Positive local two-dimensional renormalizable RG theory

Work under AC for the Hilbert and spectral constructions. A model has a specified Euclidean local action $S(g,a)={\int }_{{\mathbb{R}}^2}\sigma (g,a;x)d^{2}x$ and ultraviolet cutoff $a>0$. The coordinates $x^1,x^2$ and $a$ have units of length; the scalar density $\sigma$ has units $[\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}{]}^{-2}$; and the couplings are dimensionless.

Here “renormalizable” is a data-level condition: each declared cutoff change is represented by a map on the specified coupling space that preserves separated correlation functions with the stated operator-basis transformation. The maps and covariance equations below give this condition precisely. This terminology does not assert perturbative power-counting renormalizability or construct a continuum limit.

For finitely many couplings, the coupling space is a smooth real $N$-manifold $\mathcal{G}$ in the sense of Smooth manifolds and their smooth charts: a topological $N$-manifold equipped with a maximal smooth atlas. Choose an open chart domain $U\subseteq \mathcal{G}$ and its chart map $\phi :U\to V$, where $V\subseteq {\mathbb{R}}^N$ is open; the coordinates $g^1,\dots ,g^N$ are the component functions of $\phi$, and ${\mathrm{\partial }}_i$ denotes differentiation in the $i$th chart coordinate. The local maps $R_s$ below act on declared open subsets of this coupling chart.

For a countable family of couplings, instead specify a finite- or countable-dimensional real locally convex topological vector space $E$, its topology, and an open coupling domain $Q\subseteq E$. Here a real locally convex topological vector space means a real vector space with a Hausdorff topology for which addition $E\times E\to E$ and scalar multiplication $\mathbb{R}\times E\to E$ are continuous and $0$ has a neighborhood basis of open convex sets. A subset $C\subseteq E$ is convex when $(1-t)x+ty\in C$ for all $x,y\in C$ and $0\le t\le 1$. The same convention applies to the declared locally convex codomain below. Differentiability of maps $F_0:Q\to F$ into a declared locally convex space $F$ uses the Bastiani convention: $d{F}_0(x;h)={\lim }_{t\to 0}(F_0(x+th)-F_0(x))\mathrm{/}t$ must exist in $F$, and $(x,h)\mapsto d{F}_0(x;h)$ must be jointly continuous; higher differentiability iterates this condition for higher directional derivatives. Whenever component notation $g^i,{\mathrm{\partial }}_i,{\beta }^i$ is used, the data also specify the coordinate functionals and the topology in which the corresponding expansions converge. Without such a coordinate system, the formulas are read in their directional form, with each derivative and contraction explicitly supplied as a continuous map on $E$. In either case, all maps, derivative domains, and convergence of every sum and contraction used below are part of the model data.

Write $\mathcal{U}=U$ in the finite-dimensional case and $\mathcal{U}=Q$ in the countable locally convex case.

For a coupling point $g$, the real coupling insertion in a tangent direction $v\in T_g\mathcal{G}$ (or $v\in E$ in the countable locally convex case) is ${\mathrm{\Phi }}_g[v](x)=D_v\sigma (g,a;x)$. In a chart basis this is ${\mathrm{\Phi }}_i={\mathrm{\partial }}_i\sigma$ in a fixed renormalization scheme, with units $[\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}{]}^{-2}$. A bosonic symmetric Euclidean stress tensor ${\mathsf{T}}_{ab}$ is supplied on ${\mathbb{R}}^2$, with ${\mathrm{\partial }}_a:=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a$, Euclidean metric ${\delta }_{ab}$, and ${\mathrm{\partial }}^a:={\delta }^{ab}{\mathrm{\partial }}_b$. It satisfies ${\mathrm{\partial }}^a{\mathsf{T}}_{ab}=0$ away from contact insertions; contact terms in correlators are distributions supported where insertion points coincide, and no smoothness or conservation across those diagonals is assumed. Use $z=x^1+i{x}^2$ and define $T={\mathsf{T}}_{zz},\bar{T}={\mathsf{T}}_{\bar{z}\bar{z}},\mathrm{\Theta }=-{\mathsf{T}}_{z\bar{z}}\mathrm{.}$ Define $\mathrm{\partial }=\frac{1}{2}({\mathrm{\partial }}_{x^1}-i{\mathrm{\partial }}_{x^2})$ and $\bar{\mathrm{\partial }}=\frac{1}{2}({\mathrm{\partial }}_{x^1}+i{\mathrm{\partial }}_{x^2})$. The minus sign is explicit: tensor conservation is $\bar{\mathrm{\partial }}T-\mathrm{\partial }\mathrm{\Theta }=0$, not $\bar{\mathrm{\partial }}T+\mathrm{\partial }\mathrm{\Theta }=0$ for this symbol $\mathrm{\Theta }$. The separate source-normalized trace identity is stated after the cutoff beta vector has been defined.

For each $(g,a)$, the supplied correlator model includes a unital complex $\ast$-algebra ${\mathcal{P}}_{g,a}$ of finite polynomials in smeared local-field insertions, its antilinear reflection operation, and a normalized complex-linear functional ${\omega }_{g,a}:{\mathcal{P}}_{g,a}\to \mathbb{C}$ with ${\omega }_{g,a}(1)=1$. This functional is invariant under the declared translations and rotations of the Euclidean plane. For $f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^2;\mathbb{C})$, the components ${\mathsf{T}}_{ab}(f)=\int {\mathsf{T}}_{ab}(x)f(x)d^{2}x$ and the coupling insertions ${\mathrm{\Phi }}_i(f)=\int {\mathrm{\Phi }}_i(x)f(x)d^{2}x$ (or ${\mathrm{\Phi }}_g[v](f)$ in a coordinate-free chart) are elements of this algebra; products of these smeared insertions are defined there. For $A,B\in \{T,\bar{T},\mathrm{\Theta },{\mathrm{\Phi }}_i\}$ the bilinear maps

$$(f,h)⟼{\omega }_{g,a}(A(f)B(h))-{\omega }_{g,a}(A(f)){\omega }_{g,a}(B(h))$$

extend to connected distributions $W_{AB}^c\in {\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^2\times {\mathbb{R}}^2)$. The same requirement applies to the declared directional insertions ${\mathrm{\Phi }}_g[v]$ in a coordinate-free coupling model, continuously in the directions. Translation invariance makes each such distribution a kernel depending on $x-y$; $⟨A(x)B(y){⟩}_c$ denotes its separated-point value, taken by the reflected-mollifier limit required below. This defines the field and test-polynomial insertions, and the domain and codomain of the normalized correlation functional, used in the displayed pairings. The action/correlator framework is supplied physical data; this definition does not claim that a formal path integral constructs it.

For every pair in $C,H_i,G_{ij}$, angle brackets mean connected two-point functions, $⟨A(x)B(0){⟩}_c=⟨(A(x)-⟨A⟩)(B(0)-⟨B⟩)⟩\mathrm{.}$ We suppress the subscript $c$ in those three functions. The normalization radius $r=x_0$ satisfies $r\gg a$. Thus $C(r,g)=2z^4⟨T(x)T(0){⟩}_c,H_i(r,g)=z^2{r}^2⟨T(x){\mathrm{\Phi }}_i(0){⟩}_c,G_{ij}(r,g)=r^4⟨{\mathrm{\Phi }}_i(x){\mathrm{\Phi }}_j(0){⟩}_c,$ at $x=(r,0)$, $z=r>0$. These functions are twice continuously differentiable in couplings at each claimed point; all contractions are finite. The notation $⟨\cdot ⟩$ here uses ${\omega }_{g,a}$ and suppresses its subscripts; likewise $C,H_i,G_{ij}$ suppress the fixed current cutoff $a$. Their stipulated covariance under changing $a$ is stated below.

In the countable locally convex case without component coordinates, for $v,w\in E$ define $H(r,g;v)=z^2{r}^2⟨T(x){\mathrm{\Phi }}_g[v](0){⟩}_c,G(r,g;v,w)=r^4⟨{\mathrm{\Phi }}_g[v](x){\mathrm{\Phi }}_g[w](0){⟩}_c\mathrm{.}$ The data require these to be continuous linear and bilinear forms, respectively, on the declared coupling directions. These are the coordinate-free versions of the same contractions.

The required two-point regularity is explicit. Connected component correlators of the tensor, and the scalar pairings used above, extend to tempered distributions, in the sense of Tempered distribution on the Schwartz test space of Schwartz space and its seminorms. Their finite separated point values are joint limits of reflected smooth mollifications. Euclidean covariance includes the tensor transformation law; the tensor has the reflected-adjoint reality specified below. No parity assumption equates its two chiral sectors. Reflection is $\vartheta (x^1,x^2)=(-x^1,x^2)$. Write $\mathcal{J}$ for the supplied antilinear reflected adjoint. Each real coupling insertion is a reflection-even scalar: for every real tangent direction $v$ and $f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^2;\mathbb{C})$ require $\mathcal{J}({\mathrm{\Phi }}_g[v](f))={\mathrm{\Phi }}_g[v](\bar{f}\circ \vartheta )$, and extend antilinearly to complex combinations of directions. This law is an additional reality condition on the insertion data; it is not inferred from the notation $D_v\sigma$. The model obeys reflection positivity for the finite field/test polynomials used here. In the spin basis $(T,\bar{T},\mathrm{\Theta })$, the reflected adjoint fixes each label, so the induced two-point form is exactly ${\sum }_{A,B}\int \overline{f_A(t,x)}{S}_{AB}(t+s,y-x)f_B(s,y)dtdxdsdy\ge 0(\mathrm{supp}f\subset \{t>0\})\mathrm{.}$ Here $S_{AB}$ is the connected two-point distribution; centering is justified by adjoining the constant field to reflection positivity and subtracting its projection. This is a condition on actual correlation data, not a spectral-measure postulate or a full OS reconstruction assertion.

Additional invariant-trace exclusion. At every claimed coupling impose the two-point trace-clustering condition ${\lim }_{r\to \mathrm{\infty }}⟨\mathrm{\Theta }(r,0)\mathrm{\Theta }(0){⟩}_c=0.$ This condition is additional to the weaker reflection-positive package; it is not inferred from reflection positivity or from subtracting the one-point mean. Clustering here is weaker than a unique-vacuum assumption or clustering of the whole Schwinger hierarchy, and is not an assertion of an IR conformal fixed point.

Use squared-length RG time $s=\log (a^2\mathrm{/}{a}_0^2)$ on a specified interval $I\subseteq \mathbb{R}$ containing $0$. For each $s\in I$, $R_s:{\mathcal{U}}_s\to \mathcal{U}$ is a differentiable local map on a declared open domain ${\mathcal{U}}_s\subseteq \mathcal{U}$; $R_0$ is the identity and $R_{s+t}=R_s\circ R_t$ wherever both sides are defined. The cutoff changes to $e^{s\mathrm{/}2}a$, and ${\beta }_g={\frac{d}{ds}{R}_s(g)\mid }_{s=0}$ lies in $T_g\mathcal{G}$ in the finite-dimensional case and in $E$ in the countable locally convex case; ${\beta }^i={\frac{d}{ds}(R_{s}g{)}^i\mid }_{s=0}$ are its components whenever a coordinate system is specified. If $\mathrm{\ell }=\log (a\mathrm{/}{a}_0)$ is used instead, ${\beta }_{\mathrm{\ell }}=2\beta$; the two beta conventions are not silently identified. These maps preserve separated physical correlation functions with the derivative-of-coupling transformation of the ${\mathrm{\Phi }}_i$ basis. Simultaneous dilation of positions and cutoff uses their declared engineering units. Equivalently, after suppressing the fixed reference cutoff, their dimensionless two-point tensors obey $C(r,R_{s}g)=C(e^{s\mathrm{/}2}r,g),$ $\frac{\mathrm{\partial }{R}_s^j}{\mathrm{\partial }{g}^i}{H}_j(r,R_{s}g)=H_i(e^{s\mathrm{/}2}r,g),$ $\frac{\mathrm{\partial }{R}_s^k}{\mathrm{\partial }{g}^i}\frac{\mathrm{\partial }{R}_s^l}{\mathrm{\partial }{g}^j}{G}_{kl}(r,R_{s}g)=G_{ij}(e^{s\mathrm{/}2}r,g)\mathrm{.}$ These are the precise separated Callan–Symanzik covariance premises of the selected model: they say what cutoff equivalence and operator mixing mean, rather than asserting that every action has this property. In particular $D={\beta }^i{\mathrm{\partial }}_i$ agrees with $\dot{}=r^2{\mathrm{\partial }}_{r^2}$ on the scalar stress functions; on covectors and tensors the corresponding Lie derivatives are used. A fixed point is a coupling $g_{\ast }$ for which $R_s(g_{\ast })=g_{\ast }$ for all $s$ in a neighborhood of $0$, so ${\beta }_{g_{\ast }}=0$. No global existence of $R_s$ or limits of its trajectories is assumed.

Source Ward normalization. The model stipulates the renormalized trace identity $\mathrm{\Theta }={\beta }^i{\mathrm{\Phi }}_i,$ read as $\mathrm{\Theta }={\mathrm{\Phi }}_g[{\beta }_g]$ when no coupling-coordinate basis is used. This identity is in the chosen stress/action normalization, modulo a scalar vacuum counterterm, and is not inferred for an arbitrary proposed action. Centering preserves it in connected pairings. The contractions ${\beta }^i{H}_i$ and ${\beta }^i{\beta }^j{G}_{ij}$ mean $H(r,g;{\beta }_g)$ and $G(r,g;{\beta }_g,{\beta }_g)$ in the coordinate-free case.

The scalar physical operator basis is quotiented by null fields and completed in its reflected Gram norm. Apply reflection positivity to centered coupling insertions mollified at $(r\mathrm{/}2,0)$. The stated reflection law and translation invariance identify the limiting reflected Gram matrix at $(r\mathrm{/}2,0),(-r\mathrm{/}2,0)$ with $r^{-4}{G}_{ij}$; the stipulated joint point limits therefore give ${\bar{v}}^{,i}{G}_{ij}{v}^j\ge 0$. On the declared quotient/completion there is no nonzero null vector, and ${\beta }_g$ belongs to this space with finite norm. Thus $G$ is positive definite on the physical beta vector. Vacuum-energy/identity directions, which vanish in connected pairings within a fixed vacuum sector, are not retained as nonzero physical directions.

At a claimed conformal fixed point, the model must supply full local two-dimensional CFT data in the state-space/OPE presentation of Full local two-dimensional CFT data. In particular, this supplies a complete Hausdorff locally convex state space $\mathcal{H}$, a dense common invariant domain $\mathcal{D}$, and commuting left/right Virasoro actions $L_n,{\bar{L}}_n$ on $\mathcal{D}$ with central charges $c_L,c_R$ in the displayed $L_m$ convention. Thus, on $\mathcal{D}$, $[L_m,L_n]=(m-n)L_{m+n}+\frac{c_L}{12}(m^3-m){\delta }_{m+n,0}I,[{\bar{L}}_m,{\bar{L}}_n]=(m-n){\bar{L}}_{m+n}+\frac{c_R}{12}(m^3-m){\delta }_{m+n,0}I,[L_m,{\bar{L}}_n]=0.$ The model also supplies an identification of its stress components with chiral CFT stress fields on that common domain. With dimensionless coordinate $w=z\mathrm{/}{\mathrm{\ell }}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ for a specified reference length ${\mathrm{\ell }}_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$, their radial mode expansions in matrix coefficients on $\mathcal{D}$ are $T_{\mathrm{C}\mathrm{F}\mathrm{T}}(w)={\sum }_{n\in \mathbb{Z}}{L}_n{w}^{-n-2}$ and ${\bar{T}}_{\mathrm{C}\mathrm{F}\mathrm{T}}(\bar{w})={\sum }_{n\in \mathbb{Z}}{\bar{L}}_n{\bar{w}}^{-n-2}$ on the declared radial domains, with convergence in matrix coefficients required there, and $T_{\mathrm{C}\mathrm{F}\mathrm{T}}(w)={\mathrm{\ell }}_{\mathrm{r}\mathrm{e}\mathrm{f}}^{2}T(z)$, ${\bar{T}}_{\mathrm{C}\mathrm{F}\mathrm{T}}(\bar{w})={\mathrm{\ell }}_{\mathrm{r}\mathrm{e}\mathrm{f}}^2\bar{T}(\bar{z})$. Write $\tilde{c}:=c_L$ for the holomorphic central charge in the normalization used by the c-function. The fixed-point convention is

$$⟨T_{\mathrm{C}\mathrm{F}\mathrm{T}}(w)T_{\mathrm{C}\mathrm{F}\mathrm{T}}(0){⟩}_c=\frac{\tilde{c}}{2w^4}=\frac{c_L}{2w^4},⟨{\bar{T}}_{\mathrm{C}\mathrm{F}\mathrm{T}}(\bar{w}){\bar{T}}_{\mathrm{C}\mathrm{F}\mathrm{T}}(0){⟩}_c=\frac{c_R}{2{\bar{w}}^4}\mathrm{.}$$

Equivalently, in the lengthful coordinates used in this definition, $⟨T(z)T(0){⟩}_c=c_L\mathrm{/}(2z^4)$ and $⟨\bar{T}(\bar{z})\bar{T}(0){⟩}_c=c_R\mathrm{/}(2{\bar{z}}^4)$. The field identification and normalization are supplied fixed-point data; neither follows for an arbitrary action, and this definition does not construct such a model or prove that a flow reaches such a point.

Smooth manifolds and their smooth charts

Full local two-dimensional CFT data

For context, under the exact hypotheses of Positive stress spectral measure from two-point reflection positivity, the local spectral representation identifies the trace-clustering condition above with absence of the zero-energy, zero-momentum trace atom. This spectral interpretation is not a logical prerequisite for the definition; the clustering condition itself remains an additional model assumption.

The trace sign and RG-time conventions above are chosen to make the geometric Ward equations and the displayed coefficient−12 refer to the same quantities. Explicitly, Cardy's full trace is ${\mathrm{\Theta }}_{\mathrm{C}}=4{\mathsf{T}}_{z\bar{z}}=-4\mathrm{\Theta }$. His combination $2F-G_{\mathrm{C}}-3H_{\mathrm{C}}\mathrm{/}8$ becomes $2F+4B-6H$ with $G_{\mathrm{C}}=-4B$, $H_{\mathrm{C}}=16H$, and his radial identity becomes $\dot{c}=-12H$. Thus this is an explicit conversion of a stated source convention, rather than treating the literal component ${\mathsf{T}}_{z\bar{z}}$ as its negative. The newly explicit trace-clustering assumption is a genuine additional physical qualification. A centered random constant scalar gives a conserved reflection-positive trace with a nonzero invariant atom, so connected subtraction alone would not justify the positive c-function. Temperateness and tensor reflection positivity are also actual hypotheses of the two-point spectral construction; they are not inferred from a list of finite coupling contractions. No finite UV/IR conformal endpoint, global population of trajectories, or assumed Källén–Lehmann density is introduced.