Fermions that see only the causal order

Can a spinor field on a Poisson sprinkling have a Lorentz-covariant kinetic term built from the causal order alone? For a four-dimensional Weyl spinor attached to each element, with one spinor space and one Lorentz action at every element, the answer is no. If the quadratic form depends only on the order and is covariant, every kernel that couples $\psi^{\dagger}$ at one element to $\psi$ at another vanishes, and the kernels that couple $\psi$ to $\psi$ are scalar multiples of the antisymmetric matrix $\epsilon$ . These are Majorana-type pairings $\psi^{T}\epsilon\psi$ whose spinor structure carries no derivative index (Theorem 1). A first-order operator needs a vector-valued datum on each pair of elements, and the order supplies none. Kähler–Dirac fields, whose components are functions on chains of the order, escape this obstruction. On the span-restricted complex (section 3), which keeps the chains with at most $d-1$ elements between their ends, any even-odd operator that is self-adjoint for a degree-diagonal form has more exact zero modes than there are elements: proved in expectation for $d=2$ , measured for $d=4$ .

The components of a Kähler–Dirac field are cochains, functions on the chains $x_0\prec\dots\prec x_k$ of the order, and the coboundary $\delta$ is combinatorial, so no frame enters. Any operator that exchanges even and odd degrees and is self-adjoint for a nondegenerate form that is diagonal in the degree has at least $\lvert\chi\rvert$ exact zero modes, with $\chi$ the Euler characteristic of the complex (Theorem 5). On the span-restricted complex the expected value of $-\chi$ for a sprinkling of a two-dimensional diamond with $N$ elements is $N(\ln N+\gamma-3)+O(\ln N)$ (Proposition 7), and samples in four dimensions grow at a comparable rate (Table 1). A zero mode of a kinetic operator is a massless mode. The continuum operator $\mathrm d+\mathrm d^{\dagger}$ has as many zero modes as the sum of the Betti numbers of the region, which is $1$ for a contractible region, and a lattice version has $2^{d/2}$ tastes, whereas the count here grows faster than $N$ . Whether the zero-mode count also grows on the complex of all chains is not decided. The mean index there stays between $0$ and $2$ in two dimensions (Proposition 15), but the bound $\lvert\chi\rvert$ is not controlled by the mean, and the sampled root-mean-square of the index grows with the size (section 6).

In two dimensions, terms built from $\delta$ and its transpose, such as the Wilson term $rK^{2}$ that is added to give mass to unwanted modes, leave every harmonic cochain (one annihilated by both) at zero (Proposition 10). A mass on the degree-one cochains lifts the harmonic 1-cochains to the mass scale (Proposition 11). The locally constant 0-cochains stay at zero, and whether other light modes remain depends on the smallest nonzero eigenvalue of the vertex Laplacian, the graph Laplacian of the edges of the complex, which is only observed (section 4). On the chain of elements ordered along one null direction in 1+1 dimensions, a one-component chiral operator exists once that direction is chosen (Proposition 12), but every shift-invariant pure Gaussian state has as many upward as downward sign changes of its symbol, its eigenvalue on each Fourier mode, so its net chirality vanishes (Proposition 13).

Matter on causal sets has been developed mostly for scalar fields, starting from Johnston’s retarded propagators as sums over chains [Johnston 2008] (see also [Johnston thesis], [X–Dowker–Surya], and [Surya] for a review). For spinors, Sverdlov builds vierbeins from holonomies and then spinor fields on them [Sverdlov], which supplies frame data at each element beyond the order. Dowker and Sorkin showed that a Poisson sprinkling carries no distinguished spatial or temporal orientation, no preferred lattice of timelike directions and no distinguished lattice of points [Dowker–Sorkin]. Kähler–Dirac fermions on triangulations were developed by Catterall and collaborators, who find zero modes that follow the topology of the triangulation [Catterall–Laiho–Unmuth-Yockey] and relate the index to the Euler characteristic [Butt et al.]. A reduced version leaves mirror fermions [Catterall], and the Lorentzian version needs an operator incompatible with Lorentz transformations [Catterall–Hubisz]. Those zero-mode counts are for given triangulations; here the count is carried over to the span-restricted order complex of a sprinkling, where it grows with the number of elements (Proposition 7 and Corollary 8).

A frame cannot be supplied by a sprinkling at random: frame leakage proves that no Lorentz-invariant probability measure on frames exists, which Proposition 3 below uses. Which invariant random point sets exist besides the Poisson sprinkling is the subject of Lorentz-invariant point sets, where Haar-random lattices are invariant and not Poisson, and of quantum sprinklings, where an invariant locally normal bosonic state with finite mean counts gives Poisson counts with a random intensity. In chirality obstructions for frame-free Weyl operators the overlap index of an operator built from the order and the invariant interval alone is shown to be zero or undefined for retarded operators and for operators self-adjoint for a chirality-exchanging matrix, when the Wilson term acts alike on both chiralities. Frames, Wilson terms and reflection positivity treats what a frame costs in a range-cut link operator with a Wilson term, and why reflection positivity in every realisation is not defined for the Euclidean overlap on a Poisson process.

1. Causal sets, sprinklings and “order-only”

A causal set is a locally finite set $P$ with a strict partial order $\prec$ . For $x\prec y$ the interval is $I(x,y)=\{z:x\prec z\prec y\}$ , and $x\prec y$ is a link if $I(x,y)=\varnothing$ . A sprinkling of density $\rho$ into Minkowski space $\mathbb M^{d}$ is a Poisson point process with intensity $\rho$ , ordered by $x\prec y$ when $y$ lies in the causal future of $x$ . For finite computations we sprinkle into a causal diamond. In two dimensions, with null coordinates $(u,v)$ scaled so that the diamond is the unit square, $x\prec y$ means $u_x<u_y$ and $v_x<v_y$ .

Every orthochronous Poincaré transformation $g$ maps $\mathbb M^{d}$ to itself preserving the causal order, so it maps a point set $P$ to a point set $gP$ with an isomorphic causal order, the isomorphism being $x\mapsto gx$ . A construction that uses only the order is therefore natural: what it assigns to a pair $(x,y)$ in $P$ equals what it assigns to $(gx,gy)$ in $gP$ . This is the sense in which a causal-set construction is Lorentz invariant. If the construction also has to be covariant on spinors, naturality and covariance together constrain each value, and this is the question of the next section.

2. Weyl spinors and the Schur argument

A left-handed Weyl spinor in four dimensions is a pair $\psi\in\mathbb C^{2}$ on which $S\in SL(2,\mathbb C)$ acts as $\psi\mapsto S\psi$ . The connection with Lorentz transformations is this. Identify a vector $x=(x^{0},\vec x)$ with the Hermitian matrix $X=x^{0}\mathbb 1+\vec x\cdot\vec\sigma$ , so that $\det X=(x^{0})^{2}-\lvert\vec x\rvert^{2}$ . The map $X\mapsto SXS^{\dagger}$ preserves Hermiticity and the determinant, hence is a Lorentz transformation $\Lambda(S)$ , and $S\mapsto\Lambda(S)$ is a two-to-one cover of the proper orthochronous group $SO^{+}(1,3)$ with kernel $\pm\mathbb 1$ . A rotation by $2\pi$ corresponds to $S=-\mathbb 1$ , so spinors change sign. The matrix $\psi\psi^{\dagger}$ transforms as $X$ does, which is why the combinations $\psi^{\dagger}\bar\sigma^{\mu}\psi$ , with $\bar\sigma^{\mu}=(\mathbb 1,-\vec\sigma)$ , are the components of a vector. The kinetic term $\psi^{\dagger}\bar\sigma^{\mu}\partial_\mu\psi$ therefore requires a vector-valued object: the four Hermitian matrices $\bar\sigma^{\mu}$ form a basis of the space $\mathrm{Herm}(2)\cong\mathbb R^{1,3}$ of Hermitian $2\times2$ matrices. A mass term is a quadratic form in $\psi$ without a derivative. For anticommuting components the only candidate is $\psi^{T}\epsilon\psi$ with $\epsilon$ the antisymmetric matrix, a Majorana mass. In two dimensions a chiral Weyl spinor has one component, and a boost with rapidity $\eta$ acts as multiplication by $e^{\eta/2}$ . In three dimensions the spinor group is $SL(2,\mathbb R)$ acting on $\mathbb C^{2}$ .

We fix $d\in\{2,3,4\}$ and let $V$ be the spinor space, with $S(g)$ the matrix by which $g$ , in the double cover of $SO^{+}(1,d-1)$ , acts on it. A construction assigns to every locally finite point set $P\subset\mathbb M^{d}$ and every ordered pair $(x,y)$ of its elements a complex matrix $\Phi(P)(x,y)$ on $V$ with $\Phi(P)(y,x)=\Phi(P)(x,y)^{\dagger}$ , so that the quadratic form $\sum_{x,y}\psi_x^{\dagger}\Phi(P)(x,y)\psi_y$ with $\psi_x\in V$ is real. A bilinear construction assigns instead matrices that define $\sum_{x,y}\psi_x^{T}\Phi(P)(x,y)\psi_y$ . The construction is required to be natural for a group $G$ of Lorentz transformations, which is one of three: the full group $SO^{+}(1,d-1)$ ; the stabiliser of a region frame, here the two tips of the causal diamond in which we sprinkle (a transformation that moves the tips does not map the diamond to itself, so $gP$ would not be a sprinkling of the same diamond), which is trivial for $d=2$ , $SO(2)$ for $d=3$ , and $SO(3)$ acting through $SU(2)$ for $d=4$ ; and the trivial group.

(H1) There is one copy of $V$ at every element, and a Lorentz transformation $g$ acts on all copies by the same matrix $S(g)$ . This is a global trivialisation of the spinor bundle, that is a global frame.

(H2) $\Phi$ depends on the order alone: $\Phi(gP)(gx,gy)=\Phi(P)(x,y)$ for every $g\in G$ .

(H3) The quadratic form is covariant: replacing $P$ by $gP$ and $\psi_x$ by $S(g)\psi_x$ at the element $gx$ leaves its value unchanged, which means $S(g)^{\dagger}\Phi(gP)(gx,gy)S(g)=\Phi(P)(x,y)$ .

Combining (H2) and (H3) gives, for every $g\in G$ , every $P$ , $x$ and $y$ ,

S(g)^{\dagger}\,\Phi(P)(x,y)\,S(g)=\Phi(P)(x,y), \tag{1}

and the same with transposes for bilinear constructions. Each value of the kernel is therefore a $G$ -invariant form on a single copy of $V$ , and the problem reduces to representation theory.

Theorem 1 (order-natural covariant spinor kernels). Under (H1) to (H3) every value $\Phi(P)(x,y)$ is a form on $V$ invariant under $G$ . The real dimensions of the spaces of invariant sesquilinear forms (complex matrices $k$ with $S^{\dagger}kS=k$ ), of the Hermitian ones among them, and of invariant bilinear forms ( $S^{T}kS=k$ ) are

$d$	$G$	sesquilinear	Hermitian	bilinear
4	$SO^{+}(1,3)$	0	0	2
4	$SU(2)$ (region frame)	2	1	2
4	trivial	8	4	8
3	$SO^{+}(1,2)$	2	1	2
3	$SO(2)$ (region frame)	4	2	4
2	$SO^{+}(1,1)$	0	0	0
2	trivial (region frame)	2	1	2

In particular, in four dimensions with $G=SO^{+}(1,3)$ every order-natural covariant sesquilinear kernel vanishes, and the order-natural covariant bilinear kernels are $\Phi(P)(x,y)=b(P)(x,y)\,\epsilon$ with $b$ a complex function of the pair that depends on $P$ through the order. With a region frame, a sesquilinear kernel is a complex scalar function times $\mathbb 1$ and distinguishes no helicity. The Hermitian column counts the kernels allowed at coincident elements, where $\Phi(P)(x,x)=\Phi(P)(x,x)^{\dagger}$ .

Proof. An invariant form is an intertwiner, and the Weyl representation has no invariant Hermitian pairing with itself because its conjugate is a different representation. For $d=4$ we give the computation for each row.

Sesquilinear forms, $G=SO^{+}(1,3)$ . Write $k=k_1+ik_2$ with $k_1,k_2$ Hermitian. The map $k\mapsto S^{\dagger}kS$ preserves Hermiticity, so both parts are invariant. A Hermitian $k_j$ is a vector in $\mathrm{Herm}(2)\cong\mathbb R^{1,3}$ , and $k_j\mapsto S^{\dagger}k_jS$ is the Lorentz transformation $\Lambda(S^{\dagger})$ . As $S$ runs over $SL(2,\mathbb C)$ so does $S^{\dagger}$ , so an invariant $k_j$ is a vector fixed by all of $SO^{+}(1,3)$ . Rotations fix only vectors along the time axis, and a boost along $x^1$ moves $(t,0,0,0)$ to $(t\cosh\eta,t\sinh\eta,0,0)$ . So $k=0$ .

Bilinear forms, $G=SO^{+}(1,3)$ . Take $S=\mathrm{diag}(z,z^{-1})$ with $z\neq\pm1$ in $S^{T}kS=k$ : this gives $k_{11}z^{2}=k_{11}$ and $k_{22}z^{-2}=k_{22}$ , so $k_{11}=k_{22}=0$ . With $k=\begin{pmatrix}0&b\\c&0\end{pmatrix}$ and $S=\begin{pmatrix}1&a\\0&1\end{pmatrix}$ the $(2,2)$ entry of $S^{T}kS$ is $ab+ca$ , which must vanish, so $c=-b$ and $k=b\epsilon$ . Conversely $S^{T}\epsilon S=\epsilon$ for $\det S=1$ . One complex parameter is two real.

$G=SU(2)$ . Here $S^{\dagger}=S^{-1}$ , so $S^{\dagger}kS=k$ says that $k$ commutes with $SU(2)$ , hence is a complex scalar; the Hermitian ones are the real multiples of $\mathbb 1$ . For bilinear forms $S^{T}=\epsilon S^{-1}\epsilon^{-1}$ for $S\in SU(2)$ , so the condition says that $\epsilon^{-1}k$ commutes with $SU(2)$ , hence is a scalar, and $k\propto\epsilon$ .

Trivial group. All complex $2\times2$ matrices (8 real parameters), of which 4 are Hermitian.

For $d=2$ the space $V=\mathbb C$ carries the character $S=e^{\eta/2}$ of the boost, and invariance requires $e^{\eta}k=k$ for sesquilinear and for bilinear forms alike, so $k=0$ when $G$ contains boosts and $k$ is arbitrary otherwise. For $d=3$ the matrices $S$ are real, so sesquilinear and bilinear invariance coincide. The same computation as for $SO^{+}(1,3)$ with $S=\mathrm{diag}(z,z^{-1})$ and a real unipotent $S$ gives $k=b\epsilon$ , $b\in\mathbb C$ , which is Hermitian when $b$ is imaginary. For the rotation subgroup $SO(2)$ one has $S^{T}=S^{-1}$ , so $k$ commutes with the rotations and $k=a\mathbb 1+b\epsilon$ with $a,b\in\mathbb C$ , Hermitian when $a$ is real and $b$ imaginary. $\square$

(H1) fails if the spinor space at each element is allowed to depend on the element (local frames); in the continuum the same requirement makes a spinor kinetic term depend on a choice of tetrad in addition to the metric.

Corollary 2 (no order-natural Weyl operator in four dimensions). Let $K$ be an operator on fields given by $(K\psi)_x=\sum_y\Phi(P)(x,y)\psi_y$ with $\Phi$ satisfying (H1) to (H3) and $G=SO^{+}(1,3)$ . Then $\Phi=0$ and $K=0$ . With a region frame added, $K$ acts on each component of $\psi$ by the same scalar kernel, and no term in the expansion of $K\psi$ in derivatives of $\psi$ depends on its spin.

Proof. The first statement is Theorem 1. For the second, $\Phi(P)(x,y)=\varphi(P)(x,y)\mathbb 1$ , so $K$ commutes with the rotation of the components, whereas a Weyl operator $\vec\sigma\cdot\vec\nabla$ does not. $\square$

Example 1 (square roots). A Dirac-type operator is often sought as a square root of an order-natural scalar matrix $\Phi$ on the elements, for instance a retarded Green function. For any such matrix, with $\sigma_\pm$ the raising and lowering matrices of an auxiliary two-component space and $\kappa\neq0$ , the matrix $R_\kappa=-\rho\kappa\,\Phi\otimes\sigma_{+}+\kappa^{-1}\mathbb 1\otimes\sigma_{-}$ satisfies $R_\kappa^{2}=-\rho\,\Phi\otimes\mathbb 1$ , because $\sigma_\pm^{2}=0$ and $\sigma_+\sigma_-+\sigma_-\sigma_+=\mathbb 1$ . The roots form a one-parameter family. In 1+1 dimensions a boost acts on the two components as $\mathrm{diag}(e^{\eta/2},e^{-\eta/2})$ and maps $R_\kappa$ to $R_{\kappa e^{\eta}}$ . No root is boost invariant, and a boost moves one root along the family, so fixing a root fixes a frame.

Frames that are supplied

The obstruction is removed by data not contained in the order: a frame. Proposition 3 shows that such data cannot be random with an invariant law, and Proposition 4 shows what a given frame yields in link operators.

Proposition 3 (no random global frame). Let $\mathcal P$ be a Poisson sprinkling of $\mathbb M^{4}$ and let $F$ be a random element of $SL(2,\mathbb C)$ (a global frame) jointly distributed with $\mathcal P$ so that the pair $(\mathcal P,F)$ has a law invariant under $g:(P,F)\mapsto(gP,S(g)F)$ . There is no such law.

Proof. The map $F\mapsto FF^{\dagger}$ takes values in the positive Hermitian matrices of determinant 1, which form the hyperboloid $Y$ of future unit timelike vectors, and is equivariant: $(S F)(SF)^{\dagger}=S(FF^{\dagger})S^{\dagger}=\Lambda(S)\cdot(FF^{\dagger})$ . The law of $FF^{\dagger}$ would be a Lorentz-invariant probability measure on $Y$ , which does not exist by Proposition 6 of frame leakage. $\square$

A frame cannot therefore be a random variable attached to the sprinkling with an invariant joint law. It can be a fixed choice, which breaks Lorentz invariance, or it can be gauge data attached to each element, in which case the kernel depends on the frames at its two ends and (H1) fails.

Proposition 4 concerns the translation-invariant model operator with finitely many displacement vectors. Its symbol is what the plane-wave-averaged symbol of a link operator on a sprinkling becomes in frames, Wilson terms and reflection positivity (section 2), and a frame enters through the isotropy assumption below. Here $\bar\sigma\!\cdot\!v=\bar\sigma^{\mu}v_{\mu}$ .

Proposition 4 (link operators with non-negative weights). Let $v_n\in\mathbb M^{4}$ be a finite set of future-causal vectors (displacements between linked elements), $w_n\ge0$ weights that are not all zero, and $F(p)=-2\sum_nw_n\,\bar\sigma\!\cdot\!v_n\,\sin(p\!\cdot\!v_n)$ , which is $i$ times the symbol of the operator $\psi(x)\mapsto\sum_nw_n\,\bar\sigma\!\cdot\!v_n[\psi(x+v_n)-\psi(x-v_n)]$ . Then $T_{\mu\nu}=\sum_nw_nv_{n\mu}v_{n\nu}$ is a non-zero positive semidefinite matrix, hence is not proportional to the Minkowski metric, and $F(p)=-2\bar\sigma^{\mu}T_{\mu\nu}p^{\nu}+O(p^{3})$ . If the weights are rotation invariant about a unit timelike vector $e_0$ , so that $T=\mathrm{diag}(T_{00},T_{11},T_{11},T_{11})$ in the frame of $e_0$ , the infrared symbol has the relative sign of the time and space terms opposite to that of a covariant Weyl symbol $\bar\sigma^{\mu}\eta_{\mu\nu}p^{\nu}$ , that is the opposite helicity, and its speed is $T_{11}/T_{00}\le\tfrac13$ , with equality only if every $v_n$ is null.

Proof. $T$ is a sum of positive semidefinite rank-one matrices, and non-zero because $v_n\neq0$ for a causal vector. A multiple $\lambda\eta$ with $\lambda\neq0$ is indefinite, and $\lambda=0$ would make $T=0$ . Expanding the sine gives the leading term. In signature $(-+++)$ with $\bar\sigma^{\mu}=(\mathbb 1,-\vec\sigma)$ , $\bar\sigma^{\mu}\eta_{\mu\nu}p^{\nu}=-(p^{0}+\vec\sigma\!\cdot\!\vec p)$ , whereas $\bar\sigma^{\mu}T_{\mu\nu}p^{\nu}=T_{00}p^{0}-T_{11}\,\vec\sigma\!\cdot\!\vec p$ , so the relative sign of the two terms is reversed. The speed is $T_{11}/T_{00}$ . Since each $v_n$ is causal, $(v^{0}_n)^{2}\ge\lvert\vec v_n\rvert^{2}$ , so $T_{00}=\sum w(v^{0})^{2}\ge\sum w\lvert\vec v\rvert^{2}=3T_{11}$ . $\square$

A covariant infrared limit therefore needs an indefinite coefficient matrix, which for link operators of this form means signed weights. The proposition concerns the infrared symbol only and says nothing about doublers, whose location depends on the weights. The same statement is Lemma 1 of frames, Wilson terms and reflection positivity, proved there in any dimension for isotropic weights, with signature $(+,-,-,-)$ and $\bar\sigma\cdot\Delta=\Delta^{0}+\vec\sigma\cdot\vec\Delta$ in place of the conventions used here. That page continues with the Wilson term on random points.

3. Kähler–Dirac fields on the order complex

In flat space the exterior derivative $\mathrm d$ and its adjoint act on the space $\Omega=\bigoplus_{k=0}^{d}\Omega^{k}$ of differential forms, and $K=\mathrm d+\mathrm d^{\dagger}$ squares to the Hodge Laplacian. Exterior and interior products with a vector define a Clifford action on $\Omega$ , so $\Omega$ is a sum of $2^{d/2}$ copies of the Dirac spinor representation, the tastes, and $K$ is the Dirac operator on this sum (Kähler). The parity $\Gamma=(-1)^{k}$ of the degree anticommutes with $K$ . On a lattice or a triangulation the same construction uses cochains: $C^{k}$ is the space of functions on $k$ -simplices and the coboundary $\delta:C^{k}\to C^{k+1}$ replaces $\mathrm d$ [Catterall–Laiho–Unmuth-Yockey] [Butt et al.]. No frame is needed to define $\delta$ , which is why Kähler–Dirac fields can escape Theorem 1: their components sit on tuples of elements and not in a fibre of a spinor bundle.

For a causal set $P$ the simplices are chains $x_0\prec x_1\prec\dots\prec x_k$ , with the coboundary

(\delta\omega)(x_0,\dots,x_{k+1})=\sum_{i=0}^{k+1}(-1)^{i}\,\omega(x_0,\dots,\widehat{x_i},\dots,x_{k+1}). \tag{2}

The complex of all chains is not local, since a chain may contain elements far apart (section 6 gives its index). We restrict to chains whose first and last elements have at most $n_0$ elements between them, and call the resulting simplicial complex $\Delta_{n_0}$ . A sub-chain of such a chain has a smaller span, so $\Delta_{n_0}$ is closed under faces. A chain with $k+1$ elements has at least $k-1$ elements inside its span, so $\dim\Delta_{n_0}\le n_0+1$ and the choice $n_0=d-1$ gives a complex of dimension at most $d$ . For $d=2$ the complex $\Delta_1$ has as vertices the elements, as edges the links and the pairs $x\prec z$ with exactly one element $y$ in between, and as triangles the chains $x\prec y\prec z$ with $I(x,z)=\{y\}$ .

The cochain spaces carry a nondegenerate symmetric bilinear form $Q=\bigoplus_kQ_k$ , diagonal in the degree. The default is the counting form $Q_k=\mathbb 1$ , for which the adjoint of $\delta$ is the transpose $\delta^{T}$ . Another is the Krein form $Q_k=(-1)^{k}\mathbb 1$ , which is indefinite and for which the adjoint is $-\delta^{T}$ . We write $\delta^{*}$ for the adjoint and $K=\delta+\delta^{*}$ .

Theorem 5 (index of parity-odd operators). Let $\Delta$ be a finite simplicial complex with cochain spaces $C^{k}$ , let $Q$ be a nondegenerate symmetric bilinear form diagonal in the degree, and let $K$ be a linear operator on $C=\bigoplus_kC^{k}$ that maps even to odd degrees and odd to even degrees and is self-adjoint for $Q$ . Then

\dim\ker K|_{\mathrm{even}}-\dim\ker K|_{\mathrm{odd}}=\chi(\Delta)=\sum_k(-1)^{k}f_k, \tag{3}

where $f_k$ is the number of $k$ -simplices. Hence $\dim\ker K\ge\lvert\chi(\Delta)\rvert$ . If $Q$ is positive definite and $K=\delta+\delta^{*}$ , then $\dim\ker K=\sum_kb_k$ , the sum of the Betti numbers.

Proof. Rank–nullity applied to the two halves of $K$ gives the index, and self-adjointness makes the two ranks equal. Write $K_+=K|_{\mathrm{even}}:C^{\mathrm{ev}}\to C^{\mathrm{odd}}$ and $K_-=K|_{\mathrm{odd}}$ . For $x$ even and $y$ odd, self-adjointness gives $\langle K_+x,y\rangle_Q=\langle x,K_-y\rangle_Q$ . Since $Q$ is nondegenerate on $C^{\mathrm{ev}}$ , $y\perp\mathrm{ran}\,K_+$ if and only if $K_-y=0$ : $(\mathrm{ran}\,K_+)^{\perp}=\ker K_-$ . For a subspace $W$ of a space with a nondegenerate form $\dim W^{\perp}=\dim C^{\mathrm{odd}}-\dim W$ , so $\dim\ker K_-=\dim C^{\mathrm{odd}}-\mathrm{rk}K_+$ , while $\dim\ker K_+=\dim C^{\mathrm{ev}}-\mathrm{rk}K_+$ . Subtracting gives $\dim C^{\mathrm{ev}}-\dim C^{\mathrm{odd}}=\sum_k(-1)^{k}f_k$ , and $\ker K=\ker K_+\oplus\ker K_-$ gives the bound. For positive $Q$ , $\langle \delta\omega,\delta^{*}\eta\rangle=\langle \delta\delta\omega,\eta\rangle=0$ , so $\lVert K\omega\rVert^{2}=\lVert \delta\omega\rVert^{2}+\lVert \delta^{*}\omega\rVert^{2}$ , $\ker K=\ker \delta\cap\ker \delta^{*}$ , and Hodge theory identifies this with the cohomology. $\square$

Theorem 5 is the discrete McKean–Singer index relation. For a causal set the size of $\chi$ decides how many zero modes are forced, and the Euler characteristic of the complexes $\Delta_{n_0}$ is computed from the Möbius function $\mu$ , defined by $\mu(x,x)=1$ and $\sum_{x\preceq z\preceq y}\mu(x,z)=0$ for $x\prec y$ . Hall’s theorem says $\mu(x,y)=\sum_{k\ge1}(-1)^{k}c_k(x,y)$ , where $c_k$ is the number of chains $x=x_0\prec\dots\prec x_k=y$ [Wachs].

Corollary 6 (Euler characteristic of the span-restricted complex). With $N=\lvert P\rvert$ ,

\chi(\Delta_{n_0})=N+\sum_{\substack{x\prec y\\ \lvert I(x,y)\rvert\le n_0}}\mu(x,y), \tag{4}

and $\chi(\Delta_1)=N-\ell$ where $\ell$ is the number of links.

Proof. The $N$ one-element chains contribute $+1$ each. Group the other simplices by their first and last elements $x\prec y$ . If $\lvert I(x,y)\rvert\le n_0$ then every chain from $x$ to $y$ is a simplex of $\Delta_{n_0}$ , since all its elements lie in $I(x,y)$ , and the signed count is $\mu(x,y)$ by Hall’s theorem. A link has the single chain $x\prec y$ , so $\mu=-1$ ; a pair with one interior element $z$ has the chains $x\prec y$ and $x\prec z\prec y$ , so $\mu=-1+1=0$ . This gives (4) for $n_0=1$ . The same follows from the face counts of $\Delta_1$ : the triangles are in bijection with the one-interior pairs, because a triangle $x\prec y\prec z$ has $I(x,z)=\{y\}$ , so with $f_2$ triangles the number of edges is $f_1=\ell+f_2$ and $f_0-f_1+f_2=N-\ell$ . $\square$

Example 2 (four elements). Take the poset with four elements $x\prec a,b\prec y$ , $a$ and $b$ unrelated. It has four links, and $I(x,y)=\{a,b\}$ has two elements. Then $\chi(\Delta_1)=4-4=0$ . For $n_0=2$ the pair $(x,y)$ is admitted with $\mu(x,y)=-(1-1-1)=1$ , and $\chi(\Delta_2)=4-4+1=1$ , in agreement with the face count $4-5+2=1$ and with the fact that this complex is a cone, hence contractible.

How large is the Euler characteristic on a sprinkling?

A link is a pair with an empty interval, and in a Lorentz-invariant sprinkling an element has unboundedly many links as the density grows [Bombelli–Henson–Sorkin]. In two dimensions the expected number of links can be computed exactly.

Proposition 7 (expected links in two dimensions). Let $N$ points be independent and uniform in the unit square, ordered by the product order (a sprinkling of the causal diamond in null coordinates, with the number of points fixed). Then

\mathbb E[\ell]=N\,H_{N-1}+H_N-2N+1,\qquad H_n=\sum_{k=1}^{n}\frac1k, \tag{5}

and consequently $\mathbb E[\chi(\Delta_1)]=-N\bigl(\ln N+\gamma-3\bigr)+O(\ln N)$ , where $\gamma$ is Euler’s constant.

Proof. Count ordered pairs $(i,j)$ . For $i\neq j$ the differences $s=u_j-u_i$ and $t=v_j-v_i$ are both positive with joint density $(1-s)(1-t)$ on the unit square, and given them $(i,j)$ is a link exactly when none of the other $N-2$ points falls in the rectangle of area $st$ between them, which has probability $(1-st)^{N-2}$ . With $m=N-2$ ,

\mathbb E[\ell]=N(N-1)\int_0^1\!\!\int_0^1(1-s)(1-t)(1-st)^{m}\,ds\,dt .

Expand $(1-s)(1-t)=1-s-t+st$ . The product $p=st$ of two uniform variables has density $-\ln p$ , so $\iint(1-st)^{m}=\int_0^1(-\ln p)(1-p)^{m}dp=H_{m+1}/(m+1)$ . Next $\iint s(1-st)^{m}=\int_0^1\frac{1-(1-s)^{m+1}}{m+1}\,ds=\frac1{m+2}$ , and the same for $t$ . Finally $\iint st(1-st)^{m}=\int_0^1(-\ln p)\,p(1-p)^{m}dp=\frac{H_{m+2}-1}{(m+1)(m+2)}$ . With $m+1=N-1$ and $m+2=N$ the four terms add to

\mathbb E[\ell]=N(N-1)\Bigl[\frac{H_{N-1}}{N-1}-\frac{2}{N}+\frac{H_N-1}{(N-1)N}\Bigr]=NH_{N-1}-2(N-1)+H_N-1,

which is (5). The asymptotics follow from $H_{N-1}=\ln N+\gamma+O(1/N)$ and Corollary 6. $\square$

For $N=2$ equation (5) gives $1/2$ : the two points are related with probability $1/2$ , and then they are a link. For large $N$ the expected number of links per element is $H_{N-1}\simeq\ln N$ and grows without bound. Table 1 gives sample means of $-\chi/N$ for $\Delta_{d-1}$ on sprinklings of a causal diamond, together with the exact expectation in two dimensions. In four dimensions $\chi(\Delta_3)$ was computed from (4) on points uniform in a causal diamond. The links per element are listed because they show the unbounded valency; for $d=4$ the quantity $\chi(\Delta_3)$ is not $N-\ell$ , and its values come from (4).

$N$	2D, sample	2D, exact (5)	2D links per element	4D, sample	4D links per element
250	3.10	3.13	4.1	3.64	6.7
500	3.78	3.81	4.8	6.22	11.4
1000	4.54	4.49	5.5	9.01	17.6
2000	5.21	5.18	6.2	12.68	27.9
4000	5.90	5.87	6.9	16.85	42.1

Table 1. The quantity $-\chi(\Delta_{d-1})/N$ on sprinklings of a causal diamond; each sample entry is the mean over five independent sprinklings.

The standard deviation of $\chi/N$ between sprinklings, as a fraction of its mean, is below 7% for $N\ge500$ and below 4% for $N\ge1000$ in both dimensions.

The growth could come from the randomness of the sprinkling or from the span restriction alone, and a regular lattice separates the two. The product order on the integer grid $\{0,\dots,m\}^2$ is regular and not Lorentz invariant. It has $N=(m+1)^2$ points and $\ell=2m(m+1)$ links, so $\chi(\Delta_1)=N-\ell=1-m^{2}$ and $-\chi/N\to1$ , whereas $\chi(\Delta_2)=\chi(\Delta_3)=1$ for the sizes computed, $m=4,6,10$ . On the grid a span of two or three therefore removes the growth. On two-dimensional sprinklings span $2$ does not: $-\chi(\Delta_2)/N$ is $2.5$ and $2.9$ at $N=1000$ and $2000$ , and only for span $3$ is $\chi(\Delta_3)/N$ within $0.1$ of zero, with either sign. These two complexes have dimension larger than $d=2$ . On sprinklings of the unit square $-\chi(\Delta_1)/N$ grows like $\ln N$ instead of tending to $1$ (Proposition 7). Hence the regular order loses the growth once the span is two, whereas the sprinkling keeps it at span two, so at that span the growth belongs to the randomness of the points and not to the span restriction alone. Table 1 uses the span $d-1$ , which gives a complex of dimension $d$ .

Corollary 8 (zero modes of parity-odd operators). For every nondegenerate degree-diagonal $Q$ and every $Q$ -self-adjoint operator $K$ that exchanges even and odd degrees on $\Delta_{d-1}$ of a sprinkling, in particular for $\delta+\delta^{*}$ with the counting or the Krein form, $\dim\ker K\ge\lvert\chi(\Delta_{d-1})\rvert\ge-\chi(\Delta_{d-1})$ . In two dimensions, with $N$ uniform points, $\mathbb E[\dim\ker K]\ge N(\ln N+\gamma-3)+O(\ln N)$ . In the samples of Table 1 at $N=4000$ this is at least about $23\,600$ exact zero modes in two dimensions, and in four dimensions, with $-\chi/N=16.85$ , at least about $67\,000$ .

The lower bound in Corollary 8 is Theorem 5. The growth of $\chi$ is proved in expectation for $d=2$ (Proposition 7) and measured for $d=4$ . For a region of fixed topology the continuum operator $\mathrm d+\mathrm d^{\dagger}$ has $\sum_kb_k$ zero modes, equal to $1$ for a contractible region, and a lattice version has $2^{d/2}$ tastes; the interpretation of the extra zero modes as massless species rests on these two counts.

4. Wilson terms and a degree mass in two dimensions

The index relation (3) holds for operators that anticommute with $\Gamma$ . A Wilson term, such as $rK^{2}$ , commutes with $\Gamma$ , and the zero modes in (3) could in principle be lifted by it. We ask which terms of this kind lift them, and what is left. Take $d=2$ and the complex $\Delta_1$ with cochain spaces $C^{0},C^{1},C^{2}$ , coboundaries $\delta_0:C^{0}\to C^{1}$ and $\delta_1:C^{1}\to C^{2}$ , the counting form, $K=\delta+\delta^{T}$ , and the Hodge Laplacians $L_0=\delta_0^{T}\delta_0$ and $L_2=\delta_1\delta_1^{T}$ . We write $b_k$ for the Betti numbers of $\Delta_1$ , $\ell$ for its number of links and $\Pi_1$ for the projector on $C^{1}$ . Since $K$ has entries $0,\pm1$ whatever the density, lengths are measured in units of $\rho^{-1/2}$ . A Dirac operator on a region of linear size $R$ in these units, with $R^{2}\sim N$ , has its lowest nonzero eigenvalues of order $1/R\sim N^{-1/2}$ , and we call a mode light if its eigenvalue or singular value is of this order or smaller.

The first statement is about the structure of $\Delta_1$ and holds for every finite poset.

Proposition 9 (private faces). For every finite poset, $L_2=\delta_1\delta_1^{T}$ on $\Delta_1$ satisfies $L_2\ge\mathbb 1$ . Hence $b_2=0$ , and the set of nonzero eigenvalues of $K^{2}$ is the union of the nonzero spectrum of $L_0$ and the spectrum of $L_2$ , the latter lying in $[1,\infty)$ .

Proof. Each triangle has an edge that belongs to no other triangle, so that $\delta_1$ contains an invertible block of signed permutations. Let $t=(x,y,z)$ be a triangle, so $I(x,z)=\{y\}$ . The faces $(x,y)$ and $(y,z)$ are links, because $I(x,y)$ and $I(y,z)$ are subsets of $I(x,z)\setminus\{y\}=\varnothing$ . The face $(x,z)$ has one interior element. A second triangle containing $(x,z)$ would either have that same interior element, hence be $t$ , or have $(x,z)$ as one of its two short faces, impossible because $(x,z)$ is not a link. So one-interior edges and triangles are in bijection. Order the columns of $\delta_1$ as (one-interior edges, links). Then $\delta_1=[Z\,|\,B]$ with $Z$ a signed permutation matrix, and $L_2=ZZ^{T}+BB^{T}=\mathbb 1+BB^{T}\ge\mathbb 1$ . For the second statement, $K^{2}=L_0\oplus L_1\oplus L_2$ with $L_1=\delta_0\delta_0^{T}+\delta_1^{T}\delta_1$ , and since $\delta_1\delta_0=0$ the nonzero spectrum of $L_1$ consists of that of $\delta_0\delta_0^{T}$ (equal to that of $L_0$ ) and that of $\delta_1^{T}\delta_1$ (equal to that of $L_2$ ). $\square$

Proposition 9 says that no light mode comes from the sector of $L_2$ . The remaining sector is that of the vertex Laplacian $L_0$ , the graph Laplacian of the one-skeleton of $\Delta_1$ . Its smallest nonzero eigenvalue $q_0$ is not bounded below on all posets: for a chain of $n$ elements it decreases like $n^{-2}$ . On a sprinkling, let $v$ be an element of degree $\deg v$ in the one-skeleton, in a connected component of $n_c\ge2$ elements. The test vector $e_v-n_c^{-1}\mathbb 1_c$ , with $\mathbb 1_c$ the indicator of the component, gives $q_0\le\deg v\;n_c/(n_c-1)$ , so an element joined to only one or two others puts $q_0$ below about $1$ or $2$ . Such elements are rare. On sprinklings of the unit square the median of $q_0$ is $2.7$ , $3.1$ and $3.3$ at $N=250$ , $500$ and $1000$ (100, 40 and 20 sprinklings). The value $q_0<2$ occurs in 5 of 100 sprinklings at $N=250$ , in 1 of 40 at $N=500$ and in none of 20 at $N=1000$ , and the smallest value is $0.88$ . So $q_0\ge1$ for typical sprinklings, and it increases slowly with $N$ . This is an observation, and the statements below keep $q_0$ explicit.

Proposition 10 (Hodge-type terms do not lift). Let $W$ be any element of the non-unital algebra generated by $\delta$ and $\delta^{T}$ (it contains $rK^{2}$ ) and $\alpha,\beta\in\mathbb C$ . Then $\ker(\alpha K+\beta W)\supseteq\ker K$ , and on $\Delta_1$ $\dim\ker(\alpha K+\beta W)\ge\sum_kb_k=2b_0+\ell-N\ge\ell-N+2$ .

Proof. By Theorem 5, $h\in\ker K$ implies $\delta h=0$ and $\delta^{T}h=0$ . Every word in $\delta$ and $\delta^{T}$ acts first with one of them, hence annihilates $h$ . For the count, $b_0-b_1+b_2=\chi=N-\ell$ by Corollary 6, $b_2=0$ by Proposition 9, and $b_0\ge1$ , so $b_1=b_0+\ell-N$ and $\sum_kb_k=2b_0+\ell-N$ . $\square$

The proof uses that the Wilson term is built from the same operators as $K$ . A term that is not built from them can act on the harmonic cochains, as the following mass does. In Euclidean signature the continuum Dirac operator is anti-Hermitian and a mass term is Hermitian, so $iK$ plus a Hermitian term is the discrete analogue of $D+m$ . This operator is not normal, and its singular values are the relevant measure because the norm of its inverse is the reciprocal of the smallest one.

Proposition 11 (degree-one mass). Let $A=[\delta_0\,|\,\delta_1^{T}]$ as a map from $C^{0}\oplus C^{2}$ to $C^{1}$ , and $r>0$ . The singular values of $iK+r\Pi_1$ are $0$ with multiplicity $b_0$ , $r$ with multiplicity $\dim\mathrm{coker}A=b_1$ , and $\tfrac12(\sqrt{r^{2}+4q}\mp r)$ for each eigenvalue $q$ of $L_0\oplus L_2$ other than $0$ , counted with multiplicity. Let $q_*$ be the smallest nonzero eigenvalue of $L_0\oplus L_2$ ; then $q_*\ge\min\{q_0,1\}$ , and if $b_1>0$ the smallest nonzero singular value is $\min\{r,\tfrac12(\sqrt{r^{2}+4q_*}-r)\}$ , which is at least $\min\{r,\tfrac12(\sqrt{r^{2}+4\min\{q_0,1\}}-r)\}$ .

Proof. Since $\delta_1\delta_0=0$ , $A^{T}A=L_0\oplus L_2$ . On each singular pair of $A$ with singular value $s>0$ the operator acts as $\begin{pmatrix}0&is\\ is&r\end{pmatrix}$ in a two-dimensional space, whose singular values have product $s^{2}$ and squares summing to $2s^{2}+r^{2}$ , hence are $\tfrac12(\sqrt{r^{2}+4s^{2}}\mp r)$ . The kernel of $A$ consists of the locally constant 0-cochains ( $\delta_1^{T}$ is injective because $L_2\ge\mathbb 1$ by Proposition 9), and its cokernel is the space of harmonic 1-cochains, of dimension $b_1$ , on which the operator is $r$ . The smaller root $\tfrac12(\sqrt{r^{2}+4q}-r)$ increases with $q$ , which gives the minimum, and the spectrum of $L_2$ lies in $[1,\infty)$ . $\square$

The $b_1\ge\ell-N+1$ harmonic 1-cochains that Proposition 10 leaves at zero are lifted to $r$ . The $b_0$ locally constant 0-cochains stay at zero, as the constant modes of a Dirac operator on a connected region do, and the other singular values are at least $\tfrac12(\sqrt{r^{2}+4q_*}-r)$ . Where $q_0\ge1$ this is $\tfrac12(\sqrt{r^{2}+4}-r)$ or more, and for $r$ of order one in the units above no light mode remains besides the constant ones. Without a gap on $L_0$ , Proposition 11 gives no such lower bound.

The sum $iK+r\Pi_1$ is an anti-Hermitian Kähler–Dirac operator plus a Hermitian term, and other phases are not covered. If $W$ is Hermitian and commutes with $K$ , then on an eigenspace of $K$ with eigenvalue $\lambda$ the operator $iK+W$ is normal with singular values $\sqrt{\lambda^{2}+w^{2}}\ge\lvert\lambda\rvert$ , so such a term cannot create light modes from modes of $K$ that are not light. A non-Hermitian $\Gamma$ -even commuting $W$ is not covered, and among degree masses only a mass proportional to $\Pi_1$ is treated. All statements concern the two-dimensional complex $\Delta_1$ with the counting form. For the Krein form $K$ is replaced by $\delta-\delta^{T}$ , and Proposition 10 holds with the same proof because $\lVert(\delta-\delta^{T})h\rVert^{2}=\lVert\delta h\rVert^{2}+\lVert\delta^{T}h\rVert^{2}$ . The proof of Proposition 9 is specific to $\Delta_1$ and does not carry over to the four-dimensional complex $\Delta_3$ .

5. The order chain in 1+1 dimensions

In two dimensions the stabiliser of a causal diamond is trivial, and the $d=2$ rows of Theorem 1 allow order-natural covariant kernels once a region frame is fixed. A kinetic term for a one-component chiral spinor still needs a null direction, which is a binary choice. The kernel constructed below has entries $\pm1$ independent of the positions, so its quadratic form is invariant when the components transform as scalars and not with the weight $e^{\eta/2}$ of Theorem 1. It is therefore not an instance of (H1) to (H3), and the null direction is the extra datum.

Sprinkle $N$ points (with $N$ even) into the unit square in null coordinates and label them $1,\dots,N$ in increasing $u$ . Let $\mathcal C_{xy}=1$ if $x\prec y$ and $0$ otherwise. The antisymmetric part $\mathcal C-\mathcal C^{T}$ of the causal matrix plays the role of the commutator function in the Sorkin–Johnston construction for scalar fields [Surya]. Since $x\prec y$ means that $u$ and $v$ both increase, $(\mathcal C-\mathcal C^{T})_{xy}=\tfrac12[\mathrm{sgn}(u_y-u_x)+\mathrm{sgn}(v_y-v_x)]$ , that is $\mathcal C-\mathcal C^{T}=-\tfrac12(\Sigma_u+\Sigma_v)$ , where $(\Sigma_u)_{xy}=\mathrm{sgn}(u_x-u_y)$ is the sign matrix of the linear order in $u$ and $\Sigma_v$ that of the order in $v$ . The matrix splits into one piece for each null direction, each the sign matrix of a linear order. Choosing one of them, say $\Sigma_u$ , which in the $u$ -labelling has entries $\mathrm{sgn}(x-y)$ , selects the chiral sector of fields depending on $u$ . The choice is a binary datum: reflecting the diamond in space exchanges $u$ and $v$ and leaves the law of the sprinkling invariant, and no property of the order selects one side, in line with the absence of an intrinsic orientation [Dowker–Sorkin].

Let $U$ be the antiperiodic shift, $Ue_j=e_{j+1}$ for $j<N$ and $Ue_N=-e_1$ , so $U^{N}=-\mathbb 1$ and its eigenvalues are $e^{i\theta_j}$ with $\theta_j=(2j-1)\pi/N$ . Then

\Sigma_u=(\mathbb 1+U)(\mathbb 1-U)^{-1}, \tag{6}

because $U^{N}=-\mathbb 1$ gives $(\mathbb 1-U)^{-1}=\tfrac12\sum_{k=0}^{N-1}U^{k}$ , hence $(\mathbb 1+U)(\mathbb 1-U)^{-1}=\sum_{k=1}^{N-1}U^{k}$ , and $U^{k}e_j$ is $e_{j+k}$ for $j+k\le N$ and $-e_{j+k-N}$ otherwise, so the $(x,y)$ entry is $+1$ for $x>y$ and $-1$ for $x<y$ . The eigenvalues of the real antisymmetric matrix $\Sigma_u$ are $i\cot(\theta_j/2)$ . The matrix $\Sigma_u$ acts on a function sampled at the points as $N$ times the integral operator with kernel $\mathrm{sgn}(u-u')$ , which is the inverse of $\tfrac12\partial_u$ . So $\Sigma_u^{-1}$ discretises $\partial_u/2N$ , and we take $D_u=-2iN\,\Sigma_u^{-1}$ as the kinetic operator of a real (Majorana) one-component field with one chiral sector. Its eigenvalues are $-2N\tan(\theta_j/2)$ , which for $\theta_j\ll1$ are close to $-(2j-1)\pi$ , the spectrum of $-i\partial_u$ on antiperiodic functions of $u\in[0,1]$ .

Proposition 12 (the chain derivative). For every even $N$ and every sprinkling, $\mathrm{spec}\,D_u=\{-2N\tan(\theta_j/2)\}$ , with $\theta_j$ taken in $(-\pi,\pi)$ . The function $\theta\mapsto-2N\tan(\theta/2)$ is strictly monotone on $(-\pi,\pi)$ , so $D_u$ has a single zero of the symbol, at $\theta=0$ , and $\lambda_j=-(2j-1)\pi\,(1+O(\Lambda_0^{2}/N^{2}))$ for the eigenvalues with $\lvert\lambda_j\rvert<\Lambda_0\ll N$ . The inverse is $\Sigma_u^{-1}=\sum_{k=1}^{N-1}(-1)^{k}U^{k}$ , so every off-diagonal entry of $\Sigma_u^{-1}$ is $\pm1$ : all pairs of elements are coupled with the same weight.

Proof. $U$ is diagonalised by the Fourier vectors with eigenvalues $e^{i\theta_j}$ , and the Cayley map $e^{i\theta}\mapsto(1+e^{i\theta})/(1-e^{i\theta})=i\cot(\theta/2)$ gives the spectrum of $\Sigma_u$ . Inverting gives $-i\tan(\theta/2)$ , and monotonicity is a property of $\tan$ . For the inverse, $\Sigma_u^{-1}=(\mathbb 1-U)(\mathbb 1+U)^{-1}$ and $(\mathbb 1+U)\sum_{k=0}^{N-1}(-U)^{k}=\mathbb 1-(-U)^{N}=2\cdot\mathbb 1$ for even $N$ since $U^{N}=-\mathbb 1$ . So $\Sigma_u^{-1}=\tfrac12(\mathbb 1-U)\sum_{k=0}^{N-1}(-U)^{k}=\tfrac12\bigl[\mathbb 1+2\sum_{k=1}^{N-1}(-U)^{k}+(-U)^{N}\bigr]=\sum_{k=1}^{N-1}(-U)^{k}$ . $\square$

The operator is a non-local lattice derivative with a single zero of the symbol, at the cost of couplings of equal magnitude between all elements. The remaining question is the vacuum. A pure Gaussian state of the Majorana field is given by a real antisymmetric matrix $\mathcal J$ with $\mathcal J^{2}=-\mathbb 1$ , through $\langle\psi_r\psi_s\rangle=\tfrac12(\mathbb 1+i\mathcal J)_{rs}$ . The analogue of the Sorkin–Johnston prescription takes $i\mathcal J$ equal to the sign of the Hermitian matrix $i\Sigma_u$ . We consider every state that is invariant under the shift $U$ . A zero of the occupation function of such a state, a Fermi point, is a value of $\theta$ at which its symbol changes sign.

Proposition 13 (sign changes of shift-invariant states). Let $\mathcal J$ be real antisymmetric with $\mathcal J^{2}=-\mathbb 1$ and $[\mathcal J,U]=0$ . Then $\mathcal J$ has a symbol $f(\theta_j)\in\{+i,-i\}$ which is an odd function of $\theta$ . Consequently (i) $f$ changes sign between $\theta=-\pi/N$ and $\theta=\pi/N$ and between the two grid points adjacent to $\theta=\pi$ ; (ii) around the circle the number $n_{\uparrow}$ of upward jumps equals the number $n_{\downarrow}$ of downward jumps, so the net chirality $n_{\uparrow}-n_{\downarrow}$ , the signed number of jumps, vanishes for every such state. For a state $f=i\,\mathrm{sgn}\,\varepsilon$ built from a dispersion $\varepsilon(\theta)$ , the sign of a jump is the sign of $\varepsilon'$ at the Fermi point. For the state with $i\mathcal J=\mathrm{sgn}(i\Sigma_u)$ the symbol is $i\,\mathrm{sgn}\sin\theta$ , with exactly these two jumps and no others, which is the sign of the symbol $i\sin\theta$ of the nearest-neighbour Majorana hopping chain, whose Fermi points are at $\theta=0$ and $\theta=\pi$ . The first jump is at the zero of the symbol of $D_u$ ; the second is at $\theta=\pi$ , where $D_u$ has a pole and $\Sigma_u$ a zero.

Proof. A matrix commuting with $U$ , which has distinct eigenvalues, is a polynomial in $U$ and is diagonalised by the same Fourier vectors; call its eigenvalue on the vector with phase $\theta$ the symbol $f(\theta)$ . Reality of $\mathcal J$ gives $f(-\theta)=\overline{f(\theta)}$ , antisymmetry gives $f(-\theta)=-f(\theta)$ , so $f$ is purely imaginary and odd, and $\mathcal J^{2}=-\mathbb 1$ gives $f\in\{\pm i\}$ . The grid $\theta_j=(2j-1)\pi/N$ is symmetric under $\theta\to-\theta$ , with $\pm\pi/N$ adjacent at $0$ and $\pm(N-1)\pi/N$ adjacent at $\pi$ (using $N$ even), and oddness forces opposite signs at each pair. A cyclic sequence of $\pm1$ has as many upward as downward jumps. For the sign state, $\mathrm{sgn}\cot(\theta/2)=\mathrm{sgn}\sin\theta$ on $(-\pi,\pi)\setminus\{0\}$ . $\square$

Proposition 13 is the lattice pairing of Fermi points for every invariant pure state, and it has a consequence for entanglement. For a conformal field theory the entanglement entropy of an interval of $n$ sites grows as $\tfrac{c_L+c_R}{6}\ln n$ [Calabrese–Cardy], which is $\tfrac16\ln n$ for a non-chiral Majorana fermion ( $c_L=c_R=\tfrac12$ ) and $\tfrac1{12}\ln n$ for a single Majorana–Weyl fermion. For a block of $n$ consecutive elements in the $u$ -order, the entropy of the Gaussian state is the sum of $h\bigl(\tfrac{1+\nu}{2}\bigr)$ over the positive eigenvalues $\nu$ of $i\mathcal J$ restricted to the block, with $h(p)=-p\ln p-(1-p)\ln(1-p)$ . For the sign state at $N=1000$ and blocks of $n=20,30,\dots,400$ elements it fits $\zeta\ln[(N/\pi)\sin(\pi n/N)]+\text{const}$ with $\zeta=0.1667$ , the coefficient $1/6$ of two Fermi points. The entanglement entropy of this state therefore sees both chiralities, as Proposition 13 requires. This is a numerical observation about one state.

6. The full order complex and the Möbius function

If the span restriction is dropped, the complex $\Delta(P)$ of all chains is not local, but the index of Theorem 5 has a closed form, and in two dimensions its mean is bounded. A bounded mean does not bound $\lvert\chi\rvert$ , which is the quantity in the zero-mode bound, so the growth of section 3 is neither established nor excluded for the complex of all chains. Let $\hat0$ and $\hat1$ be the bottom and top of a causal diamond (for a sprinkling, its two tips, which are not elements), $P^{\circ}$ the set of elements strictly between them, and $\Delta(P^{\circ})$ the order complex of $P^{\circ}$ .

Proposition 14 (index of the open and closed interval). For every finite poset $P^{\circ}$ , $\chi(\Delta(P^{\circ}))=1+\mu(\hat0,\hat1)$ , so any operator as in Theorem 5 has $\dim\ker K\ge\lvert1+\mu(\hat0,\hat1)\rvert$ on $\Delta(P^{\circ})$ . On the closed interval $\Delta(P^{\circ}\cup\{\hat0,\hat1\})$ , a cone, $\ker(\delta+\delta^{*})$ is one-dimensional for the counting form.

Proof. Hall’s theorem gives $\mu(\hat0,\hat1)=\sum_{k\ge1}(-1)^{k}c_k$ , where $c_k$ is the number of chains $\hat0\prec x_1\prec\dots\prec x_{k-1}\prec\hat1$ , that is the number of $(k-1)$ -element chains of $P^{\circ}$ (for $k=1$ , the empty chain). With $a_j$ the number of $j$ -element chains of $P^{\circ}$ and $a_0=1$ , $\mu=\sum_{j\ge0}(-1)^{j+1}a_j$ while $\chi(\Delta(P^{\circ}))=\sum_{j\ge1}(-1)^{j-1}a_j=1+\mu$ . A cone has the cohomology of a point. $\square$

Proposition 15 (mean index in two dimensions). Sprinkle the diamond $[0,1]^{2}$ in null coordinates with a Poisson process of mean $\bar N$ points and take $\hat0=(0,0)$ , $\hat1=(1,1)$ . Then $\mathbb E[\chi(\Delta(P^{\circ}))]=1-J_0(2\sqrt{\bar N})$ , with $J_0$ the Bessel function. Since $\lvert J_0\rvert\le1$ , the mean index lies between $0$ and $2$ for every $\bar N$ .

Proof. By the Mecke formula, the expected number of $j$ -element subsets of a Poisson process with a given property is $\bar N^{j}/j!$ times the probability that $j$ independent uniform points have it. Sort $j$ uniform points by $u$ ; they form a chain exactly when their $v$ -ranks are increasing, with probability $1/j!$ . So $\mathbb E[a_j]=\bar N^{j}/(j!)^{2}$ , and Proposition 14 gives $\mathbb E[\mu]=-\sum_j(-\bar N)^{j}/(j!)^{2}=-J_0(2\sqrt{\bar N})$ . The bound is $\lvert J_0\rvert\le1$ . $\square$

With $\mathcal C$ the causal matrix, $(\mathbb 1+\mathcal C)^{-1}$ has the $(\hat0,\hat1)$ entry $\mu$ and is the sum over chains with weight $-1$ per step. Johnston used this geometric series to define retarded propagators and showed that in two dimensions the propagator with weight $-m^{2}/(2\rho)$ per step has the mean $\tfrac12J_0(m\tau)$ between points at proper time $\tau$ [Johnston 2008]. The same Bessel function is the two-dimensional massive Green function [X–Dowker–Surya]. The weight $-1$ corresponds to $m^{2}=2\rho$ and $m\tau=2\sqrt{\bar N}$ , so $-\mu/2$ has the mean $\tfrac12J_0(2\sqrt{\bar N})$ of Proposition 15. The index of Proposition 14 is therefore $1$ minus twice Johnston’s propagator at this mass, and Proposition 15 is the statement about its mean.

The fluctuations of the index are large. In samples of Poisson sprinklings of the diamond, the mean absolute index $\mathbb E\lvert\chi\rvert$ is $1.3$ , $2.7$ and $17$ and the root-mean-square of $\chi$ is $1.7$ , $4.2$ and $32$ at $\bar N=10,30,100$ (20000, 20000 and 8000 sprinklings; the root-mean-square at $\bar N=100$ is dominated by rare large values and is less precise). Over this range $\ln\mathrm{rms}\,\chi$ is close to affine in $\sqrt{\bar N}$ with slope about $0.4$ . This is an observation from three points and is not proved. The lower bound $\lvert\chi\rvert$ on the zero modes of the full complex is therefore not bounded in the samples, although $\mathbb E[\chi]$ is; whether it grows in proportion to $\bar N$ is not decided. On the closed interval the index is $1$ for every sprinkling, and the fluctuating index belongs to the open interval, whose complex $\Delta(P^{\circ})$ becomes a cone, with index $1$ , once the two tips are adjoined.

7. Open points: local frames, other inner products and the full complex

If each element carries its own frame, as when vierbeins are built from holonomies [Sverdlov], (H1) fails and Theorem 1 does not apply: the kernel then depends on frame data attached to the elements or pairs, as gauge data. Proposition 3 excludes an invariant random global frame. A gauge-fixed description with a frame per element that is consistent with the order would settle whether this route exists. A given global frame or region frame is covered by Theorem 1 with $G=SU(2)$ , which allows scalar kernels only, so a kinetic term then uses the frame explicitly; the transfer of such a frame into low-energy physics is the subject of frame leakage. Constructions covariant only on average, or only approximately, fall outside (H3).

The form on the cochains is the open point for Lorentzian Kähler–Dirac fields. A Lorentzian Whitney form is non-diagonal and indefinite, outside the hypotheses of Theorem 5. A computation of its zero modes on the order complex, and of whether the private-face weight of Proposition 9 stays bounded below, would show whether it yields region-scale modes. The Lorentzian construction on a manifold needs an operator that is incompatible with the Lorentz transformations [Catterall–Hubisz]; the analogue of that condition on the order complex has not been worked out.

Proposition 10 holds on any complex, but its bound on $b_2$ uses Proposition 9, which is specific to $\Delta_1$ . A replacement for Proposition 9 would extend the Wilson-term statements to the complexes $\Delta_{n_0}$ with $n_0\ge2$ in two dimensions and to $\Delta_3$ in four dimensions, which are unexamined, and a proof of the gap of the vertex Laplacian of $\Delta_1$ would turn the observation of section 4 into a theorem. On the grids of section 3, $\chi(\Delta_{n_0})=1$ for $n_0=2,3$ , at the cost of the Lorentz symmetry. For the complex of all chains, a proof that $\lvert\chi\rvert$ grows or stays bounded would decide the question that the bounded mean of Proposition 15 leaves open, and the growth law of the variance of the Möbius function of a random two-dimensional order rests on three sample values. The extension of Proposition 7 beyond two dimensions is open.

Proposition 13 settles shift-invariant pure Gaussian states on the 1+1 chain. Whether states that are not shift invariant, mixed or non-Gaussian states, or interactions that could decouple a mirror fermion give a one-handed vacuum is not decided.

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