A gravitationally induced decoherence model for photons in the context of the relational formalism

We formulate a model of gravitationally induced decoherence for photons starting from Maxwell theory coupled to linearised gravity, expressed in terms of Ashtekar-Barbero variables and treated as an open quantum field theoretic system. In contrast to quantum mechanical models, the interaction between the system (Maxwell field) and the environment (gravitational field) is not postulated phenomenologically, but is instead dictated by the underlying action in a post-Minkowskian approximation. This framework extends earlier models for a scalar field and enables a more detailed analysis of the role of dynamical reference fields (clocks) within the relational formalism. We show that, for a suitable choice of geometrical clocks together with a U(1)-Gauss clock, and by employing an appropriate combination of the observable map and its dual, the resulting Dirac observables are given directly by the transverse components of the photon field as well as the symmetric-transverse-traceless degrees of freedom of gravitational waves on the linearised phase space of the coupled system. In addition we also compare different choices of Dirac observables and their dynamics. Upon applying a Fock quantisation to the reduced system, we derive the time convolutionless (TCL) master equation, truncated at second order, and analyse its structural properties. These results provide a foundation for further investigations of the decoherence model, including its renormalisation and a detailed study of its one-particle sector, and are found to be structurally consistent with former master equations for photons derived using ADM variables and a specific gauge fixing.

💡 Research Summary

The paper presents a first‑principles derivation of a gravitationally induced decoherence model for photons, using the relational formalism to treat the coupled Maxwell‑linearised‑gravity system as an open quantum field theory. Starting from the action of Maxwell theory minimally coupled to linearised gravity, the authors rewrite the theory in terms of Ashtekar‑Barbero variables (the SU(2) connection A_i^a and its conjugate densitised triad E_a^i). This choice is motivated by possible connections to loop quantum gravity, although the present work proceeds with a standard Fock quantisation.

A central technical step is the selection of dynamical reference fields (clocks) that allow the construction of gauge‑invariant Dirac observables. The authors introduce two families of clocks: (i) geometric clocks δΞ_j associated with the geometric Gauss constraints of linearised gravity, and (ii) a U(1) Gauss clock δT_{U(1)} built from the electromagnetic sector. To turn these clocks into physical observables they employ two complementary maps: the observable map O_F, which lifts any phase‑space function to a gauge‑invariant extension once a set of clocks is fixed, and its dual O_dual_F, which interchanges the role of clocks and constraints. By applying the observable map followed by its dual, they obtain a set of mutually commuting clocks and a weakly Abelianised constraint algebra. The resulting Dirac observables are remarkably simple: the transverse components of the photon field A_T^a (and their conjugate momenta) and the symmetric‑transverse‑traceless (STT) modes of the linearised graviton h_{STT}^{ij} (with their momenta). The Poisson brackets between the electromagnetic and gravitational sectors vanish, while each sector reproduces the standard canonical algebra.

With the reduced phase space identified, the authors construct the physical Hamiltonian. It consists of the free Hamiltonians of the photon and graviton sectors plus an interaction term proportional to the contraction of the electromagnetic energy‑momentum tensor T_{μν} with the STT graviton field. This term encodes the back‑reaction of the photon on the gravitational environment and provides the source of decoherence.

Quantisation proceeds by promoting the reduced variables to operators on a Fock space. The photon field is quantised in the usual two‑polarisation basis; the graviton is quantised in its two physical STT polarisation modes. The total Hilbert space is thus a tensor product of photon and graviton Fock spaces. Treating the graviton as an environment, the authors trace it out using the projection‑operator technique. They derive a time‑convolutionless (TCL) master equation for the reduced photon density matrix, truncating the perturbative expansion at second order in the coupling (Born approximation). Unlike the standard Lindblad form, the TCL master equation retains a non‑Markovian memory kernel K(t,s) that couples the photon density matrix at earlier times to its present evolution. Nevertheless, the structural features of the master equation—such as the appearance of the energy‑momentum tensor in the dissipator—match earlier results obtained with ADM variables and explicit gauge fixing, confirming the consistency of the relational‑formalism approach.

The paper also explores alternative choices of clocks and constraint combinations, showing that the simplicity of the observable algebra strongly depends on the specific clock set. The authors argue that the selected set yields the most tractable algebra for canonical quantisation and for deriving the master equation.

Finally, the authors outline several avenues for future work: (1) explicit calculation of decoherence rates in the one‑photon sector, (2) extension to higher‑order perturbative terms and renormalisation within the relational framework, (3) systematic analysis of the conditions under which the Markov and rotating‑wave approximations become valid, and (4) comparison with a full loop‑quantum‑gravity quantisation of the Ashtekar‑Barbero variables to assess genuine quantum‑gravity corrections.

In summary, the article provides a rigorous, gauge‑invariant construction of photon‑gravity decoherence, demonstrates how relational clocks can be used to isolate physical degrees of freedom, and delivers a structurally sound master equation that bridges earlier phenomenological models with a first‑principles quantum‑field‑theoretic description. This work lays a solid foundation for both theoretical extensions and potential experimental probes of gravitationally induced decoherence in quantum optics and related fields.