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

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16.2 Physics examples 585

then the space spanned by the ∂

’s is Lagrangian, as is the space spanned by the ∂

’s, and

the space spanned by the ∂

p+iq

’s. In the last case, we have allowed the coefficients of the

vectors in V

to be complex numbers. Now we let x vary and consider the distribution

defined by the vector fields spanning the V

’s. We require this distribution to be globally

integrable so that the V

are the tangent spaces to a global foliation of M . With these

ingredients at hand, we declare a section  of the line bundle to be polarized if ∇

 = 0

for all ξ ∈ V

. Here,

ξ is the vector field whose components are the complex conjugates

of those in ξ .

We define an inner product on the space of polarized sections by using the Liouville

measure ω

/n! on the phase space. The quantum Hilbert space then consists of finite-

norm polarized sections of the line bundle. Only classical functions that give rise to

polarization-compatible vector fields will have their Poisson-bracket algebra coincide

with the quantum commutator algebra.

Quantizing spin

To illustrate these ideas, we quantize spin. The classical mechanics of spin was discussed

in Section 6. There we showed that the appropriate phase space is the 2-sphere equipped

with a symplectic form proportional to the area form. Here we must be specific about the

constant of proportionality. We choose units in which  = 1, and take ω = jd(Area).

The integrality of ω/2π requires that j be an integer or half-integer. We will assume that

j is positive.

We parametrize the 2-sphere using complex stereographic coordinates z,

z which are

constructed similarly to those in Section 12.4.3. This choice will allow us to impose

a natural complex polarization on the wavefunctions. In contrast to Section 12.4.3,

however, it is here convenient to make the point z = 0 correspond to the south pole, so

the polar coordinates θ, φ, on the sphere are related to z,

z via

cos θ =

|z|

− 1

|z|

+ 1

iφ

sin θ =

|z|

+ 1

−iφ

sin θ =

|z|

+ 1

. (16.38)

In terms of the z,

z coordinates the symplectic form is given by

ω =

2ij

(1 +|z|

)

dz ∧ dz. (16.39)

As long as we avoid the north pole, where z =∞, we can write

ω = d



z − zdz

1 +|z|



= dη, (16.40)

586 16 The geometry of fibre bundles

and so the local connection form has components proportional to

=−ij

|z|

+ 1

, η

= ij

|z|

+ 1

. (16.41)

The covariant derivatives are therefore

∇

∂

∂z

− j

|z|

+ 1

, ∇

∂

∂z

+ j

|z|

+ 1

. (16.42)

We impose the polarization condition that ∇

 = 0. This condition requires the

allowed sections to be of the form

(z,

z) = (1 +|z|

)

−j

ψ(z), (16.43)

where ψ depends only on z, and not on

z. It is natural to combine the (1 +|z|

)

−j

prefactor with the Liouville measure so that the inner product becomes

ψ|χ=

2j + 1

2πi



dz ∧ dz

(1 +|z|

)

2j+2

ψ(z)χ (z). (16.44)

The normalizable wavefunctions are then polynomials in z of degree less than or equal

to 2j, and a complete orthonormal set is given by

(z) =



2j!

(j − m)!(j + m)!

j+m

, −j ≤ m ≤ j. (16.45)

We desire to find the quantum operators =ρ(J

) corresponding to the components

= j sin θ cos φ, J

= j sin θ sin φ, J

= j cos θ , (16.46)

of a classical spin J of magnitude j, and also to the ladder-operator components J

± iJ

. In our complex coordinates, these functions become

= j

|z|

− 1

|z|

+ 1

= j

|z|

+ 1

−

= j

|z|

+ 1

. (16.47)

Also in these coordinates, Hamilton’s equations dH =−ω(v

, ) take the form

˙z = i

(1 +|z|

)

∂H

∂z

z =−i

(1 +|z|

)

∂H

∂z

, (16.48)

16.2 Physics examples 587

and the Hamiltonian vector fields corresponding to the classical phase space functions

, J

and J

−

are

= iz∂

− iz∂

=−iz

∂

− i∂

−

= i∂

+ iz

∂

. (16.49)

Using the recipe (16.33) for =ρ(H ) from the previous section, together with the fact

that ∇

 = 0, we find, for example, that

=ρ(J

)(1 +|z|

)

−j

ψ(z)



−z



∂

∂z

−

(1 +|z|

)



2jz

(1 +|z|

)



(1 +|z|

)

−j

ψ(z),

= (1 +|z|

)

−j



−z

∂

∂z

+ 2jz



ψ. (16.50)

It is natural to define operators

def

= (1 +|z|

)

=ρ(J

)(1 +|z|

)

−j

(16.51)

that act only on the z-polynomial part ψ(z) of the section (z,

z). We then have

=−z

∂

∂z

+ 2jz. (16.52)

Similarly, we find that

−

∂

∂z

, (16.53)

= z

∂

∂z

− j. (16.54)

These operators obey the su(2) Lie-algebra relations

[

]=±

[

−

]=2

, (16.55)

and act on the ψ

(z) monomials as

(z) = m ψ

(z),

(z) =



j(j + 1) − m(m ± 1)ψ

m±1

(z). (16.56)

This is the familiar action of the su(2) generators on |j, m basis states.

Exercise 16.1: Show that with respect to the inner product (16.44) we have

†

−

588 16 The geometry of fibre bundles

Coherent states and the Borel–Weil–Bott theorem

We now explain how the spin wavefunctions ψ

(z) can be understood as sections of a

holomorphic line bundle.

Suppose that we have a compact Lie group G and a unitary irreducible representation

g ∈ G (→ D

(g). Let |0 be the normalized highest (or lowest) weight state in the

representation space. Consider the states

|g=D

(g)|0, g|=0|

(g)

†

. (16.57)

The |g compose a family of generalized coherent states.

There is a continuous infinity

of the |g, and so they cannot constitute an orthonormal set on the finite-dimensional rep-

resentation space. The matrix-element orthogonality property (15.81), however, provides

us with a useful over-completeness relation:

I =

dim(J )

Vo l G



|gg|. (16.58)

The integral is over all of G, but many points in G give the same contribution. The

maximal torus, denoted by T , is the abelian subgroup of G obtained by exponentiating

elements of the Cartan algebra. Because any weight vector is a common eigenvector

of the Cartan algebra, elements of T leave |0 fixed up to a phase. The set of distinct

|g in the integral can therefore be identified with G/T . This coset space is always an

even-dimensional manifold, and thus a candidate phase space.

Consider, in particular, the spin-j representation of SU(2). The coset space G/T is

then SU(2)/U (1) * S

. We can write a general element of SU(2) as

U = exp(

) exp(θJ

) exp(γ J

−

) (16.59)

for some complex parameters

z, θ and γ which are functions of the three real coordinates

that parameterize SU(2). We let U act on the lowest-weight state |j, −j. The rightmost

factor has no effect on the lowest weight state, and the middle factor only multiplies it

by a constant. We therefore restrict our attention to the states

z=exp(zJ

)|j, −j, z|=j, −j|exp(zJ

−

) = (|z)

†

. (16.60)

These states are not normalized, but have the advantage that the z| are holomorphic in

the parameter z – i.e. they depend on z but not on

The set of distinct |

z can still be identified with the 2-sphere, and z,

z are its complex

stereographic coordinates. This identification is an example of a general property of

compact Lie groups:

G/T

∼

. (16.61)

A. Perelomov, Generalized Coherent States and their Applications (Springer-Verlag, 1986).

16.2 Physics examples 589

Here, G

is the complexification of G – the group G, but with its parameters allowed to

be complex – and B

is the Borel group whose Lie algebra consists of the Cartan algebra

together with the step-up ladder operators.

The inner product of two |

z states is

z



|z=(1 + zz



)

, (16.62)

and the eigenstates |j, m of J

and J

possess coherent state wavefunctions:

(1)

(z) ≡z|j, m=



2j!

(j − m)!(j + m)!

j+m

. (16.63)

We recognize these as our spin wavefunctions from the previous section.

The over-completeness relation can be written as

I =

2j + 1

2πi



z ∧ dz

(1 + zz)

2j+2

|zz|, (16.64)

and provides the inner product for the coherent-state wavefunctions. If ψ(z) =z|ψ 

and χ(z) =z|χ then

ψ|χ=

2j + 1

2πi



z ∧ dz

(1 + zz)

2j+2

ψ|zz|χ

2j + 1

2πi



z ∧ dz

(1 + zz)

2j+2

ψ(z)χ (z), (16.65)

which coincides with (16.44).

The wavefunctions ψ

(1)

(z) are singular at the north pole, where z =∞. Indeed, there

is no actual state ∞| because the phase of this putative limiting state would depend on

the direction from which we approach the point at infinity. We may, however, define a

second family of coherent states:

ζ 

= exp(ζ J

−

)|j, j,

ζ |=j, j|exp(ζ J

), (16.66)

and form the wavefunctions

(2)

(ζ ) =

ζ |j, m. (16.67)

These new states and wavefunctions are well defined in the vicinity of the north pole,

but singular near the south pole.

To find the relation between ψ

(2)

(ζ ) and ψ

(1)

(z) we note that the matrix identity



0 −1



z 1





−z

−1



−z 0

0 −z

−1



1 z

−1



, (16.68)

590 16 The geometry of fibre bundles

coupled with the faithfulness of the spin-

representation of SU(2), implies the relation

=w exp(zJ

) = exp (−z

−1

−

)(−z)

exp (z

−1

), (16.69)

where =w = exp(−iπ J

). We also note that

j, j|=w = (−1)

j, −j|, j, −j|=w =j, j|. (16.70)

Thus,

(1)

(z) =j, −j|e

−

|j, m

= (−1)

j, j|=w e

−

|j, m

= (−1)

j, j|e

−z

−1

−

(−z)

−1

|j, m

= (−1)

(−z)

j, j|e

−1

|j, m

= z

(2)

−1

). (16.71)

The transition function z

that relates ψ

(1)

(z) to ψ

(2)

(ζ ≡ 1/z) depends only on z.We

therefore say that the wavefunctions ψ

(1)

(z) and ψ

(2)

(ζ ) are the local components of

a global section ψ

↔|j, m of a holomorphic line bundle. The requirement that the

transition function and its inverse be holomorphic and single valued in the overlap of

the z and ζ coordinate patches forces 2j to be an integer. The ψ

form a basis for the

space of global holomorphic sections of this bundle.

Borel, Weil and Bott showed that any finite-dimensional representation of a semisim-

ple Lie group G can be realized as the space of global holomorphic sections of a line

bundle over G

. This bundle is constructed from the highest (or lowest) weight

vectors in the representation by a natural generalization of the method we have used

for spin. This idea has been extended by Witten and others to infinite-dimensional Lie

groups, where it can be used, for example, to quantize two-dimensional gravity.

Exercise 16.2: Normalize the states |

z, z|, by multiplying them by N = (1 +|z|

)

−j

Show that

z|J

|z=j

|z|

− 1

|z|

+ 1

z|J

|z=j

|z|

+ 1

z|J

−

|z=j

|z|

+ 1

thus confirming the identification of z,

z with the complex stereographic coordinates on

the sphere.

16.3 Working in the total space 591

16.3 Working in the total space

We have mostly considered a bundle to be a collection of mathematical objects and a

base-space to which they are attached, rather than treating the bundle as a geometric

object in its own right. In this section we demonstrate the advantages to be gained from

the latter viewpoint.

16.3.1 Principal bundles and associated bundles

The fibre bundles that arise in a gauge theory with Lie group G are called principal G-

bundles, and the fields and wavefunctions are sections of associated bundles. Aprincipal

G-bundle comprises the total space, which we here call P, together with the projection

π to the base space M . The fibre can be regarded as a copy of G, i.e.

π : P → M , π

−1

(x)

∼

G. (16.72)

Strictly speaking, the fibre is only required to be a homogeneous space on which G acts

freely and transitively on the right; x → xg. Such a set can be identified with G after

we have selected a fiducial point f

∈ F to be the group identity. There is no canonical

choice for f

and, if the bundle is twisted, there can be no globally smooth choice. This

is because a smooth choice for f

in the fibres above an open subset U ⊆ M makes P

locally into a product U × G. Being able to extend U to the entirety of M means that P

is trivial. We will, however, make use of local assignments f

(→ e to introduce bundle

coordinate charts in which P is locally a product, and therefore parametrized by ordered

pairs (x, g) with x ∈ U and g ∈ G.

To understand the bundles associated with P, it is simplest to define the sections of

the associated bundle. Let ϕ

(x, g) be a function on the total space P with a set of indices

i carrying some representation g (→ D(g) of G. We say that ϕ

(x, g) is a section of an

associated bundle if it varies in a particular way as we run up and down the fibres by

acting on them from the right with elements of G; we require that

(x, gh) = D

−1

)ϕ

(x, g). (16.73)

These sections can be thought of as wavefunctions for a particle moving in a gauge field

on the base-space. The choice of representation D plays the role of “charge”, and (16.73)

are the gauge transformations. Note that we must take h

−1

as the argument of D in order

for the transformation to be consistent under group multiplication:

(x, gh

) = D

−1

)ϕ

(x, gh

)

= D

−1

)ϕ

(x, g)

= D

−1

)ϕ

(x, g)

= D

((h

)

−1

)ϕ

(x, g). (16.74)

592 16 The geometry of fibre bundles

The construction of the associated bundle itself requires rather more abstraction.

Suppose that the matrices D(g) act on the vector space V . Then the total space P

of the associated bundle consists of equivalence classes of P × V under the relation

((x, g), v) ∼ ((x, gh), D(h

−1

)v) for all v ∈ V , (x, g) ∈ P and h ∈ G. The set of G-action

equivalence classes in a cartesian product A ×B is usually denoted by A ×

B. Our total

space is therefore

= P ×

V . (16.75)

We find it conceptually easier to work with the sections as defined above, rather than

with these equivalence classes.

16.3.2 Connections

Agauge field is a connection on a principal bundle. The formal definition of a connection

is a decomposition of the tangent space TP

of P at p ∈ P into a horizontal subspace

(P) and a vertical subspace V

(P). We require that V

(P) be the tangent space to

the fibres and H

(P) to be a complementary subspace, i.e. the direct sum should be the

whole tangent space

= H

(P) ⊕ V

(P). (16.76)

The horizontal subspaces must also be invariant under the push-forward induced from the

action on the fibres from the right of a fixed element of G. More formally, ifR[g] : P → P

acts to take p → pg, i.e. by R[g](x, g



) = (x, g



g), we require that

R[g]

∗

(P) = H

(P). (16.77)

Thus, we get to choose one horizontal subspace in each fibre, the rest being determined

by the right-invariance condition.

We now show how this geometric definition of a connection leads to parallel-transport.

We begin with a curve x(t) in the base-space. By solving the equation

˙g +

∂x

∂t

(x)g = 0, (16.78)

we can lift the curve x(t) to a new curve (x(t), g(t)) in the total space, whose tangent

is everywhere horizontal. This lifting operation corresponds to parallel-transporting the

initial value g(0) along the curve x(t) to get g(t). The A

= i

are a set of Lie-

algebra-valued functions that are determined by our choice of horizontal subspace. They

are defined so that the vector (δx, −A

δx

g) is horizontal for each small displacement

δx

in the tangent space of M . Here, −A

δx

g is to be understood as the displacement

that takes g → (1 −A

δx

)g. Because we are multiplying A in from the left, the lifted

curve can be slid rigidly up and down the fibres by the right action of any fixed group

element. The right-invariance condition is therefore automatically satisfied.

16.3 Working in the total space 593

The directional derivative along the lifted curve is

˙x

=˙x





∂

∂x



− A



, (16.79)

where R

is a right-invariant vector field on G, i.e. a differential operator on functions

defined on the fibres. The D

are a set of vector fields in TP. These covariant derivatives

span the horizontal subspace at each point p ∈ P, and have Lie brackets

, D

]=−F

µν

. (16.80)

Here, F

µν

is given in terms of the structure constants appearing in the Lie brackets

, R

]=f

µν

= ∂

− ∂

− f

. (16.81)

We can also write

µν

= ∂

− ∂

+[A

, A

] (16.82)

where F

µν

= i

µν

and [

]=if

Because the Lie bracket of the D

is a linear combination of the R

, it lies entirely

in the vertical subspace. Consequently, when F

µν

= 0 the D

are not in involution, so

Frobenius’ theorem tells us that the horizontal subspaces cannot fit together to form the

tangent spaces to a smooth foliation of P.

We now make contact with the more familiar definition of a covariant derivative.

We begin by recalling that right-invariant vector fields are derivatives that involve

infinitesimal multiplication from the left. Their definition is

(x, g) = lim

→0





(x, (1 + i

)g) − ϕ

(x, g)



, (16.83)

where [

]=if

As ϕ

(x, g) is a section of the associated bundle, we know how it varies when we

multiply group elements in on the right. We therefore write

(1 + i

)g = gg

−1

(1 + i

)g, (16.84)

and from this (and writing g for D(g) where it makes for compact notation) we find

(x, g) = lim

→0



−1

(1 − i

)g)ϕ

(x, g) − ϕ

(x, g)

/

=−D

−1

)(i

)

(g)ϕ

(x, g)

=−i(g

−1

. (16.85)

594 16 The geometry of fibre bundles

Here, i(

)

is the matrix representing the Lie algebra generator i

in the representation

g (→ D(g). Acting on sections, we therefore have

ϕ =



∂



+ (g

−1

g)ϕ. (16.86)

This still does not look too familiar, because the derivatives with respect to x

are being

taken at fixed g. We normally fix a gauge by making a choice of g = σ(x) for each

. The conventional wavefunction ϕ(x) is then ϕ(x, σ(x)). We can use ϕ(x, σ(x )) =

−1

(x)ϕ(x, e), to obtain

∂

ϕ =



∂





∂

−1

σϕ =



∂



−



−1

∂

ϕ. (16.87)

From this, we get a derivative

∇

def

= ∂

+ (σ

−1

σ + σ

−1

∂

σ) = ∂

+ A

(16.88)

on functions ϕ(x)

def

= ϕ(x, σ(x )) defined (locally) on the base-space M . This is the

conventional covariant derivative, now containing gauge fields

(x) = σ

−1

σ + σ

−1

∂

σ (16.89)

that are gauge transformations of our g-independent A

. The derivative has been

constructed so that

∇

ϕ(x) = D

ϕ(x, g)

g=σ(x)

, (16.90)

and has commutator

[∇

, ∇

]=σ

−1

µν

σ = F

µν

. (16.91)

Note the sign change vis-à-vis Equation (16.80).

It is the curvature tensor F

µν

that we have met previously. Recall that it provides a

Lie-algebra-valued 2-form

F =

µν

= dA + A

(16.92)

on the base-space. The connection A = A

is a 1-form on the base space, and both F

and A have been defined only in the region U ⊂ M in which the smooth gauge-choice

section σ(x) has been selected.