Information Geometry on Theory Space

Information geometry gives a local metric on a controlled family of QFT states or generating functionals. Source derivatives turn the metric into integrated connected correlators, while contact terms, redundant operators, and regulator dependence determine which directions are physical. The construction is local in a chosen family; it does not make the set of all QFTs one finite-dimensional Riemannian manifold.

Required background. Bures and Kubo–Mori metrics supplies the inequivalent monotone metrics; information measures along RG flows fixes the family and scale comparison.

Helpful background. Fidelity susceptibility under deformations supplies the quadratic response expansion.

Source families and local metrics

Let

$$I[\lambda] =I_0+\sum_i\lambda^i\int d^d x\,\mathcal O_i(x)$$

define a regulated smooth family. For normalized Euclidean probability functionals, the Hessian of the log partition function is the connected covariance,

$$\partial_i\partial_j\log Z =\int d^d x\,d^d y\, \langle\mathcal O_i(x)\mathcal O_j(y)\rangle_c,$$

up to signs fixed by the source convention. For quantum states, the Kubo–Mori metric is the Hessian of relative entropy; the Bures metric instead comes from fidelity. They agree for commuting families but weight noncommuting directions differently.

The operator-algebraic transition distance begins with Bures 1969, Theorems 1–2; the classification of contractive matrix metrics is Petz 1996, Theorem 2 and pp. 87–90.

The chapter diagram places this metric after the state family, region, and scheme have been fixed.

Theory-space information geometry is built from a declared source and state family. Integrated correlators, contact prescriptions, and redundant directions are part of the definition before RG scaling is interpreted. Schematic and not to scale.

Contacts, boundaries, and renormalization

The double integral is singular at $x=y$ and can also diverge at Euclidean-time cuts or spatial entangling surfaces. Composite-operator counterterms add contact contributions to $g_{ij}$. A finite metric therefore requires smearing, separated insertions, or a renormalization prescription. Universal logarithmic coefficients can be robust, while finite components can depend on scheme.

For a regional density matrix, the relative-entropy Hessian includes a modular kernel rather than an unrestricted Euclidean covariance. Schematically,

$$g^{\rm KM}_{ij} =\int d^d x\,d^d y\, K_R(x,y)\, \langle\mathcal O_i(x)\mathcal O_j(y)\rangle_c +g^{\rm contact}_{ij}.$$

The kernel $K_R$ depends on the region and reference state. This is why a whole-space partition-function metric and a regional distinguishability metric need not coincide.

Redundant directions and coordinate covariance

A coupling redefinition $\lambda^i\mapsto\lambda'^a(\lambda)$ changes metric components by

$$g'_{ab} =\frac{\partial\lambda^i}{\partial\lambda'^a} \frac{\partial\lambda^j}{\partial\lambda'^b} g_{ij}.$$

Scalar distances are coordinate invariant only after the same physical family and renormalization prescription are used. Directions generated by field redefinitions, equations of motion, total derivatives, or exact symmetry transformations can be redundant. They should be quotiented or appear as null directions after all contact terms are treated consistently.

The RG vector $\beta^i(\lambda)$ is a vector field in these coordinates. Quantities such as $g_{ij}\beta^i\beta^j$ are useful local diagnostics, but a general gradient-flow equation $\beta^i=-g^{ij}\partial_j C$ is an additional theorem, not a consequence of positivity of $g$ alone.

Gaussian mass and source example

For a free scalar with parameters $(m^2,J)$, the vacuum wavefunctional is Gaussian. A regulated fidelity or Kubo–Mori calculation reduces to covariance-matrix derivatives. The mass direction changes the spectrum, while a uniform source shifts the first moments. A source proportional to an equation-of-motion operator can become redundant after boundary terms and support restrictions are handled.

The benchmark should compare the metric obtained from state overlaps with the one from integrated two-point functions, including contact terms. Refining the lattice at fixed $mR$ tests the continuum scaling. A vanishing eigenvalue must be checked against numerical conditioning before it is interpreted as a redundant direction.

The fidelity-metric diagnosis of critical response is illustrated in Zanardi, Giorda, and Cozzini 2007, Eqs. (2)–(7).

Evidence boundary as of 10 August 2026

Information metrics are rigorous for specified finite-dimensional state families and have controlled algebraic extensions. In QFT, useful continuum metrics exist for particular regions, sources, and subtraction schemes. No unique metric covers all theories: Bures, Kubo–Mori, Zamolodchikov-type, and partition-function metrics answer different questions, and coordinate covariance does not remove scheme dependence.

The two-dimensional correlator metric and its role in an RG monotonicity formula appear in Zamolodchikov 1986, pp. 730–732.

Validity map for theory-space metrics. A common regulator and one smooth state/source family are required. Contractivity becomes an RG statement only if a channel acts on that family; otherwise the metric is a local response diagnostic. Schematic and not to scale.

Common pitfalls

Calling metric components observables. Components change under coupling coordinates and finite counterterms. State invariant combinations and the chosen scheme.

Ignoring redundant operators. Field redefinitions and equations of motion can create null or gauge-like directions. Quotient them before inverting the metric.

Inferring a gradient RG flow from positivity. A positive information metric does not by itself supply a potential whose gradient is the beta function.

References

• Bures, Donald. “An Extension of Kakutani’s Theorem on Infinite Product Measures to the Tensor Product of Semifinite $W^*$-Algebras.” Transactions of the American Mathematical Society 135 (1969): 199–212. DOI.
• Petz, Dénes. “Monotone Metrics on Matrix Spaces.” Linear Algebra and its Applications 244 (1996): 81–96. DOI.
• Zamolodchikov, Alexander B. “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory.” JETP Letters 43 (1986): 730–732. INSPIRE record.
• Zanardi, Paolo, Paolo Giorda, and Marco Cozzini. “Information-Theoretic Differential Geometry of Quantum Phase Transitions.” Physical Review Letters 99 (2007): 100603. DOI.