What is the physical intuition of Noether current? [closed]

Question

What is the intuition behind Noether current $$J^{\mu}_N={\Pi}^{\mu}D{\phi}-W^{\mu}$$ where

• $J^{\mu}_N$ is the Noether current.
• $D\phi$ is the change of the field $\phi$ with respect to some parameter $\lambda$.
• ${\Pi}^{\mu}$ is the generalized conjugate momentum.
• $W^{\mu}$ is some arbitrary funtion of spacetime.

Answer 1

In case of translational symmetries of spacetime, the Noether charges $\int T^{00}d^3x$ and $T^{0i}d^3x$ represent the Hamiltonian and the components of the momentum of the field, respectively. Here, the stress-energy tensor $T^{\mu\nu}$ is the Noether current. For rotational symmetries, it is the angular momentum of the field which is the Noether's charge.

For internal symmetries such as baryon and lepton number in the Standard Model, you can write down a baryon and lepton number current which is classically conserved. To understand the status of Noether's theorem, and Noether's charge for local symmetries see this.