Quantum constraint closure for Born-Infeld electrodynamics: Dirac's 1960 open problem of constructing a consistent quantum theory

In his 1960 reformulation of Born-Infeld electrodynamics, P. A. M. Dirac completed the classical theory: a single action principle for the nonlinear electromagnetic field and point charges, a full Hamiltonian formulation with first-class constraints, positive energy density, and a Lorentz-type equation of motion with finite electromagnetic self-energy. He concluded, however, that the passage to the quantum theory 'appears to be insoluble with present methods of quantization': the Hamiltonian and part of the constraint set depend on the canonical variables through square roots, so no operator ordering is known for which the quantum constraint commutators close on the constraints. No consistent quantum theory of Born-Infeld electrodynamics with an anomaly-free closed quantum constraint algebra is known, and no impossibility theorem has been proved; the exact constraint quantization of the untruncated theory remains open.

Background

Maxwell's electrodynamics with point charges has an infinite field energy: the Coulomb field of a charge $e$ carries self-energy $\sim\int E^2\,d^3x$, which diverges at the particle. In 1934 Max Born and Leopold Infeld proposed a nonlinear modification of electrodynamics whose Lagrangian density is a square root of a determinant; in flat space it can be written (up to conventions) as $\mathscr{L}=b^2\Bigl(1-\sqrt{1+\frac{B^2-E^2}{b^2}-\frac{(E\cdot B)^2}{b^4}}\Bigr)$, where $E$ and $B$ are the electric and magnetic fields and $b$ is Born's constant, a maximum admissible field strength (its inverse is sometimes called the universal length). The theory reduces to Maxwell theory for weak fields, bounds the field strength of a point charge, and makes its total field energy finite.

In 1960 Paul A. M. Dirac reformulated this theory. He started from a comprehensive action $I=I_1+I_2+I_3$, where $I_1=\int\mathscr{L}\,d^4x$ is the Born-Infeld field action with $\mathscr{L}=-\Lambda$ and $-\Lambda^2=\det(g_{\mu\nu}+f_{\mu\nu})$ ($f_{\mu\nu}$ the field tensor), $I_2$ couples the field to a point charge $e$ on a world-line, and $I_3=-M\int ds$ gives the particle a non-electromagnetic rest mass $M$. Varying both the potentials and the coordinates of the singularity, he derived the field equations with point sources and an equation of motion that reduces, for weak slowly varying external fields, to the Lorentz force law with inertial mass $M+m$, where $m$ is the finite Born-Infeld field energy of the charge. He then constructed the Hamiltonian formulation on arbitrary spacelike hypersurfaces, using flat-space coordinate fields $y_K$ as canonical variables, and worked out the constraint set: primary constraints $D^0=0$ (vanishing momentum conjugate to the scalar potential), $w_Ky_{K,r}+f_{rs}D^s=0$ (tangential momentum constraints), and $w_Kw_K+g_{rs}(D^rD^s+B^rB^s)-\Gamma^2=0$ (a quadratic constraint, $-\Gamma^2$ being the determinant of the spatial metric), together with the secondary Gauss constraint $D^r{}_{,r}=0$ (with a point-charge source term when charges are present). These constraints are first class, meaning their mutual Poisson brackets vanish on the constraint surface and each generates a gauge transformation. The energy density equals $-w_{\mathrm{nor.}}$, the negative of the momentum conjugate to the surface normal, and is positive definite, excluding runaway solutions (solutions in which a charge self-accelerates without bound).

The obstruction appears on the way to quantum theory. In Dirac's canonical quantization of constrained Hamiltonian systems, each classical constraint $\phi_n=0$ must be promoted to an operator condition $\hat\phi_n\Psi=0$ on wave functionals, with an operator ordering chosen so that the commutators close on the constraints, $[\hat\phi_n,\hat\phi_m]=\sum_k \hat c_{nmk}\hat\phi_k,$ with the coefficient operators $\hat c_{nmk}$ standing to the left; otherwise the conditions $\hat\phi_n\Psi=0$ are mutually inconsistent (an anomaly). For Born-Infeld electrodynamics the normal momentum $w_{\mathrm{nor.}}$ enters the Hamiltonian and constraints through the non-polynomial (square-root) relation $w_{\mathrm{nor.}}+\{\Gamma^2-g_{rs}(D^rD^s+B^rB^s)-e^{rs}f_{rt}D^tf_{su}D^u\}^{1/2}=0$, and no operator ordering of these square-root functionals is known that makes the closure condition hold; replacing the square-root form by the rational polynomial constraint improves but does not restore closure. Dirac's paper (full text paywalled) therefore ends with the classical theory complete but quantization blocked, and records the suggestion that a more radical modification of the quantum formalism, such as introducing anticommuting field quantities to avoid the square roots, might be required.

The problem has not been dissolved by later physics. Born-Infeld theory re-emerged from string theory as the low-energy effective action of open strings and D-branes (membrane-like extended objects on which open strings end), and in that role it is quantized only perturbatively, as an effective field theory, with quantum corrections deforming the exact square-root form; in ordinary QED the analogous low-energy quantum effects are described by the Euler-Heisenberg effective action, the standard one-loop effective Lagrangian for slowly varying electromagnetic fields. What has never been obtained is the exact, nonperturbative quantum theory of the untruncated Born-Infeld system in the canonical sense Dirac set up: well-defined constraint operators, closed under commutators, acting on a Hilbert space of physical states. The conventions of the problem are those of Dirac's paper: four-dimensional flat spacetime, gauge invariance with two photon polarizations in the weak-field limit, and a finite Born constant $b$.

Problem Statement

Construct a consistent quantum theory of Born-Infeld electrodynamics in Dirac's canonical Hamiltonian-constraint formulation, or in a reformulation demonstrably equivalent to it. The setting is the one fixed by the source: four-dimensional flat spacetime, the Born-Infeld action $\mathscr{L}=-\Lambda$ with $-\Lambda^2=\det(g_{\mu\nu}+f_{\mu\nu})$, where $g_{\mu\nu}$ is the flat spacetime metric and $f_{\mu\nu}$ is the electromagnetic field tensor, coupled to point charges carrying a non-electromagnetic rest mass $M$ through the source's comprehensive action principle, which determines both the field equations and the equations of motion of charged particles, and the first-class constraint system obtained from its Hamiltonian formulation. A solution of the constructive type must supply, and demonstrate: (i) an operator realization of the canonical variables and of every classical first-class constraint as a well-defined operator $\hat\phi_n$ (with explicit smearing of each operator against test functions, a stated operator ordering, and a dense invariant domain such as wave functionals of finitely many field modes); (ii) closure of the quantum constraint algebra, $[\hat\phi_n,\hat\phi_m]=\sum_k\hat c_{nmk}\hat\phi_k$ with the coefficient operators standing to the left and no anomalous remainder, which is precisely the condition that no known operator ordering of the untruncated square-root Hamiltonian satisfies; (iii) the existence of nontrivial physical states $\Psi$ annihilated by all constraints, or a consistent constructive procedure for them; and (iv) recovery of classical Born-Infeld electrodynamics (with its bounded field strength, finite point-charge self-energy, and Lorentz-type equation of motion) in the classical limit. The problem includes determining, by construction or by rigorous argument, whether this is possible with standard canonical quantization or requires an alternative quantization scheme or a reformulation of the variables; a rigorous demonstration that no operator ordering of the standard canonical square-root constraints closes resolves only that negative branch, namely that an alternative scheme or reformulation is required, and does not by itself resolve the problem, which requires the construction of (i)-(iv) above. The problem concerns the untruncated theory coupled to point charges, not a truncated or linearized version of the theory and not the pure (charge-free) Born-Infeld field.

References

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