What do creation operators measure?

[jjustinn]

My understanding is that in general, operators -- corresponding to observables -- act on a state (itself a member of an infinite-dimensional Hilbert space), and the eigenvalue is the value in that state (at least, if it's a pure state).

To get an expectation value, you take the dot product of the final and the operator of the initial state.

That much at least is put forward in pretty much every text on QM of any level, and is basically given as received gospel, and it's pretty clear when working with "first quantized" operators, like position, energy or linear/angular momentum.

Then, with second quantization, the state turns into a Fock space, and the operator (or state, depending on the representation) is defined over all of space-time.

Even there, the meaning of the Number operator (destruction operator followed by creation operator) seems pretty clear: N(p, x, y, z, t) [ state > has an eigenvalue telling how many particles of momentum p there are at xyzt.

But then when it comes to creation/annihilation operators, it seems like every text switches from an operator being an artifice for getting a measurement into something that actually "operates" on the state -- e.g. Create(p, x, t, z) [0> is described as "adding a particle of momentum p at position xyzt to the vacuum state"...so it's not clear at all what it "measures", if anything. My gut says that the expectation value should be the probability that a particle of momentum p is created/removed at xyzt, but I can't find anyone saying that...and even of that were the case, it seems like the operator would be redundant, since the final state would have the added(removed) particle wrt the initial one, so shouldn't just the dot product of the two give the amplitude of going from one to the other?

Now, I did see a note in the most recent text I've flipped through (Duncan's Conceptual Foundations -- great stuff btw) that the creation/annihilation operators were non-hermitian...but even if that meant they were useless for measuring, that doesn't explain the entirely new interpretation.

 

[tom.stoer]

Some remarks

1) self-adjoint operators do not measure anything in the sense that "they perform a measurement"; they do not represent the apparatus; the spectrum (eigenvalues and eigenvectors) represent possible results of measurements of the physical observable represented by the self-adjoint operator (in most texts the operator and the observable ar identified, but strictly speaking this is not true)

2) neither in QM nor in QFT does the number operator N depend on x,y,z; in QFT it's an integral over all momenta, so it does not locally "at x,y,z" but globally for R³

3) the number operator N and especially the creation and annihilation operators are abstract entities; they do not physically count or create particles (b/c the way a single operator "creates" a particle violates several conservation laws); they simply transform a state |0> into a state |1>, but this is not to be confused with a physical process (which s described by a certain combination of different operators)

4) creation and annihilation operators are not related to any measurement; first -as explained above - the relation between self-adjoint operators and measurements is rather indirect; second, this relation does not hold for creation and annihilation operators b/c they are not self-adjoint, so they do not have real eigenvalues and corresponding eigenstates representing results of measurements

5) it may be instructive to construct eigenstates for these operators for the simply harmonic oscillator; you will find that there are no eigenstates for the creation operator, but that there are eigenstates for the annihilation operator

6) switching from the harmonic oscillator to QFT is both a huge mathematical step and a step to a new level of interpretation; suppose you have a couple of creation and annihilation operators and you can write down a state like |1,2,0,0,1,0,...>; w/o specifying a context and w/o knowing the derivation you are not able to interpret the states; it's the derivation in QFT based on the association of plane waves with singe particles that allows you to interpret the state as "1 particle in state A, 2 particles in state 2, 0 particles in state C, ..."; this interpretation does neither follow from the formal definition of the operators and states (but from the context and its interpretation) nor does it follow from a measurement