1. OK, I am a bit befuddled here. I have a pretty rubbish text at home that waves its hands and says something like

"the expectation value is simply the average of all measurements made on the observable of a large number of systems all in the state "

To be fair my text does give some mathematics but it all looks a bit hollow to me. So let me follow P.A.M. Dirac (although DrRocket warned me that he is no less guilty of hand-waving than anyone else) and recast this as.......

Assume a system whose state can be given by the state vector . Let us further assume that an "observable" is the operator acting on this vector, and finally that a "measurement" of this state corresponds to an eigenvalue for the operator . As far as I recall. this is how Dirac presents it. Let us say this is all well and good (though I confess to a slight queasiness about it!).

No matter, the fault is probably mine.

But I have come across the term Vacuum Expectation Value here and there, and can make very little sense of it. I suppose it must in some way be related to the above, so I can only assume I am missing something - is it a rather refined use of the word "vacuum"? Perhaps it is the lowest possible eigenvalue for some operator acting on our state vector? Or perhaps it refers to the state itself? In the latter case, it is not all obvious to me that one say can anything meaningful about the state without making a measurement - i.e. finding the eigenvalues for some as-yet-undefined operator.  2. But I have come across the term Vacuum Expectation Value here and there, and can make very little sense of it. I suppose it must in some way be related to the above, so I can only assume I am missing something - is it a rather refined use of the word "vacuum"? Perhaps it is the lowest possible eigenvalue for some operator acting on our state vector? Or perhaps it refers to the state itself? In the latter case, it is not all obvious to me that one say can anything meaningful about the state without making a measurement - i.e. finding the eigenvalues for some as-yet-undefined operator.
I think what is meant here is a "physical" vacuum, i.e. the absence of all other fields and matter. The vacuum expectation value would then be the average expectation value that one might encounter in a given volume of vacuum for a certain operator. For example, for certain boundary conditions ( say two closely spaced uncharged plates ) the average expectation value of the QED energy operator is non-zero and finite, resulting in the Casimir Effect :

Casimir effect - Wikipedia, the free encyclopedia

In the same way one can define a vaccum expectation value for the Higgs field, which is somewhere in the region of 246 GeV.

As for how to make all this mathematically rigorous - sorry, no idea.  3. Originally Posted by Guitarist OK, I am a bit befuddled here. I have a pretty rubbish text at home that waves its hands and says something like

"the expectation value is simply the average of all measurements made on the observable of a large number of systems all in the state "

To be fair my text does give some mathematics but it all looks a bit hollow to me. So let me follow P.A.M. Dirac (although DrRocket warned me that he is no less guilty of hand-waving than anyone else) and recast this as.......

Assume a system whose state can be given by the state vector . Let us further assume that an "observable" is the operator acting on this vector, and finally that a "measurement" of this state corresponds to an eigenvalue for the operator . As far as I recall. this is how Dirac presents it. Let us say this is all well and good (though I confess to a slight queasiness about it!).

No matter, the fault is probably mine.

But I have come across the term Vacuum Expectation Value here and there, and can make very little sense of it. I suppose it must in some way be related to the above, so I can only assume I am missing something - is it a rather refined use of the word "vacuum"? Perhaps it is the lowest possible eigenvalue for some operator acting on our state vector? Or perhaps it refers to the state itself? In the latter case, it is not all obvious to me that one say can anything meaningful about the state without making a measurement - i.e. finding the eigenvalues for some as-yet-undefined operator.

There are calculations using quantum electrodynamics that attempt to quantify the energy density of the vacuum, the vacuum being the ground state. As with most calculations in quantum field theory there is a bit of an art to the calculation in selecting cutoffs for the integral to be evaluated so as to get a finite answer.

When this calculation is done in order to estimate the negative pressure term attendant to the quantum vacuum energy level, which shows up in general relativity as the cosmological constant, the result differs from the observed value by a FACTOR of
10000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 0000000000000000000000000000000.

Some people consider this a large error. NOBODY understands this. It does suggest that vacuum energy calculations are a wee bit suspect.

Understanding the quantum vacuum is a major open problem in physics. If you get to the point where you think you understand you are either mistaken or ready for an all-expense paid trip to Stockholm.  4. Originally Posted by Guitarist Perhaps it is the lowest possible eigenvalue for some operator acting on our state vector? Or perhaps it refers to the state itself?
As DrRocket pointed out the Vacuum Expectation Value (or VEV) is a measure of the energy density of the vacuum. The vacuum in perturbation theory, as far as I know, is defined as the lowest energy state of a given system and is given as a state vector, . One can then find the expectation value of this with some observable like any other state vector you may be interested in.  Posting Permissions
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