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I have been looking for a clear expository mathematical text on the relation between the theory of elementary particles and the representation theory of $U(1), SU(2), SU(3)$, I was very disappointed about The step which explains why an elementary particle should correspond to an irreducible representation! I asked physicists to explain The step and I get more confused then before asking. I was wondering if someone could explain this passage with mathematical rigor and physical principles! And pretty sure I'm not the only one in this situation :)

Thanks.

Edit Thank you very much all for your answers, unfortunately I can't choose one! It was great with lot of gratitude to all of you, still I do not understand the key step probably I need to learn more physics to feel what is really happening.

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    $\begingroup$ I asked exactly this question here: mathoverflow.net/questions/16074/… $\endgroup$ – Qiaochu Yuan Mar 16 '17 at 19:07
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    $\begingroup$ If I remember correctly, the first chapter of vol 1 of the QFT books by Steven Weinberg is good at explaining "the step" from scratch (Wigner's theorem about transition probability preserving transformations on rays being representable by unitary or antiunitary maps) all the way to intrinsic definitions of mass and spin for an elementary particle. As Qiaochu said in his nice answer below the groups $U(1),...$ appearing as gauge groups, that's further down the line. $\endgroup$ – Abdelmalek Abdesselam Mar 16 '17 at 21:50
  • $\begingroup$ @Qiaochu Yuan: Please double-check your above link. It loops to itself. $\endgroup$ – Qmechanic Mar 17 '17 at 11:49
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    $\begingroup$ Whoops. Here's another variant of this question with more physics-y answers: mathoverflow.net/questions/30480/… $\endgroup$ – Qiaochu Yuan Mar 17 '17 at 17:47
  • $\begingroup$ It's not so simple, see e.g. en.wikipedia.org/wiki/Weinberg_angle . $\endgroup$ – jjcale Mar 18 '17 at 16:58
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You can understand this philosophy as a generalization of Noether's theorem. Let me only state Noether's theorem in the quantum case because it's actually easier to understand there than in the classical case (for me, anyway).

As setup, we have a Hilbert space of states $V$ and a state vector $\psi \in V$ which evolves according to a Hamiltonian $H$, meaning (in natural units, so $\hbar = 1$) that

$$\psi(t) = e^{iHt} \psi.$$

Suppose in addition that we have a one-parameter family $g(t)$ of symmetries of our quantum system, meaning that $g(t)$ commutes with the Hamiltonian: $g(t) H = H g(t)$. In particular, $g(t)$ is a one-parameter family of unitary maps, and so by Stone's theorem $g(t)$ must have the form

$$g(t) = e^{iAt}$$

for some self-adjoint operator $A$ (which in physically relevant examples is often unbounded). (In the finite-dimensional case this is just saying that the Lie algebra of the unitary group is the Lie algebra of skew-adjoint matrices.) Noether's theorem is the observation that this means $A$ must also commute with $H$, which means that it is an observable which is conserved under time evolution in the sense that

$$e^{iHt} A e^{-iHt} = A$$

(time evolution of $A$ looks like conjugation in the Heisenberg picture). This general observation reproduces many of the familiar conserved quantities in physics. To give two examples:

  • If $g(t)$ is translation in a space direction, $A$ is momentum in that direction. For example, if $g(t)$ is translation in the $x$ direction, $A = i \frac{\partial}{\partial x}$.
  • If $g(t)$ is rotation around an axis, $A$ is angular momentum around that axis.

Because $A$ is a conserved quantity, it's natural to break up $V$ into eigenspaces of $A$ (corresponding to states where $A$ has a definite value), and the reason is that time evolution preserves all of these eigenspaces. This means that the statement "$\psi$ belongs to such-and-such eigenspace" is physically meaningful, e.g. the statement that $\psi$ has a fixed momentum.

The connection to representation theory comes from thinking of $g(t)$ as a representation of $\mathbb{R}$, so that the eigenspaces of $A$ are the isotypic components of this representation. Irreducible representations correspond to eigenvectors, which are, as above, states where $A$ has a definite and fixed value.

Now many physical systems come with a noncommutative group of symmetries, so it's natural to generalize $g(t)$ to an action of a nonabelian Lie group $G$, for example $SO(3)$, which we again posit to commute with $H$. What we might call the generalized Noether theorem is the observation that this implies that time evolution preserves the decomposition of $V$ into isotypic components of this representation, so it's again physically meaningful to say things like "$\psi$ belongs to the isotypic component corresponding to such-and-such irreducible representation" (in physics language, "$\psi$ transforms under...") because such statements are preserved by time evolution. This is the beginning of Wigner's classification (although that classification is relativistic whereas this story I've been telling is decidedly not so some tweaks need to be made). So you can think of the irrep a state belongs to as a "generalized conserved quantity."

(The reason we want to consider irreps is that they give more precise information while continuing to be physically meaningful. I could talk about e.g. particles whose momentum lies in a certain range instead of talking about particles with particular values of their momentum, but the latter is more precise so I do that first.)

The relationship to the groups $U(1), SU(2), SU(3)$ appearing in the standard model requires a bit more elaboration, because these groups don't act by physical symmetries (like the Poincare group) but by gauge symmetries. But that's a story that's a bit outside my competence to describe the physical relevance of. I can tell you that the $U(1)$ factor corresponds to charge conservation.

I should mention that I asked exactly this question awhile ago, and after thinking about the answer I got I wrote this blog post about a toy model of quantum mechanics on a finite graph that you might find helpful.

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  • $\begingroup$ My understanding is that the gauge group associated to a compact group $G$ is something like the group of smooth $G$-valued functions on $\mathbb R^n$. This immediately leads to the observation that a finite-dimensional unitary representation of this group can be constructed by evaluating the function at points $x_1,\dots,x_n$ and then applying irreducible representations $\pi_1,\dots,\pi_n$. Presumably this is how point particles arise from field theories in quantum physics. $\endgroup$ – Will Sawin Mar 23 '17 at 5:33
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The Algebra of Grand Unified Theories, by John Baez and John Huerta may well be to your liking:

A full-fledged treatment of particle physics requires quantum field theory, which uses representations of a noncompact Lie group called the Poincaré group on infinite-dimensional Hilbert spaces. This brings in a lot of analytical subtleties, which make it hard to formulate theories of particle physics in a mathematically rigorous way. In fact, no one has yet succeeded in doing this for the Standard Model. But by neglecting the all-important topic of particle interactions, we can restrict attention to finite-dimensional Hilbert spaces: that is, finite-dimensional complex inner product spaces. This makes our discussion purely algebraic in flavor.

We start from a compact Lie group, say $G$. Particles then live in representations of $G$ on a finite-dimensional Hilbert space $V$. More precisely: $V$ can always be decomposed as a direct sum of irreducible representations (irreps), and elementary particles are basis vectors of irreps. This provides a way to organize particles, which physicists have been exploiting since the 1960s.

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In quantum mechanics, state is encoded as a unit vector in a Hilbert space. If we just studied $\mathbb{C}^N$ and $L^2(\mathbb{R})$, life would be boring and we would be missing out on a lot of interesting physics and mathematics.

Here is a more general setup. Let $M$ be a manifold. Choose a principal $G$-bundle $P \to M$ and a finite dimensional $G$-representation $V$. Then $E = P \times_G V$ is a $G$-vector bundle on $M$. If $G \subseteq U(n)$ for some $n$, then we have a Hermitian product on $E$, so we can talk about $L^2(E)$. This is a Hilbert space. Sections of $E$ are called "matter fields" or quantum particles. You get different species of particles from different representations $V$. For example, protons an neutrons correspond to taking $G = SU(2)$ and $V = \mathbb{C}^2$ the first fundamental representation.

In quantum mechanics, we need to talk about observables. We can multiply sections of $E$ by sections of the trivial bundle $M \times \mathbb{R}$. This generalizes position. Momentum is more interesting. If we want to differentiate sections of $E$, we need to choose a connection on $P$. In physics, this is called a gauge field and generalizes the electromagnetic potential (the curvature is called the field strength). Things get more interesting once you realize that connections on $P$ are the same as sections of the adjoint bundle $P \times_G \mathfrak{g}$. In other words, you can work with gauge fields just like you work with the other particles.

Things get more subtle if you want to work relativistically or do quantum field theory.

EDIT: Let me talk about the relationship between sections and particles a bit more. In statistics and probability theory, we assign probabilities to logical statements. For example, if $X$ is a random variable taking values in the finite set $S = \{ a,b,c \}$, then we assign a probability to the statement $X = a$. In quantum theory, we assign amplitudes (complex numbers) to logical statements. We can talk about a quantum particle in $S$. It is specified by a function $\psi : S \to \mathbb{C}$ and $\psi(a)$ is the amplitude that the particle is in state $a$.

If we have a quantum particle in the real line, then its position is going to be specified by a function $\psi : \mathbb{R} \to \mathbb{C}$. Now suppose that we have a quantum particle in the real line which can be in one of two states $a$ or $b$. In this case, the state is specified by a function $\psi : \mathbb{R} \to \mathbb{C}\{ a,b \}$. More generally, if we are talking about quantum particles with internal state on a manifold, we could specify them as functions $M \to V$ but it is useful to also discuss sections of more general vector bundles.

I learned all this from reading QED and thinking about the analogies between statistics and quantum mechanics.

If I had to explain this in one sentence, I would say something like "Quantum particles behave more like random variables than points in space, but you use amplitudes instead of probabilities"

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    $\begingroup$ I suppose the question is mainly about why/how can sections of $E$ be viewed as particles. Concerning me, I do not have any associations whatsoever between a section of a vector bundle and a particle. Explain to a layman. You tell me to multiply $E$ by some trivial bundle, choose a connection on $P$, etc. OK. Now here is a section of $E$. And here is a proton, somewhere in ${\mathbb R}^3$, at some moment of time, moving in some direction with some speed. How to use the above gadgets to relate these two? $\endgroup$ – მამუკა ჯიბლაძე Mar 16 '17 at 19:53
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    $\begingroup$ @მამუკა ჯიბლაძე: historically the first case that motivated the use of vector bundles was (quantum) electrodynamics, where the vector bundle is a complex line bundle on spacetime and the state vector of an electron is a section of this line bundle. You might ask why not just always take it to be the trivial line bundle, so just work with complex functions instead, but Dirac found physical reasons to believe that the line bundle could be nontrivial (the "Dirac charge quantization argument"). Another interesting effect is that the line bundle comes equipped with a connection whose curvature... $\endgroup$ – Qiaochu Yuan Mar 17 '17 at 1:13
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    $\begingroup$ ...describes the electromagnetic field, so that the connection describes the potential. Now it can happen that the curvature vanishes, so the connection is flat, but nevertheless the connection is nontrivial. This has no effects classically because it means the electromagnetic field vanishes, but does have effects quantumly (en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect). $\endgroup$ – Qiaochu Yuan Mar 17 '17 at 1:15
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    $\begingroup$ A maybe more cryptic comment is that the connection / potential still only matters "up to gauge symmetry," meaning up to isomorphism as an object in the groupoid of line bundles with connection. Now "groupoid of trivial line bundles with connection" does not form a sheaf (really I mean a stack here), but it has a sheafification (stackification), which is "groupoid of line bundles with connection." So one has to admit the possibility of nontrivial bundles in order to get a coordinate-independent notion of electromagnetic potential which is "truly local" in the way that physicists like. $\endgroup$ – Qiaochu Yuan Mar 17 '17 at 1:23
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    $\begingroup$ @მამუკა ჯიბლაძე: in the simplest case, $M$ is Minkowski space $\mathbb{R}^{3, 1}$, the bundle is a trivial line bundle, and so the section is just a function $\psi : M \to \mathbb{C}$ and describes the state of, say, an electron. At a spacetime point $m \in M$, $| \psi(m) |^2$ is proportional to the probability density of finding the electron at that point in time and space. (Although I am less familiar with the relativistic than nonrelativistic setting here so I'm less confident about this than I could be.) $\endgroup$ – Qiaochu Yuan Mar 17 '17 at 8:05
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The answers given so far are all beautiful, but lack one detail which I think should be clarified. I'll try to answer the question concerning The Step in a more intuitive way, but essentially following John Baez as quoted by Carlo Beenakker.

First of all, the definition of elementary particle is not entirely rigorous. In high energy physics, the phrase elementary particle is unfortunately used for truly elementary particles (these are stable particles, which do not decay into others) and quite a lot of unstable particles.

Let me start with the stable elementary particles. The very fact that they are called elementary means that all mathematical structures needed to describe their properties should be such that these cannot be decomposed into smaller entities. Otherwise, it would be extremely likely that the particle described with these mathematical structures could also be decomposed into "smaller" particles, it would not be stable, or it could at least be fragmented by external forces. Thus, if properties of particles are related to representations of (Lie) groups, stable elementary particles should be related to the smallest possible representations, i.e. the fundamental representations (which happen to be irreducible).

Now, all the other particles, which happen to be not stable, will ultimately decay into stable elementary particles. Thus, they should be related to representations which can be decomposed into fundamental ones. A particle can only decay into other particles, if the tensor product of all the representations related to the fragments contains the representation of the original particle.

High energy experiments produce quite a lot of states which physicists like to associate with particles or excited states of particles. Since the compact Lie groups, which typically appear in physics, have the property that all of their finite dimensional representations can be decomposed into direct sums of irreducible representations, it became customary to associate the word particle in the sense of building block of physical entities to irreducible representations. But as explained above, only stable elementary particles are naturally associated with particular representations, the fundamental ones. For the others, it is merely a convenient way to get some order into the zoo of high energy physics. As Baez writes, working with the irreps is a convenient choice of basis.

Finally, physicists nowadays use the word quasiparticle for a lot of physical entities, which behave like particles, i.e. like excitations of quantum fields of some effective quantum field theories, in particular in condensed matter systems. Again, the word is defined to refer to entities which have well defined representation theoretic properties with respect to all groups involved.

I know, this is a very sloppy answer. However, I found it important to shift the focus towards the important distinction between truly elementary stable particles and all the other "elementary" particles.

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