Geometric Approach to Quantum Theory. L-functionals

Abstract: This publication consists of slides from my talk at the Simons Center for Geometry and Physics in 2024. It contains a brief review of the L-functional formalism, along with a discussion of its possible applications to QED, linearized gravity, and quenched disorder. The most interesting part is the discussion of the infrared problem in QED and a conjecture on how to construct an infrared-finite perturbation theory for QED.

• The paper develops L-functionals as representation-independent state functionals whose doubled fields naturally generate Keldysh and thermo-field Green functions across inequivalent CCR representations.
• The paper identifies the inclusive scattering matrix as the appropriate asymptotic observable when conventional scattering fails, including in QED with soft-photon emission and for quasiparticle processes.
• The paper conjectures that absorbing incoming and outgoing particle currents into QED’s free Hamiltonian removes infrared divergences, while extending the framework programmatically to linearized gravity and quenched disorder.

Overview

This paper, consisting of slides from a 2024 talk by A. Schwarz at the Simons Center for Geometry and Physics, presents the formalism of L-functionals as a geometric approach to quantum theory and applies it to three problems: QED, linearized gravity, and quenched disorder. The central technical contribution is a conjecture that an infrared-finite perturbation theory for QED can be constructed within the L-functional framework by a suitable splitting of the interaction Hamiltonian. The material builds on Schwarz's monograph "Quantum mechanics and quantum field theory from algebraic and geometric viewpoints" (Springer, 2024) and on work with I. Frolov.

Geometric versus algebraic foundations

The algebraic approach to quantum theory takes as primitive data a unital associative $*$-algebra $\mathcal{A}$; states are positive normalized linear functionals $\omega$ with $\omega(A^*A)\geq 0$. The geometric approach inverts this: the starting point is the convex set of normalized states $N$, or the cone of states $\mathcal{C}$, regarded as subsets of a Banach space $L = \mathcal{A}^\vee$. Evolution operators $T_\tau$ are automorphisms of $N$. Within this framework the paper situates several constructions: decoherence induced by interactions with adiabatic random perturbations (with probabilities derived from decoherence), classical theories with restricted sets of observables — modeling the fact that physical devices measure only part of the observable algebra — and quantum mechanics as arising from such restricted theories.

When the geometric theory admits commutative groups of time and spatial translations, one obtains quantum field theory, with particles identified as elementary excitations of the ground state and quasiparticles as elementary excitations of translation-invariant stationary states. The natural asymptotic observable is then not the conventional scattering matrix but the inclusive scattering matrix, whose matrix elements give inclusive cross-sections for processes of the form $(M,N)\to(P,Q,\ldots,R)+\text{something}$, where "something" denotes unobserved soft or collinear radiation. The paper notes that an LSZ-type formula expresses the inclusive scattering matrix in terms of generalized Green functions taken on shell, and points out that this was independently rediscovered by Caron–Huot, Giroux, Hannesdottir, and Mizera (JHEP 2024) under the question "What can be measured asymptotically?" — asymptotic observables coincide with the inclusive scattering matrix.

Two structural claims deserve emphasis. First, when a particle interpretation exists, the inclusive scattering matrix carries the same information as the conventional one. Second, and more importantly for applications, an inclusive scattering matrix can exist even when the conventional scattering matrix does not — for instance for quasiparticles. An existence theorem holds for theories with the strong cluster property (i.e., with a gap). In QED the conventional scattering matrix fails to exist because every process involving a fixed number of particles has zero probability due to soft-photon emission; nevertheless, the inclusive scattering matrix can be defined as a limit of inclusive scattering matrices of gapped theories, and infrared divergences cancel in inclusive cross-sections.

L-functionals

The L-functional formalism goes back to Schwarz's 1967 work. Quantizing a classical theory yields CCR for smeared operators $\mathcal{A}$0, $\mathcal{A}$1 with $\mathcal{A}$2 in a test-function space $\mathcal{A}$3. For infinite-dimensional systems there exist representations of the CCR inequivalent to Fock representation; the key move is to represent every state — vector or density matrix in any representation — by the functional

$\mathcal{A}$4

which is well-defined for density matrices in any CCR representation. Working with L-functionals therefore amounts to treating all CCR representations simultaneously. Since $\mathcal{A}$5 generates correlation functions but is not analytic in $\mathcal{A}$6, it is written $\mathcal{A}$7, reflecting a systematic doubling of fields. This doubling connects the formalism to the Keldysh contour and to thermo-field dynamics: generalized Green functions $\mathcal{A}$8, with $\mathcal{A}$9 a chronological and $\omega$0 an antichronological product, are precisely the Green functions natural to L-functionals.

The state space $\omega$1 is identified with the dual of the exponential form of the Weyl algebra $\omega$2 (the norm closure of the Weyl operators). Each algebra element $\omega$3 acts on $\omega$4 by left and right multiplication, $\omega$5 and $\omega$6, and evolution takes the Schrödinger-like form $\omega$7 with a "Hamiltonian" $\omega$8 built from multiplication operators $\omega$9 and variational derivative operators $\omega(A^*A)\geq 0$0 acting on the two doubled fields. For a free Hamiltonian $\omega(A^*A)\geq 0$1, the stationary quasi-free states are Gaussians

$\omega(A^*A)\geq 0$2

with $\omega(A^*A)\geq 0$3 giving equilibrium (Bose) occupation numbers. Perturbation theory for the interaction-picture operator $\omega(A^*A)\geq 0$4 proceeds exactly as in the operator formalism.

Adiabatic scattering matrices and inclusive limits

The bridge between dynamics and scattering is the adiabatic switching construction: one evolves under $\omega(A^*A)\geq 0$5 with $\omega(A^*A)\geq 0$6, $\omega(A^*A)\geq 0$7, and studies the limit $\omega(A^*A)\geq 0$8. Both the conventional and the inclusive scattering matrices are obtained from the adiabatic scattering matrix multiplied by simple factors (a result attributed to Likhachev, Tyupkin, and Schwarz). In finite volume $\omega(A^*A)\geq 0$9, the conventional matrix requires

$N$0

with phase renormalization $N$1. The inclusive version replaces this by

$N$2

where $N$3 is fixed by requiring one-particle L-functionals to be $N$4-invariant, with $N$5 expressed through the one-particle energies of the coupled Hamiltonian. The relation between the two objects is compact:

$N$6

so the inclusive scattering matrix acts on states exactly as conjugation by the conventional matrix would — whenever the latter exists. A further consequence: the limit $N$7 is a stationary state of $N$8, and the diagram technique for correlation functions in this state coincides with Keldysh and TFD diagrammatics. This gives the L-functional method direct access to non-equilibrium stationary states.

Applications

Quenched disorder. When Hamiltonian coefficients are random, non-stationary problems reduce to evolving the state in perturbation theory and averaging over coefficients, which the L-functional (or Keldysh) language accommodates directly. In the stationary case, averaging of correlation functions at fixed temperature is possible via L-functionals at $N$9; more generally, the formalism permits averaging at fixed entropy of the equilibrium state rather than fixed temperature — a flexibility not available in purely thermal formulations.

Classical currents (joint work with Frolov). For QED with the photon field treated against a prescribed divergence-free current $\mathcal{C}$0, the L-functional definition must be modified to respect the Lorenz gauge constraint $\mathcal{C}$1 (no modification is needed in Coulomb gauge), and a further modification restores manifest Lorentz invariance. The evolution equation is solved exactly by an exponential ansatz, yielding

$\mathcal{C}$2

where $\mathcal{C}$3 is the expectation value of the electromagnetic potential. The inclusive cross-section for emission of $\mathcal{C}$4 photons factorizes completely:

$\mathcal{C}$5

This exact solvability of the current-coupling sector is what makes the subsequent infrared analysis tractable.

The infrared problem in QED and the main conjecture

In Coulomb gauge the QED Hamiltonian splits as $\mathcal{C}$6, where $\mathcal{C}$7 is the instantaneous Coulomb term. The paper proposes to split the current coupling as $\mathcal{C}$8 plus a numerical piece $\mathcal{C}$9 absorbed into the free Hamiltonian. The conjecture states: with the right choice of numerical current, there are no infrared divergences in the L-functional formalism if the first line of the split is treated as the free Hamiltonian in perturbation theory. The refined version specifies the choice: $L = \mathcal{A}^\vee$0 should coincide with the current of the incoming particles as $L = \mathcal{A}^\vee$1 and with the current of the outgoing particles as $L = \mathcal{A}^\vee$2.

The proof sketch rests on a careful decomposition of the electron current using the Gordon identity,

$L = \mathcal{A}^\vee$3

Terms whose time dependence involves $L = \mathcal{A}^\vee$4 with $L = \mathcal{A}^\vee$5 are bounded away from zero frequency and cannot generate infrared divergences. The dangerous sector has frequencies $L = \mathcal{A}^\vee$6, which vanish linearly in $L = \mathcal{A}^\vee$7 — the origin of soft divergences. Writing $L = \mathcal{A}^\vee$8, the piece $L = \mathcal{A}^\vee$9 is shown not to contribute to infrared divergences after the Gordon rearrangement, leaving only $T_\tau$0 with

$T_\tau$1

Absorbing this into the free part of the doubled ("Keldysh") Hamiltonian — i.e., choosing the numerical current to track the actual charge trajectories at early and late times — is claimed to remove all infrared divergences from inclusive cross-sections. If correct, this would provide an infrared-finite perturbative scheme for QED without the usual apparatus of soft-photon resummation or coherent-state dressing, since the cancellation is built into the definition of the free Hamiltonian used in the expansion. It should be stressed that the paper provides only a sketch; a complete proof with control over higher-order diagrams is not given here.

Linearized gravity

The same machinery extends to gravitons. Writing $T_\tau$2, the L-functional is defined via $T_\tau$3. In Lorenz gauge the linearized Einstein equations read $T_\tau$4. The paper advocates the true radiation gauge of Chen and Zhu, $T_\tau$5, in which only the transverse part of the energy-momentum tensor sources the propagating components $T_\tau$6. The expected outputs are gravitational wave forms and inclusive cross-sections for graviton emission, directly analogous to the photon case. No detailed computation is presented in these slides, so the extension remains programmatic.

Limitations and open questions

Several caveats are explicit or implicit in the presentation. The infrared-finiteness claim for QED is stated as a conjecture supported by a sketch of proof; establishing it order by order in perturbation theory, and verifying that the refined choice of $T_\tau$7 (matching incoming and outgoing particle currents) suffices beyond low orders, remains open. The existence theorem for inclusive scattering matrices requires the strong cluster property, hence a gap; the treatment of massless theories such as QED proceeds instead by limiting arguments from gapped theories, whose interchange with other limits deserves scrutiny. The application to quenched disorder at fixed entropy is asserted without a worked example, and the gravity section stops at the level of gauge choices and expected quantities. Finally, whether the inclusive scattering matrix in QED, constructed via the proposed Hamiltonian splitting, reproduces the standard results of Yennie–Frautschi–Suura-type exponentiation is a natural check the slides do not perform.

Conclusion

The paper consolidates the L-functional formalism as a geometric, representation-independent formulation of quantum theory in which state doubling makes generalized (Keldysh-type) Green functions fundamental, and in which the inclusive scattering matrix — expressible through on-shell generalized Green functions — is the correct asymptotic observable. Its most substantive proposal is a concrete prescription for an infrared-finite perturbation theory of QED based on absorbing the soft current of external charges into the free Hamiltonian. Confirmation of this conjecture would place inclusive QED cross-sections on the same footing as ordinary S-matrix computations, and extending the scheme to linearized gravity and disordered systems defines the immediate open agenda.