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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?

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People are voting this up, so I think I must be missing something here (and I'm not an expert on either number theory or physics). However, this seems pretty vague to me. In what precise nontrivial sense could these two things be "analogous"? –  Andy Putman Dec 3 '13 at 23:21
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@AndyPutman That's my question! –  Bruno Joyal Dec 3 '13 at 23:32
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If gauge bosons correspond to irreducible representations, why are there finitely many gauge bosons but infinitely many irreducible representations? –  Will Sawin Dec 4 '13 at 5:52
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It seems that Joyal's interpretation of gauge bosons as irreps of the gauge group is at odds with both physics textbooks and Wikipedia - as far as I can tell, a complete set of gauge bosons for a theory is given by a basis of the Lie algebra. The usual interpretation of particles as irreps was given by Wigner, but these are irreps of the spacetime symmetry group. –  S. Carnahan Dec 5 '13 at 0:37
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@Carnahan: you are correct about Wigner (this is what gives us the spin types, e.g. scalar and vector bosons, 1/2-spinors, etc.). But the other statement is a little imprecise. It's certainly possible to split the Lie algebra using a basis and say that you have 8 different kinds of gluons. But the actual physics and math doesn't really see the (arbitrary) splitting, so it's more correct to view a gluon as an 8-dim vector in the Lie algebra. –  Marek Dec 5 '13 at 11:12

2 Answers 2

I don't know enough about Langlands reciprocity to answer your question as asked, but the following might be helpful in thinking about the connection you are asking about.

There is by now a well understood connection between the geometric Langlands correspondence and S-duality of a topologically twisted version of $N=4$ Super Yang Mills theory (SYM) formulated on four manifolds of the form $ \Sigma \times X$ with $X$ a closed Riemann surface. Taking $X$ to be "small" leads to a two-dimensional topological field theory on $\Sigma$ and the S-duality of $N=4$ SYM becomes a kind of mirror symmetry of the topological field theory that relates the theory with gauge group $G$ to one with its Langlands dual ${}^LG$. This point of view was developed by Kapustin and Witten and is explained by E. Frenkel in arXiv:0906.2747.

The connection of this to particle-wave duality in quantum mechanics is as follows. S-duality of $N=4$ Super Yang-Mills theory has its origin in the electric-magnetic duality of pure Maxwell theory in $R^4$. Maxwell's equations with vanishing charge density and current sources are invariant under the transformation $\vec E \rightarrow \vec B$, $\vec B \rightarrow - \vec E$ of the electric field $\vec E $ and the magnetic field $\vec B$. This transformation is very analogous to the following transformation on the coordinate $x$ and momentum $p$ of a one-dimensional simple harmonic oscillator (SHO) $ x \rightarrow p$, $p \rightarrow -x$ and this duality transformation is a symmetry of the simple harmonic oscillator Hamiltonian $H= p^2/2+x^2/2$. In the quantum theories this analogy can be made precise by decomposing the electromagnetic field into modes and applying canonical quantization.

Now this symmetry of the SHO should be regarded as the statement that the SHO is self-dual under particle wave duality. The particle aspects are most clearly thought of in coordinate space while the wave aspects are most obvious in momentum space obtained by Fourier transform. Put another way, a particle is localized in $x$, a wave is localized in the canonical dual variable $p$. The self-duality of the SHO under particle-wave duality is manifested in various ways. For example, the ground state wave function of the SHO is a Gaussian in coordinate space. The dual under the above transformation gives the ground state wave function in momentum space since the Fourier transform of a Gaussian is again a Gaussian.

So I claim these is a connection between (geometric) Langlands and wave-particle duality that runs as particle-wave duality of SHO-> electric-magnetic duality of Maxwell theory -> S-duality of N=4 SYM -> Langlands. I leave it someone more knowledgeable to say whether this connection means anything in the number theoretical context.

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I believe that this is an interesting question that deserves attention even though it is difficult and perhaps too vague, as noted in the comments. Needless to say that the following will not be a detailed answer but my goal is to at least give some pointers of how to view the question in a slightly different way.

The first question is whether the structural outline given in the question is the appropriate one. In my view the picture that emerges is slightly different. While on the arithmetic side the absolute Galois group $G_{\mathbb Q}$ is usually emphasized as the primary object and their dual motives (via Grothendieck and Langlands) both types are also associated to reductive groups $G$. When thinking about motives in specific geometric situations a reductive group $G$ is chosen, much like a specific Lie group $G_L$ is associated to a specific physical theory. In the context of classical modular forms of some weight $w$ this would be SL$_2({\mathbb R}$) and its congruence subgroups $\Gamma_0(N)\subset {\rm SL}_2({\mathbb Z})$, and for more general automorphic forms higher rank groups enter. In this view the focus is more on the individual reductive groups, for example GL$(n)$ or GSp$(n)$, which therefore play the role of the physical Lie groups, such as SU$(n)$ or the exceptional groups $E_n$. The analogy then is between automorphic forms on the arithmetic side and sections of vector bundles associated to the principal bundle $P(G_L)$ determined by the Lie group $G_L$. While the absolute Galois group is of course still the unifying fundamental object, in physics this would shift the focus away from individual theories to considering all theories at the same time, something that could in principle be done at the level of effective field theories. The particle-wave duality is encoded in the interpretation of this section, either quantum mechanically or quantum field theoretically, where the quantization is implemented in a different way.

The second question raised is concerned with what else can be said about this potential analogy. In physics the key question is the dynamics of the theory. Historically, more often than not fundamental theories in particle physics were constructed by first extracting the symmetry group $G_L$ from observed patterns (an example would be Gell-Mann's discovery of quarks) and after the group had been extracted the question became what the dynamics is. This is a nontrivial task because while symmetry principles put constraints on the dynamics, they usually do not fix it. At the end of the day nature tells us what the right dynamics is, which can be encoded either in some system of hyperbolic differential equations or the Lagrangian with its associated action. If one views the Lie groups $G_L$ in physics as the analog of the reductive groups of the arithmetic theory, and the physical fields defined by sections in bundles as analogs of automorphic forms associated to motives, then the next question would be "what is the "dynamics" of automorphic forms".

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