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And expansion of space is equal to expansion of vacuum?

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They are something completely different, but there is connection.

Quantum Field Theory (which underlies the Standard Model) is dependent on background. It's possible to formulate it on the curved background but usually one works in the Minkowski space-time and special relativity. Vacuum is then a special state $\left | 0 \right >$ in the Hilbert space that is annihilated by all annihilation operators and has no a priori connection with the usual vacuum (i.e. empty space) you are thinking of. There can be multiple vacuum states (so called degeneracy of the vacuum corresponding to inequivalent sets of annihilation operators connected by non-unitary transformations) so then the interpretation is obviously muddled.

Also vacuum in Standard Model is not empty in the sense that it is able to produce particle-antiparticle pairs (or in different words, that Standard Model has non-trivial field content and interactions). But the most important point is that this vacuum is static and has no effect on the curvature of the space-time whatsoever.

You can see that space in the General Relativity has none of the above properties and also that it is dynamic. What you are after is a theory of quantum gravity (or at least some approximation) that would take the Standard Model vacuum interactions into account when solving Einstein's equations. Of course you can try to do this and obtain e.g. prediction for cosmological constant in terms of properties of Standard Model vacuum. Problem is that the predicted value is completely wrong and so this is one of the major problems on the road to quantum gravity.

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is it problem for the string theory? –  voix Jan 8 '11 at 14:52
    
@voix: well, in a sense, but for different reasons. In string theory one has great number of vacuums (sorry for overloading the term even further, this has nothing to do with either empty space or Standard model; it is just a term people use to describe possible shapes of Calabi-Yau manifolds and other content of the string theory that lets you compute particle properties) and most of them don't resemble anything similar to our world. So string physicists are busy with finding the right vacuum. –  Marek Jan 8 '11 at 15:01
    
The two are related in a theory of quantum gravity. By "space" I assume you mean "spacetime," which I further assume means "solution of Einstein's field equations." By "vacuum" in a quantum field theory we mean a state of minimal energy (there needs to be time translation symmetry to make sense of this) or a state which is annihilated by generators of the symmetry group (in correlators at least). In a quantum theory which includes gravity, the vacuum should correspond to empty (Minkowski) space, with stable excited states corresponding to nontrivial solutions (cf. positive mass theorem). –  Eric Zaslow Jan 8 '11 at 19:54
    
What is the result of space expansion? The gravitational attraction becomes weaker? –  voix Jan 8 '11 at 21:06
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The space in GR and the vacuum of the SM are meant to represent the same physical object - the space out there (imagining me waving my hands and pointing to a random direction now).

However, the GR and the SM focus on different sets of phenomena that occur in that environment, so they say very different things about it. The Standard Model neglects gravity - which is the main thing that General Relativity wants to study. So in GR, the space can get curved and expanded; in the SM, it can't. In the SM, the vacuum has lots of activity and virtual particles; in GR, there is nothing going on in the vacuum.

Of course, the real vacuum/space around us has the complicated properties from both GR and SM. It can get curved and expanded and there is a lot of microscopic virtual particle activity in it, too. The only known consistent theory that incorporates both GR-like and SM-like aspects of the vacuum - and anything else - is string/M-theory.

The word "vacuum" has different meanings in GR, SM, and string theory. While the meanings in GR and SM were sketched above, in string theory, we usually mean a particular "empty space" with the maximum (Poincare or de Sitter or anti de Sitter) symmetries. There are many solutions of string theory that fit into this category; their set is sometimes referred to as the "landscape". So we often hear that we have $10^{500}$ semi-realistic vacua in string theory although the number is a guess rather than a result of a fully controllable calculation.

One of those vacua - we don't know which one - is identified with the vacuum/space we see around, and all its aspects that are included both in GR and SM can be extracted from string theory. Qualitatively, we know it to be the case - all of GR and SM effects can be extracted from the stringy vacua (or a subclass of them). We just don't know which one is exactly the right one so that we could calculate all properties of the real world, including the elementary particle masses.

Note that in the Standard Model or quantum field theory, the vacuum itself "knows" about the properties of all particles because they're constantly emerging and disappearing in the vacuum as virtual particles.

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Nice answer @Luboš Motl. I was searching for something and I arrived here. I had a question on your comment, please. When you say "In the SM, the vacuum has lots of activity and virtual particles; in GR, there is nothing going on in the vacuum", doesn't the cosmological constant in GR represent vacuum energy? Isn't the vacuum energy in GR related to virtual particles in SM?? I am not an expert on those fields but I had this question when I read your answer. Thank you very much –  Gotaquestion Nov 2 '13 at 19:44
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