Completing Dirac's 1937 complex-variable representation of quantum mechanics: real-axis pole prescriptions, singular operator representatives, boundary-condition selection, and delta-function representatives

In 1937 P. A. M. Dirac proposed reformulating quantum mechanics by letting the momentum variable of the position representation range over the complex plane: states are represented by functions analytic in a half-plane, dynamical variables act as singular integral kernels, and scalar products are contour integrals governed by a 'fundamental theorem' that fixes how poles on the real axis are passed. Dirac developed the formalism for one degree of freedom on the semi-infinite line and applied it to the hydrogen atom, but left five situations without systematic treatment: coincident simple poles of both factors of the scalar-product integrand on the real axis; higher-order poles on the real axis; the failure of the transformed equation to yield the correct origin boundary condition for the hydrogen s-state; the arbitrary integration constant in the representative of $q^{-1}$ , equivalent to adding a multiple of $\delta(q)$ ; and representatives containing positive powers of $p$ , corresponding to derivatives of the delta function. Prior treatments of the five cases are not known, so the problem of turning Dirac's case-by-case prescriptions into a systematic theory — or proving specific cases impossible within the framework — remains open.

Background

By a 'representation' of quantum mechanics one means a choice of variables in terms of which state vectors ('kets') and their duals ('bras') are represented by functions, and in which every dynamical variable (observable) is represented by a generalized matrix, i.e. a kernel carrying two such variables. In the standard position representation the variables are the real eigenvalues of the position operator $q$ , and momentum $p$ acts by differentiating the representative. In the 1937 paper 'Complex variables in quantum mechanics' Dirac asked instead that the momentum variable itself be complex: the ket factor $(p|\dots)$ of the scalar-product integrand $\int(\dots|p)\,dp\,(p|\dots)$ is taken regular in one half of the $p$ -plane bounded by the real axis, and the bra factor $(\dots|p)$ in the complementary half-plane. The scalar product is a contour integral along the real axis, and the paper's 'fundamental theorem' fixes the pairing: it assigns each factor its half-plane of regularity and thereby dictates on which side of a pole lying on the real axis the contour passes (below for a pole of the second factor, above for a pole of the first). Ordinary operators acquire singular kernels — the position operator kernel is recorded as proportional to $(p'-p'')^{-2}$ and that of $q^{-1}$ as proportional to $\log(p'-p'')$ — so the operator algebra resides entirely in these prescriptions. Dirac worked the formalism out for a single degree of freedom with the semi-infinite position interval (the setting of the radial hydrogen problem) and showed for hydrogen that requiring the representer to be regular in a half-plane, with branch points fixed by the potential, selects exactly the Bohr discrete energy levels.

Five situations were left unresolved. (1) If both factors of the integrand have a simple pole at the same point of the real axis, the prescriptions conflict and the theory breaks down; the paper says this never occurs in practice but gives no general argument. (2) Operators whose representatives involve higher-order real-axis poles are not ordinarily met, but may be; the suggested definition — the path avoids the singularity in the direction allowed by the fundamental theorem — is not developed into a theory. (3) In the hydrogen application the transformed equation does not automatically give the correct boundary condition at the origin for the s-state (the case $n=0$ in the paper's counting): the formal solution for $n=0$ is not a solution of the transformed equation, and additional physical reasoning is needed to obtain the correct wavefunction. (4) The representative of $q^{-1}$ contains an arbitrary constant of integration, corresponding to adding a multiple of the Dirac delta distribution $\delta(q)$ (the distribution concentrated at $q=0$ ); the choice is settled by boundary conditions at $q=0$ case by case, with no systematic rule. (5) Representatives containing positive powers of $p$ , which correspond to derivatives of $\delta(q)$ , are dismissed as practically unimportant, with no general prescription. In addition, the construction is given only for a single variable whose position domain is the semi-infinite interval; this is the recorded scope of the paper rather than a situation it flags as unresolved.

The subsequent history splits into lines that do not meet these cases. Dirac's own continuation — a 1943 Dublin monograph on quantum electrodynamics and a 1945 review — recast the idea in an oscillator-type setting: kets are represented by entire functions of the Bargmann type (the holomorphic representation adapted to the harmonic-oscillator state space), bras by meromorphic functions with poles, and the scalar product is a contour integral enclosing those poles. This is the form revived by Vourdas and Bishop in 1996 for thermofield dynamics (a real-time finite-temperature formalism) and developed by the Vourdas school for finite-dimensional quantum systems, coherent states, and quantum gates; Mohammed and Mageed (2024) extended it to finite-dimensional systems on the extended complex plane (the complex plane together with a point at infinity). This entire modern 'Dirac contour representation' shares the contour-integral skeleton but not the 1937 complex-momentum, half-line setting, and none of it encounters the five unresolved cases. A separate line — Galler, Canfield, and Freericks (2021, 2023) — reconstructs the Laplace method for the hydrogen atom (the complex-analytic integration technique Schrödinger used in 1926 and Dirac advocated) and documents historically that the 1937 approach 'appears to have been ignored by all (including Dirac himself)'. Finally, two rigorous frameworks border the construction without reconstructing it: the rigged Hilbert space (a distribution-theoretic refinement of the state space that justifies Dirac's bra-ket formalism and its delta-function representatives; de la Madrid 2005) and the Hardy-class theory of scattering (Hardy classes being spaces of functions analytic and square-integrable in a half-plane — the modern rigorous home of exactly the half-plane analyticity and pole-prescription structure the 1937 paper used; Gadella and Wickramasekara 2007). Neither cites the 1937 paper. The question is therefore what a serious mathematical treatment of the 1937 construction, as given, would say about the five cases.

Problem Statement

The problem is to replace the case-by-case prescriptions of Dirac's 1937 complex-variable representation with a systematic theory — or to prove that specified cases admit no consistent treatment within that framework. The common objective uniting the five cases is: for the complex-momentum representation on the semi-infinite line exactly as Dirac defined it (paired half-plane regularity of the two factors and the fundamental-theorem contour pairing), state precisely which states and operators are admissible, how each singular situation is to be defined, and how physical answers — spectra and boundary conditions — are extracted, with proofs or rigorous arguments. Concretely, the five jointly required cases are: (i) either prove that a simple pole of both factors of $\int(\dots|p)\,dp\,(p|\dots)$ at the same real point cannot arise for the admissible class of states, or characterize exactly when it arises and define the pairing in that case; (ii) give a definition of the action of operators whose representatives involve higher-order poles on the real axis, consistent with the contour theorem, and verify the consistency; (iii) derive, within the formalism rather than by ad hoc physical reasoning, the correct boundary condition at the origin for the hydrogen s-state (the $n=0$ case), whose formal solution fails to solve the transformed equation; (iv) give a rule that fixes the arbitrary integration constant in the representative of $q^{-1}$ — equivalently the multiple of $\delta(q)$ — for arbitrary problems, or characterize precisely the residual ambiguity and what boundary information resolves it; and (v) give a general prescription for representatives containing positive powers of $p$ , corresponding to derivatives of $\delta(q)$ . Each of the five cases may be settled either by proving that the problematic configuration cannot arise for admissible states or by an explicit construction, in each case with a proof or rigorous argument. The acceptance boundary is confined to the single-variable, semi-infinite-interval representation exactly as Dirac defined it; an extension to several variables or to more general domains is not a required part of the completion.

References

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P. A. M. Dirac, 'Quantum electrodynamics', Communications of the Dublin Institute for Advanced Studies, Series A, No. 1, 36 pp. (1943). zbMATH: Zbl 0060.45502, https://zbmath.org/?q=an%3A0060.45502
P. A. M. Dirac, 'On the Analogy Between Classical and Quantum Mechanics', Reviews of Modern Physics 17 (2-3), 195-199 (1945). DOI: 10.1103/RevModPhys.17.195, https://doi.org/10.1103/RevModPhys.17.195
A. Vourdas and R. F. Bishop, 'Dirac's contour representation in thermofield dynamics', Physical Review A 53 (3), R1205-R1208 (1996). DOI: 10.1103/PhysRevA.53.R1205, https://doi.org/10.1103/PhysRevA.53.R1205
A. Vourdas, 'Analytic representations in quantum mechanics', Journal of Physics A: Mathematical and General 39 (7), R65-R141 (2006). DOI: 10.1088/0305-4470/39/7/R01, https://doi.org/10.1088/0305-4470/39/7/R01
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A. Vourdas, 'Coherent spaces, Boolean rings and quantum gates', Annals of Physics 373, 557-580 (2016). arXiv:1608.03437, https://arxiv.org/abs/1608.03437, DOI: 10.1016/j.aop.2016.07.034
Aisha Faraj Abukhzam Mohammed and Ismail Mageed, 'Dirac Contour Representation for Quantum Systems with Finite-Dimensional Hilbert Space in the Extended Complex Plane', Advances in Science and Technology 144, 3-15 (2024). DOI: 10.4028/p-kbygq1, https://doi.org/10.4028/p-kbygq1
Anna Galler, Jeremy Canfield, and James K. Freericks, 'Schrödinger's original quantum-mechanical solution for hydrogen', European Journal of Physics 42 (3), 035405 (2021). arXiv:2007.14798, https://arxiv.org/abs/2007.14798, DOI: 10.1088/1361-6404/abb9ff
Jeremy Canfield, Anna Galler, and James K. Freericks, 'The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics', Quantum Reports 5 (2), 370-397 (2023). arXiv:2208.07433, https://arxiv.org/abs/2208.07433, DOI: 10.3390/quantum5020024
R. de la Madrid, 'The role of the rigged Hilbert space in Quantum Mechanics', European Journal of Physics 26 (2), 287-312 (2005). arXiv:quant-ph/0502053, https://arxiv.org/abs/quant-ph/0502053, DOI: 10.1088/0143-0807/26/2/008, https://doi.org/10.1088/0143-0807/26/2/008
M. Gadella and S. Wickramasekara, 'On the applications of Hardy class functions in scattering theory' (2007). arXiv:0707.0778, https://arxiv.org/abs/0707.0778