Remark. Equations (27)“(29) can also be derived from the action

1 µν

F Fµν ’ F µν ‚µ Aν .

d4 x

Scov =

4

This action is equivalent to SEM modulo a total divergence. Eliminating Fµν as

an independent variable gives back the usual standard action

1

S[Aµ ] = ’ d4 x F µν (A)Fµν (A),

4

with Fµν (A) given by the right-hand side of (27).

Aspects of BRST Quantization 105

1.2 Symmetries and Noether™s Theorems

In the preceding section we have presented two elementary examples of sys-

tems whose complete physical behaviour was described conveniently in terms

of one or more evolution equations plus one or more constraints. These con-

straints are needed to select a subset of solutions of the evolution equation as

the physically relevant solutions. In both examples we found, that the full set of

equations could be derived from an action principle. Also, in both examples the

additional (auxiliary) degrees of freedom, necessary to impose the constraints,

allowed non-trivial local (space-time dependent) rede¬nitions of variables leaving

the lagrangean invariant, at least up to a total time-derivative.

The examples given can easily be extended to include more complicated

but important physical models: the relativistic string, Yang“Mills ¬elds, and

general relativity are all in this class. However, instead of continuing to produce

more examples, at this stage we turn to the general case to derive the relation

between local symmetries and constraints, as an extension of Noether™s well-

known theorem relating (rigid) symmetries and conservation laws.

Before presenting the more general analysis, it must be pointed out that

our approach distinguishes in an important way between time- and space-like

dimensions; indeed, we have emphasized from the start the distinction between

equations of motion (determining the behaviour of a system as a function of

time) and constraints, which impose additional requirements. e.g. restricting the

spatial behaviour of electro-magnetic ¬elds. This distinction is very natural in

the context of hamiltonian dynamics, but potentially at odds with a covariant

lagrangean formalism. However, in the examples we have already observed that

the non-manifestly covariant treatment of electro-dynamics could be translated

without too much e¬ort into a covariant one, and that the dynamics of the rela-

tivistic particle, including its constraints, was manifestly covariant throughout.

In quantum theory we encounter similar choices in the approach to dynam-

ics, with the operator formalism based on equal-time commutation relations

distinguishing space- and time-like behaviour of states and observables, whereas

the covariant path-integral formalism allows treatment of space- and time-like

dimensions on an equal footing; indeed, upon the analytic continuation of the

path-integral to euclidean time the distinction vanishes altogether. In spite of

these di¬erences, the two approaches are equivalent in their physical content.

In the analysis presented here we continue to distinguish between time and

space, and between equations of motion and constraints. This is convenient as it

allows us to freely employ hamiltonian methods, in particular Poisson brackets

in classical dynamics and equal-time commutators in quantum mechanics. Nev-

ertheless, as we hope to make clear, all applications to relativistic models allow

a manifestly covariant formulation.

Consider a system described by generalized coordinates q i (t), where i labels the

complete set of physical plus auxiliary degrees of freedom, which may be in¬nite

in number. For the relativistic particle in n-dimensional Minkowski space the

q i (t) represent the n coordinates xµ (») plus the auxiliary variable e(») (some-

times called the ˜einbein™), with » playing the role of time; for the case of a

106 J.W. van Holten

¬eld theory with N ¬elds •a (x; t), a = 1, ..., N , the q i (t) represent the in¬nite

set of ¬eld amplitudes •a (t) at ¬xed location x as function of time t, i.e. the

x

dependence on the spatial co-ordinates x is included in the labels i. In such a

case summation over i is understood to include integration over space.

Assuming the classical dynamical equations to involve at most second-order

time derivatives, the action for our system can now be represented quite generally

by an integral

2

i

L(q i , q i ) dt,

S[q ] = ™ (31)

1

where in the case of a ¬eld theory L itself is to be represented as an integral of

some density over space. An arbitrary variation of the co-ordinates leads to a

variation of the action of the form

2

2

‚L d ‚L ‚L

’

i i

δS = dt δq + δq , (32)

‚q i dt ‚ q i ‚ qi

™ ™

1 1

with the boundary terms due to an integration by parts. As usual we de¬ne

generalized canonical momenta as

‚L

pi = . (33)

‚ qi

™

From (32) two well-known important consequences follow:

- the action is stationary under variations vanishing at initial and ¬nal times:

δq i (t1 ) = δq i (t2 ) = 0, if the Euler“Lagrange equations are satis¬ed:

dpi d ‚L ‚L

= = i. (34)

dt ‚ q i

dt ™ ‚q

i

- let qc (t) and its associated momentum pc i (t) represent a solution of the Euler“

i

Lagrange equations; then for arbitrary variations around the classical paths qc (t)

in con¬guration space: q i (t) = qc (t) + δq i (t), the total variation of the action is

i

2

δSc = δq i (t)pc i (t) . (35)

1

We now de¬ne an in¬nitesimal symmetry of the action as a set of continuous

transformations δq i (t) (smoothly connected to zero) such that the lagrangean L

transforms to ¬rst order into a total time derivative:

‚L ‚L dB

δL = δq i + δqi i =

™ , (36)

i

‚q ‚q

™ dt

where B obviously depends in general on the co-ordinates and the velocities, but

also on the variation δq i . It follows immediately from the de¬nition that

2

δS = [B]1 . (37)

Observe, that according to our de¬nition a symmetry does not require the action

to be invariant in a strict sense. Now comparing (35) and (37) we establish the

Aspects of BRST Quantization 107

result that, whenever there exists a set of symmetry transformations δq i , the

physical motions of the system satisfy

2

δq i pc i ’ Bc = 0. (38)

1

Since the initial and ¬nal times (t1 , t2 ) on the particular orbit are arbitrary,

the result can be stated equivalently in the form of a conservation law for the

quantity inside the brackets.

To formulate it more precisely, let the symmetry variations be parametrized

by k linearly independent parameters ± , ± = 1, ..., k, possibly depending on

time:

(n) ±

δq i = Ri [±] = ±

R± + ™± R± + ... +

(0)i (1)i

R± i + ...,

(n)

(39)

(n)

±

where denotes the nth time derivative of the parameter. Correspondingly,

the lagrangean transforms into the derivative of a function B[ ], with

(n) ±

±

B± + ™± B± + ... +

(0) (1) (n)

B[ ] = B± + .... (40)

With the help of these expressions we de¬ne the ˜on shell™ quantity2

G[ ] = pc i Rc [ ] ’ Bc [ ]

i

(41)

(n)

± (0) (1) ± (n)

™ ± G±

= G± + + ... + G± + ...,

(n) (n) i (n)

with component by component G± = pc i Rc ± ’ Bc ± . The conservation law

(38) can now be stated equivalently as

dG[ ] (n)

™± ™ ™

±

G(0) + ™± G(0) + G(1) + ... + ± G(n’1) + G(n) + ... = 0. (42)

= ± ± ± ±

dt

We can now distinguish various situations, of which we consider only the two

extreme cases here. First, if the symmetry exists only for = constant (a rigid

(n)

symmetry), then all time derivatives of vanish and G± ≡ 0 for n ≥ 1, whilst

for the lowest component

G(0) = g± = constant, ±

G[ ] = g± , (43)

±

as de¬ned on a particular classical trajectory (the value of g± may be di¬erent

on di¬erent trajectories). Thus, rigid symmetries imply constants of motion; this

is Noether™s theorem.

Second, if the symmetry exists for arbitrary time-dependent (t) (a local sym-

metry), then (t) and all its time derivatives at the same instant are independent.

2

An ˜on shell™ quantity is a quantity de¬ned on a classical trajectory.

108 J.W. van Holten

As a result

™ (0)

G± = 0,

™ (1) (0)

G± = ’G± ,

(44)

...

™ (n) (n’1)

G± = ’G± ,

...

Now in general the transformations (39) do not depend on arbitrarily high-order

derivatives of , but only on a ¬nite number of them: there is some ¬nite N

(n)

such that R± = 0 for n ≥ N . Typically, transformations depend at most on

(n)

the ¬rst derivative of , and R± = 0 for n ≥ 2. In general, for any ¬nite N

all quantities R(n) i , B (n) , G(n) then vanish identically for n ≥ N . But then

(n)

G± = 0 for n = 0, ..., N ’ 1 as well, as a result of (44). Therefore G[ ] = 0 at

all times. This is a set of constraints relating the coordinates and velocities on

a classical trajectory. Moreover, as dG/dt = 0, these constraints have the nice

property that they are preserved during the time-evolution of the system.

The upshot of this analysis therefore is that local symmetries imply time-

independent constraints. This result is sometimes referred to as Noether™s second

theorem.

Remark. If there is no upper limit on the order of derivatives in the transfor-

mation rule (no ¬nite N ), one reobtains a conservation law

±

G[ ] = g± (0) = constant.

(n)

To show this, observe that G± = ((’t)n /n!) g± , with g± a constant; then com-

parison with the Taylor expansion for (0) = (t ’ t) around (t) leads to the

above result.

Group Structure of Symmetries. To round o¬ our discussion of symme-

tries, conservation laws, and constraints in the lagrangean formalism, we show

that symmetry transformations as de¬ned by (36) possess an in¬nitesimal group

structure, i.e. they have a closed commutator algebra (a Lie algebra or some

generalization thereof). The proof is simple. First observe, that performing a

second variation of δL gives

‚2L 2

i ‚L ‚L

j i j

+ (δ2 δ1 q i ) i

δ2 δ 1 L = δ 2 q δ 1 q + δ2 q δ1 q

™

j ‚q i j ‚q i

‚q ‚q

™ ‚q

(45)

2 2

‚L ‚L ‚L d(δ2 B1 )

+ δ2 q j δ1 q i + δ2 q j δ1 q i j i + (δ2 δ1 q i ) i =

™ ™ ™ ™ .