SY M = ’ d4 x Fµν Fa ,

a µν

(121)

4

where Fµν is the ¬eld strength of the Yang“Mills vector potential Aa :

a

µ

Fµν = ‚µ Aa ’ ‚ν Aa ’ gfbc a Ab Ac .

a

(122)

ν µ µν

122 J.W. van Holten

Here g is the coupling constant, and the coe¬cients fbc a are the structure con-

stant of a compact Lie algebrag with (anti-hermitean) generators Ta :

[Ta , Tb ] = fabc Tc . (123)

The Yang“Mills action (121) is invariant under (in¬nitesimal) local gauge trans-

formations with parameters Λa (x):

δAa = (Dµ Λ)a = ‚µ Λa ’ gfbc a Ab Λc , (124)

µ µ

a

under which the ¬eld strength Fµν transforms as

δFµν = gfbc a Λb Fµν .

a c

(125)

To obtain a canonical description of the theory, we compute the momenta

’Ea , µ = i = (1, 2, 3);

i

δSY M

= ’Fa =

µ 0µ

πa = (126)

0, µ = 0.

a

δ‚0 Aµ

Clearly, the last equation is a constraint of the type we have encountered before;

indeed, the time component of the vector ¬eld, Aa , plays the same role of La-

0

grange mutiplier for a Gauss-type constraint as the scalar potential φ = A0 in

electro-dynamics, to which the theory reduces in the limit g ’ 0. This is brought

out most clearly in the hamiltonian formulation of the theory, with action

2

‚Aa

d3 x ’E a · ’ HY M

SY M = dt ,

‚t

1

(127)

1

(E 2 + B 2 ) + Aa (D · E)a .

d3 x

HY M = a a 0

2

Here we have introduced the notation B a for the magnetic components of the

¬eld strength:

1

a a

Bi = µijk Fjk . (128)

2

In (127) we have left out all terms involving the time-component of the momen-

tum, since they vanish as a result of the constraint πa = 0, cf. (126). Now Aa

0

0

appearing only linearly, its variation leads to another constraint

(D · E)a = ∇ · E a ’ gfbc a Ab · E c = 0. (129)

As in the other theories we have encountered so far, the constraints come in pairs:

one constraint, imposed by a Lagrange multiplier, restricts the physical degrees

of freedom; the other constraint is the vanishing of the momentum associated

with the Lagrange multiplier.

To obtain the equations of motion, we need to specify the Poisson brackets:

{Aa (x, t), Ejb (y, t)} = ’δij δb δ 3 (x’y),

a

Aa , (x, t), πb (y, t) = δij δb δ 3 (x’y),

0 a

i 0

(130)

Aspects of BRST Quantization 123

or in quasi-covariant notation

Aa (x, t), πb (y, t) = δµ δb δ 3 (x ’ y).

ν νa

(131)

µ

Provided the gauge parameter vanishes su¬ciently fast at spatial in¬nity, the

canonical action is gauge invariant:

2

‚Λa

δSY M = ’ d x ∇ · Ea

3

dt 0. (132)

‚t

1

Therefore it is again straightforward to construct the generator for the local

gauge transformations:

d3 x ’δAa · E a + δAa πa

0

G[Λ] = 0

(133)

d3 x Λa (D · E)a + πa (D0 Λ)a .

d3 x πa (Dµ Λ)a

µ 0

=

The new aspect of the gauge generators in the case of Yang“Mills theory is, that

the constraints satisfy a non-trivial Poisson bracket algebra:

{G[Λ1 ], G[Λ2 ]} = G[Λ3 ], (134)

where the parameter on the right-hand side is de¬ned by

Λ3 = gfbc a Λb Λc . (135)

12

We can also write the physical part of the constraint algebra in a local form;

indeed, let

Ga (x) = (D · E)a (x). (136)

Then a short calculation leads to the result

{Ga (x, t), Gb (y, t)} = gfabc Gc (x, t) δ 3 (x ’ y). (137)

We observe, that the condition G[Λ] = 0 is satis¬ed for arbitrary Λ(x) if and

only if the two local constraints hold:

(D · E)a = 0, 0

πa = 0. (138)

This is su¬cient to guarantee that {G[Λ], H} = 0 holds as well. Together with

the closure of the algebra of constraints (134) this guarantees that the constraints

G[Λ] = 0 are consistent both with the dynamics and among themselves.

Equation (138) is the generalization of the transversality condition (116)

and removes the same number of momenta (electric ¬eld components) from the

physical phase space. Unlike the case of electrodynamics however, it is non-linear

and cannot be solved explicitly. Moreover, the constraint does not determine in

closed form the conjugate co-ordinate (the combination of gauge potentials) to

be removed from the physical phase space with it. A convenient possibility to

impose in classical Yang“Mills theory is the transversality condition ∇ · Aa = 0,

which removes the correct number of components of the vector potential and

still respects the rigid gauge invariance (with constant parameters Λa ).

124 J.W. van Holten

1.8 The Relativistic String

As the last example in this section we consider the massless relativistic (bosonic)

string, as described by the Polyakov action

1√

’ ’gg ab ‚a X µ ‚b Xµ ,

d2 ξ

Sstr = (139)

2

where ξ a = (ξ 0 , ξ 1 ) = („, σ) are co-ordinates parametrizing the two-dimensional

world sheet swept out by the string, gab is a metric on the world sheet, with g its

determinant, and X µ (ξ) are the co-ordinates of the string in the D-dimensional

embedding space-time (the target space), which for simplicity we take to be ¬‚at

(Minkowskian). As a generally covariant two-dimensional ¬eld theory, the action

is manifestly invariant under reparametrizations of the world sheet:

‚ξ c ‚ξ d

Xµ (ξ ) = Xµ (ξ), gab (ξ ) = gcd (ξ) . (140)

‚ξ a ‚ξ b

The canonical momenta are

√

δSstr δSstr

= ’ ’g ‚ 0 Xµ ,

Πµ = πab = = 0. (141)

δ‚0 X µ δ‚0 g ab

The latter equation brings out, that the inverse metric g ab , or rather the com-

√

bination hab = ’gg ab , acts as a set of Lagrange multipliers, imposing the

vanishing of the symmetric energy-momentum tensor:

2 δSstr 1

Tab = √ = ’‚a X µ ‚b Xµ + gab g cd ‚c X µ ‚d Xµ = 0. (142)

’g δg ab 2

Such a constraint arises because of the local reparametrization invariance of the

action. Note, however, that the energy-momentum tensor is traceless:

Ta a = g ab Tab = 0. (143)

and as a result it has only two independent components. The origin of this reduc-

tion of the number of constraints is the local Weyl invariance of the action (139)

gab (ξ) ’ gab (ξ) = eΛ(ξ) gab (ξ), X µ (ξ) ’ X µ (ξ) = X µ (ξ),

¯

¯ (144)

¯

which leaves hab invariant: hab = hab . Indeed, hab itself also has only two indepen-

dent components, as the negative of its determinant is unity: ’h = ’ det hab = 1.

The hamiltonian is obtained by Legendre transformation, and taking into

account π ab = 0, it reads

√

1

’g ’g 00 [‚0 X]2 + g 11 [‚1 X]2 + π ab ‚0 gab

H= dσ

2

(145)

dσ T 0 + π ab ‚0 gab .

= 0

Aspects of BRST Quantization 125

The Poisson brackets are

{X µ („, σ), Πν („, σ )} = δν δ(σ ’ σ ),

µ

(146)

1 cd

δa δb + δa δb δ(σ ’ σ ).

cd dc

gab („, σ), π („, σ ) =

2

The constraints (142) are most conveniently expressed in the hybrid forms (using

relations g = g00 g11 ’ g01 and g11 = gg 00 ):

2

1

gT 00 = ’T11 = Π 2 + [‚1 X]2 = 0,

2 (147)

√

’g T 0 = Π · ‚1 X = 0.

1

These results imply, that the hamiltonian (145) actually vanishes, as in the case

of the relativistic particle. The reason is also the same: reparametrization invari-

ance, now on a two-dimensional world sheet rather than on a one-dimensional

world line.

The in¬nitesimal form of the transformations (140) with ξ = ξ ’ Λ(ξ) is

√

1

δX µ (ξ) = X µ (ξ) ’ X µ (ξ) = Λa ‚a X µ = ’g Λ0 Π µ + Λ1 ‚σ X µ ,

00

gg

δgab (ξ) = (‚a Λc )gcb + (‚b Λc )gac + Λc ‚c gab = Da Λb + Db Λa ,

(148)

where we use the covariant derivative Da Λb = ‚a Λb ’ “ab Λc . The generator of

c

these transformations as constructed by our standard procedure now becomes

1 0√

dσ Λa ‚a X · Π + Λ ’g g ab ‚a X · ‚b X + π ab (Da Λb + Db Λa )

G[Λ] =

2

√

dσ ’ ’g Λa T 0 + 2π ab Da Λb .

= a

(149)

which has to vanish in order to represent a canonical symmetry: the constraint

G[Λ] = 0 summarizes all constraints introduced above. The brackets of G[Λ] now

take the form

{X µ , G[Λ]} = Λa ‚a X µ = δX µ , {gab , G[Λ]} = Da Λb + Db Λa = δgab , (150)

and, in particular,