The Categorical Imperative: How to Become a Theoretical‑Physics Fugitive (Legally Emigrate, Get Funded, and Still Sleep)

# The Categorical Imperative: Roadmaps, Logic, and the Art of Running Away to Physics

So you want to do theoretical physics. Translation: you enjoy abstract wars with differential operators and prefer beautiful equations to predictable paychecks. The handbook you were handed above is a damned useful checklist. Here, I want to fold that checklist into a little mathematical and logical thinking that makes the choices less arbitrary and more like strategies in a solvable game.

Think of this as career advice with theorems. Well, not actual theorems, but lemmas you can apply between coffee breaks.

## 1. Five years as an algorithmic plan
The original five year plan reads like a dependency graph. Make that explicit.

– Nodes: courses, research experiences, software skills, tests, fund applications.
– Edges: prerequisites and opportunity costs.

This is project management, but the math helps. Consider partial orders: some things must come first, others commute. Real analysis unlocks rigorous quantum mechanics; group theory unlocks particle symmetries; Python unlocks everything that looks like a data problem. But don’t fetishize topological purity while your CV lacks a letter that says you finish projects. Think of breadth versus depth as an optimization over a budgeted resource, namely time. Use greedy algorithms early (get the core courses), then switch to dynamic programming (choose research that gives maximal future options).

Tradeoff lens: specialization buys depth and a clearer supervisor fit; breadth buys resilience in funding markets. Both are rational in different value functions.

## 2. Asking smartly is applied logic
Online forums are not magic, they are oracles with limited budgets.

Treat each post as a logical query. The probability of an answer is proportional to signal to noise. High signal means: clear statement, what you tried, and small, reproducible examples. This is Bayesian in spirit: prior expertise of the forum plus quality of the evidence yields posterior utility.

From a formal perspective, this is abductive reasoning: you present data and invite the best explanation. The better your prior work and the cleaner your post, the more you get back. If nobody answers, update your priors and reformulate. In practice: fewer idle rants, more compact LaTeX and code snippets.

## 3. QFT without existential dread via structural thinking
QFT can feel like herding cats in infinite dimensions. A categorical perspective calms the rabble.

– Fields as objects in a category, with local operators as morphisms. Quantization becomes a functor from classical structures to quantum algebras.
– Spinors and representations? Representation theory is just a language for symmetry. If you think of irreps as atoms of transformation, the Dirac equation is group theory plus a minimal dynamics constraint.

This is not handwaving. Category theory and functorial thinking clarify what is invariant and what is gauge. That helps you prioritize which technicalities you need to master now and which you can defer. You do not need to become a homological algebraist to be useful, but knowing that cohomology measures obstructions will make you less flustered when something seems impossible.

Philosophically: embrace both operator algebra rigor and physicist pragmatism. They are dual descriptions, like Schrödinger versus Heisenberg pictures. Use whichever representation simplifies the computation at hand.

## 4. Machine learning as applied logic and computational algebra
ML is not a magical talisman for tenure, but it is a sharpened instrument. From a mathematical viewpoint:

– ML models are function approximators. Think in terms of approximation theory and spectral bias. Know what you are approximating and why.
– Symmetry matters. Embed invariances into architectures rather than hoping the network will ‘learn’ them; this is the physics equivalent of a good ansatz.
– Use ML as a numerical accelerator for lattice calculations, PDE solvers, and surrogate models. The best papers are the ones where physics supplies inductive bias and ML supplies computational horsepower.

In logic terms, ML amplifies inductive reasoning. But remember: extrapolation is the devil. Your model is only as honest as your validation scheme.

## 5. Applying, getting fundable, and the little utility theory of letters
Admissions and funding decisions feel capricious because committees optimize different utility functions.

Model them. Some groups maximize research fit, others maximize throughput, others play political games. Your job is to be in the set of candidates that dominates others under most plausible committee utilities. Practically: strong letters, clean statements referencing specific faculty, and demonstrable project experience. Those are robust features.

Cultivate recommenders who can speak to your intellectual independence. A bland checklist letter is dominated by a shorter, enthusiastic letter that contains concrete examples. Think of letters as votes weighted by credibility; increase your weighted vote.

## Both sides of the coin
There are two honest debates here.

– Theory purity versus marketability. Become a pure mathematician and we will respect you; become a pragmatic computational physicist and you may be funded. Both paths are defensible. The better play is to have a core of rigorous knowledge and a portfolio of usable skills.
– Breadth versus depth. Breadth buys slack in a volatile job market; depth gets you to the hardest problems. Early on, aim for breadth to discover where your curiosity really sticks; then double down.

Both positions are rational under different priors and different risk tolerances.

## Final thought
The original checklist is excellent because it is an implicit algorithm plus heuristics. Making the heuristics explicit with a little math and logic turns anxious choices into informed gambles. You will still hit walls and ugly eigenvalues; you will still stay up all night debugging a proof or a simulation. But when things get ugly, remember you are performing a sequence of decisions under uncertainty. Know your utility function, accept irreducible risk, and stack the probabilities in your favor.

I will leave you with a question that keeps me awake in the best way: when you choose between elegance and fundability, which of the two do you want to be remembered for, and how will that choice shape the theorems you love to prove?