Apologies for the vague question, and for the many inaccuracies (I am not a physicist and barely a number theorist).

In physics, there is the notion of gauge group of a field theory. The gauge group is the symmetry group of the Lagrangian describing the theory. It is the "symmetry group of reality" in a reality dictated by the Lagrangian.

On the other hand, the Galois group $G_{\mathbf Q}$ also occurs as the symmetry group of a theory, namely of the theory of algebraic varieties over ${\mathbf Q}$. We might like to think of it as the symmetry group of a hidden "arithmetic field theory" attached to $\mathbf Q$.

Irreducible representations of the gauge group of a field theory can be interpreted as elementary particles of a certain kind (the gauge bosons of the field theory); for this reason, people have said that adequate Galois representations of $G_{\mathbf Q}$ could be thought of as elementary particles (I believe I read this from Brian Conrad).

Elementary particles, however, can also be viewed as incarnations of their wave functions $f \in \Psi$. Moreover, suppose that we have a complete set of commuting observables $\{T_n :\Psi \to \Psi\}$. Let $\mathbf T$ be the $\mathbf Z$-algebra of $End(\Psi)$ generated by the $T_i$'s. Then it is natural to think that the simultaneous eigenfunctions of $\mathbf T$ should be precisely the wave functions of the gauge bosons: if two wave functions *look* the same under every element of our *complete* set of observables $\mathbf T$, then they should actually be indistinguishible.

An analogous situation on the arithmetic side is the following: start with the Tate motive $M$ of an elliptic curve $E/\mathbf Q$. Then, according to the modularity theorem, we can attach a "wave function" to the "elementary particle" $M$, namely the Hecke eigenform $f$ associated to $E/\mathbf Q$. Here of course $\mathbf T$ is the Hecke algebra. We can reconstruct the elementary particle $M$ from its wave function, using the algebra of observables $\mathbf T$ (Eichler-Shimura theory).

Thus, it appears to me that the passage from Galois representations to automorphic forms is analogous to the passage from "particle" to "wave" in quantum mechanics.

Is this right?

If so, what more can be said about this?