group theory & lie algebra primer - university of oxford€¦ · theory (e.g. maxwell’s...

Group Theory & Lie Algebra Primer

Uli Haisch, C6, MT 2012Please send corrections to [email protected]

Role of Symmetries

Theory (e.g. Maxwell’s theory of electrodynamics)

Symmetry (e.g. Lorentz invariance of electrodynamics)

discover symmetries (historic approach)

find most general theory compatible with symmetry

(modern approach)

Symmetries in Mathematics

Symmetry Group

Objects that symmetry acts on

Representations of group

What is a Group?

Definition:

A group G is a set with a map G × G → G, a “multiplication”, satisfying three conditions

1) Associativity: g1· (g2· g3) = (g1· g2)· g3 for ∀ g1, g2, g3 ∈ G

2) Neutral element: there is e ∈ G with g· e = g for ∀ g ∈ G

3) Inverse element: for ∀ g ∈ G there is g−1 ∈ G with g· g−1 = e

Group Properties

Axioms imply

i) e is unique & e· g = g· e = g for ∀ g ∈ G

ii) g−1 is unique for each g ∈ G & g−1· g = g· g−1 = e

If in addition

4) g1· g2 = g2· g1 for ∀ g1, g2 ∈ G

G is called abelian, otherwise G is non-abelian

Well-Known Abelian Groups

Integers with respect to addition

Non-zero real & complex numbers with respect to addition &

multiplication

For a, b integer:

closure: a + b is an integerassociativity: a + (b + c) = (a + b) + ccommutativity: a + b = b + aidentity: a + 0 = ainverse: a + (−a) = 0

Why are integers not a group under multiplication?

A Finite Abelian Group

ZN = {0, 1, . . . , N − 1}

n1 · n2 = (n1 + n2)modN ∀n1, n2 ∈ ZN

The integers modulo N

form an abelian group with a finite number of elements

A Infinite Abelian Group

x

y

z1

1Consider the complex numbers of unit length

U(1) = {z ∈ C|z|=1}

with respect to ordinary multiplication

This group is infinite, continuous & one-dimensional. It corresponds to the unit circle in the complex plane

Well-Known Non-Abelian Groups

The special unitary group of degree N, denoted SU(N), is the group of complex N × N matrices U with

detU = 1U †U = UU† = 1

The group operation is the normal matrix multiplication & the identity is the N × N unit matrix. Because matrices do in general not commute

SU(N) is non-abelian

[U1, U2] = U1U2 − U2U1 �= 0 U1, U2 ∈ SU(N)

A Closer Look at SU(2)

What is the dimension of SU(2)? Or how many numbers do we need to parametrize U ∈ SU(2)?

U is a complex 2 × 2 matrix

with U†U = 1

with det U = 1

⇒

⇒

⇒

8 parameters

−4 conditions

−1 condition

3 parameters

A Closer Look at SU(2)

Easy to see that one can write

SU(2) =

��α β

−β∗ α∗

�, α,β ∈ C , |α|2 + |β|2 = 1

�

which shows that SU(2) can be thought as the unit sphere in four-dimensional Euclidean space. The explicit choice

α =�

1− |β|2 eiσ β = β1 − iβ2

with β1,2, σ real, solves the remaining constraint

Group Representations: Definition

We denote by GL(n) the group of invertible n × n matrices with real (complex) entries. These matrices can be thought as linear transformations acting on an n-dimensional real (complex) vector space V

Definition:

A representation R of a group G is a map R: G → GL(n) which satisfies R(g1· g2) = R(g1) R(g2)

Group Representations: Meaning

Mathematics: A representation assigns to each g ∈ G a matrix R(g) ∈ GL(n) such that these matrices multiply “in the same way” as the group elements. In this manner, G is “represented” as a set of n × n matrices acting on V. The dimension n of V is the dimension of the representation

Physics: The elements of V are interpreted as fields & group theory provides the appropriate mathematical formalism to describe how symmetries act on them. In particular, finding all representations is then equivalent to finding all fields on which the symmetry can act. These fields in turn represent the “building blocks” which can be used to construct any field theory that respects the symmetry

Trivial Representation

The trivial representation which exists for all groups is

R(g) = 1 ∀g ∈ G

& represents each group element by the unit matrix in a given dimension

Representations of Integers Modulo N

For the group of integers modulo N & each integer q,

Rq(n) =

�cos (2πqn/N) sin (2πqn/N)− sin (2πqn/N) cos (2πqn/N)

�∈ R2

is a representation. Restricting the “charge” q to [0, ..., N-1] provides a complete set of representations

Rq(n)Rq(m) =

�cos (2πq(n+m)/N) sin (2πq(n+m)/N)− sin (2πq(n+m)/N) cos (2πq(n+m)/N)

�= Rq(n+m)

n1 · n2 = (n1 + n2)modN

⇔

Note that:

Representations of Integers Modulo N

The same representations can also be written as

Rq(n) = exp (2πiqn/N) ∈ C

Φ → Rq(n)Φ = exp (2πiqn/N)Φ

The group acts on the elements Φ of the one-dimensional complex vector space as

Realize that q = 0 corresponds to the trivial representation

Representations of U(1)

The representations of U(1) are also easily written down:

Here α ∈ [0, 2π] & the charge has to be integer for Rq to be continuous on the circle. The representations Rq with q an arbitrary integer provide all continuous representations of U(1)

Rq(eiα) = eiqα ∈ C

Representations of SU(N)

The groups SU(N) are already given by matrices, so these matrices “represent themselves”. This representation is N-dimensional & complex & called fundamental. It action on a N-dimensional complex vector Φ is given by

Φ → UΦ U ∈ SU(N)

As we will see later, this is by far not the only representation of SU(N), but there is an infinite number of them

Constructing Representations

For each complex representation R(g) there is a complex conjugate representation R* defined by R*(g) = (R(g))*

Example: In the case of SU(N) there is a complex conjugate fundamental representation U*, which is represented by the complex conjugate of the original representation matrices


For two representations R1 & R2 with dimensions n1 & n2 one can consider the direct sum

Such a representation has dimension n1 + n2 & is referred to as reducible. A representation R(i) that cannot be split into smaller blocks, as above, is called irreducible

(R1 ⊕R2)(g) =

�R1(g) 00 R2(g)

�

Example: The representation R(e iα) = diag (e iα, e −iα) is an explicit embedding of two U(1)’s with charge 1 & −1 into the group SU(2)


Two representations R1 & R2 can also be combined into a tensor representation with dimension n1 n2 via

(R1 ⊗R2)(g) = R1(g)×R2(g)

Here × is the Kronecker product, which for a n × m matrix M & a p × q matrix N gives the n p × m q matrix

M×N =

M11N . . . M1nN

.... . .

...Mm1N . . . MmnN

=

M11N11 . . . M1nN1q

.... . .

...Mm1Np1 . . . MmnNpq


A tensor representation can typically be decomposed into a sum of irreducible representations

R1 ⊗R2 =�

i

R(i)

This is called Clebsch-Gordon decomposition

An Example: SU(3) Color

Here the three components correspond to the three colors red, green, and blue. There is also a 3* representation with the associated colors anti-red, anti-green, and anti-blue, shown in cyan, magenta, and yellow

The quark fields are classified in the three-dimensional fundamental representation of SU(3) & form color triplets

3 =

ψr

ψg

ψb

, 3∗ =

ψ̄r

ψ̄g

ψ̄b


It is easy to see that these meson states (pion, ...) transform indeed trivial under SU(3):

According to “color confinement” only colorless objects are allowed to exist in Nature. In fact, the quark colors & anti-colors can be combined to “white” objects. For example

ψ̄iψi → U∗ijUikψ̄jψk = ψ̄iψi , U†U = UU† = 1

blue anti-blue , . . .ψ̄iψi =


�ijkψiψjψk → �ijkUii�Ujj�Ukk�ψi�ψj�ψk�

= det�U

��ijkψiψjψk = �ijkψiψjψk , det

�U

�= 1

transform as singlets under SU(3). This is readily seen

Also baryons (proton, neutron, ...)

red

green blue

�ijkψiψjψk = , . . .

det(U) det(U) = 1


This example shows that group theory provides a neat way to understand important aspects of the subatomic world. It tells us that there a two types of hadrons & after digging deeper, we would see, that it also allows us to predict the multiplet structure of these mesons & baryons

In Nature one finds a nonet of pseudoscalar mesons (neutral pion, ...) & vector mesons (rho, ...) as well as spin-1/2 baryon octet (proton, neutron, ...) & a spin-3/2 baryon decuplet (Sigma, ...)

Lie Groups & Algebras

The matrices of a Lie group G form a continuous & differen-tiable family M = M(t), where t = (t1, ..., tm) are m real parame-ters. Close to the identity one can write

M(t) = 1+�

i

tiT i +O(t2) Ti =∂M(t)

∂ti

��t=0

The matrices Ti are called generators & the vector space spanned by them is called Lie algebra L(G)

The group G can be reconstructed by the exponential map

M(t) = exp�itiT

i�


M(t)−1M(s)−1M(t)M(s) = 1 + tisj [Ti, T j ] + . . .

Since the left-hand side is a group element, we conclude that the commutator of two generators must be an element of the Lie algebra

Consider now

Remember from quantum mechanics: eAeB = eBeAe[A,B]

[T i, T j ] = f ijkT

k

The coefficients fijk are called structure constants of L(G)


Definition:

A representation r of a Lie algebra L is a linear map which assigns to elements T ∈ L matrices r(T) such that one has [r(T), r(S)] = r([T, S]) for ∀ T, S ∈ L

This is equivalent to saying that the representation matrices r(Ti) commute in the same way as the generators Ti & have the same structure constants: [r(Ti), r(Tj)] = f ijk r(Tk)


In practice it is often easier to find representations of L(G) rather than G. But this is not a problem, as one can use the exponential map to reconstruct the group representations: Ti → r(Ti), exp (i tiTi) → exp (i ti r(Ti))

Recall that the dimension of a representation r is equal to the dimensionality of the vector space that the matrices r(T) act on. Hence it is given by the size of r(T). This dimension is not to be confused with the dimension of the Lie algebra, the latter being the dimension of L(G) as a vector space of matrices, which is equal to the number of generators

SU(2) Lie Algebra

Consider a SU(2) matrix U close to the identity & write

U †U =�eiT

�†eiT = 1− i

�T † − T

�+ . . . = 1

detU = det�eiT

�= etr(iT ) = 1 + i tr(T ) + . . . = 1

U = eiT = 1+ iT + . . . T ∈ L (SU(2)) = SU(2)

Then

which implies that the generators are hermitian & traceless:

T † = T tr(T ) = 0

SU(2) Lie Algebra

The vector space formed by the hermitian, traceless 2 × 2 matrices has three dimensions. In fact, this is exactly the number of parameters (β1, β2 & σ) we had in our explicit parametrization of SU(2)

A convenient basis of generators τi (i = 1, 2, 3) of L(SU(2)) is obtained from the Pauli matrices

σ1 =

�0 11 0

�σ2 =

�0 −ii 0

�σ3 =

�1 00 −1

�

SU(2) Lie Algebra

σiσj = δij1+ i�ijkσk

tr [σiσj ] = 2δij

[σi,σj ] = 2i�ijkσk

The Pauli matrices satisfy the following useful relations

which implies that L(SU(2)) is spanned by

τi =1

2σi [τi, τj ] = i�ijk τk

Here εijk is the Levi-Civita tensor (ε123 = +1). Note that τ3 corresponds to the charge (1, −1) U(1) subgroup of SU(2)

The commutation relations of SU(2) are identical to those of the angular momentum operator in quantum mechanics. Finite-dimensional representations of SU(2) can hence be labelled by a “spin” j = 0, 1/2, 1, 3/2, ...

For a given j, the representation is 2 j + 1 dimensional & the representation space is spanned by |jm , m = −j, −j+1..., 0, ..., j−1, j

The two-dimensional representation for j = 1/2 of course corresponds to the explicit representation of L(SU(2)) in terms of Pauli matrices. The complex conjugate of this fundamental representation has also two dimensions & can again be identified with the j = 1/2 representation

SU(2) Lie Algebra

〉

SU(2) Lie Algebra

2⊗ 2 = 1⊕ 3 2⊗ 3 = 2⊕ 4

From quantum mechanics we also might remember that the tensor product of two representations j1 & j2 decomposes into irreducible representations with j in the range |j1 − j2|, |j1 − j2| +1, ..., j1 + j2, which provides an explicit example of a Clebsch-Gordon decomposition

For example, denoting the j = 1/2 representation as 2 & the j = 1 representation as 3, one can show that

The first relation implies that if we have a field ϕ that is a doublet of SU(2), the term ϕ2 will contain a singlet (this will become relevant in the discussion of the Higgs mechanism)

SO(3) Lie Algebra

Another important Lie group is SO(3), the group of three-dimensional rotations, formed by real 3 × 3 matrices with

OTO = 1 detO = 1

Writing O = 1 + i T with (purely imaginary) generators T, the first relation implies that the generators are hermitian & therefore that L(SO(3)) consists out of three 3 × 3 anti-symmetric matrices (multiplied by i), which may be written as (Ti)jk = −i εijk

SO(3) Lie Algebra

[Ti, Tj ] = i�ijkTk

The corresponding commutator reads

Notice that these are the relations we already saw in the case of L(SU(2)). It follows that the generators Ti form a three-dimensional (irreducible) representation of the Lie algebra of SU(2). Clearly, this representation must fit into the classification of SU(2) in terms of integer or half-integer spin. Based on dimensional grounds we thus conclude that it can only be the j = 1 representation

Lorentz Group

The Lorentz group L is of fundamental importance for the construction of field theories, since L is associated to the symmetry of four-dimensional space-time

Using the metric η = diag(1,−1, −1, −1), L consists of the real 4 × 4 matrices Λ satisfying

or

when written out in components

ΛT ηΛ = η

ηµν = ηρσΛµρΛ

νσ

ds2 = ηµνdxµdxν → ηµνΛ

µρdx

ρΛνσdx

σ

= ηρσdxρdxσ

= ds2

Lorentz Group

Remember that the latter transformation property leaves the distance ds2 invariant, as readily seen:

xµ → Λµνx

ν

Lorentz Group

Special Lorentz transformations are the identity 14, parity P, time-reversal T & the combination of P & T:

P = diag(1,−1,−1,−1)

T = diag(−1, 1, 1, 1)

PT = −14

We note that the set {14, P, T, PT} forms a finite subgroup of the Lorentz group

(Λ00)

2 = 1 +3�

i=1

(Λi0)

2 ≥ 1

Lorentz Group

implying

It is straightforward to see that det(Λ) = ±1 & furthermore one has for ρ = σ = 0,

ηµνΛµ0Λ

ν0 = 1

So either Λ00 ≥ 1 or Λ00 ≤ −1. This sign choice combined with the sign of the determinant leads to four disconnected components of the Lorentz group

Lorentz Group4.1. SYMMETRIES 43

det(!) !00 name contains given by

+1 ! 1 L!+ 14 L!

+

+1 " #1 L"+ PT PTL!

+

#1 ! 1 L!# P PL!

+

#1 " #1 L"# T TL!

+

Table 4.1: The four disconnected components of the Lorentz group. The union L+ = L!+ $ L"

+ is also called theproper Lorentz group andL! = L!

+$L!# is called the orthochronosLorentz group (as it consists of transformations

preserving the direction of time). L!+ is called the proper orthochronos Lorentz group.

O satisfying OTO = 1 and det(O) = 1. Writing O = 1+ iT with (purely imaginary) generators T , the relationOTO = 1 implies T = T † and, hence, that the Lie-algebra of SO(3) consists of 3 % 3 anti-symmetric matrices(multiplied by i). A basis for this Lie algebra is provided by the three matrices Ti defined by

(Ti)jk = #i!ijk , (4.16)

which satisfy the commutation relations[Ti, Tj] = i!ijkTk . (4.17)

These are the same commutation relations as in Eq. (4.13) and, hence, the Ti form a three-dimensional (irreducible)representation of (the Lie algebra of) SU(2). This representation must fit into the above classification of SU(2)representations by an integer or half-integer number j and, simply on dimensional grounds, it has to be identifiedwith the j = 1 representation.

4.1.4 The Lorentz groupThe Lorentz group is of fundamental importance for the construction of field theories. It is the symmetry associatedto four-dimensional Lorentz space-time and should be respected by field theories formulated in Lorentz space-time.Let us begin by formally defining the Lorentz group. With the Lorentz metric " = diag(1,#1,#1,#1) the Lorentzgroup L consists of real 4% 4 matrices ! satisfying

!T "! = " . (4.18)

Special Lorentz transformations are the identity 14, parityP = diag(1,#1,#1,#1), time inversionT = diag(#1, 1, 1, 1)and the product PT = #14. We note that the four matrices {14, P, T, PT } form a finite sub-group of the Lorentzgroup. By taking the determinant of the defining relation (4.18) we immediately learn that det(!) = ±1 for allLorentz transformations. Further, if we write out Eq. (4.18) with indices

"µ!!µ"!

!# = ""# (4.19)

and focus on the component # = $ = 0 we conclude that (!00)2 = 1 +

!

i(!i0)2 ! 1, so either !0

0 ! 1or !0

0 " #1. This sign choice for !00 combined with the choice for det(!) leads to four classes of Lorentz

transformations which are summarised in Table 4.1. Also note that the Lorentz group contains three-dimensionalrotations since matrices of the form

! =

"

1 00 O

#

(4.20)

satisfy the relation (4.18) and are hence special Lorentz transformations as long as O satisfies OTO = 13.To find the Lie algebra of the Lorentz group we write! = 14+iT+ . . . with purely imaginary 4%4 generators

T . The defining relation (4.18) then implies for the generators that T = #"T T", so T must be anti-symmetricin the space-space components and symmetric in the space-time components. The space of such matrices is six-dimensional and spanned by

Ji =

"

0 00 Ti

#

, K1 =

$

%

%

&

0 i 0 0i 0 0 00 0 0 00 0 0 0

'

(

(

)

, K2 =

$

%

%

&

0 0 i 00 0 0 0i 0 0 00 0 0 0

'

(

(

)

, K3 =

$

%

%

&

0 0 0 i0 0 0 00 0 0 0i 0 0 0

'

(

(

)

, (4.21)

The union L+ = L+ ∪ L+ is also called the proper Lorentz group, while L↑ = L+ ∪ L− is known as the orthochronos Lorentz group (because its transformations preserve the time direction). Finally, L+ is both proper & orthochronos

↑ ↓

↑ ↑

↑

Lorentz Group

A simple rotation by the angle θ about the z-axis & a boost by v < 1 (h = c = 1) along the x-axis (γ = (1 − v2)−1/2),

while all other components are zero. With this in hand, we then obtain from ∂µF µ0 = ρ and∂µF µ1 = j1,

∂µFµ0 = ∂0F

00 + ∂iFi0 = ∇ · E = ρ ,

∂µFµ1 = ∂0F

01 + ∂iFi1 = −∂E1

∂t+

∂B3

∂x2− ∂B2

∂x3=

�∇×B − ∂E

∂t

�1

= j1 .(2.39)

Similar relations hold for the remaining components i = 2, 3. Taken together this proves thesecond inhomogeneous Maxwell equation (2.28).

Let me also derive the energy-momentum tensor T µν of electrodynamics, ignoring for themoment the source term AµJµ. Using (2.33) one finds

T µν = (∂νAµ)(∂ρAρ)− (∂µAρ)(∂

νAρ) +1

4ηµνFρσF

ρσ . (2.40)

Notice that the first term in (2.40) is not symmetric, which implies that T µν �= T νµ. In fact,this is not really surprising since the definition of the energy-momentum tensor (2.20) doesnot exhibit an explicit symmetry in the indices µ and ν. Nevertheless, there is typically a wayto massage the energy-momentum tensor of any theory into a symmetric form.5 To learn howthis can be done in the case under consideration is the objective of a homework problem.

2.4 Space-Time Symmetries

One of the main motivations to develop QFT is to reconcile QM with special relativity. Wethus want to construct field theories in which space and time are placed on an equal footingand the theory is invariant under Lorentz transformations,

xµ → (x�)µ = Λµνx

ν , (2.41)

withηµν = ηρσΛµ

ρΛνσ , (2.42)

so that the distance ds2 = ηµνdxµdxν is preserved. Here ηµν = ηµν = diag (1,−1,−1,−1)denotes the Minkowski metric. E.g., a rotation by the angle θ about the z-axis, and a boostby v < 1 along the x-axis are respectively described by the following Lorentz transformations

Λµν =

1 0 0 0

0 cos θ − sin θ 0

0 sin θ cos θ 0

0 0 0 1

, Λµ

ν =

γ −γv 0 0

−γv γ 0 0

0 0 1 0

0 0 0 1

, (2.43)

with γ = 1/√

1− v2. The Lorentz transformations form a Lie group under matrix mul-tiplication. You’ll learn more about this if you study the “group theory crash course” ofMartin Bauer, which contains some useful exercises.

5One (but not the only) reason that you might want to have a symmetric energy-momentum tensor Tµν

is to make contact with general relativity, since such an object sits on the right-hand side of Einstein’s field

equations.

14

while all other components are zero. With this in hand, we then obtain from ∂µF µ0 = ρ and∂µF µ1 = j1,

∂µFµ0 = ∂0F

00 + ∂iFi0 = ∇ · E = ρ ,

∂µFµ1 = ∂0F

01 + ∂iFi1 = −∂E1

∂t+

∂B3

∂x2− ∂B2

∂x3=

�∇×B − ∂E

∂t

�1

= j1 .(2.39)

Similar relations hold for the remaining components i = 2, 3. Taken together this proves thesecond inhomogeneous Maxwell equation (2.28).

Let me also derive the energy-momentum tensor T µν of electrodynamics, ignoring for themoment the source term AµJµ. Using (2.33) one finds

T µν = (∂νAµ)(∂ρAρ)− (∂µAρ)(∂

νAρ) +1

4ηµνFρσF

ρσ . (2.40)

Notice that the first term in (2.40) is not symmetric, which implies that T µν �= T νµ. In fact,this is not really surprising since the definition of the energy-momentum tensor (2.20) doesnot exhibit an explicit symmetry in the indices µ and ν. Nevertheless, there is typically a wayto massage the energy-momentum tensor of any theory into a symmetric form.5 To learn howthis can be done in the case under consideration is the objective of a homework problem.

2.4 Space-Time Symmetries

One of the main motivations to develop QFT is to reconcile QM with special relativity. Wethus want to construct field theories in which space and time are placed on an equal footingand the theory is invariant under Lorentz transformations,

xµ → (x�)µ = Λµνx

ν , (2.41)

withηµν = ηρσΛµ

ρΛνσ , (2.42)

so that the distance ds2 = ηµνdxµdxν is preserved. Here ηµν = ηµν = diag (1,−1,−1,−1)denotes the Minkowski metric. E.g., a rotation by the angle θ about the z-axis, and a boostby v < 1 along the x-axis are respectively described by the following Lorentz transformations

Λµν =

1 0 0 0

0 cos θ − sin θ 0

0 sin θ cos θ 0

0 0 0 1

, Λµ

ν =

γ −γv 0 0

−γv γ 0 0

0 0 1 0

0 0 0 1

, (2.43)

with γ = 1/√

1− v2. The Lorentz transformations form a Lie group under matrix mul-tiplication. You’ll learn more about this if you study the “group theory crash course” ofMartin Bauer, which contains some useful exercises.

5One (but not the only) reason that you might want to have a symmetric energy-momentum tensor Tµν

is to make contact with general relativity, since such an object sits on the right-hand side of Einstein’s field

equations.

14

are also part of the Lorentz group. In fact, any rotation

Λ =

�1 00 O

�

leaves ds2 invariant because OTO = 13 for any O ∈ SO(3)

Lorentz Lie Algebra

To find the Lie algebra of the Lorentz group, we follow the usual procedure & write

Λ = 14 + iT + . . .

T = −ηTT η

with purely imaginary generators. Inserting the linearized Lorentz transformation into the defining relation, implies

This tells us that the generators must be antisymmetric in the space-space components, but symmetric in the space-time components

Lorentz Lie Algebra

The space of such matrices has dimension six & is spanned by the set

Ji =

�0 0

0 Ti

�

K1 =

0 i 0 0

i 0 0 0

0 0 0 0

0 0 0 0

K2 =

0 0 i 0

0 0 0 0

i 0 0 0

0 0 0 0

K3 =

0 0 0 i

0 0 0 0

0 0 0 0

i 0 0 0

where Ti (i = 1,2, 3) are the generators of the rotation group or SO(3). It follows that Ji (Ki) generate rotations (boosts)

Lorentz Lie Algebra

It is straightforward to work out the commutation relations of the Lorentz Lie algebra generators. One finds

[Ji, Jj ] = i�ijkJk

[Ki,Kj ] = −i�ijkJk

[Ji,Kj ] = i�ijkKk

Lorentz Lie Algebra

The matrices Ji & Ki can also be written by means of 4 × 4 matrices σμν labelled by two antisymmetric four-indices & defined as

(σµν)ρσ = i

�ηρµηνσ − ηµση

ρν

�

Note that μ & ν label the six different σ matrices, whereas ρ & σ denote the elements of these objects

Λρσ = δρσ − i

2ωµν(σµν)

ρσ = δρσ + ωρ

σ

Lorentz Lie Algebra

Ji =1

2�ijkσjk Ki = σ0i

In terms of the σ matrices the original generators Ji & Ki

are given by

while infinitesimal Lorentz transformation can be written as

Here ωμν are six independent parameters that characterize the Lorentz transformation

Lorentz Lie Algebra

Realize that the commutation relations of the Lorentz Lie algebra look quite similar to those of L(SU(2)). To make this analogy even more explicit, we introduce

J±i =

1

2(Ji ± iKi)

These new generators satisfy

[J±i , J±

j ] = i�ijkJ±k [J±

i , J∓j ] = 0

which is precisely the form of two copies (a direct sum) of two SU(2) Lie algebras

Lorentz Lie Algebra

The latter property implies that irreducible representations of the Lorentz group can be labelled by a pair (j+, j-) of two spins & that the dimension of this representations is simply (2 j+ + 1) (2 j- + 1)

Field theories in Minkowski space usually require Lorentz invariance, which explains the fundamental importance of the Lorentz group & its Lie algebra. Since it is related to the space-time symmetries it is often referred to as a external symmetry

Lorentz Lie Algebra44 CHAPTER 4. CLASSICAL FIELD THEORY

(j+, j!) dimension name symbol(0, 0) 1 scalar !(1/2, 0) 2 left-handed Weyl spinor "L

(0, 1/2) 2 right-handedWeyl spinor "R

(1/2, 0)! (0, 1/2) 4 Dirac spinor #(1/2, 1/2) 4 vector Aµ

Table 4.2: Low-dimensional representations of the Lorentz group.

where Ti are the generators (4.16) of the rotation group. Given the embedding (4.20) of the rotation group intothe Lorentz group the appearance of the Ti should not come as a surprise. It is straightforward to work out thecommutation relations

[Ji, Jj ] = i$ijkJk , [Ki,Kj ] = "i$ijkJk , [Ji,Kj] = i$ijkKk . (4.22)

The above matrices can also be written in a four-dimensional covariant form by introducing six 4#4matrices %µ! ,labelled by two anti-symmetric four-indices and defined by

(%µ!)"# = i(&"µ&!# " &µ#&

"!) . (4.23)

By explicit computation one finds that Ji = 12$ijk%jk andKi = %0i. Introducing six independent parameters $µ! ,

labelled by an anti-symmetric pair of indices, a Lorentz transformation close to the identity can be written as

!"# $ '"# " i

2$µ!(%µ!)

"# = '"# + $"#; . (4.24)

The commutation relations (4.22) for the Lorentz group are very close to the ones for SU(2) in Eq. (4.13). Thisanalogy can be made even more explicit by introducing a new basis of generators

J±i =

1

2(Ji ± iKi) . (4.25)

In terms of these generators, the algebra (4.22) takes the form

[J±i , J±

j ] = i$ijkJ±k , [J+

i , J!j ] = 0 , (4.26)

that is, precisely the form of two copies (a direct sum) of two SU(2) Lie-algebras. Irreducible representations of theLorentz group can therefore be labelled by a pair (j+, j!) of two spins and the dimension of these representationsis (2j++1)(2j!+1). A list of a few low-dimensional Lorentz-group representations is provided in Table 4.2. Fieldtheories in Minkowski space usually require Lorentz invariance and, hence, the Lorentz group is of fundamentalimportance for such theories. Since it is related to the symmetries of space-time it is often also referred as externalsymmetry of the theory. The classification of Lorentz group representations in Table 4.2 provides us with objectswhich transform in a definite way under Lorentz transformations and, hence, are the main building blocks ofsuch field theories. In these lectures, we will not consider spinors in any more detail but focus on scalar fields!, transforming as singlets, ! % ! under the Lorentz group, and vector fields Aµ, transforming as vectors,Aµ % !µ

!A! .

4.2 General classical field theory4.2.1 Lagrangians and Hamiltonians in classical field theoryIn this subsection, we develop the general Lagrangian and Hamiltonian formalism for classical field theories.This formalism is in many ways analogous to the Lagrangian and Hamiltonian formulation of classical mechan-ics. In classical mechanics the main objects are the generalised coordinates qi = qi(t) which depend on timeonly. Here, we will instead be dealing with fields, that is functions of four-dimensional coordinates x = (xµ)on Minkowski space. Lorentz indices µ, (, · · · = 0, 1, 2, 3 are lowered and raised with the Minkowski metric(&µ!) = diag(1,"1,"1,"1) and its inverse &µ! . For now we will work with a generic set of fields !a = !a(x)before discussing scalar and vector fields in more detailed in subsequent sections. Recall that the Lagrangian in

In what follows we will not deal with the spinor fields χL,R & ψ hat describe fermions (quarks, leptons, ...). Our focus will be on scalar ϕ & vector Aμ fields, which have very simple transformation properties under the Lorentz group:

φ → φ Aµ → ΛµνAν

group theory & lie algebra primer - university of oxford€¦ · theory (e.g. maxwell’s...

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