There is nothing particular quantum about this question but I'm posting it here because I think the quantum folks are likely more familiar with the topic. Hope that's ok.(adsbygoogle = window.adsbygoogle || []).push({});

There are two ways of looking at field Lagrangian densities in relation to particle Lagrangians.

(1) A particle (one coordinate) action looks like

[tex]S=\int L\left(q(t),\dot{q}(t)\right)dt[/tex]

t is a continuous parameter. and

[tex]\dot{q}=\frac{dq}{dt}[/tex]

We can easily generalize this to 4 continuous parameters and write

[tex]S=\int L\left(q(x),\partial_{\mu} q(x)\right)d^4 x[/tex]

where now [tex]x=(x^0, x^1, x^2, x^3)[/tex] and [tex]\partial_{\mu} q[/tex] stands for all four partial derivatives. We then call q afieldand L a Lagrangiandensity. But from this viewpoint the names suggest more distinction than is warranted. It is just a matter of how many continuous parameters we consider. The first is a system with one coordinate and one parameter, the second one coordinate and four parameters.

(2) In the second approach we first expand to systems of N coordinates and write

[tex]S=\int L\left(q_1,...,q_N,\dot{q_1},...,\dot{q_N}\right)dt[/tex]

Then we suppose that N goes to (continuum) infinity. We replace the discrete [tex]q_j(t)[/tex] with [tex]q(\vec{x},t)[/tex], wherexis just a "label" to idenfity one of the infinitely many degrees of freedom. The Lagrangian becomes

[tex]L=\int \mathcal{L}\left(q(\vec{x},t),\dot{q}(\vec{x},t),\frac{\partial q}{\partial x^j}\right)d^3 x[/tex]

When we put this L into the action integral we get something like what we got in (1) above.

This second approach is common in the physics literature and is suggestive of physical signficance: fields are systems with infinitely many degrees of freedom. However, I find this approach troubling in two ways. First, L is represented as an integral, which is a limit of a sum. Yet there is no summation in the definition of L in the N degrees of freedom case. So how is one the limiting case of the other?

Secondly, how do the partial derivatives with respect to x come in? They are not analogous to anything in the discrete case.

Is this approach inherently misleading? (yet historically useful in an accidental sort of way) Or am I missing something?

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# Field Lagrangians as systems with infinte degrees of freedom?

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