---
title: 'Fermionic Projector: Spectral Splitting'
url: https://www.emergentmind.com/topics/fermionic-projector
type: topic
---

# Fermionic Projector: Spectral Splitting

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 [1401.4353] [1708.09643].

## 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
\[
S_m:\mathcal H_m\to \mathcal H_m
\]
such that
\[
\langle \phi_m\mid \psi_m\rangle=(\phi_m\mid S_m\psi_m)_m .
\]
When \(S_m\) is self-adjoint, it has a spectral decomposition and the negative spectral projector \(\chi_{(-\infty,0)}(S_m)\) is well defined; the fermionic projector is then
\[
P_m=-\,\chi_{(-\infty,0)}(S_m)\,k_m,
\]
where \(k_m\) is the causal fundamental solution, i.e. the difference of the advanced and retarded Green’s operators [1708.09643].

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 \(S\) on \(H_m\), and the negative and positive operators are
\[
P_-=-\chi_{(-\infty,0)}(S)\,k_m,\qquad
P_+=\chi_{[0,\infty)}(S)\,k_m.
\]
The range of \(P_-\) is precisely the negative-spectral subspace of \(S\), and the corresponding kernel \(P(x,y)\) is a bi-solution of the Dirac equation in both arguments [1301.5420].

The vacuum formula on Minkowski space gives the original physical interpretation: \(P_m\) projects onto the filled Dirac sea of occupied negative-frequency states, while its complement projects onto the positive-frequency, unoccupied sector [1401.4353]. 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 \(\psi=(\psi_m)_{m\in I}\) over a bounded mass interval \(I=(m_L,m_R)\) and the mass-integrated field
\[
(p\psi)(x)=\int_I \psi_m(x)\,dm .
\]
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
\[
\bigl|\langle p\psi\mid p\phi\rangle\bigr|
\le c\int_I \|\psi_m\|_m\,\|\phi_m\|_m\,dm .
\]
Under this condition, the spacetime pairing can be represented fiberwise by a bounded symmetric family \(S_m\), giving a canonical decomposition of \(\mathcal H_m\) into positive and negative spectral subspaces that is independent of any observer or time-foliation [1312.7209] [1501.05522].

The underlying mechanism is oscillatory decay in the mass parameter. In Minkowski vacuum, repeated integration by parts in \(m\) yields time decay of the mass-integrated field, and a Plancherel argument gives the sharp strong estimate. For smooth time-dependent external potentials \(\mathscr B\) with sufficient decay in time, the same scheme can be implemented via the Lippmann–Schwinger equation; if
\[
\int_{-\infty}^{+\infty} |\mathscr B(t)|_{C^0}\,dt<\infty,
\]
then an explicit formula for \(\tilde S_m\) yields the strong mass oscillation property [1501.05522].

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 [1607.02909]. 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 [1902.09144]. These examples show that the existence of \(S_m\) 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 \(R_{m,a}\) between solution spaces of different masses. This leads to modified weak and strong mass-oscillation properties, operators \(S_{a,m}\), 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 [1607.02909].

## 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 \(P\) is represented by a unique bi-distribution \(\mathcal P(x,y)\) such that
\[
\langle \phi\mid P\psi\rangle=\mathcal P(\overline\phi\otimes\psi),
\]
and the associated quasi-free state has two-point function
\[
\omega\bigl(\Psi(g)\Psi^*(f)\bigr)
=-\iint g(x)\,\mathcal P(x,y)\,f(y)\,d^4x\,d^4y .
\]
The kernel satisfies the Dirac equation in both variables and obeys the symmetry relation \(P(x,y)^*=P(y,x)\) [1501.05522] [1609.04516].

The same structure appears in the self-dual CAR formalism. A projection \(P\) on the one-particle Hilbert space satisfying \(0\le P=P^*\le 1\) together with the charge-conjugation condition defines a gauge-invariant pure quasifree state. On ultrastatic slabs, the two-point function \(\omega_2\) can be extracted directly from the fermionic-projector kernel, and Araki’s formula gives the state on the CAR algebra [1408.1645].

Gauge invariance is explicit. Under a \(U(1)\) gauge transformation, the kernel transforms locally by
\[
P(x,y)\mapsto e^{i\chi(x)}\,P(x,y)\,e^{-i\chi(y)},
\]
while the induced CAR-algebra automorphism leaves the quasifree state invariant. Equivalently, \(P\) commutes with the global \(U(1)\) action, so the resulting state is gauge invariant [1408.1645].

The spectral calculus of \(S_m\) also permits a broader class of states. For any non-negative bounded Borel function \(W\) on \(\mathbb R\), one defines
\[
P_m^W=-\,W(S_m)\,k_m,
\]
obtaining a **generalized fermionic projector state**. For orientation-preserving spacetime symmetries these generalized states are strictly invariant; time-orientation reversal changes \(W(\lambda)\) to \(W(-\lambda)\) [1708.09643]. 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 [1708.09643], Rindler space [1606.03882], plane electromagnetic-wave backgrounds [1609.04516], de Sitter spacetime [1902.09144], the exterior Schwarzschild geometry [1812.02010], and Reissner–Nordström geometry in horizon-penetrating coordinates [2605.30176].

| Background | Characterization of \(S_m\) or \(P\) | Resulting state |
|---|---|---|
| Minkowski / ultrastatic | \(S_m\) has spectrum \(\{\pm1\}\); in ultrastatic case \(S_m=\mathrm{sign}(H)\) | Usual positive-frequency splitting |
| 2D Rindler | \(\chi_{(-\infty,0)}(S)\) selects exactly the \(\Omega<0\) modes | Fulling–Rindler vacuum |
| Plane electromagnetic wave | \(S_m\) acts by multiplication with \(\mathrm{sign}(u)\) | Distinguished Hadamard FP-state |
| Closed de Sitter | Negative spectral subspace is one-dimensional modewise | Maximally symmetric spinorial Bunch–Davies state |
| Exterior Schwarzschild | \(S_m\) obtained from a mass decomposition with horizon term | FP-state coincides with the Hadamard vacuum at spatial infinity |
| Reissner–Nordström | \(S_m\) 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 \(\beta=2\pi\) one recovers the Unruh state [1606.03882]. In four-dimensional Rindler space the construction instead produces a new family of quasi-free states that mix spin indices and transverse momenta [1606.03882].

For a plane electromagnetic wave background with potential \(A_\mu(t+x)\), separation of variables in null coordinates reduces the Dirac equation to an algebraic relation and an ODE in the null variable \(s=t+x\). The fermionic signature operator then acts simply by \(\mathrm{sign}(u)\), where \(u\) is the momentum conjugate to the complementary null coordinate \(l=t-x\). The resulting state is a distinguished, covariantly defined ground state even though the background is genuinely time dependent and non-decaying [1609.04516].

In de Sitter spacetime, the situation depends on the slicing. In the closed slicing the strong mass oscillation property holds, \(S_m\) 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 [1902.09144]. 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 [1902.09144].

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 \(S_m\) and a principal-value double mass integral encoding flux across the event horizon. The spectrum of \(S_m\) can be computed explicitly in terms of asymptotic transmission coefficients, and a separate fermionic flux operator \(B_m\) captures the part of the Dirac current falling into the black hole or emerging from the white-hole region [1812.02010]. 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 [2605.30176].

## 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
\[
\WF(\mathcal P)=\{(x,\xi,y,-\xi)\mid \xi^2=0,\ \xi^0<0,\ y-x\parallel \xi\},
\]
and the proof proceeds by a careful analysis of the \(u\)-integral, showing that only negative-frequency null covectors occur [1609.04516].

Several positive results are non-perturbative. For smooth time-dependent external potentials decaying faster than quadratically for large times, if all time derivatives satisfy \(L^1\)-conditions and
\[
\int |\mathscr B|<\sqrt2-1,
\]
then the fermionic projector is of Hadamard form [1501.05522]. In the plane-wave background the FP-kernel is Hadamard despite the absence of ordinary frequency splitting in \(t\) [1609.04516]. 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 [1902.09144]. In the Reissner–Nordström geometry, the fermionic projector state is constructed and shown to satisfy the Hadamard condition [2605.30176].

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 \(C^\infty\) [1408.1645]. The same construction also gives divergent fluctuations of the renormalized energy density for the sharp cutoff. Replacing the characteristic cutoff by any nonnegative \(f\in C_0^\infty(\mathbb R)\) yields a softened family of FP-states for which the mode-mixing angles decay rapidly, \(\omega_2^{FP_f}-\omega_2^0\) converges in \(C^\infty(M\times M)\), and the resulting state is Hadamard; within this family, finiteness of all Wick-polynomial fluctuations is equivalent to the Hadamard property [1408.1645].

A related limitation appears in black-hole settings. In the Schwarzschild exterior, generalized fermionic-projector states built from arbitrary nonnegative functions \(W(S_m)\) need not reproduce the Hadamard condition in general; the special choice \(W=\chi_{(-\infty,0)}\) is the one that coincides with the Hadamard vacuum obtained by frequency splitting at spatial infinity [1812.02010]. 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 \(G\) acts on \(M\) by spin-bundle isometries and induces a unitary representation \(U(g)\) on \(\mathcal H_m\), then
\[
U(g)\,S_m=S_m\,U(g),
\]
and consequently
\[
U(g)\,\chi_{(-\infty,0)}(S_m)=\chi_{(-\infty,0)}(S_m)\,U(g),\qquad
U(g)\,P_m\,U(g)^{-1}=P_m .
\]
For Lie-group symmetries, the self-adjoint infinitesimal generators commute with \(S_m\), so the positive and negative spectral subspaces are invariant under the corresponding one-parameter groups [1708.09643]. 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
\[
P_m\,P_{m'}=\delta(m-m')\,P_m,
\]
whereas **spatial normalization** is characterized by
\[
2\pi\!\int_{\mathbb R^3}\!d^3y\;
P\bigl(x,(t,\vec y)\bigr)\gamma^0 P\bigl((t,\vec y),z\bigr)
=-P(x,z).
\]
The two normalizations agree at lowest order but differ beginning at second order in the external potential \(\mathscr B\) [1401.4353]. 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 \(x\) and \(y\), and it proves a generalized Furry theorem identifying vanishing fermion-loop diagrams [1401.4353].

The fermionic-projector kernel also underlies the construction of causal fermion systems. Starting from a finite-lifetime spacetime, one introduces regularization operators \(\mathcal R_\varepsilon\), forms evaluation maps \(e_x^\varepsilon\), and defines the local correlation operators
\[
F^\varepsilon(x)=-\,\iota_x^\varepsilon e_x^\varepsilon .
\]
Pushing forward the spacetime volume measure under \(x\mapsto F^\varepsilon(x)\) gives a measure \(\rho\) on a space of finite-rank self-adjoint operators, thereby producing a causal fermion system. The regularized kernels
\[
P^\varepsilon=-\,\mathcal R_\varepsilon\,\chi_{(-\infty,0)}(S)\,\mathcal R_\varepsilon^*\,k_m
\]
converge weakly to the unregularized fermionic projector [1301.5420]. 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 [1310.4121].

In the discrete-space-time formulation of the fermionic projector approach, the fundamental object is a rank-\(f\) projector \(P\) on an indefinite inner-product space, with kernel
\[
P(x,y)=E_x\,P\,E_y .
\]
The continuum free Dirac sea is recovered from this kernel in the Minkowski limit [1011.2162]. 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.

Source: https://www.emergentmind.com/topics/fermionic-projector