Does the vacuum fluctuate?
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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.