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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.

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The starting point is Fourier analysis on locally compact abelian groups, which can be seen as a simultaneous generalization of the Fourier transforms on $\mathbb{Z}$, $\mathbb{R}$ and $\mathbb{T}$ for the three most classical cases. It can also be seen as the theory of irreducible representation of abelian groups (since all irreps of abelian groups are $1$-dim). See Rudin's eponymous book for more on that. Also Pontryagin duality should be mentioned. –  t.b. Aug 19 '11 at 14:33
For the bare (mathematical) theory, I think the book by Dales et. al is an excellent and readable introduction (the harmonic analysis parts by Willis are great!). Look at my answer here for links to three classics: Loomis, Reiter and Hewitt-Ross and some other relevant books. Folland (mathematical) and Barut–Rączka (physical) are excellent more modern references for harmonic analysis. –  t.b. Aug 19 '11 at 14:38
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4 Answers

up vote 12 down vote accepted

"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.

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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.

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Dear Matt, Could you be so kind as to point me to some expositions of the aspects of Harish-Chandra's work you're discussing here and on MO? Thanks in advance! –  t.b. Aug 20 '11 at 9:52
@Theo: Dear Theo, I learned a lot of this from Anthony Knapp's "Representation theory of semisimple groups: an overview based on examples". Some of the cultural background, as well as a sense of the big picture, I got from Rebecca Herb's article "Harish-Chandra and his work" (in the Bulletin of the AMS) (which has the advantage of being a medium-length article, rather than a rather long book!). Best wishes, –  Matt E Aug 24 '11 at 5:57
Dear Matt, thanks a lot for the reference to Rebecca Herb's beautiful article (to the extent that I can judge that after a first reading). I was completely unaware of it and it is exactly the kind of thing that I hoping to get. I wanted to tell you many times already how well you succeed in achieving what you describe in your answer here. You may have noticed that number theory is rather foreign to me, but whenever I see an answer of yours there's something in it for a layman like me as well. Thank you so much for all your efforts! Best wishes, –  t.b. Aug 24 '11 at 21:17
@Theo: Dear Theo, Thanks for your kind words, and I'm glad that you like Rebecca Herb's article. Best wishes, –  Matt E Aug 25 '11 at 1:10
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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.

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.

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Answer for the question"What is Fourier Analysis on Groups and does it have “applications” to physics?"

I have engaged since 1990, when I was student in Ph.D program and I had published two papers in noncommutative harmonic analysis on the motion group Rⁿ⋊K , where K is a connected compact Lie group. Since then I have done a lot to enlarge my research on the Lie groups. My research area in Mathematics is the opening a new ways in noncommutative Fourier analysis (abstract harmonic analysis) on Lie groups to obtain the solution of the major problems in Fourier analysis on Lie groups. Abstract harmonic analysis is a beautiful and powerful area of pure mathematics that has connections to, theoretical physics, chemistry analysis, algebra, geometry, solving problems in robotics, image analysis, mechanics. and the theory of algorithms. In mathematics. Abstract harmonic analysis on locally compact groups is generally a difficult task. In the second half of twenty century, two points of view were adopted by the community of the mathematics: The first one is the theory of representations of Lie groups. Unfortunatly If the group G is no longer assumed to be abelian, it is not possible anymore to consider the dual group G (i.e the set of all equivalence classes of unitary irreducible representations). For a long time, people have tried to construct objects in order to generalize Fourier transform and Pontryagin,s theorem to the non abelian case. However, with the dual object not being a group, it is not possible to define the Fourier transform and the inverse Fourier transform between G and G. These difficulties of Fourier analysis on noncommutative groups makes the noncommutative version of the problem very challenging. It was necessary to find a subgroup or at least a subset of locally compact groups which were not "pathological", or "wild" as Kirillov calls them. Here are some interesting examples of these groups. So this point view to do abstract harmonic analysis on locally compact groups is generally a difficult task due to the nature of the group representations.There was little success in this theory, for example Mautner and Segal had introduced the Plancherel formula in 1950, for the type I unimodular Lie group, so that the following held


for all f∈L²(G), where G is the set of all irreducible unitary representations of G, π∈G , dμ is the the Plancherel measure on G and ‖π(f)‖_{H.S}² is the Hilbert-Schmidt norm of the operator π(f). The second is the quantum groups, which was introduced by Vladimir Drinfeld and Michio Jimbo. some little results were obtained by this theory. Professor Dr. A., Van Daele wrote in his paper, "The Fourier transform in quantum group theory, preprint (math.RA/0609502 at http://lanl.arXiv.org) 2007", wrote, I will illustrate various notions and results using not only classical Fourier theory on the circle T, but also on the additive group Qp of p-adic numbers. It should be observed however that these cases are still too simple to illustrate the full power of the more general theory. To do Fourier analysis, one needs an integral on A as well as on the dual of A. This is the case when A is a multiplier Hopf algebra with integrals. Then, the dual of A can be considered and it is again a multiplier Hopf algebra with integrals. The theory of Hopf algebras is not sufficiently general for this purpose because in this case, requiring an integral both on A and its dual, forces A to be finite-dimensional. Therefore, many interesting cases and examples can not be treated if we stick to the theory of Hopf algebras. Consequantly these programs had certain limited success.

Professor Shahn Majid wrote in his paper " What is the Quantum Group" Notices of the AMS (2006). There are three points of view leading independently of the four axioms of Hopf algebra. Each of them defines what is quantum group. As well known, the second and third points view are the same major problems in abstract harmonic analysis (Fourier anlysis on non commutative Lie groups), on non abelian locally compact Lie groups mentioned in the above, which are (I)- The first is to construct object G which will be the dual group of G in order to do the Fourier transform on G and more generalize Pontryagin's theorem to the non-abelian case. (II)-The second is to study the Fourier analysis on non abelian locally compact Lie groups. (III)- The third is to study the group algebra of non abelian locally compact Lie groups as non commutative Banach algebra, enveloping algebra, ….and their ideals. So still now neither the theory of quantum groups nor the representations theory have done to reach this goal

The important and interesting question is: One can do abstract harmonic analysis on Lie groups i.e. the Fourier transform can be defined to solve the above problems. Recently, these problems found a satisfactory solution with the papers. Therefore, I would like to attire your attention on the ideas of my research which focus on the abstract harmonic analysis (Fourier analysis on non commutative Lie groups) that I can summarized in two ways: The first way is that the solvability of Lewy operator and the invalidity of Hormander,s condition for the solvability of differential operators with coefficient variables, which established in my papers "note on the solvability of the Lewy operator" and "note on the solvability of the Mizohata operator" in International Mathematical Forum. I believe that these papers will be the business of the expertise in the theory of partial differential equations with variable coefficients as their solutions( functions or numerical) and their applications The second, I believe with trust that I can solve the three major problems for nilpotent Lie groups, and completely solvable Lie groups, Motion group ≃ Rⁿ⋊K, where K is a connected compact Lie group Galelian group ≃ H⋊ SO(3, R), Poincare group (Space time) ≃R⁴⋊SL(2, ℂ)≃R⁴⋊SO(3,1), Jacobi group J≃H⋊ SL(2, R), where H is the 3-dimentioal Heisenberg group, GL₊(n, R). More, in my recent paper "Fourier Transform and Plancherel Formula for the Lorentz Group", I have proved that the sets R_{- , }^{∗} O₋(n, R), and GL₋(n, R) , each one have a structure of group isomorphic onto R_{ , } SO(n, R), and GL₊(n, R) see my papers ( go to google and just write Kahar el Hussein to chick my contribution in this field) . So, I believe that these ways will be the business of the expertise in the theory of noncommutative Fourier analysis on Lie groups, and Physicists, and that is what I am interested. For example to achieve this goal for the Poincare group(Space time) ≃R⁴⋊SL(2, ℂ)≃R⁴⋊SO(3,1) we begin by defining the Fourier transform and establishing Plancherel Formula for the Motion group and the complex semisimple Lie group SL(2,ℂ), and then the spacetime(Poincare group).

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