Finite matrix formal inverse and gauge-covariant kernel

Let $X$ and $E$ be finite sets, let $s,t:E\to X$ give the source and target of each directed edge, and assume $X\mathrm{\ne }\mathrm{\varnothing }$. Fix a positive integer $r$, a matrix $T_e\in M_r(\mathbb{C})$ for each $e\in E$, and an external link phase ${\zeta }_e\in U(1):=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$. The finite spin space is ${\mathcal{H}}_X:={⨁}_{x\in X}{\mathbb{C}}^r$, of dimension $d=r\mathrm{\mid }X\mathrm{\mid }$.

For a real vertex phase $\theta =({\theta }_x{)}_{x\in X}\in {\mathbb{R}}^X$, define ${\zeta }_e^{\theta }:=e^{i({\theta }_{s(e)}-{\theta }_{t(e)})}{\zeta }_e,D_{\theta }:={\mathrm{diag}}_{x\in X}(e^{i{\theta }_x}{I}_r)\in G{L}_d(\mathbb{C})\mathrm{.}$ Define the link-hopping matrix $V(\zeta )\in M_d(\mathbb{C})$ by its $x,y$ block indexed by $x,y\in X$ $V(\zeta {)}_{xy}:={\sum }_{\begin{array}{c}e\in E\\ s(e)=x,t(e)=y\end{array}}{T}_e{\zeta }_e\mathrm{.}$ An empty sum is the zero $r\times r$ matrix. Additional fixed diagonal blocks belong to a separate base matrix. These data are finite-dimensional algebraic data; the phases are external parameters, not rotor multiplication operators on $L^2({\mathbb{T}}^E)$.

The graph action uses the dimensionless phase angle $\theta$ as an independent external parameter. It does not choose an embedding of $G=(X,E,s,t)$ into spacetime or assert a continuum limit.

Let $R$ be a commutative unital $\mathbb{C}$-algebra and let $d\ge 1$. Let ${\mathrm{\Lambda }}_d(R)$ be the exterior algebra over $R$ on $2d$ distinct generators ${\psi }_1,\dots ,{\psi }_d,{\bar{\psi }}_1,\dots ,{\bar{\psi }}_d$. Thus every pair of generators anticommutes, each generator squares to zero, and coefficients from $R$ commute with every generator. The bar is part of a generator's name; no complex-conjugation relation is imposed.

Order the generators as ${\bar{\psi }}_1,\dots ,{\bar{\psi }}_d,{\psi }_1,\dots ,{\psi }_d$. The corresponding ordered monomials form the $R$-basis of ${\mathrm{\Lambda }}_d(R)$. Let $M_d:={\bar{\psi }}_1\cdots {\bar{\psi }}_d{\psi }_1\cdots {\psi }_d$ and let $[M_d]P\in R$ denote the coefficient of $M_d$ in $P\in {\mathrm{\Lambda }}_d(R)$. Define the finite Grassmann integral by the $R$-linear functional ${\mathcal{I}}_d(P):=(-1{)}^{d(d+1)\mathrm{/}2}[M_d]P\mathrm{.}$ For $K\in M_d(R)$, put $B_K:={\sum }_{i,j=1}^d{\bar{\psi }}_i{K}_{ij}{\psi }_j$. Since every monomial of degree greater than $2d$ vanishes, its exponential is the finite sum $\exp (-B_K):={\sum }_{n=0}^d\frac{(-B_K{)}^n}{n!}\mathrm{.}$ The sign factor in ${\mathcal{I}}_d$ is the chosen normalization. Its Gaussian determinant identity, including the value at $K=I_d$, is proved separately.

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Use the transverse space ${\mathcal{H}}_T=\mathrm{Ran}{M}_{P_T}$ of Positive transverse photon one-particle space, with $P_T(\mathbf{k})=I_3-\mathbf{n}{\mathbf{n}}^T$ for $\mathbf{k}\mathrm{\ne }0$ and any assigned value at the null point $\mathbf{k}=0$. In the QFT length convention, assign $\mathbf{k}$ units m${}^{-1}$ and define the nonnegative measurable multiplier $\omega (\mathbf{k}):=\mathrm{\mid }\mathbf{k}\mathrm{\mid }\mathrm{.}$ Its maximal multiplication operator on ${\mathcal{H}}_T$ is $(M_{\omega }f)(\mathbf{k}):=\omega (\mathbf{k})f(\mathbf{k}),D(M_{\omega }):=\{f\in {\mathcal{H}}_T:\omega f\in L^2({\mathbb{R}}^3,d^3\mathbf{k};{\mathbb{C}}^3)\}\mathrm{.}$ Because $\omega$ is a scalar multiplier, $\mathbf{k}\cdot f(\mathbf{k})=0$ a.e. implies $\mathbf{k}\cdot (M_{\omega }f)(\mathbf{k})=0$ a.e.; thus the displayed multiplication maps its stated domain into ${\mathcal{H}}_T$. The value $\omega (0)=0$ is immaterial to the $L^2$ operator. For $\mathrm{\hslash },c>0$, the corresponding one-particle energy convention is $E_1:=\mathrm{\hslash }c{M}_{\omega }$, in joules.

The QFT-MG01 transverse-space draft is present and its exact projection-space definition was inspected, and its page manifest lists the draft supplier items. This local multiplier is conditional on that mathematical supplier. No photon polarization frame, interacting dynamics, or empirical photon-energy claim is part of this definition.

For a formal indeterminate $z$, let $\mathbb{C}\mathrm{⟦}z\mathrm{⟧}$ be the commutative ring of sequences $(a_0,a_1,\dots )$ written ${\sum }_{n\ge 0}{a}_n{z}^n$, with coefficientwise addition and Cauchy-product multiplication. No convergence of a numerical series is part of this definition. For $d\ge 1$, matrices over $\mathbb{C}\mathrm{⟦}z\mathrm{⟧}$ use the usual finite matrix addition and multiplication.

Use the finite graph phase data of Finite-graph U(1) phases for matrix kernels and the QED $U(1)$, Clifford, and natural-unit conventions of Charged gauge data and SI-natural conventions. Let $K_0(\zeta )\in G{L}_d(\mathbb{C})$ be an invertible base matrix and let $V(\zeta )\in M_d(\mathbb{C})$ be its finite link-hopping matrix. The admissible covariant families satisfy $K_0({\zeta }^{\theta })=D_{\theta }{K}_0(\zeta )D_{\theta }^{-1},V({\zeta }^{\theta })=D_{\theta }V(\zeta )D_{\theta }^{-1}\mathrm{.}$ The scalar base $K_0=m{I}_d$, $m\in \mathbb{C}\setminus \{0\}$, is an explicit admissible case. Define $K(z;\zeta ):=K_0(\zeta )+zV(\zeta ),S(z;\zeta ):={\sum }_{n=0}^{\mathrm{\infty }}(-z{K}_0(\zeta {)}^{-1}V(\zeta ){)}^n{K}_0(\zeta {)}^{-1}\mathrm{.}$ The coefficient of each power $z^n$ in $S$ is one finite matrix. The item defines this formal candidate; the following lemma proves its two-sided inverse and covariance properties. Define $\det A$ for a finite matrix $A$ over $\mathbb{C}\mathrm{⟦}z\mathrm{⟧}$ by the Leibniz formula $\det A={\sum }_{\sigma \in S_d}\mathrm{sgn}(\sigma ){\prod }_{j=1}^d{A}_{j,\sigma (j)}$.

The draft carrier for Charged gauge data and SI-natural conventions is present and fixes the $(-+++)$ spacetime convention, the gamma-matrix convention, the natural charge $g$, and $D_a={\mathrm{\partial }}_a-ig{a}_a$. The current qft-mg01 manifest lists its draft items; this definition remains conditional on their mathematical conventions. The finite graph angle $\theta$ is an independent real vertex tuple. If a real vertex tuple $\alpha$ is separately assigned, one may set ${\theta }_x=g{\alpha }_x$ numerically, but no graph-to-spacetime embedding or evaluation of a spacetime gauge function at a vertex is supplied. This definition asserts no continuum kernel, convergence, or loop-distribution claim.

Use the transverse one-particle space ${\mathcal{H}}_T$ and multiplier $\omega (\mathbf{k})$ of Free transverse-photon energy multiplier, and form its symmetric Fock space ${\mathcal{F}}_s({\mathcal{H}}_T)$. Let $H_{\mathrm{p}\mathrm{h}}:=\mathrm{\hslash }cd\mathrm{\Gamma }(\omega )$, with $\mathrm{\hslash },c>0$ and $d\mathrm{\Gamma }(\omega )$ acting on the $n$-particle sector as multiplication by ${\sum }_{j=1}^n\mathrm{\mid }{\mathbf{k}}_j\mathrm{\mid }$, as in Second-quantized multipliers, joint spectral measure and exact domains. For a coherent vector use $ϵ(f):={⨁}_{n=0}^{\mathrm{\infty }}\frac{f^{\otimes n}}{\sqrt{n!}},C(f):=e^{-\mathrm{\parallel }f{\mathrm{\parallel }}^2\mathrm{/}2}ϵ(f),$ using the tensor and direct-sum conventions of Schwartz/distribution spaces and Hilbert Fock conventions.

Let $\kappa >0$. Let $b:S^2\to {\mathbb{C}}^3$ be measurable with $b\in L^2(S^2,d\mathrm{\Omega };{\mathbb{C}}^3)$, $\mathbf{n}\cdot b(\mathbf{n})=0$ for a.e. $\mathbf{n}$, and $C_b:={\int }_{S^2}\mathrm{\mid }b(\mathbf{n}){\mathrm{\mid }}^{2}d\mathrm{\Omega }>0.$ Take $\mathbf{k},\kappa ,ϵ$ in inverse metres and $b$ dimensionless; then $f_{ϵ}$ has momentum-wavefunction units ${\mathrm{m}}^{3\mathrm{/}2}$, $C_b$ is dimensionless, and $\mathrm{\hslash }c(\kappa -ϵ)$ has energy units. For $0<ϵ<\kappa$, define $f_{ϵ}(0)=0$ and $f_{ϵ}(\mathbf{k}):={\mathbf{1}}_{\{ϵ<\mathrm{\mid }\mathbf{k}\mathrm{\mid }<\kappa \}}\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^{-3\mathrm{/}2}b\text{\hspace{0.17em}⁣}\left(\frac{\mathbf{k}}{\mathrm{\mid }\mathbf{k}\mathrm{\mid }}\right)\mathrm{.}$ Then $\mathrm{\parallel }{f}_{ϵ}{\mathrm{\parallel }}^2=C_b\log \frac{\kappa }{ϵ},⟨C(f_{ϵ}),H_{\mathrm{p}\mathrm{h}}C(f_{ϵ})⟩=\mathrm{\hslash }c{C}_b(\kappa -ϵ)\mathrm{.}$ Each $C(f_{ϵ})$ is a unit vector in $D(H_{\mathrm{p}\mathrm{h}})$. As $ϵ\downarrow 0$, $C(f_{ϵ})\rightharpoonup 0$ weakly, but the family has no strong Fock-space limit. The statement is an exact free transverse-mode construction; it asserts no interacting charged sector, continuum QED limit, or empirical observation.

Let $K(z;\zeta )=K_0(\zeta )+zV(\zeta )$ and $S(z;\zeta )$ be the finite-dimensional formal matrices of Finite matrix formal kernel and inverse series, with $d\ge 1$, $K_0(\zeta )$ invertible, and both coefficient matrices covariant under the vertex phase angles of Finite-graph U(1) phases for matrix kernels. Then $S$ is a two-sided inverse of $K$ in $M_d(\mathbb{C}\mathrm{⟦}z\mathrm{⟧})$, and $S(z;{\zeta }^{\theta })=D_{\theta }S(z;\zeta )D_{\theta }^{-1}$ coefficientwise. For a real vertex tuple $\theta$, put $\mathrm{\Theta }:={\mathrm{diag}}_{x\in X}({\theta }_x{I}_r)$, ${\zeta }_s:={\zeta }^{s\theta }$, and $D_s:=D_{s\theta }=e^{is\mathrm{\Theta }}$. Write $K_s:=K(z;{\zeta }_s)$ and $S_s:=S(z;{\zeta }_s)$, and define $\delta K:={\frac{d}{ds}{K}_s\mid }_{s=0}$ and $\delta S:={\frac{d}{ds}{S}_s\mid }_{s=0}$ coefficientwise along this differentiable phase path. The variations satisfy $\delta K=i[\mathrm{\Theta },K],\delta S=i[\mathrm{\Theta },S]=-S(\delta K)S\mathrm{.}$ For ordinary finite matrices $A,B\in G{L}_d(\mathbb{C})$, the corresponding resolvent identity is $B^{-1}(B-A)A^{-1}=A^{-1}-B^{-1}$. These are formal or finite algebraic statements; they assert no convergence of a perturbation series, continuum limit, or loop Ward identity.

Use the $(-+++)$ coordinate convention, $h=-\eta =\mathrm{diag}(1,-1,-1,-1)$, and the Clifford matrices of Dirac Clifford matrices and energy projections; the shell parameter $\mu >0$ has inverse-length units as in Scalar future mass shell, invariant measure and test restriction. For $p=(p^0,\mathbf{p})\in {\mathbb{R}}^4$, write $\mathrm{\Gamma }(p):={\gamma }^0{p}^0-{\sum }_{j=1}^3{\gamma }^j{p}_j,K(p):=\mathrm{\Gamma }(p)-\mu I_4\mathrm{.}$ When $h(p,p)\mathrm{\ne }{\mu }^2$, define the finite rational inverse $S_0(p):=\frac{\mathrm{\Gamma }(p)+\mu I_4}{h(p,p)-{\mu }^2}\mathrm{.}$ For $p,p+k$ with both denominators nonzero, $S_0(p+k)\mathrm{\Gamma }(k)S_0(p)=S_0(p)-S_0(p+k)\mathrm{.}$ For any finite chain of off-shell momenta, the corresponding insertion differences telescope with both endpoint terms retained. If $u(p),u(p^{\mathrm{\prime }})$ satisfy $(\mathrm{\Gamma }(p)-\mu )u(p)=0$, $(\mathrm{\Gamma }(p^{\mathrm{\prime }})-\mu )u(p^{\mathrm{\prime }})=0$ and $p,p^{\mathrm{\prime }}$ lie on the future shell, then $\bar{u}(p^{\mathrm{\prime }})\mathrm{\Gamma }(p-p^{\mathrm{\prime }})u(p)=0$, where $\bar{u}=u^{†}\beta$. If $v(q)$ satisfies $(\mathrm{\Gamma }(q)+\mu )v(q)=0$ on the future shell, then $\bar{v}(q)\mathrm{\Gamma }(p+q)u(p)=0$. These are finite rational/tree identities; no Feynman boundary value, singular distribution product, loop theorem, or all-diagram gauge-independence claim is included.

Let $R=\mathbb{C}\mathrm{⟦}z\mathrm{⟧}$, let $d\ge 1$, and let $K\in M_d(R)$ be a finite matrix as in Finite matrix formal kernel and inverse series. With ${\mathcal{I}}_d$ the top-coefficient functional of Finite Grassmann algebra and top-coefficient integral, ${\mathcal{I}}_d\text{\hspace{0.17em}⁣}(\exp \text{\hspace{0.17em}⁣}(-{\sum }_{i,j=1}^d{\bar{\psi }}_i{K}_{ij}{\psi }_j))=\det K\mathrm{.}$ If $D=\mathrm{diag}(u_1,\dots ,u_d)$ with each $u_i\in R^{\times }$ and $K^{\mathrm{\prime }}=DK{D}^{-1}$, then the finite Grassmann integral and determinant are gauge invariant: ${\mathcal{I}}_d(\exp (-\bar{\psi }{K}^{\mathrm{\prime }}\psi ))=\det K^{\mathrm{\prime }}=\det K$. No infinite-dimensional Grassmann measure, continuum limit, or anomaly claim is made.

Use $h=-\eta =\mathrm{diag}(1,-1,-1,-1)$ from Off-shell free Dirac Ward telescoping and tree currents for contractions: for $w\in {\mathbb{C}}^4$, set $w_a={\sum }_{b=0}^3{h}_{ab}{w}^b$, and sum repeated component indices over $0,1,2,3$. Let $p_1,p_1^{\mathrm{\prime }},p_2,p_2^{\mathrm{\prime }}$ be future-shell momenta for the same $\mu >0$ convention as that lemma, and suppose $k=p_1-p_1^{\mathrm{\prime }}=p_2^{\mathrm{\prime }}-p_2$. Define $J_1^{\mu }:=\bar{u}(p_1^{\mathrm{\prime }}){\gamma }^{\mu }u(p_1),J_2^{\nu }:=\bar{u}(p_2^{\mathrm{\prime }}){\gamma }^{\nu }u(p_2),$ with the on-shell Dirac equations in Off-shell free Dirac Ward telescoping and tree currents. If a finite rational covariant-gauge kernel changes by $\mathrm{\Delta }{D}_{\mu \nu }(k)=c(k)k_{\mu }{k}_{\nu }$, where $c(k)\in \mathbb{C}$ is finite at the chosen $k$, then $J_1^{\mu }\mathrm{\Delta }{D}_{\mu \nu }(k)J_2^{\nu }=0.$ Likewise, let $J^a=\bar{u}(p^{\mathrm{\prime }}){\gamma }^{a}u(p)$ with transfer $\mathrm{\ell }=p-p^{\mathrm{\prime }}$, or $J^a=\bar{v}(q){\gamma }^{a}u(p)$ with transfer $\mathrm{\ell }=p+q$, under the respective future-shell and Dirac-equation hypotheses of Off-shell free Dirac Ward telescoping and tree currents. If $k=\mathrm{\ell }$ or $k=-\mathrm{\ell }$, then for $ϵ\in {\mathbb{C}}^4$ and a finite $\alpha \in \mathbb{C}$, the contraction ${ϵ}_a{J}^a$ is unchanged by ${ϵ}^a\mapsto {ϵ}^a+\alpha k^a$. These statements concern only the displayed finite tree-current contractions; they assert no gauge independence for arbitrary loop diagrams or renormalization schemes.