• White Kochenspek physics T-shirt, front view (flat-lay mockup), view 1

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Kochenspek

The Kochen–Specker Theorem Physics

\[\text{18 vectors, 9 orthogonal bases, each vector in 2 bases (4 dimensions)}\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

Kochenspek wears a crown of vectors and tries to paint each one 0 or 1. Every time, one basis is left without a consistent answer.

The Physics

Can every quantum measurement have a pre-set answer, waiting to be read, no matter which other compatible measurements are made alongside it? Kochen and Specker proved in 1967 that it cannot. In a Hilbert space of dimension 3 or more, you cannot assign 0 or 1 to every direction so that each orthogonal basis gets exactly one 1. Their original proof used 117 directions in three dimensions.

Cabello, Estebaranz and García-Alcaine gave a short proof in four dimensions: 18 vectors arranged in 9 orthogonal bases, with each vector in exactly 2 bases. A parity count finishes it.

9 bases × 1 one each = 9 (odd) but each vector counted twice ⇒ even

  • Basis: a set of 4 mutually orthogonal vectors in 4 dimensions
  • 9 bases × 4 vectors = 36 = 18 vectors × 2

Kochen & Specker, J. Math. Mech. 17, 59 (1967); Cabello, Estebaranz & García-Alcaine, Phys. Lett. A 212, 183 (1996).

Honest caveat: The theorem rules out non-contextual hidden variables, not all hidden variables: contextual theories such as Bohmian mechanics survive. It is not Bell's theorem and makes no locality assumption. The mathematics is settled; what it means for interpretations is still debated.

On the shirt

Front: Kochenspek, the pixel-art character, with “NO HIDDEN LABELS”. Back: a schematic diagram of 9 bases and 18 vectors (an illustration of the counting, not Cabello's actual vectors), the parity argument and the 1967 and 1996 sources. Printed on a unisex tee.

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Para ter uma escala: o tamanho M tem 20 in (50.80 cm) de largura, cerca de 3.143 × 1034 comprimentos de Planck (lP = 1.616255 × 10−35 m). Uma fita métrica é mais rápida.

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

I wear a crown of vectors and try to paint each one 0 or 1. Every time, one basis is left without a consistent answer.

Can every quantum measurement have a pre-set answer, waiting to be read, whatever compatible measurements are made alongside it? Simon Kochen and Ernst Specker proved in 1967 that it cannot: in three or more dimensions you cannot give every direction a 0 or 1 so that each orthogonal basis gets exactly one 1. Their proof used 117 directions in three dimensions. In 1996 Cabello, Estebaranz and García-Alcaine found a short proof in four: 18 vectors in 9 bases, each vector in exactly two bases. The 9 bases need 9 ones, an odd number, but counting each vector twice always gives an even number. No labelling works. Answers depend on context.

It rules out non-contextual hidden variables, not all hidden variables; contextual theories such as Bohmian mechanics survive. It is not Bell's theorem and assumes nothing about locality. The mathematics is settled; what it means is still debated.

Equations

\[\text{18 vectors, 9 orthogonal bases, each vector in 2 bases (4 dimensions)}\]
\[9 \times 1 = 9 \ (\text{odd}) \quad \text{vs} \quad 2 \times (\text{number of vectors labelled } 1) \ (\text{even})\]
SymbolMeaningUnit
\(\text{basis}\) a set of 4 mutually orthogonal vectors in 4 dimensions \(\mathrm{1}\)
\(0,\ 1\) the answer assigned to each vector (exactly one 1 per basis) \(\mathrm{1}\)

Sources

  1. Kochen & Specker (1967) The Problem of Hidden Variables in Quantum Mechanics, J. Math. Mech. 17, 59
  2. Cabello, Estebaranz & García-Alcaine (1996) Bell-Kochen-Specker theorem: A proof with 18 vectors, Phys. Lett. A 212, 183

Field notes

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