Existence of interacting local Hamiltonian densities satisfying Dirac's surface-deformation commutation relations: are atomic systems localizable? (Dirac, 1948)

In his 1948 theory of localizable dynamical systems, Dirac reduced relativistic quantum dynamics to Hermitian operator densities $P^n(u)$ — one local Hamiltonian per point of an arbitrary spacelike surface — which must satisfy his fundamental commutation relations (50), (51): the quantum surface-deformation algebra whose structure functions are the components of the induced metric on the surface. The paper's central open question asks whether nontrivial, consistent interacting examples of such densities exist; if none can be found, one could infer that atomic systems are not localizable in Dirac's sense. The existence question — a rigorous interacting realization of the full algebra, or a theorem that none exists — remains open.

Background

Every relativistic quantum theory should describe the state of a system by data that no light signal can connect — data on a three-dimensional spacelike surface $S$ (a smooth surface in Minkowski space such that the straight line joining any two of its points lies outside the light cone; when such a surface extends to infinity it is a Cauchy surface, on which initial data determine the state everywhere). In 1948 P. A. M. Dirac asked which quantum theories admit a formulation in terms of such data. He called a dynamical variable localized if it describes physical conditions at one space-time point (a field quantity at a point, or a particle's position at a time), and a dynamical system localizable if its wave functions can be expressed entirely in terms of localized variables taken on an arbitrary spacelike surface. For a localizable system the wave function $\psi(q,S)$ depends jointly on the localized variables $q$ and on the surface, so that changing the surface plays the role that advancing the time plays in nonrelativistic quantum mechanics; there is then not one Schrödinger equation but one for each infinitesimal deformation of the surface.

A parametrized surface is specified by an embedding $x_\mu(u)$ of coordinates $u=(u_1,u_2,u_3)$. An arbitrary infinitesimal deformation moves each point by a shift $a_r(u)$ along the surface plus a displacement $a_n(u)$ along the unit normal $n_\mu(u)$. Deformations are generated by operator densities, $\Pi^r(u)$ (reparametrizations) and $\Pi^n(u)$ (normal deformations), acting on the surface dependence of $\psi$, and the local Schrödinger-type wave equations read $\Pi^\alpha(u)\psi=P^\alpha(u)\psi$ ($\alpha=1,2,3,n$), where the Hermitian densities $P^\alpha(u)$ act on the localized variables $q$. The $P^n(u)$ are the local Hamiltonians of the theory — one Hamiltonian operator for each point of $S$. The $\Pi$'s satisfy a fixed geometric algebra (the algebra of surface deformations) whose coefficients include the induced metric $\gamma^{rs}(u)$, the metric of the surface in the coordinates $u$ (Dirac's convention: the surface coordinates $u_r$ carry lower indices, and $\gamma^{rs}=-(\partial x_\mu/\partial u_r)(\partial x_\mu/\partial u_s)$ contracts with the contravariant densities $\Pi^r$, $P^r$); for example $[\Pi^n(u),\Pi^n(u')]=-(\partial\delta(u-u')/\partial u_s)\{\gamma^{rs}(u)\Pi^r(u)+\gamma^{rs}(u')\Pi^r(u')\}$, where $\delta(u-u')$ is the three-dimensional delta distribution. Under the assumption that the wave function on any one surface is arbitrary (no supplementary conditions), consistency of the many simultaneous wave equations forces the $P$'s to satisfy the mirror algebra — Eqs. (50) and (51) of the paper:

$$[P^r(u),P^n(u')]+[\Pi^n(u'),P^r(u)]-[\Pi^r(u),P^n(u')]=\frac{\partial\delta(u-u')}{\partial u_r}\,P^n(u),$$ $$[P^n(u),P^n(u')]+[\Pi^n(u'),P^n(u)]-[\Pi^n(u),P^n(u')]=\frac{\partial\delta(u-u')}{\partial u_s}\big\{\gamma^{rs}(u)P^r(u)+\gamma^{rs}(u')P^r(u')\big\}.$$

Equations (50) and (51) are preceded in the paper by the tangential relations (49) among the reparametrization densities $P^r(u)$ themselves (the reparametrization-reparametrization case of the same mirror construction), which Dirac says ought to be satisfied automatically if the $P^r$'s have been correctly calculated, and which he verifies explicitly for his Eq. (43).

Dirac emphasized that these relations are not merely necessary: any set of linear operators $P^\alpha(u)$ satisfying them furnishes a relativistic quantum theory of a localizable dynamical system, and conversely. Consistency also forces the $P$'s to depend on the surface only through its intrinsic geometry (the $\gamma$'s, the curvature quantities $\Omega^{rs}=(\partial n^\mu/\partial u_r)(\partial x_\mu/\partial u_s)$, and derivatives) — the relativistic analogue of time-independence of the Hamiltonian. The paper's only explicit construction is the reparametrization density $P^r(u)$ (his Eq. (43)) for spinless particles; no interacting $P^n(u)$ is given.

Dirac therefore posed the central question: if atomic systems are assumed localizable, constructing their relativistic theory 'becomes the problem of obtaining examples of sets of operators $P^n(u)$ satisfying (50), 51)'; if no suitable examples can be found, 'free from inconsistencies and sufficiently complicated to correspond to interaction between the elementary particles', one could infer that atomic systems are not localizable. The two known routes to candidate densities were explicitly inadequate: Weiss's canonical procedure (building Hamiltonians from a classical Lagrangian and canonical conjugate pairs $q,p$ on the surface) applies only to canonical scalar $q$'s and $p$'s, excluding fields obeying Fermi statistics (anticommutation relations, as electron fields do) and non-scalar localized variables; and Tomonaga's alternative identification of wave functions on different surfaces — identifying them by evolving with the free (interaction-free) equations, which removes the need to parametrize the surfaces — had been developed only for the case in which all Hamiltonian densities commute with one another.

A restricted, flat-surface shadow of the same structure is the Tomonaga–Schwinger equation of covariant perturbation theory: in the interaction picture (the representation in which the fields evolve with the interaction switched off and the state vector carries the interaction), the state assigned to a spacelike surface obeys $i\,\delta\Psi/\delta\sigma(x)=\mathcal{H}_{\rm int}(x)\Psi$, and its integrability — independence of the surface — requires $[\mathcal{H}_{\rm int}(x),\mathcal{H}_{\rm int}(y)]=0$ for spacelike separated points $x,y$, a condition equivalent to microcausality (local commutativity) of the interaction density. Dirac's relations (50), (51) are the general, curved-surface operator form of this requirement; Schwinger later restated their flat-surface content as the commutator condition relating the energy density to the momentum density that he used as the criterion of relativistic invariance.

Partial results bound the question without settling it: formal perturbative solutions exist (the Tomonaga–Schwinger interaction representation of QED, including Fermi fields), defined only order by order, with no known operator-theoretic sum; Haag-type results obstruct the nonperturbative interaction picture; non-Abelian gauge fields satisfy the flat-surface commutator condition only in gauges tied to a fixed time axis; classical realizations of the deformation algebra are classified and yield general relativity, while an anomaly-free quantum representation of that algebra remains a central open problem of canonical quantum gravity; and rigorous interacting examples exist only as free or low-dimensional multi-time toy models. Whether the full algebra admits nontrivial interacting operator realizations — the quantum surface-deformation algebra with structure functions rather than constants — is the unresolved target.

Problem Statement

The problem is Dirac's (1948) central open question, in its original generality: obtain examples of sets of linear operators $P^n(u)$ — together with their companion reparametrization densities $P^r(u)$ — acting on wave functions of localized dynamical variables on arbitrary parametrized spacelike surfaces, such that:

(i) the fundamental commutation relations (50), (51) hold as operator identities — with the deformation operators $\Pi^\alpha(u)$ and the induced-metric structure functions $\gamma^{rs}(u)$, exactly as stated in the Background — under the assumption of no supplementary conditions on the wave function of any one surface; the companion densities $P^r(u)$ must also satisfy the tangential relations (49) among themselves, which in Dirac's construction hold automatically for correctly calculated $P^r$'s;

(ii) the construction is free from inconsistencies — the densities are well-defined Hermitian operators, the surface-to-surface evolution is unitary, and the $P$'s depend on the surface only through its intrinsic geometry;

(iii) the dynamics is nontrivially interacting — a genuine interaction between the elementary particles in the (3+1)-dimensional field-theoretic setting, and not a free theory or one of the low-dimensional multi-time particle models of Refs. 17 and 19;

or else prove rigorously that no such interacting realization exists — in which case, by Dirac's own biconditional, one could infer that atomic systems are not localizable. The question is posed for the full algebra (50), (51) with its surface-geometric structure functions, of which the flat-surface commutator consequence $[\mathcal{H}_{\rm int}(x),\mathcal{H}_{\rm int}(y)]=0$ is the special case.

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