Kiefer C. Quantum gravity

and are compatible with the time evolution,
PARTICLE SYSTEMS 77
{φ a ,H} = d b a φ b . (3.17)
Constraints which do not obey these relations are called second-class constraints.
They play a role, for example, in supergravity; cf. Section 5.3.6.
We now add the first-class constraints to the action with Lagrange multipliers
λ a ,
∫
S = dτ (p ˙q − H − λ a φ a ) . (3.18)
(In the example given by (3.10) one has instead of −H − λ a φ a only the term
−NH S in the action, H S being the only constraint.) Therefore, the time evolution
of a function A(q, p) reads
Ȧ(q, p) ={A, H} + λ a {A, φ a } . (3.19)
The Lagrange parameters λ a , therefore, introduce an arbitrariness into the time
evolution. In fact, first-class constraints generate gauge transformations: expanding
A(q(τ),p(τ)) around τ = 0 up to order ∆τ for two different values λ (1)
a and
and performing the difference δA, one obtains the ‘gauge transformation’
λ (2)
a
δA = ɛ a {A, φ a } , (3.20)
where ɛ a =∆τ(λ (1)
a (0) − λ (2)
a (0)). The constraints (3.15) define a hypersurface
Γ c in phase space, the constraint hypersurface, and generate the gauge transformations
(3.20) on this hypersurface. 1 The sets of gauge equivalent configurations
are also called gauge orbits.
Functions A(q, p) forwhich{A, φ a }≈0 holds are often called ‘observables’
because they do not change under a gauge transformation. It must be emphasized
that there is no apriorirelation of these observables to observables in an
operational sense. This notion had been introduced by Bergmann in the hope
that these quantities might play the role of the standard observables in quantum
theory.
In order to select one physical representative amongst all equivalent configurations,
one frequently employs ‘gauge conditions’. This is important, for
example, in path-integral quantization; see Section 2.2.3. A gauge should be
chosen in such a way that there is no further gauge freedom left and that any
configuration can be transformed in one satisfying the gauge. The first condition
is sometimes violated (‘Gribov ambiguities’), but this is irrelevant for infinitesimal
gauge transformations. If one identifies all points on the same gauge orbit,
one arrives at the reduced phase space of the theory. In the general case, the reduced
phase space is not a cotangent bundle, that is, one cannot identify which
variables are the q’s and which are the p’s.
1 For second-class constraints, which do not fulfil (3.16), the Lagrange parameters can be
determined by choosing A = φ b in (3.19) and demanding ˙φ b =0.