Recently, I was reading about Hawking Radiation in A Brief History of Time. It says that at no point can all the fields be zero and so there's nothing like empty space(quantum fluctuation etc.). Now, the reason mentioned was that virtual(force-carrier) particles cannot have both a precise rate of change and a precise position(Uncertainty Principle).

So, my question is : This video says that virtual particles don't follow normal physical laws. So, how can we say that they obey the uncertainty principle?


2 Answers 2


The reason for many contradictory statements regarding the nature of virtual particles is that they are often invoked for heuristical explanations of phenomena that arise within the framework of quantum field theory. One then tries to justify those explanations by attributing certain properties to virtual particles they do not actually possess.

What virtual particles actually are:

By definition, a virtual particle is an internal line in a Feynman diagram. The latter are used in perturbative quantum field theory to make the calculation of series expansions easier. In order to do this, one draws those diagrams, for which each line and vertex (knot) has a precise corresponding mathematical expression that needs to be written down. At this level of the calculation, there is no physical interpretation of a single Feynman diagram, physical meaning is only attached to the final result of the calculation. A virtual particle, which is nothing but a line in a auxiliary diagram, is nothing physically meaningful on its own.

How they are related to physically meaningful quantities:

As mentioned above, virtual particles arise within diagrams in perturbative expansions of quantities one calculates within quantum field theory. One such quantity would be the energy of the vacuum (hence the statement "there is nothing like empty space"), others would be particle decay rates or scattering cross sections, and there are many other examples. One can think of virtual particles as mathematical contributions to the final result of the calculation, but nothing more. Be careful not to take the particle analogy too far.

Regarding the confusion about their reality:

Since certain phenomena in quantum (field) theory may appear counter-intuitive (e.g. vacuum energy), one feels more comfortable with a nice, simple picture to invoke in order to explain them. This is especially true when explaining to laymen, which is essentially what popular science does (like A Brief History of Time). This is where virtual particles enter: since they contribute mathematically to the description of these phenomena, they are also used in heuristic explanations. It is easy to imagine a particle being exchanged, or a pair of particles being created and annihilated after a short while. But this does not mean that it is actually happening in reality. It is a nice and simple picture, nothing more. But if one wants to go as far as taking their reality serious, one needs to invoke additional concepts in order to justify this. This is why energy/time uncertainty is often used to explain the existence of virtual particles.

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    $\begingroup$ So if one used non-perturbative methods to do calculations, there would be no purpose for virtual particles? $\endgroup$
    – Nick
    Commented Apr 20, 2014 at 17:07
  • $\begingroup$ Yes, that is true. $\endgroup$ Commented Apr 20, 2014 at 17:09
  • $\begingroup$ So, virtual particles are an ad hoc hypothesis, right? $\endgroup$
    – Yashbhatt
    Commented Apr 20, 2014 at 17:11
  • $\begingroup$ In this context, the uncertainty principle would be the ad hoc hypothesis. $\endgroup$ Commented Apr 20, 2014 at 17:13
  • $\begingroup$ But the uncertainty principle is experimentally verified whereas by their very definition virtual particles are impossible to detect. $\endgroup$
    – Yashbhatt
    Commented Apr 20, 2014 at 17:28

Here is a simple Feynman diagram.

feynman diagram

electron electron elastic scattering if time is the y axis,( electron positron elastics scattering if time is the x axis)

Real particles are the incoming and the outgoing that can be measured in an experiment in the lab. The exchanged photon is called virtual.

The difference between real particles and virtual particles in the mathematical definition is that virtual particles are off mass shell, i.e. they have all the quantum numbers identifying the particle by its name, but not the mass which can be positive negative or zero depending on the integration. Real particles except their quantum numbers also have an identifying mass.

This has to be so because the Feynman diagram is a symbolic shorthand for an integration which takes place over all the internal variables identifying the cross section of scattering two electrons into two electrons.

In a very real sense what is real and what is virtual depends on the boundary values of our calculation. In this diagram of electron positron annihilating into two quarks and a gluon


the electron and positron are known on mass shell real particles, the photon is virtual, and in a strict feynman diagram sense since quarks and gluons cannot be free but have to bind with other quarks and gluons, the outgoing should also be considered virtual. There we substitute the concept of a gluon jets two quark jets, which can be well measured in the lab and christen the outgoing three real.

The normal physical law that is violated by virtual particles is the mass shell as explained above. All the other quantities that identify the particles are there , that is why we can have virtual electrons and virtual photons, it is only the mass that is not respected within the bounds of the calculations for the quantities of interest in a physical measurement.

The Heisenberg Uncertainty Principle comes when we contemplate ground states in energy, and there exist diagrams where the vacuum is composed of virtual particles being created and annihilated, because we can never measure zero energy due to the Heisenberg uncertainty. There is nothing that constrains the mass for the HUP, so there is no conflict in describing situations with such virtual particles. There are few situations where the effect of vacuum fluctuations can be measurable, one of them is in the Hawking radiation. Another one is the Casimir Effect.

Edit after comments:

This question about the meaning of "virtual" comes up again and again, and I believe the confusion arises because of the tendency of most of us to mix up three different frameworks:

1) One framework is the symbolic Feynman diagrams,

2) the second is the mathematical framework of integrals within integrals in any cross section etc calculation,

3) and the third is the measurement/physical/laboratory framework.

With great ingenuity Feynman took the complicated integrations in scattering calculations before his "invention" of the diagrams, and made a one to one correspondence of the mathematical framework to a consistent system of diagrams with rules for converting to integration. This simplified enormously setting up the program for calculations .

Then comes the identification of the symbolic plots to the laboratory/measurement framework. This is done by taking the initial values from the experiment under consideration and predicting the values for the outcome of the experiment.

The initial and final states are the ones measured in the lab and nailing the mathematics to reality/experiment, and thus the incoming and outgoing lines in the diagrams are called "real".

The intermediate lines are called virtual particles because they, like a virtual optical image, are an analogue of the real particles because they carry all the quantum numbers of the real particles except the mass, they are off mass shell.

Usually the three frameworks are not logically separated because there is no necessity, there is no problem if one is sloppy on whether one is talking mathematics or diagrams or laboratory measurements. The calculations fit and that it that.

Confusion arises when thinking about the vacuum and about Hawking radiation.

We can draw Feynman diagrams which correspond to the zero point energy with virtual particles without incoming and outgoing lines. The boundary values are given by the Heisenberg Uncetainty principle which is the correspondence with physical reality ( it has not been invalidated as a postulate for the mathematical modeling of elementary particle physics with quantum field theory). Thus we have the three frameworks.

When asking if any real particles can come out, it is a question for the first and second framework, the diagrams and the mathematics calculations associated. The answer there is yes, if energy can be supplied in excess of the HUP uncertainty, and that is what allows the Hawking radiation hypothesis for black holes. It is still in the first two frameworks, a mathematical prediction, until some ingenious experiment will be able to show the radiation coming from a black hole.

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    $\begingroup$ "the outgoing should also be considered virtual" You are mixing up virtual particles with short-lived, unstable particles. The latter are on-shell, but have imaginary mass. Furthermore, the Casimir effect is not a proof of the reality of virtual particles, it is just a proof that our perturbative calculation is correct. $\endgroup$ Commented Apr 20, 2014 at 18:51
  • $\begingroup$ @FredericBrünner well perturbative calculations are all what supports the concept of virtual particles after all, it is a tautology what you say about the Casimir effect. Quarks cannot be free in our lab experiments, so a diagram without closure by other quarks can only have a virtual meaning : the mass of the quarks and the mass of the gluon in the diagram above are undefined and dependent on further intergrations which we gloss over with the jet language from the experimental observations. $\endgroup$
    – anna v
    Commented Apr 20, 2014 at 18:57
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    $\begingroup$ But within perturbative calculations, virtual particles are just mathematical tools, nothing that is supposed to have a correspondence in physical reality. Furthermore, I am not sure what you mean by "virtual meaning". $\endgroup$ Commented Apr 20, 2014 at 19:00
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    $\begingroup$ Mathematical consistency requires them to have the right quantum numbers, but that does not make them more real. Regarding real particles that do not appear as free asymptotic states: they exist, but not off-shell, and hence they are not virtual. These are poles of the scattering amplitudes with large imaginary parts, i.e. unstable particles. In the language of low energy effective field theory, they are called resonances. $\endgroup$ Commented Apr 21, 2014 at 3:51
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    $\begingroup$ Baryons can appear experimentally as resonances. Regarding your other statement: of course the fact that they have the same quantum numbers as real particles makes them look real, but nevertheless, this is just an analogy. $\endgroup$ Commented Apr 21, 2014 at 4:05

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