A mechanism for dark matter and dark energy in the theory of causal fermion systems

We begin with the general definitions.

Definition 2.1 (Causal fermion systems). Given a separable complex Hilbert space $\mathcal{H}$ with scalar product $\langle .|. \rangle_\mathcal{H}$ and a parameter $n \in \mathbb{N}$ (the spin dimension), we let ${\mathcal{F}} \subset {\textrm{L}}(\mathcal{H})$ be the set of all symmetric ³ operators on $\mathcal{H}$ of finite rank, which (counting multiplicities) have at most n positive and at most n negative eigenvalues. On ${\mathcal{F}}$ we are given a positive measure ρ (defined on a σ-algebra of subsets of ${\mathcal{F}}$). We refer to $(\mathcal{H}, {\mathcal{F}}, \rho)$ as a causal fermion system.

A causal fermion system describes a spacetime together with all structures and objects therein. The physical equations are formulated for a causal fermion system by demanding that the measure ρ should be a minimizer of the causal action principle defined as follows. For any $x, y \in {\mathcal{F}}$, the product $x y$ is an operator of rank at most 2 n. However, in general it is no longer symmetric because $(xy)^* = yx$, and this is different from xy unless x and y commute. As a consequence, the eigenvalues of the operator xy are in general complex. We denote the non-trivial eigenvalues counting algebraic multiplicities by $\lambda^{xy}_1, \ldots, \lambda^{xy}_{2n} \in \mathbb{C}$ (more specifically, denoting the rank of xy by $k \leqslant 2n$, we choose $\lambda^{xy}_1, \ldots, \lambda^{xy}_{k}$ as all the non-zero eigenvalues and set $\lambda^{xy}_{k+1}, \ldots, \lambda^{xy}_{2n} = 0$). We introduce the Lagrangian and the causal action by

$$\begin{align} \mathrm {Lagrangian}: {\mathcal{L}}(x,y) &= \frac{1}{4n} \sum_{i,j=1}^{2n} \Big( \big|\lambda^{xy}_i \big| - \big|\lambda^{xy}_j \big| \Big)^2 \end{align} \tag{ 2.1 }$$

$$\begin{align} \mathrm {causal action}: {\mathcal{S}}(\rho) = \iint_{{\mathcal{F}} \times {\mathcal{F}}} {\mathcal{L}}(x,y)\: d\rho(x)\, d\rho(y) \:. \end{align} \tag{ 2.2 }$$

The causal action principle is to minimize ${\mathcal{S}}$ by varying the measure ρ under the following constraints,

$$\begin{align} \mathrm {volume~constraint}: \rho({\mathcal{F}}) = \text{const} \quad\;\; \end{align} \tag{ 2.3 }$$

$$\begin{align} \mathrm {trace~constraint}: \int_{\mathcal{F}} \mathrm{tr}(x)\: d\rho(x) = \text{const} \end{align} \tag{ 2.4 }$$

$$\begin{align} \mathrm {boundedness~constraint}: \iint_{{\mathcal{F}} \times {\mathcal{F}}} |xy|^2 \: d\rho(x)\, d\rho(y) &\leqslant C \:, \end{align} \tag{ 2.5 }$$

where C is a given parameter, $\mathrm{tr}$ denotes the trace of a linear operator on $\mathcal{H}$, and the absolute value of xy is the so-called spectral weight,

$$\begin{align*} |xy| := \sum_{j=1}^{2n} \big|\lambda^{xy}_j \big| \:. \end{align*}$$

This variational principle is mathematically well-posed if $\mathcal{H}$ is finite-dimensional (for details see [10] or [18, Chapter 12]).

A minimizer of the causal action principle satisfies the following EL equations. For a suitable value of the parameter ${\mathfrak{s}}\gt0$, the function $\ell : {\mathcal{F}} \rightarrow \mathbb{R}_0^+$ defined by

$$\begin{align} \ell(x) := \int_M {\mathcal{L}}_\kappa(x,y)\: d\rho(y) - {\mathfrak{s}} \end{align} \tag{ 2.6 }$$

is minimal and vanishes on the support ⁴ of ρ,

$$\begin{align} \ell|_{\mathrm{supp} \rho} \equiv \inf_{\mathcal{F}} \ell = 0 \:. \end{align} \tag{ 2.7 }$$

Here the κ-Lagrangian ${\mathcal{L}}_\kappa$ is defined by

$$\begin{align} {\mathcal{L}}_\kappa \::\: {\mathcal{F}} \times {\mathcal{F}} \rightarrow \mathbb{R}\:,\qquad {\mathcal{L}}_\kappa(x,y) := {\mathcal{L}}(x,y) + \kappa\: |xy|^2 \end{align} \tag{ 2.8 }$$

with a non-negative parameter κ, which can be thought of as the Lagrange parameter corresponding to the boundedness constraint. Likewise, the parameter ${\mathfrak{s}} \geqslant 0$ in (2.6) is the Lagrange parameter corresponding to the volume constraint. For the derivation and further details we refer to [21, section 2] or [18, chapter 7].

Let ρ be a minimizing measure. Spacetime is defined as the support of this measure,

$$\begin{align*} M := \mathrm{supp} \rho \;\subset\; {\mathcal{F}} \:, \end{align*}$$

where on M we consider the topology induced by ${\mathcal{F}}$ (generated by the operator norm on ${{L}}(\mathcal{H})$). Thus the spacetime points are symmetric linear operators on $\mathcal{H}$. The restriction of the measure $\rho|_M$ gives a volume measure on spacetime.

The operators in M contain a lot of information which, if interpreted correctly, gives rise to spacetime structures like causal and metric structures, spinors and interacting fields (for details see [11, chapter 1]). All the resulting objects are inherent in the sense that we only use information already encoded in the causal fermion system. Here we restrict attention to those structures needed in what follows. We begin with the following notion of causality:

Definition 2.2 (causal structure). For any $x, y \in {\mathcal{F}}$, we again denote the non-trivial eigenvalues of the operator product xy (again counting algebraic multiplicities) by $\lambda^{xy}_1, \ldots, \lambda^{xy}_{2n}$. The points x and y are called spacelike separated if all the $\lambda^{xy}_j$ have the same absolute value. They are said to be timelike separated if the $\lambda^{xy}_j$ are all real and do not all have the same absolute value. In all other cases (i.e. if the $\lambda^{xy}_j$ are not all real and do not all have the same absolute value), the points x and y are said to be lightlike separated.

Restricting the causal structure of ${\mathcal{F}}$ to M, we get causal relations in spacetime.

The Lagrangian (2.1) is compatible with the above notion of causality in the following sense. Suppose that two points $x, y \in M$ are spacelike separated. Then the eigenvalues $\lambda^{xy}_i$ all have the same absolute value. As a consequence, the Lagrangian (2.1) vanishes. Thus pairs of points with spacelike separation do not enter the action. This can be seen in analogy to the usual notion of causality where points with spacelike separation cannot influence each other. This is the reason for the notion 'causal' in causal fermion system and causal action principle.

A causal fermion system also gives rise to spinorial wave functions in spacetime, as we now explain. For every $x \in {\mathcal{F}}$ we define the spin space S_x by $S_x = x(\mathcal{H})$; it is a subspace of $\mathcal{H}$ of dimension at most 2 n. It is endowed with the spin inner product $\prec.|. \succ_x$ defined by

$$\begin{align} \prec u | v \succ_x = -\langle u | x v \rangle_\mathcal{H} \qquad \text{(for all $u,v \in S_x$)} \:. \end{align} \tag{ 2.9 }$$

It is an important observation that every vector $u \in \mathcal{H}$ of the Hilbert space gives rise to a unique wave function denoted by ψ^u , which to every $x \in M$ associates a vector of the corresponding spin space $\psi^u(x) \in S_x$. It is obtained by orthogonal projection to the spin space,

$$\begin{align} \psi^u \::\: M \rightarrow \mathcal{H}\qquad \text{with} \qquad \psi^u(x) := \pi_x u \in S_xM \quad \text{for all~$x \in M$}\:. \end{align} \tag{ 2.10 }$$

We refer to ψ^u as the physical wave function of the vector $u \in \mathcal{H}$. Varying the vector $u \in \mathcal{H}$, we obtain a whole family of physical wave functions. This family is described most conveniently by the wave evaluation operator Ψ defined at every spacetime point $x \in M$ by

$$\begin{align} \Psi(x) \::\: \mathcal{H} \rightarrow S_x \:, \qquad u \mapsto \psi^u(x) \:. \end{align} \tag{ 2.11 }$$

It is a simple but important observation that every spacetime point operator can be recovered from its wave evaluation operator by (for the proof see for example [11, lemma 1.1.3]).

$$\begin{align} x = - \Psi(x)^* \Psi(x) \:. \end{align} \tag{ 2.12 }$$

Having constructed the spacetime point operators, we also recover all the other inherent structures of a causal fermion system. Proceeding in this way, all spacetime structures can be regarded as being induced by the physical wave functions. Moreover, restricting attention to variations of Ψ, one can understand the causal action principle as a variational principle for the family of physical wave functions. Finally, one can construct concrete examples of causal fermion systems by choosing the physical wave functions more specifically as the quantum mechanical wave functions in a classical Lorentzian spacetime. In the next section we explain this construction in more detail.

We now explain how a classical curved spacetime is described by a causal fermion system. Our starting point is Lorentzian spin geometry. Thus we let $(\mathscr{M}, g)$ be a smooth, globally hyperbolic, time-oriented Lorentzian spin manifold of dimension four. For the signature of the metric we use the convention $(+ ,-, -, -)$. We denote the corresponding spinor bundle by $S\mathscr{M}$. Its fibers $S_x\mathscr{M}$ (with $x \in \mathscr{M}$) are endowed with an inner product $\prec.|. \succ_x$ of signature $(2,2)$. Clifford multiplication is described by a mapping γ which satisfies the anti-commutation relations,

$$\begin{align*} \gamma \::\: T_x\mathscr{M} \rightarrow {{L}}(S_x\mathscr{M}) \qquad \text{with} \qquad \gamma(u) \,\gamma(v) + \gamma(v) \,\gamma(u) = 2 \, g(u,v){\unicode{x1D7D9}_{S_x(\mathscr{M})}} \:. \end{align*}$$

We also write Clifford multiplication in components with the Dirac matrices γ^j . The metric connections on the tangent bundle and the spinor bundle are denoted by $\nabla$.

We denote the smooth sections of the spinor bundle by $C^\infty(\mathscr{M}, S\mathscr{M})$. The Dirac operator ${\mathcal{D}}$ is defined by

$$\begin{align*} {\mathcal{D}} := i \gamma^{\,j} \nabla_j \::\: C^\infty(\mathscr{M}, S\mathscr{M}) \rightarrow C^\infty(\mathscr{M}, S\mathscr{M})\:. \end{align*}$$

Given a real parameter $m \in \mathbb{R}$ (the mass), the Dirac equation reads

$$\begin{align*} ({\mathcal{D}} - m) \,\psi = 0 \:. \end{align*}$$

We mainly consider solutions in the class $C^\infty_{{\textrm{sc}}}(\mathscr{M}, S\mathscr{M})$ of smooth sections with spatially compact support (i.e. wave functions whose restriction to any Cauchy surface is compact). On such solutions, one has the scalar product

$$\begin{align} (\psi | \phi)_m = \int_{\mathscr{N}} \prec\psi \,|\, \gamma(\nu)\, \phi \succ_x\: d\mu_{\mathscr{N}}(x) \:, \end{align} \tag{ 2.13 }$$

where $\mathscr{N}$ denotes any Cauchy surface and ν its future-directed normal (due to current conservation, the scalar product is in fact independent of the choice of $\mathscr{N}$; for details see [24, section 2]). Forming the completion gives the Hilbert space $(\mathcal{H}_m, (.|.)_m)$.

We next choose a closed subspace $\mathcal{H} \subset \mathcal{H}_m$ of the solution space of the Dirac equation. The induced scalar product on $\mathcal{H}$ is denoted by $\langle .|. \rangle_\mathcal{H}$. There is the technical difficulty that the wave functions in $\mathcal{H}$ are in general not continuous, making it impossible to evaluate them pointwise. For this reason, we need to introduce a regularization on the length scale ε, described mathematically by a linear

$$\begin{align} \mathrm {regularization~operator} \qquad {\mathfrak{R}}_\varepsilon \::\: \mathcal{H} \rightarrow C^{\,0}(\mathscr{M}, S\mathscr{M}) \:. \end{align} \tag{ 2.14 }$$

In the simplest case, the regularization can be realized by a convolution on a Cauchy surface or in spacetime (for details see [24, section 4] or [11, section § 1.1.2]). For us, the regularization is not merely a technical tool, but it realizes the concept that we want to change the geometric structures on the microscopic scale. With this in mind, we always consider the regularized quantities as those having mathematical and physical significance. Different choices of regularization operators realize different microscopic spacetime structures.

Evaluating the regularization operator at a spacetime point $x \in \mathscr{M}$ gives the regularized wave evaluation operator $\Psi^\varepsilon(x)$,

$$\begin{align} \Psi^\varepsilon(x) = {\mathfrak{R}}_\varepsilon(x) \::\: \mathcal{H} \rightarrow S_x \mathscr{M} \:. \end{align} \tag{ 2.15 }$$

We also take its adjoint (with respect to the Hilbert space scalar product $\langle .|. \rangle_\mathcal{H}$ and the spin inner product $\prec.|. \succ_x$),

$$\begin{align*} \big(\Psi^\varepsilon(x) \big)^* \::\: S_x \mathscr{M} \rightarrow \mathcal{H} \:. \end{align*}$$

Multiplying $\Psi^\varepsilon(x)$ by its adjoint gives the operator

$$\begin{align} F^\varepsilon(x) := - \big( \Psi^\varepsilon(x) \big)^* \,\Psi^\varepsilon(x) \::\: \mathcal{H} \rightarrow \mathcal{H} \:, \end{align} \tag{ 2.16 }$$

referred to as the local correlation operator at the spacetime point x. The local correlation operator is also characterized by the relation

$$\begin{align} (\psi \,|\, F^\varepsilon(x)\, \phi) = -\prec({\mathfrak{R}}_\varepsilon\psi)(x) | ({\mathfrak{R}}_\varepsilon \phi)(x) \succ_x \qquad \text{for all~$\psi, \phi \in \mathcal{H}$} \:. \end{align} \tag{ 2.17 }$$

Taking into account that the inner product on the Dirac spinors at x has signature $(2,2)$, it is a symmetric operator on $\mathcal{H}$ of rank at most four, which (counting multiplicities) has at most two positive and at most two negative eigenvalues. Varying the spacetime point, we obtain a mapping

$$\begin{align*} F^\varepsilon \::\: \mathscr{M} \rightarrow {\mathcal{F}} \subset {{L}}(\mathcal{H})\:, \end{align*}$$

where ${\mathcal{F}}$ denotes all symmetric operators of rank at most four with at most two positive and at most two negative eigenvalues. Finally, we introduce the measure ρ on ${\mathcal{F}}$ by taking the push-forward of the volume measure on $\mathscr{M}$ under the mapping F^ε ,

$$\begin{align} \rho := (F^\varepsilon)_* \mu_{\mathscr{M}} \end{align} \tag{ 2.18 }$$

(thus $\rho(\Omega) : = \mu_{\mathscr{M}}((F^\varepsilon)^{-1}(\Omega))$). The resulting structure $(\mathcal{H}, {\mathcal{F}}, \rho)$ is a causal fermion system of spin dimension two.

The construction of the causal fermion system in the previous section involves the Lorentzian metric g. But this Lorentzian metric does not need to satisfy the Einstein equation. Instead of postulating the Einstein equations, one can derive these equations from the causal action principle. To this end, one evaluates the EL equation (2.7) for the causal fermion system $(\mathcal{H}, {\mathcal{F}}, \rho)$ constructed in the previous section in the limit $\varepsilon \searrow 0$ when the regularization is removed. This analysis, referred to as the continuum limit, is carried out in detail in [11, chapter 4]. The main result is that the EL equations are satisfied asymptotically for small ε > 0 only if the Lorentzian metric satisfies the Einstein equations, up to possible higher order corrections in curvature (which scale in powers of $(G\: \text{Riem})$, where G is the gravitational coupling constant and Riem is the curvature tensor), i.e. (see [11, theorems 4.9.3 and 5.4.4])

$$\begin{align} R_{jk} - \frac{1}{2}\:R\: g_{jk} + \Lambda\, g_{jk} = G\, T_{jk} + {\mathcal{O}} \big( G^2 \,\text{Riem}^2 \big) \:. \end{align} \tag{ 2.19 }$$

We now briefly outline how this result is derived in [11, chapter 4]. Given a causal fermion system $(\mathcal{H}, {\mathcal{F}}, \rho)$ describing a globally hyperbolic spacetime $(\mathscr{M}, g)$ (as constructed in section 2.4), the task is to evaluate the EL equation (2.7). For the computations, it is very useful that the EL equations can be expressed in terms of the kernel of the fermionic projector $P(x,y)$. Here we do not need to enter the details (which can be found in [11, § 1.1.3 and § 1.4.1]), but it suffices to explain what the kernel of the fermionic projector is and how it encodes the gravitational field. Abstractly, the kernel of the fermionic projector is defined in terms of the wave evaluation operator (2.11) by

$$\begin{align*} P(x,y) = -\Psi(x)\, \Psi(y)^* \::\: S_y \rightarrow S_x \:. \end{align*}$$

In an orthonormal basis $(e_i)$ of $\mathcal{H}$, it can be expressed as

$$\begin{align} P(x,y) = -\sum_i |\psi^{e_i}(x) \succ\prec\psi^{e_i}(y)| \:, \end{align} \tag{ 2.20 }$$

showing that it is composed of all the physical wave functions of the system (for details see also [11, § 1.2.4]). For a causal fermion system constructed in a Lorentzian spacetime, one can identify $P(x,y)$ with the regularized kernel of the fermionic projector $P^\varepsilon(x,y)$ defined by (for details see [11, section 1.2])

$$\begin{align} P^\varepsilon(x,y) = -\Psi^\varepsilon(x)\big( \Psi^\varepsilon(y) \big)^* \::\: S_y\mathscr{M} \rightarrow S_x\mathscr{M} \:. \end{align} \tag{ 2.21 }$$

In the continuum limit, one studies the EL equations expressed in terms of $P^\varepsilon(x,y)$ asymptotically for small ε > 0. In this asymptotic regime, it suffices to analyze the behavior of $P^\varepsilon(x,y)$ near the light cone (i.e. if x and y have nearly lightlike separation). An explicit analysis is possible based on the method of integration along characteristics (see the regularized light-cone expansion in [11, section 2.2] or, more generally, the regularized Hadamard expansion [22]). Evaluating the resulting expressions in the EL equation (2.7), one gets equations involving the gravitational field and the Dirac wave functions (see [11, section 4.5]). In models containing neutrinos, these equations give rise to the Einstein equation (2.19) (see [11, section 4.9]).

Before explaining how to go beyond classical gravity, we point out that the above constructions only give a special class of examples of causal fermion systems describing classical spacetimes. In general, the measure ρ defined by (2.18) will not be a minimizer of the causal action principle. We only know that it is an approximate minimizer if the Einstein equations are satisfied. A good strategy for getting an exact minimizer is to take the causal fermion system $(\mathcal{H}, {\mathcal{F}}, \rho)$ as constructed in section 2.4 from a Lorentzian spacetime as the starting point, and to minimize the causal action principle further by varying the wave evaluation operator (2.11) of this causal fermion system. The resulting physical wave functions will no longer satisfy the Dirac equation. Even more, in general a Lorentzian metric ceases to exist. In other words, we are leaving the realm of classical gravity and differential geometry. The geometry of the resulting quantum spacetime has been explored in [15]. We also refer the reader interested in this direction of research to the survey articles [12, section 4] or [13, section 4].

A more modest approach and more computational approach for going beyond classical gravity is to work out small corrections to the Dirac and Einstein equations described by the EL equations of the causal action principle. This strategy was first used in [16], where modifications of the Dirac equations were derived from the EL equations and the corresponding conservation laws, giving rise to a mechanism of baryogenesis. In the present paper, we proceed in the same spirit to again derive modifications of the Dirac equation, but now giving a possible explanation for the dark Universe. We note that a systematic study of all corrections to the Dirac and Einstein equations (also including dynamical coupling constants as envisioned in the preliminary preprint [25]) is still lacking and seems an interesting field of future research.

We set

$$\begin{align} P^\varepsilon_m(p) := (\,p\!\!\!/+m)\: \delta \big( p^2-m^2 \big)\: \theta(-p^0)\:e^{\varepsilon p^0} \:. \end{align} \tag{ 4.1 }$$

We now 'smear out' the mass by setting

$$\begin{align} P^\varepsilon = \frac{1}{\delta m(\vec{k})} \int_{m_0+\delta n(\vec{k})}^{m_0+\delta m(\vec{k}) + \delta n(\vec{k})} P^\varepsilon_m\: dm \:, \end{align} \tag{ 4.2 }$$

where $p = (\omega, \vec{k})$. We choose $\delta m$ such that the width in ω is given by

$$\begin{align} \delta \omega = \frac{2 \pi}{T} \:, \end{align} \tag{ 4.3 }$$

where T is the lifetime of our Universe. Taking the difference of the dispersion relations

$$\begin{align*} (\omega - \delta \omega)^2 - |\vec{k}|^2 = (m_0+\delta m+\delta n)^2 \qquad \text{and} \qquad \omega^2 - |\vec{k}|^2 = (m_0+\delta n)^2 \:, \end{align*}$$

we obtain for any momentum $\vec{k}$ the condition

$$\begin{align*} (\omega - \delta \omega)^2 - \omega^2 = (m_0+\delta m+\delta n)^2 - (m_0+\delta n)^2 \:. \end{align*}$$

Expanding linearly, substituting (4.3) and solving for $\delta m$ gives

$$\begin{align} \delta m = \frac{|\omega|}{m_0}\: \frac{2 \pi}{T} + \mathcal{O} \big( (m_0 T)^{-2} \big) \:. \end{align} \tag{ 4.4 }$$

The function $\delta n(\vec{k})$ gives us the freedom to 'slightly deform' the mass shell; this will be explained in remark 4.1 below.

In order to determine the resulting contribution to the Einstein equations as obtained in the continuum limit, it suffices to compute the vector component of P^ε . In view of (4.1) and (4.2), we may pull out the Dirac matrices,

$$\begin{align} P^\varepsilon(p) = \,p\!\!\!/\: R(p)\: \theta(-p^0) \:e^{\varepsilon p^0} + (\text{scalar component}) \end{align} \tag{ 4.5 }$$

with

$$\begin{align} R(p) := \frac{1}{\delta m(\vec{k})} \int_{m_0+\delta n(\vec{k})}^{m_0+\delta m(\vec{k}) + \delta n(\vec{k})} \delta \big( p^2-m^2 \big)\: dm \end{align} \tag{ 4.6 }$$

(here by 'scalar component' we mean a multiple of the identity matrix). We next expand the δ distribution in the integrand in the mass parameter,

$$\begin{align*} \delta \big( p^2-m^2 \big) &= \delta \big( p^2-m_0^2 \big) + \delta^{^{\prime}} \big( p^2-m_0^2 \big)\: (m^2-m_0^2) \\ &\quad\; + \frac{1}{2}\: \delta^{^{\prime\prime}}\big( p^2-m_0^2 \big)\:(m^2-m_0^2)^2 + \mathcal{O} \big( (m^2-m_0^2)^3 \big) \:. \end{align*}$$

Now we can carry out the integral in (4.6),

$$\begin{align*} R&= \delta \big( p^2-m_0^2 \big) + \delta^{^{\prime}} \big( p^2-m_0^2 \big)\: \frac{1}{\delta m(\vec{k})} \int_{m_0+\delta n(\vec{k})}^{m_0+\delta m(\vec{k}) + \delta n(\vec{k})} \big(m^2-m_0^2 \big)\: dm \\ &\quad\: + \frac{1}{2}\: \delta^{^{\prime\prime}}\big( p^2-m_0^2 \big)\: \frac{1}{\delta m(\vec{k})} \int_{m_0 + \delta n(\vec{k})}^{m_0+\delta m(\vec{k}) + \delta n(\vec{k})} \big( m^2-m_0^2 \big)^2\: dm + \mathcal{O}\Big( \big(\delta m + |\delta n| \big)^3 \Big) \\ &= \delta \big( p^2-m_0^2 \big) + \delta^{^{\prime}} \big( p^2-m_0^2 \big)\: \bigg( m_0 \,\big( \delta m+2 \delta n \big) + \frac{1}{3}\, (\delta m)^{2}+ \delta m\: \delta n + (\delta n)^2 \bigg) \\ &\quad\: + \frac{1}{2}\: \delta^{^{\prime\prime}}\big( p^2-m_0^2 \big)\: \bigg( \frac{4}{3}\: m_0^2 \:\Big( \delta m^2 + 3 (\delta m) (\delta n) + 3 (\delta n)^2 \Big) \bigg) + \mathcal{O}\Big( \big(\delta m + |\delta n| \big)^3 \Big) \:. \end{align*}$$

According to (4.5), the vector component of P is obtained from this formula by multiplying with $\,p\!\!\!/$.

We next want to choose $\delta n$ in such a way that the leading contributions $\sim \delta^{^{\prime}}$ vanish. To this end, we choose

$$\begin{align} \delta n = -\frac{\delta m}{2} - \frac{1}{24}\: \frac{(\delta m)^2}{m_0} \:, \end{align} \tag{ 4.7 }$$

because then

$$\begin{align*} m_0 \,\big( \delta m+2 \delta n \big) &= -\frac{1}{12}\: (\delta m)^2 \\ \frac{1}{3}\, (\delta m)^{2}+ \delta m\: \delta n + (\delta n)^2 &= \frac{1}{3}\, (\delta m)^{2} - \frac{1}{2}\:(\delta m)^2 + \frac{1}{4}\:(\delta m)^2 + \mathcal{O}\big( (\delta m)^3 \big) \\ &= \frac{1}{12}\, (\delta m)^{2} + \mathcal{O}\big( (\delta m)^3 \big) \:, \end{align*}$$

as desired (this choice will be discussed in remark 4.1 below). We thus obtain

$$\begin{align} R =\delta \big( p^2-m_0^2 \big) + \frac{1}{6}\: m_0^2\, (\delta m)^2\: \delta^{^{\prime\prime}}\big( p^2-m_0^2 \big) + \mathcal{O}\big( (\delta m)^3 \big) \:. \end{align} \tag{ 4.8 }$$

Remark 4.1 (Choice of n). The ansatz (4.2) was already motivated in the introduction. It was also explained why δω and $\delta m$ are to be chosen according to (4.3) and (4.4) (see also (1.10) in the introduction). However, the choice of n according to (4.7) still requires a detailed explanation. The first summand in (4.7) can be understood immediately from the fact that we do not want to change the mean value of the mass. More precisely, it gives rise to a smearing

$$\begin{align*} \frac{1}{\delta m(\vec{k})} \int_{m_0 - \frac{\delta m(\vec{k})}{2}}^{m_0 + \frac{\delta m(\vec{k})}{2}} P_m\: dm + \mathcal{O}(\Delta m) \end{align*}$$

which is symmetric about m₀. This seems the natural choice.

The quadratic term in (4.7)

$$\begin{align*} -\frac{1}{24}\: \frac{(\delta m)^2}{m_0} \end{align*}$$

is less obvious. This choice can be understood from the requirement that the regularized kernel $P^\varepsilon(x,y)$ must be a minimizer of the causal action principle. The argument goes as follows: The corresponding contribution to the smeared δ-distribution is given by (here and in what follows, the symbol $\asymp$ denotes a selected contribution to the expression on the left)

$$\begin{align*} R &\asymp \delta^{^{\prime}} \big( p^2-m_0^2 \big)\: 2 \,m_0\, \delta n + \mathcal{O}\Big( \big(\delta m + |\delta n| \big)^3 \Big) \\ &= -\frac{1}{12}\: \delta^{^{\prime}} \big( p^2-m_0^2 \big)\: (\delta m)^2 + \mathcal{O}\Big( \big(\delta m + |\delta n| \big)^3 \Big) \:. \end{align*}$$

Using (4.4), we can write this contribution as

$$\begin{align*} R \asymp -\frac{\pi^2}{3}\: \frac{1}{T^{\,2}}\: \delta^{^{\prime}} \big( p^2-m_0^2 \big)\: \frac{\omega^2}{m_0^2} = -\frac{\pi^2}{6}\: \frac{1}{m_0^2 T^{\,2}}\: \omega\: \frac{\partial}{\partial \omega} \delta\big( p^2-m_0^2 \big) \:. \end{align*}$$

Taking the Fourier transform, the factor $\omega \partial_\omega$ becomes the operator $-(1+ t \partial_t)$, because

$$\begin{align*} \int_{\mathbb{R}^4} & \frac{d^4p}{(2 \pi)^4} \: \omega\: \Big( \frac{\partial}{\partial \omega} \delta\big( p^2-m_0^2 \big) \Big)\: \theta(-p^0)\: e^{ip \xi} \\ &= -i \frac{\partial}{\partial t} \int_{\mathbb{R}^4} \frac{d^4p}{(2 \pi)^4} \: \Big( \frac{\partial}{\partial \omega} \delta\big( p^2-m_0^2 \big) \Big)\: \theta(-p^0)\: e^{ip \xi} \\ &= i \frac{\partial}{\partial t} \int_{\mathbb{R}^4} \frac{d^4p}{(2 \pi)^4} \: \delta\big( p^2-m_0^2 \big)\: \theta(-p^0)\: \frac{\partial}{\partial \omega} e^{ip \xi} \\ &= -\frac{\partial}{\partial t} \bigg( t \int_{\mathbb{R}^4} \frac{d^4p}{(2 \pi)^4} \: \delta\big( p^2-m_0^2 \big)\: \theta(-p^0)\: e^{ip \xi} \bigg) \\ &= -\Big( 1 + t\: \frac{\partial}{\partial t} \Big) \int_{\mathbb{R}^4} \frac{d^4p}{(2 \pi)^4} \: \delta\big( p^2-m_0^2 \big)\: \theta(-p^0)\: e^{ip \xi} \end{align*}$$

(here we take into account only the leading contribution in ε). Thus the corresponding contribution to the regularized kernel is

$$\begin{align} P^\varepsilon_{\delta \omega}(x,y) \asymp \frac{\pi^2}{6}\: \frac{1}{m_0^2 T^{\,2}} \: i {\partial\!\!\!\!/}_x \Big( \big( 1+t \partial_t \big) {\mathcal{T}}^\varepsilon_{m_0^2}(\xi) \Big) \:, \end{align} \tag{ 4.9 }$$

where ${\mathcal{T}}^\varepsilon_{m^2}(\xi)$ is the regularized Fourier transform of the lower mass shell,

$$\begin{align} {\mathcal{T}}^\varepsilon_{m^2}(\xi) = \int_{\mathbb{R}^4} \frac{d^4p}{(2 \pi)^4} \: \delta(p^2 - m^2)\: \theta(-p^0)\: e^{\varepsilon p^0} e^{ip\xi} \:. \end{align} \tag{ 4.10 }$$

The time derivative $\partial_t$ in (4.9) changes the singularity structure on the light cone by making the order of the singularity stronger. This is inhibited by the causal action principle, because stronger singularities on the light cone give rise to extremely large contributions to the causal action. Therefore, we can argue that minimizing the causal action forces us to choose n according to (4.7). $\Diamond$

It remains to analyze the resulting formula (4.8) and to make the connection to dark matter and dark energy. To this end, we now compute the corresponding contributions to the kernel of the fermionic projector.

Proposition 4.2. The 'smearing' of the mass parameter (4.2) with $\delta m$ and $\delta n$ according to (4.4) and (4.7) gives rise to corrections to the kernel of the fermionic projector of the form

$$\begin{align} \big(P^\varepsilon - P^\varepsilon_{m_0} \big)(x,y) = \frac{\pi^2}{6}\: \frac{1}{T^{\,2}}\: t^2\,P^\varepsilon(x,y)\qquad \end{align} \tag{ 4.11 }$$

$$\begin{align} \quad\quad\qquad\qquad\quad\;&\: - \frac{i \pi^2}{3}\:\frac{1}{T^{\,2}}\: \gamma^0 \:t\,{\mathcal{T}}^\varepsilon_{m_0^2}(\xi) \end{align} \tag{ 4.12 }$$

$$\begin{align} \quad\quad\qquad\qquad\quad\;&\: +\frac{i \pi^2}{12}\: \frac{1}{T^{\,2}}\: \int_{-\infty}^\infty \epsilon(\tau)\: \gamma^0 \: {\mathcal{T}}^\varepsilon_{m_0^2}(\xi + \tau e_0) \: d\tau \end{align} \tag{ 4.13 }$$

$$\begin{align} \quad\quad\qquad\qquad\quad\;&\: -\frac{\pi^2}{12}\: \frac{1}{T^{\,2}} \int_{-\infty}^\infty \epsilon(\tau) \:(t+\tau)\: P^\varepsilon(x,y + \tau e_0) \: d\tau \nonumber\\ \qquad\quad\qquad\qquad\quad\;&\: - (\mathrm{scalar~component}) +\mathcal{O} \big( (m_0 T)^{-3} \big) \:, \end{align} \tag{ 4.14 }$$

where ${\mathcal{T}}^\varepsilon_{m_0^2}$ is the distributional kernel (4.10) (and is the sign function $\epsilon(\tau) = 1$ for $\tau \geqslant 0$ and $\epsilon(\tau) = -1$ otherwise).

Proof. Applying (4.4) to (4.8), we obtain

$$\begin{align} R =\delta \big( p^2-m_0^2 \big) + \frac{2 \pi^2}{3}\: \frac{\omega^2}{T^{\,2}}\: \delta^{^{\prime\prime}}\big( p^2-m_0^2 \big) +\mathcal{O} \big( (m_0 T)^{-3} \big) \:. \end{align} \tag{ 4.15 }$$

We rewrite the correction term in this equation as

$$\begin{align} \frac{2 \pi^2}{3}\: \frac{\omega^2}{T^{\,2}}\: \delta^{^{\prime\prime}}\big( p^2-m_0^2 \big) &= \frac{2 \pi^2}{3}\: \frac{\omega^2}{T^{\,2}}\: \bigg( \frac{1}{2 \omega} \frac{\partial}{\partial \omega} \Big( \frac{1}{2 \omega} \frac{\partial}{\partial \omega} \: \delta\big( p^2-m_0^2 \big) \Big) \bigg) \nonumber\\ &=\frac{\pi^2}{6}\: \frac{1}{T^{\,2}}\: \bigg( \frac{\partial^2}{\partial \omega^2} \delta\big( p^2-m_0^2 \big) - \frac{1}{\omega}\:\frac{\partial}{\partial \omega} \delta\big( p^2-m_0^2 \big) \bigg) \:. \end{align} \tag{ 4.16 }$$

Now we can compute the Fourier transform of the resulting terms,

$$\begin{align} &\int_{\mathbb{R}^4} \frac{d^4p}{(2 \pi)^4} \: \Big( \frac{\partial^2}{\partial \omega^2} \delta\big( p^2-m_0^2 \big) \Big)\: \theta(-p^0)\: e^{ip \xi}\nonumber\\ & \quad = -t^2 \int_{\mathbb{R}^4} \frac{d^4p}{(2 \pi)^4} \: \delta\big( p^2-m_0^2 \big)\: \theta(-p^0)\: e^{ip \xi}\nonumber\\ &\int_{\mathbb{R}^4} \frac{d^4p}{(2 \pi)^4} \: \frac{1}{\omega}\:\Big(\frac{\partial}{\partial \omega} \delta\big( p^2-m_0^2 \big) \Big)\: \theta(-p^0)\: e^{ip \xi}\nonumber\\ & \quad =-\frac{i}{2} \int_{-\infty}^\infty \epsilon(\tau) \bigg( \int_{\mathbb{R}^4} \frac{d^4p}{(2 \pi)^4} \: \Big(\frac{\partial}{\partial \omega} \delta\big( p^2-m_0^2 \big) \Big)\: \theta(-p^0)\: e^{ip \xi + i \omega \tau} \bigg) \: d\tau \end{align} \tag{ 4.17 }$$

$$\begin{align} \quad &=-\frac{1}{2} \int_{-\infty}^\infty \epsilon(\tau)\, (t+\tau)\: \bigg( \int_{\mathbb{R}^4} \frac{d^4p}{(2 \pi)^4} \: \delta\big( p^2-m_0^2 \big)\: \theta(-p^0)\: e^{ip \xi + i \omega \tau} \bigg) \: d\tau \:. \end{align} \tag{ 4.18 }$$

For clarity, we remark that in (4.17) we used the distributional formula

$$\begin{align*} \frac{1}{\omega} = -\frac{i}{2}\: \lim_{\delta \searrow 0} \int_{-\infty}^\infty \epsilon(\tau)\: e^{-\delta\, |\tau|}\: e^{i \omega \tau}\: d\tau \:. \end{align*}$$

After pulling the τ-integral outside, we can take the limit $\delta \searrow 0$, because the expression inside the brackets has suitable decay properties in τ. Using these formulas in (4.5), in analogy to (4.9) we obtain the corrections to $P^\varepsilon_{\delta \omega}(x,y)$ given by

$$\begin{align*} \frac{\pi^2}{6}\: \frac{1}{T^{\,2}}\: i {\partial\kern-0.5pt\!\!\!\!/}_x \Big( t^2 \,{\mathcal{T}}^\varepsilon_{m_0^2}(\xi) \Big) - \frac{\pi^2}{12}\: \frac{1}{T^{\,2}}\: i {\partial\kern-0.5pt\!\!\!\!/}_x \int_{-\infty}^\infty \epsilon(\tau)\, (t+\tau)\: {\mathcal{T}}^\varepsilon_{m_0^2}(\xi + \tau e_0) \: d\tau\:. \end{align*}$$

Carrying out the derivatives inside the integrand gives the result.

The corrections computed in this proposition give rise to dark matter and dark energy with the scalings (1.1) and (1.2), as we now explain. The energy-momentum tensor of a Dirac wave function ψ has the general form

$$\begin{align*} T_{jk} = \frac{1}{2} \mathrm{Re} \,\prec\psi \,|\, \big( i \gamma_j \nabla_k + i \gamma_k \nabla_j \big) \psi \succ \end{align*}$$

(for a derivation see for example [26]). Noting that the kernel of the fermionic projector is composed of all physical wave functions (see (2.20)), the total energy-momentum tensor is given formally by

$$\begin{align} T_{jk}(x) = -\frac{1}{2} \mathrm{Re} \mathrm{Tr}_{S_x \mathscr{M}} \Big( \big( i \gamma_j \nabla_k + i \gamma_k \nabla_j \big) P^\varepsilon(x,x) \Big) \:. \end{align} \tag{ 4.19 }$$

The reason why this equation holds only formally is that it involves the energy-momentum of all the sea states, which clearly must not be included in the Einstein equations. The causal action principle takes care of this issue without the need of counter terms, simply because the vacuum Dirac sea is a minimizer (as explained in section 3). Consequently, the sea states do not show up in the EL equations (more technically, for the vacuum Dirac sea configuration the positive and negative terms in the Lagrangian (2.1) cancel each other). However, changes to the vacuum Dirac sea configurations in general do not drop out of the EL equations. More precisely, the term (4.11) changes the eigenvalues of the closed chain only by a prefactor. Therefore, it does drop out of the Lagrangian. The other terms (4.12)–(4.14), however, do not drop out, and we may compute the corresponding contributions to the energy-momentum tensor simply with the help of (4.19). Since the summands (4.12) and (4.13) involves the Dirac matrix γ⁰, the contribute only to $T^{\,0}_0$ and can be understood as dark matter (1.2). The summand (4.14), on the other hand, also involves the spatial Dirac matrices, thereby contributing to both dark energy (1.1) and dark matter (1.2). In view of the factors $1/T^{\,2}$ in (4.11)–(4.14), all these contributions have the desired scaling behavior (1.1) and (1.2).

We finally point out that the contributions (4.11)–(4.14) are singular on the light cone. This means that, in order to compute how they contribute to the EL equations of the causal action principle, one needs to employ the formalism of the continuum limit (as outlined in section 2.5). In this formalism, the prefactors becomes empirical so-called regularization parameters. In particular, at present the signs of the prefactors are unknown.

In this section, we compute the effect of the periodization (1.7). The result of our analysis will be that the resulting contributions to the Einstein equations are much smaller than (1.1) and (1.2), so that the periodization is of no relevance for the dark Universe. Our starting point is again the smeared Dirac sea (4.2) and (4.1) with $\delta m$ and $\delta n$ as given by (4.4) and (4.7). But now we need to expand $P(\xi)$ for large ξ. In preparation, we analyze the distribution ${\mathcal{T}}^\varepsilon_{m^2}$ introduced in (4.10). The Fourier integral over the lower mass shell can be carried out explicitly using Bessel functions to obtain (for details see [11, proof of lemma 1.2.9])

$$\begin{align} {\mathcal{T}}^\varepsilon_{m^2}(\xi) = \frac{m^2}{(2\pi)^3} \frac{K_1 \big(m\sqrt{r^2+(\varepsilon+it)^2}\,\big)}{m\sqrt{r^2+(\varepsilon+it)^2}} \:, \end{align} \tag{ 4.20 }$$

where we again set $\xi = y-x$ as well as $t = \xi^0$ and $r = |\vec{\xi}|$. Expanding the modified Bessel function K₁ for large values of its arguments t of r, one finds that (see [30, equation (10.40.2)])

$$\begin{align} {\mathcal{T}}^\varepsilon_{m^2}(t,\vec{0}) &\simeq \frac{\sqrt{m} }{|t|^{\frac{3}{2}}} \: e^{-i m t} \Big(1+\mathcal{O}\big( |t|^{-1} \big) \Big) \end{align} \tag{ 4.21 }$$

$$\begin{align} {\mathcal{T}}^\varepsilon_{m^2}(0, \vec{\xi}) &= \frac{\sqrt{m} }{r^{\frac{3}{2}}} \: e^{-m r} \Big(1+\mathcal{O}\big( r^{-1} \big) \Big) \:. \end{align} \tag{ 4.22 }$$

Finally, the regularized kernel of the fermionic projector (1.4) is obtained by differentiation,

$$\begin{align} P^\varepsilon(x,y) = (i {\partial\!\!\!\!/}_x + m)\: {\mathcal{T}}^\varepsilon_{m^2}(\xi) \:. \end{align} \tag{ 4.23 }$$

The exponential decay in spacelike directions (4.22) explains why, as already mentioned in the introduction after (1.5), a periodization in space does not lead to any measurable physical effects. In timelike directions, however, one has only polynomial decay,

$$\begin{align*} P^\varepsilon(x,y)|_{\xi=(t,\vec{0})} \simeq \frac{m^\frac{3}{2}}{|t|^{\frac{3}{2}}} \: e^{-i m t} \Big(1+\mathcal{O}\big( |t|^{-1} \big) \Big) \:. \end{align*}$$

In order to compute the resulting effect of the periodization in time, following the consideration after (1.6) we need to take into account the smearing in the mass parameter. Therefore, our next task is to compute the effect of the smearing in the mass parameter for large times. Our starting point is again the formula (4.6) for the smeared lower mass shell. But, instead of expanding in the mass parameter, we now rewrite the smearing by a convolution in the variable ω. In position space, this convolution corresponds to multiplication by a cutoff function in time, giving us the desired information for large times.

We introduce the parameter τ by the condition

$$\begin{align*} \omega^2 - m^2 = \big(\omega+\tau \big)^2 -m_0^2 \:; \end{align*}$$

it tells us about the shift in frequency as caused by the mass change. Using that ω is negative, we can express τ and m in terms of each other by

$$\begin{align} m(\tau) &= \sqrt{m_0^2 + \omega^2 - \big(\omega+\tau \big)^2} = \sqrt{m_0^2 - 2 \omega \tau - \tau^2} \end{align} \tag{ 4.24 }$$

$$\begin{align} \tau(m) &= -\omega - \sqrt{\omega^2 - m^2 + m_0^2} \:. \end{align} \tag{ 4.25 }$$

Moreover,

$$\begin{align*} m \: dm = (-\omega-\tau)\: d\tau \:,\qquad dm = -\frac{\omega+\tau}{m(\tau)}\: d\tau \:. \end{align*}$$

This makes it possible to rewrite (4.6) as

$$\begin{align} R = \frac{1}{\delta m(\vec{k})} \int_{\tau(m_0+\delta n)}^{\tau(m_0+\delta m + \delta n)} \delta \big( (-\omega+\tau)^2-|\vec{k}|^2 -m_0^2 \big)\: \Big( - \frac{\omega+\tau}{m(\tau)} \Big) \: d\tau \:. \end{align} \tag{ 4.26 }$$

A direct computation using (4.25), (4.4) and (4.7) yields

$$\begin{align*} \tau(m_0+\delta n) &= -\frac{\pi}{T} + m_0\, \mathcal{O} \Big( (m_0 T)^{-2} \big) \\ \tau(m_0+\delta m + \delta n) &= \frac{\pi}{T} + m_0\,\mathcal{O} \big( (m_0 T)^{-2} \big) \end{align*}$$

(apart from the error terms, these relations are also obvious from (4.3) and (4.7)). Again using (4.4), we obtain

$$\begin{align} R &= \frac{m_0}{|\omega|}\: \frac{T} {2 \pi} \int_{-\frac{\pi}{T}}^{\frac{\pi}{T}} \delta \big( (-\omega+\tau)^2-|\vec{k}|^2 -m_0^2 \big)\: \Big( - \frac{\omega+\tau}{m(\tau)} \Big) \: d\tau \; \Big( 1 + \mathcal{O}\big( (m_0 T)^{-1} \big) \Big)\nonumber\\ &= \frac{T} {2 \pi} \int_{-\frac{\pi}{T}}^{\frac{\pi}{T}} \delta \big( (-\omega+\tau)^2-|\vec{k}|^2 -m_0^2 \big)\: d\tau \; \Big( 1 + \mathcal{O}\big( (m_0 T)^{-1} \big) \Big)\nonumber\\ &= \int_{-\infty}^{\infty} \delta \big( (-\omega+\tau)^2-|\vec{k}|^2 -m_0^2 \big)\: \hat{\eta}_T(\tau) \:d\tau + \mathcal{O}\big( (m_0 T)^{-1} \big) \:, \end{align} \tag{ 4.27 }$$

where $\hat{\eta}_T$ is a multiple times the characteristic function,

$$\begin{align*} \hat{\eta}_T := \frac{T} {2 \pi} \: \chi_{\big[-\frac{\pi}{T}, \frac{\pi}{T} \big]} \:. \end{align*}$$

In other words, the leading effect of the mass smearing for large $m_0 T$ can be described simply by taking the convolution in ω with the function $\hat{\eta}_T$. In position space, this simply corresponds to multiplication by the Fourier transform of $\hat{\eta}_T$ given by

$$\begin{align*} \eta_T(t) = \int_{-\infty}^\infty \hat{\eta}_T(\omega)\: e^{-i \omega t}\: d\omega = \frac{T} {\pi t}\: \sin \Big( \frac{\pi t}{T} \Big) \:. \end{align*}$$

Therefore, the leading effect of the mass smearing for large times amounts to multiplying the kernel by a function which is T-periodic and vanishes at $t = n T$ with n ≠ 0.

Having clarified the effect of the mass smearing in position space for large times, we can now perform the periodization in time. Using (4.23), similar to (1.6) we consider the series

$$\begin{align*} & \sum_{\tau \in T \mathbb{Z}} (i {\partial\!\!\!\!/}_x + m)\: \Big( \eta_T(\tau)\: {\mathcal{T}}^\varepsilon_{m^2} \big( x,y + (\tau,0) \big) \Big) \:. \end{align*}$$

In order to determine the scaling behavior, it suffices to consider the summand with $\tau = T$. Since $\eta_T(t)$ vanishes at t = T, the leading contribution scales like

$$\begin{align*} &(i {\partial\!\!\!\!/}_x + m)\: \Big( \eta_T(T + \xi^0 )\: {\mathcal{T}}^\varepsilon_{m^2} \big( x,y + (T,0) \big) \Big) \\ & \quad \sim \gamma^0 \,\big( i \eta^{^{\prime}}_T(T+\xi^0) + m\: \eta(T+\xi^0) \big) \: {\mathcal{T}}^\varepsilon_{m^2} \big( x,y + (T,0) \big) \\ & \quad \sim \gamma^0 \: \frac{\sqrt{m} }{T^{\frac{3}{2}}} \Big( \frac{1}{T} + \frac{i m t}{T} \Big) = \gamma^0 \: \frac{\sqrt{m} }{T^{\frac{5}{2}}} \big( 1 + i m t \big) \:, \end{align*}$$

where in the last line we used (4.21). Compared to the contributions computed in proposition 4.2, these contributions are by a scaling factor $(m_0 T)^{-\frac{1}{2}}$ smaller and can therefore be neglected.

By direct computation, one finds that the contributions by the error terms in (4.27) are even smaller and can also be neglected. In order to explain how this comes about, we consider exemplarily the contributions by the quadratic term in (4.7). Expanding (4.26), the resulting contribution to R takes the form

$$\begin{align*} - \frac{1}{24}\: \frac{\delta m(\vec{k})}{m_0} \int_{\tau(m_0 - \delta m/2)}^{\tau(m_0+\delta m/2)} \frac{\partial}{\partial \omega} \delta \big( (-\omega+\tau)^2-|\vec{k}|^2 -m_0^2 \big)\: \Big( - \frac{\omega+\tau}{m(\tau)} \Big) \: d\tau \:. \end{align*}$$

As explained after (4.27), the τ-integration gives rise to the multiplication by $\eta_T(t)$. Compared to (4.27), we now have the additional factor

$$\begin{align*} \frac{(\delta m)^2}{m_0} \frac{\partial}{\partial \omega} \overset{(4.4)}{\simeq} \frac{\omega^2}{m_0^3\, T^{\,2}} \:\frac{\partial}{\partial \omega} \:. \end{align*}$$

Taking the Fourier transform similar to the computation after (4.16), we obtain an additional scaling factor which is at least as small as $(m_0 T)^{-1}$, justifying that this contribution is negligible. For the other contributions one can argue similarly.

There is one point in the periodization which requires closer attention: if we periodize both in space and time (as shown in figure 1) we also get contributions when the vector $(\tau, \vec{\ell})$ in (1.7) is close to the light cone. The following consideration shows that the resulting contributions to the kernel of the fermionic projector are also negligible: The scaling behavior can be determined by considering the simple $i \varepsilon$ regularization obtained by the replacement $t \rightarrow t- i \varepsilon$. Then the Bessel function in (4.20) becomes (for details see [11, § 1.2.5] or [31, section 5.1 and appendix B])

$$\begin{align*} K_1 \big(m\sqrt{r^2+(\varepsilon+it)^2}\,\big) \:. \end{align*}$$

Evaluating this function on the light cone $t = \pm r$, one obtains exponential decay

$$\begin{align*} K_1 \big(m\sqrt{\varepsilon^2+2 i \varepsilon t}\,\big) \lesssim K_1 \big( m \sqrt{\varepsilon \,|t|} \big) \lesssim e^{- m \sqrt{\varepsilon \,|t|}} \:. \end{align*}$$

As a consequence, the functions ${\mathcal{T}}^\varepsilon_{m^2}$ and P^ε decay exponentially in lightlike directions as

$$\begin{align} e^{- m \sqrt{\varepsilon T}} \:. \end{align} \tag{ 4.28 }$$

This decay is very slow because the regularization length ε comes into play. Nevertheless, we get exponential decay on cosmological scales, as is verified as follows. Choosing m as the mass of the τ-lepton, ε as the Planck length and T according to (1.3), we obtain

$$\begin{align*} m \varepsilon \: \sqrt{m T} \approx 10 \end{align*}$$

and thus

$$\begin{align*} m^2\, \varepsilon T \approx 10\, \sqrt{mT} \approx 4 \times 10^{21} \:. \end{align*}$$

As a consequence, the exponential factor (4.28) is extremely small and may be neglected. With this in mind, in the periodization we may disregard null directions. A more detailed estimate of the kernel of the fermionic projector in a strip around the light cone can be found in [31, appendix B].