Exact Modular-Flow Examples and Convention Atlas

Exact modular-flow formulas are easy to miscompare because authors choose opposite modular-time directions, call either $-\log\rho_A$ or $-\log\Delta$ the modular Hamiltonian, and absorb $2\pi$ into the generator or the parameter. This atlas fixes one reference convention, translates the common alternatives, and records the hypotheses for wedge, ball, interval, thermal, null, and finite-dimensional examples.

Required background. The wedge theorem and conformal ball flow supply the exact geometric cases.

Helpful background. Modular KMS correlators supply the imaginary-time check that fixes signs and normalization.

Reference convention

For the standard modular operator of $(\mathcal A,\Omega)$ , use

K=-\log\Delta, \qquad \sigma_s(A)=\Delta^{is}A\Delta^{-is} =e^{-isK}Ae^{isK}.

The parameter $s$ is dimensionless. With this convention, $F(s)=\omega(A\sigma_s(B))$ extends to the lower unit strip and

F(s-i)=\omega(\sigma_s(B)A).

An author using $\widetilde\sigma_s=\Delta^{-is}(\,\cdot\,)\Delta^{is}$ has simply chosen $\widetilde\sigma_s=\sigma_{-s}$ ; the KMS strip is then the upper unit strip. An author defining $\widetilde K=\log\Delta$ writes the same unitary flow as $e^{is\widetilde K}(\,\cdot\,)e^{-is\widetilde K}$ .

For a faithful density matrix on a tensor factor,

K_A=-\log\rho_A, \qquad K_{\rm std}=K_A-K_{A^c}.

Both generate the same adjoint flow on $A$ -side observables. Adding $c\mathbf1$ to $K_A$ changes normalization but not flow; it must not be confused with changing the support of $\rho_A$ .

The structural map places Exact Modular-Flow Examples and Convention Atlas between intrinsic modular data and the additional hypotheses that permit a geometric interpretation.

A standard algebra-state pair determines the Tomita operator, modular conjugation, modular flow, and relative modular data; only additional covariance or conformal hypotheses turn the flow into wedge or ball geometry. — Tomita polar decomposition intrinsically produces $J$ and $\Delta$ . Wedge boosts and CFT ball flow are special consequences of locality, covariance, the spectrum condition, the vacuum, and—only for the ball—the conformal map. The diagram is schematic and not to scale.

Vacuum wedge

For $W_R=\{x^1>\lvert t\rvert\}$ in a Poincaré-covariant vacuum QFT satisfying the hypotheses of Bisognano and Wichmann 1975, pp. 985–1007,

\Delta_{W_R}^{is}=U(\Lambda_R(-2\pi s)).

At $t=0$ , the one-sided representative is

K_R=2\pi\int_{x^1>0}d^{d-1}x\;x^1T_{00}+c.

The full standard generator is right minus left. Reversing the definition of boost rapidity or modular time changes the displayed sign but not the $2\pi$ . The exactness boundary is sharp: arbitrary regions, excited states, and Lorentz-breaking regulators are not covered.

Vacuum ball and interval in a CFT

For a radius- $R$ ball in the CFT vacuum,

K_B=2\pi\int_{r<R}d^{d-1}x\; \frac{R^2-r^2}{2R}\,T_{00}+c.

The associated conformal Killing vector is

\zeta_B=\frac{\pi}{R} \left[(R^2-t^2-r^2)\partial_t-2tx^i\partial_i\right].

For an interval $(u,v)$ of a two-dimensional CFT vacuum,

K_{(u,v)} =2\pi\int_u^v dx\; \frac{(x-u)(v-x)}{v-u}\,T_{00}(x)+c.

All weights are positive in the region and vanish linearly at its boundary. These formulas require the conformal vacuum and the round-ball or interval geometry. A Weyl anomaly can affect partition functions and constants but does not replace the local conformal Killing generator by an arbitrary one.

Thermal equilibrium

For $\rho_\beta=Z^{-1}e^{-\beta H}$ on a regulated full algebra,

\sigma_s(A) =\rho_\beta^{is}A\rho_\beta^{-is} =e^{-i\beta sH}Ae^{i\beta sH}.

If physical Heisenberg evolution is $\tau_t(A)=e^{itH}Ae^{-itH}$ , then

\sigma_s=\tau_{-\beta s}.

Thus physical time is $t=-\beta s$ in the reference convention. Choosing $\Delta^{-is}$ instead gives $t=+\beta s$ . This identification requires that the state be KMS for the stated physical dynamics; it does not hold for a generic subregion modular flow.

Null translations and half-lines

For a wedge translated along a future null direction, isotony gives a nested pair of wedge algebras. With the vacuum and the appropriate orientation, the pair is a half-sided modular inclusion and reconstructs a positive null translation $T(a)=e^{iaP_+}$ :

\Delta_W^{is}T(a)\Delta_W^{-is} =T(e^{-2\pi s}a)

in the orientation used here. This is an exact statement for the structured inclusion. It is not a formula for an arbitrary cut of a null plane. More general null-shape modular Hamiltonians need separate locality, Markov, or deformation results.

Finite faithful and thermofield examples

For $\rho=\sum_i p_i\lvert i\rangle\langle i\rvert$ represented on Hilbert–Schmidt operators,

\Delta(\lvert i\rangle\langle j\rvert) =\frac{p_i}{p_j}\lvert i\rangle\langle j\rvert, \qquad K(\lvert i\rangle\langle j\rvert) =-\log\!\frac{p_i}{p_j}\lvert i\rangle\langle j\rvert.

For a thermofield-double purification of a Gibbs state,

\lvert{\rm TFD}\rangle =Z^{-1/2}\sum_n e^{-\beta E_n/2} \lvert n\rangle_L\lvert n\rangle_R,

the standard generator is proportional to $H_L-H_R$ , with the sign and factor fixed by which side is chosen as $\mathcal A$ :

K_{\rm std}=\beta(H_L-H_R)

for $K_L=-\log\rho_L=\beta H_L+\log Z$ . It annihilates the TFD state. Zero Schmidt coefficients invalidate the inverse on the full space and require restriction to the support.

Conversion checklist

When translating a formula, record these six items:

Is the flow $\Delta^{is}$ or $\Delta^{-is}$ ?
Is $K$ defined as $-\log\Delta$ , $\log\Delta$ , or $-\log\rho_A$ ?
Is $2\pi$ included in $K$ , in the rapidity, or in modular time?
Which algebra and which complementary orientation are used?
Has an additive normalization constant been suppressed?
Are support and operator domains unchanged by the translation?

A correct conversion preserves the adjoint action, the KMS boundary equation, and the geometric orbit. Matching only the symbol $K$ is not enough.

Common pitfalls

Comparing signs without comparing adjoint actions. Two authors can assign opposite signs to $K$ and still describe identical flow. Translate the complete exponential.

Calling every null result exact. The translated-wedge inclusion is exact under its hypotheses. An arbitrary null cut is a different problem.

Using the thermal row as a general interpretation. Modular time becomes physical time only because the chosen state is KMS for that Hamiltonian.

Before assigning geometric meaning to this modular statement, use the validity map to check standardness, spectral domains, normalization, geometric hypotheses, and approximation control independently.

A decision map separates exact modular conclusions from failures caused by missing faithfulness, uncontrolled operator domains, absent geometric theorems, or perturbative nonlocal kernels. — A modular-flow claim is only as strong as its algebra–state standardness, spectral domain, normalization, geometric hypotheses, and approximation control. Dashed branches mark conditional steps; the terminal warnings identify common out-of-domain uses. The diagram is schematic and not to scale.

References

Bisognano, Joseph J., and Eyvind H. Wichmann. “On the Duality Condition for a Hermitian Scalar Field.” Journal of Mathematical Physics 16 (1975): 985–1007. DOI.