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In physics a conserved current is a current, , that satisfies the continuity equation =.The continuity equation represents a conservation law, hence the name. Indeed, integrating the continuity equation over a volume , large enough to have no net currents through its surface, leads to the conservation law =, where = is the conserved quantity.
The formal proof of the theorem utilizes the condition of invariance to derive an expression for a current associated with a conserved physical quantity. In modern terminology, the conserved quantity is called the Noether charge, while the flow carrying that charge is called the Noether current.
According to Noether's theorem, each symmetry of a system is associated a conserved quantity. [1] [2] For example, the rotational invariance of a system implies the conservation of its angular momentum, or spacetime invariance implies the conservation of energy–momentum. In quantum field theory, internal symmetries also result in conserved ...
The term on the right is the divergence of the current density J at the same point. The equation equates these two factors, which says that the only way for the charge density at a point to change is for a current of charge to flow into or out of the point. This statement is equivalent to a conservation of four-current.
In the one-dimensional space a conservation equation is a first-order quasilinear hyperbolic equation that can be put into the advection form: + = where the dependent variable y(x,t) is called the density of the conserved (scalar) quantity, and a(y) is called the current coefficient, usually corresponding to the partial derivative in the ...
The structure which gives rise to the Goldberger–Treiman relation was called the partially conserved axial current (PCAC) hypothesis, spelled out in the pioneering σ-model paper. [5] Partially conserved describes the modification of a spontaneously-broken symmetry current by an explicit breaking correction preventing its conservation.
Weinberg and Witten proved two separate results. According to them, the first is due to Sidney Coleman, who did not publish it: . A 3 + 1D QFT (quantum field theory) with a conserved 4-vector current (see four-current) which is Poincaré covariant (and gauge invariant if there happens to be any gauge symmetry which hasn't been gauge-fixed) does not admit massless particles with helicity |h ...
In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is the rate of flow of this fluid.