Formal-Power-Series Interacting Constructions, States, and Scope

A perturbative interacting algebra and its states are exact algebraic objects over a ring of formal power series, but they are not automatically functions of a numerical coupling. Products, field equations, Ward identities, and normalization hold coefficientwise. Positivity uses the ordering of formal real series: the sign is the sign of the first nonzero coefficient. Consequently an interacting Hadamard functional can be positive as a formal state even though a finite truncation evaluated at a chosen number is negative.

Required background. The perturbative Bogoliubov map constructs interacting observables coefficientwise. Algebraic adiabatic limits explains how those observables form switching-independent local algebras. Helpful background. Existence, construction, reconstruction, and continuum claims distinguishes formal construction from analytic existence. Constructive existence by model, dimension, and observable records what a nonperturbative theorem must add.

Formal *-algebras and ordered-series positivity

Let $\mathcal A_0$ be a complex vector space of equicausal functionals, hence satisfying the standard microcausal wavefront restriction and the compact-set uniformity that makes the star product close Hawkins, Rejzner, and Visser 2026, §7, Theorem 7.4, pp. 32–36 of the open manuscript, and take

$$\mathcal A=\mathcal A_0[[\lambda,\hbar]].$$

A deformed product is a bidouble series

$$A\star B=\sum_{r,s\ge0}\lambda^r\hbar^s C_{r,s}(A,B),$$

understood coefficientwise: the coefficient at any fixed $(r,s)$ receives only finitely many contributions. Associativity means equality of every coefficient in $(A\star B)\star C$ and $A\star(B\star C)$. The involution is likewise formal, and the interacting equation of motion means that every coefficient of its series vanishes. Nothing here asserts a nonzero radius of convergence.

The real ring $\mathbb R[[\lambda]]$ is ordered lexicographically by its lowest nonzero coefficient. Thus

$$a(\lambda)=a_k\lambda^k+O(\lambda^{k+1}), \qquad a_k\ne0,$$

is positive exactly when $a_k>0$. A normalized formal state is a $\mathbb C[[\lambda,\hbar]]$-linear functional $\omega:\mathcal A\to\mathbb C[[\lambda,\hbar]]$ with $\omega(1)=1$ and

$$\omega(A^*\star A)\ge0$$

in a declared ordering of the two formal parameters, or after combining them into a single filtration. Because different multivariable orderings need not agree, the filtration is part of the statement. Bordemann and Waldmann develop the ordered Laurent-series and formal-positive-functional construction in Bordemann and Waldmann 1998, §2, pp. 553–559 and the associated formal GNS construction in §3, pp. 559–566.

Formal positivity is not coefficientwise positivity: higher coefficients may have either sign. Nor does it make every numerical truncation positive. A genuine C*-state at fixed $\lambda$ would require a normed algebra, convergence or a controlled summation, and continuity—data absent from the definition above.

First application: compactly supported curved-space φ⁴

Let $(M,g)$ be globally hyperbolic, let $\omega_0$ be a quasifree Hadamard state of the free Klein–Gordon field, and choose $g_c\in C_c^\infty(M)$. Put

$$V=\lambda V_1, \qquad V_1=\frac1{4!}\int_M g_c(x)\phi(x)^4\,\mathrm d\mu_g(x).$$

Fix locally covariant time-ordered products satisfying causal factorization, the microlocal spectral condition, unitarity, and the field equation. For a microcausal observable $A$, the retarded Bogoliubov map has the expansion

$$R_V(A)=A +\lambda R_{1,1}(V_1;A) +\frac{\lambda^2}{2}R_{2,1}(V_1,V_1;A) +O(\lambda^3).$$

On the interacting algebra, whose product is pulled back so that $R_V$ is a unital *-homomorphism, define

$$\omega_V(A)=\omega_0(R_V(A)).$$

Through second order,

$$\omega_V(A)=\omega_0(A) +\lambda\,\omega_0(R_{1,1}(V_1;A)) +\frac{\lambda^2}{2}\,\omega_0(R_{2,1}(V_1,V_1;A)) +O(\lambda^3).$$

This is the compactly supported interacting-state construction used with local S-matrices and causal factorization. The term “interacting Hadamard state” here means a normalized positive formal functional whose coefficient distributions obey the perturbative microlocal spectrum condition and whose zeroth-order two-point function is Hadamard. It does not mean a density matrix or an exact state on an already completed interacting C*-algebra.

The coefficient check has four parts.

1. Zeroth order. $\omega_0(1)=1$, positivity is ordinary free-state positivity, and $\operatorname{WF}(\omega_{0,2})$ is the Hadamard positive-frequency null relation.
2. First order. $R_{1,1}(V_1;1)=0$, so the normalization coefficient vanishes. Every contraction pairs the compact vertex with $A$ through Hadamard or retarded kernels. Hörmander’s criterion and the microcausal domain exclude a zero covector in the fiber sums, giving the allowed first-order wavefront cone.
3. Second order. $R_{2,1}(V_1,V_1;1)=0$. Subdivergences and the total diagonal have already been extended by the chosen $T_2$ and $T_3$ prescription; local counterterms preserve the declared microlocal bound. The factor $1/2$ is the symmetry factor from the exponential, not an optional diagrammatic convention.
4. Positivity. Since $R_V$ preserves the involution and interacting product, $\omega_V(A^*\star_V A)=\omega_0(R_V(A)^*\star R_V(A))$. Expanding the right side proves formal positivity in the chosen filtration.

Hollands and Wald construct local covariant time-ordered products with the requisite microlocal properties in Hollands and Wald 2002, §§3–4, pp. 318–341. Their theorem supplies the extensions used in the coefficient check; it does not supply convergence of the resulting interacting series.

An independent normalization check differentiates $R_V(1)=1$. Every retarded coefficient with the final entry $1$ must vanish, so $\omega_V(1)=1$ at all orders. An independent causal check changes $g_c$ outside a causal neighborhood of $\operatorname{supp}A$: the relative-S cocycle intertwines the two representatives, so the abstract local state assignment changes only by that declared identification.

Adversarial test: a negative truncation

Suppose for a particular $A$ one obtains

$$\omega(A^*\star A)=1-2\lambda+3\lambda^2+O(\lambda^3).$$

The formal series is positive because its first nonzero coefficient is $+1$. The first-order truncation evaluated at $\lambda=1$ equals $-1$. That numerical value neither refutes formal positivity nor defines a physical probability: the omitted terms are uncontrolled there, and evaluation at $\lambda=1$ is not a homomorphism on arbitrary formal series.

The converse error also matters. If several low-order truncations happen to be positive over a range of numerical couplings, that does not prove the full series converges or defines a positive exact state. Such a claim needs remainder bounds, summability, or a separate nonperturbative construction.

Exercises

1. Leading-order sign. For $a(\lambda)=\lambda^4(2-7\lambda)$ and $b(\lambda)=-\lambda^6+O(\lambda^7)$, determine their formal signs.

Solution

The first nonzero coefficient of $a$ is $+2$, so $a>0$; the first nonzero coefficient of $b$ is $-1$, so $b<0$. Their values at a chosen numerical $\lambda$ are irrelevant to this formal ordering.

2. Second-order normalization. Use unitality of $R_V$ to find the coefficients of $\lambda$ and $\lambda^2$ in $\omega_V(1)$.

Solution

$R_V(1)=1$ implies $R_{1,1}(V_1;1)=0$ and $R_{2,1}(V_1,V_1;1)=0$. Hence $\omega_V(1)=\omega_0(1)=1$ through second order, and the same argument holds at every order.

References

• Bordemann, Martin, and Stefan Waldmann. “Formal GNS Construction and States in Deformation Quantization.” Communications in Mathematical Physics 195 (1998): 549–583. DOI; Open manuscript.
• Hawkins, Eli, Kasia Rejzner, and Berend Visser. “A Novel Class of Functionals for Perturbative Algebraic Quantum Field Theory.” Revised 2026. Open manuscript, arXiv:2312.15203.
• Hollands, Stefan, and Robert M. Wald. “Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 231 (2002): 309–345. DOI; Open manuscript.