Perspective

 About $\blacktriangleright$ Perspective $\blacktriangleright$ On reading notes on physical theories . note On physical units . note

Note

(This is a perspective on mathematical content as it currently develops in my head, shaped by the stuff I learn.)

Three major chunks of “doing physics” are learning, computing and modeling. Of course, whatever you're focusing on at the moment, the other two are usually part of it too.

The point that I mainly want to comment on here is computing.

1. Learning

As I experience it, learning is reading until an insight is achieved, and then cross-checking that new perspective with a whole lot of other things. Sometimes (and I think that's the addictive part) understanding one thing enables one to break off a chain of aha!'s regarding items one previously failed to understand.

The section On Reading below is also on Learning.

2. Computing

todo: try to remember and then draw a diagram of the definition/problem-scheme I drew

In practice, doing a physics calculation quite often means you're dropped somewhere in the following chain of tasks:

• You must solve a differential equation.
• This amounts to computing an $A$ valued integral over a space $X$.
• This amounts to computing an infinite sum.
• This amounts to computing the limit of a sequence.
• This amounts to computing a function $f:\omega\to A$ at large values and with reference to a metric $d$ on $A$.

Of course, you're not necessarily dropped at the top of these bullet points: A limit can arise from something else than an infinite sum, and an infinite sum can arise from something else than a (Riemann) integral.

And you don't have to work down that list either: Instead of attempting to solve a differential equation by the corresponding analytic integral, the solution might also be rewritten as a recursively quantity. Or the integral might be computed not as a limit but via a Monte-Carlo method.

Furthermore, knowledge of mathematical theorems also may let you just skip items along the list - for example, you know the solution of $h'(t)=3\,h(t)$ without doing an integral. For a similar purpose, what many theorems in algebraic theories do is to collapse expressions. It's inefficient (sad, even) if a programmer let's a computer do the calculation $\sum_{k=0}^n\frac{n!}{k!(n-k)!}$ instead of $2^n$ in a subroutine of an application. Or similarly, when he computes $\sum_{k=0}^7\frac{7!}{k!(7-k)!}(-1)^k$, not knowing it's zero. Clearly, it's valuable know that generally $\sum_{k=0}^n\frac{n!}{k!(n-k)!}a^k b^{n-k} = (a+b)^n$ and to know generalizations of that (e.g. $a,b$ may denote commutative matrices, or elements of a more general ring, or we might replace the sum by some other function and add a correction term, and so on). Similarly, $\sum_{n=0}^\infty x^n$ should, when it appears, be translate to $\dfrac{1}{1-x}$, which is simple to compute for any $x$. Indeed, the Point here is to choose the normal forms w.r.t. computational complexity. A more elaborate translation would be ergodic theorems, which tell you that particular infinite sums (time averages following paths of your dynamics) equals an integral (state space average).

In any case, it's clear that being able to solve the problems above in a suitable way is mandatory for doing physics and so the concepts in that list are a focal point in what follows. Given the top down hierarchy of this wiki, our emphasis is on the bottom of the list and goes up.

Remark: It's interesting how an expression $\sum_{n=0}^\infty \frac{1}{n!}x^n=:\exp(x)$ (similar to $\sum_{n=0}^\infty x^n$) doesn't permit a normal form that drastically reduced the complexity of it' evaluation. However in a context with an operator $\frac{{\mathrm d}}{{\mathrm d}x}$ it's an expression that fulfills an algebraic purpose in $\frac{{\mathrm d}}{{\mathrm d}x}\exp(A\,x)=A\exp(A\,x)$. Here $\exp$ instead takes the role of something like an $x$-indexed basis. However, if $x=\ln(y)\sim\exp^{-1}(y)$, the expression $\exp(x)$ can indeed by evaluated in a very simple way.

3. Modeling

In most cases, one uses the math to set up a representation of some notion of physical space $X$, some notion of physical quantity $A$ and some form of dynamics $F$ posing a task. Modeling is mostly trial and error, really.

Knowledge of and experience with existing mathematical structures, other models and computational tools, as well as the courage to come up with new ones are important here. That's true, even when the process “merely” consists of coming up with good approximations of existing models.

PS I think probability theory proper should be discouraged. E.g. the quantum state and density operator framework, especially with it's non-commutative observables and measurements, speaks of “propability” but doesn't fit into the frameowork so well. 