Meta Theory of the Zeckendorf-Hilbert Universe

11.3 First Law of Entanglement: Modular Hamiltonian and Energy-Momentum Tensor

In Section 11.2, we defined generalized entropy $S_{\text{gen}}$ as the sum of geometric area term and matter entanglement entropy term. To construct the dynamical equations of gravity, we need to know how these entropy terms evolve when the system state undergoes small changes. In classical thermodynamics, energy change and entropy change are related through the first law $dE=TdS$. In quantum information theory, there exists a strictly corresponding law—the First Law of Entanglement.

This section will prove that for any small perturbation of a quantum state, the change in entanglement entropy precisely equals the change in expectation value of a specific operator—the Modular Hamiltonian. More crucially, for the vacuum state in a small causal diamond, this abstract information operator directly corresponds to the physical energy-momentum tensor flux. This establishes the most solid mathematical bridge between “information (entropy)” and “matter (energy)”.

11.3.1 Algebraic Definition of Modular Hamiltonian

Consider a density matrix $\rho$ in Hilbert space. Since $\rho$ is a positive definite Hermitian operator (assuming full rank), we can write it in exponential form.

Definition 11.3.1 (Modular Hamiltonian)

For any density matrix $\rho$, its modular Hamiltonian $K_{\rho }$ is defined as:

$$K_{\rho }\equiv -\ln \rho$$

This makes $\rho =e^{-K_{\rho }}$ (usually normalized such that $\text{Tr}(e^{-K_{\rho }})=1$, or the normalization factor is absorbed into the constant term of $K$, i.e., $\rho =e^{-K}/Z$).

Physical Significance:

Although $K_{\rho }$ is called a “Hamiltonian,” it is usually not the physical Hamiltonian $H$ that controls the system’s time evolution. It is an operator intrinsically defined by the system state $\rho$, describing the “energy” weight of that state in the sense of information geometry. Only in thermal equilibrium states (Gibbs state $\rho =e^{-\beta H}/Z$) does $K_{\rho }$ become proportional to the physical Hamiltonian ($K_{\rho }=\beta H+\ln Z$).

11.3.2 Rigorous Derivation of the First Law of Entanglement

Now consider a system state deviating slightly from a reference state $\sigma$ (e.g., vacuum state) to $\rho =\sigma +\delta \rho$. We need to calculate the change $\delta S$ in entanglement entropy $S(\rho )=-\text{Tr}(\rho \ln \rho )$.

Theorem 11.3.2 (First Law of Entanglement)

Let $\sigma$ be an arbitrary reference state, and $\rho =\sigma +\delta \rho$ be its perturbed state (satisfying $\text{Tr}(\delta \rho )=0$ to maintain normalization). To first order in $\delta \rho$, the change in von Neumann entropy $\delta S$ equals the change in expectation value of the modular Hamiltonian $K_{\sigma }$:

$$\delta S=\delta ⟨K_{\sigma }⟩\equiv \text{Tr}(K_{\sigma }\delta \rho )$$

Proof:

The relative entropy (Relative Entropy) $S(\rho \Vert \sigma )$ is defined as:

$$S(\rho \Vert \sigma )=\text{Tr}(\rho \ln \rho )-\text{Tr}(\rho \ln \sigma )=-S(\rho )+\text{Tr}(\rho K_{\sigma })$$

i.e., $S(\rho \Vert \sigma )=⟨K_{\sigma }{⟩}_{\rho }-S(\rho )$.

For small perturbations $\rho =\sigma +ϵ\Delta \rho$, relative entropy has second-order minimality (i.e., $S(\sigma +ϵ\Delta \rho \Vert \sigma )\sim \mathcal{O}({ϵ}^2)$), because $\sigma$ makes $S(\cdot \Vert \sigma )$ vanish to first order in variation (similar to free energy taking minimum at equilibrium).

Therefore, to first order:

$$0=\delta ⟨K_{\sigma }⟩-\delta S⟹\delta S=\delta ⟨K_{\sigma }⟩$$

$\square$

Physical Interpretation:

This formula is the quantum generalization of the thermodynamic first law $dS=dE/T$. Here, the modular Hamiltonian $K_{\sigma }$ plays the role of “energy/temperature.” It tells us that to change the entanglement entropy of a quantum state, we must inject “modular energy” in the conjugate direction of the modular Hamiltonian.

11.3.3 Geometrization: Bisognano-Wichmann Theorem

For arbitrary quantum systems, the modular Hamiltonian $K$ is often non-local and complex. However, in the context of quantum field theory and small causal diamonds, $K$ has remarkable geometric simplicity.

Consider the reduced density matrix ${\sigma }_{\Sigma }$ of the Minkowski vacuum state $|\Omega ⟩$ restricted to a small causal diamond (or Rindler wedge) $\Sigma$. According to the Bisognano-Wichmann theorem, the flow generated by the modular Hamiltonian $K_{\Sigma }$ is not only an algebraic automorphism, but also a conformal Killing flow in spacetime geometry.

Theorem 11.3.3 (Geometric Form of Modular Hamiltonian)

For a small causal diamond $\Sigma$, its vacuum modular Hamiltonian $K_{vac}$ is given by the integral of the energy-momentum tensor $T_{\mu \nu }$ along the conformal Killing vector ${\zeta }^{\mu }$:

$$K_{vac}=\frac{2\pi }{\hslash }{\int }_{\Sigma }{T}_{\mu \nu }{\zeta }^{\mu }d{\Sigma }^{\nu }+\text{const}$$

where ${\zeta }^{\mu }$ is a vector field that keeps the diamond boundary $\partial \Sigma$ invariant. In the inertial frame near the diamond center, ${\zeta }^{\mu }$ approximates the Lorentz boost generator, with its modulus proportional to the distance from the center.

Physical Corollary:

Combining the first law of entanglement $\delta S=\delta ⟨K⟩$ with the geometric form, we obtain:

$$\delta S_{\text{mat}}=\frac{2\pi }{\hslash }{\int }_{\Sigma }\delta ⟨T_{\mu \nu }⟩{\zeta }^{\mu }d{\Sigma }^{\nu }$$

This shows that the change in matter entanglement entropy $\delta S_{\text{mat}}$ directly corresponds to the energy flux (Energy Flux) passing through the causal diamond.

The coefficient $2\pi /\hslash$ actually contains information about the Unruh temperature ($T_U=\hslash a/2\pi$), making the above equation dimensionally consistent with $dS=dE/T$.

11.3.4 Generalized Entropy Balance and Preview of Gravitational Field Equations

At this point, we have all the components to derive the gravitational equations:

1. Geometric Side: According to Section 11.1, spacetime curvature causes diamond area deficit $\delta A\propto -G_{00}$.

2. Matter Side: According to this section, matter energy flux causes entanglement entropy increase $\delta S_{\text{mat}}\propto T_{00}$.

If we assume that spacetime follows the generalized entropy balance principle (i.e., total entropy $S_{\text{gen}}=A/4G+S_{\text{mat}}$ remains extremal or balanced under vacuum perturbations), then the decrease in geometric entropy must be compensated by the increase in matter entropy:

$$\delta S_{\text{grav}}+\delta S_{\text{mat}}=0$$

Substituting the respective expressions, we will see that there must be a linear relationship between $G_{00}$ and $T_{00}$. This is what Chapter 12 will rigorously prove.

Summary

This section established the first law of entanglement $\delta S=\delta ⟨K⟩$ and used the Bisognano-Wichmann theorem to identify the modular Hamiltonian as the integral of the energy-momentum tensor. This step is crucial—it “translates” abstract quantum information (entropy) into concrete physical entities (energy), allowing Einstein’s equations to emerge from purely information-theoretic principles.

In the next section 11.4, we will introduce the Raychaudhuri equation describing geometric focusing effects, which is the final geometric step connecting geometric changes (area changes) with energy conditions (QNEC).