- #1
Max Renn
- 5
- 0
Hi,
I have a problem in classical field theory.
I have a Lagrangian density [tex]\mathcal{L}=\frac{1}{2}\partial_\lambda \phi \partial^\lambda \phi + \frac{1}{3}\sigma\phi^3 [/tex]. Upon solving the Euler-Lagrange equation for this density, I get an equation of motion for my scalar field [tex]\phi (x)[/tex], where [tex]x = x^\mu[/tex] is a space-time coordinate. I figured this is [tex]\Box \phi - \sigma \phi^2 = 0 [/tex]. Now, the problem begins.
I have to calculate the following stress tensor:
[tex]T^{\mu \nu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial^\nu \phi - g^{\mu \nu}\mathcal{L} [/tex].
The metric tensor [tex]g^{\mu \nu}[/tex] is the "energy-momentum" type, [tex]\mathrm{diag}(1, -1, -1, -1)[/tex]. Then I have to find its 4-divergence [tex]\partial_\mu T^{\mu \nu}[/tex] and show that it's conserved when [tex]\phi(x)[/tex] obeys its equation of motion, i.e. that [tex]\partial_\mu T^{\mu \nu} = 0[/tex].
Now, if I didn't know any better, I'd say that
[tex]T^{\mu \nu} = \frac{1}{2}\partial^\mu \phi \partial^\nu \phi + \frac{1}{3} g^{\mu \nu} \sigma \phi^3[/tex].
I have some serious doubts, however. If this is correct, I have another problem in that I can't seem to find a zero 4-divergence.
I'm quite new to this sort of thing and I have a feeling it's just a lack of practice with tensors and indices.
I have a problem in classical field theory.
I have a Lagrangian density [tex]\mathcal{L}=\frac{1}{2}\partial_\lambda \phi \partial^\lambda \phi + \frac{1}{3}\sigma\phi^3 [/tex]. Upon solving the Euler-Lagrange equation for this density, I get an equation of motion for my scalar field [tex]\phi (x)[/tex], where [tex]x = x^\mu[/tex] is a space-time coordinate. I figured this is [tex]\Box \phi - \sigma \phi^2 = 0 [/tex]. Now, the problem begins.
I have to calculate the following stress tensor:
[tex]T^{\mu \nu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial^\nu \phi - g^{\mu \nu}\mathcal{L} [/tex].
The metric tensor [tex]g^{\mu \nu}[/tex] is the "energy-momentum" type, [tex]\mathrm{diag}(1, -1, -1, -1)[/tex]. Then I have to find its 4-divergence [tex]\partial_\mu T^{\mu \nu}[/tex] and show that it's conserved when [tex]\phi(x)[/tex] obeys its equation of motion, i.e. that [tex]\partial_\mu T^{\mu \nu} = 0[/tex].
Now, if I didn't know any better, I'd say that
[tex]T^{\mu \nu} = \frac{1}{2}\partial^\mu \phi \partial^\nu \phi + \frac{1}{3} g^{\mu \nu} \sigma \phi^3[/tex].
I have some serious doubts, however. If this is correct, I have another problem in that I can't seem to find a zero 4-divergence.
I'm quite new to this sort of thing and I have a feeling it's just a lack of practice with tensors and indices.
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