Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 301 / 1 / i

Table of contents
- Solution i

Solution

Translations produce the conserved symmetric tensor

T^{μν} = \partial^{μ} ϕ \partial^{ν} ϕ - η^{μν} L, \partial_{μ} T^{μν} = 0.

The conserved charges are

E = \int_{R^{3}} d^{3} x T^{00} = \int d^{3} x {\frac{1}{2} \dot{ϕ}^{2} + \frac{1}{2} ∣\nabla ϕ ∣^{2} + V (ϕ)},

and

P^{i} = \int_{R^{3}} d^{3} x T^{0 i} .

Indeed,

\frac{d}{d t} \int d^{3} x T^{0 ν} = - \int d^{3} x \partial_{i} T^{i ν} = - lim_{R \to \infty} \int_{S_{R}} T^{i ν} n_{i} d S = 0

when the field and its derivatives decay sufficiently rapidly.

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