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I think I have found a majot error in Neuenschwander's book on Noether's Theorem, but I'd like some confirmation from someone familiar with the book or with the Rund_Trautman identity for fields. As far as I can see the extension of the R-T identity for fields seems to be Neuenschwander's work, so there's no a lot about it on line.
We have a Lagrangian Density:
$$\mathcal{L}(x^{\mu}, \phi, \phi_{\mu})$$
The proof of the R-T identity is on page 105-106 and gives that the functional of ##\mathcal{L}## is invariant under the infinitesimal transformation:
$$x'^{\mu} = x^{\mu} + \epsilon \tau^{\mu}(x^{\rho}, \phi); \ \ \phi' = \phi + \epsilon \zeta(x^{\rho}, \phi)$$
Iff
$$\frac{\partial \mathcal{L}}{\partial \phi}\zeta + p^{\rho}\frac{\partial \zeta}{\partial x^{\rho}} + \frac{\partial \mathcal{L}}{\partial x^{\rho}} \tau^{\rho} - \mathcal{H}_{\rho}^{\ \nu} \frac{\partial \tau_{\rho}}{\partial x^{\nu}} = 0$$
Where ##\mathcal{H}## is the Hamiltonian density:
$$\mathcal{H}_{\rho}^{\ \nu} = \phi_{\rho}p^{\nu} - \mathcal{L}\delta_{\rho}^{\ \nu}$$
And ##p^{\nu}## is the canonical momentum ##\frac{\partial \mathcal{L}}{\partial \phi_{\nu}}##
I believe there is a term missing, which is:
$$p^{\rho}\frac{\partial \zeta}{\partial \phi} \phi_{\rho}$$
The critical step in the proof is (6.5.15), where he exapnds:
$$\frac{\partial \phi'}{\partial x^{\nu}} = \phi_{\nu} + \epsilon \frac{\partial \zeta}{\partial x^{\nu}}$$
But, I believe this should be:
$$\frac{\partial \phi'}{\partial x^{\nu}} = \phi_{\nu} + \epsilon \frac{\partial \zeta}{\partial x^{\nu}} + \epsilon \frac{\partial \zeta}{\partial \phi} \phi_{\nu}$$
And this results in the extra term.
I got suspicious about this when looking at transformations of the field only, and not the coordinates. It seemed there ought to be term that reflected the function ##\zeta## in terms of the field ##\phi##. And, several examples and problems were not working out and, suspiciously perhaps, for the rest of the chapter the author jumps straight to the implied conservation law with the stated assumption that the functional was invariant.
I know this book has been recommended by several people on PF (@bhobba). I'm not just flicking through this, but I'm trying to work most of the examples and problems and, to be honest, this is the 2nd or 3rd major error, not to mention all the minor ones I think I've found.
Any confirmation of this would be welcome, although I'm fairly convinced that the book must be in error.
We have a Lagrangian Density:
$$\mathcal{L}(x^{\mu}, \phi, \phi_{\mu})$$
The proof of the R-T identity is on page 105-106 and gives that the functional of ##\mathcal{L}## is invariant under the infinitesimal transformation:
$$x'^{\mu} = x^{\mu} + \epsilon \tau^{\mu}(x^{\rho}, \phi); \ \ \phi' = \phi + \epsilon \zeta(x^{\rho}, \phi)$$
Iff
$$\frac{\partial \mathcal{L}}{\partial \phi}\zeta + p^{\rho}\frac{\partial \zeta}{\partial x^{\rho}} + \frac{\partial \mathcal{L}}{\partial x^{\rho}} \tau^{\rho} - \mathcal{H}_{\rho}^{\ \nu} \frac{\partial \tau_{\rho}}{\partial x^{\nu}} = 0$$
Where ##\mathcal{H}## is the Hamiltonian density:
$$\mathcal{H}_{\rho}^{\ \nu} = \phi_{\rho}p^{\nu} - \mathcal{L}\delta_{\rho}^{\ \nu}$$
And ##p^{\nu}## is the canonical momentum ##\frac{\partial \mathcal{L}}{\partial \phi_{\nu}}##
I believe there is a term missing, which is:
$$p^{\rho}\frac{\partial \zeta}{\partial \phi} \phi_{\rho}$$
The critical step in the proof is (6.5.15), where he exapnds:
$$\frac{\partial \phi'}{\partial x^{\nu}} = \phi_{\nu} + \epsilon \frac{\partial \zeta}{\partial x^{\nu}}$$
But, I believe this should be:
$$\frac{\partial \phi'}{\partial x^{\nu}} = \phi_{\nu} + \epsilon \frac{\partial \zeta}{\partial x^{\nu}} + \epsilon \frac{\partial \zeta}{\partial \phi} \phi_{\nu}$$
And this results in the extra term.
I got suspicious about this when looking at transformations of the field only, and not the coordinates. It seemed there ought to be term that reflected the function ##\zeta## in terms of the field ##\phi##. And, several examples and problems were not working out and, suspiciously perhaps, for the rest of the chapter the author jumps straight to the implied conservation law with the stated assumption that the functional was invariant.
I know this book has been recommended by several people on PF (@bhobba). I'm not just flicking through this, but I'm trying to work most of the examples and problems and, to be honest, this is the 2nd or 3rd major error, not to mention all the minor ones I think I've found.
Any confirmation of this would be welcome, although I'm fairly convinced that the book must be in error.