Quantum dimensions, by hand
Quantum dimensions are the numbers that tell you how much an anyon counts. In a modular tensor category, every particle-like excitation carries one, and fusion multiplies them: if quantum dimensions are d1 and d2, fusing gives something whose dimension is d1 times d2. That multiplicative structure is why the total quantum dimension in the last post was a clean product. Here is how to compute an individual one by hand, and how the same computation once caught this programme publishing a wrong table.
- Also see: Quantum Entanglement in Exoplanet Detection
- Also see: Quantum Dimensions by Hand
The sine formula
For an SU(2) theory at level k, the anyons are labelled j = 0, one half, 1, up to k over 2, and the quantum dimension of anyon j is a ratio of sines. The formula: d(j) = sin of (2j+1) times pi over (k+2), divided by sin of pi over (k+2). Nothing in that needs a computer.
Take SU(2) at level 4, the piece that enters the 2592 calculation. Here k = 4, so the denominator is sin of pi over 6, which is one half. The anyons have j = 0, one half, 1, three halves, 2.
j = 0: numerator sin of pi over 6, which is one half. Over one half: d = 1. Trivial anyon, as expected. j = one half: numerator sin of 2 pi over 6 = sin of pi over 3 = root three over two. Over one half: d = root 3, about 1.732. j = 1: numerator sin of 3 pi over 6 = sin of pi over 2 = 1. Over one half: d = 2. j = three halves: by the symmetry of the sine, d = root 3 again. j = 2: sin of 5 pi over 6 = one half, over one half: d = 1.
Now the check that matters. The total quantum dimension squared is the sum of the squares of all the d’s. Squares: 1, 3, 4, 3, 1. Sum: 12. And SU(2) level 4 should contribute 12 to a total built as 12 times 36 times 6. It does. The sine formula and the product structure agree, which is exactly the kind of cross-check that is boring when it passes and priceless when it fails.
The table that summed wrong
An earlier version of this programme printed a quantum dimension table for the SU(3) piece that did not sum to 36. The failure was not exotic. The values had been carried through several documents as decimals, rounded independently, and nobody added the column. When the audit recomputed each entry from the SU(3) level-3 fusion rules, the values came out as integers, the sum landed on 36 exactly, and the corrected table is what now enters 2592.
The lesson is not that quantum dimensions are fragile. It is that a column of numbers nobody adds is not a calculation, it is a decoration. The sine formula here, and the fusion-rule recomputation in the audit, are both one step away from anyone with a calculator. Decorations do not survive that proximity; results do.
Why dimensions matter downstream
The quantum dimensions are not internal bookkeeping. The total quantum dimension sets the topological entropy, which sets how much information a boundary of the category can carry, which is the quantity the capacity argument uses. The individual dimensions set fusion multiplicities, which the anyon functor uses to model how branch states compose. If any d in the table is wrong, the errors propagate multiplicatively, which is why a sum wrong by a few per cent was never a few per cent problem.
So the discipline stands: every table in this archive gets summed, every sum gets traced to a formula, and every formula gets checked against an independent route. The sine formula against the fusion rules is the model for what that means. One number, two derivations, and a sum that has to close.