Consistent quantization of general classical constrained Hamiltonian systems: operator ordering and the quantum Dirac bracket (Dirac, 1950)

Dirac's 1950 paper 'Generalized Hamiltonian Dynamics' introduced the first/second-class classification of constraints and the Dirac bracket, and stated that the passage from classical constrained systems to quantum theory is not mathematically well defined: operator-ordering ambiguities may make it impossible to choose orderings that keep all quantum equations consistent, the second-class (Dirac-bracket) sector was given no commutator counterpart, and no general quantization method for arbitrary constrained systems exists. Later literature transformed parts of this question into precise obstruction theorems (the Groenewold no-go theorem and its rigorous modern versions) and built powerful frameworks for wide classes of systems (BRST/BFV for first-class gauge systems, conversion of second-class constraints into first-class ones in an extended phase space, deformation quantization under geometric restrictions), but no general, mathematically well-defined, internally consistent quantization for arbitrary constrained systems, and no general obstruction classification, exists. The problem posed here is the surviving source-faithful core of Dirac's question: construct a general quantization procedure - including a generally valid quantum Dirac bracket with proven closure of the full quantum constraint algebra and a globally consistent operator-ordering prescription - applicable to arbitrary classical constrained Hamiltonian systems (mixed first- and second-class, including field-theoretic cases), or prove a general obstruction theorem characterizing exactly which such systems admit a consistent quantization.

Background

In Hamiltonian mechanics a dynamical system is described by coordinates $q_n$ and conjugate momenta $p_n$, with time evolution generated by a Hamiltonian through the Poisson bracket $\{\xi,\eta\}=\sum_n(\partial\xi/\partial q_n\,\partial\eta/\partial p_n-\partial\xi/\partial p_n\,\partial\eta/\partial q_n)$, a bilinear operation on phase-space functions that is antisymmetric and obeys the Jacobi identity. Quantization starts from this form: Dirac's correspondence principles state that (i) a Poisson-bracket relation between classical variables goes over to a commutation relation $\{\xi,\eta\}\leftrightarrow(\xi\eta-\eta\xi)/(i\hbar)$ (written with the convention $2\pi(\xi\eta-\eta\xi)/ih$ in the 1950 paper) between the corresponding self-adjoint operators, and (ii) a classical condition that holds on the physical part of phase space becomes a linear condition on the allowed quantum state vectors.

A Lagrangian is singular when the velocities cannot be solved for from the defining relations $p_n=\partial L/\partial\dot q_n$. Consistency then forces relations $\phi_m(q,p)\approx 0$ (weak equations, valid on the physical region of phase space), called constraints; iterating the time evolution of the constraints along the equations of motion produces further constraints until the system closes. A constraint is first class if its Poisson bracket with every constraint vanishes on the constraint surface; first-class constraints generate gauge freedom, i.e. the arbitrary functions appearing in the general solution (for example the arbitrariness of the time parameter in relativistic dynamics). Constraints failing this test are second class: they do not generate gauge freedom but restrict the physical degrees of freedom. For an even number $s$ of second-class constraints $\theta_s$, the 1950 paper introduced a modified bracket, now called the Dirac bracket,

$\{f,g\}_D=\{f,g\}-\{f,\theta_s\}C^{ss'}\{\theta_{s'},g\},\qquad C^{ss'}\{\theta_{s'},\theta_{s''}\}=\delta^s_{\,s''},$

where $C^{ss'}$ is the inverse of the antisymmetric matrix of mutual brackets of the second-class constraints (eq. (36) of the paper, up to index conventions). Under the Dirac bracket every second-class constraint has vanishing bracket with anything, so the constraints can be promoted to identities ('strong' equations) and the theory reduces to a smaller number of effective degrees of freedom.

In Section 11 of that paper Dirac stated the unresolved quantum issues. First, the classical-to-quantum map is not mathematically well defined: whenever a classical quantity is a product of two factors whose Poisson bracket does not vanish, the order of the corresponding non-commuting operators is ambiguous; for simple systems the correct ordering can be guessed, but for complicated constrained systems it may be impossible to choose orderings that make all quantum equations consistent, and then 'one would not know how to quantize the theory'. Second, quantum consistency already forces all constraints to be first class (applying two constraint conditions to a state vector and using the commutator shows that a second-class pair would annihilate the whole state space), so second-class constraints must first be converted into strong equations through the Dirac bracket - a route Dirac only sketched, leaving the quantum, commutator version of that bracket undeveloped. Third, supplementary conditions involving the auxiliary velocity variables $v_m$ (which fix undetermined velocities, as in gauge fixing) have no operator image and can spoil the classical integrability of the gauge orbits. Dirac concluded that the quantization methods of the day were 'of the nature of practical rules' and gave no general procedure; the requirement that all constraints be first class is necessary but not sufficient for a consistent quantization.

Subsequent work sharpened this landscape without closing it. The no-go theorem originating with Groenewold (1946), usually called the Groenewold-van Hove theorem, states that no quantization can satisfy the full Poisson-bracket-to-commutator correspondence for all observables; Gotay (1999) made the obstruction rigorous for the polynomial Poisson algebra on $\mathbb{R}^{2n}$ (filling a gap in Groenewold's proof) and, in dimension one, determined the largest consistently quantizable Lie subalgebras, while Gotay, Grundling and Hurst (1996) proved the analogous non-existence result for the Poisson algebra of the sphere $S^2$. Orderings are therefore only manageable on restricted observable classes: the Born-Jordan rule (which averages the orderings of non-commuting factors) satisfies the Dirac correspondence exactly for symbols of the form $f(x)+g(\xi)$ and is characterized by that property (de Gosson and Nicola, 2018). For first-class gauge systems, BRST quantization (named after Becchi, Rouet, Stora and Tyutin) - a construction that adds fermionic ghost variables encoding the gauge symmetry cohomologically, together with its phase-space BFV (Batalin-Fradkin-Vilkovisky) version (Batalin and Fradkin, 1986; the monograph of Henneaux and Teitelboim, 1992) - provides a systematic operator and path-integral framework, and the equivalence of reduced-phase-space, Dirac, and BRST quantization has been proved for reducible gauge systems whose constraints are linear in the momenta (Ferraro, Henneaux and Puchin, 1993). Reduced-phase-space quantization is under mathematical control only in special settings, e.g. the Guillemin-Sternberg conjecture that quantization commutes with symplectic reduction (the operation of quotienting phase space by a symmetry group and retaining only the group-invariant part), proved for compact group actions by Tian and Zhang (1998). For second-class systems the standard practice is to promote the Dirac bracket to $(i\hbar)^{-1}$ times the commutator, but general constructions exist only through conversion of second-class constraints into first-class ones in an extended phase space (Batalin and Fradkin, 1987; the conversion is not fully general for nonscalar, non-Abelian constraint sets - Batalin, Grigoriev and Lyakhovich, 2005), or as deformation quantization (star products, i.e. associative products on functions deforming the pointwise product so that the first-order commutator reproduces the bracket), available only under geometric restrictions such as (almost) Kähler constraint surfaces (Lyakhovich and Sharapov, 2002). What no later work supplies is the object Dirac asked for: a general, mathematically well-defined and internally consistent quantization applicable to arbitrary constrained systems, with a generally valid quantum version of the Dirac bracket, or a theorem delimiting exactly where the obstruction lies.

Problem Statement

Settle, in general, the question Dirac raised in Section 11 of 'Generalized Hamiltonian Dynamics' (1950): the procedure of passing from a classical constrained Hamiltonian system to its quantum theory is not mathematically well defined, because 'whenever a classical quantity involves a product of two factors whose P.b. does not vanish, there is an ambiguity in the order in which the two factors should appear in the corresponding quantum expression', and 'with complicated examples it may be impossible to choose the order in each case so as to make all the quantum equations consistent, and then one would not know how to quantize the theory'. (In these quotations 'P.b.' is Dirac's abbreviation for the Poisson bracket, defined in the background above.)

At the generality of the source (systems with both first-class and second-class constraints, including field-theoretic instances; no restriction to special geometries, constraint structures, or method classes), the question has two resolution modes:

Both resolution modes share the same admissibility floor. The bracket-to-commutator correspondence that is retained under (1) or declared under (2) must include the constraint functions and the Hamiltonian of the classical system, and in the classical limit must reproduce the Dirac bracket of the second-class sector - on these observables the commutator divided by $i\hbar$ must tend to the corresponding Dirac bracket at leading order in $\hbar$ - together with a globally consistent operator-ordering prescription. This floor is fixed at the generality stated above: neither the retained bracket relations nor the declared axiom set may be narrowed to special geometries, constraint structures, or method classes.

(1) A general, mathematically well-defined quantization procedure. Given the classical data - the phase-space variables, the Hamiltonian, and the complete constraint set with its first/second-class split as produced by Dirac's consistency algorithm - the procedure must assign a quantum theory: a state space, self-adjoint operators for the classical observables, and an implementation of the constraints, together with a definite operator-ordering prescription for products of non-commuting factors. It must be proved that all quantum equations are mutually consistent across the declared class: the constraint conditions on state vectors (Dirac's principle for first-class constraints) and the operator identities replacing constraints made strong through the Dirac bracket must be compatible with the commutation relations the construction retains from the bracket-to-commutator correspondence - the construction must delimit exactly which bracket relations it preserves and prove that these weakened requirements still pin down a non-trivial quantum theory faithful to the classical limit - and must include a consistent quantum counterpart of the Dirac bracket for second-class constraints, with the correct classical limit.

(2) A general obstruction or classification theorem. Fix an axiom set for 'consistent quantization' faithful to Dirac's correspondence principles - at minimum retaining the constraint-condition principle and a non-trivial subset of the bracket-to-commutator correspondence, so that quantizations satisfying the axioms are non-trivial quantum theories faithful to the classical limit - and characterize exactly which classical constrained systems admit a quantization satisfying it, proving both that every system in the characterized class can be quantized consistently and that no system outside it can.

The question covers the mixed first-/second-class case; constructions valid only for conversion-friendly, linear, Abelian, Kähler-type, or otherwise special constraint structures, or only for finite-dimensional systems, fall outside its scope. The component of the source question concerning supplementary conditions that involve the velocity variables has been absorbed into modern gauge-fixing methodology and is context, not a separate deliverable.

References

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