Exact compact U1 rotor plus finite CAR data

Assume Countable Choice. Let $\mathcal{K}$ be a finite-dimensional complex Hilbert space with a finite ordered orthonormal basis $(e_j{)}_{j\in J}$, and use the antisymmetric Fock construction and second-variable-linear convention of Schwartz/distribution spaces and Hilbert Fock conventions. Assume its fermionic operators satisfy the CAR of Bosonic and fermionic Fock operators with exact domains. For $S=\{j_1<\cdots <j_k\}\subseteq J$, set ${\xi }_S=c^{†}(e_{j_1})\cdots c^{†}(e_{j_k})\mathrm{\Omega },{\xi }_{\mathrm{\varnothing }}=\mathrm{\Omega }\mathrm{.}$ Then $({\xi }_S{)}_{S\subseteq J}$ is an orthonormal basis of ${\mathcal{F}}_a(\mathcal{K})$. For each $j\in J$, the bounded number operator $N_j=c^{†}(e_j)c(e_j)$ is an orthogonal projection and $N_j{\xi }_S={\mathbf{1}}_{j\in S}{\xi }_S\mathrm{.}$ Consequently, for any partition $J={⨆}_{x\in V}{J}_x$ into finite site-mode sets, $n_x={\sum }_{j\in J_x}{N}_j$ is bounded self-adjoint and acts on ${\xi }_S$ by the occupation count $N_x(S)=\mathrm{\mid }S\cap J_x\mathrm{\mid }\in \{0,\dots ,\mathrm{\mid }{J}_x\mathrm{\mid }\}$.

Assume Countable Choice. Let $\mathrm{\Gamma }=(V,E,s,t,\mathcal{P})$ be a finite oriented graph with nonempty finite vertex set $V$, nonempty finite link set $E$, endpoint maps $s,t:E\to V$, and a finite list $\mathcal{P}$ of oriented simple cycles. Parallel links and self-loops are allowed. An oriented simple cycle $p\in \mathcal{P}$ is a finite closed path using each link at most once; write ${\sigma }_{p,e}\in \{-1,0,1\}$ for its signed use of link $e$, so its signed incidence at every vertex is zero. Fix enumerations $E=({\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_N)$ and $V=(v_1,\dots ,v_M)$, where $N=\mathrm{\mid }E\mathrm{\mid }$ and $M=\mathrm{\mid }V\mathrm{\mid }$. Write ${\mathbb{T}}_1=\mathbb{R}\mathrm{/}\mathbb{Z}$ and ${\mathbb{T}}_{2\pi }=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$, and set $\mathbb{T}={\mathbb{T}}_{2\pi }$ with normalized Haar measure $d\theta \mathrm{/}(2\pi )$ and product measure $m_E$ on ${\mathbb{T}}^E$.

The coordinatewise map ${\mathrm{\Phi }}_E:{\mathbb{T}}_{2\pi }^E\to {\mathbb{T}}_1^E,([{\theta }_e]{)}_{e\in E}\mapsto ([{\theta }_e\mathrm{/}(2\pi )]{)}_{e\in E},$ is a well-defined measure-preserving group isomorphism for the normalized product Haar measures. The fixed enumeration of $E$ identifies ${\mathbb{T}}_1^E$ with ${\mathbb{T}}_1^N$; pulling back the Fourier characters $x\mapsto \exp (2\pi i{\sum }_{e\in E}{n}_e{x}_e)$ along ${\mathrm{\Phi }}_E$ gives $e_{\mathbf{n}}(\theta )=\exp (i{\sum }_{e\in E}{n}_e{\theta }_e)$. Thus the Fourier basis theorem The Fourier basis and Parseval's identity on the finite torus, whose torus is $\mathbb{R}\mathrm{/}\mathbb{Z}$ and whose characters are $\exp (2\pi ikx)$, gives an orthonormal basis $\{e_{\mathbf{n}}:\mathbf{n}\in {\mathbb{Z}}^E\}$ of $L^2({\mathbb{T}}^E)$. The second-variable-linear $L^2$ pairing used below is the conjugate of the first-variable-linear pairing in that theorem, so orthonormality and completeness are unchanged.

Fix an integer $s\ge 1$ and let $\mathcal{K}={⨁}_{x\in V}{\mathbb{C}}^s$ with orthonormal modes $e_{x\alpha }$, $1\le \alpha \le s$. This is finite-dimensional and hence separable. Fix the lexicographic order on $J=V\times \{1,\dots ,s\}$ induced by the chosen order on $V$ and the usual order on $\{1,\dots ,s\}$. Use the antisymmetric Fock space and second-variable-linear inner product convention of Schwartz/distribution spaces and Hilbert Fock conventions. The current draft supplier Bosonic and fermionic Fock operators with exact domains states that, assuming Countable Choice, ${\mathcal{F}}_a(\mathcal{K})$ is a Hilbert space and its creation and annihilation operators are bounded and satisfy the CAR; that supplier is still a draft carrier, not an accepted result in this run. For $\mathbf{m}=(m_{x\alpha })\in \{0,1{\}}^J$, put $S(\mathbf{m})=\{(x,\alpha ):m_{x\alpha }=1\}$ and write ${\xi }_{\mathbf{m}}={\xi }_{S(\mathbf{m})}$ for the ordered wedge basis vector of Finite fermion-mode occupation basis. This is a complete orthonormal basis of ${\mathcal{F}}_a(\mathcal{K})$; changing the chosen order changes individual wedge vectors only by signs. Write $c_{x\alpha }=c(e_{x\alpha })$, $c_{x\alpha }^{†}=c^{†}(e_{x\alpha })$, and $n_x={\sum }_{\alpha =1}^s{c}_{x\alpha }^{†}{c}_{x\alpha }$. On ${\xi }_{\mathbf{m}}$, $n_x$ acts by $N_x(\mathbf{m})={\sum }_{\alpha =1}^s{m}_{x\alpha }$; it is bounded self-adjoint with spectrum contained in $\{0,\dots ,s\}$.

Use the second-variable-linear $L^2$ pairing $⟨f,g{⟩}_2={\int }_{{\mathbb{T}}^E}\overline{f(\theta )}g(\theta )d{m}_E(\theta )$. The kinematical Hilbert space is ${\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}=L^2({\mathbb{T}}^E)\widehat{\otimes }{\mathcal{F}}_a(\mathcal{K}),$ where $\widehat{\otimes }$ has the same completion norm as Hilbert tensor construction: inner product, completion and product bases and the second-variable-linear pairing obtained by conjugating its first-variable-linear tensor pairing, as in Schwartz/distribution spaces and Hilbert Fock conventions. Concretely, the tensor supplier is applied to the conjugate first-variable-linear pairings $(f,g{)}_1=⟨g,f{⟩}_2$ and $(\mathrm{\Phi },\mathrm{\Psi }{)}_1=⟨\mathrm{\Psi },\mathrm{\Phi }{⟩}_2$, and the resulting tensor pairing is conjugated; this preserves the norm and the product-basis assertion. Consequently $\{e_{\mathbf{n}}\otimes {\xi }_{\mathbf{m}}:\mathbf{n}\in {\mathbb{Z}}^E,\mathbf{m}\in \{0,1{\}}^J\}$ is a complete orthonormal basis of ${\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}$ by the product-basis assertion of Hilbert tensor construction: inner product, completion and product bases. This is the joint basis used for all diagonal multipliers below. Write $\psi ={\sum }_{\mathbf{n},\mathbf{m}}{\psi }_{\mathbf{n},\mathbf{m}}{e}_{\mathbf{n}}\otimes {\xi }_{\mathbf{m}}$. All matter operators previously defined on ${\mathcal{F}}_a(\mathcal{K})$ act on ${\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}$ as identity on the rotor factor tensor that operator. On the full kinematical basis define $E_e(e_{\mathbf{n}}\otimes {\xi }_{\mathbf{m}})=n_e(e_{\mathbf{n}}\otimes {\xi }_{\mathbf{m}}),D(E_e)=\{\psi \in {\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}:{\sum }_{\mathbf{n},\mathbf{m}}{n}_e^2\mathrm{\mid }{\psi }_{\mathbf{n},\mathbf{m}}{\mathrm{\mid }}^2<\mathrm{\infty }\}\mathrm{.}$ Thus $E_e$ is the maximal real diagonal multiplier, self-adjoint with integer spectrum, and on its domain its coefficients are $n_e{\psi }_{\mathbf{n},\mathbf{m}}$. Define the bounded unitary $U_e$ as multiplication by $e^{i{\theta }_e}$ on the rotor factor and identity on the matter factor. Equivalently, $U_e(e_{\mathbf{n}}\otimes {\xi }_{\mathbf{m}})=e_{\mathbf{n}+{\delta }_e}\otimes {\xi }_{\mathbf{m}},(U_e\psi {)}_{\mathbf{n},\mathbf{m}}={\psi }_{\mathbf{n}-{\delta }_e,\mathbf{m}}\mathrm{.}$ Each $U_f$ preserves $D(E_e)$, since $(n_e+{\delta }_{ef}{)}^2\le 2n_e^2+2$ and the coefficient sequence is square summable; its inverse preserves that domain by the same estimate. Computing on coefficients gives $[E_e,U_f]={\delta }_{ef}{U}_f$ on $D(E_e)$.

Fix background integers $b_x\in \{0,\dots ,s\}$. In the joint basis define the real integer multiplier ${\gamma }_x(\mathbf{n},\mathbf{m})={\sum }_{e:s(e)=x}{n}_e-{\sum }_{e:t(e)=x}{n}_e-({\sum }_{\alpha =1}^s{m}_{x\alpha }-b_x),$ and let $G_x$ be its maximal diagonal operator, with domain $D(G_x)=\{\psi :{\sum }_{\mathbf{n},\mathbf{m}}\mathrm{\mid }{\gamma }_x(\mathbf{n},\mathbf{m}){\mathrm{\mid }}^2\mathrm{\mid }{\psi }_{\mathbf{n},\mathbf{m}}{\mathrm{\mid }}^2<\mathrm{\infty }\}$. On the common link-electric domain this is the finite sum ${\sum }_{s(e)=x}{E}_e-{\sum }_{t(e)=x}{E}_e-(n_x-b_{x}I)$. These are the Gauss operators; the operator constraint defines ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:={\bigcap }_{x\in V}\ker G_x$. For $\alpha \in {\mathbb{T}}^V$ define the diagonal multiplier $U(\alpha )=\exp (i{\sum }_x{\alpha }_x{G}_x)$; the next lemma proves that these operators form a strongly continuous representation and that their normalized Haar average is the orthogonal projection onto ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$.

For each $x$, choose a Hermitian matrix $M_x\in M_s(\mathbb{C})$, and for each link $e$ choose a matrix $T_e\in M_s(\mathbb{C})$. Choose real coefficients ${\lambda }_E>0$ and ${\lambda }_p\in \mathbb{R}$ for $p\in \mathcal{P}$. Write $W_p={\prod }_{e\in E}{U}_e^{{\sigma }_{p,e}},V_B=-\frac{1}{2}{\sum }_{p\in \mathcal{P}}{\lambda }_p(W_p+W_p^{†}),$ and $V_F={\sum }_{x\in V}{\sum }_{\alpha ,\beta =1}^s{c}_{x\alpha }^{†}(M_x{)}_{\alpha \beta }{c}_{x\beta }+{\sum }_{e\in E}(A_e+A_e^{†}),A_e={\sum }_{\alpha ,\beta =1}^s{c}_{s(e),\alpha }^{†}(T_e{)}_{\alpha \beta }{U}_e{c}_{t(e),\beta }\mathrm{.}$ The operator $H_0={\lambda }_E{\sum }_{e\in E}{E}_e^2$ is the diagonal multiplier with domain $D(H_0)=\{\psi :{\sum }_{\mathbf{n},\mathbf{m}}{\left({\lambda }_E{\sum }_{e\in E}{n}_e^2\right)}^2\mathrm{\mid }{\psi }_{\mathbf{n},\mathbf{m}}{\mathrm{\mid }}^2<\mathrm{\infty }\},$ where $\mathbf{m}=(m_{x\alpha })$ ranges over occupation configurations with each $m_{x\alpha }\in \{0,1\}$, and $N_x(\mathbf{m})={\sum }_{\alpha =1}^s{m}_{x\alpha }$ is the eigenvalue of $n_x$ on that basis vector. The coefficients ${\lambda }_E,{\lambda }_p$ and entries of $M_x,T_e$ are ordinary real or complex scalars here; no physical units are assigned. Set $H=H_0+V_B+V_F$ on $D(H_0)$. This is a mathematical finite-graph regulator definition. It asserts no physical adoption, continuum limit, Lorentz invariance, or identification with a particular continuum QED model.

For sign comparison only, if a smooth path ${\gamma }_e:[0,1]\to X$ in a smooth manifold $X$, a real $C^1$ connection one-form $a$, and a real $C^1$ function $\lambda$ are supplied, take $U_e^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}=\exp (-ig{\int }_0^1{a}_{{\gamma }_e(r)}({\dot{\gamma }}_e(r))dr)$ for real $g$. Under $a\mapsto a+d\lambda$, the fundamental theorem along the supplied path gives the endpoint factor $e^{ig(\lambda ({\gamma }_e(0))-\lambda ({\gamma }_e(1)))}$. This is an edgewise sign comparison: it has the same source-minus-target angle pattern as the finite vertex-phase convention. Neither $s(e)$ nor $t(e)$ is evaluated by $\lambda$; they are vertices in $V$, whereas ${\gamma }_e(0),{\gamma }_e(1)$ are points of $X$. No map $V\to X$ or compatible paths for the whole graph are supplied, and no continuum limit is asserted.

The statements involving ${\mathcal{F}}_a(\mathcal{K})$ and its CAR operators are conditional on Bosonic and fermionic Fock operators with exact domains as it appears on the current batch-37 QFT-MF01 manifest. I inspected its actual statement and full draft proof: it assumes Countable Choice, a complex Hilbert space and the second-variable-linear convention, and states bounded CAR operators on the whole Fock space. The actual interface matches this finite-dimensional $\mathcal{K}$ and the uses above. Its item and the Fock definition remain drafts without a current Step-1 readiness record in this run; the arguments here remain conditional and do not treat draft status as proof acceptance. The finite occupation basis is established locally by Finite fermion-mode occupation basis.

Assume the conditional compact-rotor/CAR data of Exact compact U(1) rotor plus finite CAR data. In the joint Fourier/occupation basis write ${\gamma }_x(\mathbf{n},\mathbf{m})={\sum }_{e:s(e)=x}{n}_e-{\sum }_{e:t(e)=x}{n}_e-{\sum }_{\alpha =1}^s{m}_{x\alpha }+b_x\in \mathbb{Z}\mathrm{.}$ For every vertex $x$, let $G_x$ be multiplication by ${\gamma }_x$ on its maximal diagonal domain $D(G_x)=\{\psi :{\sum }_{\mathbf{n},\mathbf{m}}\mathrm{\mid }{\gamma }_x(\mathbf{n},\mathbf{m}){\mathrm{\mid }}^2\mathrm{\mid }{\psi }_{\mathbf{n},\mathbf{m}}{\mathrm{\mid }}^2<\mathrm{\infty }\}\mathrm{.}$ Then each $G_x$ is self-adjoint, the $G_x$ strongly commute, and $U(\alpha )=\exp (i{\sum }_x{\alpha }_x{G}_x)$ is a strongly continuous unitary representation of ${\mathbb{T}}^V$. Its action obeys $U(\alpha )c_{x\beta }U(\alpha {)}^{†}=e^{i{\alpha }_x}{c}_{x\beta },U(\alpha )U_{e}U(\alpha {)}^{†}=e^{i({\alpha }_{s(e)}-{\alpha }_{t(e)})}{U}_e\mathrm{.}$ The normalized Haar average $P=\frac{1}{(2\pi {)}^{\mathrm{\mid }V\mathrm{\mid }}}{\int }_{[0,2\pi {]}^{\mathrm{\mid }V\mathrm{\mid }}}U(\alpha )d\alpha$ is the orthogonal projection onto ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\bigcap }_x\ker G_x$. This subspace is nonzero for every admissible fixed background $b$: it contains the normalized state with link Fourier labels $n_e=0$ and site occupations $m_{x\alpha }$ chosen so ${\sum }_{\alpha }{m}_{x\alpha }=b_x$.

Assume the model data of Exact compact U(1) rotor plus finite CAR data. The nonnegative electric operator $H_0={\lambda }_E{\sum }_e{E}_e^2$ is self-adjoint on the stated maximal diagonal domain and finite Fourier/occupation sums are a graph core. The finite plaquette and fermion interaction $V=V_B+V_F$ is bounded and self-adjoint. Therefore $H=H_0+V$ is self-adjoint on exactly $D(H_0)$. Moreover $H$ commutes with every gauge transformation $U(\alpha )$ on this domain. No continuum-limit assertion follows.

Assume the model data of Exact compact U(1) rotor plus finite CAR data and the Axiom of Choice. Let $H$ be the self-adjoint Hamiltonian on the exact domain supplied by Rotor interacting Hamiltonian with exact domain, and let $P$ be the physical projection of Gauss operators, gauge group and nonempty physical projection. For any scalar $\mathrm{\hslash }>0$, the family $\mathcal{U}(t)=e^{-itH\mathrm{/}\mathrm{\hslash }},t\in \mathbb{R},$ is a strongly continuous unitary group on ${\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}$, preserves $D(H)$, and satisfies ${\lim }_{h\to 0}\frac{\mathcal{U}(h)\psi -\psi }{h}=-\frac{i}{\mathrm{\hslash }}H\psi (\psi \in D(H))\mathrm{.}$ It commutes with every gauge transformation $U(\alpha )$ and with $P$. Hence ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=\mathrm{ran}P$ is a reducing subspace and $\mathcal{U}(t){\mathrm{\mid }}_{{\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}$ is a strongly continuous unitary group with skew-adjoint infinitesimal generator $-iH{\mathrm{\mid }}_{D(H)\cap {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}\mathrm{/}\mathrm{\hslash }$; its self-adjoint Stone operator is $-H{\mathrm{\mid }}_{D(H)\cap {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}\mathrm{/}\mathrm{\hslash }$. Equivalently, the restricted physical Hamiltonian $H{\mathrm{\mid }}_{D(H)\cap {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}$ is self-adjoint. This is an exact fixed-graph result; no continuum dynamics are asserted.

Assume the data of Exact compact U(1) rotor plus finite CAR data and let $R\subseteq V$. Define $\mathrm{\partial }R=\{e\in E:\text{exactly one endpoint of }e\text{ lies in }R\},$ with $ϵ(e,R)=1$ when $s(e)\in R,t(e)\mathrm{\notin }R$ and $ϵ(e,R)=-1$ when $t(e)\in R,s(e)\mathrm{\notin }R$. On the domain ${\mathcal{D}}_R={\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\cap {\bigcap }_{e\in \mathrm{\partial }R}D(E_e),$ the region number $N_R={\sum }_{x\in R}(n_x-b_{x}I)$ obeys the exact identity $N_R\psi =({\sum }_{e\in \mathrm{\partial }R}ϵ(e,R)E_e)\psi (\psi \in {\mathcal{D}}_R)\mathrm{.}$ Thus for $R=V$, where $\mathrm{\partial }R=\mathrm{\varnothing }$, every physical vector has total occupation ${\sum }_x{n}_x={\sum }_x{b}_x$. This finite-graph identity is exact and makes no continuum-limit claim.

Assume the finite-graph model of Exact compact U(1) rotor plus finite CAR data and the Axiom of Choice. Let $\psi \in {\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ be a normalized vector. A bounded operator $O$ has gauge charge $q=(q_x{)}_{x\in V}\in {\mathbb{Z}}^V$ when $U(\alpha )OU(\alpha {)}^{†}=e^{i{\sum }_x{q}_x{\alpha }_x}O\mathrm{.}$ Then $⟨\psi ,O\psi ⟩=0$ whenever $q\mathrm{\ne }0$, and any finite product of bounded gauge-charge eigenoperators has zero expectation when the sum of its charge vectors is nonzero.

For each $x$, if $O$ maps $D(G_x)$ into itself and $[G_x,O]$ extends to a bounded operator, then $⟨\psi ,[G_x,O]\psi ⟩=0.$ The same assertions apply to $O(t)=\mathcal{U}(t{)}^{†}O\mathcal{U}(t)$: the finite-regulator time evolution preserves the gauge charge and the domains $D(G_x)$. These are exact operator identities for the stated regulator; no trace or path-integral identity and no continuum Ward–Takahashi statement is asserted.