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Functorial evolution of quantum fields

Stefano Gogioso, Maria E. Stasinou, Bob Coecke

arXiv:2003.13271v1quant-ph

TL;DR

The paper addresses how to formulate quantum-field evolution on discretized causal structures without relying directly on topology or analysis. It develops a functorial, theory-independent framework, connects it to AQFT and QCA, and establishes no-signalling and continuous-limit examples within the framework. Its scope is limited by the additional structure required to express quantum unitarity and by some AQFT generalizations being left for future work.

  • Problem

    The paper addresses the open characterization of causal orders arising from Lorentzian manifolds and the need for a unified order-theoretic framework for quantum fields.

    Source: §1 · pp. 2–3

  • Method

    It defines causal field theories functorially over causal orders, connects regional state spaces to AQFT through sheaf-theoretic constructions, and models cellular automata within the same framework.

    Source: §1 · p. 3, §4.2 · p. 15, §6.3 · p. 29

  • Results

    The framework yields no-signalling field evolution, models traditional and generalized QCA, and sketches a continuous-time Dirac-QCA limit.

    Source: §4.3 · p. 16, §7 · p. 30, §6.3 · p. 30

  • Takeaways & Limitations

    The framework is proposed as a flexible starting point for further developments in AQFT, QCA, and quantum field theory.

    Source: §1 · p. 3, §7 · p. 30

    Expressing unitarity requires additional structure on the field category, and a more general AQFT setting is left for future work.

    Source: §5 · p. 21, §6.1 · p. 24

Abstract

from arXiv · show

We present a compositional algebraic framework to describe the evolution of quantum fields in discretised spacetimes. We show how familiar notions from Relativity and quantum causality can be recovered in a purely order-theoretic way from the causal order of events in spacetime, with no direct mention of analysis or topology. We formulate theory-independent notions of fields over causal orders in a compositional, functorial way. We draw a strong connection to Algebraic Quantum Field Theory (AQFT), using a sheaf-theoretical approach in our definition of spaces of states over regions of spacetime. We introduce notions of symmetry and cellular automata, which we show to subsume existing definitions of Quantum Cellular Automata (QCA) from previous literature. Given the extreme flexibility of our constructions, we propose that our framework be used as the starting point for new developments in AQFT, QCA and more generally Quantum Field Theory.

1 Introduction

The paper develops an order-theoretic, functorial framework for quantum fields that unifies algebraic, topological, and cellular-automata perspectives without directly using topology or analysis.

1 Introduction

  • Causal-set research instead studies locally finite posets and asks how continuum spacetime properties might emerge from discrete structure.Whether causal sets can be suitably embedded into Lorentzian manifolds remains incompletely answered.Source: §1 · p. 2
  • Algebraic approaches study local fields through observable algebras over spacetime regions, while topological approaches impose invariance under large-scale deformations.AQFT and TQFT both admit categorical formulations, but they use different kinds of spacetime pieces.Source: §1 · p. 2
  • Quantum Cellular Automata approximate full quantum fields using manageable discrete-lattice theories with discrete time evolution.The paper positions QCA alongside algebraic and topological approaches to quantum field theory.Source: §1 · pp. 2–3
  • The framework models causally well-behaved spacetimes through causal orders and seeks an operational, process-theoretic account of quantum theory and relativistic causality.Future-and-past–distinguishing Lorentzian manifolds are determined up to conformal equivalence by their causal orders, but the converse characterization remains open.Source: §1 · pp. 2–3
  • The paper proposes a theory-independent functorial field theory based solely on causal-order structure, connects it to AQFT through sheaf-theoretic states, and generalizes existing QCA notions.These constructions are presented as a flexible basis for developments in AQFT, QCA, and quantum field theory.Source: §1 · p. 3

2 Causal orders

This section reconstructs causal paths, dependence, slices, and regions using only poset structure, then organizes slices into a partially monoidal category for field evolution.

2 Causal orders

  • Causal orders are posets whose elements are events, with causal precedence given by the partial order; this supplies an abstract model of causally well-behaved spacetimes.The framework works in Pos with monotone maps and uses causal-order language for partial orders.Source: §2 · p. 3

2.1 Causal Paths

  • A causal path is a maximal totally ordered subset between two events, and causal diamonds are unions of all such paths.Causal paths form a category whose morphisms are paths, identities are singleton paths, and composition is set-theoretic union.Source: §2.1 · pp. 3–4

2.2 Space-like Slices

  • Domains of dependence select events whose every relevant causal path intersects a subset, yielding the inclusions D+(A) ⊆ J+(A) and D−(A) ⊆ J−(A).Further propositions constrain the causal futures and pasts of subsets lying in a domain of dependence.Source: §2.2 · pp. 4–5
  • Space-like slices are antichains, and Σ ↠ Γ holds exactly when Γ lies entirely in the future domain of dependence of Σ.Disjoint space-like slices combine by a partial tensor product given by disjoint union, with the empty slice as unit; this relation is chosen so later field data is determined by earlier data and marginalization.Source: §2.2 · pp. 5–6

2.3 Diamonds and Regions

  • Regions are convex causal sub-orders containing every causal path between their events, and the region between Σ ↠ Γ is J+(Σ) ∩ J−(Γ).Every closed bounded region, including every finite region, is the region between its minimal and maximal slices.Source: §2.3 · pp. 7–8

3 Categories of slices

The paper restricts slices into usable partially monoidal categories and defines causal-order morphisms that preserve the relevant order structure, including region, refinement, and quotient factorizations.

3 Categories of slices

  • Categories of slices retain enough selected slices to reconstruct causal-order structure while permitting restrictions that exclude irregular or physically unsuitable slices.They must satisfy event coverage, closure under regional restriction, and partial monoidal-subcategory conditions.Source: §3 · pp. 8–9
  • Cauchy slices intersect every bi-infinite causal path exactly once, and any foliation by pairwise disjoint Cauchy slices generates a category of Cauchy slices.The resulting category contains slices lying within the foliation’s Cauchy slices.Source: §3 · p. 10

3.1 The category of causal orders

  • The category CausOrd uses causal orders as objects and specialized order-preserving maps as morphisms, with disjoint union as a symmetric monoidal product.Its morphisms require x < y in the source whenever f(x) < f(y) in the target.Source: §3.1 · p. 11
  • Every causal sub-order inclusion factors essentially uniquely into a region inclusion followed by a refinement.The factorization isolates the convex region containing the included events from the additional events inserted between them.Source: §3.1 · pp. 11–12
  • Every CausOrd morphism also admits an essentially unique epi-mono factorization into a causal quotient followed by a causal sub-order.The paper illustrates a causal quotient from the honeycomb lattice to the diamond lattice and defines pullbacks of slices along such maps.Source: §3.1 · p. 12, Figure 5 · p. 13

4 Causal field theories

The framework equips causal orders with theory-independent quantum fields as monoidal functors, supporting field evolution, locality, causality, and no-signalling across flexible process theories.

4.1 Categories for quantum fields

  • The framework is agnostic to the symmetric monoidal category and physical theory used to model quantum fields.It accommodates causal process theories generally, including finite- and infinite-dimensional examples.Source: §4.1 · pp. 13–14

4.2 Causal field theories

  • A causal field theory is a monoidal functor from a category of slices on a causal order to a symmetric monoidal field category.Objects represent field spaces over slices, while morphisms encode evolution between slices.Source: §4.2 · pp. 14–15
  • Functorial evolution maps initial field states to final states, while monoidality assigns disjoint slices the tensor product of their field spaces.This tensor-product requirement is stronger than AQFT’s commutation condition for spacelike-separated regions.Source: §4.2 · p. 15
  • For spacelike-separated slice pairs, monoidality makes the joint evolution factor into the product of the two individual evolutions.The factorisation expresses locality or clustering in the causal field theory.Source: §4.2 · pp. 15–16

4.3 Causality and no-signalling

  • With injectivity imposed, the induced effects form an environment structure, so states on a slice do not depend on future or spacelike-separated field states.These effects generalize discarding maps and partial traces in quantum theory.Source: §4.3 · p. 16
  • The asymmetric ordering of slices breaks time symmetry and yields causal and no-signalling constraints from functorial evolution.The conditions defining forward and reversed slice arrows are not generally equivalent.Source: §4.3 · pp. 16–17

5 Connection with Algebraic Quantum Field Theory

The framework connects causal field theories to AQFT through decompositional, region-based presheaves of states rather than algebras. It establishes restriction, global-state, and reversibility results for these constructions.

AQFT connection

  • Causal field theories are decompositional: they evolve fields over one fixed spacetime, unlike the compositional construction of arbitrary spacetimes in TQFTs.This perspective is shared with AQFT and motivates moving from functors on slices to presheaves on regions.Source: §5 · p. 17

Regions and restrictions

  • Regions are unions of bounded diamonds, and restricting a category of slices to any such region again yields a category of slices.This closure property supports defining field theories and state spaces locally over regions.Source: §5 · p. 18

Spaces of states

  • A causal field theory restricts to each region, while StatesΨ(R) collects slice-state families stable under the field theory's evolution.States on objects of the field category are morphisms from the monoidal unit, StatesD(Ψ(∆)) = HomD[I, Ψ(∆)].Source: §5 · pp. 18–19

Presheaf structure

  • The regional state spaces form a presheaf StatesΨ: Regions(C)^op → Set, with restriction maps for inclusions and compatible families defining global states.Global states restrict to components on every region, and a state over a region Ω corresponds to a global compatible family.Source: §5 · p. 19

Scope

  • The construction does not develop enriched state spaces, and a more general succinct description is left for settings where every region has a suitable Cauchy slice and the theory is reversible.The authors state that enrichment can be added readily, while the broader AQFT connection remains future work.Source: §5 · pp. 20–21

Causal reversibility

  • For reversible causal field theories on foliated causal orders, the state on any Cauchy slice uniquely determines—and can reconstruct—the global state.CauchySlices(F) is causally reversible, and the result applies to every slice in the foliation, not only an initial one.Source: §5 · p. 21

6 Connection to quantum cellular automata

The paper places quantum cellular automata within the standard lattice-based picture of local, homogeneous discrete-time quantum evolution. It notes that the framework is designed to capture and generalize such constructions.

Cellular automata

  • A classical cellular automaton evolves values on a lattice through a fixed local and homogeneous update rule.Each site at the next time step depends only on a finite neighborhood of sites at the current step.Source: §6 · p. 22

Quantum cellular automata

  • A QCA replaces lattice values with pure states in an infinite tensor product of finite-dimensional Hilbert spaces and replaces the update rule with a local, homogeneous unitary.The Hilbert spaces associated with lattice sites are finite-dimensional and isomorphic.Source: §6 · p. 22

Relation to prior QCA work

  • The framework is intended to encompass and greatly generalize existing QCA constructions, including the construction presented in [1].Earlier work showed that certain free quantum-field evolutions can arise as continuous limits of QCAs.Source: §6 · p. 22

6.1 Causal cellular automata

Causal cellular automata reformulate QCA requirements using causal orders, foliations, slice symmetries, and causal field theories. The resulting definition retains locality and homogeneity while treating spacetime evolution decompositionaly and leaving reversibility explicit.

Framework

  • The framework treats all of spacetime at once, recovering states over slices and regions through decomposition rather than step-by-step updates.Homogeneity is expressed as invariance under suitable spacetime symmetries.Source: §6.1 · pp. 22–23

Symmetry

  • A symmetry is a group action by causal-order automorphisms extended to slices, preserving objects, causal morphisms, and defined tensor products.Symmetry-invariant field theories carry natural isomorphisms Ψ ⇒ Ψ◦g satisfying the group law.Source: §6.1 · p. 23

Causality

  • Causality is automatic: a slice state never depends on a spacelike-separated slice state.This replaces the finite-neighborhood locality condition used in the usual lattice formulation.Source: §6.1 · pp. 23–24

Unitarity and reversibility

  • Unitarity is not built into the general framework because it is process-theory-specific and ordinary QCA formulations omit restrictions, whereas the framework treats restrictions as integral to evolution.Reversibility is identified as the foundational feature of interest but remains an explicit desideratum rather than part of the basic definition.Source: §6.1 · p. 24

Causal cellular automata

  • A causal cellular automaton consists of a foliation, a category of Cauchy slices, and a group action transitive on the foliation's Cauchy slices.These ingredients generalize discrete time, equal-time hypersurfaces, and spatial and temporal homogeneity in QCAs.Source: §6.1 · p. 24

QCA correspondence

  • The usual QCA requirements map to the new framework: foliations represent time steps, slices support fixed-time states, and symmetry invariance represents homogeneity.The symmetry action is transitive on the foliation and the field theory is G-invariant.Source: §6.1 · pp. 24–25

6.2 Partitioned causal cellular automata

Partitioned causal cellular automata are constructed as G-invariant causal field theories over a discrete Minkowski causal order, with slice morphisms combining restriction, evolution, and marginalisation.

6.2 Partitioned causal cellular automata

  • Partitioned CCAs generalise partitioned QCAs by allowing arbitrary scattering processes and state restriction to non-Cauchy equal-time surfaces.The construction is presented as a large family of causal cellular automata derived from partitioned QCAs.Source: §6.2 · p. 25

6.2.1 Causal order

  • The causal order uses a discrete (1+d)-dimensional Minkowski lattice whose events have 2d immediate predecessors and successors determined by the neighbourhood N={±1}^d.For d=1, this yields the diamond lattice.Source: §6.2.1 · p. 25

6.2.2 Foliation and category of slices

  • The foliation consists of constant-time Cauchy slices, with finite same-time subsets as objects and morphisms defined through iterated neighbourhoods.The iterated neighbourhood N^(k) is formed by adding k copies of N, with parity-dependent lattice points.Source: §6.2.2 · pp. 25–26

6.2.3 Symmetry

  • The slice category is reversible, and its symmetry group G=Z^N is generated by translations from each event to one of its immediate causal successors.The generators act transitively on the foliation by sending each Cauchy slice to the next one.Source: §6.2.2 · p. 26, §6.2.3 · p. 26

6.2.4 Causal field theory - field over slices

  • A causal field theory assigns H^⊗N to each event, with each factor encoding the contribution from one immediate causal predecessor, and uses a scattering map U:H^⊗N→H^⊗N.The field category is a symmetric monoidal process theory with discarding maps generalising partial trace and CPTP structure.Source: §6.2.4 · pp. 26–27

6.2.5 Causal field theory - restriction and evolution

  • Functorial evolution acts on restrictions by discarding excluded events and on one-step evolutions by applying U followed by marginalisation of outputs not reaching the target slice.General slice morphisms factor into these restriction and evolution-marginalisation components, yielding a G-invariant causal field theory.Source: §6.2.5 · p. 27
  • When U is an isomorphism, the resulting partitioned causal cellular automaton is reversible through the corresponding construction on the reversed slice category.The framework therefore supplies both a G-invariant causal field theory and a reversibility condition.Source: §6.2.5 · p. 28

6.3 Sketch of the continuous limit for the Dirac QCA

The paper sketches a continuous Dirac-QCA limit by rescaling the lattice to infinitesimal spacing and using non-standard hyperfinite-dimensional Hilbert spaces. In this limit, infinitesimal scattering steps accumulate into continuous-time evolution following the Dirac equation.

6.3 Sketch of the continuous limit for the Dirac QCA

  • The continuous-limit construction rescales the discrete causal lattice to infinitesimal mesh ε and permits slices with infinitely many points approximating equal-time partial Cauchy hypersurfaces.Non-standard hyperfinite-dimensional Hilbert spaces handle the resulting infinite tensor products.Source: §6.3 · pp. 28–29
  • The scattering map uses the Dirac-QCA unitary with the real parameter ε replaced by an infinitesimal, so each update is infinitesimally close to the identity.Across infinitely many such slices, the first-order approximations become legitimate infinitesimal differentials.Source: §6.3 · p. 29
  • After taking the standard part, the rescaled lattice covers Minkowski spacetime and the functor becomes a continuous-time field evolution following the Dirac equation.The construction turns the discrete-limit argument into a formal sketch within the causal-field framework.Source: §6.3 · pp. 28, 30

7 Conclusions and Future work

The paper concludes that its order-theoretic, functorial framework supports causal field theories, AQFT-related state spaces, and both discrete and continuous QCA constructions. It identifies stronger AQFT connections, continuous-QCA analysis, and broader causal-order generalisations as future work.

7 Conclusions and Future work

  • The framework defines theory-independent causal field theories from causal order alone and models QCA in both traditional discrete and continuous formulations.It also connects to AQFT through spaces of states over spacetime regions.Source: §7 · p. 30
  • Future work includes strengthening the AQFT connection, characterising continuous QCA limits for perturbative quantum field theory, and extending the framework to indefinite causal order and enrichment.The authors also propose restricted classes of causal paths and more general slice categories.Source: §7 · p. 30

9 Funding

The work acknowledges support from the John Templeton Foundation and an EPSRC grant supporting MS.

9 Funding

  • The paper reports support from the John Templeton Foundation and an EPSRC grant, reference EP/P510270/1.It notes that the opinions expressed are those of the authors and may not reflect the Templeton Foundation’s views.Source: §9 · p. 30
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