On the geometry of quantum constrained systems

The gist The use of geometric methods has proved useful in the hamiltonian description of classical constrained systems, and this work puts forward first steps toward the description of quantum constrained systems by comparing classical and quantum formalisms.

Classical Constrained Systems

The classical treatment of constrained Hamiltonian systems benefits from symplectic geometry, where a physical system is represented by a phase space with a symplectic two-form. A vector field V generates infinitesimal canonical transformations if it Lie drags the symplectic form, i.e., £V omega = 0. The Poisson bracket is defined as the change of f given by the motion generated by (the HVF of) g, i.e.,

How it works

The constraints are manifested through M constraint functions Ci, and if they form a first class system, their Hamiltonian vector fields Xa i are closed under the commutator. The standard ‘Dirac conjecture’ states that points along the orbits of the constraint vector fields represent physically indistinguishable configurations and are thus regarded as gauge. This classical two step process consists of restriction to the constraint surface and then quotient by the gauge orbits.

Observables in Classical Theory

An observable is defined as a function f on the phase space Γ such that its HVF (when restricted to Γ) is tangent to the constraint surface. This means that the finite symplectic transformations generated by the observables preserve the constraint surface, which corresponds to equation (6) in terms of Poisson brackets. An observable O is such that its HVF (and its integral curve) is tangent to Γ.

Quantum Constrained Systems

The quantum state space P forms a K¨ahler manifold endowed with both a symplectic structure responsible for the implementation of unitary transformations and a Riemannian structure relevant for the uncertainties and probabilistic aspects of the quantum theory. The Dirac quantum conditions, for the simplest case where the physical Hilbert space is a sub-vector space, can be cast in the same form as the classical conditions: as the inverse image of zero for some functions Ci on the quantum state space P.

Quantum Constraints and Observables

The condition (21) is equivalent to (20) since hΨCˆ2Ψi = Cˆ Ψi squared = 0 ⇔ Cˆ Ψi = 0. The constraint function C:= C squared = hΨCˆ 2Psi i on P is precisely the quantum equivalent of the classical constraint function and the condition (21) is the analogue of the classical C = 0 condition. The Hamiltonian vector field of an operator Fˆ, at the point Ψ, is given by (the projection to P of) the vector −J F Ψ.

Quantum Observables and Master Constraint

The condition for a physical observable Oˆ is to require that the one parameter family of unitary transformations a physical observable Oˆ generates leaves the quantum state space Pphy invariant. This means that the one parameter family of states Φ(λ):= e iλ Oˆ· Ψphy, for all λ and Ψphy ∈ Pphy, must also be a physical state. The condition on the function hOˆi is that its n-Poisson brackets with C¯2 vanish. This is the quantum analogue of the double commutator condition of the master constraint program [5].

Conclusion

The geometric formulation suggests that the relevant quantities to consider when looking at the motions generated on the quantum state space are the squares of the constraints and not the constraints themselves. The requirement for general classical and quantum observables to become physical observables is that they preserve, under the motion they generate, the submanifold of physical states. This in turn implies that the condition on quantum observables is given by the double commutator with the square of the constraint(s) and thus, the contact with the master equation of the master constraint program arises in a natural fashion. The geometric formulation seems to suggest that there are no degenerate directions of the symplectic structure associated to the constraint. This answers the question posed above.

How it works

The difference is that in the classical case, any point of the orbit is a physically admissible configuration, since it satisfies all the constraints, so it is rather natural to identify them as ‘the same’ state. In the quantum case, however, generic states in that orbit are not physical states. The difference is that in the classical case, the physics (as seen by the true observables) is the same for any element of the equivalence class, so it is rather natural to identify them as ‘the same’ state. In the quantum case, however, it is only after the true projection as defined by the group average mapping η that the physical state arises. The difference is that in the classical case, there are no degenerate directions of the symplectic structure associated to the constraint.

Discussion

This could point to the conclusion that there is a fundamental difference between the motions generated by the constraints and those motions generated by observables of the theory, that might generate symmetries (i.e. as asymptotic Poincare generators, or canonical transformations generated by boundary terms). This could point to the conclusion that there is a fundamental difference between possible gauge anomalies and regular (symmetry) anomalies deserves further attention. The possibility that there is a fundamental difference between possible gauge anomalies and regular (symmetry) anomalies deserves further attention.

Master Constraint

What we have shown here is that the apparent difficulties in the constraint algebra and the so-called ‘Master Equation’ do not arise and that, rather, they are rather natural from the standard Dirac procedure, when analyzed from the geometric perspective. The Master Constraint Program as put forward by Thiemann has several objectives. We have no particular comments on this possibilities.

Off-shell Algebra

The natural objects to consider in this case are the vector fields XCi = XC2 i that correspond to the functions defining the physical state space Pphy (by requiring the vanishing of all Ci). The commutator of two such fields is given by [XCbar i, XCbar j] = Xh[Cˆ2 i,Cˆ2 j] i. Thus, within the geometric formulation, the relevant quantities in order to compute the commutator of vector fields are the commutators of the squares of the constraint operators [Cˆ2 i, Cˆ2 j]. The relevant quantity for computing observables is again the Poisson bracket defined by omega, which contains information regarding the commutator of the vector fields corresponding to the observable Oˆ and Cˆ2 i (which in turn is obtained from [Cˆ2 i, Oˆ]). The relevant quantities in order to compute the commutator of vector fields are again the commutators of the squares of the constraint operators [Cˆ2 i, Cˆ2 j].

Summary

The paper explores a new perspective on quantum constrained systems by translating classical reduction into a geometric language and showing that physical observables are characterized by their invariance properties under the generated unitary transformations. The core finding is that the condition on quantum observables is given by the double commutator with the square of the constraint(s). This suggests that in contrast to classical theory, there are no degenerate directions of the symplectic structure associated to the constraint.

1. The AI system can implement constraint satisfaction through expectation value conditions, specifically by requiring hΨCˆ2Ψi = 0 instead of the naive hΨCˆΨi = 0, which is described as being too weak.

2. The system can determine physical observables by verifying the invariance property: the one parameter family of unitary transformations a physical observable Oˆ generates leaves the quantum state space Pphy invariant, which is mathematically equivalent to requiring that expectation values satisfy specific conditions involving the double commutator, such as hΨphy [O, ˆ Cˆ2] Ψphyi = 0.

3. The AI system can distinguish between classical and quantum reduction by recognizing that in the quantum case, there are no degenerate directions of the symplectic structure associated to the constraint, unlike the classical case where one must perform a second reduction along the ‘quantum gauge orbits’ which would be in contradiction with the fact that no further reduction is necessary.

4. The system can employ a geometric approach to state space analysis by modeling quantum states as rays in Hilbert space, acquiring the structure of a "Kähler manifold endowed with both a symplectic structure responsible for the implementation of unitary transformations, and a Riemannian structure relevant for the uncertainties and probabilistic aspects of the quantum theory."

5. The AI system can interpret measurement probabilities using geodesic distance on this Kähler manifold: The probability of measuring fi is then, Pi(Ψ) = cos 2[d(Ψ, φi)], where d(Ψ, φi) is the geodesic distance given by g from the point Ψ to the point φi.