Particles as representations of groups

[tulip]

Hello everyone. I need someone to explain a concept to me.

I'm confused about how a type of particle can be a representation of a lie group. For example, I read that particles with half-integer spin j are a representation of the group SU(2), or that particles with charge q are a representation of U(1). But what is it about the group that the particle actually represents? Do the particles correspond to the generators? Or the group elements? What's the defining feature that a type of particle must have to be a representation of the group?

If anyone can explain this in a way I can understand, I'd be eternally grateful!

 

[haushofer]

How I understand it is the following: a representation of a group acts on a vector space V. This V depends ofcourse on the representation. People often say things like "a particle is an representation of the Poincaré group", by which they mean that a particle is something living in a certain vector space on which the representation of the Poincare group acts.

For instance, the electronfield psi which you plug in the Dirac equation has four components, which you can quite easily derive by using the Clifford algebra. This field appears to be in the (1/2,0)+(0,1/2) representation of the Poincare group. This means that this particular representation of the Poincare group acts on the fermionfield to generate translations and Lorentz transformations.

However, this representation is reducible; it's the sum of two irreps. These irreps correspond to electrons and anti-electrons (an irrep means that there isn't an invariant subspace of V).

Another example is a vector field which lives in the (1/2,1/2) representation of the Poincare group.

 

[xepma]

Hi tulip, and welcome to the forums.

The particles carry a representation space of the (Lie) group around. This representation space can either correspond to the "real" physical space or some internal space of the particle. The elements of the group act on this representation space. The defining feature is, well, sometimes a bit ad hoc and sometimes has a physical reasoning. I'll come back to this, but it's better to give some examples first:

Example: translational invariance corresponds to the (non-compact) group of the manifold R^3 (recall that any Lie group corresponds to some manifold). For translations in 1 direction you simply have the numbers of R, where each number corresponds to the distance you translate a particle over.

If we act with a group element of R on a particle the particle is translated in the corresponding direction and over the corresponding distance. To put this in mathematical terms, we say that the wavefunction of the particle forms a representation of the group. The group elements act on the particle through their action on the representation space. That's it really: the particle carries a representation of the translational group, therefore we can translate it by performing a symmetry operation (=action of a group element)!

Example 2: Particles carrying a spin are said to carry a representation of the group SU(2). For integer values of the spin such a representation is also a representation of the rotational group SO(3). The action of an element of SO(3) corresponds to a physical rotation. The response of the (point!) particle is a rotation in the internal space (the representation space) of the particle. So even though we are dealing with a point particle which has no structure there is still "something happening" when we rotate the particle. The response is even measurable, since we can measure the spin of the particle.

The spin itself is actually a label: it defines the representation space completely (i.e. its dimensionality and the way the group acts on it is completely determined by the spin).

You may wonder why we talk about SU(2) instead of SO(3). There are a few ways to explain this -- the "mathematical" way is by saying that SU(2) is the double cover of SO(3), meaning that any representation of SO(3) is also one of SU(2) (besides, SU(2) is easier to work with since it is simply connected). Another reason is that ordinary representations of SO(3) do not completely exhaust all possibilities (you would not get half-integer spin). But let's not get lost in these technicalities.

Apart from these examples let me also make some comments:
--The thee "inputs" that you need for this structure of physical group representations are:
(a) A (Lie) group, with elements (let's call them a,b,c,...
(b) A representation space V
(c) a representation mapping F

The wavefunction of the particle is then an element of V (call it w). The wavefunction usually has a lot more structure than that, but that's ok. The action of a group element is given by F(a)w (think in terms of vectors and matrices). That's it, really.

Generators and lie algebra can come into play because the construction of F(a) in terms of matrices is not always that trivial.

--The reason why we deal with groups so much is the relative simple structure a group has. Frequently, the action of a group element corresponds to a physical operation (translations, rotations, Lorentz boosts). Such physical operations have: an inverse (just perform the same operation in reverse), an identity element (just do nothing) and are associative (you'll need to convince yourself on this part). If some physical action satisfies these three things you immediatly know that the action corresponds to a group representation! On top of this: a lot the physical actions, like translations and rotations, correspond to continious groups. I.e. we can make infinitesimal transformations, and the resulting group is therefore a Lie group. Lie groups are frequently used because they are easier to study than other infinite dimensional groups. For instance, the representation theory for a special class of Lie groups (the compact Lie groups) is completely solved making live much easier (well, to some degree ofcourse).

--From this it also follows that we can use the representations of the groups to impose certain restrictions on physical particles. For instance, special relativity is in fact completely determined by its corresponding group structure, which is the Poincare group. Since we know that any physical theory has to be compatible with special relativity we can automatically demand that any theory we write down must carry a representation of the Poincare group. The Poincare group is quite a remarkable piece of work, and a b*tch to work through, but in the end it certainly pays off. For instance, the type of Quantum Field Theories you can write down is severely restricted because of the structure of the Poincare group. Since all particles are essentially described by Quantum Field Theories you can imagine the significance of the representation theory of the Poincare group! (which I can assure you, is not a pleasent one).

--You also mention gauge groups (the charge of a particle: U(1)). This is really a field on it's own. The idea is that in order to describe the (internal) structure of the particle (i.e. the full wavefunction) we sometimes need to resort to an overcomplete description. For instance, we can specifiy the global phase of a wavefunction although we know that the global phase never comes into play and is therefore not physical. To be specific [tex]\Psi[/tex] and [tex]e^{i\theta}\Psi[/tex] correspond to the same probability distribution (because you take the absolute square) but are mathematically distinct.

The description that we use is redundant: multiple "representatives" of the wavefunction correspond to the same physical system. Such a redundant structure is called a gauge structure. If two wavefunctions correspond to the same physical system, then we can map one wavefunction to the other via a gauge transformation. On the level of the wavefunction, such a gauge transformation corresponds to the action of a group. This group is called the gauge group. The internal, redundant (non-physical!) space "of redundancies" is called the gauge structure. The overall combination of things is called a gauge "symmetry", although this is not to be confused with a true symmetry of the system.

This is still not the complete story though -- I can explain the rest, but I'll leave it with this for now. In case you have any questions: shoot!

 

[Gerenuk]

I'm also interested in this topic, but I don't think I can learn that to my satisfaction from a forum.
Could you recommend specific books to finally understand it?

 

[Bob_for_short]

...Do the particles correspond to the generators? Or the group elements? What's the defining feature that a type of particle must have to be a representation of the group?

Look at this from an elementary point of view. Take, for example, air temperature. Its measured vaue does not depend on your orientation. They say, T is a scalar of rotational group. Take a wind velocity. Its measured "vaule" depends on your orientation. They say it is a vector of rotational group. The wind velocity in one frame of reference can be recalculated into the wind velocity in another RF with help of some linear transfromation including the rotation angle. The velocity is a group element. All possible velocities form a group called rotational group representation. Generators and the angle give you the way how to recalculate the wind velocity from one RF to another. The same is valid in a general case. There are scalar, spinor, vector particles, and even tensor ones. Their states transform while RF changing.

 

[George Jones]

I'm also interested in this topic, but I don't think I can learn that to my satisfaction from a forum.
Could you recommend specific books to finally understand it?

Try the book is Relativity, Groups, and Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics by Roman U. Sexl and Helmuth K. Urbantke, starting with chapter 6.

https://www.amazon.com/dp/3211834435/?tag=pfamazon01-20.

 

[Fredrik]

I'm confused about how a type of particle can be a representation of a lie group.

That's not quite right. Each particle species corresponds to a different irreducible representation of the Poincaré group (or rather its universal covering group...an annoying mathematical technicality). I'm not sure how to deal with charges or other "intrinsic" properties (that aren't mass or spin). I guess we just start with a larger symmetry group. Perhaps someone else can clear that up.

Do the particles correspond to the generators? Or the group elements? What's the defining feature that a type of particle must have to be a representation of the group?

The generators are observables. The group elements are symmetries of the theory. The irreducible representations of the Poincaré group are characterized by different values of mass and spin.

I'm also interested in this topic, but I don't think I can learn that to my satisfaction from a forum.
Could you recommend specific books to finally understand it?

"The quantum theory of fields", by Steven Weinberg. Specifically, volume 1, chapter 2. This post can perhaps give you an idea about what it's about.

 

[tulip]

Thank you everyone for the replies, they've got things a bit clearer in my head. I'll take a look at the book recommendations when I get to the library tomorrow.

xepma, I really liked the way you described things, but there's just one thing I'm still confused about. If U(1) is a group that acts to change the phase of the wavefunction, and the phase is a non-physical property of a particle, doesn't that mean that systems of any type of particle will have a symmetry under U(1)? Noether's theorem tells us that symmetries lead to conserved charges - so what is the conserved charge associated with this phase change symmetry?

As I understood it, a U(1) symmetry exists in the theory of QED (which applies to charged particles only) where it gives rise to the conservation of electric charge - is this correct?

 

[xepma]

Thank you everyone for the replies, they've got things a bit clearer in my head. I'll take a look at the book recommendations when I get to the library tomorrow.

xepma, I really liked the way you described things, but there's just one thing I'm still confused about. If U(1) is a group that acts to change the phase of the wavefunction, and the phase is a non-physical property of a particle, doesn't that mean that systems of any type of particle will have a symmetry under U(1)? Noether's theorem tells us that symmetries lead to conserved charges - so what is the conserved charge associated with this phase change symmetry?

A very astute observation! The U(1) symmetry associated with the global phase invariance is indeed valid for all sorts of particles. The conserved quantity associated with this symmetry is the number of particles.

As I understood it, a U(1) symmetry exists in the theory of QED (which applies to charged particles only) where it gives rise to the conservation of electric charge - is this correct?

My first comment is that this type of symmetry is a different type of symmetry then the U(1) global phase invariance. This really is the domain of gauge theory.

The idea is as follows: the representations I mentioned above are associated with the global properties of the wavefunction / particle.

Recall that when we act on the wavefunction with a symmetry operation h, we actuall mean that we act on the representation space V through the mapping F(h) (where F is the representation of the action of h on V). The wavefunction is said to live in the representation space V. We talk of a global symmetry, since the entire wavefunction is an element of one representation space.

In gauge theory this idea is extended. The global symmetry is turned into a local symmetry. You can imagine that we can assign a representation space V to each spatial point x, giving V(x). Performing a symmetry operation means that we perform a symmetry operation at each point x. We can perform a different symmetry operation at each point x, i.e. the group "element" is x-dependent: h(x). For instance, a local U(1) phase transformation now looks like [tex]e^{ie\phi(x)}[/tex].

I assume here that although we have an infinite number of representation spaces, V(x), we still have "the same" type of representation at each point x (i.e. same dimension etc). This representation is labeled by the charge e of the particle with respect to the symmetry group G. This is the mathematical meaning of the charge: it labels the kind of representation the particle carries (for both local and global symmetries).

For a U(1) symmetry we simply have the electrical charge. For non-Abelian local symmetries, like SU(3), it leads to the notion of color charge. Quarks carry red, green or blue charge, meaning they live in the 3-dimensional representation of SU(3), called the fundamental representation. Gluons, on the other hand, live in the 8-dimensional representation space of SU(3), called the adjoint representation.

The fact that we now have a local symmetry has major consequences. We know now that we change the phase of the wavefunction in a space-dependent way, i.e.

[tex]\Psi(x,t) \rightarrow e^{ie\phi(x)}\Psi(x,t)[/tex]

If we have an action of the field [tex]\Psi[/tex] we always deal with the kinetic term. The kinetic term compares the value of the field at different points x. Put differently, the kinetic term is evaluated through the derivative of the field. To make sense of this term we take the function [tex]\phi(x)[/tex] to be a smooth and differentiable.

If you evaluate a kinetic term such as: [tex]\partial_{\mu}\Psi^* \partial^{\mu}\Psi(x,t)[/tex] you'll notice that the differential operator creates a term proportional to [tex]\partial_\mu \phi(x)[/tex]. This means that the action of a free field, containing a kinetic term is not invariant under local gauge transformations! Local gauge symmetry does not exist in a free theory because of the kinetic term.

To solve this one introduces another field, [tex]A[/tex], which also transforms under gauge transformations (but differently). The transformation is "defined" such that the overall action is invariant under local gauge transformations again. This field is called a gauge field, and for a gauge group U(1) this field is identified with the Electromagnetic vector potential.

The moral of the story is that a non-interacting theory is not locally gauge invariant, only globally. However, we can impose local gauge invariance at the cost of introducing a coupling to another field, the gauge field. The gauge field cancels the anomalous terms, thus generating local gauge invariance. In addition, this new gauge field can have dynamics as well.

A final remark: At first instance it looks like that we have changed the spacetime manifold M into the larger product manifold, MxV, where V is the representation space. But this structure is too simple and does not encapture all the physics. For that you need the notion of a fiber bundle. The particle fields then live in the space E, which is a fiber bundle. Locally, the bundle looks like the product space MxV, but globally the space may have non-trivial topological properties (this is the geometrical origin of the Aharanov-Bohm effect).

 

[Atakor]

Hello,

nice presentation xepma !

A final remark: At first instance it looks like that we have changed the spacetime manifold M into the larger product manifold, MxV, where V is the representation space. But this structure is too simple and does not encapture all the physics. For that you need the notion of a fiber bundle. The particle fields then live in the space E, which is a fiber bundle. Locally, the bundle looks like the product space MxV, but globally the space may have non-trivial topological properties (this is the geometrical origin of the Aharanov-Bohm effect).

Any reference with this geometrical 'interpretation' of gauge theories please ?

something troubles me, do gauge bosons necessarily belong to the adjoint representation ?

 

[xepma]

Any reference with this geometrical 'interpretation' of gauge theories please ?

The book Gauge Fields, Knots, and Gravity by Baez and Muniain is precisely what you're looking for.

something troubles me, do gauge bosons necessarily belong to the adjoint representation ?

More specifically: the gauge field is actually a connection -- similar to, but not entirely the same as the connection you find in gravity. In particular, the gauge field has an associated curvature, which is interpreted as the field strength. The components of the connection A take on values in the Lie algebra g of the Lie group G. In some sense you can argue that this is the "adjoint rep".

 

[Atakor]

thanks, but from a purely group-theory or physical point of view..how do you explain it ?

 

[tulip]

xepma - thanks again for a very helpful and clear response. I don't know anything about fiber bundles, but the rest of what you said made sense.