What is Fourier Analysis on Groups and does it have “applications” to physics?

Question

I am trying to be as specific as possible, but I am extremely unclear about this topic (Fourier Analysis on Groups).

In Reed-Simon Vol II (Fourier Analysis, Self-Adjointness) there is some discussion about semigroups (those that are holomorphic, hypercontractive, $L^p$-contractive, strongly continuous). There is also mention of pseudo-differential operators in the noncommutative case. Furthermore, I know that infinte-dimensional group representations are very valuable in physics as well, but is that even related to the the group representations that arise in "Fourier Analysis on Groups"?

I know I am asking an extremely superficial question, but I don't know anything about Fourier Analysis on Groups and I am trying to understand what it is about.

Answer 1

"Fourier Analysis on Groups" is, as Theo already pointed out, usually understood to be about the irreducible representations of locally compact abelian groups. I would like to add to the existing references:

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995, MR1397028

How is this important for physics? I will pick the most important example that I know from axiomatic quantum field theory (also known as algebraic QFT or Haag-Kastler axiomatic approach).

Reed & Simon talk about semigroups because in quantum mechanics there is often a relationship of observables = essentially selfadjoint operators A and the semigroup of symmetry transformations that an observable generates via $\exp(i t A)$ (this is Stone's theorem).

In quantum field theory one usually talks about Minkowski spacetime which has as symmetry group the Poincaré group. This group has a subgroup, the abelian group of translations. For a quantum system in Minkowski spacetime, we have a Hilbert space $H$ of possible states of the system and a strongly continuous representation of the Poincaré group in the group of unitary operators of $H$. The generators of the groups of translations $T$ correspond to energy (time translations) and momentum (spatial translations), this is the quantum version of the energy-momentum-vector in classical special relativity.

A result of harmonic analysis on locally compact groups is the SNAG theorem (SNAG = Stone-Naimark-Ambrose-Godement).

In the special case of the translation group $T$ it says that there is a spectral measure $P$ such that we have for the translations $U(t)$ this relation:

$$ \mathcal{U}(t) = \int_{k\in \mathbb{R}^n} e^{i \langle t, k\rangle} \mathcal{P}(d k) \qquad \forall t \in \mathcal{T} $$ In this special case we can identify the support of the spectral measure $P$ as a subset of $\mathbb{R}^n$ (which we view as Minkowski spacetime here. For the Minkowski spacetime n is usually set to 4, but I don't see any harm in keeping the n here.)

Now, one part of the famous Haag-Kastler axioms is the positivity of energy: This axiom is needed in order to prevent theories where e.g. the vacuum is instable because it decayes to states with lower and lower energies. This axiom can be formulated, thanks to the SNAG theorem, by stating that the support of the spectral measure $P$ should be contained in the closed positive light cone of Minkowski spacetime. This implies that there cannot be states with a negative eigenvalue of the energy operator.

So, this is an example where a rather deep theorem from harmonic analysis, the SNAG-theorem, is needed in order to understand one of the axioms of an axiomatic approach to quantum field theory.

Answer 2

I believe that some of the deepest results in the Fourier analysis of groups are Harish-Chandra's results on the Plancherel formula for semi-simple Lie groups.

If $G$ is a semi-simple Lie group (e.g. the special linear groups $SL_n(\mathbb R)$, the special orthogonal groups $SO(p,q)$, etc.) then $G$ has a unique (up to scaling) translation invariant measure, the so-called Haar measure, and the $L^2$-space with respect to this measure, i.e. $L^2(G)$, is naturally a representation of $G$ under the (left or right, it's your choice) translation action of $G$.

The problem of the Plancherel formula is to decompose $L^2(G)$ into irreducible representations.

Here are some examples (neither group is semi-simple, actually, but they will give the idea):

If $G$ is the circle group $S^1$, then $L^2(S^1)$ is a (completed) direct sum of the characters $z \mapsto z^n$ (for $n \in \mathbb Z$); this is the theory of Fourier series.

If $G = \mathbb R$ then $L^2(G)$ is the direct integral of the characters $x\mapsto e^{ix y}$ ($y\in \mathbb R$); this is the theory of the Fourier transform.

When $G$ is a non-abelian group like $SL_2(\mathbb R)$ the situation is more complicated, because (a) such groups will admit infinite-dimensional irreducible representations, which can appear in $L^2$, so $L^2$ won't just decompose into one-dimensional characters anymore; (b) unlike in the abelian case, where the space of characters is again a group, and hence homogenous, the collection of representations of a semi-simple group won't be homogenous in any sense, and correspondingly, the decomposition of $L^2(G)$ won't be homogeneous — it can contain both direct sum parts (like in the Fourier series case) and direct integral parts (like in the Fourier transform case).

Harish-Chandra determined the Plancherel formula by first finding the direct sum part for every semi-simple group, and then making an inductive argument on the dimension of the group to understand the direct integral part.

Some more details of this picture can be found in this MO answer.

As for applications to physics, I'm not the right person to comment seriously on that . However, it seems worth noting that the unitary irreducible representations of $SL_2(\mathbb R)$ were first classified by Bargmann, who I think was a physicist, and that Harish-Chandra, who was a student of Dirac, began his work by trying to generalize Bargmann's classification to other semi-simple groups.

Answer 3

The usual Fourier analysis of periodic functions generalizes to finite symmetry groups by decomposing functions $f:G\to\mathbb{C}$ into underlying "representational" components, similar to how waves are mixtures or superpositions of their modes. In the abelian case, where representations are group characters, the basic idea is that the function is a linear combination of unique contributions from each element $\chi$ of the dual group $\hat{G}$, arguably analogous to sound waves being created out of specific contributions from harmonics of various frequencies.

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From what I've read, I gather that Fourier analysis on finite groups is indispensible in quantum algorithms. Specifically in relation to the hidden subgroup problem:

Definition 3.1 (Separates Cosets). $\text{ }$ Given a group $G$, a subgroup $H\le G$, and a set $X$, we say that a function $f:G\to X$ separates cosets of $H$ if for all $g_1,g_2\in G$, $f(g_1)=f(g_2)$ if and only if $g_1H=g_2H$.

Definition 3.2 (The Hidden Subgroup Problem). $\text{ }$ Let $G$ be a group, $X$ a finite site, and $f:G\to X$ a function such that there exists a subgroup $H<G$ for which $f$ separates cosets of $H$. Using information gained from evaluations of $f$, determine a generating set for $H$.

Wikipedia discusses a couple computational motivations behind the problem:

• Shor's quantum algorithm for integer factorization and discrete logarithms (as well as several of its extensions) relies on the ability of quantum computers to solve the HSP for finite abelian groups.
• The existence of efficient quantum algorithms for HSPs for certain non-Abelian groups would imply efficient quantum algorithms for two major problems: the graph isomorphism problem and certain shortest vector problems in lattices. More precisely, an efficient quantum algorithm for the HSP for the symmetric group would give a quantum algorithm for the graph isomorphism. An efficient quantum algorithm for the HSP for the dihedral group would give a quantum algorithm for the poly(n) unique SVP

The exponential speedup found in most quantum algorithms comes from solving HSP efficiently, that is in $O(\mathrm{poly}(\log|G|))$ time. The setup is designed so that a quantum computer's qubits behave as elements of $G$ under the action of certain operators, and the measurement operator behaves as an oracle giving uniformly random evaluations of $f$. The standard algorithm (see HSP link) is:

[...] Quantum Fourier Sampling, or QFS for short. It is the process of preparing a quantum state in a uniform superposition of states indexed by a group, then performing an oracle function, then a quantum Fourier transform, and finally sampling the resulting state to gather information about subgroups hidden by the oracle.

I'm a bit fuzzy on the details myself, as this stuff is mostly out of my league. But quantum Fourier sampling for the hidden subgroup problem is the first and only real application that comes to my mind when I think of Fourier analysis on finite groups, and I think it looks beautiful.