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8

I can't give an answer using fiber bundles, but I don't think it is important as the confusion is at a much simpler level. A field can be in different representations for different symmetry group. The Higgs field is in the trivial representation of the Poincarre group, that is, under Lorentz transformations, $\phi(x)\to \phi(\Lambda x)$, but in non-trivial ...


5

Let us consider an example and take the Weinberg-Salam Lagrangian: $$ \mathscr{L} = i\bar{\psi}\gamma\cdot\partial\psi - m\bar{\psi}\psi $$ and let us adapt it to the case describing electrons and neutrinos as $$ \mathscr{L} = i\bar{\textrm{e}}_R\gamma\cdot\partial\textrm{e}_R + i\bar{\textrm{e}}_L\gamma\cdot\partial\textrm{e}_L + i\bar{\nu}_L\gamma\cdot\...


3

Although I think this is a good question - the finding of meaning and relationships between physics notions is always worthwhile - "physical meaning" is not a good choice of words here. This is because gauge invariance is a redundancy in the mathematical description of a system; it means that we can partition the solutions of the description into equivalence ...


3

The term gauge transformation refers to two related notions in this context. Let $P$ be a principal $G$-bundle over a manifold $M$, and let $\cup_i U_i$ be a cover of $M$. A connection on $P$ is specified by a collection of $\mathfrak{g}=\mathrm{Lie}(G)$ valued 1-forms $\{A_i\}$ defined in each patch $\{U_i\}$, together with $G$-valued functions $g_{ij} : ...


3

The problem lies in what we learn about good old constrained dynamics from traditional Dirac approach is not complete and is somehow inconsistent, and the above is one example of this. This was the message of Pitts' paper mentioned in the question above, who reviewed a bunch of previous work on this very matter. I will mention couple of references from that ...


2

For $D=d+A$,with respect to the usual inner product on $\mathbb{R}^2$ and the ones induced by it on differential forms, one has $D^{*}_{A}=-*D_{A} *$ where $*$ stands for the hodge star operator. For example, $$D^{*}_A (f_1dx_1+f_2dx_2)=-*D_{A} *(f_1dx_1+f_2dx_2)=-*D_{A} (f_1dx_2-f_2dx_1)=-*(\frac{\partial f_1}{\partial x_1}+\frac{\partial f_2}{\partial x_2}+...


2

When we say scalar, spinor, vector, and so on, field, we mean which representation of the frame bundle the field belongs to. Or in index notation, which spacetime indices the field has: none, spinor, vector, and so on. We can combine this with internal symmetries which are $G$-bundles for some gauge group $G$, for example $SU(2)$. In indices this is some ...


2

Expanding on my comment, I think the Rarita Schwinger field (spin 3/2) has exactly the gauge symmetry you want: https://books.google.be/books?id=KFUhAwAAQBAJ&lpg=PA96&ots=vh0WtWM5rg&dq=rarita%20schwinger%20fermionic%20gauge%20symmetry&pg=PA95#v=onepage&q&f=false This gauge symmetry removes the spin 1/2 component of the field so only ...


2

The Lorenz gauge condition is nice if you want to have electromagnetism mediated by a massive force carrier. Of course the mass would have to be very very tiny to not be immediately contradicted by the good experimental fit of the inverse square law. But since experimental results always have nonzero error bars, there will always be a nonzero mass ...


1

Hamiltonian gauge symmetries usually come up in the context of lattice gauge theory, in which the system is defined on a discrete lattice. Such a Hamiltonian is defined to have a gauge symmetry if it commutes with some extensively-scaling set of local (i.e. finitely-supported) unitary operators $\{ U_i = e^{-i Q_i} \}$. Operators that commute with the ...


1

You should probably read up on the Stueckelberg action and the Affine Higgs mechanism it sends you to. Your boldface supposition "I'm not supposing that my matter has any global symmetry here, that I might be able to gauge" is unwarranted for the specific model you propose, $J_\mu=\partial_\mu\phi$. There is a global symmetry, $\phi \to \phi+\alpha$, whose ...



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