Milestones and self-assessment

Finishing a page is different from being able to use it. A useful milestone leaves you with a calculation you can reproduce and an explanation of what its answer does—and does not—establish. Begin with the starting-knowledge questions if the ordinary field-theory tools need refreshing. The following method then works throughout the course.

Three questions for any result

What is being studied? State the spacetime dimension and signature, fields or states, symmetries, and quantity to be calculated. A supersymmetry label does not specify a gauge group, its rank, or a background. For a curved-space problem, say which geometric and background fields are held fixed.

Can you reproduce the result? Work through the consequential steps without copying the final formula. Fix the ordering, units, and normalizations before comparing answers. Include a check capable of detecting a mistake: a matrix product, a charge balance, a known limit, or an independent numerical evaluation with its precision stated.

What changes when an assumption changes? Alter one input and identify which steps survive. Separate a derived answer from a plausible conjecture. A successful calculation in one example does not establish a result for every theory with the same number of supercharges.

A worked self-assessment

Use the two-state example from What a supercharge does. There is one time dimension, no spatial spinor index, and an orthonormal pair of states: the even state $|b\rangle$ and odd state $|f\rangle$. In the ordered basis $(|b\rangle,|f\rangle)$, let

$$H=E I_2,\qquad \Gamma=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad Q=\sqrt{2E}\begin{pmatrix}0&0\\1&0\end{pmatrix}, \qquad E>0.$$

Here $I_2$ is the identity, $\Gamma$ is fermion parity, and the dagger is the Hilbert-space adjoint. This is a single positive-energy block, not a full oscillator spectrum or a relativistic field theory.

Attempt the check. For $|v\rangle=a|b\rangle+b|f\rangle$ with $|a|^2+|b|^2=1$, calculate

$$\|Qv\|^2+\|Q^\dagger v\|^2.$$

Then decide whether doubling $Q$ while keeping $H$ fixed preserves the algebra $\{Q,Q^\dagger\}=2H$.

Hint

First act on each basis state. For the changed operator, both factors in each product acquire the rescaling.

Complete answer and diagnosis

The actions are

$$Q|v\rangle=\sqrt{2E}\,a|f\rangle,\qquad Q^\dagger|v\rangle=\sqrt{2E}\,b|b\rangle.$$

Orthonormality gives

$$\|Qv\|^2+\|Q^\dagger v\|^2 =2E(|a|^2+|b|^2)=2E =2\langle v|H|v\rangle.$$

This checks the anticommutator on an arbitrary normalized vector, rather than only on one selected basis state. The matrices also give $Q^2=0$ and $\Gamma Q=-Q\Gamma$.

If $Q'=2Q$, then

$$\{Q',Q'^\dagger\}=4\{Q,Q^\dagger\}=8H.$$

Thus the original normalization fails with $H$ fixed. It is restored by changing the Hamiltonian to $H'=4H$. More generally, $Q'=cQ$ requires $H'=|c|^2H$ for a complex number $c$. A pure phase preserves $H$.

If you obtained $4H$ after doubling $Q$, you rescaled only one factor. If you concluded that every supersymmetric field theory contains only these two states, you extended the example beyond its assumptions. If you obtained $2E$ but cannot explain why the two norms are nonnegative, review the role of the Hilbert-space adjoint.

The first answer demonstrates a calculation. Identifying the dimension and the restricted energy block demonstrates that you know what was calculated. Predicting the rescaling demonstrates a controlled extension. All three matter.

Later milestones

Use the same questions as the objects become more elaborate:

Milestone	A concrete output	A claim that needs additional work
Component supersymmetry	Apply transformations with dimensions, statistics, and charges checked	Invariance of an action or closure on every field
A curved-space action	Specify the background, solve the required spinor equation, and verify the action’s variation	Preservation of the same symmetry on a different manifold
A proposed duality	Define both theories and compare an operator map, anomalies, or a deformation	Equivalence of all observables from one matching number
An exact observable	State the operator, normalization, parameters, and regime, then reproduce the result	A complete solution of the theory
A numerical experiment	Give the input, method, uncertainty, and an independent benchmark	An exact identity inferred only from finite-precision agreement

These are checks of understanding, not a requirement to master every branch before progressing. Your chosen route determines which milestone comes next.

Decide the next action

If the model is unclear, return to the definitions and write the missing assumptions. If the model is clear but the derivation fails, find the first unjustified step and do one smaller example. If the derivation works but the interpretation is too broad, change one assumption and locate the point where the argument stops applying.

Keep a short note containing the question, your worked answer, the check, and the remaining uncertainty. An uncertainty stated precisely is useful: “I lose the sign when moving the odd parameter past the fermion” points to Grassmann calculus; “I do not understand supersymmetry” does not yet identify a next calculation.