Brief introduction to groups and group theory
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1 Brief introduction to groups and group theory In physics, we often can learn a lot about a system based on its symmetries, even when we do not know how to make a quantitative calculation Relevant in particle physics, crystollography, atomic and molecular physics, etc. Symmetry leads to degeneracy (We can predict that degeneracies should exist even if we do not know how to compute them, and if the degeneracies are broken there is interesting new physics!) Group theory is useful and important. Often considered hard by students, but it s easier than you might think at first! Yet entire books are devoted to the subject... brief intro here
2 Definition of a group Any set of elements {A, B, C,...} with the following properties forms a group: Any product of two elements yields another element in the group (closure) We can make a table to characterize the group. For example, consider the elements ±1, ±i. This has the identity element 1, and yields a table below
3 Definition of a group Any set of elements {A, B, C,...} with the following properties forms a group: Any product of two elements yields another element in the group (closure) The products of elements is associative (AB)C = A(BC) (associative law) We can make a table to characterize the group. For example, consider the elements ±1, ±i. This has the identity element 1, and yields a table below
4 Definition of a group Any set of elements {A, B, C,...} with the following properties forms a group: Any product of two elements yields another element in the group (closure) The products of elements is associative (AB)C = A(BC) (associative law) There is an identity element I with the property AI = IA = A for each element (unit element) We can make a table to characterize the group. For example, consider the elements ±1, ±i. This has the identity element 1, and yields a table below
5 Definition of a group Any set of elements {A, B, C,...} with the following properties forms a group: Any product of two elements yields another element in the group (closure) The products of elements is associative (AB)C = A(BC) (associative law) There is an identity element I with the property AI = IA = A for each element (unit element) Every element has an inverse, for example AB = BA = I (inverses) We can make a table to characterize the group. For example, consider the elements ±1, ±i. This has the identity element 1, and yields a table below
6 Multiplication table for a group 1 i -1 -i 1 1 i -1 -i i i -1 -i i 1 i -i -i 1 i -1 Completely defines group Satisfies all properties required of a group Also could have used e iπ/2, e iπ. e 3iπ/2, and e 2iπ for the elements... rotations? (We ll see connection!)
7 Same group... different representation We can define a different set of objects with the same multiplication table! Same group, different representation of the group Consider the 2 2 matrices below, and find group multiplication table ( ) 1 0 I = 0 1 ( ) 0 1 A = 1 0 ( ) 1 0 B = 0 1 ( ) 0 1 C = 1 0
8 Multiplication table for the group I A B C I I A B C A A B C I B B C I A C C I A B Same as previous if we took 1 I, i A, 1 B, and i C I is identity or rotation by 2π, A is rotation by π 2, B is rotation by π, and C is rotation by 3π 2 The elements 1 = e 2iπ, i = e πi 2, 1 = e πi, and i = e 3iπ 2 can be used to rotate x + iy = e iφ in the complex plane Same group, different representation
9 Example: symmetry operations of an equilateral triangle For example, each vertex could represent an atom in a molecule Identity I ( ) 1 0 I = 0 1 Rotation by 120, A = ( ) 1/2 3/2 3/2 1/2 Rotation by 240, B = ( 1/2 3/2 3/2 1/2 )
10 Symmetry of an equilateral triangle; reflections Reflection through y-axis F = ( ) Refelection through line G G = ( 1/2 3/2 3/2 1/2 ) Reflection through line H H = ( ) 1/2 3/2 3/2 1/2
11 Symmetry of an equilateral triangle; multiplication table Simple matrix multiplication yields the table below (also in book) I A B F G H I I A B F G H A A B I G H F B B I A H F G F F H G I B A G G F H A I B H H G F B A I We can identify the inverses from wherever there are elements I in the table In particular, A 1 = B, B 1 = A, F 1 = F, G 1 = G, and H 1 = H
12 Conjugate elements and class If there is a relationship C 1 AC = B, then A and B are conjugate elements From the table above, we see that I 1 AI = A, A 1 AA = A, B 1 AB = AAA 1 = A, F 1 AF = FG = B, G 1 AG = GH = B, and H 1 AH = HF = B This can be done for B as well, and we find that A, B form a set of conjugate elements We define a set of all conjugate elements as a class We likewise find F, G, and H form a class
13 Character, reducible and irreducible representations The character of a representation is found from the trace of the matrix For example, the trace of A and B for the symmetry group above A = B = ( ) 1/2 3/2 3/2 1/2 ( 1/2 3/2 3/2 1/2 The character of F, G, H is zero. The character of I is two The number of elements in the group is equal to the sum of the characters squared. For our 6-element group, we find = 6 )
14 Character, reducible and irreducible representations To avoid ambiguity, we define the character only for irreducible representation. If we can block diagonalize all elements simultaneously (which amounts to a change of basis), the representation is said to be reducible. The character depends on representation! We can sometimes find one, two, three, etc dimensional representations of a group. For the group above with 6 elements, no three dimensional representations exist because 3 2 = 9 > 6 (The trace of I is equal to the dimensionality of the representation).
15 Chapter 4: Partial differentiation It is generally the case that derivatives are introduced in terms of functions of a single variable. For example, y = f (x), then dy dx = df dx = f. However, most of the time we are dealing with quantities that are functions of several variables. For example, we usually want physical quantities in three dimensional space. For example, the electric field at each point in space might depend on x, y, and z, E E(x, y, z). Or, it might be convenient in some cases to use spherical coordinates, and then E E(r, φ, θ). We hence have to think about partial differentiation in physics.
16 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables
17 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials
18 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation
19 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation Use the chain rule for differentiation of a function of a function
20 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation Use the chain rule for differentiation of a function of a function Use partial differentiation in maximum/minimum problems
21 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation Use the chain rule for differentiation of a function of a function Use partial differentiation in maximum/minimum problems Use Lagrange multipliers in maximum/minimum problems with constraints
22 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation Use the chain rule for differentiation of a function of a function Use partial differentiation in maximum/minimum problems Use Lagrange multipliers in maximum/minimum problems with constraints Make changes of variables, including using spherical and cylindrical coordinate systems
23 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation Use the chain rule for differentiation of a function of a function Use partial differentiation in maximum/minimum problems Use Lagrange multipliers in maximum/minimum problems with constraints Make changes of variables, including using spherical and cylindrical coordinate systems Take derivatives of integrals
24 Introduction and notation For example, say we have z = f (x, y), we then need partial derivatives z x = f x = f x The key is we take derivative with respect to x, while keeping y fixed For example, z = f (x, y) = x 2 cos y, then z x = 2x cos y Then we can take another derivative, 2 z y x Does the order matter? Notice that 2 z x y We see 2 z. This is most often true. x y = 2 z y x = 2x sin y = 2x sin y. Sometimes we explicitly note that one variable is fixed (for example, in thermodynamics) ( ) z = 2x cos y x y
25 Power series in two variables We can take a Taylor series expansion about the point x = a, y = b of the function f (x, y) The power series can be represented by f (x, y) = a 00 +a 10 (x a)+a 01 (y b)+a 20 (x a) 2 +a 02 (y b) 2 +a 11 (x a)(y We see that a 00 = f (a, b) We can take partial derivatives with respect to x and y of the power series f x = f x = a a 20 (x a) + a 11 (y b) +... f y = f y = a a 02 (y b) + a 11 (x a) +... We evaluate the derivatives at x = a, y = b, and obtain the infinite Taylor series
26 Power series continued a 10 = f x (a, b), a 01 = f y (a, b), a 20 = 1 2 f xx(a, b), a 02 = 1 2 f yy(a, b), a 11 = f xy (a, b) = f yx (a, b), etc. We can express with h = x a and k = y b, f (x, y) = n=0 Where we mean y f (a, b) = f y (a, b) ( 1 h n! x + k ) n f (a, b) y
27 Example: Section 2, Problem 2 Find the Mclaurin series (expansion about x = 0,y = 0) of f (x, y) = cos(x + y) We see f x = f y = sin(x + y), and f xx = f yy = f xy = f yx = cos(x + y), etc. sin 0 = 0 and cos 0 = 1, so with h = x and k = y, f (x, y) = cos(x + y) n=0 ( 1 h n! x + k ) n f (a, b) y cos(x+y) = 1 1 2! (x 2 +2xy+y 2 )+ 1 4! (x 4 +4x 3 y+2x 2 y 2 +4xy 3 +y 4 )+...
28 Total differential for y = f (x) For y=f(x), we have y = dy dx = df dx We can treat dx = x as an independent variable In the limit x 0, then dy dx = lim y x 0 x If x finite, then dy is not exactly y
29 Total differential for z = f (x, y) and for many independent variables For a function of two variables, z = f (x, y), we can define the total differential dz = z x dx + y dy We can have dx and dy independent variables Then dz is the change in z along the tangent plane at x,y As with the previous example, dz is not equal to z for finite dx and dy For a function of many variables u = f (x 1, x 2,..., x N ), we define the total differential du = N n=1 u x n dx n
30 Thermodynamics In thermodynamics, we have quantities that might pressure p, volume V, temperature T, entropy S, particle number N, and chemical potential µ. These are not all independent, so if we know p then V is determined, hence we describe quantities in terms of some subset of all the possible variables (In fact, p and V are conjugate pairs, as are T and S, and also N and µ.) The total energy U(S, V, N), so du = U S We define T = U S, p = U V U U ds + dv + V N dn, and µ = U N du = TdS pdv + µdn
31 Legendre transformations Construct a new function F = U TS, then df = du TdS SdT = SdT pdv + µdn We see that F (T, V, N), different independent variables! This is an example of a Legendre transformation Consider another example, G = F + pv, so dg = df + pdv + Vdp = SdT + Vdp + µdn The thermodynamics function G(T, p, N) is quite convenient because in experiments it is easy to control and measure T and p, as opposed to S and V (entropy and volume)
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