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I have a question regarding Noether's theorem. In our introductory QFT class (which is based on the book by Michele Maggiore) we have derived the Noether currents in the same form as displayed in this post: Question about Noether theorem In this formula, there are contributions from two different kinds of transformations: a transformation of the field alone and a transformation of the coordinates.

My problem is: I don't understand the meaning of the transformation of coordinates. I have tried to understand the derivation from different QFT books (and I haven't found the same derivation twice, which doesn't make it easier) in the hope that I then would better understand the premises, but unfortunately I have not succeeded so far.

Also Peskin/Schröder for example only discuss transformations of fields and don't mention the transformation of coordinates at all. Poincare symmetry, which is in most books treated as a transformation of the coordinates, can be treated also as a transformation of fields, as shown in the answer to the following question for pure translations: Noether's Theorem: Foundations. Like the guy who asked that question, I think that the coordinates entering the action are only dummy variables. So what is then the meaning of the coordinate transformations in the prevalent formulation of Noether's theorem? Maybe someone can give a concrete example to illustrate the idea.

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Classical Lagrangian field theory deals with fields $\phi: M \to N$, where $M$ is spacetime and $N$ is the target-space of the fields. We shall for convenience call $M$ and $N$ the horizontal and the vertical space, respectively. OP is in this terminology essentially asking

Q: What is the meaning of horizontal transformations?

A: It is a (horizontal) flow in spacetime $M$. Infinitesimally, it is generated by a (horizontal) vector field $X\in\Gamma(M)$.

Q: How can the horizontal/spacetime coordinates be important if they are just dummy variables in the action $S=\int_{\Omega} \!d^4x~{\cal L}$?

A: Well, as Phoenix87 points out in his answer, there can be a flow in and out of the integration region $\Omega\subseteq M$ which may create boundary contributions. Moreover, $\Omega$ is often considered to be an arbitrary integration region.

Already Noether herself considered both horizontal and vertical transformations in her seminal 1918 paper. There are many examples on Phys.SE where horizontal transformations play a role. See e.g. this and this Phys.SE posts.

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  • $\begingroup$ Thank you very much so far for your answer. It is still not clear to me why Peskin/Schröder don't need the horizontal transformations at all when deriving momentum and angular momentum as conserved charges. But I have also found a nice and thorough treatment in the book by Sexl/Urbantke which could make things clearer. I will definitely look closer into this within the next couple of days. $\endgroup$ – LLang Jul 26 '15 at 11:05
  • $\begingroup$ @LLang: Which pages in P&S? $\endgroup$ – Qmechanic Jul 26 '15 at 11:57
  • $\begingroup$ In the edition I have, the equation for Noether currents is Eq. 2.12 on page 18. They arrive at the energy-momentum tensor on page 19 (Eq. 2.17 - 2.19). Angular momentum is discussed for the specific example of a Dirac field on page 60 (Eq. 3.111). $\endgroup$ – LLang Jul 26 '15 at 12:14
  • $\begingroup$ It seems that both mentioned cases use both horizontal & vertical transformations. $\endgroup$ – Qmechanic Jul 26 '15 at 13:00
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When you integrate the Lagrangian density over a certain region $\Omega$, this is in principle allowed to change and this gives you a "boundary" term in the variation. This is well discussed in, e.g., the book of Goldstein (3rd edition), where the correct proof of the Noether theorem is given.

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I think that after 1,5 years I can finally appreciate Qmechanic's answer. Let me try to formulate what I think would have been the ideal answer on my question and correct me if I am wrong. I'm using the symbols as defined in Weinberg's Quantum Theory of Fields.

A field is a function $\Psi: M \rightarrow N$, where $M$ is Minkowski space (which we call for convenience the horizontal space) and $N$ is the target space of the fields (which we call for convenience the vertical space). $N$ can e.g. be the space of scalars, vectors, Dirac spinors, antisymmetric tensors, etc. In order to define an infinitesimal transformation $X \rightarrow X$ (where $X$ is the space of fields), we can consider separate infinitesimal transformations in the horizontal and vertical space:

a) horizontal transformation: Transformation of the type $h:M\rightarrow M$. An example is $x \mapsto x + \omega x$ (if $\omega$ with both indices up or down is antisymmetric, this is an infinitesimal Lorentz transformation).

b) vertical transformation: Transformation of the type $v:N \rightarrow N$. An example is $\Psi \mapsto \Psi + \frac{i}{2}\omega^{\mu\nu} \mathcal J_{\mu\nu} \Psi$, where $J_{\mu\nu}:N\rightarrow N$ are the infinitesimal generators in the irreducible representation of the Lorentz group under which the elements of $N$ transform. $\omega$ is defined as above.

We can now define the transformation of fields $X \rightarrow X$ by combining a horizontal and a vertical transformation: $\Psi \mapsto \Psi'$ with $\Psi'(h(x)) := v(\Psi(x))$. The action is only a functional of the fields themselves, but as we see the transformed field depends on both a vertical and a horizontal transformation.

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