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6.2 Effective constraints

In Section 6.1 we describe how effective equations for an unconstrained system are obtained. Gravity as a fully constrained system requires an extension of the methods to dynamics provided by a Hamiltonian constraint ˆC rather than a Hamiltonian Hˆ. In this case, the expectation value CQ = ⟨ ˆC ⟩, again interpreted as a function on the infinite dimensional quantum phase space, plays a role similar to HQ. Thus, the main constraint equation of the effective theory is obtained through expectation values of the constraint operator, which will play a role in applications to loop quantum cosmology. However, several new features arise for constrained systems.

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 ⟨ ˆC ⟩ but also ⟨Cˆn ⟩ and even expressions such as ⟨O ˆCˆn ⟩ for arbitrary powers n > 0 and operators ˆ O 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 Ga,n and for canonical variables take the form

2 G0,2G2,2 − (G1,2)2 ≥ ℏ-. (74 ) 4
Some of the effective constraints will constrain the quantum variables and could violate the original uncertainty relations, corresponding to the fact that the physical inner product would define a new Hilbert space and thus reduced quantum phase space structure. Alternatively, it is often more convenient to respect uncertainty relations, but allow complex solutions to the quantum constraints. By definition, quantum variables refer to expectation values of totally symmetric operators, which should be real. For a constrained system, however, this would refer to the kinematical Hilbert space structure. Violations of kinematical reality then indicate that the physical Hilbert space structure differs from the kinematical one. In fact, one can ensure the physical inner product by requiring that all quantum variables left after solving the quantum constraints are real. This has been shown to provide the correct results in examples, and is much easier to implement than finding an explicit inner product for states in a Hilbert space representation.

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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