Exact Modular-Flow Examples and Convention Atlas

Exact modular-flow formulas often look contradictory because authors reverse modular time, use either $-\log\rho$ or $-\log\Delta$, and place $2\pi$ in different parameters. This atlas freezes one convention, distinguishes the intrinsic standard-representation generator from regulated one-sided representatives, and tests wedge, ball, and thermal flow on the same operator.

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.

The chapter overview places the examples in the route from standard pairs to geometric flow, compares representative modular flows and their boundaries, and gives the checklist for deciding when a modular-flow claim is licensed.

Frozen convention and the KMS test

For a standard pair $(\mathcal A,\Omega)$, use

$$K=-\log\Delta, \qquad \sigma_s(A)=\Delta^{is}A\Delta^{-is} =e^{-isK}Ae^{isK}. \tag{1}$$

Modular time $s$ is dimensionless. If

$$F_{A,B}(s)=\langle\Omega,A\sigma_s(B)\Omega\rangle,$$

then, for the usual analytic elements, $F_{A,B}$ extends to the lower unit strip and has boundary value

$$F_{A,B}(s-i) =\langle\Omega,\sigma_s(B)A\Omega\rangle. \tag{2}$$

This is the KMS boundary relation in the sign convention (1); compare Takesaki 1970, Chapter III, §§1–2. Replacing $\Delta^{is}$ by $\Delta^{-is}$ sends $s\mapsto-s$ and moves the convenient analytic strip to the upper half-plane.

Fix also the geometric action convention

$$U(g)\mathcal O(x)U(g)^*=\mathcal O(gx) \tag{3}$$

for a scalar field. If a charge $Q$ generates the positive-parameter geometric vector $\xi$, then $e^{-isQ}$ in (1) follows the vector $-\xi$. This observation fixes every sign in the common-operator test below.

Standard representation versus a physical complement

Let $\rho>0$ be a faithful density matrix on a finite-dimensional Hilbert space and represent $B(\mathcal H)$ on the Hilbert–Schmidt space. Then

$$\Delta(X)=\rho X\rho^{-1}, \qquad K_{\rm std}(X)=(-\log\rho)X-X(-\log\rho). \tag{4}$$

It is convenient to write (4) as

$$K_{\rm std}=K_L-K_R, \qquad K_\rho=-\log\rho, \tag{5}$$

where $K_L$ and $K_R$ mean left and right multiplication by $K_\rho$. The right action is the commutant action in the standard representation; it is not automatically a physical complementary subsystem.

Only when a specified bipartite pure state supplies a Schmidt representation may one identify the right action with a physical factor and write

$$K_{\rm std}=K_A\otimes\mathbf1-\mathbf1\otimes K_{A^c}. \tag{6}$$

Zero Schmidt coefficients require restriction to the joint support before inverse powers in $\Delta$ are used. In a sharp continuum QFT, local algebras are typically type III: there is no trace-class regional $\rho_A$ and generally no tensor-factor complement. The exact object is $\Delta_{\mathcal A,\Omega}$, while formulas such as $K_A=-\log\rho_A+c$ are regulated representatives or quadratic-form expressions.

An additive $c\mathbf1$ changes neither a regulated adjoint flow nor density-matrix normalization after $c$ is fixed. It is not an ambiguity of the normalized standard modular operator itself: $\Delta\Omega=\Omega$ fixes its scale.

Atlas of exact and conditional examples

Example	Exact object and action in (1)	Essential hypotheses	Boundary of the claim
Finite faithful algebra	Standard-representation $\Delta$ acts by left $\rho$ and right $\rho^{-1}$; $K$ is (4)	Faithful normal state and declared standard representation	A physical complement requires extra bipartite data; restrict zero modes to support
Vacuum right wedge	$\Delta^{is}$ implements a Lorentz boost of rapidity $-2\pi s$	The Wightman-field, vacuum, covariance, locality, and spectrum hypotheses of Bisognano–Wichmann	Arbitrary regions, excited states, and Lorentz-breaking regulators are outside the theorem
Vacuum CFT ball or interval	$\Delta^{is}$ implements the conformal transformation preserving the causal diamond	CFT vacuum and round ball; interval is the two-dimensional specialization	Generic shapes or nonconformal theories do not inherit the local weight
Thermal full algebra	$\sigma_s=\tau_{-\beta s}$	Faithful Gibbs density matrix in a type-I regulator, or a $\beta$-KMS GNS representation for the algebraic identity	Generic subregion flow is not thermal time
Half-sided null translation	Modular dilation rescales a reconstructed positive translation by $e^{-2\pi s}$	Common cyclic/separating vector, oriented half-sided inclusion, and positive-generator theorem	Arbitrary null cuts require separate Markov, locality, or deformation results

The table compares automorphisms and hypotheses rather than bare symbols. The detailed formulas follow.

Vacuum wedge

Let

$$W_R=\{(t,x^1,\mathbf x_\perp):x^1>|t|\}$$

and define the right-wedge boost by

$$\begin{aligned} t'&=t\cosh\eta+x^1\sinh\eta,\\ x^{1\prime}&=x^1\cosh\eta+t\sinh\eta. \end{aligned} \tag{7}$$

For the field-theoretic settings covered by the Bisognano–Wichmann results,

$$\Delta_{W_R}^{is}=U(\Lambda_R(-2\pi s)). \tag{8}$$

The scalar-field theorem is Bisognano–Wichmann 1975, Theorem 1 and equations (3.1)–(3.4); the companion treatment of general Wightman fields and duality is Bisognano–Wichmann 1976, §§2–4.

On the $t=0$ slice, a regulated one-sided representative is

$$K_R =2\pi\int_{x^1>0}d^{d-1}x\;x^1T_{00}(0,\mathbf x)+c. \tag{9}$$

The exact continuum statement is the modular-unitary identity (8); the stress-tensor expression (9) is understood through the boost charge, as a regulated operator or quadratic form on its natural energy domain. The full standard generator is the right representative minus its left-wedge counterpart.

Vacuum ball and interval in a CFT

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

$$K_B =2\pi\int_{r<R}d^{d-1}x\; \frac{R^2-r^2}{2R}T_{00}(0,\mathbf x)+c, \tag{10}$$

and the conformal Killing vector generated by $K_B$ is

$$\xi_B =\frac{\pi}{R} \left[ (R^2-t^2-r^2)\partial_t -2t x^i\partial_i \right]. \tag{11}$$

The conformal map, flow, and local generator are Casini–Huerta–Myers 2011, §2.1, equations (15), (20), and (22). The density-matrix notation used there presupposes a regulator; the induced continuum algebra automorphism is the exact content.

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

$$K_{(u,v)} =2\pi\int_u^v dx\; \frac{(x-u)(v-x)}{v-u}T_{00}(0,x)+c. \tag{12}$$

The weights in (10) and (12) are positive inside the region and vanish linearly at the entangling boundary. Neither formula extends unchanged to a generic shape or nonconformal theory.

Thermal equilibrium

In a faithful type-I regulator with a trace-class Gibbs density matrix

$$\rho_\beta=Z^{-1}e^{-\beta H},$$

direct functional calculus gives

$$\sigma_s(A) =\rho_\beta^{is}A\rho_\beta^{-is} =e^{-i\beta sH}Ae^{i\beta sH}. \tag{13}$$

If physical Heisenberg evolution is

$$\tau_t(A)=e^{itH}Ae^{-itH},$$

then

$$\sigma_s=\tau_{-\beta s}. \tag{14}$$

More generally, a $\beta$-KMS state need not be represented by any trace-class density matrix. In its GNS standard representation, the surviving algebraic identity is still $\sigma_s=\tau_{-\beta s}$ in the frozen sign convention; it follows from modular KMS uniqueness Takesaki 1970, Chapter III, §2, not from the first equality in (13). The equilibrium conditions and representation-level qualifications are developed in Haag–Hugenholtz–Winnink 1967, §§2–3. Equation (14) is not a generic interpretation of subregion modular time.

Half-sided null translations

In the negative half-sided convention used on the preceding page, the completed structure theorem gives

$$T(a)=e^{iaP}, \qquad P\ge0,$$

and

$$\Delta_{\mathcal M}^{is}T(a) \Delta_{\mathcal M}^{-is} =T(e^{-2\pi s}a). \tag{15}$$

The generator and domain statement are Araki–Zsidó 2005, Theorem 2.1, equations (2.20)–(2.25). Reversing the half side reverses the associated orientation; it does not permit retaining an inconsistent mixture of signs.

Thermofield double

For

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

and the left algebra in the Schmidt support,

$$\Delta =e^{-\beta(H_L-H_R)}, \qquad K_{\rm std}=\beta(H_L-H_R). \tag{16}$$

The generator annihilates the TFD vector. Choosing the right algebra instead inverts $\Delta$ and reverses $K_{\rm std}$. Equation (16) is a special physical realization of the abstract left-minus-right action (5), not a model for every type-III local algebra.

First application: one operator, three exact flows

Choose a scalar primary local field $\mathcal O$, smear it with a smooth compactly supported test function strictly inside the relevant region, and work as quadratic forms on the common invariant Wightman domain $\mathcal D$. At $t=0$, the divergence of (11) vanishes, so the conformal flow has no additional scaling term there. Equations (3), (8), (11), and (14) give

$$\left.\frac{d}{ds} \sigma_s^{\rm wedge}(\mathcal O(0,\mathbf x)) \right|_{s=0} =-2\pi x^1\partial_t\mathcal O(0,\mathbf x), \tag{17}$$ $$\left.\frac{d}{ds} \sigma_s^{\rm ball}(\mathcal O(0,\mathbf x)) \right|_{s=0} =-\frac{\pi}{R}(R^2-r^2) \partial_t\mathcal O(0,\mathbf x), \tag{18}$$

and

$$\left.\frac{d}{ds} \sigma_s^{\rm thermal}(\mathcal O(0,\mathbf x)) \right|_{s=0} =-\beta\partial_t\mathcal O(0,\mathbf x). \tag{19}$$

These distributional formulas mean their smeared matrix elements between vectors in $\mathcal D$. They are exact infinitesimal actions under their respective theorem hypotheses: there is no approximation parameter or numerical uncertainty. Their coefficients vanish or scale in physically diagnostic ways—the wedge coefficient vanishes at $x^1=0$, the ball coefficient at $r=R$, while the thermal coefficient is spatially constant.

Adversarial convention test

Deliberately use $e^{+isK_R}$ for the wedge while still labeling the result as the reference flow (1). Equation (17) becomes

$$\left.\frac{d}{ds}\widetilde\sigma_s(\mathcal O) \right|_{s=0} =+2\pi x^1\partial_t\mathcal O,$$

the opposite orbit. Define the relative coefficient residual

$$r_c=\frac{|c_{\rm test}-c_{\rm ref}|}{|c_{\rm ref}|}. \tag{20}$$

For the sign error, $c_{\rm test}=+2\pi$ and $c_{\rm ref}=-2\pi$, so $r_c=2$. If the sign is correct but $2\pi$ is omitted, $c_{\rm test}=-1$ and

$$r_c=1-\frac1{2\pi}\approx0.840845. \tag{21}$$

Both failures are visible before any model-dependent matrix element is evaluated. The KMS check is independent: replacing $\Delta^{is}$ by $\Delta^{-is}$ while retaining the lower-strip boundary (2) puts the analytic continuation on the wrong side. The strongest surviving statement after either error is merely that the proposed generator traces the same geometric orbits up to a reversed direction or rescaled parameter; it is not the frozen modular flow.

Conversion checklist

For every imported formula, record:

1. whether the adjoint action uses $\Delta^{is}$ or $\Delta^{-is}$;
2. whether $K$ means $-\log\Delta$, $\log\Delta$, or a regulated $-\log\rho_A$;
3. whether $2\pi$ is in the generator, rapidity, or modular-time parameter;
4. which algebra, commutant, or physical complement is used;
5. whether an additive constant belongs only to a regulated representative;
6. the support and domain on which inverse powers and commutators are defined;
7. which theorem supplies geometric action and where its hypotheses stop.

A correct conversion preserves the full adjoint action, the KMS boundary, and the geometric orbit—not merely the symbol $K$.

Common pitfalls

Writing $K_A-K_{A^c}$ without a tensor product. In standard form the intrinsic statement is left action minus commutant action. A physical complement is extra structure.

Treating a type-III algebra as a density-matrix factor. Use $\Delta$ intrinsically; label one-sided stress-tensor formulas as regulated representatives or forms.

Comparing signs without comparing exponentials. Translate the entire adjoint action and then test (2), (17), or both.

Exercises

1. Compute the finite standard modular spectrum

Let $\rho=\sum_i p_i|i\rangle\langle i|$ with all $p_i>0$. Compute $\Delta$ and $K_{\rm std}$ on $E_{ij}=|i\rangle\langle j|$. What changes if $p_j=0$?

Solution

Left and right multiplication give

$$\Delta(E_{ij}) =\rho E_{ij}\rho^{-1} =\frac{p_i}{p_j}E_{ij}.$$

Therefore

$$K_{\rm std}(E_{ij}) =-\log\!\left(\frac{p_i}{p_j}\right)E_{ij} =( -\log p_i+\log p_j)E_{ij}.$$

This equals $(K_L-K_R)E_{ij}$. If $p_j=0$, $\rho^{-1}$ does not exist on the full space and the displayed eigenvalue is divergent. The modular construction must be restricted to the faithful support; one may not silently set the inverse to zero.

2. Check the interval weight and its normalization

Set $u=-R$ and $v=R$ in (12). Recover the ball weight in one spatial dimension, locate its maximum, and calculate its slope at both endpoints.

Solution

Substitution gives

$$2\pi\frac{(x+R)(R-x)}{2R} =\frac{\pi}{R}(R^2-x^2),$$

which is the coefficient multiplying $T_{00}$ in (10) for $d=2$. It is maximal at $x=0$, where the value is $\pi R$. Its derivative is

$$-\frac{2\pi x}{R}.$$

The slope is $+2\pi$ at $x=-R$ and $-2\pi$ at $x=R$, so the weight vanishes linearly with the universal local Rindler normalization at each endpoint.

3. Expose a mixed wedge convention

An author writes $\widehat\sigma_s(A)=e^{+isK_R}Ae^{-isK_R}$ but quotes the geometric orbit $\Lambda_R(-2\pi s)$. Using (3), determine the actual orbit, the coefficient residual (20), and the KMS strip appropriate to $\widehat\sigma$.

Solution

Because $e^{+isK_R}$ is the inverse of the unitary in (1), its actual orbit is

$$\Lambda_R(+2\pi s),$$

not $\Lambda_R(-2\pi s)$. At $t=0$ the derivative coefficient is $+2\pi$ rather than $-2\pi$, so

$$r_c=\frac{|2\pi-(-2\pi)|}{2\pi}=2.$$

The inverse flow is $\widehat\sigma_s=\sigma_{-s}$. Accordingly the convenient KMS analytic continuation uses the upper unit strip; retaining the lower-strip equation (2) would be a second, independent convention error.

References

• Araki, Huzihiro, and László Zsidó. “Extension of the Structure Theorem of Borchers and Its Application to Half-Sided Modular Inclusions.” Reviews in Mathematical Physics 17 (2005): 491–543. DOI. Open HTML, Theorem 2.1.
• 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.
• Bisognano, Joseph J., and Eyvind H. Wichmann. “On the Duality Condition for Quantum Fields.” Journal of Mathematical Physics 17 (1976): 303–321. DOI.
• Casini, Horacio, Marina Huerta, and Robert C. Myers. “Towards a Derivation of Holographic Entanglement Entropy.” Journal of High Energy Physics 2011, no. 5 (2011): 036. DOI. Open HTML, §2.1.
• Haag, Rudolf, Nico M. Hugenholtz, and Marius Winnink. “On the Equilibrium States in Quantum Statistical Mechanics.” Communications in Mathematical Physics 5 (1967): 215–236. DOI.
• Takesaki, Masamichi. Tomita’s Theory of Modular Hilbert Algebras and Its Applications. Lecture Notes in Mathematics 128. Berlin: Springer, 1970. DOI.