Does the vacuum fluctuate? -------------------------- Quantum fluctuations are a popular buzzword for the statistical triviality that the variance of a random variable A with zero mean is typically not zero - except that A is now an operator. Some people therefore think that this deserves a much more mysterious name. Vacuum polarization is a physical concept with observable consequences (e.g., the Casimir effect). But the vacuum fluctuations frequently associated with vacuum polarization are not fluctuations in time, of virtual particles popping in and out of existence from the vacuum (as popular sources on the internet assert), but formal properties of the quantum formalism. And ''observable consequences'' does not imply a cause and effect relationship -- Vacuum fluctuations cause nothing, hence have no effect. It is their presence in the equations that has some observable consequences. In the presence of vacuum fluctuations, nothing more pops in and out of existence than in the case where you repeat a measurement multiple times and get a nonzero mean square deviation. Fluctuations have a much better ontological status than virtual particles. Their properties are indeed computable nonperturbatively, hence are properties of the system under study and (unlike virtual particles) not only of the approximation method used. They are properties of the system, whether or not somebody measures it. In this sense they exist independent of measurement, like a tree exists no matter whether someone looks at it. But they are not what conventional story-telling claims they are: They are not changes in time. Instead, they describe uncertainties about what one gets when one tries to measure something. They describe the fluctuations in the measurement results when one repeats them under identical conditions - not fluctuations in what is measured. The meaning of the quantum fluctuation of a quantity Q is not the value of a measurement of Q but - by a form of Heisenberg's uncertainty relation - the intrinsic uncertainty of the measurement result in any attempt to measure Q. (What's the latitude of Europe? It fluctuates in a similar - but classical - sense as quantum fluctuations in that it has an intrinsic uncertainty - different measurement protocols give widely different answers.) There may be additional uncertainty due to limitations of the particular equipment used - but this is not a property of the system but of the equipment.) Thus quantum fluctuations reflect something about the limits of measurement processes, not something about rapid changes in time. In QED, the theory of photons and electrons, the renormalized photon propagator Delta_ren(q) is a nonperturbative object, defined without reference to virtual particles. The vacuum polarization tensor is defined nonperturbatively in terms of it, as (q^2 eta - q tensor q)Pi(q^2) := Delta_free(q) - Delta_ren(q), which is equivalent to Dyson's equation (cf. Weinberg, Vol. I, p.451). Its scalar part Pi(p^2) is related to the running fine structure constant. This contains all the physics of vacuum polarization, and is completely independent of virtual particles. (The relation between vacuum polarization and the fine structure constant is also described in http://en.wikipedia.org/wiki/Vacuum_polarization But the talk there about short-living virtual particles is nonsense: There hasn't been a single publication about the life-time of virtual particles - there is no such concept.) To compute the running fine structure constant in perturbation theory, one has to sum an infinite number of integrals described in terms of Feynman diagrams. The simplest of these is the one depicted in the wikipedia reference. It depicts a pair of internal electron-positron lines, which (as all internal lines) are commonly referred to as virtual particles. This name is of historical origin, but does not imply that they are more than lines on paper, illustrations of formal properties of how the integral is composed. Trying to give sense to them in reality requires the introduction of lots of complementary nonsense that pollutes the imagination and inhibits rational thinking about the matter. In the following, I'll describe the background that gives rise to the idea of vacuum fluctuations, starting with a simple analogy (in fact the special case when space-time is 1-dimensional) which shows why it is misguided to interpret it as a real process. The expectation sigma^2 = <(q-q_0)^2> (where q_0=) of a quantum harmonic oscillator in the ground state is nonzero, because the ground state is not an eigenstate of q. Does this mean that the ground state of a harmonic oscillator fluctuates? No. It just means that one cannot assign it a definite position -- that measurements of the position fluctuate for a particle in the ground state. According to the Heisenberg uncertainty relation, the variance simply tells one something about the theoretical limit accuracy in trying to measure the oscillator position rather than about any fluctuation of the oscillator itself. It is the measurement results that fluctuate, not the underlying object. The generalization of this observation to an infinite number of degrees of freedom gives quantum field theory, with Phi(x) replacing q, and the vacuum replacing the oscillator ground state. Vacuum fluctuations are just informal talk for the formal fact that sigma^2(x) = <(Phi(x)-phi(x))^2> (where phi(x)=) and more complex expressions of the same sort involving a factor of Phi(x)-phi(x) do not vanish. Does this mean that the vacuum fluctuates? No -- by the same reasoning as for the oscillator. On the other hand, the fact that sigma^2 resp. sigma^2(x) and similar expectations do not vanish shows in nontrivial physics, for example, a nontrivial zero-point energy. The zero-point energy can often, but not always be neglected. It can be utilized for the derivation of observable consequences. One of them is the Casimir effect. But the Casimir effect can also be derived without reference to the zero-point energy. Some people take the zero-point energy as resulting from the motion of virtual photons, a concept as unreal as the fluctuating vacuum. Those with this view then cite the Casimir effect as proof of existence of virtual photons.