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

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- Solution a

Solution

Differentiating

Λ^{T} η Λ = η

at the identity gives

X^{T} η + η X = 0

. The six matrices

(M^{μν})^{α}_{β} = η^{μα} δ_{β}^{ν} - η^{να} δ_{β}^{μ}, M^{μν} = - M^{νμ},

obey this condition and form a basis of the Lorentz algebra. With cyclic indices,

J_{1} = M^{23}, J_{2} = M^{31}, J_{3} = M^{12}, K_{i} = M^{0 i} .

The

J_{i}

mix the two spatial coordinates perpendicular to

x^{i}

and therefore generate rotations about that axis;

K_{i}

mixes

x^{0}

with

x^{i}

and generates a Lorentz boost in the

x^{i}

direction.

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