Noether's Theorem

When discussing spacetime symmetry, we fix the field \(\phi \) such that it is unaffected by variations (\(\Psi=0\)). We are left with: $$\partial _\mu \left[\left({\partial \mathcal{L}\over\partial \left(\partial _\mu \phi\right) } \partial _\alpha \phi - \delta ^\mu _\alpha L \right) Χ^\alpha \right]=0$$ We define the energy-momentum tensor \(T^{\mu\nu}\): $$T^{\mu\nu}\equiv {{\partial L}\over{\partial \left( \partial _\nu \phi \right)}} \partial ^\mu \phi -\eta ^{\mu\nu} L$$ The conservation current \(\partial _\nu T^{\mu\nu}=0\) represents: $$\partial _\nu T^{0\nu}=0 \text{ Energy conservation}$$ $$\partial _\nu T^{i\nu}=0 \text{ Momentum conservation}$$ In relativity, the energy-momentum tensor for a free particle is: $$T^{\mu\nu}\equiv mU^\mu U^\nu$$ For a classical particle, returning to non-relativistic terms, the trajectory \(q\) only depends on \(t\): $$T^{\mu\nu}=T^{00}={\partial L\over \partial \dot{q} } \dot{q} -L=H$$ \(T^{00}\) is the Hamiltonian, representing energy conservation: $$\partial _\nu T^{0\nu}=\partial _0 T^{00}={dH \over dt}=0$$ The energy does not change with time!

Classical Noether theorem can be stated as:

$$Classical Non-relativistic$$	$$Relativistic$$
$$S=\int Ldt $$	$$S=\int \mathcal{L}d^4 x$$
$$N\equiv {\partial L\over\partial \dot{q} _i } Q_i-\left({\partial L\over\partial \dot{q} _i } \dot{q} _i-L\right)T$$	$$N^\mu \equiv {\partial \mathcal{L}\over\partial (\partial _\mu \alpha^\nu ) } \Psi^\nu-\left({\partial \mathcal{L}\over\partial (\partial _\mu \alpha^\nu ) } \partial _\alpha \alpha^\nu- \delta ^\mu _\alpha L\right) Χ^\alpha $$

In classical mechanics, time \(t\) plays the role corresponding to spacetime \(x^\mu\), so time symmetry represents energy conservation. Other symmetries \(Q_i\) can represent spatial \(q_i\) or angular \(\theta_i\) coordinates, with: $${\partial L\over\partial \dot{q} _i } \text{ Momentum conservation}$$ $${\partial L\over \partial \dot{\theta}_i } \text{ Angular momentum conservation}$$

Originally written in Chinese by the author, these articles are translated into English to invite cross-language resonance.