Group Theory Robert Gilmore Physics Department, Drexel University, Philadelphia, Pennsylvania 19104, USA (Dated: October 3, 2012) Printed from:Wiley-Mathematical/GroupChapter/group.tex on October 3, 2012 I. INTRODUCTION in Physics. As a latest tour de force in the develop- ment of Physics, they play a central roll in the creation Symmetry has sung its siren song to Physicists since of Gauge Theories, which describe the interactions be- the beginning of time, or at least since before there were tween Fermions and the Bosons that are responsible for Physicists. Today the ideas of symmetry are incorpo- the interactions among the Fermions, and which lies at rated into a subject with the less imaginative and sug- the heart of the Standard Model. We provide the sim- gestive name of Group Theory. This Chapter introduces plest example of a gauge theory, based on the simplest many of the ideas of group theory that are important in compact one parameter Lie group U(1), in Sec. VIII. the natural sciences. For an encore, in Sec. IX we show how the theory of Natural philosophers in the past have come up with the special functions of mathematical physics (Legendre many imaginative arguments for estimating physical and associated Legendre functions, Laguerre and associ- quantities. They have often used out-of-the-box methods ated Laguerre functions, Gegenbauer, Chebyshev, Her- that were proprietary to pull rabbits out of hats. When mite, Bessel functions, and others) are subsumed under these ideas were made available to a wider audience they the theory of representations of some low-dimensional Lie were often improved upon in unexpected and previously groups. The classical theory of special functions came to unimaginable ways. A number of these methods are pre- fruition in the mid 19th century, long before Lie groups cursors of group theory. These are Dimensional Analysis, and their representations were even invented. Scaling Theory, and Dynamical Similarity. We review these three methods in Sec. II. In Sec. III we get down to the business at hand, intro- II. PRECURSORS TO GROUP THEORY ducing the definition of group and giving a small set of important examples. These range from finite groups to Barenblatt has given a beautiful derivation of Pythago- discrete groups to Lie groups. These also include trans- ras' Theorem that is out-of-the-box and suggests some of formation groups, which played an important if under- the ideas behind Dimensional Analysis. The area of the 1 recognized roll in the development of classical physics, in right triangle ∆(a; b; c) is 2 ab (Fig. 1). Dimensionally, particular the theories of Special and General Relativity. the area is proportional to square of any of the sides, mul- The relation between these theories and group theory is tiplied by some factor. We make a unique choice of side indicated in Sec. IV. by choosing the hypotenuse, so that ∆(a; b; c) = c2×f(θ), Despite this important roll in the development of θ is one of the two acute angles, and f(θ) 6= 0 unless θ = 0 Physics, groups existed at the fringe of the Physics of or π=2. Equating the two expressions the early 20th century. It was not until the theory of the linear matrix representations of groups was invented 1 a b 1 b a symmetry π that the theory of groups migrated from the outer fringes f(θ) = = = f( − θ) to play a more central roll in Physics. Important points 2 c c 2 c c 2 in the theory of representations are introduced in Sec. (1) V. Representations were used in an increasingly imag- This shows (a) that the same function f(θ) applies for th π inative number of ways in Physics throughout the 20 all similar triangles and (b) f(θ) = f( 2 − θ). The latter century. Early on they were used label states in Quantum result is due to reflection `symmetry' of the triangle about systems with a symmetry group: for example, the rota- the bisector of the right angle: the triangle changes but tion group SO(3). Once states were named, degeneracies its area does not. We need (a) alone to prove Pythagoras' could be predicted and computations simplified. Such ap- Theorem. The proof is in the figure caption. plications are indicated in Sec. VI. Later, they were used websearch: Pythagoras' theorem Barenblatt when symmetry was not present, or just the remnant of a broken symmetry was present. When used in this sense, they are often called \dynamical groups." This type of use grossly extended the importance of Group Theory 2 We can invert this matrix to find −1 2 1 1 1 3 2 1 −1 −1 3 4 0 3 2 5 = 4 0 −1 −2 5 (4) 0 −2 −1 0 2 3 This allows us to determine the values of the exponents which provide the appropriate combinations of impor- tant physical parameters to construct the characteristic atomic length: 2 a 3 2 1 −1 −1 3 2 0 3 2 −1 3 FIG. 1: The area of the large right triangle is the sum b = 0 −1 −2 1 = −1 (5) of the areas of the two similar smaller right triangles: 4 5 4 5 4 5 4 5 c 0 2 3 0 2 ∆(a; b; c) = ∆(d; f; a) + ∆(f; e; b), so that c2f(θ) = 2 2 a f(θ) + b f(θ). Since f(θ) 6= 0 for a nondegenerate This result tells us that right triangle, a2 + b2 = c2. −1 2 −1 2 2 2 −8 a0 ∼ m (e ) (~) = ~ =me ∼ 10 cm (6) A Dimensional Analysis To construct a characteristic atomic time, we can re- place the vector col[0; 1; 0] in Eq. (5) by the vector 3 2 2 How big is a hydrogen atom? col[0; 0; 1], giving us the result τ0 ∼ ~ =m(e ) . Fi- The size of the electron `orbit' around the proton in nally, to get a characteristic energy, we can form the the hydrogen atom ought to depend on the electron mass combination E ∼ ML2T −2 = m(~2=me2)2(~3=me4)−2 = me, or more precisely the electron-proton reduced mass me4=~2. Another, and more systematic, way to get this µ = meMP =(me + MP ). It should also depend on the result is to substitute the vector col[1; 2; −2] in Eq. (5). value of Planck's constant h or reduced Planck's constant Note that our estimate would be somewhat different if ~ = h=2π. Since the interaction between the proton and we had used h instead of ~ = h=2π in these arguments. electron is electromagnetic, of the form V (r) = −e2=r We point out that this method is very useful for estimat- (Gaussian units), it should depend on e2. ing the order of magnitude of physical parameters and Mass is measured in gm. The dimensions of the charge usually gets the prefactor within a factor of 10. coupling e2 are determined by recognizing that e2=r is a websearch: dimensional analysis (potential) energy, with dimensions M 1L2T −2. We will use capital letters M, L, and T to characterize the three independent dimensional `directions'. As a result, the B Scaling charge coupling e2 has dimensions ML3T −2 and is mea- 3 2 sured in gm(cm) =sec . The quantum of action ~ has Positronium is a bound state of an electron e with a 2 −1 dimensions [~] = ML T . positrone ¯, its antiparticle. How big is positronium? Constant Dimensions Value Units To address this question we could work very hard and µ M 9:10442 × 10−28 gm 2 −1 −27 2 −1 solve the positronium Hamiltonian. Or we could be lazy ~ ML T 1:05443 × 10 gm cm sec 2 3 −2 −19 3 −2 and observe that the hydrogen atom radius is inversely e ML T 2:30655 × 10 gm cm sec proportional to the reduced electron-proton mass, so the a L ? cm 0 positronium radius should be inversely proportional to Can we construct something with the dimensions of the reduced electron-positron mass: µ = m m =(m + length from m, e2, and ? To do this, we introduce three pos: e e¯ e ~ m ) = 1 m since the electron and positron have equal unknown exponents a, b, and c and write e¯ 2 e masses me = me¯. Since the reduced electron-proton mass a 2 b c a 3 −2 b 2 −1 c is effectively the electron mass, the positronium atom is a0 ' m (e ) ~ = (M) (ML T ) (ML T ) = (M)a+b+c L0a+3b+2c T 0a−2b−c approximately twice as large as the hydrogen atom. (2) In a semiconductor it is possible to excite an electron and set this result equal to the dimensions of whatever from an almost filled (valence) band into an almost empty we would like to compute, in this case the Bohr orbit (conduction) band. This leaves a `hole' behind in the valence band. The positively charged hole in the valence a0 (characteristic atomic length), with [a0] = L. This results in a matrix equation band interacts with the excited electron in the conduction band through a reduced Coulomb interaction: V (r) = 2 2 1 1 1 3 2 a 3 2 0 3 −e /r. The strength of the interaction is reduced by screening effects which are swept into a phenomenological 4 0 3 2 5 4 b 5 = 4 1 5 (3) 0 −2 −1 c 0 dielectric constant . In addition, the effective masses ∗ ∗ me of the excited electron and the left-behind hole mh 3 are modified from the free-space electron mass values by many-particle effects.