As an experimentalist I know that there exist elementary particles, particles that we measure with our detectors and are similar to macroscopic particles ( billiard ball like) sometimes , and sometimes show a statistical behavior that has sinusoidal/wave-like probability density distributions ( i.e. when measuring many of them in the same boundary conditions the distributions of their spatial behavior show wave like interferences) .
Here is a link which shows bubble chamber photos of the tracks ofelementary particles .
A large accumulation of measurements allowed the formulation of Quantum mechanics ( in contrast to classical mechanics) and the development of mathematical models that can describe the scattering and decay behaviors of elementary particles and be able to predict new ones. These models depend on expansions in series and integrals over functions in a convoluted manner, and were simplified with the iconal representations of Feynman diagrams. There is a one to one correspondence between a Feynman diagram amd a term in the sum to get the crossection etc.
An example of the scattering of e+e- going to a mu+ mu- Feynman diagram shows the real particles measurable in a detector as incoming and outgoing lines, and a virtual photon exchanging energy and momentum and quantum numbers . It is virtual because an integrations is implied over the variables so though it has the photon quantum numbers its mass is not zero nor fixed because it plays within the limits of integration. There are many higher order with many more exchanges diagrams, but this is the tree level with the higher coefficient in the perturbative expansion.
Thus the word "virtual" implies "integration within limits" , and contrasted with "real" where the particle has a fixed unchanged mass and can be measured in the detectors.
Now field theory has developed in a one to one correspondence with the Feynman diagram representations, where all of space has operators called creation and annihilation operators representing the fields. It is a mathematically satisfying representation for most theoretically inclined physicists but the ground on which field theory is validated is the Feynman diagram integrations bounded by the wavefunctions of a specific quantum mechanical equation ( Schrodinger, Klein Gordon, Dirac).
On the simple Feynman representation one can understand the vacuum as arising from the Heisenberg uncertainty condition . Within the integration bounds of Heisenberg uncertainty virtual particle antiparticle pairs can exist, but it means that they are not real, i.e. not on mass shell, because the vacuum has very little energy available and energy is needed to materialize them into real. That this is true comes from higher order calculations than the example above, where the contribution to crossections etc of these possible virtual particles is measurable.
In a blog post Lubos Motl is discussing the vacuum, starting with the history, and asking :" is the vacuum empty and boring?" emphasizing conservation laws and concluding :
Questions about values of local fields may produce uncertain and fluctuating answers in quantum field theory (these predictions only become physical if you actually measure these observables: you may never assume that particular preferred values "objectively exist" without any measurement). But if you are an observer who cares about the total energy or the total momentum (or the total angular momentum etc.) of the vacuum, the situation is very different. Quantum field theory unambiguously says (and there is no contradiction with the previous sentence) that the vacuum is empty and boring, after all.
After this preamble:
- Is the vacuum empty or is it not?
It depends on the particular observation, and observation means energy exchanges and Feynman diagrams.
- Are there particles in the vacuum and can they create a universe?
Depending on the experiment , yes, there will be measurable effects of virtual particles in the vacuum. For example the Hawking radiation where the energy for getting one on shell is provided by the gravitational field of the black hole.
Creating a universe is not within the present day knowledge of physics. It may be a hypothesis within General Relativity where energy conservation is sometimes moot.
- But when virtual particles are just a mathemacical "trick" to calculate something, what does Lawrence Krauss mean?
Extrapolating in the case where energy is not a conserved quantity and hoping his proposal is right.
- What is the matter about the vacuum?
As quoted above
if you are an observer who cares about the total energy or the total momentum (or the total angular momentum etc.) of the vacuum, the situation is very different. Quantum field theory unambiguously says (and there is no contradiction with the previous sentence) that the vacuum is empty and boring, after all.
Energy has to be provided by interactions for the vacuum to be populated.