Noether First and Second Theorem

Question

I have this question related to the the Noether's Theorems. I want to know a rigorous enough enunciation of this theorem, the context is Classical Field Theory without fancy geometrical structures but the usual stuff you need to know to do QFT and the use of Lie Groups(without being too abstract, I need a sensible connection with particle physics).

For what I read around in standard classical mechanics texts, Peskin, Brading and Brown and the thesis of one those authors,it is not really clear to me what a symmetry transformation and Noether's theorems are if viewed with a group theoretic perspective and the knowledge constraints mentioned above.

For what I read in other posts on the site, the group that acts on the Lagrangian and gives the conservation of currents(conservation law) and Noether's theorems is the group of transformations on the space of fields $ \mathfrak{F(\mathcal{M})}$ that is $\mathcal{G} = \{ \Lambda: \mathcal{M} \rightarrow G \} $ where $\mathcal{M}$ is a manifold(for now let just say Minkowski space or Euclidean ), $\Lambda(x)$ is the transformation and $G$ is a group usually compact as $SU(N)$ or Poincare/Galileo. But the problem is that there is a big distance in understanding between this fact and what I have read about Noether's theorems using the literature mentioned above .

Following what they do in the paper, lets define the total variation of the action as $\hat{\delta}S\equiv S(\phi'(x'), \partial_{\mu},\phi(x'),x') -S(\phi(x),\partial_{\mu}\phi(x),x) $. They also define a generic transformation of the action as $\Delta S \equiv \tilde{S}(\phi'(x'), \partial_{\mu},\phi(x'),x') -S(\phi(x),\partial_{\mu}\phi(x),x) $. The transformations that give both variations are "infinitesimal"(what are they more rigorously?). Question a) Are these elements of $\mathcal{G}$? I think they are.

In the thesis they define a symmetry as the transformations (I suppose elements of $\mathcal{G}$) that give $\hat{\delta} S = 0$. Here I think they are again talking about the action of some infinitesimal transformations of $\mathcal{G}$. Then they proceed to derive the so called Noether relations without imposing the Lagrange Euler conditions. Those are: \begin{equation} \sum_{i=1}^{N} \left( \partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi_i)} - \frac{\partial \mathcal{L}}{\partial \phi_i} \right)\delta \phi_i= \sum_{i=1}^N \partial_{\mu}\left( \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}\phi_i)}\delta \phi_i + \mathcal{L}\delta x^{\mu} \right) \end{equation} Then they do following. To enunciate Noether's First Theorem they restrict to "finite dimensional continuous group of transformations depending smoothly on $\rho$ independent parameters $\omega_{i}, (i= 1, \cdots, \rho)$ " that give $\hat{\delta}S = 0$. Then they proceed to expand $\delta \phi$ around the parameters and impose Lagrange Euler equations and get the conservation theorem. For what I understand is that they restrict to the "infinitesimal group action" of some group of transformations that depends on finite parameters. Q b) This is not a Lie group, right? What is the relation with the usual physicist defnition of global symmetries as "the infinitesimal action of a finite dimensional Lie group that leaves $S$ invarant"

Question c) Are they talking about a subgroup of $\mathcal{G}$ that is finite? how is that if $\mathcal{G}$ is infinite dimensional? Are they talking about $G$? or is that the subgroup of $\mathcal{G}$ is somehow isomorphic to G? They seem to act the same way.

In the paper they just refer to transformation that do not act on coordinates ("they defined them as gauge transformations") but in the thesis the same approach is done with one that changes coordinates.

For Noether's second theorem they consider the infinite dimensional group of transformations with finite parameters that depend on x (i.e. functions). I really don't understand this. How is that having the parameters depending explicitly on spacetime changes your the dimension of the group of transformations. How this Second Theorem relates to usual local symmetries as defined in textbooks of physics is even muddier at least for me.

Thanks in advance.

Answer 1

This question (v1) asks many questions. Let us here make some general remarks, which OP hopefully will find useful.

1. Noether's theorem only needs infinitesimal transformations to work. Hence the important object is not the set $G$ of finite transformations, but rather the set $\mathfrak{g}$ of infinitesimal transformations.

2. In general, the set $\mathfrak{g}$ does not have to constitute a Lie algebra or even a Lie algebroid. The "Lie bracket" of two infinitesimal transformations might only close on-shell, i.e. modulo Euler-Lagrange equations. (This is known as an open algebra.)

3. A horizontal infinitesimal transformation $\delta x^{\mu}$ changes the spacetime point $x^{\mu}$, while a vertical infinitesimal transformation $\delta_0 \phi^{\alpha}(x)$ changes the fields $\phi^{\alpha}(x)$ without moving the spacetime point $x$. A general infinitesimal transformation is a combination of horizontal and vertical infinitesimal transformations.

4. A vertical infinitesimal transformation is typically of the form $$\tag{1} \delta_0 \phi^{\alpha}(x) ~=~\varepsilon^a(x) ~Y^{\alpha}_a(\phi(x),\partial\phi(x),x) + d_{\mu}\varepsilon^a(x)~ Y^{\alpha, \mu}_a(\phi(x),\partial\phi(x),x),$$ where $\varepsilon^a(x)$ are infinitesimal transformation parameters, which are coordinates of a section $\varepsilon(x)$ in a vector bundle $E$ over spacetime.

5. To apply Noether's first theorem for a finite subspace of global$^1$ infinitesimal transformations, one identifies a finite-dimensional subspace of sections $\varepsilon_{(1)}(x)$,$\ldots,$ $\varepsilon_{(m)}(x)$, in $E$. Thus the global infinitesimal transformations are of the form $$\tag{2} \varepsilon(x)~=~ \sum_{r=1}^m \omega^{(r)}~\varepsilon_{(r)}(x), $$ where the parameters $\omega^{(1)}$, $\ldots$, $\omega^{(m)}$, are $x$-independent. In coordinates, $$\tag{3} \varepsilon^a(x)~=~ \sum_{r=1}^m \omega^{(r)}~\varepsilon^a_{(r)}(x). $$

--

$^1$ A global (local) transformation refers in this physics context to an $x$-independent ($x$-dependent) transformation, respectively. What are $x$-independent are here really the $\omega^{(r)}$ parameters, not necessarily the basis elements $\varepsilon_{(r)}(x)$. Thus the notion of global transformations depends in principle on the choice of section basis $\varepsilon_{(1)}(x)$,$\ldots,$ $\varepsilon_{(m)}(x)$. [Local and global transformation in physics should not be confused with the mathematical notion of locally and globally defined objects. All transformations in this answer (local as well as global) are assumed to be globally defined on the entire spacetime. Locally defined transformations take us to the realm of gerbes.]

Answer 2

Let's see if we can make sense out of this. For the following, we'll introduce manifolds along with the coordinates naturally suited for them.

The base space is the manifold $M:\left(x^μ:0≤μ<n\right)$ with $\text{dim}(M) = n$. In the field equations $x^μ$ play the role of independent variables. The field components reside in a space $F:\left(q^a:0≤a<f\right)$ with $\text{dim}{F} = f$. At each $x ∈ M$ is attached a copy $F_x:\left(q^a_x\right)$ of $F$. The connections are smoothly done, resulting in a manifold $$Y:\left(x^μ,q^a_x\right) = ⋃_{x∈M}\left(\{x\}×F_x\right)$$ that is a Fiber Bundle or Fibered Manifold over $M$.

Reading the definitions in the links, fiber bundles require the surjection $π: (x,q)∈Y ↦ x∈X$ to be continuous, while fibered manifolds require their differentials to be surjective, too (i.e. that $Tπ:T_{(x,q)}Y → T_xX$ be onto, for each $(x,q)∈Y$). So: fibered manifolds, then.

The field components play the role of dependent variables, and the space of fields is given by the sections, $q: x ∈ M ↦ (x,q(x)) ∈ Y$, which is denoted $Γ(Y)$ - your $(M)$.

For actions with first-order Lagrangians, you also want the coordinates $v^a_μ$ for the gradients $∂_μq^a$, resulting in the manifold $J^1Y:(x,q,v)$ - the first order Jet Bundle of $Y$. The interpretation of $v^a_μ = ∂_μq^a$ (denoting $∂/∂x^μ$ by $∂_μ$) is enforced by identifying each section $q ∈ Γ(Y)$ with its first order jet prolongation $j^1q: x ∈ M ↦ \left(x^μ,q^a(x),∂_μq^a(x)\right)$. This enforces what I call the “Kinematic Constraint”: $v^a_μ = ∂_μq^a$. Variationals on $q$ are done on the sections and their prolongations, so this constraint entails the same condition $δv^a_μ = ∂_μδq^a$ on the variationals, too.

As a geometric object, $v^a = v^a_μ dx^μ$, so that the kinematic constraint could just as well be written $v^a = dq^a$, with the corresponding condition on the variational being $δv^a = d\left(δq^a\right)$. In here, and the following, I will be using the summation convention with the indexed notation.

I will make a distinction between “total variations”, which allows the $x$ coordinates to move - denoting them by $Δ$ - versus “local variations”, which keep the $x$ coordinates fixed - denoting them by $δ$. In particular, $δx^μ = 0$. The transforms on $x$ are associated with a vector field $Δx = Δx^μ ∂_μ$. The flows of such fields $Δx$ are assumed to generate Local Diffeomorphisms, $x ∈ M ↦ x' = X(x) ∈ M$, which they are then the infinitesimal forms of. Implied in this is that $X$ is invertible, so $x = X^{-1}(x')$ is well-defined.

Total variations, on $Y$, involve coordinate transitions of the form $(x,q) ∈ Y ↦ (X(x),Q(x,q)) ∈ Y$. The key feature is that fibers map to fibers, i.e. there is no $q$-dependence in $X(x)$; while the transitions for local variations are restricted further to the forms $(x,q) ∈ Y ↦ (x,Q(x,q)) ∈ Y$ that also keep $x$ fixed; and therefore also keep each fiber in place. A section $q$ transforms into $$q': X(x) ∈ Y ↦ (X(x),Q(x,q(x))) ∈ Y,$$ i.e. $$q': x' ∈ Y ↦ (x',Q(X^{-1}(x'),q(X^{-1}(x')))) ∈ Y.$$ For internal transforms, the points are fixed $x' = x$, so these reduce to the form: $$q': x ∈ Y ↦ (x,Q(x,q(x))) ∈ Y.$$

The coordinate transforms are all invertible, so they form infinite-dimensional Lie groups. I'm not totally clear on what the names are. I think $\text{Aut}(Y)$ (for “automorphism”) comprises all the “total” coordinate transforms on $Y$ - the ones that move points in $X$ - while $\text{Gau}(Y)$ (for “gauge”) make up the “local” coordinate transforms - the one that keep points in $X$ fixed. Besides the name terminology, what I'm not clear about is whether they are local diffeomorphisms, or whether they are required to be Diffeomorphisms, which is a stronger condition.

If $Y$ is a Natural Bundle then coordinate transforms $x$ on $M$ are naturally associated with transforms on $F$ and $Y$. The archetype of this is the Tangent Bundle $Y = TM: \left(x^μ,\dot{x^μ}∂_μ\right)$, where ${x'}^{μ'} = X^{μ'}(x)$ yields the transform classically associated with vector fields $\dot{x'}^{μ'} = ∂_{μ'}X^μ(x) \dot{x}^μ$. For natural bundles, I believe you have a clean-separation of internal transforms from total transforms by the formula: $$Δ = δ + _{Δx},$$ where $_{Δx}$ denotes the Lie Derivative associated with the vector field $Δx^μ ∂_μ$. Otherwise, if it's not a natural bundle, then Lie derivatives have to be replaced by some reduced version of it. Compare with the Generalizations Of Lie Derivatives section of the Lie Derivatives link.

At a pedestrian level, for a Lagrangian $n$-form $L = (x,q,v)d^nx$ and its corresponding Lagrangian density $(x,q,v)$, I would proceed as follows. First, for convenience, define the partial derivatives: $$_μ = \frac{∂}{∂x^μ},\quad _a = \frac{∂}{∂q^a},\quad ^μ_a = \frac{∂}{∂v^a_μ}.$$ A distinction must then be made between the partial derivative $_μ$ and "total" partial $∂_μ$, which is applied via the chain rule on the section $j^1q$, which for convenience, will be denoted $∂_μ((x,q(x),v(x)))$, or just $∂_μ()$, for short. By the chain rule, this yields the relation: $$∂_μ((x,q(x),v(x))) = _μ + ∂_μq^a _a + ∂_μ v^a_ν ^ν_a.$$ Under the kinematic constraint: $$∂_μq^a = v^a_μ,\quad ∂_μ v^a_ν = ∂_μ∂_νq^a = ∂_ν∂_μq^a = ∂_νv^a_μ.$$

The total variational takes place through the product rule and chain rule: $$ΔL = Δ(x,q,v)d^nx + (x,q,v)Δd^nx.$$ The action on $d^nx$ is a pure Lie derivative: $$Δd^nx = _{Δx}\left(d^nx\right) = ∂_μΔx^μ d^nx.$$ The action on $(x,q,v)$ uses the chain rule: $$Δ = _μΔx^μ + Δq^a _a + Δv^a_ν ^ν_a.$$ For natural-object fields, there is then the decomposition: $$Δq^a = δq^a + _{Δx}q^a,\quad Δv^a_ν = δv^a_ν + _{Δx}δv^a_ν = ∂_νδq^a + _{Δx}δv^a_ν.$$ If we're talking about fields with scalar-valued components (as opposed - say - to fields where each component $q^a$ is a differential form), then we have: $$_{Δx}q^a = Δx^μ∂_μq^a,\quad _{Δx}δv^a_ν = Δx^μ∂_μv^a_ν.$$ Therefore, combining results and using integration by parts (and doing a little index-switching): $$\begin{align} Δ &= _μΔx^μ + \left(δq^a + Δx^μ∂_μq^a\right) _a + \left(∂_νδq^a + Δx^μ∂_μv^a_ν\right) ^ν_a\\ &= \left(_μ + ∂_μq^a _a + ∂_μv^a_ν ^ν_a\right)Δx^μ + δq^a _a + ∂_μδq^a ^μ_a\\ &= Δx^μ∂_μ() + δq^a _a + ∂_μ\left(δq^a ^μ_a\right) - δq^a ∂_μ^μ_a\\ &= Δx^μ∂_μ() + ∂_μ(δq^a ^μ_a) + δq^a \left(_a - ∂_μ^μ_a\right). \end{align}$$

All of this combines to yield the following expression for the variational on the Lagrangian $n$-form: $$\begin{align} ΔL &= \left(Δx^μ∂_μ() + ∂_μ\left(δq^a ^μ_a\right) + δq^a \left(_a - ∂_μ^μ_a\right)\right)d^nx + \left(∂_μΔx^μ d^nx\right)\\ &= \left(∂_μ\left(Δx^μ + δq^a ^μ_a\right) + δq^a \left(_a - ∂_μ^μ_a\right)\right)d^nx. \end{align}$$

There is a reference out there somewhere, by Hehl, who worked out the math for the case where $q^a$ is a form-valued field. If I find it, I'll add it as an edit or comment.