same actual history of the system.

The BRST-formalism [1,2] has been developed speci¬cally to deal with such

situations. The roots of this approach to constrained dynamical systems are

found in attempts to quantize General Relativity [3,4] and Yang“Mills theo-

ries [5]. Out of these roots has grown an elegant and powerful framework for

dealing with quite general classes of constrained systems using ideas borrowed

from algebraic geometry.1

In these lectures we are going to study some important examples of con-

strained dynamical systems, and learn how to deal with them so as to be able

to extract relevant information about their observable behaviour. In view of the

applications to fundamental physics at microscopic scales, the emphasis is on

quantum theory. Indeed, this is the domain where the full power and elegance

of our methods become most apparent. Nevertheless, many of the ideas and

1

Some reviews can be found in [6“14].

J.W. van Holten, Aspects of BRST Quantization, Lect. Notes Phys. 659, 99“166 (2005)

http://www.springerlink.com/ c Springer-Verlag Berlin Heidelberg 2005

100 J.W. van Holten

results are applicable in classical dynamics as well, and wherever possible we

treat classical and quantum theory in parallel. Our conventions and notations

are summarized at the end of these notes.

1.1 Dynamical Systems with Constraints

Before delving into the general theory of constrained systems, it is instructive to

consider some examples; they provide a background for both the general theory

and the applications to follow later.

The Relativistic Particle. The motion of a relativistic point particle is spec-

i¬ed completely by its world line xµ („ ), where xµ are the position co-ordinates

of the particle in some ¬xed inertial frame, and „ is the proper time, labeling

successive points on the world line. All these concepts must and can be properly

de¬ned; in these lectures I trust you to be familiar with them, and my presen-

tation only serves to recall the relevant notions and relations between them.

In the absence of external forces, the motion of a particle with respect to an

inertial frame satis¬es the equation

d2 xµ

= 0. (1)

d„ 2

It follows that the four-velocity uµ = dxµ /d„ is constant. The complete solution

of the equations of motion is

xµ („ ) = xµ (0) + uµ „. (2)

A most important observation is, that the four-velocity uµ is not completely

arbitrary, but must satisfy the physical requirement

uµ uµ = ’c2 , (3)

where c is a universal constant, equal to the velocity of light, for all particles

irrespective of their mass, spin, charge or other physical properties. Equivalently,

(3) states that the proper time is related to the space-time interval travelled by

c2 d„ 2 = ’dxµ dxµ = c2 dt2 ’ dx 2 , (4)

independent of the physical characteristics of the particle.

The universal condition (3) is required not only for free particles, but also

in the presence of interactions. When subject to a four-force f µ the equation of

motion (1) for a relativistic particle becomes

dpµ

= f µ, (5)

d„

where pµ = muµ is the four-momentum. Physical forces “ e.g., the Lorentz force

in the case of the interaction of a charged particle with an electromagnetic ¬eld

“ satisfy the condition

p · f = 0. (6)

Aspects of BRST Quantization 101

This property together with the equation of motion (5) are seen to imply that

p2 = pµ pµ is a constant along the world line. The constraint (3) is then expressed

by the statement that

p2 + m2 c2 = 0, (7)

with c the same universal constant. Equation (7) de¬nes an invariant hypersur-

face in momentum space for any particle of given rest mass m, which the particle

can never leave in the course of its time-evolution.

Returning for simplicity to the case of the free particle, we now show how

the equation of motion (1) and the constraint (3) can both be derived from a

single action principle. In addition to the co-ordinates xµ , the action depends on

an auxiliary variable e; it reads

2

1 dxµ dxµ

m

’ ec2 d».

µ

S[x ; e] = (8)

2 e d» d»

1

Here » is a real parameter taking values in the interval [»1 , »2 ], which is mapped

by the functions xµ (») into a curve in Minkowski space with ¬xed end points

(xµ , xµ ), and e(») is a nowhere vanishing real function of » on the same interval.

1 2

Before discussing the equations that determine the stationary points of the

action, we ¬rst observe that by writing it in the equivalent form

2

dxµ dxµ

m

’ c2 ed»,

µ

S[x ; e] = (9)

2 ed» ed»

1

it becomes manifest that the action is invariant under a change of parametriza-

tion of the real interval » ’ » (»), if the variables (xµ , e) are transformed simul-

taneously to (x µ , e ) according to the rule

x µ (» ) = xµ (»), e (» ) d» = e(») d». (10)

Thus, the co-ordinates xµ (») transform as scalar functions on the real line R1 ,

whilst e(») transforms as the (single) component of a covariant vector (1-form)

in one dimension. For this reason, it is often called the einbein. For obvious

reasons, the invariance of the action (8) under the transformations (10) is called

reparametrization invariance.

The condition of stationarity of the action S implies the functional di¬erential

equations

δS δS

= 0, = 0. (11)

δxµ δe

These equations are equivalent to the ordinary di¬erential equations

2

1 dxµ 1 dxµ

1d

= ’c2 .

= 0, (12)

e d» e d» e d»

The equations coincide with the equation of motion (1) and the constraint (3)

upon the identi¬cation

d„ = ed», (13)

102 J.W. van Holten

a manifestly reparametrization invariant de¬nition of proper time. Recall, that

after this identi¬cation the constraint (3) automatically implies (4), hence this

de¬nition of proper time coincides with the standard geometrical one.

Remark. One can use the constraint (12) to eliminate e from the action; with

the choice e > 0 (which implies that „ increases with increasing ») the action

reduces to the Einstein form

2 2

dxµ dxµ

SE = ’mc ’ d» = ’mc2 d„,

d» d»

1 1

where d„ given by (4). As a result one can deduce that the solutions of the equa-

tions of motion are time-like geodesics in Minkowski space. The solution with

e < 0 describes particles for which proper time runs counter to physical labora-

tory time; this action can therefore be interpreted as describing anti-particles of

the same mass.

The Electro-magnetic Field. In the absence of charges and currents the

evolution of electric and magnetic ¬elds (E, B) is described by the equations

‚E ‚B

= ∇ — B, = ’∇ — E. (14)

‚t ‚t

Each of the electric and magnetic ¬elds has three components, but only two of

them are independent: physical electro-magnetic ¬elds in vacuo are transverse

polarized, as expressed by the conditions

∇ · E = 0, ∇ · B = 0. (15)

The set of the four equations (14) and (15) represents the standard form of

Maxwell™s equations in empty space.

Repeated use of (14) yields

‚2E

= ’∇ — (∇ — E) = ∆E ’ ∇∇ · E, (16)

‚t2

and an identical equation for B. However, the transversality conditions (15)

simplify these equations to the linear wave equations

2E = 0, 2B = 0, (17)

with 2 = ∆ ’ ‚t2 . It follows immediately that free electromagnetic ¬elds satisfy

the superposition principle and consist of transverse waves propagating at the

speed of light (c = 1, in natural units).

Again both the time evolution of the ¬elds and the transversality constraints

can be derived from a single action principle, but it is a little bit more subtle

than in the case of the particle. For electrodynamics we only introduce auxiliary

Aspects of BRST Quantization 103

¬elds A and φ to impose the equation of motion and constraint for the electric

¬eld; those for the magnetic ¬eld then follow automatically. The action is

2

SEM [E, B; A, φ] = dt LEM (E, B; A, φ),

1

(18)

1 ‚E

d3 x ’ E2 ’B2 +A· ’∇—B ’ φ∇ · E .

LEM =

2 ‚t

Obviously, stationarity of the action implies

δS ‚E δS

’ ∇ — B = 0, = ’∇ · E = 0,

= (19)

δA ‚t δφ

reproducing the equation of motion and constraint for the electric ¬eld. The

other two stationarity conditions are

δS ‚A δS

= ’E ’ + ∇φ = 0, = B ’ ∇ — A = 0, (20)

δE ‚t δB

or equivalently

‚A

E=’ + ∇φ, B = ∇ — A. (21)

‚t

The second equation (21) directly implies the transversality of the magnetic ¬eld:

∇ · B = 0. Taking its time derivative one obtains

‚B ‚A

=∇— ’ ∇φ = ’∇ — E, (22)

‚t ‚t

where in the middle expression we are free to add the gradient ∇φ, as ∇—∇φ = 0

identically.

An important observation is, that the expressions (21) for the electric and

magnetic ¬elds are invariant under a rede¬nition of the potentials A and φ of

the form

‚Λ

A = A + ∇Λ, φ =φ+ , (23)

‚t

where Λ(x) is an arbitrary scalar function. The transformations (23) are the

well-known gauge transformations of electrodynamics.

It is easy to verify, that the Lagrangean LEM changes only by a total time

derivative under gauge transformations, modulo boundary terms which vanish

if the ¬elds vanish su¬ciently fast at spatial in¬nity:

d

LEM = LEM ’ d3 x Λ∇ · E. (24)

dt

As a result the action SEM itself is strictly invariant under gauge transformations,

provided d3 xΛ∇·E|t1 = d3 xΛ∇·E|t2 ; however, no physical principle requires

such strict invariance of the action. This point we will discuss later in more detail.

104 J.W. van Holten

We ¬nish this discussion of electro-dynamics by recalling how to write the

equations completely in relativistic notation. This is achieved by ¬rst collecting

the electric and magnetic ¬elds in the anti-symmetric ¬eld-strength tensor

«

0 ’E1 ’E2 ’E3

¬ E 0 B3 ’B2 ·

=¬ 1 ·

Fµν E2 ’B3 0 B1 , (25)

E3 B2 ’B1 0

and the potentials in a four-vector:

Aµ = (φ, A). (26)

Equations (21) then can be written in covariant form as

Fµν = ‚µ Aν ’ ‚ν Aµ , (27)

with the electric ¬eld equations (19) reading

‚µ F µν = 0. (28)

The magnetic ¬eld equations now follow trivially from (27) as

µµνκ» ‚ν Fκ» = 0. (29)

Finally, the gauge transformations can be written covariantly as

Aµ = Aµ + ‚µ Λ. (30)

The invariance of the ¬eld strength tensor Fµν under these transformations fol-