Modular Hamiltonians: Definitions and Domains

A modular Hamiltonian is the self-adjoint generator of modular flow for a specified algebra and state. In a finite tensor factor it can be written as $-\log\rho$, but a local continuum QFT algebra generally has no reduced density matrix and no trace. The definition that survives is spectral: construct the modular operator $\Delta$ from the algebra–state pair and set $K=-\log\Delta$ on its natural domain.

Required background. Use bounded and unbounded operators for domains and closure, self-adjointness for unitary generation, standard form for cyclic and separating vectors, relative entropy in QFT for state comparison, and factorization failure to avoid assuming a local density matrix.

Three generators that should not be conflated

Let a regulated bipartite Hilbert space be $\mathcal H_A\otimes\mathcal H_{A^c}$ and let $\rho_A$ be faithful. The one-sided entanglement Hamiltonian is

$$K_A=-\log\rho_A.$$

It is defined only on the support of $\rho_A$ if zero eigenvalues are present. Replacing $K_A$ by $K_A+c\mathbf 1$ changes the normalization of the density matrix but not the adjoint action $e^{isK_A}Ae^{-isK_A}$.

Purify the state as $\lvert\Omega\rangle=\sum_i\sqrt{p_i}\,\lvert i\rangle_A\lvert i\rangle_{A^c}$. The modular operator of the left algebra is

$$\Delta_\Omega=\rho_A\otimes\rho_{A^c}^{-1}, \qquad K_\Omega=-\log\Delta_\Omega=K_A-K_{A^c}.$$

This two-sided generator annihilates the purifying vector and implements the same adjoint flow on left operators as $K_A$. It is the finite-dimensional model of the continuum object. Finally, comparing two states $\psi$ and $\phi$ gives a relative modular operator $\Delta_{\psi\mid\phi}$; its logarithm is not obtained by subtracting two noncommuting one-sided logarithms. That comparison is developed on the relative modular page.

The structural map places Modular Hamiltonians: Definitions and Domains between intrinsic modular data and the additional hypotheses that permit a geometric interpretation.

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.

Spectral definition and natural domain

For a cyclic, separating vector $\Omega$ of a von Neumann algebra $\mathcal A$, Tomita–Takesaki theory supplies a positive self-adjoint operator $\Delta_\Omega$. If $E_\Delta(\lambda)$ is its projection-valued spectral measure, then

$$K_\Omega=-\log\Delta_\Omega =-\int_{(0,\infty)}\log\lambda\,dE_\Delta(\lambda).$$

The corresponding domain is not all of $\mathcal H$:

$$\mathcal D(K_\Omega)= \left\{\xi\in\mathcal H: \int_{(0,\infty)}\lvert\log\lambda\rvert^2 \,d\langle\xi,E_\Delta(\lambda)\xi\rangle<\infty\right\}.$$

The unitary group $\Delta^{is}=e^{-isK_\Omega}$ is bounded for every real $s$ even though $K_\Omega$ is unbounded. Thus modular flow can be perfectly well defined while the formal commutator $[K_\Omega,A]$ is not. A derivative at $s=0$ requires $A$ to preserve an appropriate core and the vector under study to lie in the generator domain.

This is the practical hierarchy:

1. use $\Delta^{is}A\Delta^{-is}$ for exact flow;
2. use spectral integrals for functions of $K$;
3. differentiate only on a stated common invariant domain;
4. treat an integral formula for $K$ as an additional theorem or approximation, not as the definition.

Support, faithfulness, and the continuum limit

In matrices, faithfulness means $\rho>0$. If $p=s(\rho)$ is the support projection, the logarithm is naturally an operator on $p\mathcal H$ and modular statements belong to the reduced algebra $p\mathcal A p$. Assigning $+\infty$ to $-\log0$ is sometimes useful in variational formulas, but it does not create a densely defined generator on the discarded null space.

In continuum QFT, the vacuum is commonly cyclic and separating for a local algebra, yet that algebra is typically type III. There is then no trace-class $\rho_A$ whose logarithm generates the flow; Witten 2018, §§2.5–2.6 explains this continuum distinction. A UV regulator may produce a matrix $\rho_{A,\epsilon}$, but its separate entropy and one-sided generator contain regulator-dependent boundary terms. The algebraic modular automorphism group can have a meaningful continuum limit even when those quantities do not.

The best-known controlled exception is geometric. For a vacuum Rindler wedge, the Bisognano–Wichmann theorem identifies modular flow with boosts. On the $t=0$ half-space $x^1>0$, a conventional one-sided expression is

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

where $c$ enforces normalization in a regulated density-matrix description. The formula is local because a theorem supplies the geometric action; it is not representative of a generic region.

A finite Gaussian check

A faithful fermionic Gaussian state with correlation matrix $C$ on a finite subsystem has

$$\rho_A=Z^{-1}\exp(-c^\dagger h c), \qquad h=\log\!\left[(\mathbf1-C)C^{-1}\right].$$

Diagonalizing $C=U\operatorname{diag}(\nu_a)U^\dagger$ gives single-particle entanglement energies $\epsilon_a=\log[(1-\nu_a)/\nu_a]$. This is a useful regulator test: eigenvalues approaching $0$ or $1$ send $\epsilon_a$ to $\pm\infty$, exposing the support and conditioning problem before one attempts a continuum extrapolation. The discrete levels should not be mistaken for the spectral type of a local type-III modular operator.

Common pitfalls

Writing $K=-\log\rho_A$ before naming a factorization. A spatial region in continuum QFT is naturally assigned an algebra, not automatically a tensor factor. Start from $(\mathcal A,\Omega)$ and use a density matrix only when a regulator or type-I setting has been stated.

Dropping domains because the exponential is unitary. Stone’s theorem makes $e^{-isK}$ globally bounded; it does not make $K\xi$ or $[K,A]\xi$ meaningful for every vector. State the domain whenever a generator or derivative appears.

Treating additive constants like support projections. Constants cancel from adjoint flow. Removing a null space changes the representation on which the logarithm exists and cannot be done silently.

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 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

• Witten, Edward. “Notes on Some Entanglement Properties of Quantum Field Theory.” Reviews of Modern Physics 90 (2018): 045003. DOI; arXiv.

Further reading

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.