What is the solution to the 1=0 paradox involving self-adjoint operators?

[ShayanJ]

I know this is raised several times in this forum but I still don't see what's the solution! So I want to discuss it again.
Consider two self-adjoint operators A and B which obey $[A,B]=c I$. Now I take the normalized state $|a\rangle$ such that $A|a\rangle=a |a\rangle$. Now I can write:

$1=\langle a | a \rangle=\frac 1 c \langle a | cI |a \rangle=\frac 1 c \langle a | [A,B] |a \rangle=\frac 1 c [\langle a | AB |a \rangle-\langle a | BA |a \rangle]=\\ \frac 1 c [(A|a\rangle)^\dagger B|a\rangle-a\langle a |B|a\rangle]=\frac 1 c [(a|a\rangle)^\dagger B|a\rangle-a\langle a |B|a\rangle]=\frac a c [\langle a | B |a \rangle-\langle a | B |a \rangle]=0$

So there should be something wrong with the reasoning. There were several suggestions before but I don't think they work.P.S.

1) Please DO NOT specialize to momentum and position!(Or any other pair, unless you can prove there is only a few pairs of operators that satisfy the requirements and you can say what is wrong about each pair.)

2) I know that the commutation relation can't be realized in a finite dimensional Hilbert space. So take an infinite-dimensional Hilbert space! Can this be used here?

3) I know that at least one of the operators should be unbounded. Can this be used here?

4) I know domains are important. But I can still find a vector in the intersection of the domains of A and B. Can you prove their domains are disjoint? Or can you prove there is no eigenvector of A in the intersection of the domains of A and B?

5) I know that here we should consider a rigged Hilbert space. But a rigged Hilbert space is actually a triplet like $D \subset L^2(\mathbb R,dx) \subset D'$. So I can still choose vectors from D which are actually ordinary nice vectors. Can you prove there is no eigenvector of A in D? Or can you prove for such an eigenvector, B necessarily gives ill-defined results so that $\langle a | B |a \rangle-\langle a | B |a \rangle$ is indeterminate instead of 0?

6) An important point here is the symmetry between A and B. So even if we can use one of the above suggestions, we then can swap the role of A and B and retain the argument. Can you break this symmetry?

7) Hey @Greg Bernhardt , can you set a prize for solving this?:D

I hope this time I'll get a definite solution for this and get this case closed.
Thanks

 

[mfb]

To make this a paradox, you have to show the existence of self-adjoint operators A, B with [A,B]=cI and with real eigenvalues for A. And all the operations above have to be well-defined, of course.
Just saying "consider" is not enough: "Consider a number x that satisfies both x=0 and x=1. Then 1=x=0".

 

[ShayanJ]

To make this a paradox, you have to show the existence of self-adjoint operators A, B with [A,B]=cI and with real eigenvalues for A. And all the operations above have to be well-defined, of course.
Just saying "consider" is not enough: "Consider a number x that satisfies both x=0 and x=1. Then 1=x=0".

Yeah, that's true but I want to know exactly which assumption is wrong or which assumptions contradict each other.
I mean...I can't show that such a pair actually exists. But this doesn't prove it doesn't exist. I want to know how can I prove that there is no such a pair.
Or, to put it another way, this can be seen as a proof by contradiction that such a pair does not exist. But which assumption should I drop so that I can have a pair satisfying those weakened assumptions?
Or maybe such a pair actually exist but some part of the calculation is not correct. Which part is that?

Also $A=\frac{\hbar}{i} \frac{\partial}{\partial \varphi}$ and $B=\varphi$ constitute such a pair because we have $A e^{i\frac m \hbar \varphi}=m e^{i\frac m \hbar \varphi}$. So there exists such a pair.

 

[Matterwave]

You can basically use your "proof" that 1=0 to show that somewhere the logic failed. So at some step, there must have been some undefined quantity. As it turns out, the undefined quantity is $\langle a|B|a\rangle$ (as such, of course $\langle a|[A,B]|a\rangle$ is also undefined, the commutator is defined only on the intersection of the domains of A and B). See my previous thread on this matter: https://www.physicsforums.com/threads/commutator-expectation-value-in-an-eigenstate.747434/

 

[ShayanJ]

You can basically use your "proof" that 1=0 to show that somewhere the logic failed. So at some step, there must have been some undefined quantity. As it turns out, the undefined quantity is $\langle a|B|a\rangle$ (as such, of course $\langle a|[A,B]|a\rangle$ is also undefined, the commutator is defined only on the intersection of the domains of A and B). See my previous thread on this matter: https://www.physicsforums.com/threads/commutator-expectation-value-in-an-eigenstate.747434/

Thank you very much man. The paper pointed to by dextercioby, was exactly the thing I was looking for all this time.
So it turned out that

there is no eigenvector of A in the intersection of the domains of A and B

is the the solution.
Case Closed!

 

[micromass]

I know this is raised several times in this forum but I still don't see what's the solution! So I want to discuss it again.
Consider two self-adjoint operators A and B which obey $[A,B]=c I$. Now I take the normalized state $|a\rangle$ such that $A|a\rangle=a |a\rangle$. Now I can write:

$1=\langle a | a \rangle=\frac 1 c \langle a | cI |a \rangle=\frac 1 c \langle a | [A,B] |a \rangle=\frac 1 c [\langle a | AB |a \rangle-\langle a | BA |a \rangle]=\\ \frac 1 c [(A|a\rangle)^\dagger B|a\rangle-a\langle a |B|a\rangle]=\frac 1 c [(a|a\rangle)^\dagger B|a\rangle-a\langle a |B|a\rangle]=\frac a c [\langle a | B |a \rangle-\langle a | B |a \rangle]=0$

So there should be something wrong with the reasoning. There were several suggestions before but I don't think they work.P.S.

1) Please DO NOT specialize to momentum and position!(Or any other pair, unless you can prove there is only a few pairs of operators that satisfy the requirements and you can say what is wrong about each pair.)

http://en.wikipedia.org/wiki/Stone–von_Neumann_theorem

2) I know that the commutation relation can't be realized in a finite dimensional Hilbert space. So take an infinite-dimensional Hilbert space! Can this be used here?

It can't be realized as bounded operators either. They need to be unbounded.

3) I know that at least one of the operators should be unbounded. Can this be used here?

4) I know domains are important. But I can still find a vector in the intersection of the domains of A and B. Can you prove their domains are disjoint? Or can you prove there is no eigenvector of A in the intersection of the domains of A and B?

I think your argument proves exactly that there is no eigenvector in the intersection of the domains of A and B. As matterwave pointed out, something like $\langle a|B|a\rangle$ needs not be well-defined.

5) I know that here we should consider a rigged Hilbert space. But a rigged Hilbert space is actually a triplet like $D \subset L^2(\mathbb R,dx) \subset D'$. So I can still choose vectors from D which are actually ordinary nice vectors. Can you prove there is no eigenvector of A in D? Or can you prove for such an eigenvector, B necessarily gives ill-defined results so that $\langle a | B |a \rangle-\langle a | B |a \rangle$ is indeterminate instead of 0?

Rigged Hilbert spaces are not important here.

 

[ShayanJ]

http://en.wikipedia.org/wiki/Stone–von_Neumann_theorem

Yeah, I saw it before but its vague to me. Because I know about unitary equivalence in terms of representation theory of groups and there, its about the similarity transformation. But I don't know how can I perform a similarity transformation on something like $\frac{\hbar}{i} \frac{\partial}{\partial x}$!
Also...does it mean that the operators $\frac{\hbar}{i} \frac{\partial}{\partial \varphi}$ and $\varphi$ defined on the space $L^2([0,2\pi),d\varphi)$ are unitarily equivalent to momentum and position operator? In what sense?

I think your argument proves exactly that there is no eigenvector in the intersection of the domains of A and B. As matterwave pointed out, something like ⟨a|B|a⟩\langle a|B|a\rangle needs not be well-defined.

I knew it. I just wanted to get some understanding of this and see why it should be the case. I mean an ordinary mathematical way of understanding this.