Vacuum expectation value

In quantum field theory, the vacuum expectation value (VEV) of an operator is the operator's average (or expectation value) in the vacuum (i.e. in the quantum vacuum state). When explaining the concept of a VEV, physicists often cite the Casimir effect as an important example of an phenomenon resulting from the VEV of of an operator.

Quotes

• I remember a lunch in which Schwinger began by saying to Weisskopf, “Now I will make you a world.” The “world” was written down on a few paper napkins, one of which I saved. In any event, one of the things that he said, which has stuck with me ever since, was that scalar particles were the only ones that could have nonvanishing vacuum expectation values. He then went on to say that if you couple one of these to a fermion $\Psi$ by a coupling of the form $\Phi {\overline {\Psi }}$$\Psi$, then this vacuum expectation value would act like a fermion mass. This sort of coupling is how mass generation is done in principle for the fermions. All particles in this picture would acquire their masses from the vacuum.
  • Jeremy Bernstein, (January 2011)"A question of mass". American Journal of Physics 79 (1): 25–31. DOI:10.1119/1.3487939. (quote from p. 30)

• A new conceptual foundation for renormalizing T_μν on locally flat space-time—to obtain the so-called Casimir effect—is presented. The Casimir ground state is viewed locally as a (nonvacuum) state on Minkowski space-time and the expectation value of the normal-ordered T_μν is taken. The same ideas allow us to treat, for the first time, self-interacting fields for arbitrary mass in perturbation theory—using traditional flat-space-time renormalization theory. First-order results for zero-mass λφ⁴ theory agree with those recently announced by Ford. We point out the crucial role played by the simple renormalization condition that the vacuum expectation value of T_μν must vanish in Minkowski space-time, and in a critical discussion of other approaches, we clarify the question of renormalization ambiguities for T_μν in curved space-times.
  • Bernard S. Kay (1979). "Casimir effect in quantum field theory". Physical Review D 20 (12): 3052–3062.

• Vacuum expectation values of products of neutral scalar field operators are discussed. The properties of these distributions arising from Lorentz invariance, the absence of negative energy states and the positive definiteness of the scalar product are determined. The vacuum expectation values are shown to be boundary values of analytic functions. Local commutativity of the field is shown to be equivalent to a symmetry property of the analytic functions. The problem of determining a theory of a neutral scalar field given its vacuum expectation values is posed and solved.
  • Arthur Wightman, (1956). "Quantum Field Theory in Terms of Vacuum Expectation Values". Physical Review 101 (2): 860–866. DOI:10.1103/PhysRev.101.860.

External links

Encyclopedic article on Vacuum expectation value on Wikipedia