The 1966 Titulaer–Glauber paper 'Density Operators for Coherent Fields' proved, under two explicitly stated hypotheses — a denumerable set of field modes (as in a finite enclosure, where a minimum frequency excludes infrared divergences) and a finite mean total photon number ⟨N⟩\langle N\rangle — that every first-order coherent field reduces to the excitation of a single, generally nonmonochromatic mode; that the higher-order coherence degrees g(n)g^{(n)} are equivalent to factorial-moment conditions on the photon-number distribution of that mode; and that full coherence holds if and only if that distribution is Poissonian, while the off-diagonal density-matrix elements BmnB_{mn} remain entirely unconstrained by the coherence conditions. The open problem asks whether these structure theorems remain valid, or must be modified, when the mode set is a continuum (free-space fields) or when ⟨N⟩\langle N\rangle diverges (infrared-divergent constructions such as soft-photon dressing clouds), and asks for a characterization of the off-diagonal freedom beyond positivity and unit trace, in particular under positive Glauber–Sudarshan PP representability.

Background

Optical coherence theory classifies radiation fields through the factorization properties of their quantum correlation functions. The state of the quantized electromagnetic field is described by a density operator ρ\rho, a positive trace-class operator of unit trace (trace-class meaning that its eigenvalues are absolutely summable, so that traces of field operators are well defined) whose off-diagonal elements in the photon-number basis — the basis of eigenstates of the total photon-number operator — encode coherences between number states. The normally ordered correlation functions are G(n)=tr{ρE(−)(x1)⋯E(−)(xn)E(+)(xn)⋯E(+)(x1)}G^{(n)}=\mathrm{tr}\{\rho E^{(-)}(x_1)\cdots E^{(-)}(x_n)E^{(+)}(x_n)\cdots E^{(+)}(x_1)\}, where E(+)E^{(+)} and E(−)E^{(-)} are the positive- and negative-frequency parts of the electric-field operator and each xix_i abbreviates position, time, and a polarization index. A field possesses first-order coherence when G(1)(x1,x2)=E∗(x1)E(x2)G^{(1)}(x_1,x_2)=\mathcal{E}^*(x_1)\mathcal{E}(x_2) for some positive-frequency solution E(x)\mathcal{E}(x) of the wave equation (a complex solution built only from the e−iωte^{-i\omega t} time dependence); this is the condition of maximum fringe visibility in interferometry. The normalized higher-order degrees g(n)g^{(n)} describe photon-correlation (Hanbury Brown–Twiss-type) experiments, and a field is fully coherent when g(n)=1g^{(n)}=1 for all nn, the idealization of a perfect laser field (Glauber 1963; Sudarshan 1963).

A mode of the field is a solution of the wave equation used as a basis function for expanding the field operators; for a field in a finite enclosure the monochromatic modes form a denumerable (countable) set. Titulaer and Glauber ('Correlation Functions for Coherent Fields', 1965, and the source paper 'Density Operators for Coherent Fields', 1966) asked what the coherence conditions impose on ρ\rho itself. Introducing nonmonochromatic modes — arbitrary wave-packet superpositions of the monochromatic modes, with annihilation operator bb — they proved three structure theorems: (a) every field with first-order coherence has a density operator that reduces to the excitation of a single (generally nonmonochromatic) mode,

ρ=∑m,nBmn(b†)mm!∣vac⟩⟨vac∣bnn!,\rho=\sum_{m,n}B_{mn}\frac{(b^\dagger)^m}{\sqrt{m!}}|\mathrm{vac}\rangle\langle\mathrm{vac}|\frac{b^n}{\sqrt{n!}},

where ∣vac⟩|\mathrm{vac}\rangle is the vacuum; (b) once first-order coherence holds, the higher-order coherence degrees g(n)g^{(n)} are completely determined by, and determine, the factorial moments of the distribution of the photon number in that mode (BmmB_{mm} is the probability of mm photons); and (c) full coherence (g(n)=1g^{(n)}=1 for all nn) holds if and only if BmmB_{mm} is a Poisson distribution. The eigenstates of the annihilation operators (the coherent states, b∣β⟩=β∣β⟩b|\beta\rangle=\beta|\beta\rangle) are the special case Bmn=e−nˉβm(β∗)n/m!n!B_{mn}=e^{-\bar n}\beta^m(\beta^*)^n/\sqrt{m!n!}; but the class of fully coherent fields is strictly larger, because the coherence conditions constrain only the diagonal elements BmmB_{mm} and leave the off-diagonal elements BmnB_{mn} entirely free, subject only to positivity of the matrix BB and unit trace. The Glauber–Sudarshan PP representation writes a density operator as a mixture of coherent states, ρ=∫P(α)∣α⟩⟨α∣ d2α\rho=\int P(\alpha)|\alpha\rangle\langle\alpha|\,d^2\alpha; states for which PP is a genuine probability density are regarded as classical, and the source paper already notes (in a footnote) a fully coherent pure state with a nonlinear phase pattern θm\theta_m that cannot be represented by means of the PP representation, so the fully coherent class straddles the classical/nonclassical boundary.

The proofs rest on two assumptions that the source paper states explicitly: the mode set is denumerable — as for a field in any finite enclosure, where a minimum frequency exists and infrared divergences do not occur — and the mean total photon number ⟨N⟩\langle N\rangle is finite, which guarantees that the expansion coefficients βk\beta_k of the excited mode are square-summable (i.e. ∑k∣βk∣2\sum_k|\beta_k|^2 converges). Two physically important situations lie outside both hypotheses. First, the free-space field has a continuum of modes. Second, in quantum electrodynamics, charged particles are accompanied by soft-photon clouds (photons of arbitrarily low frequency) whose coherent dressing states have a divergent mean photon number; such infrared-divergent constructions were developed by Kibble (1968) and by Kulish and Faddeev (1970) and are central to the modern infrared-sector program. States with divergent ⟨N⟩\langle N\rangle typically cease to live on Fock space — the Hilbert space generated by repeatedly applying creation operators to the vacuum, on which every state has finite total photon number — and are instead defined as weak limits (limits of matrix elements rather than of vectors) or as states in non-Fock representations of the field algebra. In that regime even the photon-number distribution, on which the Poisson characterization rests, may fail to normalize, and the status of all three theorems is unknown.

Problem Statement

The objective is to complete the Titulaer–Glauber characterization of density operators for coherent fields beyond the two hypotheses under which it was proved — a denumerable mode set and a finite mean total photon number ⟨N⟩\langle N\rangle — and to characterize the off-diagonal freedom that the coherence conditions leave in the single-mode density matrix. The question has three linked parts, and all three are required.

Part 1 (continuum of modes). For a field quantized over a continuum of modes (free space), retaining the finite mean total photon number ⟨N⟩<∞\langle N\rangle<\infty assumed by the source theorem so that the density operator exists, establish or refute the analogs of the three structure theorems: (a) every field whose first-order correlation function factorizes, G(1)(x1,x2)=E∗(x1)E(x2)G^{(1)}(x_1,x_2)=\mathcal{E}^*(x_1)\mathcal{E}(x_2) with E\mathcal{E} a positive-frequency solution of the wave equation, has a density operator that reduces to the excitation of the single (nonmonochromatic) mode built from E\mathcal{E}; (b) once first-order coherence holds, the higher-order coherence conditions on the normalized correlations g(n)g^{(n)} are equivalent to the corresponding conditions on the factorial moments of the photon-number distribution of that mode; (c) full coherence (g(n)=1g^{(n)}=1 for all nn) holds if and only if that distribution is Poissonian. Each of (a)–(c) is a separate target: proving or refuting any one of them is a partial result, and Part 1 is fully resolved only when all three have been established or refuted.

Part 2 (divergent mean photon number, infrared regime). For infrared-divergent constructions over an arbitrary mode set (denumerable or continuous) — states or asymptotic limits with ⟨N⟩=∞\langle N\rangle=\infty, such as soft-photon dressing clouds — determine in what sense, if any, the three theorems survive: what mathematical object replaces the trace-class density operator and the photon-number distribution, and whether an analog of the Poisson characterization of full coherence can be constructed or provably cannot exist.

Part 3 (off-diagonal freedom). Characterize the off-diagonal elements BmnB_{mn} of the single-mode density matrix of a first-order or fully coherent field: beyond positivity of the matrix BB and unit trace, determine which choices are admissible under positive Glauber–Sudarshan PP representability, giving necessary and sufficient conditions together with boundary examples on both sides of that boundary.

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