Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The four-current of a particle moving along a worldine $X^\nu(s)$ is defined as $$j^\mu(x^\nu) = ec \int u^\mu(s)\, \delta^4(x^\nu - X^\nu(s)) \, ds$$

So here's my proof that this is conserved: \begin{align} \partial_\mu j^\mu&= ec\partial_\mu \int u^\mu(s) \,\delta^4(x^\nu - X^\nu(s)) \, ds \\ &= ec \int u^\mu(s) \,\partial_\mu \delta^4(x^\nu - X^\nu(s)) \, ds \\ &= ec \int \frac{\partial x^\mu}{\partial s} \frac{\partial s}{\partial x^\mu}\frac{\partial (\delta^4(x^\nu - X^\nu(s)))}{\partial s} \, ds \\ &= ec \int \frac{\partial}{\partial s} \delta^4(x^\nu - X^\nu(s)) \, ds \\ &= ec \bigg|_{-\infty}^\infty \delta^4(x^\nu - X^\nu(s)) \\ &= 0 \end{align} Each step sort of makes sense but working with delta functions makes me feel deeply uneasy and I'm not sure whether this is "valid" by the physicist's standard of rigour or just some hack that appears to give the correct answer.

share|cite|improve this question
Possible duplicate: – Qmechanic Feb 16 '14 at 10:27
up vote 2 down vote accepted

A "kosher" way to do this employs test functions. Consider a test function $\phi:\mathbb R^4\to \mathbb R$. Notice that \begin{align} \int_{\mathbb R^4} d^4 x\, \partial_\mu j^\mu(x) \phi(x) &= ec\int_{-\infty}^{\infty} ds\,u^\mu(s)\int_{\mathbb R^4} d^4x\,\partial_\mu\delta^4(x - X(s))\phi(x) \\ &= -ec\int_{-\infty}^{\infty} ds\, u^\mu(s)\int_{\mathbb R^4} d^4 x\,\delta^4(x-X(s))\partial_\mu\phi(x) \\ &= -ec \int_{-\infty}^{\infty} ds\, u^\mu(s)\partial_\mu\phi(X(s)) \\ &= -ec\int_{-\infty}^{\infty} ds \frac{d}{ds}\phi(X(s)) \\ &= -ec \left[\lim_{s\to\infty}\phi(X(s))-\lim_{s\to-\infty}\phi(X(s))\right] \end{align} Now, if we assume that \begin{align} \lim_{s\to\pm\infty}|X(s)|\to\infty \end{align} namely that the particle starts and ends at infinity, then since, by definition, test functions vanish at infinity, we find that \begin{align} \int_{\mathbb R^4} d^4 x\, \partial_\mu j^\mu(x) \phi(x) = 0 \end{align} for all test functions $\phi$, and therefore, by definition \begin{align} \partial_\mu j^\mu = 0 \end{align} in the sense of distributions.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.