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Inphysics aconserved current is acurrent,${\displaystyle j^{\mu }}$, that satisfies thecontinuity equation${\displaystyle {\mathrm{\partial }}_{\mu }{j}^{\mu }=0}$. The continuity equation represents a conservation law, hence the name.

Indeed, integrating the continuity equation over a volume${\displaystyle V}$, large enough to have no net currents through its surface, leads to the conservation law$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}Q=0,}$$where$Q={\int }_V{j}^{0}dV$ is theconserved quantity.

Ingauge theories the gauge fields couple to conserved currents. For example, theelectromagnetic field couples to theconserved electric current.

Conserved current is the flow of thecanonical conjugate of a quantity possessing acontinuoustranslational symmetry. Thecontinuity equation for the conserved current is a statement of aconservation law. Examples of canonical conjugate quantities are:

• Time andenergy - the continuous translational symmetry of time implies theconservation of energy
• Space andmomentum - the continuous translational symmetry of space implies theconservation of momentum
• Space andangular momentum - the continuousrotational symmetry of space implies theconservation of angular momentum
• Wave functionphase andelectric charge - the continuous phase angle symmetry of the wave function implies theconservation of electric charge

Conserved currents play an extremely important role intheoretical physics, becauseNoether's theorem connects the existence of a conserved current to the existence of asymmetry of some quantity in the system under study. In practical terms, all conserved currents are theNoether currents, as the existence of a conserved current implies the existence of a symmetry. Conserved currents play an important role in the theory ofpartial differential equations, as the existence of a conserved current points to the existence ofconstants of motion, which are required to define afoliation and thus anintegrable system. The conservation law is expressed as the vanishing of a 4-divergence, where the Noethercharge forms the zeroth component of the4-current.

Theconservation of charge, for example, in the notation ofMaxwell's equations,$${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \mathbf{J}=0}$$

where

• ρ is thefree electric charge density (in units of C/m³)
• J is thecurrent density$${\displaystyle \mathbf{J}=\rho \mathbf{v}}$$ withv as the velocity of the charges.

The equation would apply equally to masses (or other conserved quantities), where the wordmass is substituted for the wordselectric charge above.

The Klein-Gordon Lagrangian density$${\displaystyle \mathcal{L}={\mathrm{\partial }}_{\mu }{\varphi }^{\ast }{\mathrm{\partial }}^{\mu }\varphi +V({\varphi }^{\ast }\varphi )}$$of a complex scalar field$\varphi :{\mathbb{R}}^{n+1}\mapsto \mathbb{C}$ is invariant under the symmetry transformation$${\displaystyle \varphi \mapsto {\varphi }^{\prime }=\varphi e^{i\alpha }.}$$ Defining$\delta \varphi ={\varphi }^{\prime }-\varphi$ we find the Noether current$${\displaystyle j^{\mu }:=\frac{d\mathcal{L}}{d\dot{\mathbf{q}}}\cdot {\mathbf{Q}}_r=\frac{d\mathcal{L}}{d({\mathrm{\partial }}_{\mu })\varphi }\frac{d(\delta \varphi )}{d\alpha }{|}_{\alpha =0}+\frac{d\mathcal{L}}{d({\mathrm{\partial }}_{\mu }){\varphi }^{\ast }}\frac{d(\delta {\varphi }^{\ast })}{d\alpha }{|}_{\alpha =0}=i\varphi ({\mathrm{\partial }}^{\mu }{\varphi }^{\ast })-i{\varphi }^{\ast }({\mathrm{\partial }}^{\mu }\varphi )}$$which satisfies the continuity equation. Here${\displaystyle {\mathbf{Q}}_r}$ is the generator of the symmetry, which is${\displaystyle \frac{d(\delta \mathbf{q})}{d{\alpha }_r}}$ in the case of a single parameter${\displaystyle \alpha }$.

• Conservation law (physics)
• Noether's theorem

• Goldstein, Herbert (1980).Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. pp. 588–596.ISBN 0-201-02918-9.

• David J Griffiths (1999).Introduction to electrodynamics (Third ed.). Prentice Hall. pp. 356–357.ISBN 978-0-13-805326-0.

• Peskin, Michael E.; Schroeder, Daniel V. (1995). "Chapter I.2.2. Elements of Classical Field Theory".An Introduction to Quantum Field Theory. CRC Press.ISBN 978-0-201-50397-5.

• Electromagnetism
• Theoretical physics
• Conservation equations
• Symmetry

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