Information Geometry on Theory Space

Information geometry gives a local quadratic form on a declared smooth family of states, probability measures, or generating functionals. Those are not interchangeable inputs: Bures, Kubo–Mori, normalized Fisher, partition-function, and conformal two-point metrics can differ in normalization, contact prescription, and even operational meaning. The result is a local geometry of one controlled family—not a unique finite-dimensional manifold containing every QFT.

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

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

The chapter overview separates cutoff, region, and deformation scales, compares the candidate observables in its comparison table, and states the independent validity gates. An information metric is licensed only after its input family, tangent domain, regulator, and subtraction rule pass those gates.

Several metrics answer several questions

For a faithful finite-dimensional quantum-state family $\rho_\lambda$, the local expansion

$$D(\rho_{\lambda+d\lambda}\Vert\rho_\lambda) =\frac12 g^{\rm BKM}_{ij}\,d\lambda^i d\lambda^j +O(\lVert d\lambda\rVert^3)$$

defines the Bogoliubov–Kubo–Mori quadratic form, with the ordering and base point shown explicitly. The Bures metric instead comes from the root fidelity. Both are monotone under channels, but they weight noncommuting tangent directions differently; Petz’s classification makes this nonuniqueness precise Petz 1996, Theorem 2 and pp. 87–90. On a commuting family they reduce to the classical Fisher metric up to the conventional factor used in defining Bures distance. Bures’s operator-algebraic transition probability is developed in Bures 1969, Theorems 1–2.

For a normalized Euclidean probability family $p_\lambda[\phi]$, the Fisher metric is

$$g^{\rm F}_{ij} =\int {\cal D}\phi\,p_\lambda[\phi] \,\partial_i\log p_\lambda[\phi] \,\partial_j\log p_\lambda[\phi].$$

If the source enters the action affinely and differentiation under the regulated integral is justified, this equals a connected covariance. By contrast, a raw Hessian $\partial_i\partial_j W$ of $W=\log Z$ is naturally tied to those affine source coordinates. It can shift under local source-only counterterms and does not transform tensorially under a nonlinear reparameterization because the chain rule also produces a first-derivative term.

Construction	Typed input	Local conclusion	Continuum requirement	Typical failure mode
Normalized Fisher	A common dominated probability family and square-integrable scores	Positive coordinate-covariant distinguishability form	Regulated/smeared scores with a finite limiting quadratic form	Mutually singular measures or divergent contact covariance
BKM	Faithful density matrices, or faithful normal states with finite tangent form	Relative-entropy Hessian	Common algebra; support and form-domain control	Changing support, nonclosable tangent form, or confusing it with Bures
Bures	States in one representation and differentiable root fidelity	Transition-probability geometry	A specified algebra/representation and finite fidelity susceptibility	Orthogonality catastrophe or convention-dependent factors
Partition Hessian	A renormalized generating functional in affine source coordinates	Integrated connected response plus contacts	Source counterterms and boundary prescription fixed	Scheme shift or nonlinear-coordinate Hessian treated as a tensor
CFT two-point metric	Exactly marginal operators with fixed normalization	Local metric on a conformal manifold	Separated correlators and contact/connection prescription	Extending it to arbitrary relevant directions or off-critical states

In a $d$-dimensional QFT, a source for an operator of scaling dimension $\Delta_i$ has mass dimension $d-\Delta_i$. Metric components therefore carry compensating units; the invariant object is $g_{ij}d\lambda^i d\lambda^j$. Dimensionless coordinates such as $\widehat\lambda^i=\mu_{\rm R}^{\Delta_i-d}\lambda^i$ require a declared renormalization scale $\mu_{\rm R}$.

A finite Gaussian benchmark

Consider a two-variable regulated Euclidean Gaussian in lattice units,

$$p_{\mu,J_1,J_2}(q)=\frac1{Z} \exp\!\left[-\frac12q^{\mathsf T}A(\mu)q +(J_1+J_2)q_1\right],$$

with $q\in\mathbb R^2$, $A(\mu)=\operatorname{diag}(1+\mu,4+\mu)$, and domain $\mu>0$, $J_1,J_2\in\mathbb R$. The two source coordinates deliberately couple to the same operator. At the base point $\mu=2$, $J_1=J_2=0$, the scores are

$$\partial_\mu\log p =\frac12\operatorname{Tr}A^{-1}-\frac12q^{\mathsf T}q, \qquad \partial_{J_1}\log p=\partial_{J_2}\log p=q_1.$$

Gaussian Wick contractions then give

$$g_{\mu\mu}=\frac12\operatorname{Tr}A^{-2}=\frac5{72}, \qquad g_{J_aJ_b}=(A^{-1})_{11}=\frac13, \qquad g_{\mu J_a}=0.$$

Thus

$$g= \begin{pmatrix} 5/72&0&0\\ 0&1/3&1/3\\ 0&1/3&1/3 \end{pmatrix},$$

whose eigenvalues are $0$, $5/72$, and $2/3$. The normalized null vector $(0,1,-1)/\sqrt2$ is not a mysterious zero-norm operator: it changes $J_1-J_2$ while leaving the probability distribution exactly unchanged. This is an explicit redundant coordinate.

The benchmark is finite, positive definite in field space, and analytic, so it has no ultraviolet, infrared, Monte Carlo, or fit uncertainty. As a reproducibility check, use

$$W(\mu,0,0)=\text{constant} -\frac12\log[(1+\mu)(4+\mu)]$$

and a centered finite difference for $W''(2)=5/72$. The residual against the exact result separates truncation error from floating-point cancellation:

Step $h$	Centered estimate of $W''(2)$	Estimate minus $5/72$
$10^{-2}$	$0.0694447723748$	$+3.28\times10^{-7}$
$10^{-3}$	$0.0694444477478$	$+3.30\times10^{-9}$
$10^{-4}$	$0.0694444279858$	$-1.65\times10^{-8}$

The $h=10^{-3}$ row is the best of these three in double precision; decreasing $h$ further amplifies subtraction roundoff. The analytic covariance result, not the smallest displayed finite difference, is the benchmark target.

Coordinate covariance and null directions

Let $u=\log(\mu/\mu_0)$. Since $d\mu/du=\mu$, the Fisher component at $\mu=2$ is

$$g_{uu}=\mu^2g_{\mu\mu}=\frac5{18},$$

and $g_{uu}\,du^2=g_{\mu\mu}\,d\mu^2$. Metric components changed, but the line element did not. The raw generating-functional Hessian behaves differently:

$$\frac{\partial^2 W}{\partial u^2} =\mu^2\frac{\partial^2 W}{\partial\mu^2} +\mu\frac{\partial W}{\partial\mu}.$$

The second term is why “the Hessian of $W$” is a coordinate-dependent phrase unless the affine source structure is retained.

In QFT, field redefinitions, equation-of-motion insertions, total derivatives, and exact symmetry directions may be redundant, but a raw regulated correlator need not make them null. Boundary terms and contact counterterms can give them nonzero components. One should identify the physical quotient or project the tangent space using the relevant Ward identities before inverting the metric. The exact duplicated-source null vector above is a diagnostic control, not evidence that every redundant QFT operator is automatically null in every scheme.

Contact-counterterm adversarial test

The same finite family cleanly exposes scheme dependence. Modify only the unnormalized generating functional by the local source-only term

$$W_c(\mu,J)=W(\mu,J)+\frac c2\mu^2, \qquad c=0.3.$$

Multiplying the unnormalized weight by $e^{c\mu^2/2}$ leaves the normalized probability $p$ unchanged: the factor cancels between numerator and partition function. Therefore the Fisher matrix, its null vector, and every normalized expectation value remain unchanged. The raw Hessian does not:

Quantity at $\mu=2$	Original	After $c=0.3$	Operational meaning
Normalized Fisher $g_{\mu\mu}$	$0.06944444$	$0.06944444$	Same distinguishability family
Raw $\partial_\mu^2W$	$0.06944444$	$0.36944444$	Shifted by the local source counterterm
Source-block eigenvalues	$0,\ 2/3$	$0,\ 2/3$	Duplicate-source redundancy unchanged
Normalized observable change	$0$	$0$	Counterterm is field independent

This adversarial test does not say all finite metric terms are meaningless. It identifies which construction was used. A separated two-point coefficient, a relative-entropy Hessian, or a counterterm-subtracted universal logarithm may be robust under a specified class of schemes; the raw finite partition Hessian is not. Contact terms also act as the connection on conformal-theory space in the analysis of Kutasov 1989, pp. 154–156.

For regional density matrices there is an additional distinction. Their relative-entropy Hessian contains modular evolution or a modular resolvent kernel and is not generally an unrestricted Euclidean double integral. The region, reference state, support/faithfulness assumptions, and operator form domain are part of the definition.

RG vectors and restricted monotonicity

The beta functions form a coordinate vector, so $g_{ij}\beta^i\beta^j$ is a scalar once the metric, quotient, and scheme are fixed. Positivity of $g$ alone does not imply a gradient formula $\beta^i=-g^{ij}\partial_j C$, nor does it choose a unique monotone function.

In two-dimensional unitary QFT, Zamolodchikov’s theorem uses a particular stress-tensor/correlator construction and its own hypotheses Zamolodchikov 1986, pp. 730–732. It should not be replaced by the statement “every positive information metric makes RG a gradient flow.” Away from such theorems, a metric remains a local response diagnostic whose regulator and operational interpretation must be carried along.

Common pitfalls

Calling every Hessian a metric tensor. A Hessian is tensorial only with additional affine structure; normalized Fisher information transforms correctly by construction.

Inverting before removing null directions. Exact redundancy makes the matrix singular. Numerical near-null directions also require conditioning and regulator-refinement checks.

Equating Euclidean and quantum metrics. An integrated Euclidean covariance, a BKM form, and a Bures fidelity susceptibility coincide only under additional commuting or path-integral identifications and convention choices.

Exercises

1. Reproduce the Gaussian metric and its null vector

Starting from the displayed two-variable probability density, derive all entries of $g$ at $\mu=2$, $J_1=J_2=0$, and verify the null direction.

Solution

At zero source the covariance is $\Sigma=A^{-1}=\operatorname{diag}(1/3,1/6)$. For a centered Gaussian,

$$\operatorname{Var}(q^{\mathsf T}q)=2\operatorname{Tr}\Sigma^2,$$

so the factor $1/2$ in the mass score gives

$$g_{\mu\mu}=\frac14\operatorname{Var}(q^{\mathsf T}q) =\frac12\left(\frac19+\frac1{36}\right)=\frac5{72}.$$

Odd centered moments vanish, hence $g_{\mu J_a}=0$. Both source scores equal $q_1$, so every source-block entry is $\langle q_1^2\rangle=1/3$. Multiplication by $(0,1,-1)^{\mathsf T}$ gives zero. The orthogonal source direction $(0,1,1)/\sqrt2$ has eigenvalue $2/3$.

2. Transform the mass coordinate

For $u=\log(\mu/\mu_0)$, derive the transformations of the Fisher component and the raw Hessian. Evaluate the Fisher component at $\mu=2$.

Solution

Scores obey $\partial_u\log p=(d\mu/du)\partial_\mu\log p=\mu\partial_\mu\log p$. Their covariance therefore gives

$$g_{uu}=\mu^2g_{\mu\mu}.$$

At $\mu=2$, this is $4(5/72)=5/18$. Since $d\mu=\mu\,du$, the line element is unchanged.

For an ordinary scalar function $W$,

$$\partial_uW=\mu\partial_\mu W, \qquad \partial_u^2W=\mu^2\partial_\mu^2W+\mu\partial_\mu W.$$

The inhomogeneous first-derivative term shows that the raw Hessian components do not obey the metric transformation law under this nonlinear coordinate change.

3. Separate a contact shift from a state change

Add $c\mu^2/2$ to $W$ with arbitrary real $c$. Show that the normalized Gaussian family and Fisher metric are unchanged while the raw mass Hessian shifts. What would be different if the counterterm contained $q^2$?

Solution

The source-only term multiplies every unnormalized weight at fixed $\mu$ by the same factor $e^{c\mu^2/2}$. The partition function acquires the same factor, so their ratio $p$ is unchanged. Every score $\partial_i\log p$, normalized expectation, and Fisher component is therefore unchanged. Direct differentiation gives

$$\partial_\mu^2W_c=\partial_\mu^2W+c.$$

A term proportional to $\mu q^2$, by contrast, changes the stiffness matrix $A$ and hence the normalized covariance. That is a change of the physical family (or of the map between renormalized and bare couplings), not merely a field-independent contact shift.

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.
• Kutasov, David. “Geometry on the Space of Conformal Field Theories and Contact Terms.” Physics Letters B 220 (1989): 153–158. 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.