Dumb question about representation theory?

[pellis]

It seems that there’s a loose way of discussing group representations that’s fine for the initiated, but confusing for the neophyte:

I understand that the objects that represent group operations are usually (or always?) square matrices, which naturally can be described as “matrix representations” (of the group operations).

In which case, what exactly (in natural language) is something described as e.g. “a spinor rep” or “a tensor rep”?

Does that mean:

- The matrix representation of a group (symmetry) operation acting ON something that is a spinor, or (…whatever)?

OR

- A group operation REPRESENTED BY a spinor (i.e. the representation of a group (symmetry) operation is itself a spinor? (But then, acting on what?))

What confuses this slightly more is that a spinor can itself be expressed as a square matrix, as can a tensor, so it looks like it is possible for a matrix representation of an operation to also be itself a spinor or tensor? Can this be so?

Could some natural-language-capable person please clarify.

Thank you.

 

[kloptok]

I understand your confusion, as I have encountered it myself. If I understood the answer I got correctly it all stems from a misuse of the word "representation" in some texts. A representation is a group of matrices (actually operators which can be represented my matrices I think) that obey the group axioms under matrix multiplication, where each group element is represented by a matrix. So you seem to have got that right. Now, what some texts do is that they refer to the vector space that the group acts on as the representation instead of the group elements, as represented by matrices/operators.

So for example the spinor space is the space that the spinor representation acts on, and confusingly enough some authors refer to the spinor space as the representation.

 

[pellis]

Many thanks - that helps a lot.

 

[HallsofIvy]

I might point out that "group representations" is a mathematics concept that is used in physics, not a directly physics concept.

 

[pellis]

Thank you for the reminder. Even so, Wu-Ki Tung, Group Theory in Physics, Vol 1, states: “The natural correspondence between the basis vectors of unitary irreducible representations of P [the Poincare’ Group] and quantum mechanical states of elementary physical systems stands out as one of the remarkable monuments to unity between mathematics and physics” (my italics.)

So can you take my query a step further? Have you a sense of what, in physical terms, an irreducible representation corresponds to? I do know what an irrep is mathematically, but I don’t fully understand it, and think I would understand it better if I could recognise some examples of the physical aspect.

The nearest I can to get to it comes from theoretical chemistry: in which atomic and molecular orbitals appear to ‘correspond to’ particular irreps. But while the number of irreps equals the number of symmetry classes, the character tables show that irreps are not identical to symmetry classes. In which case it seems that each distinct irrep embodies a “basic” symmetry ‘type’ (?) that relates to (or depends on?) all the symmetry operations of the system, and which is in some sense physically independent of (mathematically: orthogonal to) the other irreps of the system.

This is illustrated in Wikipedia: http://en.wikipedia.org/wiki/Molecular_symmetry
“Consider the example of water (H2O), which has the C2v symmetry. The 2px orbital of oxygen is oriented perpendicular to the plane of the molecule and switches sign with a C2 and a σv'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, −1, 1, −1}, corresponding to the B1 irreducible representation. Likewise, the 2pz orbital is seen to have the symmetry of the A1 irreducible representation, 2py B2, and the 3dxy orbital A2.”.

So it would appear that an irrep, in a molecular system, characterises the spatial symmetries of orbitals with respect to all the spatial symmetry operations of the whole molecule? Is there a more physical/visualisable way to express this?

But where this simplistic view gets more interesting is when it’s stated that fundamental particles correspond in some way to the irreps of the Poincare’ group (as quoted from Wu-Ki Tung, above).

- Presumably the symmetry operations are then in parameter spaces rather than the physical space spanned by atomic orbitals?
- And how does one visualise (in any sense) what characteristics of the fundamental particles correspond to their irreps?

 

[pellis]

Does no-one feel able to answer my (possibly dumb) query about what, in physical terms, an irreducible representation corresponds to, in the cases of:

a) molecular systems (is it as outlined in my preceding post? If it is “the spatial symmetries of orbitals with respect to all the spatial symmetry operations of the whole molecule” then is there a more physical/visualisable way to express this?”

b) fundamental particles: how can they be seen as irreps of the Poincare' Group?

a. In what space(s) do the symmetry operations act?
b. Does each particle have it's own unique irrep?
c. In which case how many classes of symmetry operations does the Poincare' group have (?) as it seems fundamental also that the number of irreps = the number of classes of the group
d. So does (c) suggest an upper limit to the number of fundamental particles?