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I read a lot a posts on how to verify what are the symmetries of a given Lagrangian but I really can't find what I need and can't even get it by myself, this because I don't actually understand how groups of transformation act on the fields.

My question is pretty general but probably an example will let me and others undersand the answers better. Consider for simplicity a lagrangian describing a massive vector field and a spinor, with no interaction term

$$\mathcal L = - \frac{1}{2} W^{\mu\nu}W_{\mu\nu}^\dagger + M_W^2W^{\mu}W_{\mu}^\dagger+ \bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi$$

and let us consider internal transformation only.

First question: I want to find the symmetries of that lagrangian. As said before I don't understand how field transform when we apply a transformation. For example a $U(1)$ transformation on a spinor $\psi$ will just give $e^{i\alpha}\psi$ for some $\alpha$ (right?). Is it the same for a vector field, i.e. $$W^{\mu} \stackrel{U(1)}{\longmapsto} e^{i\beta}W^{\mu} ~ ?$$

Second question: What about others gruops as $U(n), SU(n), O(n), SO(n)$, etc? Are they symmetry groups for $\mathcal L$ and how can calculate how different fields transforms under these groups?

Third question: What is the link between a $U(1)$ transformation and a $gauge$ transformation $W_{\mu}\to W_\mu + \partial_\mu\varphi(x)$? (In this case we should not have gauge symmetry because of the $M$-term, right?)

Last question: Are group representations playing a game in all this?

I hope my questions are clear.

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To answer the last question first, yes, this is about group representations. There are ways to transform the quantum field that obey the laws of particular groups. These are called representations of the group in Fock space. Fock space is the space associated with states in QFT. So certain transformations of the vectors and operators in Fock space will form these group representations.

Transformations can be separated into two categories. Gauge transformations do not change any observable property of the system, where as a other transformations (like physically rotating the system) can change observable properties. Gauge transformations and symmetries of the Hamiltonian/Lagrangian are two different concepts. A particle in a spherical potential well has a spherical symmetry which is not a gauge symmetry. Examples of gauge symmetry is the circle group U(1) symmetry of changing quantum phase $$\psi \mapsto e^{i\phi}\psi$$ as well as the ability to adjust (gauge freedom) the magnetic vector potential.

The the group Lorentz group SO(1, 3) and the Poincare group are of particular interest in QFT.

I learned to understand these transformations by first looking at the transformations of free scalar fields. Let $U(\Lambda)$ be the Fock space representation of a Lorentz transformation. That is $U(\Lambda)$ is an element of the Fock space representation of SO(1,3). $\Lambda$ is the corresponding from the familiar 4-vector Lorentz transformations. On free fields the action is like you would expect. It transforms a state of a single particle of momentum to produce a new momentum that is just the conventional Lorentz transformation applied to the original momentum

$$U(\Lambda) \mid \mathbf{p} \rangle = \mid \Lambda \mathbf{p} \rangle $$ You can figure out how it works on operators by considering what happens to a creation operator. $$U(\Lambda) a_+(\mathbf{p}) U(\Lambda)^\dagger =a_+(\Lambda\mathbf{p}) $$

All the operators in the Fock space, such as the operators in the Lagrangian, can be expressed in terms of the creation and annihilation operators. This should give you the general idea how free scalar fields transform. Next we need to

a) observe the effect of adding interactions

b) consider non-scalar fields such as the Dirac bispinor field. The Dirac field has its own representation associated with its spin

a) We often describe how groups act on fields via the generators of the group transformations. For example $$U_\text{boost} = \exp(i\beta \cdot \mathbf{K}). $$ The generators of a particular group representation obey a set of commutation relations. For example with the Poincare group, all representations obey $$ \begin{align*} \big[K^{i},\, \frac{1}{c}H\big] &= iP^i \\ \left[K^{i},\, P^{j}\right] &= -i\delta_{ij}H/c \end{align*}$$ Because $H$ is involved here, when we add interactions to our field it changes the resulting boost generator in order to obtain a Lorentz invariant theory. Briefly $$ \begin{align*} H = H_0 \to H_0 + \int d^3x \mathcal{V}(\Psi(\mathbf{x})) \\ K^i = K^i_0 \to K^i_0 + \frac{1}{\hbar c}\int d^3x\, x^i\, \mathcal{V}(\Psi(\mathbf{x})) \end{align*}$$ b) This partly describes the Poincare group representation for a scalar field. The fields you have presented, such Dirac field is a bit more complicated. The Dirac field is a bi-spinor field. When you have spin, you have a corresponding representation for the spin. In the case of the Dirac equation (bispinor field) the generators for the Lorentz group representation can be expressed as $$J^{\rho\sigma}=\frac{i}{4}\big[\gamma^{\rho} ,\,\gamma^{\sigma}\big]$$ With the boost generators K given by $K^{a}= J^{a0}$. The resulting transformation can be written in terms of left and right spinors and the Pauli matrices $\sigma$. $$\begin{bmatrix} \psi_{L} \\ \psi_{R} \end{bmatrix} \to \begin{bmatrix} e^{-i\boldsymbol{\theta} \cdot \frac{\boldsymbol{\sigma}}{2}-\boldsymbol{\phi} \cdot \frac{\boldsymbol{\sigma}}{2}} & 0 \\ 0& e^{-i\boldsymbol{\theta} \cdot \frac{\boldsymbol{\sigma}}{2}+\boldsymbol{\phi} \cdot \frac{\boldsymbol{\sigma}}{2}} \end{bmatrix} \begin{bmatrix} \psi_{L} \\ \psi_{R} \end{bmatrix}$$

So to rotate or boost the entire field we would apply this $J^{\rho\sigma}=\frac{i}{4}\big[\gamma^{\rho} ,\,\gamma^{\sigma}\big]$ to rotate the bispinor "internally" and also transform the spinor to the appropriate rotated or boosted momentum, like we discussed for a scalar field.

Each non-scalar field has its own representation describing how to transform its spin.

The conventional 3D rotations (SO(3) for vectors or SU(2) for spinors) are a subset of Lorentz transformations.

So these rules should enable one to transform the Lagrangian according to various group representations which apply to the field. The Lagrangian may or may not be invariant under some transformation.

There are books on this subject, such as The Theory of Groups and Quantum Mechanics by Hermann Weyl and it is also covered in The Quantum Theory of Fields, Volume 1 by Steven Weinberg. This post is long enough and my expertise is limited so I will leave question 3 for someone else I guess.

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