the field we find it has an infinite number of states. These states are called Fock states and these Fock states are vectors in a Fock space, just as the states for regular QM are vectors in a Hilbert space. Each Fock state has a well defined number of particles, and there is a number operator $\hat{n}$ that returns the number of particles for a state. There is a vacuum state $\vert 0 \rangle$ that has no particles i.e. $\hat{n}\vert 0\rangle = 0$.

Suppose we consider a state of the scalar field that is a superposition of Fock states with different numbers of particles:

$$ \vert X\rangle = a_0\vert 0\rangle + a_1\vert 1\rangle +\, … $$

If we apply the number operator it will randomly collapse the superposition to one of the Fock states and return the number of particles in that state. But because this is a random process, if we repeat the experiment we will get a different number of particles each time and it looks as if the number of particles in the state is fluctuating. But there is nothing fluctuating about our state $\vert X\rangle$ and the apparent fluctuations are just a consequence of the random collapse of a superposition.

And by now you’ve probably guessed where I’m going with this, though we need to be clear about a few points. The free field is a convenient mathematical object that doesn’t exist in reality - all real fields are interacting. The states of interacting fields are not Fock states and don’t live in a Fock space. In fact we know very little about these states. However we can attempt to represent the vacuum of an interacting field $\vert \Omega\rangle$ as a sum of free field Fock states, and if we do this then applying the number operator to $\vert \Omega\rangle$ will return an effectively random value, just as it would do for a superposition of free field states.

And this is what we mean by vacuum fluctuations for an interacting field. There is nothing fluctuating about the vacuum state, however measurements we make of it will return random values giving the appearance of a time dependent fluctuation. It is the measurement that is fluctuating not the state.

I’ve used the example of the number operator here, but it’s hard to see how the number operator corresponds to any physical measurement so take this just as a conceptual example. However the process I’ve described affects real physical measurements and happens whenever the vacuum is not an eigenstate of the observable measured. For an example of this have a look at Observation of Zero-Point Fluctuations in a Resistively Shunted Josephson Tunnel Junction, Roger H. Koch, D. J. Van Harlingen, and John Clarke, Phys. Rev. Lett. 47, 1216 available as a PDF here.

  • I somehow missed seeing this answer until just now. This is a fascinating and illuminating way to look at it; thanks! – Nathan Reed Jun 27 '17 at 19:19
  • @NathanReed: better late than never :-) – John Rennie Jun 27 '17 at 19:25
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    Well this answer didn't include Heisenberg's uncertainty principle which is said to be the source of vacuum energy. I recently watched an interview of Leonard Susskind that even said that. – Bill Sep 21 '17 at 11:33

Vacuum fluctuations exist, but they are not happening. The whole popular imagery surrounding the notion of vacuum fluctuations (and the associated virtual particles) is completely unsupported by the mathematics behind quantum field theory. It is solely created for the purpose of illustrating abstract concepts for an audience that likes imagery and mystery but has no understanding of the substance of quantum mechanics. Taking this imagery seriously leads to a host of unsurmountable difficulties. See my essay ''The Vacuum Fluctuation Myth''.

It's true that the vacuum ought to be an eigenstate of the full interacting Hamiltonian. But as seen from the perspective of the Hamiltonian of the free theory (all interactions being treated as perturbations around this free theory) the actual ground state is "dressed" by many vacuum fluctuations on top of the free ground state.

Vacuum fluctuations do exist, but they are not a statement about the dynamics (the time evolution) of the system. This is true for quantum fluctuations in general. The state of the system may very well be stationary, still quantum fluctuations will be present. A more correct statement is: quantum fluctuations arise if the observable measured is such, that the state of the system does not have a definite value of that observable (in mathematical terms it is not an eigenstate of the operator representing the observable).

The best way to get a handle on odd quantum effects is to look at what the effect would mean physically.

One prediction made from the idea of vacuum fluctuation is that a strong enough electric field should polarize those fluctuations. In this case we are talking about polarizing virtual electron-positron pairs. This effect is called vacuum polarization.

Another implication is the idea that two conducting plates placed close enought together should exclude some of the quantum fluctuations. In this case we are talking about virtual photons being restricted. This is called the Casimir Effect

One of the oddities of quantum mechanics is that (in a sense) the possibility of something happening can have an influence on what actually does happen. Feynman has a great description of this idea called the path integral formulation of quantum mechanics.

I'm a big fan of the popular level explanation of the path integral approach in his book QED: A strange theory of light and matter.

It is not needed for the vacuum to fluctuate all the time, but one can say the the probability of having a vacuum fluctuation at this point $x,t$ is non-zero

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