Tag Archives: representation theory

The Representation of group G on vector space V is really a left action of the group on the vector space

Say what? What does this have to do with quantum mechanics? Quite a bit. Practically everything in fact. Most chemists learn quantum mechanics because they want to see where atomic orbitals come from. So they stagger through the solution of the Schrodinger equation where the quantum numbers appear as solution of recursion equations for power series solutions of the Schrodinger equation.

Forget the Schrodinger equation (for now), quantum mechanics is really written in the language of linear algebra. Feynman warned us not to consider ‘how it can be like that’, but at least you can understand the ‘that’ — e.g. linear algebra. In fact, the instructor in a graduate course in abstract algebra I audited opened the linear algebra section with the remark that the only functions mathematicians really understand are the linear ones.

The definitions used (vector space, inner product, matrix multiplication, Hermitian operator) are obscure and strange. You can memorize them and mumble them as incantations when needed, or you can understand why they are the way they are and where they come from. So if you are a bit rusty on your linear algebra I’ve written a series of 9 posts on the subject — here’s a link to the first https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/– just follow the links after that.

Just to whet your appetite, all of quantum mechanics consists of manipulation of a particular vector space called Hilbert space. Yes all of it.

Representations are a combination of abstract algebra and linear algebra, and are crucial in elementary particle physics. In fact elementary particles are representations of abstract symmetry groups.

So in what follows, I’ll assume you know what vector spaces, linear transformations of them, their matrix representation. I’m not going to explain what a group is, but it isn’t terribly complicated. So if you don’t know about them quit. The Wiki article is too detailed for what you need to know.

The title of the post really threw me, and understanding requires significant unpacking of the definitions, but you need to know this if you want to proceed further in physics.

So we’ll start with a Group G, its operation * and e, its identity element.

Next we have a set called X — just that a bunch of elements (called x, y, . . .), with no further structure imposed — you can’t add elements, you can’t mutiply them by real numbers. If you could with a few more details you’d have a vector space (see the survival guide)

Definition of Left Action (LA) of G on set X

LA : G x X –> X

LA : ( g, x ) |–> (g . x)

Such that the following two properties hold

l. For all x in X LA : (e, x) |–> (e.x) = x

2. For all g1 and g2 in G LA ( g1 * g2), x ) |–> ( g1 . (g2 . x )

Given vector space V define GL(V) the set of invertible linear transformations (LTs) of vector space. GL(V) becomes a group if you let composition of linear transformations become its operation (it’s all in the survival guide.

Now for the definition of representation of Group G on vector space V

It is a function

rho: G –> GL(V)

rho: g |–> LTg : V –> V linear ; LTg == Linear Transformation labeled by group element g

The representation rho defines a left group action on V

LA : (g, v) |–> LTg (V) — this satisfies the two properties above of a left action given above — think about it.

Now you’re ready for some serious study of quantum mechanics. When you read that the representation is acting on some vector space, you’ll know what they are talking about.

Book recommendation

Tired of reading books about physics?  Want the real McCoy”?  Well written and informal?  Contains stuff whose names you know but don’t understand — Jones Polynomial, Loop Quantum Gravity, Quantum field theory, Gauge groups and transformations —  etc. etc.

Up to date?  Well no, it’s 25 years old but still very much worth a read, so very unlike molecular biology, chemistry, computer science etc. etc.

Probably you should know as much physics and math as a beginning chemistry grad student. If you studied electromagnetism through Maxwell’s equations it would be a plus.  I stopped at Coulomb’s Law, and picked up enough to understand NMR.

This will give you a sample of the way it is written

“Much odder is that we are saying the vector field v is the linear combination of . .  partial derivatives.  What we are doing might be regarded as rather sloppy, since we are identifying two different although related things: the vector  field and the operator v^i * d-/dx^i which takes a directional derivative in the direction of v.”

“Now let us define vector fields on a manifold M. .. . these will be entities whose sole ambition in life is to differentiate functions”

The book is “Gauge FIelds, Knots and Gravity” by John Baez and Javier P. Muniain.

The writing, although clear has a certain humility.  “Unfortunately understanding these new ideas depends on a through mastery of quantum field theory, general relativity, geometry, topology and algebra.  Indeed, it is almost certain that nobody is sufficiently prepared to understand these ideas fully.”

I’m going to take it with me to the amateur chamber music festival.  As usual, at least 2 math full professors will be there to help me out.  Buy it and enjoy



Willock pp. 51 – 104

This is a continuation of my notes, as I read  Molecular Symmetry” by David J. Willock.  As you’ll see, things aren’t going particularly well.  Examples of concepts are great once they’ve been defined, but in this book it’s examples first, definitions later (if ever).

p. 51 — Note all the heavy lifting  required to produce an object with only (italics) C4 symmetry (figure 3.6)  First,  you need 4 objects in a plane (so they rotate into each other), separated by 90 degrees.  That’s far from enough objects as there are multiple planes of symmetry for 4 objects in a plane (I count 5 — how many do you get?)  So you need another 4 objects in a plane parallel to the first.  These objects must be a different distance from the symmetry axis, otherwise the object will have A C2 axis of symmetry, midway between the two planes. Lastly no object in the second plane can lie on a line parallel to the axis of symmetry which contains an object in the first plane — e.g. the two groups of 4 must be staggered relative to each other.    It’s even more complicated for S4 symmetry.  

p. 51 — The term classes of operation really hasn’t been defined (except by example).   Also this is the first example of (the heading of) a character table — which hasn’t been defined at this point.

p. 52 — Note H2O2 has C2 symmetry because it is not (italics) planar.   Ditto for 1,2 (S, S) dimethyl cyclopropane (more importantly this is true for disulfide bonds between cysteines forming cystines — a way of tying parts of proteins to each other. 

p. 55 — Pay attention to the nomenclature: Cnh means that an axis of degree n is present along with a horizontal plane of symmetry.  Cnv means that, instead, a vertical plane of symmetry is present (along with the Cn axis)

p. 57 — Make sure you understain why C4h doesn’t  have vertical planes of symmetry.

p. 59 — A bizarre pedagogical device — defining groups whose first letter is D by something they are not (italics) — which itself (cubic groups) is at present undefined.  

Willock then regroups by defining what Dn actually is.

It’s a good exercise to try to construct the D4 point group yourself. 

p. 61 — “It does form a subgroup” — If subgroup was ever defined, I missed it.  Subgroup is not in the index (neither is group !).  Point group is in the index, and point subgroup is as well appearing on p. 47 — but point subgroup isn’t defined there.  

p. 62 — Note the convention — the Z direction is perpendicular to the plane of a planar molecule.

p. 64 — Why are linear molecules called Cinfinity ? — because any rotation around the axis of symmetry (the molecule itself) leaves the molecule unchanged, and there are an infinity of such rotations.

p. 67 — Ah,  the tetrahedron embedded in a cube — exactly the way an organic chemist should think of the sp3 carbon bonds.  Here’s a mathematical problem for you.  Let the cube have sides of 1, the bonds as shown in figure 3.27, the carbon in the very center of the cube — now derive the classic tetrahedral bond angle — answer at the end of this post. 

p. 67 — 74 — The discussions of symmetries in various molecules is exactly why you should have the conventions for naming them down pat.  

p. 75 — in the second paragraph affect should be effect (at least in American English)

p. 76 — “Based on the atom positions alone we cannot tell the difference between the C2 rotation and the sigma(v) reflection, because either operation swaps the positions of the hydrogen atoms.”   Do we ever want to actually do this (for water that is)? Hopefully this will turn out to be chemically relevant. 

p. 77 — Note that the definition of character refers to the effect of a symmetry operation on one of an atom’s orbitals (not it’s position).  Does this only affect atoms whose position is not (italics) changed by the symmetry operation?  Very important to note that the character is -1 only on reversal of the orbital — later on, non-integer characters will be seen.  Note also that each symmetry operation produces a character (number) for each orbital, so there are (number of symmetry operations) * (number of orbital) characters in a character table

p. 77 – 78 — Note that the naming of the orbitals is consistent with what has gone on before.  p(z) is in the plane of the molecule because that’s where the axis of rotation is.

Labels are introduced for each of the possible standard sets of characters (but standard set really isn’t defined).  A standard set (of sets of characters??) is an irreducible representation for the group.  

Is one set of characters an irreducible representation by itself or is it a bunch of them? The index claims that this is the definition of irreducible represenation, but given the amiguity about what a standard set of characters actually is (italics) we don’t really know what an irreducible representation actually is.   This is definition by example, a pedagogical device foreign to math, but possibly a good pedagogical device — we’ll see.  But at this point, I’m not really clear what an irreducible represenation actually is.

p. 77 — In a future edition, it would be a good idea to lable the x, y and z axes (and even perhaps draw in the px, py and pz orbitals), and, if possible, put figure 4.2 on the same page as table 4.2.  Eventually things get figured out but it takes a lot of page flipping. 

p. 79 — Further tightening of the definition of a representation — it’s one row of a character table.

p. 79 — Nice explanation of orbital phases, but do electrons in atoms know or care about them?

p. 80 — Note that in the x-y axes are rotated 90 degrees in going from figure 4.4a to figure 4.4b  (why?).   Why talk about d orbitals? — they’re empty in H20 but possibly not in other molecules with C2v symmetry.  

p. 80 — Affect should be effect (at least in American English)

p. 81 — B1 x B2 = A2 doesn’t look like a sum to me.  If you actually summed them you’d get 2 for E, -2 for C2, and 0 for the other two.  It does look like the product though.

pp. 81 – 82 — Far from sure what is going on in section 4.3

p.82 — Table 4.4b does look like multiplication of the elements of B1 by itself. 

p. 82 — Not sure when basis vectors first made their appearance, possibly here.  I slid over this on first reading since basis vectors were quite familiar to me from linear algebra (see the category https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/ ).  But again, the term is used here without really being defined.  Probably not to confuse, the first basis vectors shown first are at 90 degrees to each other (x and y), but later on (p. 85 they don’t have to be — the basis 0vectors point along the 3 hydrogens of ammonia).

p. 83 — Very nice way to bring in matrices, but it’s worth nothing that each matrix stands for just one symmetry operation.  But each matrix lets you see what happens to all (italics) the basis vectors you’ve chosen. 

p. 84 — Get very clear in your mind that when you see an expression of the form

symmetry_operation1 symmetry_operation2 

juxtaposed to each other — that you do symmetry_operation2  FIRST.

p. 87  — Notice that the term character is acquiring a second meaning here — it no longer is the effect of a symmetry operation on one of an atom’s orbitals (not the atom’s position), it’s the effect of a symmetry operation on a whole set of basis elements.

p. 88 — Notice that in BF3, the basis vectors no longer align with the bonds (as they did in NH3), meaning that you can choose the basis vectors any way you want.  

p.89 — Figure 4.9 could be markedly improved.  One must distinguish between two types of lines (interrupted and continuous), and two types of arrowheads (solid and barbed), making for confuion in the diagrams where they all appear together (and often superimposed).  

Given the orbitals as combinations of two basis vectors, the character of symmetry operation and a basis vector, acquires yet another meaning — how much of the original orbital is left after the symmetry operation. 

p. 91 — A definition of irreducible representations as the ‘simplest’ symmetry behavior.  Simplest is not defined.  Also for the first time it is noted that symmetries can be of orbitals or vibrations.  We already know they can be of the locations of the atoms in a molecule.  

Section 4.8 is extremely confusing.

p. 92 — We now find out that what was going on with a character sum of 2 on p. 81 — The sums  were 2 and 0 because the representations were reducible.  


p. 93 (added 29 Jan ’12) — We later find out (p. 115) that the number of reducible representations of a point group is the number of classes.  The index says that class is defined an ‘equivalent set of operations’ — but how two distinct operations are equivalent is never defined, just illustrated.

p. 100 — Great to have the logic behind the naming of the labels used for irreducible representations (even if they are far from intuitive)

p. 101 — There is no explanation of the difference between basis vector and basis function. 

All in all, a very difficult chapter to untangle.  I’m far from sure I understand from p. 92 – 100.  However, hope lies in future chapters and I’ll push on.  I think it would be very difficult to learn from this book (so far) if you were totally unfamiliar with symmetry.  

Answer to the problem on p. 67.  Let the sides of the cube be of length 1.  The bonds are all the same length, so the carbon must be in the center of the cube.  Any two of the bonds point to the opposite corners of a square of length 1.  Therefore the ends of the bonds are sqrt(2) apart.   Now drop a perpendicular to the middle of this line to get to the carbon in the center.  This has length 1/2.  So we have a right triangle of side 1/2 and ( sqrt(2))/2.  So the answer is 2 * arctan(1.414).  Arctan(1.414 is) 54.731533 degrees giving the angle as 109.46 degrees.