Meta Theory of the Zeckendorf-Hilbert Universe

6.4 Time Delay Matrix and Trace Formula Applications in Multi-Channel Scattering

In Sections 6.2 and 6.3, we established the time delay theory for single-channel scattering, proving that the phase derivative on the boundary (EWS operator) is strictly equivalent to the probability density integral in the bulk (dwell time). However, the real physical universe—whether quantum dots in condensed matter systems, particle collisions in high-energy physics, or black holes as scatterers—are essentially multi-channel systems. Particles not only experience phase shifts but also undergo mixing and transformation of quantum states.

This section generalizes the time delay theory to general $N$-channel systems, introducing the Wigner-Smith time delay matrix $\mathsf{Q}$. We will derive the famous trace formula, which establishes a bridge between microscopic scattering time and macroscopic thermodynamic density of states (DOS). This connection is the core hub of Volume II, revealing that “time” at the statistical mechanics level is “state counting.”

6.4.1 From Scalar Phase to Matrix Geometry: Construction of the $\mathsf{Q}$ Matrix

Consider a scattering system with $N$ open channels. The scattering matrix $S(E)$ is no longer a complex number $e^{i\varphi }$, but an $N\times N$ unitary matrix, whose elements $S_{nm}(E)$ describe the probability amplitude for incident from channel $m$ and outgoing from channel $n$.

For multi-channel systems, a simple “phase derivative” ${\phi }^{\prime }$ is no longer sufficient to describe time behavior, because the off-diagonal elements of the $S$ matrix contain entanglement and mixing between channels. We need an operator to describe the generalized delay of wave packets in the entire channel space ${\mathbb{C}}^N$.

Definition 6.4.1 (Wigner-Smith Time Delay Matrix)

Generalizing the definition in Section 6.2, the Wigner-Smith time delay matrix $\mathsf{Q}(E)$ is defined as an $N\times N$ Hermitian matrix on channel space:

$$\mathsf{Q}(E)=-i\hslash S(E{)}^{†}\frac{dS(E)}{dE}$$

In component form:

$${\mathsf{Q}}_{nm}=-i\hslash \sum _{k=1}^N{S}_{kn}^{\ast }\frac{d{S}_{km}}{dE}$$

Geometric Meaning:

The $\mathsf{Q}$ matrix describes the connection structure on the scattering manifold. If we view $S(E)$ as a curve in the unitary group $U(N)$, then $\mathsf{Q}(E)$ is precisely the projection of the tangent vector of this curve onto the Lie algebra $\mathfrak{u}(N)$ (specifically, the Hermitian elements in $\mathfrak{u}(N)$). It measures the “rotation speed” of scattering states as they vary with energy.

6.4.2 Eigen-Time Delays

Since $\mathsf{Q}$ is a Hermitian matrix ($\mathsf{Q}={\mathsf{Q}}^{†}$), it can always be diagonalized. This means there exists a special set of incident states that not only maintain orthogonality during scattering but also have definite time delays.

Definition 6.4.2 (Eigenchannels and Eigen-Time)

Let $|{\tau }_a⟩$ be the eigenvector of $\mathsf{Q}$, and ${\tau }_a$ be the corresponding eigenvalue:

$$\mathsf{Q}|{\tau }_a⟩={\tau }_a|{\tau }_a⟩,a=1,\dots ,N$$

The eigenvalues ${\tau }_a$ are called the system’s proper time delays.

Physical Interpretation:

1. Time Spectrum: For a complex quantum system (such as an atomic nucleus or chaotic cavity), the question “how long does it take to pass through it” has no single answer. The system possesses a time spectrum $\{{\tau }_1,\dots ,{\tau }_N\}$.

2. Metastable Lifetime: If the system has resonances, some ${\tau }_a$ become extremely large, corresponding to the lifetime of particles trapped in metastable states.

3. Causality Constraint: Although individual ${\tau }_a$ may be negative under certain extreme interference conditions (Wigner precession), in causal systems, the average dwell time must be positive.

6.4.3 Smith-Kreĭn Trace Formula

The most important property of the $\mathsf{Q}$ matrix lies in its trace. The trace is a basis-independent invariant that reflects the overall properties of the system. We will prove that the trace of $\mathsf{Q}$ directly corresponds to the change in density of states (DOS) inside the system.

Theorem 6.4.3 (Time-Density Relation)

For short-range interaction potentials, the trace of the time delay matrix and the change in local density of states $\Delta \rho (E)$ satisfy the following trace formula:

$$\text{Tr}[\mathsf{Q}(E)]=2\pi \hslash \Delta \rho (E)$$

where $\Delta \rho (E)={\rho }_{int}(E)-{\rho }_{free}(E)$ is the density of states increment caused by the interaction.

Proof:

1. Using Determinant Identity:

   From linear algebra, we know $\text{Tr}(A^{-1}dA)=d(\ln \det A)$. Since $S^{†}=S^{-1}$, we have:

   $$\text{Tr}[\mathsf{Q}]=-i\hslash \text{Tr}\left(S^{-1}\frac{dS}{dE}\right)=-i\hslash \frac{d}{dE}\ln (\det S)$$

2. Introducing Spectral Shift Function:

   According to Birman-Kreĭn theory, the determinant of the scattering matrix is related to the spectral shift function $\xi (E)$:

   $$\det S(E)=e^{-2\pi i\xi (E)}$$

   Substituting into the above:

   $$\text{Tr}[\mathsf{Q}]=-i\hslash \frac{d}{dE}(-2\pi i\xi (E))=-2\pi \hslash {\xi }^{\prime }(E)$$

3. Spectral Shift and Density of States:

   The derivative ${\xi }^{\prime }(E)$ of the spectral shift function is exactly the negative of the change in density of states (note the sign convention, usually defined as $\rho (E)={\xi }^{\prime }(E)$ or $-{\xi }^{\prime }(E)$ depending on the step direction of $\xi$. In Kreĭn’s definition, $\xi$ counts bound states, ${\xi }^{\prime }\sim \rho$). More directly, using the dwell time theorem (Section 6.3):

   $$\text{Tr}[\mathsf{Q}]=\sum _a{\tau }_a={\tau }_{dwell}^{tot}-{\tau }_{free}^{tot}$$

   And the total dwell time ${\tau }_{dwell}^{tot}$ is the integral of probability density over all space. Under energy normalization, the modulus-squared integral of the wave function $\int |{\psi }_E{|}^2$ is exactly the density of states $\rho (E)$ (for discrete spectrum it is $\delta (E-E_n)$, for continuous spectrum it is smooth density).

   Therefore:

   $$\text{Tr}[\mathsf{Q}(E)]=2\pi \hslash \left({\rho }_{int}(E)-{\rho }_{free}(E)\right)$$

$\square$

6.4.4 Physical Applications: From Chaotic Cavities to Holographic Entropy

The trace formula $\text{Tr}[\mathsf{Q}]=2\pi \hslash \Delta \rho$ is the Rosetta Stone connecting dynamics (time) and thermodynamics (density of states/entropy). It has profound applications in multiple frontiers of physics:

1. Mesoscopic Physics and Chaotic Scattering:

   In quantum dots or chaotic microwave cavities, the eigenvalue distribution of the $\mathsf{Q}$ matrix follows random matrix theory (RMT). The trace formula tells us that measuring transport properties (through the $S$ matrix and its energy derivative) can directly probe energy level density fluctuations inside the cavity. This is the main experimental method for verifying quantum chaos.

2. Gravity and Black Hole Entropy:

   In the context of the holographic principle, if we view a black hole as an extremely dense scatterer with extremely high density of states inside $\rho (E)\approx e^{S_{BH}(E)}$.

   According to the trace formula, the total time delay of a black hole for external probe particles will be enormous:

   $${\tau }_{delay}\sim \hslash \rho (E)\sim \hslash e^{S_{BH}}$$

   This explains why physical processes near the black hole horizon appear “frozen” to external observers—because the time delay grows exponentially with density of states (entropy). This is the Heisenberg time, the time scale required for a quantum system to traverse its entire Hilbert space.

3. Prelude to the Unified Time Identity:

   The trace formula in this section is the direct source of this book’s core formula $\kappa (E)={\rho }_{rel}(E)=\frac{1}{2\pi }\text{Tr}\mathsf{Q}$. It establishes that: the rate of time flow $\kappa$ is proportional to the information capacity (density of states) of physical reality. Time slows down (delay) because there is too much information (high density).

Summary

By introducing the $\mathsf{Q}$ matrix and its trace formula, we generalize the definition of time from single-particle trajectories to many-body statistical systems. We find that time is not only the change of phase but also the counting of states. This profound insight will be further elevated in Chapter 7 “Spectrum-Scattering Duality,” where we will see that the entire thermodynamics can be reconstructed as the geometry of scattering time delays.