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What Hermitian operators can be observables?

+ 7 like - 0 dislike

We can construct a Hermitian operator $O$ in the following general way:

  1. find a complete set of projectors $P_\lambda$ which commute,
  2. assign to each projector a unique real number $\lambda\in\mathbb R$.

By this, each projector defines an eigenspace of the operator $O$, and the corresponding eigenvalues are the real numbers $\lambda$. In the particular case in which the eigenvalues are non-degenerate, the operator $O$ has the form $$O=\sum_\lambda\lambda|\lambda\rangle\langle\lambda|$$

Question: what restrictions which prevent $O$ from being an observable are known?

For example, we can't admit as observables the Hermitian operators having as eigenstates superpositions forbidden by the superselection rules.

a) Where can I find an exhaustive list of the superselection rules?

b) Are there other rules?


c) Is the particular case when the Hilbert space is the tensor product of two Hilbert spaces (representing two quantum systems), special from this viewpoint?

This post has been migrated from (A51.SE)
asked Feb 13, 2012 in Theoretical Physics by Cristi Stoica (255 points) [ no revision ]
retagged Mar 18, 2014 by dimension10

2 Answers

+ 9 like - 0 dislike

I think the fundamental object is quantum mechanics is not the Hilbert space and operators on it but the C*-algebra of observables. In this picture the Hilbert space appears as a representation of the algebra. Different irreducible representations are different superselection sectors. The answer to "which operator is observable" is thus simple: the observable operators are those that come from the algebra. Indeed its better to think of observables as self-adjoint elements of the algebra rather than as operators

You might ask where do we get the algebra from. Well, this already should be supplied by the particular model. For a quantum mechanical particle moving on a manifold $M$, the C*-algebra consists of all bounded operators on $L^2(\hat{M})$ commuting with $\pi_1(M)$ where $\hat{M}$ is the universal cover of $M$. The superselection sectors correspond to irreducible representations of $\pi_1(M)$. For QFTs the problem of constructing the algebra of observables is in general open however certain cases (such as free QFT and I believe rational CFT as well) were solved. An approach emphasizing the algebra point of view is Haag-Kastler axiomatic QFT

From the point of view of deformation quantization the quantum observable algebra is a non-commutative deformation of the algebra of continuous (say) functions on the classical phase space. This point of view is not fundamental but it's useful. For example it allows to understand different values of spin and different quantum statistics as superselection sectors

This post has been migrated from (A51.SE)
answered Feb 14, 2012 by Squark (1,705 points) [ no revision ]
Thank you, this is very useful.

This post has been migrated from (A51.SE)
+ 6 like - 0 dislike

Without superselection rules to restrict the observables, any Hermitian operator is an admissible observable. The case of multiple identical systems is very important. Indeed, if the systems are really identical, only observables that are symmetric under the exchange of the systems are admissible. In such a case, technically speaking you should only consider observables that commute with all possible permutation operators (i.e., with the elements of the representation of the permutation group on the Hilbert space of the systems).

This post has been migrated from (A51.SE)
answered Feb 13, 2012 by Marco (260 points) [ no revision ]
Thank you for the answer. Indeed, for identical particles one takes as the Hilbert space as the quotient of the tensor product by the appropriate ideal. Do you know some bibliographic references showing that the only restrictions to a Hermitian operator to be observable are only these?

This post has been migrated from (A51.SE)
You can impose other superselection rules, typically based on some symmetry, not necessarily the exchange symmetry (that applies only for multi-particle systems). I cannot recommend a reference based on my direct knowledge, but I'd say you might start from http://rmp.aps.org/abstract/RMP/v79/i2/p555_1

This post has been migrated from (A51.SE)
Thank you, this helps.

This post has been migrated from (A51.SE)

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