Scattering Theory: Mathematical Foundation of S-Matrix

“Nature doesn’t care about intermediate processes, only asymptotic states.” — Werner Heisenberg

🎯 Goals of This Article

In the core ideas section, we intuitively understood scattering theory. This article will delve into its mathematical foundation:

• Rigorous definition of S-matrix
• Asymptotic completeness
• Derivation of Wigner-Smith matrix
• Møller wave operators
• LSZ reduction formula

📐 Mathematical Setup of Scattering Problem

Hilbert Space Decomposition

Consider Hamiltonian:

$$H=H_0+V$$

where:

• $H_0$: Free Hamiltonian (known eigenstates)
• $V$: Interaction potential (scattering potential)

Define three Hilbert spaces:

1. $\mathcal{H}$: Hilbert space of complete system
2. ${\mathcal{H}}_{\text{in}}$: Past asymptotic state space
3. ${\mathcal{H}}_{\text{out}}$: Future asymptotic state space

graph LR
    I["Past<br/>t → -∞"] --> |"Evolution U(t)"| M["Scattering Region<br/>Interaction"]
    M --> |"Evolution U(t)"| O["Future<br/>t → +∞"]

    I -.-> |"Asymptotically Free"| F1["Free State ψ_in"]
    O -.-> |"Asymptotically Free"| F2["Free State ψ_out"]

    style I fill:#e1f5ff
    style M fill:#fff4e1
    style O fill:#ffe1e1

Asymptotic Conditions

Assumption: When $|t|\to \infty$, interaction $V$ becomes negligible.

Mathematically, there exists a limit:

$$\underset{t\to \pm \infty }{\lim }\Vert U(t)\psi -U_0(t){\varphi }_{\pm }\Vert =0$$

for some free states ${\varphi }_{\pm }$.

🌊 Møller Wave Operators

Definition

Møller wave operators define mapping between asymptotic states:

$${\Omega }_{\pm }=\underset{t\to \mp \infty }{\lim }{U}^{†}(t)U_0(t)$$

Or equivalently:

$${\Omega }_{\pm }=s\text{-}\underset{t\to \mp \infty }{\lim }{e}^{iHt}{e}^{-i{H}_{0}t}$$

($s$-lim denotes strong limit)

Physical meaning:

• ${\Omega }_{-}$: Maps free states to scattering states ($t\to -\infty$)
• ${\Omega }_{+}$: Maps scattering states to free states ($t\to +\infty$)

graph TB
    F["Free State Space<br/>ℋ₀"] --> |"Ω₋"| S["Scattering State Space<br/>ℋ_scatt"]
    S --> |"Ω₊"| F2["Free State Space<br/>ℋ₀"]

    style F fill:#e1f5ff
    style S fill:#fff4e1
    style F2 fill:#e1ffe1

Properties

1. Partial isometry: ${\Omega }_{\pm }^{†}{\Omega }_{\pm }=\mathbb{I}$ (on appropriate subspace)
2. Intertwining: $H{\Omega }_{\pm }={\Omega }_{\pm }{H}_0$
3. Completeness (asymptotic completeness assumption): ${\Omega }_{\pm }{\Omega }_{\pm }^{†}=P_{\text{sc}}$ (projection onto scattering states)

⚡ Definition of S-Matrix

Defined via Wave Operators

S-matrix (scattering matrix) is defined as:

$$\boxed{S={\Omega }_{+}^{†}{\Omega }_{-}}$$

Physical meaning:

S-matrix connects past and future asymptotic free states:

$$|\text{out}⟩=S|\text{in}⟩$$

Defined via Time Evolution

Equivalently, can be written as:

$$S=\underset{t_{+}\to +\infty ,t_{-}\to -\infty }{\lim }{U}_0^{†}(t_{+})U(t_{+},t_{-})U_0(t_{-})$$

Properties of S-Matrix

1. Unitarity: $$S^{†}S=S{S}^{†}=\mathbb{I}$$ (Probability conservation)

2. Causality: $S$ only connects past with future, doesn’t violate causality

3. Energy conservation: $$[S,H_0]=0$$

4. Lorentz covariance: In relativistic case, $S$ is Lorentz scalar

📊 S-Matrix in Energy Representation

Fourier Transform

In energy representation, $S$ depends on energy $\omega$:

$$S(\omega )=\mathbb{I}-2\pi i\delta (\omega -H_0)T(\omega )$$

where $T(\omega )$ is T-matrix (transition matrix).

Optical Theorem

Optical theorem (consequence of unitarity):

$$\text{Im}{T}_{ii}(\omega )=\pi \sum _f|T_{fi}(\omega ){|}^2$$

Physical meaning:

Total scattering cross-section (imaginary part) = Sum of all outgoing channels (probability conservation)

🕰️ Wigner-Smith Time Delay Matrix

Derivation

Consider average residence time of wave packet in scattering region.

Define Wigner-Smith matrix:

$$\boxed{Q(\omega )=-iS(\omega {)}^{†}\frac{\partial S(\omega )}{\partial \omega }}$$

Physical Interpretation

Theorem (Wigner 1955, Smith 1960):

Eigenvalues ${\tau }_n(\omega )$ of $Q(\omega )$ are time delays of $n$-th channel.

Total time delay:

$${\tau }_W(\omega )=\text{tr}Q(\omega )$$

Eisenbud-Wigner Formula

For single-channel scattering $S(\omega )=e^{2i\delta (\omega )}$:

$${\tau }_W(\omega )=\frac{\partial \delta (\omega )}{\partial \omega }$$

(Derivative of phase shift w.r.t. energy)

Connection to Birman-Kreĭn

Combining with Birman-Kreĭn formula:

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

We get:

$$\frac{1}{2\pi }\text{tr}Q(\omega )=-{\xi }^{\prime }(\omega )={\rho }_{\text{rel}}(\omega )$$

This is the mathematical foundation of unified time scale!

🔬 LSZ Reduction Formula

Scattering in Field Theory

In quantum field theory, LSZ (Lehmann-Symanzik-Zimmermann) reduction formula gives:

$$⟨f|S|i⟩={\text{i}}^{n+m}\int \prod _k\frac{d^4{x}_k}{\sqrt{Z}}{e}^{-i{p}_k\cdot x_k}({\square }_k+m^2)⟨\Omega |T\{\varphi (x_1)\cdots \varphi (x_{n+m})\}|\Omega ⟩$$

Physical meaning:

S-matrix element = Amputated asymptotic particle legs × Time-ordered correlation function

Feynman Rules

LSZ formula is the foundation for deriving Feynman rules:

1. External lines: Plane wave factor $e^{ip\cdot x}$
2. Internal lines: Propagator ${\Delta }_F(x-y)$
3. Vertices: Interaction $-i\lambda$
4. Integration: $\int d^{4}x$

🌐 Multi-Channel Scattering

Channel Decomposition

For $N$ channels, $S(\omega )$ is $N\times N$ unitary matrix:

$$S(\omega )=\left(\begin{array}{cccc}{S}_{11}& S_{12}& \cdots & S_{1N}\\ S_{21}& S_{22}& \cdots & S_{2N}\\ ⋮& ⋮& \ddots & ⋮\\ S_{N1}& S_{N2}& \cdots & S_{NN}\end{array}\right)$$

Spectrum of Wigner-Smith Matrix

$Q(\omega )$ is $N\times N$ Hermitian matrix with $N$ real eigenvalues:

$${\tau }_1(\omega ),{\tau }_2(\omega ),\dots ,{\tau }_N(\omega )$$

These are time delays of $N$ orthogonal channels.

🔗 Applications in GLS Theory

1. Ontological Foundation

In GLS’s matrix universe theory:

Universe is modeled as Huge family of S-matrices $\mathbb{S}(\omega )$

All physics is considered to emerge from S-matrix data.

2. Time Scale

Unified time scale is defined by $Q(\omega )$:

$$\tau (\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )$$

3. Causal Structure

Causality of S-matrix is considered to ensure time arrow is consistent with causal arrow.

📝 Key Formulas Summary

🎓 Further Reading

• Classic textbook: J.R. Taylor, Scattering Theory (Wiley, 1972)
• Original paper: E.P. Wigner, “Lower limit for the energy derivative of the scattering phase shift” (Phys. Rev. 98, 145, 1955)
• GLS application: 04-scattering-is-evolution_en.md
• Next: 04-modular-theory_en.md - Modular Theory

🤔 Exercises

1. Conceptual Understanding:

   • Why do Møller wave operators need strong limit?
   • How does unitarity of S-matrix guarantee probability conservation?
   • Why is time delay the derivative of phase shift?

2. Calculation Exercises:

   • For $S=e^{2i\delta }$, calculate $Q$
   • Verify $S^{†}S=\mathbb{I}$ for $2\times 2$ unitary matrix
   • Prove optical theorem

3. Physical Applications:

   • Scattering interpretation of Shapiro gravitational time delay
   • Time delay in resonant scattering
   • Levinson’s theorem and number of bound states

4. Advanced Thinking:

   • How to handle scattering of long-range potential (Coulomb potential)?
   • What’s different in relativistic scattering theory?
   • Can S-matrix have complex eigenvalues?

Next Step: After mastering scattering theory, we will learn Modular Theory—how quantum states define their own “time flow”!