Physical axioms for the decoherence functional: composition, the real landscape, and the higher levels of the Sorkin hierarchy

In path-integral (histories-based) formulations of quantum theory, the decoherence functional $D(A,B)$ is the primitive object, and Dowker and Wilkes proved that the strongly positive systems form the unique class that is closed under the product composition rule and Galois self-dual — an argument, but not a physical proof, that strong positivity is the correct positivity condition. This problem collects the questions their argument leaves open: whether the product composition rule and compose-ability (tensor-closure) are physically correct, including in quantum-gravitational settings; why uniqueness fails in the landscape of real decoherence functionals and what the physically correct restriction there is; how to define the level-3 and higher (super-quantum) analogue of the decoherence functional, its composition law and positivity condition, and whether the maximality/uniqueness result extends there; and which additional axioms select the physically realised decoherent subalgebra in generalized quantum mechanics or the physically allowed co-events in quantum measure theory.

Background

Histories-based formulations of quantum theory take the set $\Omega$ of histories (for example particle trajectories or spacetime field configurations) as primitive. Physical propositions correspond to events, that is subsets of $\Omega$ forming a Boolean event algebra $\mathfrak{A}$ closed under complementation and finite union. Two frameworks of this kind are quantum measure theory (QMT), developed by Sorkin, and generalized quantum mechanics (GQM), developed by Hartle; both are pursued as relativity-friendly routes toward quantum gravity. In both, the central object is the decoherence functional $D:\mathfrak{A}\times\mathfrak{A}\to\mathbb{C}$ — called the double path integral in the QMT literature — satisfying Hermiticity $D(\alpha,\beta)=D(\beta,\alpha)^*$, bi-additivity on disjoint events, normalisation $D(\Omega,\Omega)=1$, and weak positivity $D(\alpha,\alpha)\geq 0$. Its diagonal defines the quantal measure $\mu(\alpha)=D(\alpha,\alpha)$.

Sorkin organized measure theories into a hierarchy by sum rules. Level 1 is classical (Kolmogorov) probability: $\mu(\alpha\sqcup\beta)=\mu(\alpha)+\mu(\beta)$ for disjoint events. Level 2 is quantum: $\mu$ obeys the level-2 sum rule

$\mu(\alpha\cup\beta\cup\gamma)-\mu(\alpha\cup\beta)-\mu(\beta\cup\gamma)-\mu(\alpha\cup\gamma)+\mu(\alpha)+\mu(\beta)+\mu(\gamma)=0$

for pairwise disjoint $\alpha,\beta,\gamma$, while violating level 1. Level 3, the first super-quantum level, satisfies the analogous four-event sum rule while violating level 2, and so on; each level is contained in all higher levels, so classicality of quantum theory becomes the question of why nature sits at level 2.

A decoherence functional is strongly positive if for every finite list of events $A_1,\dots,A_n$ the event matrix $M_{ij}=D(A_i,A_j)$ is positive semi-definite — a strictly stronger condition than weak positivity. In ordinary quantum mechanics $D$ is essentially a Gram matrix of history state vectors (the double path integral, i.e. the closed-time-path or Schwinger–Keldysh form of the path integral), hence strongly positive; conversely, strong positivity allows a history Hilbert space to be reconstructed from $D$, which is one motivation for adopting it as an axiom. Strong positivity of a joint quantal measure also underlies tightened bounds on nonlocal correlations in histories-based analyses.

For a composite of two non-interacting, uncorrelated subsystems, the product composition rule $\odot$ is used: the history space is the Cartesian product $\Omega_1\times\Omega_2$, the composite event algebra is generated by product events $E_1\times E_2$, and $(D_1\odot D_2)(A,B)=\sum_{i,j}D_1(A_{1i},B_{1j})D_2(A_{2i},B_{2j})$ where $A=\bigsqcup_i(A_{1i}\times A_{2i})$ and $B=\bigsqcup_j(B_{1j}\times B_{2j})$ are disjoint-product decompositions. Boës and Navascués showed that weak positivity is not preserved by $\odot$, so the class $\mathcal{W}$ of quantum measure systems is not closed under composition, and that the finite strongly positive systems form a maximal tensor-closed class (a class of systems closed under the composition $\odot$). Dowker and Wilkes extended maximality to arbitrary systems and proved uniqueness. Maximality is formalized with the Galois dual: for $\mathcal{A}\subseteq\mathcal{W}$, $\widehat{\mathcal{A}}=\{\Psi_1\in\mathcal{W}:\Psi_1\odot\Psi_2\in\mathcal{W}\ \forall\Psi_2\in\mathcal{A}\}$; a tensor-closed class that equals its Galois dual is maximal. Their Theorem 3 states that the strongly positive systems $\mathcal{S}$ are the unique tensor-closed Galois-self-dual class, and their Theorem 4 shows every tensor-closed class lies inside $\mathcal{S}$ or inside $\mathcal{R}^+$, the positive-entry systems with $D(A,B)$ real and non-negative for all $A,B$.

Two further pieces of context are needed for the open questions. First, Sorkin proved a one-to-one correspondence between quantal measures and real symmetric decoherence functionals $D^{\mu}$, so a landscape $\mathcal{W}_R$ of real decoherence functionals is the natural home for theories specified by their measures alone; there $\mathcal{R}^+$ is itself tensor-closed and Galois self-dual, and the uniqueness theorem fails. Second, the interpretive fork: in GQM one restricts to a decoherent subalgebra on which the measure is approximately classical and interprets it probabilistically, but a system typically admits many incomparable decoherent subalgebras and no axiom selects one, which makes GQM, in the authors' words, a theory of many worlds; in QMT one keeps the full algebra and supplements the Law of Preclusion (events of zero or very small measure do not happen) with axioms for physically allowed co-events — maps from the event algebra to $\mathbb{Z}_2$, the two-element field — which should be classical (algebra homomorphisms, playing the role of truth valuations on classical propositions) on the subalgebra of localized, quasi-classical macroscopic events. The identity of those axioms remains open.

Problem Statement

The problem is to settle the questions about the physical axioms of the decoherence functional that Dowker and Wilkes left open after proving that the strongly positive systems form the unique tensor-closed, Galois-self-dual class of quantum measure systems. Concretely, an answer must resolve, with the same standard of rigour as the source theorem, at least one of the following four independently posed strands:

(1) Composition and closure. Determine whether the product composition rule $\odot$ for non-interacting, uncorrelated subsystems is the correct physical composition law in a path-integral-based theory — including possible quantum-gravitational contexts where a different law may be required — and whether compose-ability (tensor-closure) of the physically allowed class is a genuine physical requirement. This means deriving these assumptions from more primitive principles, or replacing them by better-justified ones and re-deriving the resulting classification of physical systems. The two sub-demands of this strand — correctness of the composition law and the status of compose-ability (tensor-closure) — are independent: settling either one resolves the strand.

(2) The real landscape. Explain the failure of uniqueness among real decoherence functionals: in $\mathcal{W}_R$, the positive-entry systems $\mathcal{R}^+$ are themselves tensor-closed and Galois self-dual, so the uniqueness theorem fails. Determine the physically correct attitude to the imaginary part of $D$: either establish that the complex decoherence functional is primitive (so the real landscape is physically irrelevant and this can be shown, not merely assumed), or establish that the quantal measure is primitive and find the structural characterisation that then singles out the physical subclass.

(3) Level 3 and above. Construct the higher-level analogue of the decoherence functional: for level 3, a functional $E(A,B,C)$, additive in each argument, with an agreed analogue of Hermiticity, a composition law for composite level-3 systems, and a positivity condition playing the role of strong positivity (including how level-2 decoherence functionals sit inside level 3). Determine whether, in the resulting landscape, there is a unique tensor-closed Galois-self-dual (maximal) class — the analogue of the source uniqueness theorem — or prove that uniqueness fails. Extension or delimitation of the programme to all higher levels of the hierarchy is an optional further direction: resolving the level-3 construction and the level-3 maximality question above already resolves this strand.

(4) The classical regime. For GQM, supply a physical criterion (or additional axiom) selecting the physically realised decoherent subalgebra among the many incomparable ones, or show that no consistent such criterion exists; for QMT, supply a complete axiom set for the physically allowed co-events that goes beyond the Law of Preclusion and satisfies all three of the following desiderata — (i) it determines which co-events are physically allowed rather than presupposing them, (ii) classical behaviour on the subalgebra of localized, quasi-classical macroscopic events, namely that the allowed co-events restrict to $\mathbb{Z}_2$-algebra homomorphisms there, follows from the axioms and is not stipulated in their definition, and (iii) the allowed co-events reproduce the statistical predictions of established level-2 quantum theory — or show that no such axiom set exists. The GQM and QMT sub-demands are independent: settling either one resolves this strand.

Each strand stands on its own: resolving any single one of (1)-(4), with either alternative inside a strand counting as a resolution of that strand, satisfies the completion standard, and the other strands may be left open. These four strands share one objective: to convert the structural uniqueness argument of Dowker and Wilkes into a determination of the physically correct axioms — positivity and composition — for histories-based quantum theory at the quantum level and above, and to connect those axioms to the emergence of the classical world. Any answer must remain consistent with established level-2 quantum theory and with the experimental bounds on third-order interference.

References

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