Help We have forgotten how to write math stuff

[Danger]

Start by turfing anything that I can't express on a manual typewriter.

 

[micromass]

I think everybody has to admit that physicists are often very informal with math. But regardless of that, it is rather amazing that they still get correct results by applying math that isn't really rigorous. Fair enough, they also get contradictions. But I still am in awe by the fact that rather informal math actually works. For example, the dirac delta function was clearly nonsense when they first used it, but they did get the right results. It's only later that mathematicians found out why.

I think that's the key point. Physicists do things that aren't always justified, but do get the right result. Mathematicians can use these things to develop new math (such as distributions). Without physicists, there would be much less advancements in mathematics.

What I want to say is that physicists and mathematicians shouldn't be throwing mud at each other. In fact, we should benifit from each other and work together.

 

[Fredrik]

Dirac notation is only useful if they also teach rigged Hilbert spaces. Without that, it's a pretty awful notation. When I read something in Dirac notation, then I always get confused. If I then read the same thing in ordinary math notation, then I understand it immediately.

Furthermore, I think that Dirac notation tends to obfuscate domain issues. So you're more prone to errors.

But sometimes it's really nice. Consider e.g. the proof that if $\rho$ is a projection operator for the 1-dimensional subspace spanned by a unit vector f (written as |f> when we use bra-ket notation), and A is self-adjoint, then $\operatorname{Tr}(\rho A)=\langle f,Af\rangle$.

"Ordinary math notation" (with the convention to have the inner product linear in the second variable):

\begin{align}
\operatorname{Tr}(\rho A) &=\sum_n\langle e_n,\rho A e_n\rangle =\sum_n\left\langle e_n,\langle f,Ae_n\rangle f\right\rangle =\sum_n\langle \langle f,Ae_n\rangle^* e_n,f\rangle =\sum_n\langle \langle Ae_n,f\rangle e_n,f\rangle\\
&=\sum_n\langle \langle e_n,Af\rangle e_n,f\rangle =\langle Af,f\rangle =\langle f,Af\rangle
\end{align}
Bra-ket notation:
\begin{align}
\operatorname{Tr}(\rho A) &=\sum_n\langle n|f\rangle\langle f|A|n\rangle =\sum_n\langle f|A|n\rangle\langle n|f\rangle=\langle f|A|f\rangle.
\end{align}

 

[strangerep]

What I want to say is that physicists and mathematicians shouldn't be throwing mud at each other. In fact, we should benifit from each other and work together.

My feelings exactly.

 

[micromass]

"Ordinary math notation" (with the convention to have the inner product linear in the second variable):

\begin{align}
\operatorname{Tr}(\rho A) &=\sum_n\langle e_n,\rho A e_n\rangle =\sum_n\left\langle e_n,\langle f,Ae_n\rangle f\right\rangle =\sum_n\langle \langle f,Ae_n\rangle^* e_n,f\rangle =\sum_n\langle \langle Ae_n,f\rangle e_n,f\rangle\\
&=\sum_n\langle \langle e_n,Af\rangle e_n,f\rangle =\langle Af,f\rangle =\langle f,Af\rangle
\end{align}

Or: expand $f$ to an orthonormal basis $(e_n)_n$. So $e_1=f$. Then
$$Tr(\rho A) = \sum_n \langle e_n,\rho A e_n\rangle = \sum \langle \rho e_n, A e_n\rangle = \langle f,Af\rangle$$

 

[Fredrik]

That's a cool trick. I still think that bra-ket notation makes it easier to see some of these things quickly.

 

[tfr000]

The only thing that needs to change is
$$\sin^2(x)$$

This needs to die in a fire

Or even worse... $$\sin^{-1}(x)$$.
The "logic" in going from one of these to the other... like wow, man. Not going to cause any confusion there...