Dirac-Coulomb Vacua: Spectral and Vacuum Reconstruction

• Dirac-Coulomb vacua are vacuum constructions defined via spectral projectors of Dirac operators in external Coulomb and singular backgrounds, establishing in-, out-, charged, and no-pair states.
• They address key aspects such as spectral separation, self-adjointness at the Coulomb singularity, vacuum polarization, and renormalized charge density, essential for relativistic atomic and molecular theories.
• Supercritical regimes induce vacuum reconstruction with level diving and shell formation, while subcritical settings ensure stable polarized states and practical computational projections like those used in graphene studies.

Dirac-Coulomb vacua are vacuum constructions associated with Dirac operators in external Coulomb fields and closely related singular backgrounds. In the cited literature, these constructions include the static negative-energy spectral projector of a Dirac-Coulomb Hamiltonian, the in- and out-vacua for Hamiltonians that are Coulombic outside a finite time interval, the charged vacuum that emerges when a Coulomb source becomes supercritical and bound levels dive into the lower continuum, and the projected positive-energy “no-pair” sector used in relativistic atomic and molecular computations. Across these settings, the central issues are spectral separation, self-adjointness at the Coulomb singularity, vacuum polarization, renormalized charge density, and the extent to which a filled negative-energy sea can be regarded as a genuine ground state (Baskin et al., 21 Jul 2025, Davydov et al., 2017, Jeszenszki et al., 2021).

1. Basic vacuum constructions in Coulomb backgrounds

For a static Dirac-Coulomb Hamiltonian, the natural vacuum is the spectral projection onto negative energies,

$\gamma=\mathbf 1_{(-\infty,0)}(H).$

This is the Dirac-Coulomb vacuum in the strict external-field sense. When the Hamiltonian is static only in the remote past and future, the natural replacements are the in- and out-vacua, obtained by propagating the negative spectral projections of the asymptotic Hamiltonians forward or backward in time (Baskin et al., 21 Jul 2025).

In the strong-field literature, the same vacuum concept acquires a non-perturbative interpretation. For subcritical Coulomb fields, the vacuum is polarized but remains neutral after renormalization. For overcritical fields, discrete levels cross the lower continuum threshold at $E=-1$, the vacuum reconstructs, and the physically relevant state becomes a charged vacuum with vacuum shells and nonzero induced charge (Davydov et al., 2017, Davydov et al., 2017, Davydov et al., 2017).

In relativistic many-electron computation, a different but related construction is used. The bare Dirac-Coulomb spectrum contains positive-energy electronic states, negative-energy positronic states, and mixed electron-positron continuum states; to avoid Brown-Ravenhall continuum dissolution, the Hamiltonian is projected to the positive-energy subspace before diagonalization,

$$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$

This “no-pair” sector is a vacuum-defined electronic subspace rather than a dynamical pair-producing vacuum (Jeszenszki et al., 2021).

Setting	Vacuum prescription	Characteristic feature
Static Coulomb Hamiltonian	$\gamma=\mathbf 1_{(-\infty,0)}(H)$	Filled negative-energy spectral subspace
Asymptotically static $H(t)$	In/out vacua from propagated negative spectral projections	Hadamard for $r\neq 0$
Supercritical Coulomb source	Vacuum including dived levels	Charged vacuum with vacuum shells
Many-electron no-pair theory	$$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$	Positive-energy projected sector

These constructions are not interchangeable. The static spectral projector is a one-body definition; the charged vacuum of supercriticality is a non-perturbatively reconstructed many-body state; the no-pair sector is a computational restriction; and the microlocal Hadamard vacuum is defined by its short-distance singularity structure rather than by an energy minimization principle alone.

2. One-particle Dirac-Coulomb operator and spectral decomposition

The operator-theoretic basis for Dirac-Coulomb vacua is the one-particle Dirac equation with minimal electromagnetic coupling,

$$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$

with a time-independent Coulomb field

$$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$

After separation of variables under spherical symmetry, the problem reduces to the standard radial Dirac-Coulomb system with spin-orbit quantum number

$\kappa=\pm 1,\pm 2,\dots$

A unitary transformation

$E=-1$0

is then used to decouple the first-order radial equations into a Schrödinger-like second-order equation for one spinor component, with

$E=-1$1

and the other component reconstructed by a kinetic-balance relation (Alhaidari et al., 2012).

A particularly explicit representation is obtained on a Laguerre-type $E=-1$2 basis,

$E=-1$3

with wavefunction expansion

$E=-1$4

Projection onto this basis yields a symmetric tridiagonal matrix representation and a three-term recursion relation,

$E=-1$5

with

$E=-1$6

The coefficients are identified with Pollaczek polynomials after the parameter matching

$E=-1$7

This turns the spectral problem into the asymptotic analysis of orthogonal polynomials (Alhaidari et al., 2012).

For $E=-1$8, the asymptotics are oscillatory and encode relativistic scattering amplitudes and phase shifts, including the logarithmic $E=-1$9-dependence characteristic of the Coulomb tail. For $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$0, square-integrability requires the dominant asymptotic term to vanish, leading to the quantization condition $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$1 and the standard relativistic Coulomb spectrum

$$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$2

The bound-state condition also requires the Coulomb interaction to be attractive,

$$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$3

The negative-energy sector is obtained from the positive-energy one by

$$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$4

This spectral separation is the formal input behind vacuum projectors: one first identifies the negative-energy subspace, then defines the vacuum by filling it (Alhaidari et al., 2012).

3. Static, in-, and out-vacua as Hadamard states

For a three-dimensional Dirac field in an external Coulomb potential with possible additional smooth time dependence supported in a finite time interval, the Hamiltonian is

$$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$5

with scalar potential

$$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$6

and

$$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$7

The condition $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$8 ensures that $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+.$$9 is essentially self-adjoint on the standard domain. The main analytic complication is the Coulomb singularity at $\gamma=\mathbf 1_{(-\infty,0)}(H)$0, which produces diffractive propagation of singularities rather than the standard smooth-microlocal picture (Baskin et al., 21 Jul 2025).

In the static case, the vacuum density matrix is the negative spectral projector

$\gamma=\mathbf 1_{(-\infty,0)}(H)$1

If the Hamiltonian is static in the past, the in-vacuum is

$\gamma=\mathbf 1_{(-\infty,0)}(H)$2

and an analogous definition gives the out-vacuum from $\gamma=\mathbf 1_{(-\infty,0)}(H)$3. Under the assumptions $\gamma=\mathbf 1_{(-\infty,0)}(H)$4 and

$\gamma=\mathbf 1_{(-\infty,0)}(H)$5

the in-vacuum is Hadamard for $\gamma=\mathbf 1_{(-\infty,0)}(H)$6; the same argument applies to the out-vacuum (Baskin et al., 21 Jul 2025).

The Hadamard condition is formulated microlocally: a pair of two-point functions $\gamma=\mathbf 1_{(-\infty,0)}(H)$7 is Hadamard if

$\gamma=\mathbf 1_{(-\infty,0)}(H)$8

Because of the Coulomb singularity, the statement is weakened to Hadamard regularity away from the center. The paper proves more sharply that

$\gamma=\mathbf 1_{(-\infty,0)}(H)$9

where $H(t)$0 means that the points are connected by a diffractive bicharacteristic. Thus the Coulomb center does generate extra singular behavior, but in a controlled form compatible with the vacuum construction (Baskin et al., 21 Jul 2025).

A second central result concerns renormalized charge density. For two regular Hadamard density matrices $H(t)$1 and $H(t)$2, the relative charge density

$H(t)$3

is well-defined and satisfies

$H(t)$4

with near-origin behavior

$H(t)$5

This establishes that the renormalized vacuum polarization density remains locally integrable even at the Coulomb singularity. The paper also states that, for $H(t)$6, the Epstein--Glaser--Brunetti--Fredenhagen perturbative fermionic QFT based on $H(t)$7 is well-defined for any polynomial interaction (Baskin et al., 21 Jul 2025).

4. The minimum-energy question and the status of the filled sea

The identification of the vacuum with the filled negative-energy sea is older than the modern operator-theoretic formulation, and its status is not uniform across frameworks. In Dirac hole theory, the vacuum is postulated to be the state in which all negative-energy states are occupied and all positive-energy states are empty. The many-electron vacuum is built as a Slater determinant of orthogonal one-particle solutions, and expectation values of one-particle operators reduce to sums of one-particle expectation values. This provides a formal many-electron realization of the Dirac sea (Solomon, 2012).

A direct challenge to the standard claim that the filled sea is the minimum-energy state is given in a $H(t)$8-dimensional model with external static inverse square well

$H(t)$9

and Hamiltonian

$r\neq 0$0

The analysis is restricted to $r\neq 0$1, so there are no negative-energy bound states. Filling all negative-energy continuum states defines the vacuum energy

$r\neq 0$2

to be compared with the free vacuum energy $r\neq 0$3. Assuming the Dirac sea to be the minimum-energy state yields an inequality equivalent to

$r\neq 0$4

or, writing

$r\neq 0$5

to the requirement $r\neq 0$6 for $r\neq 0$7 (Solomon, 2012).

The vacuum polarization density is

$r\neq 0$8

and the integrated charge in the well,

$r\neq 0$9

is evaluated mode by mode in a finite form suitable for numerics. For the tested cases with $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$0, $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$1 is negative in all reported cases; the tabulated values include $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$2, $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$3, $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$4, $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$5, $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$6, $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$7, $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$8, $$\bar{\mathcal H}_\mathrm{DC}=\Lambda^+\mathcal H_\mathrm{DC}\Lambda^+$$9, and $$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$0. Hence

$$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$1

contradicting the inequality required by the minimum-energy assumption. The conclusion is that the standard Dirac sea is not the minimum-energy state in hole theory for this model (Solomon, 2012).

This result does not concern a Coulomb source, but it is directly relevant to the conceptual status of Dirac-Coulomb vacua. It shows that “filled negative-energy states” and “absolute ground state” need not coincide in every formulation. A plausible implication is that modern spectral and microlocal definitions of the Dirac-Coulomb vacuum should be distinguished from older hole-theoretic energy-minimization intuitions.

5. Supercritical Coulomb fields and charged vacuum reconstruction

The strongest departure from a neutral polarized vacuum occurs in the supercritical regime. In a $$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$2-dimensional Dirac-Coulomb system with an extended finite-size Coulomb source,

$$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$3

the first critical charge is

$$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$4

and the next threshold is

$$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$5

Below $$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$6, the renormalized induced charge vanishes. When a bound level dives through $$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$7, the vacuum acquires an additional shell, and because of the double degeneracy in the representation used, each dived level changes the induced charge by

$$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$8

Accordingly, the renormalized induced charge is $$\left(i\gamma^\mu \partial_\mu - e\gamma^\mu A_\mu - mc\right)\Psi = 0,$$9 at $$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$0, $$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$1 at $$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$2, and $$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$3 at $$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$4 (Davydov et al., 2017).

The density is treated non-perturbatively by a Wichmann-Kroll contour representation of the Green function. Renormalization proceeds by subtracting the linear-in-$$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$5 Born term and adding back the renormalized perturbative density. This isolates the universal ultraviolet-divergent piece and leaves a finite remainder encoding the non-perturbative vacuum rearrangement. The large-$$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$6 asymptotics

$$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$7

ensures convergence of the partial-wave expansion for the renormalized density (Davydov et al., 2017).

The corresponding vacuum energy is defined spectrally, decomposed into partial waves, and renormalized by a fermionic-loop counterterm fixed by matching to perturbation theory. Before renormalization, each partial-wave contribution is finite and the divergence appears only in the sum over $$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$8; after renormalization, the partial terms behave as

$$A^\mu=(V,\mathbf 0), \qquad V(r)=\frac{Ze}{4\pi\varepsilon_0\,r}.$$9

so the series converges. Over a wide parameter range in the overcritical region,

$\kappa=\pm 1,\pm 2,\dots$0

The paper interprets the rapid decrease of $\kappa=\pm 1,\pm 2,\dots$1 as the energetic signature of vacuum-shell formation and strong screening by the vacuum, though it does not claim complete screening in $\kappa=\pm 1,\pm 2,\dots$2 dimensions (Davydov et al., 2017).

A closely related $\kappa=\pm 1,\pm 2,\dots$3-dimensional model uses a smoothed Coulomb source

$\kappa=\pm 1,\pm 2,\dots$4

with, for $\kappa=\pm 1,\pm 2,\dots$5,

$\kappa=\pm 1,\pm 2,\dots$6

Each diving level changes the induced charge by $\kappa=\pm 1,\pm 2,\dots$7, and the renormalized vacuum energy can deviate strongly from perturbative quadratic growth, even decreasing to large negative values when the quadratic renormalization coefficient is negative. In that setting, the overcritical vacuum is explicitly interpreted as the new ground state, with spontaneous positron emission accompanying shell formation (Davydov et al., 2017).

For massless planar fermions in $\kappa=\pm 1,\pm 2,\dots$8 dimensions with Coulomb and Aharonov-Bohm potentials, the singular radial Hamiltonian requires a one-parameter self-adjoint extension. In the overcritical regime

$\kappa=\pm 1,\pm 2,\dots$9

the spectrum develops virtual or quasistationary bound states with complex energies, and the paper interprets their appearance as vacuum restructuring: if the emergent virtual level is empty, an electron-hole pair is created, the electron occupies the resonance and screens the Coulomb center, and the hole is expelled to infinity (Khalilov et al., 2013).

Taken together, these results define the charged Dirac-Coulomb vacuum not merely as a polarized sea but as a non-perturbatively reconstructed state whose charge and energy track level diving across the lower continuum.

6. Subcritical realizations and positive-energy projections

The subcritical regime provides a complementary perspective because it isolates vacuum response without level diving. In graphene, low-energy quasiparticles obey a two-dimensional Dirac equation and a single tunable Coulomb impurity realizes an experimental Dirac-Coulomb problem. For a $E=-1$00 impurity charge state, the effective charge is

$E=-1$01

with

$E=-1$02

Within the intrinsic screening length

$E=-1$03

the response is dominated by graphene’s interband screening. The experiment directly observes the expected electron-hole asymmetry: for a positive impurity, empty-state LDOS above the Dirac point is enhanced, while filled-state LDOS below it is suppressed. No sharp impurity resonances or bound-state peaks are observed off the trimer, consistent with the predicted subcritical Coulomb impurity regime. Fitting the asymmetry yields

$E=-1$04

close to the stated RPA estimate

$E=-1$05

The paper emphasizes that, unlike the nonrelativistic hydrogen problem, the subcritical graphene impurity does not produce quasi-bound states because the centrifugal barrier and Coulomb potential scale in the same way for linear dispersion (Wang et al., 2012).

In relativistic atomic and molecular theory, an alternative way of controlling the vacuum problem is to exclude pair sectors from the outset. The many-electron Dirac-Coulomb Hamiltonian

$E=-1$06

is projected onto the positive-energy subspace of the noninteracting problem. The projector is

$E=-1$07

and the no-pair Hamiltonian is

$E=-1$08

This eliminates negative-energy contamination and prevents Brown-Ravenhall continuum dissolution. Within an explicitly correlated Gaussian basis with restricted kinetic balance, the resulting no-pair Dirac-Coulomb energies converge to parts-per-billion precision; the reported uncertainties are about $E=-1$09 for H$E=-1$10, H$E=-1$11, and the helium excited state, and about $E=-1$12 for the helium ground state (Jeszenszki et al., 2021).

These subcritical and projected settings clarify a basic distinction. In graphene, the vacuum or filled valence sector responds physically to a Coulomb center but remains subcritical, so no vacuum shells form. In no-pair atomic and molecular calculations, the vacuum problem is controlled by construction through $E=-1$13, and pair creation is excluded. Both cases illuminate Dirac-Coulomb vacua, but they represent controlled regimes on opposite sides of the supercritical, pair-producing phenomenon.