Confusion surrounding Noether's Theorem

Question

As an example of Noether's Theorem, my QFT textbook gave the example of how the conservation of momentum and energy arises from symmetry in space-time translations. The book arrives at the conclusion that the Noether Current is the Energy-Momentum Tensor and that conserved charges from this current can be written as: $$P^{a}=\int_{ }^{ }d^{3}x\ \ T^{0a}$$ (Where $T$ is the energy momentum Tensor), My first Question is: How did they arrive at this conclusion, and why is the conserved quantity called a 'charge'? The book then demonstrates how Energy is conserved for when $a=0$, however when $a$, is non-zero, the book says that the conserved 'charge' is momentum, stating: $$P^{a}=\int_{ }^{ }d^{3}x\ \ T^{0a}=\int_{ }^{ }d^{3}x\ \ Π^{0}∂^{a}ϕ$$ Which is said to be the momentum. My second Question is: How is the above expression equal to the momentum?

I am not studying QFT professionally, so there may be some gaps in my knowledge!

Answer 1

The continuity equation for electric charge can be written $$\frac{\partial \rho}{\partial t} + \nabla \cdot \vec J = 0$$

where $\rho$ is the electric charge density and $\mathbf J$ is the electric current density. In integral form, this expresses the idea that if the amount of charge in some arbitrary volume changes, it is because some charge passed through the boundary; in other words, charge does not simply appear or disappear.

If we define the current 4-vector as $\mathbf J = (c\rho, \vec J)$ and the position coordinates $x^\mu = (ct,\vec r)$, this can be expressed as

$$\frac{\partial }{\partial x^\mu} J^\mu = 0$$

where the Einstein summation convention is used. In particular, note that $J^0$ is identified with the charge (density) and the spatial components $J^a$, $a=1,2,3$ are identified with the current (density). In some volume $V$, the total charge is given by $Q = \int_V \mathrm d^3 x\ J^0$.

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From Noether's theorem, we find that spacetime translation invariance yields $\frac{\partial}{\partial x^\mu} T^{\mu\nu} = 0$. That is, translation invariance in the $x^0$ direction (i.e. a time translation) yields the continuity equation $\frac{\partial}{\partial x^\mu} T^{\mu 0}= 0$, with total "charge" given by $\int \mathrm d^3x \ T^{00}$. Similarly, translation invariance in the $x^3$ direction yields $\frac{\partial}{\partial x^\mu} T^{\mu 3} = 0$ which has total charge $\int \mathrm d^3x \ T^{03}$.

Noether's theorem takes a continuous symmetry as an input and spits out a continuity equation of the form $\partial_\mu J^\mu = 0$, where $J^0$ is the conserved Noether charge density (and $\int \mathrm d^3 x \ J^0$ is the total charge). In this framework, the definition of energy is the Noether charge corresponding to time translations, while momentum is the Noether charge corresponding to spatial translations.