The Born-Weyl quantization of field equations: a covariant, polymomentum-based quantization reproducing canonical commutation relations and the standard probabilistic interpretation

In 1934 Max Born proposed to quantize field equations using Hilbert's independence theorem of the calculus of variations, and Hermann Weyl's accompanying Physical Review analysis fixed the questions that define the problem to this day: the scheme lacks the coordinate-momentum commutation relations needed to interpret the electromagnetic field as a superposition of oscillators (photons); the quantum field equations should take the manifestly covariant four-component form $d\Psi/dt^{i}=T_{i}\Psi$ with energy and momentum operators rather than a single scalar evolution equation; matching the Heisenberg-Pauli canonical formalism forces one to abandon integrability of the extremal field; and the resulting amplitudes do not carry the usual probabilistic interpretation. Born himself abandoned the independence-integral route within the year. The question was revived from the late 1990s as the 'Born-Weyl quantization' program - quantization based on the covariant polymomentum (De Donder-Weyl) Hamiltonian formalism, realized as precanonical quantization with a Clifford-valued wave function and a covariant Schrödinger-type equation. That program supplies structural elements (graded Poisson brackets on differential forms, covariant commutation relations) but explicitly leaves open the positive-definite inner product, the interpretation of the wave function, and a rigorous equivalence with standard quantum field theory, including the electromagnetic case central to Weyl's objection. The problem is to complete such a construction, or prove a precise obstruction theorem for the whole class.

Background

The variational background. For a classical field theory one starts from a Lagrangian density $L(t^{i}, z^{a}, z^{a}_{i})$ - a function of the independent variables $t^{i}$ (in physics the $r=4$ spacetime coordinates), the field components $z^{a}$, and their first derivatives $z^{a}_{i}=\partial z^{a}/\partial t^{i}$ - and requires the action $\int L\, dt^{1}\cdots dt^{r}$ to be stationary. Weyl's analysis works with the associated vector field $z^{a}_{i}(t,z)$ of gradient data, which is called integrable when it actually is the derivative field of a family of surfaces $z^{a}(t)$, i.e. when the compatibility conditions $\partial z^{a}_{i}/\partial t^{j}=\partial z^{a}_{j}/\partial t^{i}$ hold. Hilbert's independence theorem concerns extremal vector fields, that is, vector fields satisfying the Euler-Lagrange equations in first-order form: performing the Legendre transformation to the polymomenta $p^{i}_{a}=\partial L/\partial z^{a}_{i}$ (one momentum for each field component and each independent variable - the field-theoretic generalization of the canonical momenta of mechanics), a special extremal field is one whose momenta are the field-space gradient $p^{i}_{a}=\partial S/\partial z^{a}$ of a function $S$ satisfying the Jacobi-Hamilton equation (the covariant Hamilton-Jacobi equation). For such fields the flux of a certain vector density through a closed curve defines the independence integral, whose value depends only on the boundary curve and not on the surface spanning it - the field-theoretic analogue of Hamilton's principal function. The clean form of this construction is available only for special extremal fields; the paradigm case is the 'problem without forces', in which the covariant Hamiltonian depends only on the momenta, the polymomenta are constant, and the resulting plane waves are what Born's construction uses.

Born's proposal and Weyl's critique. Born's procedure forms the independence integral $S$ through a closed curve, uses $e^{iS/\hbar}$ as a plane-wave-type amplitude, superposes such waves, and interprets the squared modulus of the resulting amplitude as a probability. Weyl's analysis - the source of the present problem - makes four points. First, the probability Born attaches to such amplitudes, a probability for the values of certain integrals over a spacetime region, 'does not coincide with the usual well-proved interpretation of quantum mechanics even in the one-dimensional case', where quantum mechanics predicts probabilities for the values of quantities at an instant; Weyl modifies the construction (fluxes through open cross-sections instead of closed curves) to restore the usual interpretation on spacelike slices, but a comparison with the canonical theory of Heisenberg and Pauli - the treatment of fields as mechanical systems with infinitely many degrees of freedom and a distinguished time coordinate, quantized by imposing commutation relations on coordinates and momenta - becomes possible only 'under the simplest circumstances', and 'Born's scheme proves to be too narrow'. Second, that comparison itself forces one to allow canonical momenta that are arbitrary functions of the space coordinates, and the corresponding extremal vector field 'is not integrable!'; so one must renounce integrability, and it is left open what mathematical structure then guarantees the coherence and physical interpretability of the quantization. Third, 'the characteristic commutation rules of coordinates and corresponding momenta, q and p, are missing in Born's theory', and they 'seem to be essential for the possibility of considering the electromagnetic ether as a superposition of oscillators (photons)'. Fourth, Weyl argues that the quantum field equations should not be a single scalar Schrödinger-type equation but 'four equations $d/dt^{i}=T_{i}$' in which differentiation with respect to each spacetime coordinate is performed by an operator - energy for $i=0$ and the three momentum components for $i=1,2,3$ - so that relativistic covariance is manifest from the start; in the Heisenberg-Pauli construction these operators exist, but their recipe is complicated and their four-vector character needs a special proof, and Weyl expresses the hope that 'a way similar to that followed by Born will lead to an essentially simpler formulation'.

The modern framework. The classical structure behind Born's proposal is today called the De Donder-Weyl covariant Hamiltonian formalism: with polymomenta conjugate to each spacetime derivative of the fields, the field equations take a first-order Hamiltonian form generated by a single De Donder-Weyl Hamiltonian $H(y^{a},p^{i}_{a})$, with a covariant Hamilton-Jacobi equation for a function $S(y,x)$ on field values and spacetime points. Its geometric home is multisymplectic (or polysymplectic) geometry - the generalization of the symplectic geometry of Hamiltonian mechanics in which the phase space is replaced by a finite-dimensional space of field values, spacetime points, and polymomenta, and the symplectic form by a differential form of higher degree. Since the late 1990s I. V. Kanatchikov has explicitly revived Weyl's question under the name 'Born-Weyl quantization': he constructed a graded Poisson bracket on differential forms over the polymomentum phase space (a bracket generalizing the Poisson bracket, with antisymmetry properties graded by the degree of the forms), quantized the resulting covariant brackets $\{p_{a},y^{b}\}=\delta_{a}{}^{b}$ by Dirac's correspondence rule (the standard prescription that Poisson brackets of classical quantities become commutators of the corresponding quantum operators), realized the polymomenta as operators built from Clifford-algebra generators (anticommuting matrix units $\gamma^{i}$ with $\gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2g^{ij}$, the algebra underlying Dirac's relativistic wave equation), and proposed a covariant analogue of the Schrödinger equation

$i\hbar\kappa\,\gamma^{i}\partial_{i}\Psi=\hat{H}\Psi,$

where $\kappa$ is a constant of dimension inverse spatial volume and $\Psi(y,x)$ is a Clifford-valued wave function on the finite-dimensional space of field values and spacetime points. This is called precanonical quantization; the left-hand side, a Dirac operator bundling the four spacetime derivatives into one covariant object, is the modern realization of Weyl's four operators $T_{i}$. The standard theory it must be compared with is most directly the functional Schrödinger picture of quantum field theory, in which the state is a wave functional over field configurations on space, evolving in time.

Problem Statement

Determine whether a quantization of classical field theory can be based on the covariant polymomentum Hamiltonian structure underlying Born's 1934 independence-integral proposal - in Weyl's terms, whether a Born-inspired covariant quantization can be made coherent. The question, raised by Weyl's 1934 analysis of Born's scheme and revived unchanged by the modern De Donder-Weyl (polysymplectic) quantization program, is to be settled by a construction satisfying all of the following, or by a rigorous obstruction theorem showing that none exists:

(i) Manifest covariance: the quantum evolution is generated by four operators $T_{i}$ (energy and the three momentum components) through $d\Psi/dt^{i}=T_{i}\Psi$, or by a single covariant Dirac-type operator bundling them, provided that the four operators $T_{i}$ acting on the state are recovered from it and the claimed equivalence, including the mutual consistency of the four evolutions, is demonstrated rather than asserted, so that relativistic covariance is built into the formalism rather than imposed and verified a posteriori as in the Heisenberg-Pauli construction.

(ii) Canonical adequacy: the construction reproduces the characteristic commutation relations between coordinates and conjugate momenta, so that the quantized electromagnetic field admits the standard interpretation as a superposition of oscillators (photons).

(iii) No integrability or 'no forces' restriction: the construction is formulated on the general (non-integrable) extremal structures that the comparison with Heisenberg-Pauli quantization forces - canonical momenta that are arbitrary functions of the space coordinates - and does not require the special extremal fields or the momentum-only ('no forces') Hamiltonian on which Born's original plane-wave construction rested.

(iv) Probabilistic interpretability: the wave amplitudes carry the usual probability interpretation for field configurations on spacelike slices, with a positive-definite inner product.

The construction in addition defines the state space, the operator algebra (including the treatment of gauge constraints for the electromagnetic field), and demonstrates the correspondence with standard quantum field theory at least for the free electromagnetic field - for example, explicit recovery of the functional Schrödinger representation, its vacuum functional (the ground-state wave functional), and the photon/oscillator structure. Any obstruction theorem for (i)-(iv) is meaningful and non-vacuous only if the class of quantizations extending Born's scheme that it addresses covers the constructions the conditions target - in particular quantizations formulated on the general non-integrable extremal structures of condition (iii), with Born's original independence-integral scheme and the covariant polymomentum (precanonical) constructions described in the Background as members or limiting cases; a no-go theorem confined to a subclass that could not satisfy (iii) in the first place is vacuous.

References

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