In 1957 P. A. M. Dirac set up quantum electrodynamics in the Schrödinger picture with the filled negative-energy electron sea and constructed successive approximations to the physical vacuum: a first-approximation ket obtained by a Coulomb-field unitary dressing of the free ket, which enforces the Gauss-law supplementary condition, and a second-approximation ket ∣V2⟩|V_2\rangle whose additional photon-dependent dressing removes all order-ee pair-creating terms, so that deviations from stationarity begin only at order e2e^2. He left open three linked questions: whether ∣V2⟩|V_2\rangle is Lorentz invariant; whether it is, or approximates, the minimum-energy ground state of the full Hamiltonian, given that correlations between the filled sea and the field might lower the energy without bound; and how to obtain an accurate expression for the exact vacuum ket, either by enlarging the dressing ansatz or by modifying the Hamiltonian in a way that preserves agreement with experiment and is Lorentz-covariant.

Background

Quantum electrodynamics (QED) is the quantum field theory of electrons, positrons and photons. By the late 1950s practical QED was organized in the interaction representation (also called the Dirac picture), in which field operators evolve with the free part of the Hamiltonian and the interaction generates transitions between free-field states. Dirac regarded this framework as inadequate for the most basic stationary-state question of the theory, the vacuum: applying the full Hamiltonian HH to the free vacuum generates states containing an electron–positron pair (with or without an accompanying photon) with amplitudes proportional to the charge ee multiplying divergent momentum integrals, so the free vacuum is not stationary, and the standard renormalized perturbation series never delivers a closed stationary state vector. The alternative perturbative route — defining the interacting vacuum by adiabatic switching in the interaction representation (turning the interaction on infinitely slowly so that the free vacuum is continuously deformed into the interacting one), as in Gell-Mann and Low's bound-state construction of 1951 — yields a series whose individual vacuum diagrams diverge, and Dirac explicitly set that whole framework aside.

Dirac's 1957 paper therefore works in the Schrödinger picture, in which states are fixed kets (state vectors) on which the full Hamiltonian acts, and asks directly for a vacuum ket ∣V⟩|V\rangle with

H∣V⟩=C∣V⟩,H|V\rangle = C|V\rangle,

where CC is a constant. The formulation uses the historical sea description of the electron–positron field: every single-electron state of negative energy is regarded as filled (the Dirac sea), a positron is a hole in the sea, and the free reference ket ∣S⟩|S\rangle is the state in which the sea is completely filled and no photons are present. The theory is quantized in the Coulomb gauge, in which the scalar potential is eliminated in favour of an instantaneous Coulomb interaction between charges and only the two transverse photon polarizations remain dynamical. In this gauge the longitudinal part of Gauss's law does not vanish as an operator equation but survives as a supplementary condition, that is, an operator constraint that every physical ket must satisfy:

(Erxr+e {ψˉϱxψ}) ∣ ⋅ ⟩=0,\big(E_{rx}{}^r + e\,\{\bar{\psi}\varrho_x\psi\}\big)\,|\,\cdot\,\rangle = 0,

where ErxrE_{rx}{}^r is the divergence of the electric field operator at the point xx, ψ\psi is the electron field operator, and {ψˉϱxψ}\{\bar{\psi}\varrho_x\psi\} denotes the charge density at xx with its vacuum diagonal part subtracted (braces indicate this subtraction throughout). The free ket ∣S⟩|S\rangle fails this condition, so it is not even a physical state of the interacting theory.

The paper's first approximation repairs this by a unitary dressing: ∣V1⟩=exp⁡(−ie{ψˉVψ})∣S⟩|V_1\rangle = \exp(-ie\{\bar{\psi}V\psi\})|S\rangle, where VV is the instantaneous Coulomb potential kernel. This operator attaches to every charge its own Coulomb field; the dressed ket satisfies the supplementary condition exactly, and the conjugated Hamiltonian H∗=exp⁡(ie{ψˉVψ})Hexp⁡(−ie{ψˉVψ})H^* = \exp(ie\{\bar{\psi}V\psi\})H\exp(-ie\{\bar{\psi}V\psi\}) acting on ∣S⟩|S\rangle contains no order-ee terms that violate Gauss's law. However, H∗H^* still contains terms bilinear in the electron operators that connect ∣S⟩|S\rangle to states with one electron–positron pair, so ∣V1⟩|V_1\rangle is not stationary. The second approximation therefore multiplies the dressing by a further unitary factor built from an operator-valued matrix λ\lambda that depends only on the photon creation operators (the operators that add photons to a state):

∣V2⟩=e−ie{ψˉVψ} e−ψˉλψ ∣S⟩.|V_2\rangle = e^{-ie\{\bar{\psi}V\psi\}}\,e^{-\bar{\psi}\lambda\psi}\,|S\rangle.

The matrix λ\lambda is determined only implicitly, by a nonlinear operator equation arranged so that conjugation of H∗H^* by this factor cancels all non-constant bilinear terms; writing the unknown matrix e−λe^{-\lambda} column by column, each column satisfies a Dirac-type wave equation in which the photon creation operators play the role of variables, describing an electron accompanied by a cloud of photons. With this choice the deviations of ∣V2⟩|V_2\rangle from stationarity first appear at order e2e^2, as quartic terms that create two electron–positron pairs simultaneously — terms whose momentum integrals still diverge, so the ultraviolet infinity is postponed to higher order rather than removed. Acting on ∣V2⟩|V_2\rangle with a photon creation operator or an electron creation operator then gives approximate one-photon and one-electron stationary kets; the one-electron ket describes a dressed electron carrying its Coulomb field and its near-zone photon cloud rather than a bare particle.

A relativistic theory should also have a Lorentz-invariant vacuum: a state that every Lorentz transformation (boost or rotation) maps to itself up to a phase. Because the Schrödinger picture singles out a time direction, Lorentz covariance of a constructed vacuum ket is not automatic and must be verified. The paper closes by leaving precisely this question open, together with the energy question — a unitary dressing of the free ket does not by itself minimize the energy, and classical electrodynamics admits arbitrarily large negative interaction energies (runaway-type behaviour), so quantum correlations between the filled sea and the field could in principle lower the true ground-state energy without bound. For the exact ket Dirac left the method of construction open as well, describing two possible routes: enlarging the dressing ansatz to more complicated operator forms and searching for kets satisfying the eigenket equation to higher accuracy, with the deviation from stationarity controlled and systematically improvable; or modifying the Hamiltonian by adding terms, chosen so that agreement with experiment is not spoiled and the vacuum ansatz becomes an exact eigenket. Gell-Mann and Low (1951) is included here as the standard interaction-representation alternative whose framework Dirac rejected; the infraparticle literature of the 1970s–80s (infrared dressing of charged states — an infraparticle being a charged particle whose unavoidable cloud of very low energy photons prevents it from being a state with a definite photon number) addresses the charged sectors, not the vacuum ket. Within this formulation the construction was subsequently taken up by G. Rosen (1968) and the Rosen–Brett series (1968–1978), who made the dressing well defined and accurate to order e2e^2 in the functional Schrödinger representation; the line continues in modern variational Schrödinger-picture quantum electrodynamics.

Problem Statement

Within Dirac's 1957 Schrödinger-picture formulation of quantum electrodynamics — the full Hamiltonian HH of the transverse photon field plus the electron–positron field with the filled negative-energy sea, quantized in the Coulomb gauge with the supplementary Gauss-law condition, together with the dressing construction that produces ∣V1⟩|V_1\rangle and ∣V2⟩|V_2\rangle from the free ket ∣S⟩|S\rangle — resolve the three linked questions left open by that work:

  1. Exact vacuum ket (the principal remaining problem). Obtain an accurate expression for the exact vacuum ket, that is, a ket ∣V⟩|V\rangle satisfying H∣V⟩=C∣V⟩H|V\rangle=C|V\rangle for the full Hamiltonian, within Dirac's formulation.

  2. Lorentz covariance. Establish whether the resulting vacuum ket is Lorentz invariant — mapped into itself up to a phase by Lorentz transformations — rather than singling out a preferred frame. A Hamiltonian modification invoked to make the vacuum ansatz an exact eigenket should be formulated in a Lorentz-covariant way, and the difficulty of doing so is part of the problem.

  3. Ground-state status. Establish whether the vacuum ket is, or approximates, the minimum-energy eigenket of the (possibly modified) Hamiltonian — in particular whether the spectrum is bounded below — or whether quantum correlations between the filled negative-energy sea and the electromagnetic field lower the true ground-state energy without bound, in analogy with the large negative interaction energies of classical electrodynamics.

The three questions are interdependent: obtaining the exact vacuum ket does not by itself establish its Lorentz covariance or its ground-state status.

References

  1. P. A. M. Dirac, "The vacuum in quantum electrodynamics", Il Nuovo Cimento 6 (Suppl. 1), 322–339 (1957). DOI 10.1007/BF02724789, https://doi.org/10.1007/bf02724789. Full text paywalled.
  2. G. Rosen, "The dressed vacuum in quantum electrodynamics", Il Nuovo Cimento A 57, 870–874 (1968). DOI 10.1007/BF02751391, https://doi.org/10.1007/BF02751391.
  3. Gerald Rosen, "Dressed-Electron Stationary States in Quantum Electrodynamics", Physical Review 172 (5), 1632–1634 (1968). DOI 10.1103/PhysRev.172.1632, https://doi.org/10.1103/PhysRev.172.1632.
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