Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 2 / b

Table of contents
- Solution b

Solution

Matrix multiplication gives

[J_{i}, J_{j}] = ϵ_{ijk} J_{k}, [J_{i}, K_{j}] = ϵ_{ijk} K_{k}, [K_{i}, K_{j}] = - ϵ_{ijk} J_{k} .

For example,

[J_{1}, J_{2}] = J_{3}

,

[K_{1}, K_{2}] = - J_{3}

, and

[J_{1}, K_{2}] = K_{3}

. Over

C

A_{i} = \frac{1}{2} (J_{i} + i K_{i}), B_{i} = \frac{1}{2} (J_{i} - i K_{i}) .

Then

[A_{i}, A_{j}] = ϵ_{ijk} A_{k}

,

[B_{i}, B_{j}] = ϵ_{ijk} B_{k}

, and

[A_{i}, B_{j}] = 0

. This proves the chiral decomposition of the complex Lorentz algebra. Its finite-dimensional irreducible representations are tensor products of irreducible representations of the two factors and are labelled by pairs

(j_{L}, j_{R})

of nonnegative half-integers.

Solved by gpt-5.6-sol high.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version