Symmetries, Groups, Algebras,
Connections, and Gauge Fields.
1 Symmetries and Groups
Many laws of Nature are invariant under certain transformations, like translations,
reflexions, rotations, time reversal, charge conjugation, Lorentz transformations,
space-time diffeomorphisms, . . . In classical mechanics the wellknown
Noether theorems link symmetries of space and time to conservation
laws, for example homogeneity and isotropy are related to momentum and
angular momentum conservation.
Obviously, the composition of two symmetry transformations must result
in a symmetry transformation again, moreover, it turns out that the axioms
of group theory provide a natural framework for symmetry transformations.
Group axioms: The combination (G, ◦) of a set G and a binary operation
◦, usually called multiplication, defined on G is called a group, if
1. g ◦ h ∈ G ∀ g, h ∈ G (closure),
2. (g ◦ h) ◦ f = g ◦ (h ◦ f) (associativity),
3. there exists a unit element e so that g ◦ e = e ◦ g = g ∀ g ∈ G,
4. ∀ g ∈ G there exists an inverse element g−1
, so that g◦g−1
= g−1
◦g = e.
If, in addition to 1. - 4., a commutative law g ◦h = h◦g ∀g, h ∈ G holds, the
group is called commutative or Abelian. In the following we shall mostly use
the shorthand notation G for (G, ◦), with the operation ◦ implicitly included.
Example: The group O(3) of orthogonal transformations in 3-dimensional
Euclidean space R3
, subgroup of the Galilei and the Lorentz group. The
group elements act as matrices M on the elements x = (x, y, z) of R3
, leaving
the inner product - and, in consequence, the norm - invariant, i. e. |Mx| = |x|.
In index notation this reads
Mi
k xk
Mi
l xl
= δkl xk
xl
, (1)
from which we may conclude
Mi
k Mi
l = δkl or MT
M = 1, (2)
1
where MT
is the transposed matrix and 1 is the unit matrix. In other words,
the transposed of an orthogonal matrix is its inverse. There is an immediate
consequence of this property for the determinant
detM = detMT
= detM−1
=
1
detM
, (3)
which means that detM = ±1. The two values ±1 of the determinant in O(3)
have an important geometric meaning: matrices, whose determinant is equal
to +1 leave the orientation of an arbitrary basis in R3
unchanged, whereas
matrices with determinant -1 invert the orientation. The former ones, often
called “even” transformations, form the subgroup of rotations, denoted by
SO(3), the group of special orthogonal transformations. A subgroup is a subset
of a group, which satisfies the group axioms. The “odd” transformations
with determinant -1 do not form a subgroup, because they do not contain the
unit element. They can be decomposed into a rotation and a space inversion,
the latter one represented by the matrix P = diag(−1, −1, −1).
The rotation group SO(3) Because of the symmetry of the indices k and
l, (2) is a set of 6 equations for the 9 components of a 3 × 3 matrix, so
that there remain 3 independent components of a rotation matrix. Therefore
SO(3) is a three-parameter group. A convenient choice of the parameters
are the so-called Euler angles, 0 ≤ α, γ < 2π, 0 ≤ β < π with the following
meaning: An arbitrary rotation matrix R ∈ SO(3) can be written as the
composition of three subsequent rotations by an angle α around the axis e3,
followed by a rotation by an angle β around the rotated axis k = R(α)e1 and
finally by an angle γ around the new axis e3,
R(α, β, γ) = Re3 (γ) · Re1 (β) · Re3 (α). (4)
More explicitly,
R =



cos γ − sin γ 0
sin γ cos γ 0
0 0 1






1 0 0
0 cos β − sin β
0 sin β cos β






cos α − sin α 0
sin α cos α 0
0 0 1


 .
(5)
Group manifolds Every group element can be represented as a point (α, β, γ)
in the parameter space. On the other hand, every R ∈ SO(3) is equivalent to
a single rotation around a certain axis e. With this interpretation in mind,
we can visualize the group as a solid sphere: e is characterized by the angles ϑ
and ϕ in spherical coordinates, and the rotation angle 0 ≤ χ ≤ π determines
2
the radial coordinate, so we have a sphere of radius π. As rotations around
an axis e by an angle π are equivalent to rotations by π around the antiparallel
axis −e, the antipode points of the sphere must be identified. This
is the compact group manifold of SO(3). Manifolds are typical for groups,
which depend in a continuous way on their parameters, so that the group
multiplication is a continuous operation (differentiable). Groups of this kind
were studied first 1893 by Sophus Lie.
Exercises:
1. Verify the group axioms for orthogonal matrices!
2. Consider two matrices A and B, describing a rotation by an angle α
around the x-axis and by an angle β around the y-axis. Calculate AB
and BA and their respective rotation axes!
Subgroups, Cosets As mentioned above, a subgroup K ⊂ G is a subset of
G that satisfies the group axioms. The left coset of the element g ∈ G with
respect to the subgroup K is the set
gK := {gk : g ∈ G, k ∈ K ⊂ G}, (6)
where k runs over all elements of K. Analogously the right cosets are defined
as
Kg := {kg : g ∈ G, k ∈ K ⊂ G}. (7)
From a left (right) coset of an element g an arbitrary element gk (kg) can be
chosen as a representative. The cosets of a group are disjoint, i. e. gK∩hK =
∅, when g and h are representatives of different cosets, and the union of all
cosets is the group G, g gK = G.
If for a subgroup N ⊂ G the left and the right cosets are identical,
gN = Ng ∀ g ∈ G, the subgroup is called normal subgroup or invariant
subgroup. In this case a multiplication of cosets can be introduced. Let
νi = giN and νj = gjN be two cosets. In the product of two representatives,
gin1 gjn2, with n1, n2 ∈ N, n1gj can be written as gjn3, with n3 being
another element of N, leading to
νi νj = giN gjN = gigj NN = gigj N = νij
in shorthand notation, where νij is the coset of gigj. It is easy to see that the
other group axioms are satisfied with e = N and ν−1
i = g−1
i N. The group of
cosets of a group G with respect to a normal subgroup N is called the factor
3
group or quotient group G/N. The centre of a group, the set of all elements
that commute with everything,
Z = {z ∈ G : gz = zg ∀ g ∈ G}, (8)
is the maximal Abelian normal subgroup.
Example: O(3) and SO(3) are non-abelian groups, SO(3) ⊂ O(3). Convenient
representatives of the cosets are the unity matrix 1 and the parity
transformation P = −1. Both of them commute with all rotation matrices
R ∈ SO(3), so the latter one is a normal subgroup and the factor group
O(3)/SO(3) = {1, P} is isomorphic to Z2 = {1, −1}.
2 Lie Groups
Lie groups, which O(3) and SO(3) are representative examples of, are described
by
1. (G, ◦) is a group.
2. G is an analytic manifold of dimension dG = n, n is the number of
parameters, there are local coordinates xi
, g = g(x1
, x2
, . . . , xn
), with
g depending analytically on them.
3. The mapping (g(xi
), g(yi
)) → g(xi
) g−1
(yi
) is analytical, i. e. together
with g(xi
) also g−1
(xi
) is analytical, and the multiplication is analytical.
(3.) means that when a group element f(p1, p2, . . . , pn) is composed with
g(q1, q2, . . . , qn) to give an element f ◦ g(r1, r2, . . . , rn), then the parameters
ri are analytic functions of pi and qi, and the parameters of the inverse
element f−1
are analytic functions of pi.
Usually the parametrization is chosen in such a way that the unit element
lies in the coordinate origin, e = g(0, 0, . . . , 0). This is the case for SO(3)
and the Euler angles, where R(0, 0, 0) = 1 and further R(α, β, γ)−1
= R(π −
γ, β, π − α) (modulo 2π).
Another example: The Lorentz group as a matrix group is the set of
4 × 4 matrices which are orthogonal in the sense of the Minkowski metric
ηik = ±diag(−1, 1, 1, 1), leaving inner products of the form xi
yk
ηik invariant.
This is equivalent with the relations
Li
m Lk
n ηik = ηmn, or LT
ηL = η (9)
4
for Lorentz matrices L. (9) is a set of 10 equations for the total of 16 matrix
elements, so there are 6 independent parameters, namely a velocity vector v
and 3 rotation angles. A matrix satisfying (9) can be written in the form
L =
γ aT
b M
,
where b is a column vector, aT
a row vector, and M is a 3 × 3 matrix. From
the invariance property of L and its inverse follows
γ2
− a2
= γ2
− b2
= 1, Ma = γb, bT
M = γaT
,
MT
M = aaT
+ 1, MMT
= bbT
+ 1.
Pure rotations form the subgroup SO(3), pure velocity transformations do
not. The matrix form of the latter ones can be derived from the well-known
form of a Lorentz transformation with velocity v in a certain coordinate
direction, say x:
Li
k =





γ γv 0 0
γv γ 0 0
0 0 1 0
0 0 0 1





(10)
with the usual γ = (1−v2
)−1/2
. Here time transforms as t → γt+γvx, in the
case of a transformation by an arbitrary velocity v it is almost obvious that
this transformation generalises to t → γt+γvx, this determines the first row
and the first column of the general Lorentz matrix. For the generalization of
the spatial submatrix we observe that in (10) the coordinate in the direction
of motion is multiplied by γ, whereas the coordinates orthogonal to it are
unchanged. In the general case we decompose any coordinate vector x into
a component along the velocity vector and an orthogonal component and
multiply the former one by γ,
x → γ
xv
|v|2
v + x −
xv
|v|2
v + γvt.
In consequence, the matrix of an arbitrary velocity transformation has the
form
Li
k =
γ γvT
γv 1 + γ2
1+γ
vvT .
Addition of velocities: Consider a particle moving at a speed ¯w in an inertial
system ¯I. What is its speed in an inertial system I, moving at a speed v
5
w. r. to ¯I? The coordinates x and t in I are related to ¯x and ¯t in ¯I in the
following way
x = ¯x +
γ2
γ + 1
(¯xv)v + γv¯t, (11)
t = γ¯t + γ(v¯x). (12)
Inserting ¯x = ¯w¯t, we obtain for u = x/t =: ¯w ◦ v
u =
¯w
γ
+ γ
γ+1
(v¯w)v + v
1 + v¯w
.
This is the general form of the addition theorem of 3-velocities, the generalisation
of the well-known formula u = v+w
1+vw
.
Now we can analyse the composition of two velocity transformations:
The product of two matrices Lv and L¯w is not of the symmetric form of
a pure velocity transformation, so it obviously contains some rotation. To
calculate the latter one, we carry out a transformation by −u = −¯w ◦ v on
the combined Lorentz transformation and find a rotation matrix:
R(α) = L¯w Lv L−u,
according to a rotation angle α
cos α =
(1 + γu + γv + γ¯w)2
(1 + γu)(1 + γv)(1 + γ¯w)
− 1 =
γv + γ¯w
1 + γvγ¯w
.
This is the Thomas rotation, arising when two non-parallel pure velocity
transformations are composed.
Exercise: Decompose the product LwLv of two velocity transformations with
w = (w, 0, 0) and v = (0, v, 0) into a product LRLu of a rotation and a pure
velocity transformation. Calculate u and the rotation angle.
Components of the Lorentz group manifold: The transformations with
determinant equal to +1 and L0
0 > +1 form the subgroup L↑
+, the group
of proper, orthochronous Lorentz transformations, without space or time
reflexion. The cosets L↓
+ = TL↑
+ (det = −1, L0
0 < −1, L↑
− = PL↑
+ (det = −1,
L0
0 > 1) and L↓
− = PTL↑
+ (det = +1, L0
0 < −1), constructed by application
of space and time reflexion P and T, are all disjoint, the Lorentz group
manifold has four non-connected components. Furthermore, the absolute
value of the velocity being strictly less than one (i. e. |vi| < 1, i = 1, 2, 3),
the Lorentz group is an example of a non-compact Lie group.
6
3 Representations
Under a representation of a group we understand its action on a linear space.
The examples considered so far are matrix representations of SO(3) in 3dimensional
euclidian space and of the Lorentz group SO(3,1) in Minkowski
space. But, beside on vectors, Lorentz transformations can be applied to
other objects, like tensors and spinors. To make notations more precise,
consider the elements of a group G as points on the group manifold, with a
“multiplication rule” g1 ◦ g2 = g3. A (linear) representation is an association
of a transformation L(g) on a certain space, the representation space, to each
g ∈ G, which reproduces the abstract calculation rules of the group.
L(g1 ◦ g2) = L(g1) L(g2), L(g−1
) = L(g)−1
, L(e) = id.
Examples: (1) Contragredient representation: Consider a matrix L acting
on the components of a vector with respect to a basis {ei} in the form
¯vi
= Li
k vk
.
When ¯vk
are to be the components of the original vector in a different basis
{¯ei} (passive transformation), we may write
v = vi
ei = ¯vi
¯ei = Li
k vk
¯ei
and express the “old” basis in terms of the “new” one,
ek = Li
k ¯ei.
Assuming a basis transformation by a matrix ˜L,
¯ei = ˜L k
i ek,
we find
Li
k
˜L j
i = δj
k,
in matrix notation
˜L = L−1 T
.
This is called the contragredient representation to the representation by the
matrices L.
Under a change of bases, {ei} → {¯ei = Si
k
ek} the transformation matrix
L goes over into ˜L = S L S−1
. Matrix representations related in this form
are called equivalent.
7
(2) Tensor representations, Kronecker product. Take two representations
g → Tg and g → Tg in two vector spaces V and V . For the Kronecker
product acting on V ⊗V we assume that tensor components Tik
transform
like products of vector components v i
v k
, where v ∈ V and v ∈ V ,
v
¯ı
v
¯α
= Tg
¯ı
k
Tg
¯α
β
v
k
v
β
= (Tg ⊗ Tg )¯ı¯α
kβ
v
k
v
β
.
(3) Direct sum arrange the components v i
and v α
to column vectors which
transform according to
v ¯ı
v ¯α =
Tg
¯ı
k
0
0 Tg
¯α
β
vk
vβ .
The block matrix is written as Tg ⊕ Tg .
3.1 Reducible and irreducible representations
In example (3) the representation acts in both vector spaces V and V
independently, each one is invariant under the action of the transformations.
A representation is called reducible, if there are invariant subspaces of the
representation space, otherwise it is called irreducible. Note that not every
reducible representation splits into a direct sum of representations, when it
does, it is called fully reducible and its matrix has block diagonal form. The
matrix of a not fully reducible representation has the typical form
Tg Ag
0 Tg
.
Obviously the vectors (v , 0) are transformed among themselves, i.e. V is an
invariant subspace, but not so V .
A common task is the reduction of the Kronecker product of two representations,
the result is the Clebsch-Gordan series.
Example: Transformation of the electromagnetic field tensor. We may arrange
the components as a 6-vector (E, B). Under spatial rotations E and
B transform separately, SO(3) is represented as a direct sum. Boosts, on the
other hand, do not leave the spaces spanned by (E, 0) and (0, B) invariant.
3.2 Schur’s lemma
In many cases, particularly in higher dimensions, an index-free notation is
more convenient. In this notation a representation g → Tg of a group G =
8
{g, . . .} in a vector space V over the complex numbers, is a linear, nonsingular
mapping of V onto itself, characterized by
Tg(αv + βw) = αTgv + βTgw, v, w ∈ V, α, β ∈ C,
with the representation property
Tg1 Tg2 = Tg1g2 , Te = idV.
Reducibility means the existence of a nontrivial linear subspace V1 ⊂ V, so
that TgV1 ⊂ V1. Two representations g → Tg and g → Tg in V and V are
equivalent, Tg Tg, if there exists a one-to-one linear mapping S : V → V ,
such that
Tg = S−1
TgS ∀g ∈ G.
The relation STg = TgS can be visualized in a commutative diagram.
If S is a not necessarily one-to-one mapping V → V with TgS = STg,
called an intertwiner, the image SV = V1 ⊆ V is a linear subspace of V ,
which is invariant under Tg;
TgV1 = TgSV = STgV = SV = V1.
Also the set V0 ⊂ V of vectors, which are mapped to the zero vector
by S is an invariant subspace of V, because from {0 } = TgSV0 = STgV0
follows TgV0 ⊂ V0.
From this we deduce a theorem, the first part of Schur’s lemma.
For two irreducible representations g → Tg and g → Tg in the spaces V and
V and S being a linear mapping V → V with STg = TgS, either S vanishes
identically or it is one-to-one and the representations are equivalent.
When Tg and Tg are supposed to be irreducible, V1 or V0 must coincide
with {0 } or V , or with {0} or V, respectively. V0 = V or V = {0 } means
S ≡ 0, V0 = {0} and V1 = V implies invertibility of S and equivalence of
g → Tg and g → Tg.
The second part of Schur’s lemma is
If g → Tg is a representation in V and S : V → V is a linear mapping that
commutes with all Tg, then S is either a multiple of the identity mapping or
Tg is reducible.
Proof: Consider the linear subspace Vs ⊂ V of eigenvectors v of S with
eigenvalue s, (Sv = sv). Due to STg = TgS we have STgv = TgSv = sTgv ∈
Vs, and Vs is invariant under all Tg. As an invariant subspace Vs must
coincide with V if Tg is irreducible; Sv = sv ∀v ∈ V means S = s·idV. Note
that here we need a vector space over the complex numbers, in the reals the
eigenvalue s need not exist!
9
4 Lie algebras
4.1 Infinitesimal rotations
Consider group elements close to the unit element. Finite elements can be
constructed by composition of infinitesimal ones. To do this, we make use
of the fact that we can compose group elements and that we can do calculus
on a Lie group.
We write R = 1 + Ω for a small rotation, where the elements of the
matrix Ω are small of first order, i. e. higher orders are negligible. From
orthogonality R RT
= 1 we obtain Ω + ΩT
= 0, Ω is antisymmetric. Such a
matrix can be written in the form Ω = αΛ with the vector α = (α1, α2, α3)
pointing into the direction of the rotation axis and a length equal to the
rotation angle, and Λ = (Λ1, Λ2, Λ3) being a formal 3-vector of the matrices
Λ1 =



0 0 0
0 0 −1
0 1 0


 , Λ2 =



0 0 1
0 0 0
−1 0 0


 , Λ3 =



0 −1 0
1 0 0
0 0 0


 .
In this basis of the linear space of antisymmetric 3 × 3 matrices we have for
the µν element of the matrix Λλ
Λλµν = (Λλ)µν = −ελµν.
In first order the transformation equation x = Rx goes over to x µ
=
xµ
− ελµν
αλ
xν
or x = x + α × x.
To establish the relation between infinitesimal and finite transformations
we write a finite rotation R(α) as
R(α) = R
α
N
R
α
N
= . . . = R
α
N
N
,
for N large enough we can write R(α/N) ≈ 1 + αΛ/N and in the limit
N → ∞
R(α) = e(αΛ)
= eαΛ/N
N
.
Example: Calculate the matrices eαΛ1
, eβΛ2
, and eγΛ3
.
Rotations R(τα0) = exp(τα0Λ) for fixed α0 and variable τ form a oneparameter
subgroup; for τ = 0 we get the unity, for τ = 1 the rotation
matrix exp(α0Λ). Each matrix of the form α0Λ is the generator of such a
one-parameter subgroup, with sums and real multiples of generators being
generators as well. Thus the generators form a 3-dimensional vector space
10
with Λ1, Λ2, Λ3 providing a basis. (They are the generators of the rotations
around the coordinate axes.)
Matrix multiplication leads out of this vector space, because the product
of two antisymmetric matrices is not antisymmetric in general. The commutator,
however,
[A, B] = AB − BA = −[B, A]
is antisymmetric, because
[A, B]T
= [BT
, AT
] = −[AT
, BT
] = −[A, B].
The commutator of two generators is a generator again and may be written
in the form αΛ. For A = mΛ and B = nΛ we get
[mΛ, nΛ] = (m × n)Λ,
or, for the above basis,
[Λµ, Λν] = εµνλΛλ.
With A◦B = [A, B] we have a product on the vector space of generators.
Due to the antisymmetry and the Jacobi identities for commutators,
[[A, B], C] + [[C, A], B] + [[B, C], A] = 0,
we have the rules
A ◦ B = −B ◦ A, (A ◦ B) ◦ C + (C ◦ A) ◦ B + (B ◦ C) ◦ A = 0.
A vector space with a product satisfying these rules is called a Lie algebra.
In general, for a basis {XA} the relation
XA ◦ XB = CD
AB XD
defines the algebra. The structure constants CD
AB (structure tensor) must
satisfy the rules
CD
AB = −CD
BA, CD
AB CE
CD + CD
CA CE
BD + CD
BC CE
AD = 0.
The generators of the defining representation of SO(3) form a threedimensional
Lie algebra over the reals with structure tensor ελµν.
11
4.2 Lie algebra and representations of SO(3)
Consider the orthogonal transformations of R3
as special representation of the
Lie group SO(3). It is obviously irreducible, because there are no invariant
subspaces.
Assume a representation g → Tg of SO(3) in a vector space V. To a onedimensional
subgroup g(τ) with g(0) = e a matrix group Tg(τ) is associated.
For small τ we have
Tg(τ) ≈ idV + τt,
where
t =
∂
∂τ
Tg(τ)
τ=0
is the generator of the considered subgroup. We will show that the generators
of such one-dimensional subgroups form a vector space and find a basis of
three generators tµ, generating the rotations around the coordinate axes and
satisfying the commutation relations
[tµ, tν] = εµνλ tλ.
The generators form also a Lie algebra, the structure of which is isomorphic
to the already known one (with the exception of the trivial representation,
where tµ = 0).
Thus the problem, how to find all irreducible representations of SO(3), is
reduced to the determination of the representations of the three generators
of the Lie algebra.
To prove this claim we consider a rotation, given by the matrix R(α)
in the basis eµ. In relation to the basis ¯eµ, which originates from eµ by
the rotation S, ¯eµ = Sµν eν, the considered rotation is given by the matrix
S R(α) S−1
. As α becomes α = Sα in the new basis,
S R(α) S−1
= R(Sα)
must hold. With R(α) =: Sg(α) and S =: Sh considered as representation
matrices of the abstract group elements g(α) and h in the defining representation,
we have
h g(α) h−1
= g(Shα).
For an arbitrary representation in a vector space V this means
Th Tg(α) Th−1 = Tg(Shα). (13)
Replacing α by τα and considering small τ, we obtain
Tg(τα) ≈ idV + τt, t :=
∂
∂τ
Tg(τα)
τ=0
.
12
With
tµ :=
∂
∂αµ
Tg(α)
α=0
and t := (t1, t2, t3)
we have
t = αµtµ = α t.
The generators tµ of rotations around the axes form indeed a vector space
spanned by t1, t2, and t3.
Inserting the expansion of Tg(τα) into (??) and replacing α by τα, τ 1
yields
Th αt T−1
h = (Shα)t.
Now we assume also h close to the unit element, i. e. in the form h(τβ) with
τ 1, so that
Th ≈ idV + τβt, T−1
h ≈ idV − τβt
and, according to the behaviour of a vector under infinitesimal rotations,
Sh α = α + τβ × α.
Inserting this we get
[βt, αt] = (β × α)t
or
[tµ, tν] = εµνλtλ.
These commutation relations hold for any representation, this proves the
above claim.
The key relation can be written in a different form. α being arbitrary, we
can specialize to
Th tµ T−1
h = (Sh)νµ tν,
or, with h → h−1
and S−1
h νµ
= (Sh)µν, to
T−1
h t Th = Sh t.
Generally a triple v of operators on V satisfying the relation
T−1
h v Th = Sh v
is called a vector operator. Another relation is
[v, βt] = v × β.
13
The square v2
= vµvµ is invariant under the representation, as
v2
= (Shv)2
= T−1
h v Th T−1
h v Th = T−1
h v2
Th,
and from this follows
Th v2
= v2
Th.
v2
commutes with all operators Th of the representation. If the representation
is irreducible in V , then v2
is a multiple of the unit idV according to Schur’s
lemma.
For v = t in particular, we get the Casimir operator
C := t2
,
which commutes with all representation operators. For SO(3) we have
Λ2
1 + Λ2
2 + Λ2
3 = −2 · 1.
4.3 Lie algebras of Lie groups
Consider an arbitrary n-dimensional Lie group G with n parameters βA and
with the elements g(βA) = g(β1, . . . , βn) and g(0) = e. g(τ) := g(β(τ))
defines a curve in the group manifold. For curves through the unit element,
g(βA(0)) = e, in the vicinity of e the representation g → Tg is approximated
by
Tg(τ) ≈ idV + τt,
where
t :=
∂
∂τ
Tg(τ)
τ=0
=
∂βA
∂τ τ=0
∂
∂βA
Tg(β)
β=0
.
t is the generator of a one-parameter subgroup in the representation. The
finite transformations of this subgroup are given by exp(τt), with the multiplication
in the subgroup defined by exp(τ1t) exp(τ2t) = exp[(τ1 + τ2)t]. An
arbitrary curve g(τ) with g(0) = e is not a subgroup in general.
If g(τ), g1(τ) are two curves through e, then also the products g(τ)g1(τ)
form a curve through e. In the representation Tg we have for infinitesimal τ
g(cτ)g1(τ) → Tg(cτ) Tg1(τ) ≈ idV + τ(ct + t1),
t1 is a generator like t. The generators thus form a vector space LV, spanned
by
tA :=
∂
∂βA
Tg(β)
β=0
, A = 1, . . . , n.
14
The products g(τ)g1(τ) constructed from two one-parameter subgroups
do not form a subgroup in general, rather only a curve through e. This is
reflected by the representation of the two subgroups, generally
exp(τt) exp(τt1) = exp(τt + τt1).
To show that the generators build up a Lie algebra with the commutator,
we consider, in addition to a curve g(τ) through e the elements h g(τ) h−1
with an arbitrary h ∈ G, forming a curve through e, too. For small τ we have
Th Tg(τ) T−1
h ≈ idV + τThtT−1
h ,
so, together with t ∈ LV also ThtT−1
h ∈ LV is a generator. If we write
h = g1(τ) and assume a small τ, so that
Tg1(τ) ≈ idV + τt1, T−1
g1(τ) ≈ idV − τt1,
then the ensuing relation
LV (Th t T−1
h − t)/τ ≈ [t1, t]
shows that LV is indeed a Lie algebra.
The association t → Th t T−1
h is a one-to-one mapping of LV onto itself,
the adjoint action of h on LV . If the representation is faithful (The relation
g ◦ h ↔ Tg Th is an isomorphism), this operator is denoted by Adh and
h → Adh is a representation of G in the n-dimensional space LV, the adjoint
representation. For SO(3) it is accidentally equal to the defining one.
With h = g1(τ), τ 1 we can consider the derivative of the adjoint map,
Th t Th
−1
≈ t + τ[t1, t].
The derivative
lim
τ→0
Th t Th
−1
− t
τ
= [t1, t]
is for each t1 ∈ LV a mapping LV → LV, t → [t1, t], called the adjoint action
of t1 and denoted by
adt1 t = [t1, t].
It satisfies the Leibniz rule as a consequence of the following commutator
relation,
adt[t1, t2] = [adtt1, t2] + [t1, adtt2].
With the aid of the adjoint representation of a Lie algebra we can construct
the symmetric, second-order Killing-Cartan tensor,
gAB := Tr(adXA
adXB
) = CC
DA CD
CB,
15
which may be expressed in terms of the structure constants. Like the latter
ones, it is invariant under the adjoint representation. For semisimple groups
(each Abelian normal subgroup is discrete - a simple group ia a group, whose
only normal subgroups are the trivial group and the group itself) the matrix
gAB is invertible and in each representation the Casimir operator
C := gAB
tAtB
is invariant, i. e. T−1
g CTg = C. With tA
:= gAB
tB the further operators
Tr(adXA
adXB
. . . adXC
)tA
tB
. . . tC
= CE
DA CF
EB . . . CD
GC tA
tB
. . . tC
are invariant. In irreducible representations they are all multiples of the unit
operator.
4.4 Unitary irreducible representations of SO(3)
are formulated in Hilbert spaces, which play a fundamental role in quantum
mechanics and quantum field theory. (Conservation of scalar products,
norms, probabilities,. . . )
A Hilbert space H is a (finite or infinite dimensional) vector space over
the complex numbers with a scalar product
x, αy1 + βy2 = α x, y1 + β x, y2 , α, β ∈ C
x, y = y, x ∗
||x||2
:= x, x > 0 ∀x = 0.
A representation g → Tg of a Lie group in a Hilbert space is called unitary,
if ∀g ∈ G, ∀x, y ∈ H
Tgx, Tgy = x, y ,
i. e. when the operators Tg leave the scalar product invariant. Infinitesimal
unitary operators are given by
Tg(τ) ≈ idH + τt,
where from unitarity follows the relation
tx, y + x, ty = 0.
This means the generators are antihermitian operators, whereas ±it are her-
mitian,
±itx, y = x, ±ity .
16
They are called the hermitian generators of the associated one-parameter
unitary subgroup of the representation.
The adjoint (hermitian conjugate) operator A†
to an operator A is given
by
A†
x, y = x, Ay .
Hermitian operators satisfy A†
= A, antihermitian ones A†
= −A, and
unitary ones A†
= A−1
. The eigenvalues of hermitian, antihermitian, and
unitary operators are real, imaginary, or lie on the unit circle, respectively.
Assume a finite group and a scalar product , 0 in the representation
space. Then
x, y :=
g∈G
Tgx, Tgy 0
is an invariant scalar product:
Tg x, Tg y =
g
TgTg x, TgTg y 0 =
g
Tgg x, Tgg y 0 =
g
Tg x, Tg y 0 = x, y .
For compact Lie groups the sum can be replaced by an integral. From this
follows the (specialization of the) theorem (to finite groups) that
Each irreducible representation of a compact Lie group is equivalent to a
unitary representation.
According to a further theorem all the irreducible representations of a
compact Lie group in a Hilbert space are finite dimensional, so we can restrict
the search for all representations of SO(3) to finite dimensional ones.
The generators tµ are antihermitian operators in a finite dimensional
Hilbert space, the hermitian generators are
Jµ := itµ = J†
µ
with the commutation relations
[Jµ, Jν] = iεµνλJλ
and ∀x ∈ H and µ = 1, 2, 3
x, J2
µx = Jµx, Jµx ≥ 0.
According to Schur’s lemma in irreducible representations (H is a complex
vector space) the Casimir J
2
is a multiple of the unit operator,
J
2
= λ idH, λ ≥ 0.
17
The irreducible representations can be found with the aid of the eigenvalue
spectrum of one of the hermitian generators, say J3. It is convenient to pass
over to the combinations
J± := J1 ± iJ2 and J3
with
[J+, J−] = 2J3, [J3, J±] = ±J±, J
2
= J±J J3 + J2
3 .
Now consider the eigenvectors of J3, forming a complete orthogonal system
in H. Assume xm to be a normalized eigenvector to the eigenvalue m,
J3 xm = m xm, ||xm|| = 1.
Then
J3 J± xm = (m ± 1) J± xm,
J± xm, J± xm = xm, J J± xm = λ m − m2
.
From this follows that either J± xm is the zero vector or m±1 is an eigenvalue
of J3, too. The representation being finite dimensional, there are only finitely
many eigenvalues, the largest of which may be denoted by j. For a normalized
eigenvector xj
J+xj = 0 and λ = j + j2
must hold. In the series of eigenvectors J−xj, J2
−xj, . . . to the eigenvalues j,
j −1, . . . after N −1 applications of J− a least eigenvalue j must be reached,
J3(J−)N−1
xj = j (J−)N−1
xj,
so that
(J−)N
xj = 0.
From the above relations follows
λ = j2
+ j = j
2
− j , j − j + 1 = N
or (j + j )(j − j + 1) = 0, i. e. j = −j and thus 2j + 1 = N, a positive
integer. For j and λ the possible values
j = 0, 1
2
, 1, 3
2
, 2, . . . λ = j(j + 1) = 0, 3
4
, 2, 15
4
, 6, . . .
arise. The eigenvalues and eigenvectors of J3 are
j, j − 1, . . . , −j + 1, −j, xj, J−xj, . . . , (J−)2j
xj.
18
These vectors are orthogonal, i. e. linearly independent, so they span a 2j +1
dimensional subspace of H, which is invariant under J±, J3, J
2
. In irreducible
representations this subspace must be equal to H.
Up to equivalence, an irreducible representation of SO(3) is determined
uniquely by the maximal eigenvalue j of the operator J3 or the eigenvalue
j(j + 1) of the Casimir operator J
2
, j can assume only the values 0, 1/2, 1,
. . .
Normalized eigenvectors xm with ||xm|| = 1 are determined up to a phase
factor. For them
J3xm = mxm, J±xm = ρ±(m)xm±1
with
|ρ±(m)|2
= j(j + 1) m − m2
.
From J†
± = J and the orthogonality of xm follows
ρ±(m) = xm±1, J±xm = J xm±1, xm = xm, J xm±1
∗
= ρ∗
(m ± 1).
This is compatible with the above relation, so we can choose the phase
ρ±(m) = + j(j + 1) m − m2,
the associated basis {xm} is called a canonical basis. In this basis the operators
are represented by the following matrices
J3 = diag(j, j − 1, . . . , −j + 1, −j), J
2
= j(j + 1) 1,
J+ =











0 ρ+(j − 1) 0
0 ρ+(j − 2)
0
...
ρ+(−j)
0
...
0











J− =









0
ρ−(j) 0 0
ρ−(j − 1) 0
...
...
0 ρ−(−j + 1) 0









19
The simplest cases:
j = 0: trivial representation
j = 1: 3-dimensional representation,
J3 =



1 0 0
0 0 0
0 0 −1


 , J1 =
1
√
2



0 1 0
1 0 1
0 1 0


 , J2 =
1
i
√
2



0 1 0
−1 0 1
0 −1 0


 ,
J
2
= 2 · 1.
This representation is equivalent to the defining representation, J3 is the
diagonalized form of iΛ3.
j = 2: 5-dimensional representation, equivalent to the representation in the
space of traceless symmetric tensors Tµν
,
j = 1/2: 2-dimensional representation with the generators J = 1/2 σ (spinor
representation), where
σ1 =
0 1
1 0
, σ2 =
0 −i
i 0
, σ3 =
1 0
0 −1
are the Pauli spin matrices with the commutation relations
[σµ, σν] = 2iεµνλσλ
and the anticommutator (Clifford algebra) relation
{σµ, σν} = σµσν + σνσµ = 2δµνid.
Together this gives the product
σµσν = δµνid + iεµνλσλ
(formally the multiplication rules of Hamilton’s quaternions). This representation
is also called the fundamental one.
Knowing the representations of the Lie algebra so(3), what remains to do
is to construct the representation matrices of finite rotations. To represent a
group element g(α) we consider the one-parameter group g(τα). With t = αt
being the generator of this subgroup the desired representation is
g(α) → Tg(α) = exp(αt) = exp(−iαJ).
j = 0: trivial, j = 1: defining representation.
j = 1/2:
αJ =
1
2
ασ =
1
2
αµσµ,
20
(ασ)2
= αµσµανσν =
1
2
αµαν(σµσν + σνσµ) = α 2
· 1.
Writing α = αn with n 2
= 1 we get
(−iαJ )2
= −
α
2
2
· 1, (−iαJ )3
=
α
2
3
nσ, . . .
Therefore from the expansion
exp(−iαJ) = 1 cos
α
2
− inσ sin
α
2
=: U(α)
we find
U(α) =


cos α
2
− in3 sin α
2
−i(n1 − in2) sin α
2
−i(n1 + in2) sin α
2
cos α
2
+ in3 sin α
2

 .
This matrix is unitary with determinant equal to one
(detU = exp[Tr(−iασ/2)] = 1).
The “representation” by U(α) has the property that, in contrast to the
tensor representations (integer j), the composition of two rotations by an
angle π does not lead to unity, but to −id. So the matrices U(α) form a
group only when the underlying range of α is extended to 0 ≤ |α| ≤ 2π, thus
covering the set of rotations twice – to each rotation g(α) two matrices U(α)
and −U(α) = U(α + 2πα
α
) are associated and this is not a representation in
the strict sense.
4.5 The group SU(2)
For α varying over 0 ≤ |α| ≤ 2π the matrices U(α) form the group SU(2) of
all unitary, unimodular (det=1) matrices. Namely, for each complex 2 × 2
matrix
U =
a b
c d
unitarity demands c = −λb∗
, d = λa∗
, |a|2
+ |b|2
= 1, |λ| = 1, unimodularity
restricts to λ = 1, so that
U =
a b
−b∗
a∗ , |a|2
+ |b|2
= 1. (14)
From this follows |a| ≤ 1, so that there is exactly one α, 0 ≤ α ≤ 2π, for which
Re a = cos α
2
. A unique n arises from Im a = −n3 sin α
2
, Re b = −n2 sin α
2
,
Im b = −n1 sin α
2
.
If we interpret the real and imaginary parts of a and b as cartesian coordinates
in R4
, we see that the group manifold of SU(2) is the unit sphere S3
21
in R4
. U ∈ SU(2) and −U belonging to one and the same rotation, we can
consider SO(3) as S3
with antipode points identified. After the restriction
0 ≤ |α| ≤ π the identification can be dropped, except at the boundary sphere
|α| = π.
After the identification leading to SO(3) a curve g(τ) going from α = 0
to α = 2π becomes closed and not contractible to e, in contrast to curves
restricted to the lower half sphere. When this occurs, the manifold is called
multiply connected. Here we have two classes of curves, one being continuously
contractible to a point, the other not. SO(3) is therefore twofold
connected. The transition to SU(2) - which undoes the identification - makes
every curve continuously contractible, SU(2) is simply connected, it is called
the universal covering of SO(3).
A Lie group and its universal covering have the same Lie algebra and in
a sufficiently small neighborhood of the unit they are isomorphic, at a large
scale there is a homomorphism SU(2) → SO(3), where the discrete invariant
subgroup Z2 = {1, −1} is mapped to e ∈ SO(3), SO(3) SU(2)/Z2.
4.6 Tensor and spinor formulations
To construct a local isomorphism between SO(3) and SU(2), we associate
to each 3-vector x a traceless hermitian 2 × 2 matrix X = xσ. x being real
and σµ traceless, we have
X = X†
and tr X = 0.
Reversely each traceless hermitian matrix can be written in the form X = xσ
with real x. x can be retrieved from X by
x =
1
2
tr Xσ,
because tr σµσν = 2δµν. Further
X2
= x 2
· 1, det X = −x 2
.
Consider the adjoint action of U ∈ SU(2) on the element X of the Lie
algebra su(2) (linear combination of the generators σi),
X = UXU−1
= UXU†
.
The matrix X is hermitian and traceless again,
(X )†
= (UXU†
)†
= UX†
U†
= UXU†
= X ,
22
tr X = tr UXU−1
= tr U−1
UX = tr X = 0
and defines a linear transformation x → x = Rx, which is orthogonal due
to
(x )2
· 1 = X
2
= UXU−1
UXU−1
= UX2
U−1
= x 2
· 1
or
−(x )2
= det X = det U det X det U†
= det X = −x 2
.
The corresponding orthogonal matrix of the transformation
xµ = Rµν xν
can be found by comparison of
X = xµσµ = Rµνxνσµ
with UxνσνU†
:
Rµνσµ = UσνU†
= UσνU−1
.
Multiplication by σρ from the left and taking the trace gives explicitly
Rρν =
1
2
tr σρUσνU†
=
1
2
tr σρUσνU−1
.
Also U can be expressed explicitly in terms of R, namely, for each 2 × 2
matrix the identity
σµMσµ = 2 tr M · 1 − M
holds. By multiplication of Rµνσµ by σν we find
Rµνσµσν = UσνU†
σν = U(2 tr U†
· 1 − U†
) = (2 tr U)U − 1.
Taking the trace gives
2(tr U)2
= 2(1 + tr R),
hence
U = ±
1 + Rµνσµσν
2
√
1 + tr R
.
These formulae show the local equivalence of the adjoint representation of
SU(2) with the defining representation of SO(3).
A spinor u transforms under a rotation according to
u → u = U(α) u.
The scalar product in the sense of the unitary geometry, invariant under this
transformation, is
u, v = u∗
1v1 + u∗
2v2,
23
where u = (u1, u2) in the canonical basis. In analogy to the ε tensor there is
an ε spinor, defining an invariant bilinear form
εAB
uAvB = u1v2 − u2v1.
In contrast, the scalar product is sesquilinear.
Like tensor representations in arbitrary vector spaces, we may investigate
higher-order spinors and their transformations. To this end we consider
Kronecker products U(α) ⊗ U(α) ⊗ . . . and their reduction.
A simple example is the reduction of the representation g(α) → U(α) ⊗
U(α). With u = (u1, u2) and v = (v1, v2) the components of u ⊗ v are
(u1v1, u1v2, u2v1, u2v2).
If further u = Uu and v = Uv, then, with U given by (??).





u1v1
u1v2
u2v1
u2v2





=





a2
ab ab b2
−ab∗
|a|2
−|b|2
a∗
b
−ab∗
−|b|2
|a|2
a∗
b
b∗2
−a∗
b∗
−a∗
b∗
a∗2










u1v1
u1v2
u2v1
u2v2





. (15)
From this we may read off that for the antisymmetric part
u1v2 − u2v1 = |a|2
+ |b|2
(u1v2 − u2v1) = u1v2 − u2v1,
it transforms according to the trivial representation. In the subspace of the
symmetric second-order spinors u(AvB) we choose the basis (u1v1, (u1v2 +
u2v1)/
√
2, u2v2), then (??) becomes






(u1v2 − u2v1)/
√
2
u1v1
(u1v2 + u2v1)/
√
2
u2v2






=






1 0 0 0
0 a2
√
2 ab b2
0 −
√
2 ab∗
|a|2
− |b|2
√
2 a∗
b
0 b∗2
−
√
2 a∗
b∗
a∗2












(u1v2 − u2v1)/
√
2
u1v1
(u1v2 + u2v1)/
√
2
u2v2






.
This is already the complete reduction, for it is easy to see that for infinitesimal
rotations around the 3-axis (b = 0, a ≈ 1−iα/2) the generator J3 of the
arising 3-dimensional representation acquires the form diag(1, 0, −1), which
characterizes the irreducible representation of weight 1. Moreover, the form
of the generators J± shows that the chosen basis is a canonical one.
In the same way one can form the symmetric part u(AuB . . . uC) of a
higher-order spinor. To see that the space of symmetric spinors of a certain
order p is irreducible, we first find out its dimension by counting the independent
components of a spinor. We choose a basis in which the first p1
24
indices are equal to 1 – the remaining p2 = p − p1 are then equal to 2. As p1
can be 0, 1, . . . p, there are p + 1 independent components, the space is p + 1
dimensional. Now we investigate the eigenvalues of J3. In the space of all
spinors of order p an infinitesimal rotation around the 3-axis has the form
U(τe3) ⊗ . . . ⊗ U(τe3) ≈
1 ⊗ . . . ⊗ 1 −
iτ
2
(σ3 ⊗ 1 ⊗ . . . ⊗ 1 + . . . + 1 ⊗ . . . ⊗ 1 ⊗ σ3),
J3 =
1
2
(σ3 ⊗ . . . ⊗ 1 + . . . + 1 ⊗ . . . ⊗ σ3).
(16)
If u±
is an eigenspinor of J3 to the eigenvalue ±1/2, then u±
⊗. . .⊗u±
belongs
to the subspace of totally symmetric spinors of order p and is an eigenspinor
to the eigenvalue ±p/2. From the known spectrum of J3 follows that also the
eigenvalues p/2−1, . . . , −p/2+1 and the associated eigenspinors must occur.
(It is easy to see that u+
(Au+
B . . . u−
C) with p1 factors u+
and p2 factors u−
is an
eigenspinor to the eigenvalue (p1 − p2)/2.) It follows that the dimension of
the representation in the space of totally symmetric spinors must be at least
equal to 2(p/2) + 1 = p + 1. Indeed it is an irreducible representation with
the weight j = p/2.
Now we can construct the explicit form of the representation matrix of
finite rotations for each weight j. Symmetric spinors of order p (p = 2j)
transform like uAuB . . . uC, the independent components are
(u1)p
, (u1)p−1
u2, . . . , u1(u2)p−1
, (u2)p
. (17)
A rotation transforms them to the expressions (u1)p
, . . . , (u2)p
, where
u1 = au1 + bu2, u2 = −b∗
u1 + a∗
u2 (18)
and simple multiplication yields the elements of the representation matrix.
To obtain it in unitary form, we must introduce an invariant scalar product
and a norm and normalize the above expressions in such a way that the square
of the norm of an element of uAuB . . . uC appears as a sum of absolute squares
of the monomials in (??) (with normalizing denominators Ni). Unitarity
means conservation of the norm
(u1)p
N1
2
+
(u1)p−1
u2
N2
2
+ . . . =
(u1)p
N1
2
+
(u1)p−1
u2
N2
2
+ . . . (19)
The scalar product u, u = u∗
AuA is invariant, and so is
1
p!
u∗
Au∗
B . . . u∗
CuAuB . . . uC =
1
p!
(u∗
AuA)p
=
1
p!
u, u p
(20)
25
and gives a suitable square of a norm in the space of symmetric spinors. To
express it in the above form as sum of absolute squares of monomials we
make use of the binomial theorem (p2 = p − p1).
1
p!
(u∗
1u1 + u∗
2u2)p
=
1
p!
p
p1=0
p
p1
(u∗
1u1)p1
(u∗
2u2)p2
=
p1
(u∗
1)p1
(u∗
2)p2
√
p1!p2!
(u1)p1
(u2)p2
√
p1!p2!
. (21)
From this we can directly read off the normalization of the monomials, up
to phase factors. The matrix elements of the representations of rotations are
got by expanding
(u1)p1
(u2)p2
√
p1!p2!
=
(au1 + bu2)p1
(−b∗
u1 + a∗
u2)p2
√
p1!p2!
=
1
√
p1!p2! q
p1
q
(au1)q
(bu2)p1−q p2
(−b∗
u1) (a∗
u2)p2−
= (22)
p1!p2!
q
aq
uq
1 bp1−q
up1−q
2
q! (p1 − q)!
(−b∗
) u1 (a∗
)p2−
up2−
2
! (p2 − )!
and reading off the coefficients of (u1)q1
(u2)q2
/
√
q1!q2!.
After reordering according to the canonical basis, p1 = j +m, p2 = j −m,
m = −j, . . . , j, we have
(j + m)!(j − m)!
q,
aq
bj+m−q
(−b∗
) (a∗
)j−m−
q!(j + m − q)! !(j − m − )!
uq+
1 u2j−q−
2 . (23)
In terms of j and m, on the left-hand side of (??) we have the element
(u1)j+m
(u2)j−m
. An analogous enumeration uj+n
1 uj−n
2 on the right-hand side
is achieved by the substitution q = j + n − , n = −j, . . . , j, so that
(u1)j+m
(u2)j−m
(j + m)!(j − m)!
= (j + m)!(j − m)! × (24)
n,
(−1)
aj+n−
(a∗
)j−m−
bm−n+
(b∗
)
(j + n − )!(m − n + )! !(j − m − )!
uj+n
1 uj−n
2 .
From this comparison we can read off the matrix elements
D(j)
mn(α) = (25)
(−1)
(j + m)!(j − m)!(j + n)!(j − n)!
(j − m − )!(j + n − )!(m − n + )! !
aj+n−
(a∗
)j−m−
bm−n+
(b∗
) .
26
In the sum over the integer ∈ N, 0 ≤ ≤ j −m, all the values, which would
lead to factorials of negative numbers, have to be omitted. In the canonical
basis the transformation is
Tg|jmα =
n
D(j)
nm(g)|jnα , (26)
where α enumerates multiply occurring m’s in reducible representations. By
reduction of the Kronecker “powers” of the two dimensional spinor representation
we may get all the irreducible representations of SO(3), therefore the
denotation “fundamental representation”.
4.7 Representations of the Lorentz group
4.7.1 The adjoint action of the Lorentz group
In the framework of the Lorentz group the generators of the subgroup of
rotations appear as the three 4 by 4 matrices
Mi =
0 0
0 Λi
, i = 1, 2, 3. (27)
with Λi being the well-known generators of SO(3). In the Lie-Algebra there
are three further elements, generating boosts in the direction of the axes,
N1 =





0 1 0 0
1 0 0 0
0 0 0 0
0 0 0 0





, N2 =





0 0 1 0
0 0 0 0
1 0 0 0
0 0 0 0





, N3 =





0 0 0 1
0 0 0 0
0 0 0 0
1 0 0 0





.
(28)
For the commutators we find
[Mi, Mj] = ij
k
Mk, [Ni, Nj] = − ij
k
Mk, [Ni, Mj] = ij
k
Nk. (29)
In the six-dimensional basis {Ni, Mi} the adjoint action of the generators
(= commutators with the other generators) can be written in form of the
following 6 by 6 matrices
adMi
=
Λi 0
0 Λi
, adNi
=
0 Λi
−Λi 0
. (30)
For the Cartan-Killing metric gij = Tr(adti
adtj
) with ti ≡ Ni for i =
1, 2, 3 and ti = Mi for i = 4, 5, 6 we find
gij = 4
1 0
0 −1
, (31)
27
where 1 is the 3 by 3 unit matrix. Accordingly there is a Casimir operator
C1 = N 2
− M 2
. In the four-dimensional representation C1 is a multiple of
the unit matrix, C1 = 3·1, in the adjoint representation it is 4·1. Beside this,
there is a further matrix commuting with all the matrices in (??), namely
C2 =
0 −1
1 0
. (32)
Consider a finite element exp(χN1) of the adjoint representation of the
Lorentz group, generated by the boost generator in the x direction N1
exp(χN1) =










1 0 0 0 0 0
0 chχ 0 0 0 −shχ
0 0 chχ 0 shχ 0
0 0 0 1 0 0
0 0 shχ 0 chχ 0
0 −shχ 0 0 0 chχ










. (33)
If we write as usual chχ = γ and shχ = vγ and apply this matrix to a 6vector
(E, B), formed by a combination of the electric and magnetic field,
we find the usual Lorentz transformation formulae for the fields:
E
B
→










Ex
γEy − vγBz
γEz + vγBy
Bx
γBy + vγEz
γBz − vγEy










. (34)
In the representation space of electromagnetic 6-vectors the second Casimir
operator generates a “duality transformation” (E, B) ↔ (−B, E). Under
this transformations the Maxwell equations of the free electromagnetic field
are invariant, the existence of electric charges and non-existence of magnetic
ones, however, breaks this duality invariance.
The considered real representation is irreducible, but the spaces of real
linear combinations of the complex vectors (E ± iB) transform separately,
and thus span invariant subspaces.
Consider the complex basis
M±
=
1
2
M ± iN (35)
of the Lorentz Lie algebra with the commutation relations
[M±
µ , M±
ν ] = µνλM±
λ , [M+
µ , M−
ν ] = 0. (36)
28
The Lie algebra decomposes into the direct sum of two three-dimensional
Lie algebras L±
, L = L+
⊕ L−
. Both L+
and L−
have the structure of the
algebra so(3) of the rotation group.
The Lie algebra of the real Lorentz group is made up of the real superpositions
of M and N. The split is possible only by making use of i, i. e.
of coefficients that would appear in the Lie algebra of the complex Lorentz
group, which is isomorphic to SO(4, C), as the sign does not play a role in
the complex group. Locally the latter one is isomorphic to the product of
two complex rotation groups SO(3, C).
The real linear combinations of M±
build up the Lie algebra of SO(4, R),
which in turn is locally isomorphic to SO(3, R) × SO(3, R). Irreducible representations
of the Lorentz group can thus be characterized by first complexifying
it, making use of the isomorphism to SO(4, C) and the local product
decomposition of the latter one, and finally restricting to SO(4, R) and its
decomposition:
L → Lc ∼= SO(4, C) ∼= SO(3, C) × SO(3, C)
→ SO(4, R) ∼= SO(3, R) × SO(3, R).
Accordingly, the irreducible representations of the Lorentz group by matrices
D(j,j )
can be classified by two indices j and j of two copies of SO(3)
representation matrices. D(j,j )
are (2j + 1)(2j + 1) dimensional.
From the Casimir operators
(M±
)2
= M2
− N2
± 2iMN (37)
of the two rotation groups we can read off the Casimir operators
C1 = M2
− N2
and C2 = MN (38)
of the Lorentz group in the representations D(j,j )
.
In this formalism infinitesimal Lorentz transformations are given by
L(v, α) ≈ 1+α(M+
+M−
)−iv(M+
−M−
) = 1+(α−iv)M+
+(α+iv)M−
,
(39)
finite transformations by
L(v, α) = D(j,j )
(v, α) = D(j)
(α − iv) ⊗ D(j )
(α + iv). (40)
Note that
L(v, α) = exp[(α − iv)M+
+ (α + iv)M−
] !
The one-parameter subgroup connecting L(v, α) with the unit element e is
not given by the curve (v(τ), α(τ)) = (τv, τα), because rotations and boosts
do not commute and vi
are not additive parameters.
29
The representation matrix of L(v, α) is
D(j,j )
(v, α) = D(j)
(α) D(j)
(−iu) ⊗ D(j )
(α) D(j )
(iu), (41)
where u is
u = arth |v| ·
v
|v|
, (42)
D(j)
(−iu) and D(j )
(iu) are rotations by an imaginary angle.
Example: The subgroup SO(3) (v = 0)
D(j,j )
(0, α) = D(j)
(α) ⊗ D(j )
(α) = D(j+j )
(α) ⊕ . . . ⊕ D(j−j )
(α). (43)
Here the representation of the Lorentz group becomes reducible with the
exception of j = 0 or j = 0. j = 1, j = 0 and j = 0, j = 1 denote the
defining representation of SO(3).
4.7.2 Spinorial representations and local isomorphism to SL(2, C)
The simplest nontrivial representations are j = 1
2
, j = 0 and j = 0, j = 1
2
,
they are a system of fundamental representations. Choose j = 1
2
, j = 0:
D( 1
2
,0)
(v, α) = e− i
2
ασ
e
1
2
uσ
. (= e−1
2
(α−iv)σ
!) (44)
With
uσ =
u3 u1 − iu2
u1 + iu2 −u3
, and (uσ)2
= u2
· 1,
where u = (u2
1 + u2
2 + u2
3)
1
2 , we find
e−1
2
uσ
=



chu
2
− u3
u
shu
2
−u1
u
+ iu2
u
shu
2
−u1
u
− iu2
u
shu
2
chu
2
+ u3
u
shu
2


 = D( 1
2
)
(−uσ).
Introducing the unit vector ni
= ui
/u we can write
D( 1
2
)
(u, n) =


chu
2
− n3 shu
2
(−n1 + in2) shu
2
−(n1 + in2) shu
2
chu
2
+ n3 shu
2

 (45)
The determinant of this matrix is equal to one, so D( 1
2
)
(u, n) ∈ SL(2, C), the
group of complex 2 by 2 matrices with unit determinant.
30
A four-vector xi
can be uniquely transformed into a 2 × 2 matrix X in
the following way
X := x0
· 1 + xσ =


x0
+ x3
x1
− ix2
x1
+ ix2
x0
− x3

 . (46)
X is hermitian for real xi
, tr X = 2x0
. We can introduce the formal four-
vectors
{σi} = {1, σ} and ˜σi
= σi (= ηik
σk), (47)
then
xi
=
1
2
tr(X˜σi
). (48)
The determinant is
det X = (x0
)2
− x2
= xi
xi. (49)
When we multiply a hermitian X by an arbitrary complex unimodular
matrix A ∈ SL(2, C) in the following way,
X = AXA†
, (50)
then X is also hermitian and we can associate to it the real four-vector
x
i
=
1
2
tr(X ˜σi
), (51)
and
x
i
xi = det X = det X = xi
xi, (52)
because det A = 1. The norm of xi
is thus conserved.
Now the Lorentz matrix can be expressed in terms of A:
Li
k =
1
2
tr A σk A†
˜σi
, (53)
and vice versa
A = ±
Li
k σi ˜σk
√
detLi
k σi ˜σk
. (54)
One Lorentz matrix L corresponds to the matrices ± A ∈ SL(2, C), so
SL(2, C) is a double covering of the proper orthochronous Lorentz group.
31
5 Group Manifolds
5.1 Manifolds
A manifold M is a topological Hausdorff space with local diffeomorphic mappings
to Rn
– coordinates {xi
}, i = 1, . . . , n.
Curve xi
(s), tangent vector vi
= dxi
(s)/ds, v = vi
(s)∂i in the coordinate basis.
Tangent space TpM, cotangent space T∗
p M, diffeomorphism f : M → N
or M → M, f : xi
→ yi
. A diffeomorphism induces mappings in the tangent
and cotangent spaces.
Push-forward f∗ : Tp → Tf(p). f maps a curve xi
(s) to yi
(xj
(s)), f∗ maps the
corresponding tangent vectors:
dxi
ds
∂
∂xi
→
∂yi
∂xj
dxj
ds
∂
∂yi
= ¯vi ∂
∂yi
,
in components
vi ∂
∂xi
→ vi ∂yj
∂xi
∂
∂yj
. (55)
In T∗
M, the space of covectors or one-forms, the duals of vectors, f
induces a mapping T∗
f(p)M → T∗
p M, the pull-back, defined by
f∗
ω, v = ω, f∗v . (56)
For the components this means
f∗
(ωidxi
), vj ∂
∂xj
= ωi dyi
, vk ∂yj
∂xk
∂
∂yj
= ωi vj ∂yi
∂xj
,
so that
(f∗
ω)i = ωj
∂yj
∂xi
. (57)
5.2 Invariant vector fields on group manifolds
Define a left translation La : G → G and a right translation Ra : G → G for
a certain element a of a Lie group G by
Lag = ag, Rag = ga ∀g ∈ G. (58)
The diffeomorphic maps La and Ra induce maps in the tangent spaces, La∗ :
TgG → TagG, Ra∗ analogous.
Left-invariant vector fields X on a group manifold G are defined by
La∗X|g = X|ag , (59)
32
in coordinates
La∗X|g = Xµ
(g)
∂xν
(ag)
∂xµ(g)
∂
∂xν
ag
= Xν
(ag)
∂
∂xν
ag
. (60)
A vector v ∈ TeG defines a unique left-invariant vector field Xv throughout
G by
Xv|g = Lg∗v, g ∈ G
and a left-invariant vector field X defines a unique vector v = X|e ∈ TeG.
In consequence there is a vector space isomorphism between the set of
left-invariant vector fields g and TeG. To establish an algebra isomorphism
we have to find an equivalent to the commutator in the language of vector
fields.
Tangent vectors on a manifold act as differential operators on functions,
calculating their derivative in a certain direction. In a coordinate basis the
action of a vector (field) V (x) = V i
(x)∂i on a function is given by
V [f] = V i
(x)∂if(x) ∈ R. (61)
This is again a scalar field which can be acted upon once more by a vector
field, so we can define in a natural way a commutator
[U, V ][f] = UV [f] − V U[f] = (V i
,k Uk
− Ui
,k V k
)f,i, (62)
so the commutator is again a vector field, called the Lie bracket of U and V .
In terms of coordinates this is
[U, V ]i
= V i
,k Uk
− Ui
,k V k
. (63)
Left-invariant vector fields are closed under the Lie bracket,
La∗[X, Y ]g = [La∗X|g , La∗Y |g] = [X, Y ]ag, (64)
so g can be identified with the Lie algebra of G and the latter one can be
understood as an algebra of infinitesimal group elements in the neighborhood
of any finite group element. The construction of left-invariant vector fields
from a basis in TeG manages to attach the Lie algebra to every element of
G.
Example: Take the curve through the unity of SO(3) given by P(s) :=
exp(sΛi), in fact a one-parameter subgroup, and act on it with a fixed group
element g, given by the rotation matrix R(α) from the left, then
Q(s) := LgP(s) = R(α)P(s) (65)
33
is a curve passing through the element g. To find the tangent vector at g,
ti = Lg∗Λi, we write Q(s) as g ≡ R(α), followed by a rotation S(s) and
consider the limit s → 0. From Q(s) = R(α)P(s) = S(s)R(α) we find
S(s) = R(α)P(s)R(α)−1
. (66)
For s 1, i. e. close to unity on the curve P(s) and close to R(α) on Q(s),
exp(sΛi) ≈ 1 + sΛi and S(s) ≈ 1 + sti, where the generator ti represents
the tangent vector to Q(s) at R(α). From (??) we find that ti = AdgΛi and
further that [ti, tj] = εij
k
tk everywhere on the group manifold.
5.3 Frames and structure equation
Take a basis {V1, V2, . . . , Vn} of TeG of an n-dimensional group manifold. By
push-forward we can construct n linearly independent left-invariant vector
fields {X1, X2, . . . , Xn} at each point of G, Xµ|g = Lg∗Vµ. {Xµ} form a
basis of every tangent space TgG. The Lie bracket = commutator is again
an element of TgG, so it can be expanded in terms of {Xµ},
[Xµ, Xν] = cµν
λ
Xλ. (67)
Due to the invariance of the Lie bracket = commutator under the left action
Lg, the structure constants are independent of g.
Define a basis of left-invariant one-forms {θµ
} dual to {Xµ}, θµ
, Xν =
δµ
ν . The exterior derivative is a two-form with the following action on two
basis vectors
dθµ
(Xν, Xλ) = Xν[θµ
(Xλ)] − Xλ[θµ
(Xν)] − θµ
([Xν, Xλ])
= Xν[δµ
λ] − Xλ[δµ
ν ] − θµ
(cνλ
κ
Xκ) = −cνλ
µ
.
From this follows the Maurer-Cartan structure equation
dθµ
= −
1
2
cνλ
µ
θν
∧ θλ
. (68)
The Maurer-Cartan form or canonical form
is defined as a Lie algebra-valued one-form θ : TgG → TeG by
θ : X → (Lg−1 )∗X = (Lg)−1
∗ X, X ∈ TgG. (69)
Take a basis {Vµ} of TeG, a basis {Xµ} of TgG, generated by Xµ|g = Lg∗Vµ,
a basis {θµ
} of T∗
g G, and a vector Y = Y µ
Xµ ∈ TgG. Then
θ(Y ) = Y µ
θ(Xµ) = Y µ
Lg
−1
∗ [Lg∗Vµ] = Y µ
Vµ, (70)
34
θ simply replaces the basis vectors Xµ by Vµ, so we may write it explicitly as
θ = Vµ ⊗ θµ
. (71)
From the structure equation follows
dθ +
1
2
[θ ∧ θ] = −
1
2
Vµ ⊗ cνλ
µ
θν
∧ θλ
+
1
2
cνλ
µ
Vµ ⊗ θν
∧ θλ
= 0. (72)
A straightforward way to introduce the Maurer-Cartan form for matrix
groups is to define it as
θ = g−1
dg (73)
at every point g of the group manifold. Take, as an example, the oneparameter
subgroup of elements
g(φ) = exp(φΛ3) =



cos φ − sin φ 0
sin φ cos φ 0
0 0 1



of rotations around the 3-axis of SO(3). We see that
g−1
(φ) dg(φ) =



cos φ sin φ 0
− sin φ cos φ 0
0 0 1






− sin φ − cos φ 0
cos φ − sin φ 0
0 0 0


 dφ
=



0 −1 0
1 0 0
0 0 0


 dφ = Λ3 ⊗ dφ
is independent of φ. When it acts on a tangent vector to the curve g(φ) at
φ, it gives Λ3, the tangent vector at e.
From the Maurer-Cartan form g−1
dg we may read off left-invariant oneforms
for every g ∈ G and having found a basis {θµ
} at every point we can
construct a (left-invariant) metric
ds2
= θµ
⊗ θν
δµν (74)
on all of the group. It extends the Cartan-Killing metric from the Lie-algebra
TeG to the whole manifold. With the aid of it one constructs further a left
invariant measure, the left Haar measure dµH for compact groups. It satisfies
G
dµH(g) f(g) =
G
dµH(g) f(ag) ∀a. (75)
35
Analogously one can construct a right Haar measure. For SU(2), for example,
the Haar measure is the usual measure on the sphere S3
(round metric).
Example: An SO(3) matrix eΛ
, where
Λ := αΛ = α



0 0 0
0 0 −1
0 1 0


 + β



0 0 1
0 0 0
−1 0 0


 + γ



0 −1 0
1 0 0
0 0 0


 .
Explicitly
Λ =



0 −γ β
γ 0 −α
−β α 0


 and Λ2
=



−β2
− γ2
αβ αγ
αβ −α2
− γ2
βγ
αγ βγ −α2
− β2


 ,
(76)
and with the definition
n2
:= α2
+ β2
+ γ2
(77)
we find for the higher powers of Λ
Λ3
= −n2
Λ, Λ4
= −n2
Λ2
, Λ5
= n4
Λ, Λ6
= n4
Λ2
, . . .
so that
g = eΛ
= 1 +
sin n
n
Λ +
1 − cos n
n2
Λ2
. (78)
Further
g−1
= 1 −
sin n
n
Λ +
1 − cos n
n2
Λ2
(79)
and
dg =
n sin n + 2 cos n − 2
n3
Λ2
dn +
1 − cos n
n2
dΛ2
+
n cos n − sin n
n2
Λ dn +
sin n
n
dΛ.
(80)
After some calculations we find the Maurer-Cartan form in the coordinates
(α, β, γ):
g−1
dg =
n − sin n
n2
dn αi +
sin n
n
dαi −
1 − cos n
n2
(α × dα)i ⊗ Λi. (81)
In an analogous way we can define a “square” of left-invariant vector
fields, called the Laplacian,
∆ = δµν
XµXν. (82)
With the vectors interpreted as differential operators this is indeed the usual
Laplace operator associated with the left-invariant metric.
36
5.4 Bundles
A bundle E is locally a Kronecker product of a basis manifold M and a
fibre F. If F is a vector space, the bundle is called vector bundle. In local
coordinates an element u of E has coordinates (p, f), where pi
are coordinates
on M and fk
are components of a vector in F.
The canonical projection π is a mapping E → M, πu = p, π−1
p = F. A
section σ is a mapping M → E, p → (p, f(p)), such that πσ = id. Sections
determine vector fields f(p).
The structure group G of a bundle is a Lie group that acts from the left on
the fibres and has the following property. Take a set {Ui} of open coverings
of M with diffeomorphisms φi : Ui × F → π−1
(Ui) with πφi(p, f) = p, i. e. a
local trivialization of the bundle, mapping π−1
(Ui) onto Ui × F. If we write
φi(p, f) = φi,p(f), then φi,p : F → Fp is a diffeomorphism. On Ui ∩ Uj = ∅
tij(p) := φ−1
i,p ◦ φj,p : F → F is an element of G. φi and φj are related by a
smooth map
tij : Ui ∩ Uj → G, φj(p, f) = φi(p, tij(p)f). (83)
tij is called the transition function, G the structure group. Bundles are
denoted by (E, π, M, F, G). In addition to the left action there is also a right
action of G.
A simple example: A cylinder can be described as a product bundle of S1
and the interval I = [−1, 1] with the structure group consisting only of the
unit element e. On the overlap of arbitrary intervals the transition function
is the identity. A Moebius strip is locally a product of an interval U ⊂ S1
and the interval I, but it has no global product structure. The transition
functions are the identity or a reflection P of I. In this case of a nontrivial
bundle the structure group is {e, P} Z2.
A principal bundle is a fibre bundle with the fibre F identical to the
structure group. Principal bundles with structure group G are also called
G-bundles and denoted by P(M, G).
6 Connections on principal bundles
Definition: Assume u ∈ P(M, G), p = π(u). The vertical subspace VuP of
the tangent space TuP is the subspace of vectors tangent to G.
Construction: Take an element of the Lie algebra, A ∈ g. The right
action
Rexp(tA)u = u exp(tA)
37
defines a curve in P through u. As
π(u) = π (u exp(tA)) = p,
the curve lies in G. Define a vector A#
∈ TuP by
A#
f(u) =
d
dt
f (u exp(tA))
t=0
, f : P → R. (84)
A#
is tangent to the fibre Gp at u, A#
∈ VuP.
In this way we construct a vector field A#
, the so-called fundamental
vector field, generated by A. There is a vector space isomorphism
# : g → VuP, A → A#
. We have
(i) π∗X = 0 ∀X ∈ VuP,
(ii) [A#
, B#
] = [A, B]#
.
The horizontal subspace HuP of TuP is a complement. A connection on
P is a unique separation of the tangent space TuP into the vertical subspace
VuP and the horizontal subspace HuP, such that
(i) TuP = HuP ⊕ TuP,
(ii) a smooth vector field X on P is separated into smooth vector fields
XH
∈ HuP and XV
∈ VuP as X = XH
+ XV
,
(iii) HugP = Rg∗HuP for arbitrary u ∈ P, g ∈ G.
(iii) states that the horizontal subspaces HuP and HugP on the same fibre
are related by Rg∗. Thus the subspace HuP at u generates all the horizontal
subspaces on the same fibre.
The connection one-form is a Lie-algebra-valued one-form ω ∈ g⊗T∗
P
that defines as a projection of TuP onto the vertical subspace VuP g.
(i) ω(A#
) = A, A ∈ g,
(ii) Rg
∗
ω = Adg−1 ω,
that is Rg
∗
ωug(X) = ωug(Rg∗X) = g−1
ωu(X)g. The pullback acts on the
form index, the adjoint action on the Lie-algebra indices.
To show consistency with the projection property of a connection, define
the horizontal subspace HuP as the kernel of ω,
HuP = {X ∈ TuP|ω(X) = 0}. (85)
38
Show that Rg∗HuP = HuP (consistence with the connection property): Take
X ∈ HuP, construct Rg∗X ∈ TugP.
ω(Rg∗X) = Rg
∗
ω(X) = g−1
ω(X)g = 0,
as ω(X) = 0. Accordingly Rg∗X ∈ HugP. This connection is called Ehresmann
connection. The connection one-form ω thus separates TuP into
HuP ⊕ VuP in accordance with the connection axioms.
Given an open covering {Ui} of M and a local section σi in each chart
Ui, a local connection form is a Lie-algebra-valued one-form Ai on Ui,
Ai = σi
∗
ω ∈ g ⊗ Ω1
(Ui). (86)
Conversely, from a given local one-form Ai we can construct a connection
one-form ω, such that Ai = σi
∗
ω. Define a g-valued one-form ωi in P,
ωi = g−1
i π∗
Ai gi + g−1
i dP gi, (87)
dP is the exterior derivative on P and gi is the local trivialization defined by
φ−1
i (u) = (p, gi) for u = σi(p)gi.
First show that σ∗
i ωi = Ai. Take X ∈ TP M.
σ∗
i ωi(X) = ωi(σi∗X) = π∗
Ai(σi∗X) + dP gi(σi∗X)
= Ai(π∗σi∗X) + dP gi(σi∗X).
We have made use of σi∗X ∈ Tσi
P and gi = e at σi∗. We further note that
π∗σi∗ = idTp(M), dP gi(σi∗X) = 0, since g ≡ e along σi∗X. Thus σi∗ω(X) =
Ai(X).
Next show that ωi satisfies the axioms of a connection one-form.
(i) X = A#
∈ VuP, A ∈ g ⇒ π∗X = 0.
ωi(A#
) = g−1
i dP g(A#
) = g(u)−1 dg(u exp(tA))
dt t=0
= g(u)−1
g(u)
d exp(tA)
dt t=0
= A.
(ii) X ∈ TuP, h ∈ G.
R∗
hωi(X) = ω(Rh∗X) = g−1
iuh Ai(π∗Rh∗X)giuh + g−1
iuhdP giuh(Rh∗X).
Since giuh = giuh and π∗Rh∗X = π∗X (πRh = π), we have
R∗
hωi(X) = h−1
g−1
iu Ai(π∗X)giuh + h−1
g−1
iu dP giu(X)h = h−1
ωi(X)h,
39
where we have noted that
g−1
iuhdP giuh(Rh∗X) = g−1
iuh
d
dt
giγ(t)h
t=0
=
h−1
g−1
iu
d
dt
giγ(t)
t=0
h = h−1
g−1
iu dP giu(X)h.
Here γ(t) is a curve through u = γ(0), whose tangent vector at u is X.
Hence the g-valued one-form ωi indeed satisfies Ai = σ∗
i ωi and the axioms
of a connection one-form.
For uniqueness of ω throughout P the relation ωi = ωj on the overlaps
Ui ∩Uj must hold. From this follow the transformation properties of the local
forms Ai.
Lemma: Consider the local sections σi and σj on the neighborhoods Ui
and Uj of the principal bundle P(M, G). For X ∈ TP M, p ∈ Ui ∩ Uj, σi∗X
and σj∗X satisfy
σj∗X = Rtij ∗
(σi∗X) + t−1
ij dtij(X)
#
. (88)
Proof: From σj(p) = σi(p)tij(p) we derive (γ is a curve with γ(0) = p,
˙γ(0) = X.)
σj∗(X) =
d
dt
σj(γ(t))
t=0
=
d
dt
{σi(t)tij(t)}
t=0
=
d
dt
σi(t) · tij(p) + σi(p)
d
dt
tij(t)
t=0
= Rtij ∗
(σi∗X) + σj(p)t−1
ij (p)
d
dt
tij(t)
t=0
.
(Assuming G is a matrix group, we have Rg∗X = Xg.) Further
t−1
ij (p)dtij(X) = t−1
ij (p)
d
dt
tij(t)
t=0
=
d
dt
t−1
ij (p)tij(t)
t=0
∈ Te(G) g.
(Note that t−1
ij (p)tij(γ(t)) = e at t = 0.) The second term of σj∗X represents
the vector field t−1
ij dtij(X)
#
at σj(p).
The compatibility condition is obtained by applying the connection form
ω to (??):
σ∗
j ω(X) = R∗
tij
ω(σi∗X) + t−1
ij dtij(X) = t−1
ij ω(σi∗X)tij + t−1
ij dtij(X),
40
which yields
Aj = t−1
ij Ai tij + t−1
ij dtij. (89)
Example: A U(1) bundle over M. Assume Ai and Aj to be local connection
forms on the overlapping charts Ui and Uj and a transition function
tij(p) = exp[iχ(p)], χ(p) ∈ R
in Ui ∩ Uj. Then
Aj(p) = tij(p)−1
Ai tij(p) + tij(p)−1
dtij(p) = Ai(p) + i dχ(p).
This is the usual gauge transformation law for the electromagnetic vector
potential Aµ = −iAµ,
(Aj)µ = (Ai)µ + ∂µχ.
7 Curvature
7.1 The covariant derivative
of a vector-valued r-form φ ∈ Ωr
(P) ⊗ V , φ : TP ⊗ . . . ⊗ TP → V is a
mapping Ωr
(P) → Ωr+1
, defined as
Dφ(X1, . . . , Xr+1) ≡ dP φ(XH
1 , . . . , XH
r+1) (90)
for X1, . . . , Xr+1 ∈ TuP.
7.2 The curvature two-form
Ω is the covariant derivative of the connection one-form ω,
Ω ≡ Dω ∈ Ω2
⊗ g. (91)
Behavior under right-translation:
R∗
aΩ = a−1
Ω a, a ∈ G. (92)
Proof: Note that (Ra∗X)H
= Ra∗(XH
). Ra preserves the horizontal subspaces,
by virtue of the definition of the latter as kernel of the connection
and dP R∗
a = R∗
a dp for XH
, Y H
.
R∗
a Ω(X, Y ) = Ω(Ra∗X, Ra∗Y ) = dP ω (Ra∗X)H
, (Ra∗Y )H
= dP ω(Ra∗XH
, Ra∗Y H
) = R∗
adP ω(XH
, Y H
)
= dP R∗
aω(XH
, Y H
) = dP (a−1
ω a)(XH
, Y H
)
= a−1
dP ω(XH
, Y H
) a = a−1
Ω(X, Y ) a.
41
(a is a constant element and hence dP a = 0).
Cartan’s structure equation for ω and Ω.
Ω(X, Y ) = dP ω(X, Y ) + [ω(X), ω(Y )], (93)
Ω = dP ω + ω ∧ ω. (94)
Proof: There are three cases:
1) X, Y ∈ HuP.
ω(X) = ω(Y ) = 0. Ω(X, Y ) = dP ω(XH
, Y H
) = dP ω(X, Y ).
2) X ∈ HuP, Y ∈ VuP: Ω(X, Y ) = 0.
ω(X) = 0, so we need to prove dP ω(X, Y ) = 0.
dP ω(X, Y ) = Xω(Y )−Y ω(X)−ω([X, Y ]) = Xω(Y )−ω([X, Y ]). Y ∈ VuP,
so there is an element V ∈ g, s. t. Y = V #
. Then ω(Y ) = V is constant,
hence Xω(Y ) = X · V = 0.
The vector field Y is generated by g(t) = exp(tV ), so
[X, Y ] = lim
t→0
1
t
(Rg(t)∗
X − X).
Since Rg∗HuP = HugP, Rg∗X is horizontal and so is [X, Y ] ⇒ ω([X, Y ]) = 0.
3) X, Y ∈ VuP: Ω(X, Y ) = 0.
In this case dP ω(x, Y ) = Xω(Y ) − Y ω(X) − ω([X, Y ]) = −ω([X, Y ]) =
−[ω(X), ω(Y )].
7.3 The local form of the curvature
is the pull-back
F = σ∗
Ω (95)
for a local section σ on a chart U. Expressed in terms of the gauge potential
A, F becomes
F = dA + A ∧ A, (96)
where d is the exterior derivative on M.
The action on vectors of TM:
F(X, Y ) = dA(X, Y ) + [A(X), A(Y )]. (97)
Proof: With A = σ∗
ω, σ∗
dP ω = dσ∗
ω, and σ∗
(ζ ∧ η) = σ∗
ζ ∧ σ∗
η we find
from Cartan’s structure equation
F = σ∗
(dP ω + ω ∧ ω) = dσ∗
ω + σ∗
ω ∧ σ∗
ω = dA + A ∧ A, (98)
42
in components
Fµν = ∂µAν − ∂νAµ + [Aµ, Aν]. (99)
F is called the (Yang-Mills) field strength. In the overlap of two coordinate
neighborhoods Ui ∩ Uj
Fj = Adtij
Fi = t−1
ij Fi tij. (100)
7.4 Bianchi identities
When ω and Ω are given in terms of the basis {Ti} of g, ω = ωi
Ti and
Ω = Ωi
Ti, then
Ωi
= dP ωi
+ fi
jk ωj
∧ ωk
. (101)
Exterior differentiation yields
dP Ωi
= fi
jk dpωj
∧ ωk
− fi
jk ωj
∧ dpωk
. (102)
From ω(X) = 0 for a horizontal vector X we have
DΩ(X, Y, Z) = dP Ω(XH
, Y H
, ZH
) = 0,
where X, Y, Z ∈ TuP. This is the Bianchi identity
DΩ = 0, (103)
in local form
dF + [A, F] = 0. (104)
This is the form of the homogenous Maxwell and Yang-Mills equations.
To show this relation, operate with σ∗
on (??). Left-hand side
σ∗
dP Ω = d · σ∗
Ω = dF.
Right-hand side
σ∗
(dP ω ∧ ω − ω ∧ dPω) = dσ∗
ω ∧ σ∗
ω − σ∗
ω ∧ dσ∗
ω = dA ∧ A − A ∧ dA.
From this follows eq. (??)
DF := dF + A ∧ F − F ∧ A = dF + [A, F] = 0,
where the action of D on a g-valued p-form η on M is defined by
Dη = dη + [A, η]. (105)
Note that DF = dF for G = U(1) (Maxwell).
43
8 Covariant differentiation and curvature on
vector bundles
8.1 Covariant differentiation
Take a vector bundle with the fibre F being a vector space V of arbitrary
dimension, independent of the dimension of the basis manifold. The covariant
derivative Dψ of a section ψ of the bundle (= vector field) is a bundle-valued
one-form. Acting on a tangent vector u ∈ T(M) it gives another section,
(Dψ)(u) = Duψ, (106)
the covariant derivative of ψ along u. At a point p of the bundle
Dψp ∈ V ⊗ T∗
p . (107)
A concrete differentiation or connection is specified by its action on a
section basis {bI} (in the following capital indices denote vector indices in
V, lower-case indices are tangent space indices)
DbI = ωJ
I ⊗ bJ , DubI = ωJ
I(u) bJ . (108)
In a basis {ei} of tangent vectors and a co-basis {ei
} we can write
ωJ
I = AJ
Ii ei
, ωJ
I(ei) = AJ
Ii, Dei
bI = AJ
Ii bJ . (109)
For an arbitrary section ψ with ψ = ψI
bI the coefficients ψI
are ordinary
functions,
Dψ = (Dψ)I
⊗ bI, (110)
and
(Dψ)I
:= dψI
+ ωI
J ψJ
(111)
are ordinary one-forms with
Dei
ψ = ψI
;i bI, ψI
;i = ei(ψI
) + AI
Ji ψJ
. (112)
From the action on {bI} follows the action on the dual basis {bI
}:
D bI
, bJ = DbI
, bJ + bI
, DbJ = D δI
J = 0
⇒ DbI
, bJ = − bI
, DbJ = − bI
, ωK
J ⊗ bK = −ωK
J bI
, bK
= −ωI
J = −ωI
K δK
J = −ωI
K bK
, bJ = −ωI
K ⊗ bK
, bJ
44
From this we get
DbI
= −ωI
K ⊗ bK
. (113)
In the special case F = V ⊗ V∗
, when the fibre is the space of tensors of
type (1, 1), an element of this fibre defines a mapping Vp → Vp by contraction.
In the case of the tangent bundle the elements of T1
1 = T(M) ⊗ T∗
(M)
define a mapping Tp → Tp and can be also interpreted as vector-valued oneforms
and we can also define an exterior covariant derivative D (only for the
tangent bundle!). For a section ψ of T1
1
• Dψ is a T1
1 - valued one-form,
• D ∧ ψ is a T1
- valued two-form.
For a bundle-valued n-form ψ ⊗α with ψ being a section of the vector bundle
and α an n-form, the exterior covariant derivative is the bundle-valued n + 1
- form
D ∧ (ψ ⊗ α) = (Dψ) ∧ α + ψ ⊗ dα. (114)
The section
δ = δI
J
bI
⊗ bJ (115)
gives rise to the identical map Vp → Vp and its covariant derivative is equal
to zero. In the case of tangent bundles the section δ = ei
⊗ ei can be seen as
a section of T1
1 as well as a vectorial one-form with Dδ = 0 but
D ∧ δ = (Dei) ∧ ei
+ ei ⊗ dei
= ωj
i ⊗ ej ∧ ei
+ ei ⊗ dei
= (116)
ei ⊗ ωi
j ∧ ej
+ ei ⊗ dei
= ei ⊗ (dei
+ ωi
j ∧ ej
) =: ei ⊗ θi
= θ.
θ := D ∧ δ (117)
is a vector-valued two-form, called the torsion form and (??) is called the
first Cartan structure equation. The action of the torsion form on a pair of
tangent vectors is
θ(u, v) = Duv − Dvu − [u, v]. (118)
Generally a section ψ of V ⊗ V∗
has components ψI
J , ψ = ψI
J eI ⊗ eJ
.
In the case of the tangent bundle the indices can be written as i, j, so that
ψI
J;k ≡ ψi
j;k and j and k can be antisymmetrized. When φ is a vectorial
one-form,
D ∧ φ = Alt Dφ + C(φ ⊗ θ), (119)
where “Alt” means antisymmetrization and “C” means contraction. The
explicit action of a bundle-valued one form φ on a pair of vectors is
(D ∧ φ)(u, v) = Du(φ(v)) − Dv(φ(u)) − φ([u, v]). (120)
45
8.2 Parallel transport and curvature
A connection allows to transport a vector in Vp to an infinitely close fibre
Vq. A section ψ with values ψp, ψq, which are parallel in the sense of the
connection, satisfies Duψ = 0, where u is an infinitesimal vector from p to q
(value in p = value in q, transported to p). Dψ = 0 globally means that ψ
is covariantly constant. This equation does not have nontrivial solutions in
general, because it gives rise to the nontrivial integrability conditions
D ∧ Dψ = 0. (121)
Interestingly the V-valued two-form D ∧ Dψ depends at every point homogeneously
linearly only on ψ, not on its derivatives. Take a function f
D ∧ D(fψ) = D ∧ (ψ ⊗ df + f Dψ) =
Dψ ∧ df + ψ ⊗ d ∧ df + df ∧ Dψ + f D ∧ Dψ = f D ∧ Dψ,
because the first and the third term cancel and the second is zero. thus D∧D
defines a two-form Ω with values in V ⊗V∗
. In each point the values Ω(u, v)
belong to Vp ⊗ V∗
p in such a way that the contraction of Ω with ψ is equal
to Ω(ψ) = D ∧ Dψ. Ω is called the curvature form of the connection. It is
antisymmetric in its tangent vector arguments Ω(u, v) = −Ω(v, u) and from
(??) follows
Ω(u, v)(ψ) = Du(Dvψ) − Dv(Duψ) − D[u,v]ψ. (122)
This relation is called the Ricci identity.
With respect to the section basis {bI
⊗ bJ } Ω has the decomposition
Ω = ΩJ
I ⊗ bI
⊗ bJ ,
where ΩJ
I are ordinary two-forms, forming the curvature matrix. For Ω(bI) =
ΩJ
I ⊗ bJ holds
Ω(bI) = D ∧ D bI = D ∧ (ωJ
I ⊗ bJ ) = (d ∧ ωJ
I) ⊗ bJ − ωJ
I ∧ ωK
J ⊗ bK.
From this follows Cartan’s second structure equation
ΩJ
I = d ∧ ωJ
I + ωJ
K ∧ ωK
I or Ω = d ∧ ω + ω ∧ ω. (123)
For Ω holds the second Bianchi identity
D ∧ Ω ≡ 0 or d ∧ Ω + ω ∧ Ω − Ω ∧ ω ≡ 0. (124)
46
For a tangent bundle from Cartan’s first structure equation
θi
= d ∧ ei
+ ωi
j ∧ ej
(125)
follows
D ∧ θ = C(Ω ∧ δ), i. e. (D ∧ θ)i
= Ωi
j ∧ ej
, (126)
the first Bianchi identity.
Zero torsion leads to a symmetry property of the Riemann tensor, namely
Rijkl + Riklj + Riljk = 0.
8.3 Fibre metric, G-structures
In Riemsnnian geometry there is a metric in the tangent bundle of a manifold.
This can be generalized to arbitrary vector bundles in form of a fibre metric.
A metric is a section of the bundle (M, V∗
⊗ V∗
), which allows to introduce
an inner product, a norm, a notion of orthogonality,. . .
Alternatively, a metric can be constructed by introduction of orthogonal
bases {bA}, so that γ(bA, bB) = γAB = ±δAB. Two orthogonal bases {bA}
and {bB} at one point of M are related by a transformation bA = S
¯B
A b ¯B. S
lies in a (pseudo-)orthogonal subgroup G of GL(n), the group of nonsingular
n × n matrices, defined by the invariance of the scalar product. S can vary
from point to point in M. Thus a fibre metric can be defined by declaring
at every point a basis as orthonormal, then all the other orthonormal bases
are obtained by application of local transformations S ∈ G.
As a generalization, G need not be an orthogonal group, but an arbitrary
Lie subgroup of GL(n). If we choose at every point p a basis {bA} of Vp and
apply to it all transformations S ∈ G, we obtain a class of G-bases in each
fibre Vp. The totality of these bases defines a G-structure. When we start
with a different basis system, not belonging to the above class, we obtain a
different G-structure.
In Riemannian geometry the connection does not violate orthogonality.
The generalization to arbitrary vector bundles with fibre metric γ is the
postulate
Dγ := dγ − ωT
γ − γ ω = 0. (127)
This is equivalent to the Leibniz rule
u(γ(φ, ψ)) = γ(Duφ, ψ) + γ(φ, Duψ). (128)
Proof: The right-hand side is
γAB(ui
(∂iφA
+ ω A
i CφC
), ψB
) + γAB(φA
, ui
(∂iψB
+ ω B
i CψC
)) =
47
ui
γAB[(∂iφA
, ψB
) + ω A
i C(φC
, ψB
) + (φA
, ∂iψB
) + ω B
i C(φA
, ψC
)] =
ui
[∂i(γAB(φA
, ψB
)) − (∂iγAB)(φA
, ψB
) + ω C
i AγCB(φA
, ψB
) + ω C
i BγAC(φA
, ψB
)]
= ui
[∂i(γ(φ, ψ)) − (∂iγ)(φ, ψ) + ((ωT
i γ)AB + (γωi)AB)(φA
, ψB
)] =
u(γ(φ, ψ)) − ui
[∂iγ − ωT
i γ − γωi](φ, ψ) = u(γ(φ, ψ)) − Duγ(φ, ψ).
The integrability condition for covariant constance of the metric is
D ∧ Dγ = 0, ΩT
γ + γ Ω = 0. (129)
With the definition
ΩAB = γAC ΩC
B
this gives rise to another symmetry property of the Riemann tensor.
For arbitrary G-structures the connection should be compatible with it.
Given a G-basis field {bA}, translation of {bA|p} at the point p to the point q
should lead to a G-basis {bA|q} in q. {bA|p + DbA} should differ from {bA|q}
only infinitesimally, namely by a transformation S = 1+ω ∈ G. From this we
derive a postulate for the covariant derivative D: When DubA = ωB
A(u) bB,
then the matrix ω(u) belongs for each u and for each G-basis to the Lie
algebra of G. A G-connection ω is a Lie algebra valued one-form w. r. to
G-bases.
• If G = GL(n), there is no restriction, the Lie algebra contains all
nonsingular n × n matrices.
• When G is orthogonal, the Lie algebra consists of antisymmetric ma-
trices.
• In the case G = {1} the Lie algebra vanishes, ω ≡ 0. There is only one
G basis in a G−structure and Ω = 0.
Remark: A {1}-connection defines a parallelism at a distance, each bA is
covariantly constant. Vice versa, however, Ω ≡ 0 does not uniquely determine
a {1}-structure: {bA} and {¯bA = SB
A bB} with a constant matrix SB
A are
both {1}-structures. A connection with Ω ≡ 0 can have nonvanishing torsion.
8.4 Curvature and torsion of G-connections
G connections have values in the Lie algebra {Ga} with [Ga, Gb] = Cc
ab Gc,
Ω = Ωa
Ga, Ωa
= dωa
+
1
2
Ca
bc ωb
∧ ωc
. (130)
On the tangent bundle also torsion can be defined. One may ask, whether
for a given G there is a torsion-free G-connection.
48
• For (pseudo-)orthogonal groups vanishing torsion determines the Gconnection
uniquely.
• For some groups vanishing torsion is not possible - “essential torsion”.
• For other groups there are more than one torsion-free G-connections, for
example for the symplectic group, which is characterized by conserving
the symplectic form.
Among the groups with S ∈ G, det S = 1 for n > 2 (pseudo-)orthogonal
groups are the only ones with unique torsion-free G-connection. (For n = 2
G = {eΛ
, Λ ∈ R} is a counterexample.)
9 Gauge theories
Recall that in a certain basis {bI} the covariant derivative D on a vector
bundle is determined by the connection matrix ω. Consider a change of
bases
¯bI = SJ
I bJ . (131)
Then
DbI = D(SJ
I
¯bJ ) = (dS)J
I + SK
I ¯ωJ
K ⊗ ¯bJ . (132)
On the other hand
DbI = ωK
I bK = ωK
I SJ
K ⊗ ¯bJ . (133)
Comparison of (??) and (??) leads to
(dS)J
I + ¯ωJ
KSK
I = SJ
K ωK
I,
multiplication with (S−1
)I
L gives the typical behavior of a connection under
a change of basis,
¯ωJ
L = SJ
K ωK
I (S−1
)I
L − (dS)J
I(S−1
)I
L,
in matrix notation
¯ω = S ω S−1
− (dS) S−1
. (134)
For the curvature the corresponding relation is
¯Ω = S Ω S−1
= (AdS)(Ω).. (135)
Ω transforms under the adjoint representation of G, it is gauge covariant,
but ω is not, it can be made locally equal to zero at every point by certain
49
transformations S. For Ω = 0 there is of course always a gauge with ω ≡ 0
everywhere.
Thus for the covariant derivative of a section ψ holds
D ψ = dψ + ωψ, (136)
it commutes with changes of bases,
¯D Sψ = S Dψ,
whereas
d Sψ = S dψ + (dS)ψ = S dψ, (137)
which means that the ordinary exterior derivative does not commute with
transformations S ∈ G.
ω is a “compensation field”, it compensates (dS)ψ in (??). In theories the
physical content of which does not depend on x-dependent transformations,
S(x) are called local gauge transformations, G is called the gauge group.
Theories of this kind are called Gauge theories. The coefficients ωI
k and
ΩI
J kl of ωI
= ωI
k dxk
and ΩI
J = ΩI
J kl dxk
∧ dxl
are called gauge potentials
and gauge field strengths, respectively.
The difference quotient
1
t
[S(x(t)) − S(x(0))] S−1
(x(0))
corresponding to (dS)S−1
describes a curve passing through the unit element
of G, so the matrix (dS)S−1
represents the tangent vector at unity, i. e. an
element of the Lie algebra.
Example 1: M = space-time, E = tangent bundle, G = R+
×O(n) = {eΛ
O :
Λ ∈ R, O ∈ O(n)}, the Lie algebra
LG = {a · 1 + M : a ∈ R, M = −MT
} (138)
consists of the antisymmetric matrices and multiples of the unit matrix.
Denote a G basis by {ea} and a co-basis by {ωa
}. Scaling transformations
act in the following way:
ea = eΛ(x)
¯ea, ωa
= e−Λ(x)
¯ωa
. (139)
For the metric tensor
g = δab ωa
⊗ ωb
, or ¯g = δab ¯ωa
⊗ ¯ωb
(140)
50
follows the conformal transformation
¯g = e2Λ(x)
g. (141)
The G-structure distinguishes a metric, with respect to which the G-bases
are orthogonal, up to a conformal factor (conformal structure on M). A
choice of the conformal factor means a gauge of measuring rods and clocks.
A G-connection is defined with respect to {ea} by the connection matrix
ω = α · 1 + ω (142)
with curvature matrix
Ω = ϕ · 1 + Ω (143)
(unique split into an antisymmetric matrix and a multiple of the unit matrix)
ϕ = d ∧ α, Ω = dω + ω ∧ ω . (144)
• Action of pure rotations ∈ O:
ω → ¯ω = O ω O−1
− dO · O−1
, Ω → ¯Ω = O Ω O−1
(145)
(left action of the gauge group). From this follows
¯α = α, ¯ω = O ω O−1
− dO · O−1
, ¯ϕ = ϕ, ¯Ω = O Ω O−1
. (146)
• Action of scale transformations eΛ(x)
1:
ω → ¯ω = ω − dΛ · 1, Ω → ¯Ω = Ω (147)
This leads to
¯α = α − dΛ, ¯ω = ω , ¯ϕ = ϕ, ¯Ω = Ω . (148)
α and ϕ = d ∧ α transform formally like the four-potential and the electromagnetic
field tensor. 1918 H. Weyl attempted to “geometrize” electromagnetism
in this way, but for ϕ = 0 parallel transport of vectors does not
conserve length. This would lead to a “second clock effect”: after an accelerated
travel also the rate of a clock could be different than before.
Example 2: M = space-time, E = (M, C), G = U(1) = {eiΛ
: Λ ∈ R}.
Sections of this bundle are complex functions. The Lie algebra consists of
purely imaginary numbers,
ω = iα, Ω = iϕ = id ∧ α. (149)
51
Gauge transformations of a section Φ are Φ → eiΛ
Φ, the corresponding transformations
of connection and curvature are
ω → ¯ω = ω − idΛ, Ω → ¯Ω = Ω, ¯α = α − dΛ, ¯ϕ = ϕ. (150)
This reflects the transformation of the electromagnetic four-potential and
intensities; the non-integrability of the parallel transport of the phase of a
wave function Φ describes the Aharonov-Bohm effect.
Now consider the dynamics of the system: Without electromagnetic interaction
the complex scalar field Φ satisfies the free Klein-Gordon equation,
which is invariant under gauge transformations Φ → eiΛ
Φ with constant Λ,
like the Lagrangian ∂kΦ∗
∂k
Φ−m2
Φ∗
Φ. To admit local gauge transformations,
we must introduce a covariant derivative
D = d + iα, Dk = ∂k + ieAk (151)
(“minimal substitution”). From the coupled Lagrangian
L = (DkΦ)∗
Dk
Φ − m2
Φ∗
Φ −
1
4
FikFik
, (152)
respectively the form
L d4
x = (DΦ)∗
∧ DΦ − m2
Φ∗
Φ −
1
2e2
ϕ ∧ ϕ (153)
with
ϕ =
e
2
Fik dxi
∧ dxk
= dα (154)
we may derive the equations for the scalar field
ηik
DiDkΦ + m2
Φ = 0 or D ∧ DΦ + m2
Φ = 0 (155)
and the electromagnetic field
Fkl
;l = −jk
:= −ie[(Dk
Φ∗
)Φ − Φ∗
Dk
Φ], (156)
or
d ∧ ϕ = −ie2
[(DΦ∗
)Φ − Φ∗
DΦ] (157)
in terms of forms. In curved space ηik
has to be replaced by gik
, but Fik =
Ak,i − Ai,k, corresponding to ϕ = dα. Ai;k − Ak;i is gauge variant, when the
space-time connection has torsion.
In the case of a nontrivial U(1) bundle there is no global α, so ϕ is closed,
but not exact. An example is the interaction of Φ with a magnetic monopole
in flat space. In the rest-frame of a point-like monopole the fields are
E = 0, B = g
x
r3
, e. g. ϕ = eg sin ϑ dϑ ∧ dφ. (158)
52
For the four-potential in the neighbourhood of a monopole we make the
axisymmetric ansatz
α = a(r, ϑ) dφ, (159)
so that dα = ∂a
∂r
dr ∧ dφ + ∂a
∂ϑ
dϑ ∧ dφ. From this follows ∂a
∂ϑ
= eg sin ϑ with
the solution
a = eg (K − cos ϑ), (160)
corresponding to the vector potential
A = g
K − cos ϑ
r sin ϑ
eφ. (161)
A has singularities, the locations of which depend on the choice of K. A
singularity must occur, otherwise the magnetic charge would be equal to
zero:
4πeg =
S2
ϕ =
∂S2
α = 0.
Therefore for g = 0 the two-form ϕ is closed, but not exact.
The location of the singularity is without physical significance. For K =
±1, for example, there are two potentials α± with singularities at the south/
north pole of each sphere around the monopole. In the overlap they differ
by a gauge transformation α+ − α− = 2eg dφ = dΛ.
If we consider iα± indeed as connection forms of a vector bundle, they
must be related to two section bases {b±} which, when the bundle is restricted
to S2
, are defined everywhere except the south/north pole. In the rest of S2
the gauging b+ = eiΛ(x)
b− = e2iegφ
b− must be uniquely possible. Due to the
non-uniqueness of the azimuth φ this is the case only for integer values of
2eg. Including ¯h and c, this leads to the Dirac quantization condition
2eg
¯hc
∈ Z. (162)
The existence of one single magnetic monopole in the universe would give a
reason for the discrete quantization of electric charge.
Example 3: M = space-time, G = SU(2), E = (M, V), V = C2
, the
representation space of the fundamental representation of SU(2). When introduced
1954 by Yang and Mills for the description of isospin, this was
the first gauge theory on a vector bundle which is not the tangent bundle
and has a non-abelian gauge group. The motivation was the following: The
electromagnetic interaction left aside, the proton and the neutron are indistinguishable,
they can be seen as two states of the “nucleon”, related by
“rotations” in an internal space, isomorphic to the space of states of a spin
53
1/2 particle. The strong interaction is invariant under “isospin rotations”,
like the electromagnetic interaction it should be mediated by a compensation
field related to an SU(2)-covariant derivative.
The connection and curvature on the isospin bundle are the su(2) ele-
ments
ω = ω τ, Ω = Ω τ, (163)
where τi = − i
2
σi and ω and Ω are triples of coefficients. Each curvature
coefficient is expressed in the usual way by its corresponding connection
coefficient,
Ωc
= d ∧ ωc
+
1
2
abc ωa
∧ ωb
, (164)
sometimes written as
Ω = dω +
1
2
ω ∧ ω. (165)
When the forms ωa
are expressed in terms of their space-time components
Aa
µ eµ
and analogously Ωa
in terms of Fa
µν eµ
∧eν
, we get the relation between
the Yang-Mills field strengths and their potentials (see (??))
Fµν = ∂µAν − ∂νAµ + [Aµ, Aν], (166)
with the free field equations (= Bianchi identities)
D ∧ Ω = 0, (167)
in components (see (??))
dF + [A, F] = 0. (168)
The dynamics can be derived from a Lagrangian, whose expression in
terms of forms is
LY M d4
x =
1
2
Tr(Ω ∧ Ω) =
1
4
δab Ωa
∧ Ωb
, (169)
or, more generally,
LY M d4
x = fab Ωa
∧ Ωb
. (170)
The ensuing equations of motion with the isospin current one-form J are
D ∧ Ωa
= (f−1
)ab
Jb. (171)
A Yang-Mills type theory with gauge group G = SU(3) × SU(2) × U(1)
describes the contemporary standard model of particle physics.
54
10 Is the theory of gravity a gauge theory?
Beginning in 1917, Levi-Civita, Weyl, Schouten separated the notion of a
connection from the metric. 1922 ´E. Cartan laid the ground to the calculus
of differential forms. So the question, whether or not in the theory of gravity
the connection should be Levi-Civita, came up in a natural way. As the form
calculus is coordinate-free, the issue of general covariance shifted from coordinate
transformations to invariance under vierbein transformations. The
metric can be conceived as an O(3, 1) structure on the tangent bundle. Is
thus general relativity a gauge theory of the Lorentz group? If so, the independent
field variables would be the six connection forms ωik = −ωki. This
is not true, because ωik are derived from the four orthonormal basis forms ei
by d ∧ ei
= −ωi
k ∧ ek
. So ωik cannot be varied independently. The action
integral in terms of forms
W = Ωik ∧ (ei
∧ ek
) (172)
contains beside Ω, which is derived from ω, explicitly ei
.
According to the Palatini variation principle, one can ignore the dependence
of ω on e and vary independently with respect to ωik and ei
.
δW = δΩik ∧ (ei
∧ ek
) + Ωik ∧ δ (ei
∧ ek
) =
(D ∧ δωik) ∧ (ei
∧ ek
) +
1
2
Ωik
ik
lm δ(el
∧ em
) =
d ∧ (δωik ∧ (ei
∧ ek
)) + δωik ∧ D ∧ (ei
∧ ek
) + Ωik ∧ ik
lmel
∧ δem
The resulting field equations are
ik
lm Ωik ∧ el
= 0 (or equal to a source term) (173)
and
D ∧ (ei
∧ ek
) = 0. (174)
(??) are Einstein’s equations in terms of forms, (??) are equivalent to
D ∧ (ei
∧ ek
) = 0 (175)
( ik
lm is covariantly constant) and D ∧ ei
= θi
is the torsion. So (??) means
θi
∧ ek
− θk
∧ ei
= 0. (176)
55
For n ≥ 4 this is equivalent to
θi
= d ∧ ei
+ ωi
k ∧ ek
= 0. (177)
This is the equation relating ω to e. Other Lagrangians and matter Lagrangians
containing ωik (for example, the Dirac field Lagrangian) lead to
nonvanishing torsion.
The Palatini principle does not a priori assume zero torsion, torsion is
determined dynamically. If matter has zero spin density, there is no torsion
and the Palatini principle is equivalent to the usual Einstein-Hilbert principle,
spin however, if present, couples to torsion. The theory of gravity based on
this principle is called the Einstein-Cartan theory.
Another approach is the assumption of a larger gauge group, containing
ei
as connection components - the Poincar´e group. In the five-dimensional
matrix representation
(L, a) ↔
L a
0 1
(178)
the Poincar´e group appears isomorphic to a subgroup of GL(5). The corresponding
connection matrix is
ω =
ωi
k ωl
0 0
, (179)
where the basis form el
is denoted as ωl
. Gauge transformations act in the
usual way on the connection, ¯ω = SωS−1
−(dS)S−1
, see equation (??), where
S is of the form (??). We consider two cases
• S (L, 0). This is a Lorentz transformation of vierbeine for which the
connection is an O(3, 1) connection,
• S (1, a) (a translation):
¯ωi
k = ωi
k, ¯ωi
= ωi
− dai
− ωi
k ak
. (180)
In the second case ωi
k do not react to gauge translations, but ωi
do react
nontrivially. Can ωi
be identified with the basis forms ei
? For the latter
ones no translational gauge transformation was defined, so the ei
can be
identified with the connection forms belonging to the translation subgroup
of the Poincar´e group only in the case of linear independence and in a certain
translation gauge. In other words, gauge covariance under the Poincar´e gauge
group is broken at the kinematic level and is valid only under the subgroup
O(3, 1).
56
According to the two types of connection forms, there are the curvature
forms Ωi
k and Ωi
, forming the curvature matrix
Ω =
Ωi
k Ωl
0 0
. (181)
Ωi
satisfies the structure equation for torsion,
Ωi
= d ∧ ωi
+ ωi
k ∧ ωk
, (182)
but Ωi
= θi
only in the gauge ωi
= ei
. The Bianchi identities for the Poincar´e
curvature summarizes the Bianchi identities for the O(3, 1) curvature Ωi
k and
for torsion.
The behavior under translations is
¯Ωi
k = Ωi
k, ¯Ωi
= Ωi
− Ωi
k ak
, (183)
this means that translation invariance is broken also at the dynamical level, as
the action integral contains ei
; it is broken even for Yang-Mills type integrals
Ω∧ Ω, because gauge translations act on the operation. Gauge invariance
under translations would correspond to a conserved Noether current of energy
and momentum as generators of space and time translations, but as it is wellknown,
in general relativity there is no local energy-momentum conservation.
The probably most important application of gauge theory ideas to gravity
is the formulation of Ashtekar variables. In this approach space-time is split
into space and time to make a canonical formulation in terms of spatial field
variables and conjugate momenta possible, the latter ones containing time
derivatives. The canonical variables on a space manifold are orthonormal
bases (dreibeine, triads) E and (partially independent) connection components
A. In these variables the theory works as a gauge theory, the missing
dependence of the connection on the metric is imposed by a set of additional
conditions, the constraints. The local gauge group is SU(2), the universal
covering of the group of local dreibein rotations. The purpose for this reformulation
of general relativity is that the canonical pairs (A, E) are suitable
for quantization.
57