## Group Theory and Physics

Symmetry is important in the world of atoms, and Group Theory is its mathematics

Quantum mechanics showed that the elementary systems that matter is made of, such as electrons and protons, are truly identical, not just very similar, so that symmetry in their arrangement is exact, not approximate as in the macroscopic world. Systems were also seen to be described by functions of position that are subject to the usual symmetry operations of rotation and reflection, as well as to others not so easily described in concrete terms, such as the exchange of identical particles. Elementary particles were observed to reflect symmetry properties in more esoteric spaces. In all these cases, symmetry can be expressed by certain operations on the systems concerned, which have properties revealed by Group Theory, a rather obscure branch of mathematics that had previously been mainly a curiosity without practical application.

Physics uses that part of Group Theory known as the theory of representations, in which matrices acting on the members of a vector space is the central theme. It allows certain members of the space to be created that are symmetrical, and which can be classified by their symmetry. It is found that all the observed spectroscopic states of atoms and molecules correspond to such symmetrical functions, and can be classified accordingly. Among other things, it gives selection rules that specify which transitions are observed, and which are not. These matters are so commonplace in spectroscopy that the fact that they are extraordinary and wonderful is hardly realized.

In introducing a subject, especially one as abstract as Group Theory, it is important to begin with concrete, explicit examples and not with general principles. The lofty and general approach may look elegant to its author who already has a good mental appreciation of the subject, but is totally opaque to the beginner. Let us start, then, with a function f(x). If f(-x) = f(x) we have an even function. Graphing it, its symmetry is evident. A function such that f(-x) = -f(x) is also symmetric; it is an odd function. We can express any function as the sum of an even and an odd part: f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2 = f(even) + f(odd). We are doing pretty well here without any Group Theory, and this is not unusual. Anything that Group Theory does can also be done without it, and in many places physicists and chemists have gone ahead algebraically instead of learning Group Theory, often proudly. However, not using Group Theory is like not using a map--you never see the big picture and may go down many blind paths. Group theory can give you a lot of information with very little input.

### Introduction to Groups and Representations

Well, then, into Group Theory. Let the symbol σ stand for the transformation x = -x. In three dimensions, this would be a reflection in the yz-plane. We can use σ as an operator: σf(x) = f(-x). Then, σf(-x) = f(x) as well. Indeed, σσf(x) = f(x) for any f(x), so it looks like σ2 = E, where E is the identity operator, such that Ef(x) = f(x) for any f(x). E stands for Einheit, and the letter I is also used. The product of two operations simply means to do the two transformations one after the other. It has nothing to do with arithmetic multiplication in this case. Now, we see that if we make any products we like from the pair E, σ we never get anything other than E or σ. This means that the set {E,σ} is closed with respect to their products in the current sense.

If you look at transformations in general, you notice that there is always the unit transformation that changes nothing, like E above. Also, if you make a transformation, there should be a way to reverse it. Otherwise, you will lose information that you cannot get back, and this is not good. We see that σ reverses itself; it is its own inverse, or σ-1 = σ. The exponent -1 is used to denote an inverse, by analogy with the usual rules for exponents. Finally, and here is something you might overlook, the effect of a series of transformations is the same however they are grouped, just so long as the order is not changed. This may be expressed by (AB)C = A(BC) = ABC. This is rather obvious for transformations, but might not be with other kinds of operations. Therefore, the group product must be associative. This happens automatically when we begin with a concrete representation of a group, but is very restrictive if we are just building up groups abstractly.

A group is a set of members {R} that is (1) closed under the group operation, so that the product of any two members is again a member of the group; (2) the group operation is associative; (3) {R} includes the unit element E; and (4) includes the inverse R-1 to any element R, such that RR-1 = R-1R = E. Closure is the main property that is used. The others just ensure that the corners are tucked in. The number of members g is the order of the group. Groups may be finite, if g is finite, or infinite. Infinite groups are not very different from finite groups, but have some extra interesting properties. The set of operations {E, σ} that we looked at form a group of order 2 that we can call Ci. This group is too small to show some of the more interesting aspects of groupp structure, but it is indeed a group, and an important one.

 E A A E
Now consider some other sets, such as the signed integers {1, -1} under ordinary multiplication, or the inversion transformation I f(x,y,z) - f(-x, -y, -z), or the exchange operation X f(x,y) = f(y,x), together with the identity E in the latter two cases. If we call the elements E and A, we see that all obey the same multiplication table. This table defines the abstract group Ci, which has many representations, or concrete realizations. The members of the group are listed in the first row and first column. We can agree that the result of any product is found at the intersection of the row corresponding to the first factor and the column of the second factor. As operators, we consider them applied from right to left. That is, EA means we do A first, then E, and the result is found at the intersection of the E row and the A column in the multiplication table. I am sorry for this complication, but things have to be kept straight, and many authors do not make clear what conventions they observe.

One group structure, as defined by the multiplication table, can have many representations, which are concrete groups isomorphic to each other. Isomorphism implies a one-to-one correspondence of distinct elements of the groups. The group properties of isomorphic groups are exactly the same.

As an exercise, investigate the set of numbers that arise when all powers of the imaginary unit i are considered, and show that they form a group of order 4 under ordinary multiplication, writing down the multiplication table. This is the abstract group C4. If A and B are any two members of this group, then AB = BA, which means that the order of the members does not matter (this is, a fundamental property of multiplying numbers, after all). A and B are said to commute, and the group is said to be commutative, or Abelian. We noticed that {1, -1} was a representation of Ci as well as {E, σ}, and that these numbers appeared before an odd f(x) when operated on by E or σ. Also, {1, 1} obeys the multiplication table and appeared before an even f(x). All Abelian groups have representations by numbers in this way, with 1 corresponding to E. One of the representations consists only of 1's, and the number of distinct representations is equal to the order of the group. You already know two of the representations of C4. Can you find the other two? One has +i and -i interchanged; the other has two -1's. Considered as g-dimensional vectors, these representations all have the squares of their lengths equal to g, and the scalar product (sum of products of the elements) of two different ones is zero. The representations containing i's of C4 must be massaged a little to make them orthogonal to each other, but it can be done.

This may be mildly interesting so far, but the really interesting stuff happens when, for some members of a group, AB ≠ BA. That is, they do not commute, and the group is then called noncommutative. As Schensted points out, the fact that the smallest noncommutative group has g = 6 is something of an instructional disaster. However, we can make an important point simply from the fact of noncommutativity. We have seen that we can find numerical representations of Abelian groups, which is natural, because numbers commute. To find a representation of a noncommutative group that preserves the noncommutativity, we have to go beyond numbers to something that can be noncommutative. Now, matrices have this property, so we require matrices to form some representations of noncommutative groups. If we consider operation on functions, this means that we operate on sets of functions that act like vectors, for matrices operate on vectors. This occurrence of functions in natural families is exciting: they represent degenerate states (states with the same energy) in quantum mechanics. An example of noncommuting of matrices is shown at the left. If you have forgotten how to multiply matrices, the ij element in the product is the sum of the products of elements in the ith row of the first matrix with elements in the jth column of the second matrix, each by each. For example the 11 element in the top product (row 1, column 1) is 0 x 1 + 1 x 0 = 0, the 12 element 0 x 0 + 1 x -1 = -1, and so on. These matrices are members of a group of order 8, and you can find all the other members by multiplying matrices until you get no new ones. Each of the matrices multiplied in the Figure is its own inverse, which can easily be seen by multiplication. The fourth powers of the product matrices are the identity, and their product is the identity, so they are inverses of each other. The group of these matrices is called C4v, and we will come back to it later.

Symmetry was first studied mathematically with reference to crystal symmetry. If you rotate a crystal by certain angles about certain axes, or reflect it in certain planes, you find equivalent faces in equivalent places. It took genius to realize that under the imperfect symmetry of actual crystals there was an exact symmetry reflected in the orientations of crystal faces, not their apparent shapes, which are greatly affected by accidents of growth. A symmetry operation took a face of a certain orientation into an equivalent face at a symmetrical orientation. Crystals are an interesting, but somewhat difficult case to study, so we will introduce spatial symmetry by operations that keep one point fixed, which make up the point groups. The symmetry operations are rotations about an axis, and reflections in a plane (both of which pass through the one point). Cn will be the symbol for a rotation of 2π/n radians about a certain axis, σh for a reflection in a plane perpendicular to an axis, and σv for a reflection in a plane containing the axis. A symmetry operation takes a symmetrical figure into itself; the result is indistinguishable from the original. The symmetry operations of a given figure form a group, since the operations satisfy all four group postulates. Let us suppose that a rotation C3 around a certain axis, and a σv containing this axis, are symmetry elements of a certain figure, which may be, for example, an ammonia molecule, NH3. By multiplying these elements (performing them successively) we find that the set of six operations E, C3, C32, σ, σ', σ" is closed, and forms a finite group of order 6, which we shall call C3v. It is easier to appreciate the group operations using the diagram on the right. We are looking directly down the 3-fold axis, and the original symmetry plane is denoted by σ. Rotations carry this plane into equivalent symmetry planes σ' and σ". A general point P is carried into the other two solid points by rotations, and into the open points by reflections in the three symmetry planes. From this diagram, it should be easy to write down the multiplication table for the group, and to verify that it is noncommutative, since rotations and reflections will be found not to commute.

There is a curious interaction between the rotations and reflections in this group. Suppose we reflect P in σ, then rotate clockwise by C3, and finally reflect again in σ. The result is the same as rotating anticlockwise by C32. Algebraically, we can write σ-1 C3σ = C32, since σ is its own inverse. If we do the same thing with a C instead of a σ, we always get a C back, never a σ That is, if we consider the products A-1CA, where C is either of the rotations, and A is any member of the group, we get a C. In an Abelian group, this result is not interesting, since we always get back the same individual element: A-1BA = A-1AB = B. Here, however, we put in C3 and got back C32. All the members of a group that are associated in this way form a class. The members of a class are all of a similar nature, simply in a different orientation. For our group, now consider the products C-1σC, and show that the three reflections also fall into a single class. E is obviously always in a class by itself, since it commutes with every member. C3v, therefore, has three classes, E, 2C, and 3σ. It turns out to be very important to group these similar elements.

Let us now generate some representations of the group. We can set up rectangular coordinates x,y,z, perhaps with z along the symmetry axis and x along one of the symmetry planes, with y making a right-handed system. Then point P can be expressed as (x,y,z). Any symmetry operation will take these coordinates into (x',y',z') which are linear functions of the original coordinates. Explicitly, the transformation can be expressed as 3 x 3 matrices acting on 3-dimensional column vectors. There will be six matrices, and E will correspond to the indentity matrix. The matrices are easily found from the formulas for rotation of coordinates. We notice that z is not affected. It is multiplied by 1 in each transformation and the values of x,y are never mixed in. Therefore, z is the basis of a representation all by itself, the unit representation where each member corresponds to 1. The values of x and y do get mixed under the transformations, and it is easy to see that no choice of coordinates could ever change this. They form the basis of a two-dimensional matrix representation. The representation in terms of (x,y,z) is said to be reducible to the two smaller representations, and each of these is irreducible to smaller representations. It is the irreducibe representations that give the essential information; reducible representations can be formed ad libitum, and are a dime a dozen. A 1-dimensional representation is naturally irreducible, since there is nowhere lower to go.

It was easy to find the reducibility of the vector representation because we chose the coordinates wisely. If we had taken the coordinate axes in some arbitrary position, we would have obtained six rather full matrices and the reducibility with a fortunate selection of axes would not be at all obvious. Any representation changes its explicit form if we make a different choice of basis, but it is still the same at heart. How can we find out what it is really like? The answer is in the realization that choosing a different basis is a linear transformation y = Ax, under which the matrices R of a representation change into R' = A-1RA. This is called a similarity transformation. Is there something invariant under such a transformation? Yes, indeed: the diagonal sum or trace of the matrix R. That is, Tr R' = Tr R, and this trace is called the character of the member in these equivalent representations, often denoted by χ(R). Two equivalent representations differ only in the choice of basis functions use to express them explicitly; their basic nature is the same. The converse of the theorem we have just established is the important result: if the characters of two representations are the same, then the representations are equivalent, and we can look for a matrix A connecting them with hope of success. This result, and others that we will quote, apply to unitary matrices, and are certainly true for rotation and reflection matrices.

If a representation is reducible, it can be expressed with the matrices for irreducible representations along the diagonal by a suitable choice of coordinates. Therefore, the character of the reducible representation is the sum of the characters for the irreducible representations that comprise it.

 C3v E 2C 3σ A1 1 1 1 A2 1 1 -1 E 2 -1 0
The characters of matrices representing the members of a class will be the same, as we know that the members of a class are related by similarity transformations. The number of irreducible representations of a group is equal to the number of classes. This is one of the remarkable theorems about group representations whose proof we will leave to the references. We can construct a table showing the characters of its irreducible representations that turns out to be very useful. For C3v, the table is shown at the left. The letter A usually designates a one-dimensional representation with characters 1 for the rotations, and E a two-dimensional representation. Note that the sum of the squares of the characters, summed over all elements, is equal to the order of the group. For representation E, this is 2x2 + 1x1 + 1x1 + 0x0 + 0x0 + 0x0 = 6. Also, considered as group vectors with 6 components, the characters are orthogonal to each other. These surprising relations are the reason why there is a finite number of irreducible representations, and why there is one for each class.

We can now show one of the real uses of group theory. Consider the vector representation of C3v that we constructed above. Its characters were 3, 0, 1. The sum of squares is 3x3 + 2x0x0 + 3x1x1 = 12, greater than g = 6. Therefore, it is reducible. If we take the scalar product of this group vector with one of the vectors of an irreducible representation, we will find the number of times this representation is contained in the reducible representation, times the order of the group. For A1, we get 3x1 + 2x0x1 + 3x1x1 = 6. For A2, we have 3x1 + 2x0x1 + 3x1x(-1) = 0. For E, we find 3x2 + 2x(-1)x0 + 3x1x0 = 6. Therefore, our reducible vector representation is reducible to A1 + E. We could have determined this even with a bad choice of coordinates, since the characters would be unaffected. It is often easier to find the characters of a representation than the explicit matrices.

 C4v E 2C4 C2 2σ 2σ' A1 1 1 1 1 1 A2 1 1 1 -1 -1 B1 1 -1 1 1 -1 B2 1 -1 1 -1 1 E 2 0 -2 0 0
Let's now return to the group C4v, which we mentioned briefly above. To generate this group, start with a 4-fold axis and one reflection plane. Draw a diagram like the one for C3v. As you consider the symmetrical positions into which a general point P is carried by the members of the group, you will see diagonal symmetry planes appear as if by magic, arising from products like C4σ. The rotations now fall into two classes, one containing C42 = C2 alone, and one the other two rotations. The reflections are also divided into two classes, one class in the right angles between members of the other class. There are five classes, and so five irreducible representations. The new one-dimensional representations B1 and B2 appear. The character table is shown at the left.

Find the vector representation for this group, and show that it reduces to A1 + E, as for C3v, but the matrices are completely different. In fact the matrix elements are only 0, +1 and -1, and it is not difficult to find matrices for the E representation if the coordinates are chosen wisely. We have already seen some of them above. Consider the function x2 - y2. Show that it is transformed into itself by all group operations, so that it is a basis for a one-dimensional representation. This can be done without algebra, merely by considering the effects of the operations on x and y. To which representation does it belong? Do the same for the functions xy and x2 + y2.

A subset of the members of the group may form a group by themselves. This subset must contain the identity E, of course, and the inverses of each member. It is called a subgroup. A subgroup made up of whole classes is called an invariant subgroup. The order of a subgroup is a factor of the order of the group (Lagrange's Theorem). Improper subgroups are the identity E alone and the whole group; all others are proper subgroups. The members of a group that commute with all members of the group form an Abelian subgroup called the center of the group. There are many algebraic results such as these that aid an understanding of the structure of groups, but are not directly applicable to representations. Any representation of a group also gives a representation of a subgroup, but an irreducible representation of the group may give a reducible representation of a subgroup.

The n! permutations of n identical objects also form a group, the symmetric group Pn. You should be able to convince yourself that P3 is isomorphic to C3v by setting up a one-to-one correspondence of the members of each group. The diagrams that were presented above may help, if the point P and its images are numbered. Subgroups of Pn are also called permutation groups. For example, the even permutations in Pn (those that can be accomplished by an even number of interchanges) form a group of order n!/2 called the alternating group An. Any finite group is isomorphic to a certain permutation group (Cayley's Theorem). In fact, permutation groups were the first ones studied, and there is considerable lore on their representations.

### The Four Major Theorems

Most of the results of matrix representation theory that are useful in physics are derived from the following four theorems. We have already used some of them. Proofs are given in References 1-4.

I. Any matrix representation of a group is equivalent to some representation by unitary matrices.

II. A matrix that commutes with all matrices of an irreducible representation can only be a multiple of the identity matrix.

III. If A and B are the matrices of two irreducible representations of a group, of dimension n and m respectively, then if there is a matrix M with m rows and n columns such that MA = BM for all members of the group, it is the null matrix if m ≠ n, and if m = n, it is either a null matrix or a matrix with nonvanishing determinant. In the latter case, the inverse M-1 exists, and the representations A, B are equivalent.

IV. If A and B are two inequivalent, irreducible representations of a group, then Σ A*ijBkl = 0, and for a single unitary irreducible representation A we have Σ A*ijAkl = (g/n)δikδjl, where g is the order of the group, and n the dimension of the representation. The sum is over all the members of the group. The deltas are zero when their indices are different, unity when their indices are equal.

The scalar product of two vectors is the sum of the products of the individual elements, symbolically written (a,b), and is a number, not a vector. A unitary transformation or matrix U is one that preserves this scalar product, (Ua,Ub) = (a,b). This will be so if U-1ij = U*ji, if the inverse equals the hermitian conjugate (complex conjugate transposed) of the matrix. The concept can be extended to function spaces where the scalar product is an integral.

### Applications to Quantum Mechanics

The Hamiltonian operator H is a function of the coordinates and momenta of a system. In the Schrödinger representation, solutions of the eigenvalue equation Hψ = Eψ exist for certain values of the number E that can be cast into an orthonormal set of functions of the coordinates {ψ}, the eigenfunctions. If more than one state ψ corresponds to the same energy E, this energy level is said to be degenerate (not the states, since there is nothing degenerate about the states involved--they simply have the same energy). In general, this does not occur except when there is some reason forcing the states to have the same energy, as we shall see. When it occurs without a compelling reason, it is called accidental degeneracy.

Multiply the eigenvalue equation by some operator Q from the left, and insert Q-1Q between H and ψ, to get (QHQ-1)Qψ = QEψ = EQψ. If QHQ-1 = H, or what is the same thing, QH = HQ, then HQψ = EQψ, or the states ψ and Qψ belong to the same value of the energy E; that is, the level is degenerate. If Q represents some change of basis functions or coordinates, QHQ-1 is the operator H in the new frame of reference. Should Q be a symmetry operation, then we must have QHQ-1 = H. The set of operators commuting with H is a group, called the symmetry group of the Hamiltonian.

Every state can be classified by the irreducible representation to which it belongs, and any coincidence in energy is regarded as accidental, unless the representation is multidimensional, in which case the coincidence in energy is necessary. Spectroscopic classifications are essentially identifications of the irreducible representation to which the state belongs.

It is often useful to express H as the sum of H0, an approximate Hamiltonian that is simple and corresponds to most of the energy, and a perturbation Hamiltonian H' that includes finer details. If the symmetry group of H' is smaller than the symmetry group of H0, then the greater symmetry is broken by the perturbation. In this case, previously degenerate levels may split into distinct levels, some of which may still be degenerate. Group theory will provide suitable functions for this calculation that can greatly reduce the effort involved.

As an example, the states of an isolated atom are classified by the total angular momentum J, and belong to irreducible representations of the rotation group in three dimensions with dimension 2J + 1. As long as there is spherical symmetry, these states all have exactly the same energy. If you apply a magnetic field, the spherical symmetry is broken, and there is now only symmetry about an axis in the direction of the magnetic field. The 2J + 1 levels now acquire different energies, and the nondegenerate states can be classified by the magnetic quantum number M, M = J, J - 1, ..., -J. Group theory can determine these states in advance, so that the splitting is given by a simple diagonal matrix element.

As a second example, the energy levels of the hydrogen atom depend only on the principal quantum number N. For any N, there are states with J = 0, 1, ..., N-1. As far as the rotation group is concerned, this degeneracy is accidental. However, the Coulomb 1/r potential gives the Hamiltonian more symmetry than just rotation in 3-space. In fact, the Hamiltonian is invariant under the four-dimensional rotation group (actually, a group isomorphic to the four-dimensional rotation group), and its irreducible representations explain the added degeneracy, which is really not accidental at all. The degeneracy is lifted when the potential has a different radial dependence in more complex atoms, although the spherical symmetry is still there.

As still another example, it was noted in the late 1960's that the nucleons (neutron and proton) and the &Lambda were not greatly different in mass. It was conjectured that they were members of an irreducible representation of dimension 3 of a group called SU3, and that the differences in the observed masses was a splitting due to a perturbation related to the property called strangeness. There was some success in arranging the known particles in irreducible representations of SU3, but later a theory of the internal structure of heavy elementary particles superseded this idea. However, group theory is still quite important in the field.

Group theory is good for more than classification, however. The probability amplitude for a transition between two states ψ and φ is given by some integral like ∫ψ*Pφ dτ, where P is an operator characteristic of the mechanism involved in the transition (electric dipole, etc), and dτ is the differential volume element. If ψ, φ and P are all classifed according to their irreducible representations, then their product belongs to a representation whose characters are the product of the characters of each of the representations involved. This representation is almost always reducible, and the component irreducible representations can be found by character analysis. Unless the unit representation (all matrices +1) is there, the integral must vanish, since otherwise it changes under some symmetry operation, which by definition cannot change the integral. This gives selection rules for the nonvanishing of certain transitions or of certain matrix elements.

An example from spectroscopy might show how this operates. The electric dipole moment is a vector, and corresponds to the irreducible representation J = 1 of the rotation group. Operating on a state of angular momentum J, it gives representations corresponding to J + 1, J, and J - 1. The other state involved must have one of these values of J, or the result will not contain the unit representation J = 0. Therefore, we have the selection rule that J changes only by ±1 or 0 in an electric dipole transition.

### Continuous Groups

The finite groups are useful with crystals, molecular spectra, and identical particles, and give clear examples of the applications of group theory, but it will be noticed that we mentioned several continuous groups above in our examples. In fact, such groups play a large role in quantum mechanics and elementary particle physics, and some facts about them should be noted here. Their representations share many of the properties of the representations of finite groups, but the methods of working with the groups is somewhat different.

If you have studied quantum mechanics, you have certainly studied angular momentum. This is group theory, but is not usually presented as such in elementary treatments, since everything can be worked out with algebra from first principles. What you find are the irreducible representations of the rotation group in three dimensions, O3, and the rules for combining them. The rotation group O3 is an excellent example of a continuous group, though it is a little too simple to show all the wrinkles. A continuous group is of infinite order, so the sum over finite group elements becomes an integration over continuous group elements, suitably parametrized. For O3, we could use the two angles specifying the orientation of the axis of rotation, and the angle of rotation about this axis.

Angular momentum operators Jx, Jy and Jz are usually introduced to begin the study of angular momentum, and these operators satisfy the relation [Jx, Jy] = iJz, where angular momentum is measured in units of h/2π to simplify things. The [] is the commutator of the two operators: [a,b] = ab - ba. As you probably know, the whole theory can be worked out from this. You find that J2 has eigenvalues j(j + 1), where j is integral or half-integral, that Jz commutes with J2 so that states can be labelled with the quantum numbers j,m, that Jx ± iJy are the step-up and step-down operators such that Jx ± iJy|jm> = √[j(j + 1) - m(m + 1)]|jm±1>, and that there are 2j + 1 states |jm> for each value of j corresponding to the values m = j, j-1, ..., -j. All this comes from the commutation relation!

Now the J operators are not the members of a group. Rather, they form a Lie (pron. lee) Algebra closed under the [] operation. Their relation to the group members of O3 is that they are the generators of the group in the sense that D(α) = (1 + iαJz) is the operator for a rotation through an infinitesimal angle α about the z-axis. A finite rotation can be expressed as D(α) = exp(iαJz). The Lie algebra expresses the structure of the group in a concise and usable form. For many applications of angular momentum, the explicit rotation matrices are not needed. The conservation of angular momentum is a consequence of the O3 symmetry. Quite generally, any continuous symmetry is associated with a conservation law. Linear momentum is similarly related to symmetry under displacement of the system in space, and energy to displacement in time.

Finite symmetries are not associated with conserved quantities. Two important symmetries in quantum mechanics are inversion symmetry or parity, and time reversal symmetry. These are both isomorphic to the simple two-member groups that opened our discussion. Time reversal is unusual in that it involves the complex conjugation of the wave function, not simply a linear combination or a multiplication by a constant.

### Conclusion

This paper has covered about the same material as in Schensted, but with different examples and in a somewhat different way. We both have used the groups C2 and C3v as examples, as nearly all accounts do. The references below provide fuller information.

With fewer students of advanced physics, and worse-prepared students at that, this material is becoming less and less known, a definite reversal of the trends of the 60's. No one pays people to do group theory or quantum mechanics. In modern mathematics departments, it seems as unfashionable as complex variables (another field that is too difficult these days). The really hard work that is required to master quantum electrodynamics also seems beyond today's students and tomorrow's faculty. It can only be the hobby of a few, and I hope that there are enough minds fascinated by it to preserve it as a living science. No one who is unprepared is likely to make anything of the literature in the field. Much of the recent literature is not worth reading. There is still much to be known, but very little progress has been made recently. Collapse of the foundations will make further progress very difficult. There is a search now for the Higgs boson; contemplate how few people have the faintest idea of what this means. The book by Kane will give you an idea, if you don't have one already. It involves symmetry breaking, incidentally.

### References

This is very restricted set of many references on this subject, containing only those that I regard as the most informative. Further references may be found in these works. All of these authors clearly desired to communicate their knowledge and tried their best to do it, unlike most more recent authors.
1. I. V. Schensted, A Short Course on the Application of Group Theory to Quantum Mechanics (Ann Arbor, MI: NEO Press, 1967). Probably now unavailable, this was a remarkably clear and economical exposition.
2. E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (New York: Academic Press, 1959). A classic reference, but not easy reading for the uninitiated.
3. D. Schonland, Molecular Symmetry (London: D. Van Nostrand, 1965). Very clearly written with many examples and few divergences from the task of explaining the application to molecular spectra. Many chemists are mathematically challenged, so this book had to be written with them in mind.
4. S. Bhagavantam and T. Venkatarayudu, Theory of Groups and its Application to Physical Problems (New York: Academic Press, 1969). Includes lattices and crystals, as well as spectra.
5. W. Ledermann, The Theory of Finite Groups, 3rd ed. (Edinburgh: Oliver and Boyd, 1957). Algebraic properties with no matrix representation theory.
6. D. M. Brink and G. R. Satchler, Angular Momentum, 2nd ed. (Oxford: Clarendon Press, 1968). The rotation group in three dimensions, with considerable group theory and applications to quantum mechanics.
7. H. J. Lipkin, Lie Groups for Pedestrians (Amsterdam: North-Holland Publ. Co., 1966). An account of the use of groups in elementary particle theory from the heyday of SU3.
8. G. Kane, Modern Elementary Particle Physics (Reading, MA: Addison-Wesley, 1993). An account of the Standard Model, with brief, rather shallowly treated applications of group theory.