Download Groups, Representations, and Particle Physics

Groups, Representations, and
Particle Physics
Cayle Castor
Review of Groups
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Consider a set of elements G equipped with a
binary operation ( , ). G is a group under this
operation if:
0.)
1.)
2.)
3.)
G is closed under ( , )
( , ) is associative
Identity
Inverses
Quickly... examples of Groups
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ℤ
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ℤn under ( + )mod(n)
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SU(N) under matrix multiplication
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GL(2,F) under matrix multiplication
under ( + )
-Invertible <--> non-zero determinant
-If determinant = 1 we have SL(2,F) (a subgroup).
In group theory, “multiplication” really means binary
operation. It's essentially a mapping.
Consider the group Z3, the integers under addition
modulo 3.
(0,1) ---> 1
(1,0) ---> 1
(0,2) ---> 2
(2,0) ---> 2
(1,2) ---> 0
(2,1) ---> 0
There's an easier way to display this information...
(Z3)
Z3
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
...organizing the binary mapping in this way is
quite suggestive.
What does the multiplication table of a nonAbelian group look like?
Homomorphisms
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A homomorphism Φ: G → G' is a mapping from
one group, G, to another G' that preserves the
group operation.
Φ(a.b) = Φ(a).Φ(b)
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There is a subtle point about the binary operation
on the right hand side of this condition.
Classic examples
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SU(2) and SO(3)
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Φ: G⊕H → G via Φ(g,h) → g.
-What is the kernel of this operation?
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Is Φ: Z → 2Z via Φ(z)→ 2z a
homomorphism?
Representation Theory
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A representation of a group G is a mapping to a set
of linear operators on a vector space ...
D: G –> GL(V)
… such that …
1.) D(e) = I (identity)
2.) D(g1g2) = D(g1)D(g2) ∀ g1,g2
This is just a group homomorphism.
∈
G
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Consider a group (G) and some representation D of G.
g∈G
D
g1
-------------------------->
g2
--------------------------->
g3
--------------------------->
g4
--------------------------->
D(g)∈GL(V)
...etc
Regular Representation
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Take the group elements to form an orthonormal basis for
some vector space...
V={| g>| g∈G}
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The representation, D, is defined as the mapping to matrices
which give the proper multiplication rules on the states.
(Permutation Matrices for finite discrete groups)
Dimension(V) = |G|
Equivalent Representations
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We can always construct a new representation from an old one, by
performing a similarity transformation to each element...
SD(g)(S^-1)
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This clearly preserves the multiplication rules of D(g) ∈ D, (as well as
g∈G)
D and D' are said to be equivalent representations.
Faithful Representations
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Representations (homomorphisms to matrices) that are to one-to-one.
Notice that the kernel of a homomorphism is always a normal subgroup
of the first group (the domain of the mapping).
Unfaithful representations “blur” structure by “dividing” out a normal
subgroup. This “lumps” elements of G together by an equivalence class
induced by the mapping.
G/ker(D) = {g ker(D) | g∈G} is isomorphic to the image of G under D,
D(G).
ex.)
Z4={0,1,2,3}
Normally, <2>= {0,2,4,6,...}, however a coset must be a subgroup. For the operation of addition
modulo four, <2>={0,2}.
Z4/<2> = {z + <2> | z∈Z4} = {z+ {0,2} | z∈Z4}
...where z+<2> is a left coset multiplication on <2>.
0+<2>= 0+<2>
1+<2>=1+<2>
2+<2>=0+<2>
3+<2>= (1+2) + <2> = 1 + (2+<2>) = 1+<2>
… we are effectively absorbing “factors” of two into the coset <2>. The only remaining
elements are 0+<2> and 1+<2>.
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Therefore Z4/<2> is isomorphic to Z2={0,1}.
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Φ:Z4 → Z2 given by Φ(z)= z + <2> is a homomorphism.
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{0,2} are in an equivalence class, and so are {1,3}. If you “blur your eyes” elements of these
sets look equivalent.
Reducible Representations
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A representation is reducible if it has an invariant
subspace V' . In other words,
P D(g) P = D(g) P
∀g∈G
Where P is a projection operator onto the subspace.
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Can we make this intuitive?
It has something to do with our typographical method
containing redundant information.
Example:
Parity in Quantum Mechanics
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P^2 = e
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Parity
e
p
e
e
p
p
p
e
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Let's try the regular
representation...
Result
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Thus there are two irreducible representations.
1.) D(e)=1
D(p)= -1
2.) D(e)=1
D(p)=1 (Trivial Representation)
If D(p) commutes with the Hamiltonian we can simultaneously diagonalize
and assign energy eigenstates definite values of p.
Energy Eigenstates that transform as D(p)|E> = 1|E> are said to transform
as the trivial representation.
Energy eigenstates that transform as D(p)|E> = -1|E> transform nontrivially.
Lie Groups: Instructive Example
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Consider a differential sub-manifold in ℝ²
parameterized by:
ɣ(t) = ( cos(t), sin(t) )
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Now consider a ket state
defined over ℝ².
Ψ = Ψ(x,y)
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Now we want to restrict Ψ to it's values over this
manifold.
Ψ = Ψ(ɣ(t))
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Let's search for a unitary
operator acting on the
Hilbert space with the
following action on kets.
exp[itA] Ψ(ɣ(0)) = Ψ(ɣ(t))
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Here, A is some self-adjoint operator
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Let's take a derivative with respect to the manifold parameter (t).
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∂/∂t ( exp[itA] Ψ(x(0),y(0)) ) = ∂/∂t ( Ψ(x(t), y(t)) )
=> iA exp[itA] Ψ(x(0),y(0)) = (∂x/∂t)(∂Ψ/∂x) + (∂y/∂t)(∂Ψ/∂y)
=>
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iA Ψ(x(t),y(t)) = (ɣ'(t)·∇) Ψ
Recall: ɣ(t) = (rcos(t), rsin(t)) => ɣ'(t) = ( -rsin(t), rcos(t) )
∴ iA ≡ - r sin(t)(∂/∂x) + r cos(t)(∂/∂x)
= x(∂/∂y) – y(∂/∂x)
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A = x·Py – y·Px = Lz
So we see the form of the A operator is fixed by
the topology and parameterization of the submanifold of the domain of Ψ.
The above example is rather simple, due to
Stone's Theorem.
“...one-to-one correspondence between self-adjoint
operators on a Hilbert space H and one-parameter
families of unitary operators”
ex.) H ↔ {exp(iHt) | t∈ℝ}
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Notice that the set {exp(iHt) | t∈ℝ} forms an Abelian group under
multiplication...
-Identity ✓
-Inverse ✓
- Associativity ✓
-Closure: ✓
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exp(iHs)*exp(iHt)=exp(iHt+iHs)=exp(iH(t+s))
where (t+s)∈ℝ.
The closure property would fail for some t and s, if ℝ itself were not
closed under addition.
Things are not so simple when the dimension of our manifold is greater
than 1. There will be a self adjoint operator associated with each
parameter, and they may not commute.
Lie Groups, Lie Algebras
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Abstract:
A Lie group is a group which is also a finite dimensional real smooth manifold.
Practical:
Suppose we find a representation, D, of some Lie group G. This means we have...
{ D(g(t)) ∈GL(n,ℂ )| D is a homomorphism from G and t parameterizes the group elements}
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Abstract:
A Lie algebra, A, is a vector space over some field F equipped with a binary operation called a Lie bracket...
[ , ]: AxA → A
… which is Bilinear and Alternating, and satisfies the Jacobi identity. (a.k.a Commutator)
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Practical:
A Lie Algebra, A, associated with some Lie group, G, is the set of matrices...
{X ∈ GL(n, ℂ) | exp(t X) ∈ G}
(Definitions differ for the Lie Algebra. Some sources refer to the generators, and some sources refer to the vector space V spanned by
the generators For our purposes, take X as some linear combination of generators, where the coefficients are elements of a group which
can be indexed some manifold parameter.)
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Let's refer back to and expand upon the circle example (to a 3 sphere). Recall that t is the
manifold parameter.
Lie group:
{exp(it abAb) | Ab ∈ {Lx, Ly, Lz} and ab ∈ F}
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Lie Algebra:
{axLx + ayLy + azLz | equipped with [ , ] }
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The vector space is three dimensional, spanned by the angular momentum operators.
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{Lx, Ly, Lz} are called the generators of the Lie Algebra.
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Rigorously speaking, the generators are determined by looking for Lie group elements which
differ infinitesimally from the identity.
This Lie group is called SO(3) and the Lie algebra is called so(3).
A few things...
1.) Lie Group is Abelian <=> [Xi,Xj] = 0 ∀ Xi,j∈A
2.) Lie Group is non-Abelian => [Xi,Xj] = ∑Cijk Xk
-Cijk are called structure constants.
Here, we have assumed a unitary representation of the symmetry group:
U(t,ai) = exp(t ai Xi)
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If the Hamiltonian of a physical system is invariant under
some symmetry group G, then all members of a multiplet
belonging to an irreducible unitary representation have the
same energy (spectrum).
A multiplet is the set of basis vectors spanning the
vector space V upon which the representation matrices
act.
H Ψn= EnΨn
UaHΨn = (UaHUˉa) UaΨn = EnUaΨn
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H′= UaHUˉa
Ψ'n = UaΨn
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Our equation becomes...
H'Ψ'n = EnΨ'n
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However, if U is a symmetry of our physical system, then...
H'=H (notice that this implies [H,U]=0 as we expect)
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Finally, we arrive at....
HΨ'n = EnΨ'n
(The transformed states share the same energy)
Gauge Theory?
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Gauge Theory:
“In physics, a gauge theory is a type of field theory in which the
Lagrangian is invariant under a continuous group of local
transformations.”
~Wikipedia
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Non-Abelian Gauge Theory:
Just introduce non-zero structure constants!
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Cayle, this slide is to remind you to consult
your notebook and talk about non-abelian lie
groups.
The Standard Model
SU(3) × SU(2) × U(1)