First, it is not sufficient to consider a single effective constraint for every constraint operator. This can
easily be seen because one (first class) constraint removes one pair of canonical variables. The quantum
phase space, however, has infinitely many quantum variables for each classical canonical pair. If only one
effective constraint were imposed, the quantum variables of the constrained pair would remain in the system
and possibly couple to other variables, although they correspond to gauge degrees of freedom. Additional
effective constraints are easy to find: if the quantum phase space is to be restricted to physical states, then
not only but also
and even expressions such as
for arbitrary powers
and
operators
must vanish. This provides infinitely many constraints on the quantum phase space, from
which one should select a complete subset. Doing so can depend on the precise form of the constraint
and may be difficult in specific cases, but it has been shown to give the correct reduction in
examples.
Secondly, for constraints, we have to keep in mind the anomaly issue. If one has several constraint operators, which are first class, then the quantum constraints obtained as their expectation values are also first class. Moreover, one can define complete sets of higher-power constraints, which preserve the first-class nature. Thus, the whole system of infinitely many quantum constraints is consistent. However, this does not automatically extend to the effective constraints obtained after truncating the infinitely many quantum variables. For the truncated, effective constraints one still has to make sure that no anomalies arise to the order of the truncation. Moreover, in complicated theories such as loop quantum gravity it is not often clear if the original set of constraint operators was first class. In such a case one can still proceed to compute effective constraints, since potential inconsistencies due to anomalies would only arise when one tries to solve them. After having computed effective constraints, one can then analyze the anomaly issue at this phase space stage, which is much easier than looking at the full anomaly problem for the constraint operators. One can then consistently define effective theories and see whether anomaly freedom allows non-trivial quantum corrections of a certain type. By proceeding to higher orders of the truncation, tighter and tighter conditions will be obtained in approaching the non-truncated quantum theory. (See Section 6.5.4 for applications.)
The third difference is that we now have to deal with the physical inner-product issue. Since the original
constraints are defined on the kinematical Hilbert space, which is used to define the unconstrained quantum
phase space, solving the effective constraints could change the phase-space structure. This is most easily
seen for uncertainty relations, which provide inequalities for the second-order quantum variables and
for canonical variables take the form
Thus, there are promising prospects for the usefulness of effective techniques in quantum gravity, which can even address anomaly and physical inner-product issues.
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