Understanding Noether Theorem & Transformations

[facenian]

I've been looking at the original work of Noether and I'm confused about this point. The transformation of fields and coordinates are supossed to form a group, then how the inverse of
$$B^{\mu}=B^{\mu}(A^{\mu},\partial A^{\mu}/\partial x^{\nu},x^{\mu},\epsilon) $$
$$y^{\mu}=y^{\mu}(A^{\mu},\partial A^{\mu}/\partial x^{\nu},x^{\mu},\epsilon) $$
is supposed to be obtained?
For the sake of simplicity we suppose that $\epsilon$ is a single parameter and only first derivatives of the field appear.

 

[stevendaryl]

I've been looking at the original work of Noether and I'm confused about this point. The transformation of fields and coordinates are supossed to form a group, then how the inverse of
$$B^{\mu}=B^{\mu}(A^{\mu},\partial A^{\mu}/\partial x^{\nu},x^{\mu},\epsilon) $$
$$y^{\mu}=y^{\mu}(A^{\mu},\partial A^{\mu}/\partial x^{\nu},x^{\mu},\epsilon) $$
is supposed to be obtained?
For the sake of simplicity we suppose that $\epsilon$ is a single parameter and only first derivatives of the field appear.

Could you write a little bit of the context? I don't know what $B^\mu$ is, or what kind of transformation you are talking about.

For a simple scalar field $\phi$, we assume a transformation of the form: $\phi \rightarrow \phi + \epsilon \psi$. This change will leave the action unchanged if its effect on the lagrangian density is a divergence:

$\mathcal{L} \rightarrow \mathcal{L} + \epsilon \partial_\mu \Lambda^\mu$

for some vector field $\Lambda^\mu$. In that case, there is a conserved current:

$J^\mu = \Lambda^\mu - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \psi$

For this simple transformation, the inverse is pretty simple:

• The forward transformation: $T(\phi) = \phi + \epsilon \psi$
• The inverse transformation: $T^{-1}(\phi) = \phi - \epsilon \psi$

 

[facenian]

$B^\mu$ are the transformed components of the field and $y^\mu$ are new coordinates, $\epsilon$ are parameters.
The problem are the derivatives of the field components. If they were not present we could have inverted the original equations obtaining:
$$ A^\mu=A^\mu(B^\mu,x^\mu,\epsilon)$$
$$ y^\mu=A^\mu(B^\mu,x^\mu,\epsilon)$$
However the appearance of the filed derivatives seem to create a problem for the inversion process.

 

[MathematicalPhysicist]

Is there a link to a scanned copy of her paper you are addressing?

 

[facenian]

I was talking about the original paper in the book "The Noether Theorems : Invariance and Conservation Laws in the Twenty Century" but there is also the paper
by Barbashov and Nesterenko "Continous Symmetries in Field Theory" Fortschr. Phys. 31 (1983) 10, 535-567

 

[MathematicalPhysicist]

I might give a look at it, can't guarantee it though.

 

[facenian]

Title is "INVARIANT VARIATIONAL PROBLEMS
(For F. Klein, on the occasion of the fiftieth anniversary of his doctorate)
by Emmy Noether in Gottingen
Presented by F. Klein at the session of 26 July 1918∗"
in page four she only mentions that that the deriatives occur in the the transformations.
In Babashov and Nesterenko paper it is written explicitly

 

[fresh_42]

Is there a link to a scanned copy of her paper you are addressing?

Emmy Noether's original papers:
http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002504979
http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN00250510X