/sci/ - Science & Math

No, both are in the physics department.

- A theoretical physicist generally builds theory that is tied to contemporary experiment (think theoreticans making models to make predictions for a particle accelerator, or how to design a quantum computation chip, or how to compose gas for this and that theomodynamcal behavior.)
- A mathematical physicist is more interested in the theories themselves. This often means putting working but mathematicall ill-defined theories on more solid ground or, when trying to embed stronger frameworks for models, reframing theories borrowing heavy formal machinery that a theoretical physicist wouldn't want to spend much time on (e.g. being very formal w.r.t. operator analysis involved). A mathematical physicist is generally more willing to spend a few years on theories that are well established and considered being long beyond having low hanging fruits.

>>11004336
I'd love to disagree, but I also never got warm with fuzzy math. If you look at library stacks, it's clear that there was a lot of hope around the 80's and 90's, but I suppose it's not much more than another way to look at things already treated otherwise. I'd love to be wrong on that too.
I don't know why modular arithmetic made it there. Just like BRST-Cohomology. As I said, it had a physicists bias.

>>11006614
I suppose there's also no such a nice and handy formula as the Taylor expansion for products.

At one point I also found that (unless something goes very wrong I guess), you have that

[math] \prod_{k=0}^\infty b_k = \prod_{k=0}^{M-1} b_k + \sum_{n=M}^\infty(b_n-1)\,\prod_{k=0}^{n-1}b_k [/math]

That's derived in video on the channel I linked above somewhere, but the clip is not PG13.

I do think there must be a point of speaking more often in terms of the Gamma functions product representation (in terms of rationals).
There's also the Weierstrass expression with the exp upfront and I couldn't get warm with it yet.

I'd be interested to see what your building up of algebra looks like if you e.g. use

[math] \dfrac{1}{1-x} = \prod_{n=0}^{\infty}\left(1 + x^{2^n}\right) [/math]

instead of

[math] \dfrac{1}{1-x} = \sum_{n=0}^{\infty} x^n [/math]

Also there's

[math] \lim_{N\to\infty} \prod_{n=0}^{N-1} \left(1+\frac{x_n}{N}\right)=
\exp\left(\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}x_n\right) [/math]

and you get to those expression a lot in QFT.
Similarly (more exotically),

[math] \lim_{\Delta x \to 0} \prod{f(x_i)^{\Delta x}} = \exp\left(\int_a^b \ln f(x) \,dx\right) [/math]

and other product integrals.

Now, to guess "why" more morally from an exotic (topos theoretic) standpoint, it's maybe worth pointing out that [math]\sum[math] resp. [math]\prod[math] (just like [math]\exists[math] and [math]\forall[math]) arise from...