Fermionic Projector: Spectral Splitting

• The fermionic projector is a spectral and distributional construct from the Dirac equation that splits solutions into negative and positive energy states.
• It uses the causal fundamental solution and the fermionic signature operator to define quasi-free states and maintain gauge invariance across diverse backgrounds.
• Its applications in curved and external-field spacetimes ensure precise state construction, Hadamard properties, and enhanced control in quantum field theory.

The fermionic projector is a spectral and distributional object associated with the Dirac equation that encodes the occupied sector of fermionic states. In Minkowski vacuum it is the operator with momentum-space kernel

$$P_m(k)=(\!\not k+m)\,\delta(k^2-m^2)\,\Theta(-k^0),$$

so that only negative-energy solutions are selected; in globally hyperbolic Lorentzian spin manifolds it is defined more generally by composing the causal fundamental solution $k_m$ with the projection onto the negative spectral subspace of the fermionic signature operator $S_m$,

$P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$

This construction gives a canonical splitting of the solution space of the Dirac equation and, through Araki’s construction, a distinguished pure quasi-free state of the CAR algebra of the Dirac field (Finster et al., 2014, Finster et al., 2017).

1. Functional-analytic definition

For a globally hyperbolic, time-oriented Lorentzian spin manifold $(M,g)$ and fixed mass $m$, one considers the Hilbert space $\mathcal H_m$ of smooth, spatially compact solutions of the Dirac equation

$(i\!\not\!\nabla + B - m)\,\psi_m=0$

with the usual Cauchy-surface inner product. If the Lorentz-invariant spacetime pairing

$$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$

is bounded on $\mathcal H_m$, the Riesz representation theorem yields a unique densely defined symmetric operator

$k_m$0

such that

$k_m$1

When $k_m$2 is self-adjoint, it has a spectral decomposition and the negative spectral projector $k_m$3 is well defined; the fermionic projector is then

$k_m$4

where $k_m$5 is the causal fundamental solution, i.e. the difference of the advanced and retarded Green’s operators (Finster et al., 2017).

In the finite-lifetime setting this construction appears in a bounded form already at fixed mass. There one has a bounded self-adjoint signature operator $k_m$6 on $k_m$7, and the negative and positive operators are

$k_m$8

The range of $k_m$9 is precisely the negative-spectral subspace of $S_m$0, and the corresponding kernel $S_m$1 is a bi-solution of the Dirac equation in both arguments (Finster et al., 2013).

The vacuum formula on Minkowski space gives the original physical interpretation: $S_m$2 projects onto the filled Dirac sea of occupied negative-frequency states, while its complement projects onto the positive-frequency, unoccupied sector (Finster et al., 2014). In this sense, the modern operator-theoretic construction replaces a coordinate-dependent frequency splitting by a covariantly defined spectral splitting.

2. Mass oscillation and the fermionic signature operator

For spacetimes of infinite lifetime, the spacetime inner product on single-mass solutions generally diverges. The key device is to consider families $S_m$3 over a bounded mass interval $S_m$4 and the mass-integrated field

$S_m$5

The weak and strong mass oscillation properties control the spacetime pairing of such mass-integrated solutions. In the strong form, one պահանջs an estimate of the form

$S_m$6

Under this condition, the spacetime pairing can be represented fiberwise by a bounded symmetric family $S_m$7, giving a canonical decomposition of $S_m$8 into positive and negative spectral subspaces that is independent of any observer or time-foliation (Finster et al., 2013, Finster et al., 2015).

The underlying mechanism is oscillatory decay in the mass parameter. In Minkowski vacuum, repeated integration by parts in $S_m$9 yields time decay of the mass-integrated field, and a Plancherel argument gives the sharp strong estimate. For smooth time-dependent external potentials $P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$0 with sufficient decay in time, the same scheme can be implemented via the Lippmann–Schwinger equation; if

$P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$1

then an explicit formula for $P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$2 yields the strong mass oscillation property (Finster et al., 2015).

The mass-oscillation mechanism is not universal. In Rindler space, the original weak and strong mass oscillation properties fail because the boundary term on the horizon does not vanish in general; this is traced to the nonvanishing trace of solutions on the horizon (Drago et al., 2016). In the flat slicing of de Sitter spacetime, the strong mass oscillation property likewise fails due to boundary effects at the cosmological horizon, and one is led instead to a mass decomposition containing an additional boundary term (Dappiaggi et al., 2019). These examples show that the existence of $P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$3 in the simplest bounded form is sensitive to global causal structure.

To address such obstructions, Drago and Murro introduced a modified construction based on Møller-type intertwining operators $P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$4 between solution spaces of different masses. This leads to modified weak and strong mass-oscillation properties, operators $P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$5, and a corresponding modified fermionic projector state. When only the modified weak property holds, one passes to the Friedrichs extension of the resulting symmetric operator before applying spectral calculus (Drago et al., 2016).

3. Kernel, CAR quantization, and generalized fermionic-projector states

Once the negative spectral projector is available, the fermionic projector has a distributional kernel. In the external-potential setting one proves that $P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$6 is represented by a unique bi-distribution $P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$7 such that

$P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$8

and the associated quasi-free state has two-point function

$P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m .$9

The kernel satisfies the Dirac equation in both variables and obeys the symmetry relation $(M,g)$0 (Finster et al., 2015, Finster et al., 2016).

The same structure appears in the self-dual CAR formalism. A projection $(M,g)$1 on the one-particle Hilbert space satisfying $(M,g)$2 together with the charge-conjugation condition defines a gauge-invariant pure quasifree state. On ultrastatic slabs, the two-point function $(M,g)$3 can be extracted directly from the fermionic-projector kernel, and Araki’s formula gives the state on the CAR algebra (Fewster et al., 2014).

Gauge invariance is explicit. Under a $(M,g)$4 gauge transformation, the kernel transforms locally by

$(M,g)$5

while the induced CAR-algebra automorphism leaves the quasifree state invariant. Equivalently, $(M,g)$6 commutes with the global $(M,g)$7 action, so the resulting state is gauge invariant (Fewster et al., 2014).

The spectral calculus of $(M,g)$8 also permits a broader class of states. For any non-negative bounded Borel function $(M,g)$9 on $m$0, one defines

$m$1

obtaining a generalized fermionic projector state. For orientation-preserving spacetime symmetries these generalized states are strictly invariant; time-orientation reversal changes $m$2 to $m$3 (Finster et al., 2017). This framework includes ground-state and thermal constructions, notably KMS-type states in Rindler space.

4. Explicit realizations in curved and external-field backgrounds

Representative constructions have been worked out in Minkowski and ultrastatic spacetimes (Finster et al., 2017), Rindler space (Finster et al., 2016), plane electromagnetic-wave backgrounds (Finster et al., 2016), de Sitter spacetime (Dappiaggi et al., 2019), the exterior Schwarzschild geometry (Finster et al., 2018), and Reissner–Nordström geometry in horizon-penetrating coordinates (Finster et al., 28 May 2026).

Background	Characterization of $m$4 or $m$5	Resulting state
Minkowski / ultrastatic	$m$6 has spectrum $m$7; in ultrastatic case $m$8	Usual positive-frequency splitting
2D Rindler	$m$9 selects exactly the $\mathcal H_m$0 modes	Fulling–Rindler vacuum
Plane electromagnetic wave	$\mathcal H_m$1 acts by multiplication with $\mathcal H_m$2	Distinguished Hadamard FP-state
Closed de Sitter	Negative spectral subspace is one-dimensional modewise	Maximally symmetric spinorial Bunch–Davies state
Exterior Schwarzschild	$\mathcal H_m$3 obtained from a mass decomposition with horizon term	FP-state coincides with the Hadamard vacuum at spatial infinity
Reissner–Nordström	$\mathcal H_m$4 and flux operator are bounded symmetric	FP-state is Hadamard

In two-dimensional Rindler space, the fermionic signature operator is essentially self-adjoint and, after diagonalization in rapidity space, yields a fermionic projector state that coincides with the Fulling–Rindler vacuum. The same spectral framework produces thermal states by replacing the sharp projector with the Fermi–Dirac function; for $\mathcal H_m$5 one recovers the Unruh state (Finster et al., 2016). In four-dimensional Rindler space the construction instead produces a new family of quasi-free states that mix spin indices and transverse momenta (Finster et al., 2016).

For a plane electromagnetic wave background with potential $\mathcal H_m$6, separation of variables in null coordinates reduces the Dirac equation to an algebraic relation and an ODE in the null variable $\mathcal H_m$7. The fermionic signature operator then acts simply by $\mathcal H_m$8, where $\mathcal H_m$9 is the momentum conjugate to the complementary null coordinate $(i\!\not\!\nabla + B - m)\,\psi_m=0$0. The resulting state is a distinguished, covariantly defined ground state even though the background is genuinely time dependent and non-decaying (Finster et al., 2016).

In de Sitter spacetime, the situation depends on the slicing. In the closed slicing the strong mass oscillation property holds, $(i\!\not\!\nabla + B - m)\,\psi_m=0$1 can be computed explicitly mode by mode, and the induced state is maximally symmetric and of Hadamard form; by uniqueness it coincides with the spinorial Bunch–Davies state (Dappiaggi et al., 2019). In the flat slicing, by contrast, boundary effects obstruct the strong mass oscillation property and lead to a mass decomposition with a non-local boundary contribution (Dappiaggi et al., 2019).

Black-hole geometries introduce an additional structural ingredient: horizon flux. In the exterior Schwarzschild geometry, the spacetime inner product decomposes into a single mass integral involving $(i\!\not\!\nabla + B - m)\,\psi_m=0$2 and a principal-value double mass integral encoding flux across the event horizon. The spectrum of $(i\!\not\!\nabla + B - m)\,\psi_m=0$3 can be computed explicitly in terms of asymptotic transmission coefficients, and a separate fermionic flux operator $(i\!\not\!\nabla + B - m)\,\psi_m=0$4 captures the part of the Dirac current falling into the black hole or emerging from the white-hole region (Finster et al., 2018). In Reissner–Nordström geometry a closely related mass-decomposition theorem involves both the fermionic signature operator and the fermionic flux operator, now up to the Cauchy horizon (Finster et al., 28 May 2026).

5. Hadamard property, non-Hadamard examples, and microlocal issues

A central question is whether the fermionic-projector kernel has Hadamard form, equivalently whether its wave-front set has the standard microlocal structure of a physically admissible Dirac two-point function. In the plane-wave analysis this condition is written as

$(i\!\not\!\nabla + B - m)\,\psi_m=0$5

and the proof proceeds by a careful analysis of the $(i\!\not\!\nabla + B - m)\,\psi_m=0$6-integral, showing that only negative-frequency null covectors occur (Finster et al., 2016).

Several positive results are non-perturbative. For smooth time-dependent external potentials decaying faster than quadratically for large times, if all time derivatives satisfy $(i\!\not\!\nabla + B - m)\,\psi_m=0$7-conditions and

$(i\!\not\!\nabla + B - m)\,\psi_m=0$8

then the fermionic projector is of Hadamard form (Finster et al., 2015). In the plane-wave background the FP-kernel is Hadamard despite the absence of ordinary frequency splitting in $(i\!\not\!\nabla + B - m)\,\psi_m=0$9 (Finster et al., 2016). In the closed slicing of de Sitter, maximal symmetry and the explicit mode analysis lead to a Hadamard state identified with the spinorial Bunch–Davies state (Dappiaggi et al., 2019). In the Reissner–Nordström geometry, the fermionic projector state is constructed and shown to satisfy the Hadamard condition (Finster et al., 28 May 2026).

At the same time, the fermionic-projector prescription is not automatically Hadamard in every formulation. On ultrastatic slabs with compact spatial section, the unsoftened choice corresponding to a sharp time window generically fails the Hadamard condition: the operator measuring the difference between the FP two-point function and a reference Hadamard ground state is not compact unless a fine-tuned condition holds, and the remainder is not $$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$0 (Fewster et al., 2014). The same construction also gives divergent fluctuations of the renormalized energy density for the sharp cutoff. Replacing the characteristic cutoff by any nonnegative $$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$1 yields a softened family of FP-states for which the mode-mixing angles decay rapidly, $$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$2 converges in $$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$3, and the resulting state is Hadamard; within this family, finiteness of all Wick-polynomial fluctuations is equivalent to the Hadamard property (Fewster et al., 2014).

A related limitation appears in black-hole settings. In the Schwarzschild exterior, generalized fermionic-projector states built from arbitrary nonnegative functions $$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$4 need not reproduce the Hadamard condition in general; the special choice $$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$5 is the one that coincides with the Hadamard vacuum obtained by frequency splitting at spatial infinity (Finster et al., 2018). This clarifies a frequent misconception: the fermionic-projector framework is compatible with Hadamard states, but Hadamardness depends on the detailed spectral prescription and on global geometric or boundary effects.

6. Symmetries, perturbative formulations, and relation to causal fermion systems

Spacetime symmetries act naturally on the fermionic signature operator. If a local symmetry group $$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$6 acts on $$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$7 by spin-bundle isometries and induces a unitary representation $$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$8 on $$\langle \phi_m\mid \psi_m\rangle=\int_M \langle \phi_m(p)\mid \psi_m(p)\rangle_p\,d\mu_g$$9, then

$\mathcal H_m$0

and consequently

$\mathcal H_m$1

For Lie-group symmetries, the self-adjoint infinitesimal generators commute with $\mathcal H_m$2, so the positive and negative spectral subspaces are invariant under the corresponding one-parameter groups (Finster et al., 2017). This explains, for example, the Poincaré invariance of the Minkowski vacuum splitting and the symmetry properties of FP-states in ultrastatic, Friedmann–Robertson–Walker, Rindler, and plane-wave backgrounds.

A distinct but related line of work studies the fermionic projector perturbatively in external fields. In that setting the interacting projector is generated from the causal fundamental solution by a contour-integral calculus. Two normalization prescriptions arise. Mass normalization is characterized by

$\mathcal H_m$3

whereas spatial normalization is characterized by

$\mathcal H_m$4

The two normalizations agree at lowest order but differ beginning at second order in the external potential $\mathcal H_m$5 (Finster et al., 2014). The same perturbative formalism yields causal light-cone expansions in which every singular coefficient depends only on the external field along the bounded line segment joining $\mathcal H_m$6 and $\mathcal H_m$7, and it proves a generalized Furry theorem identifying vanishing fermion-loop diagrams (Finster et al., 2014).

The fermionic-projector kernel also underlies the construction of causal fermion systems. Starting from a finite-lifetime spacetime, one introduces regularization operators $\mathcal H_m$8, forms evaluation maps $\mathcal H_m$9, and defines the local correlation operators

$k_m$00

Pushing forward the spacetime volume measure under $k_m$01 gives a measure $k_m$02 on a space of finite-rank self-adjoint operators, thereby producing a causal fermion system. The regularized kernels

$k_m$03

converge weakly to the unregularized fermionic projector (Finster et al., 2013). In the broader causal-fermion-systems continuum-limit analysis, the fermionic projector describes the occupied states of the Dirac sea plus particles and holes, tree-level and bosonic loop diagrams agree with standard perturbative quantum field theory, and fermion loops are encoded by finite kernels rather than ultraviolet-divergent integrals (Finster, 2013).

In the discrete-space-time formulation of the fermionic projector approach, the fundamental object is a rank-$k_m$04 projector $k_m$05 on an indefinite inner-product space, with kernel

$k_m$06

The continuum free Dirac sea is recovered from this kernel in the Minkowski limit (Finster, 2010). This suggests a broad conceptual continuity between the spectral construction of distinguished Dirac states on curved backgrounds and the causal-action framework in which the fermionic projector is taken as primary.