Stone M., Goldbart P. Mathematics for Physics: A Guided Tour for Graduate Students

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15.4 Further exercises and problems 575

(a) Show that

ijk

= D

(g)D

(g)f

lmn

ijk

= D

(g)D

(g)d

lmn

and so f

ijk

and d

ijk

are invariant tensors in the same sense that δ

and 

...i

are

invariant tensors for SO(n).

(b) Let w

= f

ijk

. Show that if u

→ D

(g)u

and v

→ D

(g)v

, then w

→

(g)w

. Similarly for w

= d

ijk

. (Hint: show first that the D

matrices are real

and orthogonal.) Deduce that f

ijk

and d

ijk

are Clebsh–Gordan coefficients for the

8 ⊕ 8 part of the decomposition

8 ⊗ 8 = 1 ⊕ 8 ⊕ 8 ⊕ 10 ⊕

10 ⊕ 27.

αβ

and the entries in the lambda matrices (

)

αβ

can be regarded

as Clebsch–Gordan coefficients for the decomposition

3 ⊗ 3 = 1 ⊕ 8.

(d) Use the graphical method of plotting weights and peeling off irreps to obtain the

tensor product decomposition in part (b).

The geometry of fibre bundles

In earlier chapters we have used the language of bundles and connections, but in a rela-

tively casual manner. We deferred proper mathematical definitions until now, because,

for the applications we meet in physics, it helps to first have acquired an understanding

of the geometry of Lie groups.

16.1 Fibre bundles

We begin with a formal definition of a bundle and then illustrate the definition with

examples from quantum mechanics. These allow us to appreciate the physics that the

definition is designed to capture.

16.1.1 Definitions

Asmooth bundle comprises three ingredients: E, π and M , where E and M are manifolds,

and π : E → M is a smooth surjective (onto) map. The manifold E is the total space,

M is the base space and π is the projection map. The inverse image π

−1

(x) of a point

in M (i.e. the set of points in E that map to x in M)isthefibre over x.

We usually require that all fibres be diffeomorphic to some fixed manifold F. The

bundle is then a fibre bundle, and F is “the fibre” of the bundle. In a similar vein, we

sometimes also refer to the total space E as “the bundle”. Examples of possible fibres are

vector spaces (in which case we have a vector bundle), spheres (in which case we have

a sphere bundle) and Lie groups. When the fibre is a Lie group we speak of a principal

bundle. A principal bundle can be thought of as the parent of various associated bundles,

which are constructed by allowing the Lie group to act on a fibre. A bundle whose fibre

is a one-dimensional vector space is called a line bundle.

The simplest example of a fibre bundle consists of setting E equal to the cartesian

product M × F of the base space and the fibre. In this case the projection just “forgets”

the point f ∈ F, and so π : (x, f ) (→ x.

Amore interesting example can be constructed by taking M to be the circle S

equipped

with coordinate θ , and F as the one-dimensional interval I =[−1, 1]. We can assemble

these ingredients to make E into a Möbius strip. We do this by gluing the copy of I over

θ = 2π to that over θ = 0 with a half-twist so that the end −1 ∈[−1, 1] is attached to

+1, and vice versa.

A bundle that is a cartesian product E = M × F is said to be trivial. The Möbius

strip is not a cartesian product, and is said to be a twisted bundle. The Möbius strip is,

576

16.2 Physics examples 577



–1

02 



Figure 16.1 Möbius strip bundle, together with a section φ.

however, locally trivial in that for each x ∈ M there is an open retractable neighbourhood

U ⊂ M of x in which E looks like a product U ×F. We will assume that all our bundles

are locally trivial in this sense. If {U

} is a cover of M (i.e. if M =

) by such

retractable neighbourhoods, and F is a fixed fibre, then a bundle can be assembled out

of the collection of U

× F product bundles by giving gluing rules that identify points

on the fibre over x ∈ U

in the product U

× F with points in the fibre over x ∈ U

× F for each x ∈ U

∩ U

. These identifications are made by means of invertible

maps ϕ

(x) : F → F that are defined for each x in the overlap U

∩ U

. The ϕ

are known as transition functions. They must satisfy the consistency conditions

(x) = Identity,

(x) = φ

−1

(x),

(x)ϕ

= Identity, x ∈ U

∩ U

=∅. (16.1)

A section of a fibre bundle (E, π, M ) is a smooth map φ : M → E such that φ(x) lies

in the fibre π

−1

(x) over x. Thus π ◦ φ = Identity. When the total space E is a product

M × F this φ is simply a function φ : M → F. When the bundle is twisted, as is the

Möbius strip (see Figure 16.1), then the section is no longer a function as it takes no

unique value at the points x above which the fibres are being glued together. Observe that

in the Möbius strip the half-twist forces the section φ(x) to pass through 0 ∈[−1, 1]. The

Möbius bundle therefore has no nowhere-zero globally defined sections. Many twisted

bundles have no globally defined sections at all.

16.2 Physics examples

We now provide three applications where the bundle concept appears in quantum

mechanics. The first two illustrations are re-expressions of well-known physics. The

third, the geometric approach to quantization, is perhaps less familiar.

578 16 The geometry of fibre bundles

16.2.1 Landau levels

Consider the Schrödinger eigenvalue problem

−



∂

∂x

∂

∂y



= Eψ (16.2)

for a particle moving on a flat two-dimensional torus. We think of the torus as an L

×L

rectangle with the understanding that as a particle disappears through the right-hand

boundary it immediately reappears at the point with the same y coordinate on the left-

hand boundary; similarly for the upper and lower boundaries. In quantum mechanics we

implement these rules by imposing periodic boundary conditions on the wavefunction:

ψ(0, y) = ψ(L

, y), ψ(x,0) = ψ(x, L

). (16.3)

These conditions make the wavefunction a well-defined and continuous function on the

torus, in the sense that after pasting the edges of the rectangle together to make a real

toroidal surface the function has no jumps, and each point on the surface assigns a unique

value to ψ. The wavefunction is a section of an untwisted line bundle with the torus as

its base-space, the fibre over (x, y) being the one-dimensional complex vector space C

in which ψ(x, y) takes its value.

Now try to carry out the same programme for a particle of charge e moving in a uniform

magnetic field B perpendicular to the xy-plane. The Schrödinger equation becomes

−



∂

∂x

− ieA



ψ −



∂

∂y

− ieA



ψ = Eψ , (16.4)

where (A

, A

) is the vector potential. We at once meet a problem. Although the magnetic

field is constant, the vector potential cannot be chosen to be constant – or even periodic.

In the Landau gauge, for example, where we set A

= 0, the remaining component

becomes A

= Bx. This means that as the particle moves out of the right-hand edge

of the rectangle representing the torus we must perform a gauge transformation that

prepares it for motion in the (A

, A

) field it will encounter when it reappears at the left.

If Equation (16.4) holds, then it continues to hold after the simultaneous change

ψ(x, y) → e

−ieBL

ψ(x, y)

−ieA

→−ieA

+ e

−iBL

∂

∂y

+ieBL

=−ie(A

− BL

). (16.5)

At the right-hand boundary x = L

this gauge transformation resets the vector potential

back to its value at the left-hand boundary. Accordingly, we modify the boundary

conditions to

ψ(0, y) = e

−ieBL

ψ(L

, y), ψ(x,0) = ψ(x, L

). (16.6)

16.2 Physics examples 579

The new boundary conditions make the wavefunction into a section

of a twisted line

bundle over the torus. The fibre is again the one-dimensional complex vector space C.

We have already met the language in which the gauge field −ieA

is called a connec-

tion on the bundle, and the associated ieB field is the curvature. We will explain how

connections fit into the formal bundle language in Section 16.3.

The twisting of the boundary conditions by the gauge transformation seems innocent,

but within it lurks an important constraint related to the consistency conditions in (16.1).

We can find the value of ψ(L

, L

) from that of ψ(0, 0) by using the relations in (16.6)

in the order ψ(0, 0) → ψ(0, L

) → ψ(L

, L

), or in the order ψ(0, 0) → ψ(L

,0) →

ψ(L

, L

). Since we must obtain the same ψ(L

, L

) whichever route we use, we need

to satisfy the condition

ieBL

= 1. (16.7)

This tells us that the Schrödinger problem makes sense only when the magnetic flux

through the torus obeys

eBL

= 2πN (16.8)

for some integer N . We cannot continuously vary the flux through a finite torus. This

means that if we introduce torus boundary conditions as a mathematical convenience in

a calculation, then physical effects may depend discontinuously on the field.

The integer N counts the number of times the phase of the wavefunction is twisted as we

travel from (x, y) = (L

,0) to (x, y) = (L

, L

) gluing the right-hand edge wavefunction

back to the left-hand edge wavefunction. This twisting number is a topological invariant.

We have met this invariant before, in Section 13.6. It is the first Chern number of the

wavefunction bundle. If we permit B to become position dependent without altering the

total twist N , then quantities such as energiesand expectation values can change smoothly

with B.IfN is allowed to change, however, these quantities may jump discontinuously.

The energy E = E

solutions to (16.4) with boundary conditions (16.6) are given by



n,k

(x, y) =

∞



p=−∞



x −

− pL



i(eBpL

+k)y

. (16.9)

Here, ψ

(x) is a harmonic oscillator wavefunction obeying

−

mω

= E

, (16.10)

with ω = eB/m the classical cyclotron frequency, and E

= ω(n +1/2). The parameter

k takes the values 2πq/L

for q an integer. At each energy E

we obtain N independent

That the wave “function” is no longer a function should not be disturbing. Schrödinger’s ψ is never

really a function of space-time. Seen from a frame moving at velocity v , ψ(x, t) acquires factor of

exp(−imvx − mv

t/2), and this is no way for a self-respecting function of x and t to behave.

580 16 The geometry of fibre bundles

eigenfunctions as q runs from 1 to eBL

/2π. These N -fold degenerate states are

the Landau levels. The degeneracy, being of necessity an integer, provides yet another

explanation for why the flux must be quantized.

16.2.2 The Berry connection

Suppose we are in possession of a quantum-mechanical Hamiltonian

H (ξ ) depending

on some parameters ξ = (ξ

, ξ

, ...)∈ M , and know the eigenstates |n; ξ that obey

H (ξ )|n; ξ =E

(ξ)|n; ξ . (16.11)

If, for fixed n, we can find a smooth family of eigenstates |n; ξ , one for every ξ in the

parameter space M , we have a vector bundle over the space M . The fibre above ξ is

the one-dimensional vector space spanned by |n; ξ. This bundle is a sub-bundle of the

product bundle M × H where H is the Hilbert space on which

H acts. Although the

larger bundle is not twisted, the sub-bundle may be. It may also not exist: if the state

|n; ξ  becomes degenerate with another state |m; ξ  at some value of ξ, then both states

can vary discontinuously with the parameters, and we wish to exclude this possibility.

In the previous paragraph we considered the evolution of the eigenstates of a time-

independent Hamiltonian as we varied its parameters. Another, more physical, evolution

is given by solving the time-dependent Schrödinger equation

i∂

|ψ(t)=

H (ξ(t))|ψ(t) (16.12)

so as to follow the evolution of a state |ψ(t) as the parameters are slowly varied. If the

initial state |ψ(0) coincides with the eigenstate |0, ξ(0), and if the time evolution of

the parameters is slow enough, then |ψis expected to remain close to the corresponding

eigenstate |0; ξ(t) of the time-independent Schrödinger equation for the Hamiltonian

H (ξ(t)). To determine exactly how “close” it stays, insert the expansion

|ψ(t)=



(t)|n; ξ(t)exp



−i



(ξ(t)) dt



(16.13)

into (16.12) and take the inner product with |m; ξ . For m = 0, we expect that the

overlap m; ξ |ψ(t) will be small and of order O(∂ξ/∂t). Assuming that this is so, we

read off that

˙a

+ a

0; ξ |∂

|0; ξ 

∂ξ

∂t

= 0, (m = 0) (16.14)

= ia

m; ξ |∂

|0; ξ 

− E

∂ξ

∂t

, (m = 0) (16.15)

16.2 Physics examples 581

up to first-order accuracy in the time-derivatives of the |n; ξ(t). Hence,

|ψ(t)=e

iγ

Berry

(t)

⎧

⎨

⎩

|0; ξ +i



m=0

|m; ξ m; ξ|∂

|0; ξ 

− E

∂ξ

∂t

+···

⎫

⎬

⎭

−i



(t)dt

(16.16)

where the dots refer to terms of higher order in time-derivatives.

Equation (16.16) constitutes the first two terms in a systematic adiabatic series expan-

sion. The factor a

(t) = exp{iγ

Berry

(t)}is the solution of the differential equation (16.14).

The angle γ

Berry

is known as Berry’s phase, after the British mathematical physicist

Michael Berry. It is needed to take up the slack between the arbitrary ξ-dependent

phase-choice at our disposal when defining the |0; ξ , and the specific phase selected by

the Schrödinger equation as it evolves the state |ψ(t). Berry’s phase is also called the

geometric phase because it depends only on the Hillbert-space geometry of the family

of states |0; ξ , and not on their energies. We can write

Berry

(t) = i



0; ξ |∂

|0; ξ 

∂ξ

∂t

dt, (16.17)

and regard the 1-form

Berry

def

=0; ξ |∂

|0; ξ dξ

=0; ξ |d|0; ξ  (16.18)

as a connection on the bundle of states over the space of parameters. The equation



∂

∂ξ

+ A

Berry ,µ



|ψ=0 (16.19)

then identifies the Schrödinger time evolution with parallel transport. It seems reasonable

to refer to this particular form of parallel transport as “Berry transport”.

In order for corrections to the approximation |ψ(t)≈(phase)|0; ξ(t) to remain

small, we need the denominator (E

−E

) to remain large when compared to its numer-

ator. The state that we are following must therefore never become degenerate with any

other state.

Monopole bundle

Consider, for example, a spin-1/2 particle in a magnetic field. If the field points in

direction n, the Hamiltonian is

H (n) = µ|B|

σ · n. (16.20)

There are are two eigenstates, with energy E

=±µ|B|. Let us focus on the eigenstate

|ψ

, corresponding to E

. For each n we can obtain an E

eigenstate by applying the

582 16 The geometry of fibre bundles

projection operator

P =

(I + n ·

σ ) =



1 + n

− in

+ in

1 − n



(16.21)

to almost any vector, and then multiplying by a real normalization constant N . Applying

P to a “spin-up” state, for example, gives

(I + n ·

σ )







cos θ/2

iφ

sin θ/2



. (16.22)

Here, θ and φ are spherical polar angles on S

that specify the direction of n.

Although the bundle of E = E

eigenstates is globally defined, the family of states

|ψ

(1)

(n) that we have obtained, and would like to use as a base for the fibre over n,

becomes singular when n is in the vicinity of the south pole θ = π. This is because the

factor e

iφ

is multivalued at the south pole. There is no problem at the north pole because

the ambiguous phase e

iφ

multiples sin θ/2, which is zero there.

Near the south pole, however, we can project from a “spin-down” state to find

|ψ

(2)

(n)=N

(I + n ·

σ )







−iφ

cos θ/2

sin θ/2



. (16.23)

This family of eigenstates is smooth near the south pole, but is ill-defined at the north

pole. As in Section 13.6, we are compelled to cover the sphere S

by two caps, D

and

−

, and use |ψ

(1)

 in D

and |ψ

(2)

 in D

−

. The two families are related by

|ψ

(1)

(n)=e

iφ

|ψ

(2)

(n) (16.24)

in the cingular overlap region D

∩ D

−

. Here, e

iφ

is the transition function that glues

the two families of eigenstates together.

The Berry connections are

(1)

=ψ

(1)

|d|ψ

(1)

=

(cos θ − 1)dφ

(2)

=ψ

(2)

|d|ψ

(2)

=

(cos θ + 1)dφ. (16.25)

In their common domain of definition, they are related by a gauge transformation:

(2)

= A

(1)

+ idφ. (16.26)

The curvature of either connection is

dA =−

sin θdθ dφ =−

d(Area). (16.27)

16.2 Physics examples 583

Being the area 2-form, the curvature tells us that when we slowly change the direction

of B and bring it back to its original orientation the spin state will, in addition to the

dynamical phase exp{−iE

t}, have accumulated a phase equal to (minus) one-half of

the area enclosed by the trajectory of n on S

. The 2-form field dA can be thought of as

the flux of a magnetic monopole residing at the centre of the sphere. The corresponding

bundle of one-dimensional vector spaces, spanned by |ψ

(n), over n ∈ S

is therefore

called the monopole bundle.

16.2.3 Quantization

In this section we provide a short introduction to geometric quantization . This idea,

due largely to Kirilov, Kostant and Souriau, extends the familiar technique of canonical

quantization to phase spaces with more structure than that of the harmonic oscillator. We

illustrate the formalism by quantizing spin, and show how the resulting Hilbert space

provides an example of the Borel–Weil–Bott construction of the representations of a

semisimple Lie group as spaces of sections of holomorphic line bundles.

Prequantization

The passage from classical mechanics to quantum mechanics involves replacing the

classical variables by operators in such a way that the classical Poisson-bracket algebra

is mirrored by the operator commutator algebra. In general, this process of quantization

is not possible without making some compromises. It is, however, usually possible to

prequantize a phase space and its associated Poisson algebra.

Let M bea2n-dimensional classical phase space with its closed symplectic form

ω. Classically a function f : M → R gives rise to a Hamiltonian vector field v

via

Hamilton’s equations

df =−i

ω. (16.28)

We saw in Section 11.4.2 that the closure condition dω = 0 ensures that the Poisson

bracket

{f , g}=v

g = ω(v

, v

) (16.29)

obeys

, v

]=v

{f ,g,}

. (16.30)

Now suppose that the cohomology class of (2π)

−1

ω in H

(M , R) has the property

that its integrals over cycles in H

(M , Z) are integers. Then (it can be shown) there exists

a line bundle L over M with curvature F =−i

−1

ω. If we locally write ω = dη, where

η = η

, then the connection 1-form is A =−i

−1

η and the covariant derivative

∇

def

= v

(∂

− i

−1

), (16.31)

584 16 The geometry of fibre bundles

acts on sections of the line bundle. The corresponding curvature is

F(u, v)=[∇

, ∇

]−∇

[u,v]

=−i

−1

ω(u, v). (16.32)

We define a prequantized operator =ρ(f ) that, when acting on sections (x) of the line

bundle, corresponds to the classical function f :

=ρ(f )

def

=−i∇

+ f . (16.33)

For Hamiltonian vector fields v

and v

we have

[∇

+ if , ∇

]=∇

]

− iω(v

, v

) + i[f , ∇

]

= ∇

]

− i(i

ω + df )(v

)

= ∇

]

, (16.34)

and so

[−i∇

+ f , −i∇

+ g]=−

∇

]

− i v

=−i(−i∇

]

+{f , g})

=−i(−i∇

{f ,g}

+{f , g}). (16.35)

Equation (16.35) is Dirac’s quantization rule:

i [=ρ(f ), =ρ(g)]= =ρ({f , g}). (16.36)

The process of quantization is completed, when possible, by defining a polarization.

This is a restriction on the variables that we allow the wavefunctions to depend on.

For example, if there is a global set of Darboux coordinates p, q we might demand that

the wavefunction depend only on q, only on p, or only on the combination p + iq.

Such a restriction is necessary so that the representation f (→ =ρ(f ) be irreducible. As

globally defined Darboux coordinates do not usually exist, this step is the hard part of

quantization.

The general definition of a polarized section is rather complicated. We sketch it here,

but give a concrete example in the next section. We begin by observing that, at each

point x ∈ M , the symplectic form defines a skew bilinear form. We seek a Lagrangian

subspace of V

⊂ TM

for this form. A Lagrangian subspace is one such that V

= V

⊥

For example, if

ω = dp

∧ dq

+ dp

∧ dq

{

d(p

− iq

) ∧ d(p

+ iq

) + d(p

− iq

) ∧ d(p

+ iq

)

}

(16.37)