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Lagrangian density for fields

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spookyfish
#1
Oct24-13, 11:34 PM
P: 53
This is probably a minor point, but I have seen in some QFT texts the Euler-Lagrange equation for a scalar field,

[tex] \partial_{\mu} \left(\frac{\delta \cal{L}}{\delta (\partial_{\mu}\phi)}\right) - \frac{\delta \cal L}{\delta \phi }=0 [/tex]

i.e. [itex] \cal L [/itex] is treated like a functional (seen from the [itex] \delta [/itex] symbol). But why would it be a functional? Functonals map functions into numbers, and in our case [itex] \cal L [/itex] is a function of the fields (and their derivatives).
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fzero
#2
Oct25-13, 12:01 AM
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If you want to be completely rigorous, the action is the true functional. The variational derivatives of the Lagrangian (density) should be considered distributions.
spookyfish
#3
Oct25-13, 08:46 AM
P: 53
But why should there be a functional derivative of [itex] \cal L [/itex]? we have [itex] \cal L [/itex] which is a function of [itex] (\phi, \partial_\mu \phi) [/itex] and we differentiate (as a function) with respect to [itex] \partial_\mu \phi [/itex]

Bill_K
#4
Oct25-13, 09:18 AM
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Lagrangian density for fields

Quote Quote by spookyfish View Post
This is probably a minor point, but I have seen in some QFT texts the Euler-Lagrange equation for a scalar field,

[tex] \partial_{\mu} \left(\frac{\delta \cal{L}}{\delta (\partial_{\mu}\phi)}\right) - \frac{\delta \cal L}{\delta \phi }=0 [/tex]

i.e. [itex] \cal L [/itex] is treated like a functional (seen from the [itex] \delta [/itex] symbol). But why would it be a functional? Functonals map functions into numbers, and in our case [itex] \cal L [/itex] is a function of the fields (and their derivatives).
If they wrote it that way it's a misprint. The derivatives should be ∂'s, not δ's.
fzero
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Oct25-13, 10:46 AM
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Quote Quote by spookyfish View Post
But why should there be a functional derivative of [itex] \cal L [/itex]? we have [itex] \cal L [/itex] which is a function of [itex] (\phi, \partial_\mu \phi) [/itex] and we differentiate (as a function) with respect to [itex] \partial_\mu \phi [/itex]
It's common to abuse the notation and use ##\delta## for these derivatives in order to distinguish them from the coordinate derivatives ##\partial_\mu##.


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