If Black holes slowly evaporate over time is there a point where they stop being a black hole?

[joigus]

1 hour ago, Mordred said:

[...] is the directional derivative taking the previous Hamilton statement under spectral decomposition.

see here in regards to Hermitean directional derivatives

Ok. I don't see how that can signify a directional derivative. I assume you mean a derivative with respect to a matrix when evaluated at a particular matrix value? I'm no one-trick pony, I'm very familiar with derivatives with respect to a matrix (something that only makes sense when the function to differentiate is diagonal in the chosen matrix variable, which yes, happens to be the case if you're differentiating with respect to the Hamiltonian itself, which is trivial, or any of the spectral projectors, which renders the corresponding eigenvalue times the projector). But what I'm absolutely sure of is that, provided the coefficients are finally evaluated as the corresponding numerical functions of the Hamiltonian eigenvalues \( \lambda_{j} \), their values can be no other than, \( e^{-it\lambda_{j}} \)

The calculation produces, 

\[ U=\sum_{j=1}^{d}e^{-it\lambda_{j}}\left|j\right\rangle \left\langle j\right| \]

(assuming \( \hbar=1 \).

This is in keeping with the more general result of spectral analysis (for certain kind of operators, compact, etc) that, provided a certain observable \( Q \) admits the spectral expansion,

\[ Q=\sum_{q\in\sigma\left(Q\right)}q\left|q\right\rangle \left\langle q\right| \]

then, for "any" (again, "good enough"=compact) \( f\left( Q \right) \) we must have,

\[ f\left(Q\right)=\sum_{q\in\sigma\left(Q\right)}f\left(q\right)\left|q\right\rangle \left\langle q\right| \]

So the least I can say is that the notation is unnecessarily confusing. If those exponentials have the meaning of certain matrix directional derivatives and are expressed as \( e^{\lambda_j t} \), while evaluated as numbers they are \( e^{-i\lambda_j t} \) (as they surely are by the calculation you suggest I'm more familiar with), then that's notational mayhem, IMO.

And I'm sorry this discussion is drifting farther and farther apart as per OP.

[MigL]

I wasn't disagreeing with you Mordred; just trying to relate unitarity to the BH information paradox, as it pertains to the OP, in terms Airbrush would understand, without taking an advanced course in QM.

[Mordred]

1 hour ago, MigL said:

I wasn't disagreeing with you Mordred; just trying to relate unitarity to the BH information paradox, as it pertains to the OP, in terms Airbrush would understand, without taking an advanced course in QM.

Noted I didn't think you were arguing with me though. I however did want to add detail beyond the rough and tumble earlier post which I couldn't do at work.

1 hour ago, joigus said:

Ok. I don't see how that can signify a directional derivative. I assume you mean a derivative with respect to a matrix when evaluated at a particular matrix value? I'm no one-trick pony, I'm very familiar with derivatives with respect to a matrix (something that only makes sense when the function to differentiate is diagonal in the chosen matrix variable, which yes, happens to be the case if you're differentiating with respect to the Hamiltonian itself, which is trivial, or any of the spectral projectors, which renders the corresponding eigenvalue times the projector). But what I'm absolutely sure of is that, provided the coefficients are finally evaluated as the corresponding numerical functions of the Hamiltonian eigenvalues λj , their values can be no other than, e−itλj

The calculation produces, 

 

U=∑j=1de−itλj|j〉〈j|

 

(assuming ℏ=1 .

This is in keeping with the more general result of spectral analysis (for certain kind of operators, compact, etc) that, provided a certain observable Q admits the spectral expansion,

 

Q=∑q∈σ(Q)q|q〉〈q|

 

then, for "any" (again, "good enough"=compact) f(Q) we must have,

 

f(Q)=∑q∈σ(Q)f(q)|q〉〈q|

 

So the least I can say is that the notation is unnecessarily confusing. If those exponentials have the meaning of certain matrix directional derivatives and are expressed as eλjt , while evaluated as numbers they are e−iλjt (as they surely are by the calculation you suggest I'm more familiar with), then that's notational mayhem, IMO.

And I'm sorry this discussion is drifting farther and farther apart as per OP.

yes what you have is correct I suspect the notational differences is from the usage of spectral decomposition. I don't know how familiar you are with spectral decompositions but in essence \(\lambda\) is the eugenvalue with orthonormal vectors eugenvectors U. So your recasting a symmetric d x d matrix M

\[u_i \cdot u_j=\epsilon _{ij}\]

the lambda term are diagonal under matrix \(\Lambda\) values \(\lambda_1, \lambda_2....\lambda_d\) you also have matrix Q where U_d is on columns  and matrix Q^T where the U_d are row  vectors.

\[M=\sum^d_{i=1}\lambda_iu_iu_i^T\]

where any U is linearly independent. For any i \(Q\Lambda Q^Tu_i=Q\Lambda=Mu_i\)

with U_I being orthonormal \(QTq=I\) thus Q is invertible so for any j

\[(\sum_i\lambda u_iu_i^T)u_j=\lambda_ju_j=Mu_j\]

so \[M=\sum_i\lambda_i u_iu_i^T\]

https://www.stat.ucdavis.edu/~xdgli/Xiaodong_Li_Teaching_files/135Note1.pdf

the notation of this article is a bit different but essentially the same relations I have in this post.

Hopefully though this will help however as we don't want to get too sidetracked from your last post I am in agreement that's its more a notational mayhem likely through the use of spectral index notations.

course it also doesn't help that even with spectral decompositions no two articles use the same nomenclature.

 

Edited August 16 by Mordred

[joigus]

1 hour ago, MigL said:

I wasn't disagreeing with you Mordred; just trying to relate unitarity to the BH information paradox, as it pertains to the OP, in terms Airbrush would understand, without taking an advanced course in QM.

Absolutely agree.

[Mordred]

5 minutes ago, joigus said:

Absolutely agree.

ditto this can easily get sidetracked.