Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

Please welcome our new moderators!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

122 submissions , 103 unreviewed
3,497 questions , 1,172 unanswered
4,544 answers , 19,342 comments
1,470 users with positive rep
408 active unimported users
More ...

What Hermitian operators can be observables?

+ 7 like - 0 dislike
109 views

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?

Update:

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 (1,700 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, this helps.

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 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)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverflo$\varnothing$
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...