How to investigate

Begin with the phenomenon, then choose the explanation target

Before choosing a theory, state what needs explaining. Is it an observed correlation, a measured spectrum, an unexplained parameter value, a missing construction theorem, or a conflict between ways of interpreting the same equations? These are different kinds of problems. A successful phenomenological model can fit data without identifying a unique microscopic mechanism. A rigorous theorem can settle a model property without selecting a physical realization.

Write one sentence about the phenomenon and a second about the desired output. For example: “A closed spin chain seems to lose memory in its central magnetization. I want to determine whether its late-time value is fixed by energy alone for this family of initial states.” That is substantially more answerable than “Why does time have a direction?” It still illuminates part of the larger problem.

Keep the status of each claim visible

A claim can carry more than one description. An exactly solved toy model may support only a heuristic proposal about a material. A numerical result can be highly reliable within its discretized setting while leaving a mathematical continuum-existence question open.

Use the three question types correctly

Calibration questions reproduce established results. Their value is to make the quantities and factors your own. A correct answer should recover a benchmark and explain it without relying on a slogan.

Diagnostic questions test an implication inside a chosen model or compare two specified descriptions. Many have known answers in the literature. The goal is to decide whether the mechanism or inference you want to use is sound in its intended regime.

Extension questions propose bounded further work. Their phrasing does not certify novelty. Before treating one as research, identify the closest existing calculation, the precise change, and the likely obstacle. If the exact extension is already solved, reproduce the result and refine the next question rather than treating that discovery as wasted work.

The minimum investigation record

Keep the following information together, in connected prose or the notebook. The detail should scale with the calculation.

1. Question and model. State the Hamiltonian, equations, state, geometry, and observable. Specify which physical feature the model preserves and which it discards.
2. Expected output. Name the derivation, inequality, spectrum, parameter dependence, counterexample, or comparison you will produce.
3. Control. Identify the exact limit, small parameter, numerical convergence route, or published benchmark. Record how the approximation can fail.
4. Discriminating check. State an outcome that would weaken the proposed mechanism or expose a mistaken assumption.
5. Conclusion and next question. Say what was established, what remains untested, and which next calculation is now justified.

This record should also say which quantities were chosen to fit an observation. A prediction made after fixing parameters from other information carries different force from a quantity used in the fit.

An example of a completed learning question

For QM-2, prepare the Bell state $(|00\rangle+|11\rangle)/\sqrt2$ and compare it with the equal mixture of $|00\rangle$ and $|11\rangle$. Both one-qubit reduced states are $I/2$. Consequently every one-qubit observable agrees. However, $\langle X\otimes X\rangle$ is one for the pure Bell state and zero for the mixture.

The result is exact. It proves that identical local states need not describe the same global preparation. It does not select a measurement interpretation. A useful next question is which joint measurement remains available after some environment degrees of freedom become inaccessible. The model, observable, check, limitation, and next step are all explicit.

Common failures that can be caught early

A toy model reproduces a curve. Remove or vary the proposed causal ingredient and see what changes. Then ask whether another mechanism gives the same curve. Agreement demonstrates possibility; discrimination requires a comparison.

A numerical pattern looks asymptotic. Change the resolution, system size, boundary conditions, or time window separately. A finite calculation should not quietly acquire an infinite-volume or infinite-time conclusion.

A new degree of freedom fits everything. Count the newly adjustable parameters and identify an observable not used in fitting. If no such output is available, describe the result as a parametrization rather than an explanatory success.

An entropy or bound is large. Identify the operational task and its resources. A mathematical ceiling need not be reachable with the allowed states, energies, or measurements.

A formal identity seems to settle a physical question. Check its hypotheses in the physical system. For example, a current-relaxation formula derived with explicit momentum loss cannot be applied unchanged to a clean momentum-conserving continuum model.

How to read a source for a question

Read the abstract to locate the claimed result, then the setup and the equations actually carrying it. Find the approximation or limiting argument, the closest benchmark, and any discussion of failure. Read the original data or derivation before borrowing a strong conclusion from a review. Reviews remain valuable maps, particularly when entering a new field.

For a recent result, record its release date, version, and whether it is a preprint or a published analysis. A statistical preference is conditional on a comparison, dataset, and treatment of uncertainty. A quoted significance is not the probability that a named mechanism is true.

What to do after a negative answer

A negative result can rule out a proposed implication, reveal that a toy reduction discards the necessary physics, or show that two mechanisms cannot be distinguished with the chosen observable. Each gives a useful next question. Preserve the calculation and the failed claim explicitly. Changing the question is part of the investigation; silently changing what the earlier result was supposed to prove is not.