brief introduction to groups and group theory - ucf …schellin/teaching/phz3113_2011/...brief...

Brief introduction to groups and group theory

• In physics, we often can learn a lot about a system based on itssymmetries, even when we do not know how to make aquantitative calculation• Relevant in particle physics, crystollography, atomic andmolecular physics, etc.• Symmetry leads to degeneracy (We can predict that degeneraciesshould exist even if we do not know how to compute them, and ifthe degeneracies are broken there is interesting new physics!)• Group theory is useful and important. Often considered hard bystudents, but it’s easier than you might think at first!• Yet entire books are devoted to the subject... brief intro here

Patrick K. Schelling Introduction to Theoretical Methods

Definition of a group

• Any set of elements {A,B,C , ...} with the following propertiesforms a group:

I Any product of two elements yields another element in thegroup (closure)

I The products of elements is associative (AB)C = A(BC )(associative law)

I There is an identity element I with the property AI = IA = Afor each element (unit element)

I 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, andyields a table below


Multiplication table for a group

1 i -1 -i

1 1 i -1 -ii i -1 -i 1

-1 -1 -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. e3iπ/2, and e2iπ for theelements... rotations? (We’ll see connection!)


Same group... different representation

• We can define a different set of objects with the samemultiplication table!• Same group, different representation of the group• Consider the 2× 2 matrices below, and find group multiplicationtable

I =

(1 00 1

)

A =

(0 −11 0

)

B =

(−1 00 −1

)

C =

(0 1−1 0

)Patrick K. Schelling Introduction to Theoretical Methods

Multiplication table for the group

I A B C

I I A B CA A B C IB B C I AC 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 rotationby π, and C is rotation by 3π

2

• The elements 1 = e2iπ, i = eπi2 , −1 = eπi , and −i = e

3iπ2 can be

used to rotate x + iy = e iφ in the complex plane• Same group, different representation


Example: symmetry operations of an equilateral triangle

• For example, each vertex could represent an atom in a molecule• Identity I

I =

(1 00 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

)


Symmetry of an equilateral triangle; reflections

• Reflection through y-axis

F =

(−1 00 1

)• 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

)


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 HA A B I G H FB B I A H F GF F H G I B AG G F H A I BH H G F B A I• We can identify the inverses from wherever there are elements Iin the table• In particular, A−1 = B, B−1 = A, F−1 = F , G−1 = G , andH−1 = H


Conjugate elements and class

• If there is a relationship C−1AC = B, then A and B areconjugate elements• From the table above, we see that I−1AI = A, A−1AA = A,B−1AB = AAA−1 = A, F−1AF = FG = B, G−1AG = GH = B,and H−1AH = HF = B• This can be done for B as well, and we find that A, B form a setof conjugate elements• We define a set of all conjugate elements as a class• We likewise find F , G , and H form a class


Character, reducible and irreducible representations

• The character of a representation is found from the trace of thematrix• For example, the trace of A and B for the symmetry group above

A =

(−1/2 −

√3/2√

3/2 −1/2

)

B =

(−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 thecharacters squared. For our 6-element group, we find22 + 12 + 12 = 6


Character, reducible and irreducible representations

• To avoid ambiguity, we define the character only for irreduciblerepresentation. If we can block diagonalize all elementssimultaneously (which amounts to a change of basis), therepresentation is said to be reducible.• The character depends on representation! We can sometimesfind one, two, three, etc dimensional representations of a group.For the group above with 6 elements, no three dimensionalrepresentations exist because 32 = 9 > 6 (The trace of I is equalto the dimensionality of the representation).


Chapter 4: Partial differentiation

It is generally the case that derivatives are introduced in terms offunctions of a single variable. For example, y = f (x), thendydx = df

dx = f ′. However, most of the time we are dealing withquantities that are functions of several variables. For example, weusually want physical quantities in three dimensional space. Forexample, the electric field at each point in space might depend onx , y , and z , ~E → ~E (x , y , z). Or, it might be convenient in somecases to use spherical coordinates, and then ~E → ~E (r , φ, θ). Wehence have to think about partial differentiation in physics.


Chapter 4 goals

By the end of this chapter, you should be able to:

I Work with power series in two or more variables

I Use total differentials

I Use total differentials for approximation

I Use the chain rule for differentiation of a function of afunction

I Use partial differentiation in maximum/minimum problems

I Use Lagrange multipliers in maximum/minimum problemswith constraints

I Make changes of variables, including using spherical andcylindrical coordinate systems

I Take derivatives of integrals


Introduction and notation

• For example, say we have z = f (x , y), we then need partialderivatives

∂z

∂x=∂f

∂x= fx

• The key is we take derivative with respect to x , while keeping yfixed• For example, z = f (x , y) = x2 cos y , then ∂z

∂x = 2x cos y

• Then we can take another derivative, ∂2z∂y∂x = −2x sin y

• Does the order matter? Notice that ∂2z∂x∂y = −2x sin y .

• We see ∂2z∂x∂y = ∂2z

∂y∂x . This is most often true.• Sometimes we explicitly note that one variable is fixed (forexample, in thermodynamics)(

∂z

∂x

)y

= 2x cos y


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) = a00+a10(x−a)+a01(y−b)+a20(x−a)2+a02(y−b)2+a11(x−a)(y−b)+...

• We see that a00 = f (a, b)• We can take partial derivatives with respect to x and y of thepower series

fx =∂f

∂x= a10 + 2a20(x − a) + a11(y − b) + ...

fy =∂f

∂y= a01 + 2a02(y − b) + a11(x − a) + ...

• We evaluate the derivatives at x = a, y = b, and obtain theinfinite Taylor series


Power series continued

• a10 = fx(a, b), a01 = fy (a, b), a20 = 12 fxx(a, b), a02 = 1

2 fyy (a, b),a11 = fxy (a, b) = fyx(a, b), etc.• We can express with h = x − a and k = y − b,

f (x , y) =∞∑

n=0

1

n!

(h∂

∂x+ k

∂

∂y

)n

f (a, b)

• Where we mean ∂∂y f (a, b) = fy (a, b)


Example: Section 2, Problem 2

• Find the Mclaurin series (expansion about x = 0,y = 0) off (x , y) = cos(x + y)• We see fx = fy = − sin(x + y), andfxx = fyy = fxy = fyx = − 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

n!

(h∂

∂x+ k

∂

∂y

)n

f (a, b)

cos(x+y) = 1− 1

2!(x2+2xy+y 2)+

1

4!(x4+4x3y+2x2y 2+4xy 3+y 4)+...


Total differential for y = f (x)

• For y=f(x), we have y ′ = dydx = df

dx• We can treat dx = ∆x as an independent variable• In the limit ∆x → 0, then

dy

dx= lim

∆x→0

∆y

∆x

• If ∆x finite, then dy is not exactly ∆y


Total differential for z = f (x , y) and for many independentvariables

• For a function of two variables, z = f (x , y), we can define thetotal differential

dz =∂z

∂xdx +

∂

∂ydy

• 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 dxand dy• For a function of many variables u = f (x1, x2, ..., xN), we definethe total differential

du =N∑

n=1

∂u

∂xndxn


Thermodynamics

• In thermodynamics, we have quantities that might pressure p,volume V , temperature T , entropy S , particle number N, andchemical potential µ.• These are not all independent, so if we know p then V isdetermined, hence we describe quantities in terms of some subsetof 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

∂SdS +

∂U

∂VdV +

∂U

∂NdN

• We define T = ∂U∂S , p = − ∂U

∂V , and µ = ∂U∂N

dU = TdS − pdV + µdN


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 convenientbecause in experiments it is easy to control and measure T and p,as opposed to S and V (entropy and volume)


brief introduction to groups and group theory - ucf …schellin/teaching/phz3113_2011/...brief...

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