Noether's theorem: Difference between revisions

Revision as of 14:59, 12 May 2025

Revision as of 13:59, 23 May 2025

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which expresses the idea that the amount of a conserved quantity within a sphere cannot change unless some of it flows out of the sphere. For example, [[electric charge]] is conserved; the amount of charge within a sphere cannot change unless some of the charge leaves the sphere.

==== Examples ====

'''I. The [[stress–energy tensor]]'''

For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above; in other words, <math>L \left(\boldsymbol\varphi, \partial_\mu{\boldsymbol\varphi}, x^\mu \right)</math> is constant in its third argument. In that case, ''N''&nbsp;=&nbsp;4, one for each dimension of space and time. An infinitesimal translation in space, <math>x^\mu \mapsto x^\mu + \varepsilon_r \delta^\mu_r</math> (with <math>\delta</math> denoting the [[Kronecker delta]]), affects the fields as <math>\varphi(x^\mu) \mapsto \varphi\left(x^\mu - \varepsilon_r \delta^\mu_r\right)</math>: that is, relabelling the coordinates is equivalent to leaving the coordinates in place while translating the field itself, which in turn is equivalent to transforming the field by replacing its value at each point <math>x^\mu</math> with the value at the point <math>x^\mu - \varepsilon X^\mu</math> "behind" it which would be mapped onto <math>x^\mu</math> by the infinitesimal displacement under consideration. Since this is infinitesimal, we may write this transformation as

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:<math>\Lambda^\mu_r = -\delta^\mu_r \mathcal{L}</math>

and thus Noether's theorem corresponds<ref name="Goldstein1980" />{{rp|592}} to the conservation law for the [[stress–energy tensor]] ''T''_''μ''^''ν'', where we have used <math>\mu</math> in place of <math>r</math>. To wit, by using the expression given earlier, and collecting the four conserved currents (one for each <math>\mu</math>) into a tensor <math>T</math>, Noether's theorem gives

:<math>

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(we relabelled <math>\mu</math> as <math>\sigma</math> at an intermediate step to avoid conflict). (However, the <math>T</math> obtained in this way may differ from the symmetric tensor used as the source term in general relativity; see [[Stress–energy tensor#Canonical stress.E2.80.93energy tensor|Canonical stress–energy tensor]].)

'''I. The [[electric charge]]'''

The conservation of [[electric charge]], by contrast, can be derived by considering ''Ψ'' linear in the fields ''φ'' rather than in the derivatives.<ref name="Goldstein1980"/>{{rp|593–594}} In [[quantum mechanics]], the [[probability amplitude]] ''ψ''('''x''') of finding a particle at a point '''x''' is a complex field ''φ'', because it ascribes a [[complex number]] to every point in space and time. The probability amplitude itself is physically unmeasurable; only the probability ''p'' = |''ψ''|² can be inferred from a set of measurements. Therefore, the system is invariant under transformations of the ''ψ'' field and its [[complex conjugate]] field ''ψ''^* that leave |''ψ''|² unchanged, such as