S-matrix and T-matrix for Multi-channel Scattering

Hello! Let's continue our journey into scattering theory.

In the last lesson, we explored the optical theorem, which revealed a deep connection between the total scattering cross-section and the forward scattering amplitude. We saw that this relationship is a direct consequence of probability conservation, which is formally expressed by the unitarity of the S-matrix.

So far, we have mostly considered elastic scattering, where the set of particles in the final state is identical to the initial state (e.g., $A+B\to A+B$). The S-matrix we've used was for a single "channel". Today, we will generalize this powerful formalism to handle much richer physical situations, such as inelastic scattering ($A+B\to A^{\ast }+B$), particle reactions ($A+B\to C+D$), or decays. This is the domain of multi-channel scattering.

Our goal is to formulate the S-matrix and the related T-matrix for systems where the final state can consist of a different set of particles or states than the initial one. This extension is crucial for applying scattering theory to nearly all realistic problems in particle physics, nuclear physics, and condensed matter physics.

1. From Single-Channel to Multi-Channel Scattering

Let's start by recalling the concrete picture of the S-matrix in one dimension that we've seen before. For a potential on a line, we have incoming and outgoing waves on both sides.

We can describe the relationship between the amplitudes of the "in-going" waves ($A,D$) and the "out-going" waves ($B,C$) with a 2x2 matrix, the S-matrix:

Here, $S_{11}$ represents reflection, $S_{21}$ represents transmission from left to right, and so on. The unitarity of this matrix ensures that probability is conserved—whatever flux goes in must come out.

This works perfectly for elastic scattering. But what if the interaction can change the state of the particle or create new ones?

• Inelastic Scattering: An incoming electron scatters off an atom, leaving the atom in an excited state.
• Reaction: A deuteron ($d$) and a tritium nucleus ($t$) fuse to produce a helium-4 nucleus ($\alpha$) and a neutron ($n$): $d+t\to \alpha +n$.
• Condensed Matter Analogy: A neutron scatters from a crystal lattice. It can scatter elastically, or it can scatter inelastically by creating a phonon (a quantum of lattice vibration).

These are all multi-channel processes. Each distinct set of particles and their internal states in the asymptotic (t -> ±∞) region is called a channel. The process $d+t\to \alpha +n$ involves a transition from the "d+t" channel to the "α+n" channel.

To handle this, we need to upgrade our S-matrix from a simple 2x2 matrix to a more general operator that connects states across different channel subspaces.

2. The Multi-Channel S-Matrix Formalism

The key challenge is that the "free" Hamiltonian describing the asymptotic states is different for each channel. In the $d+t$ channel, the Hamiltonian must describe a free deuteron and a free triton. In the $\alpha +n$ channel, it must describe a free alpha particle and a free neutron. The interaction that binds the deuteron is part of the channel Hamiltonian for the initial state but not for the final state!

The following reading develops the formalism needed to accommodate this.

Let's break down the essential ideas from the text:

• Channel Hamiltonian ($H_{\alpha }$): For each channel $\alpha$, we define a channel Hamiltonian $H_{\alpha }$. This Hamiltonian describes the evolution of the particles within that channel when they are far apart and not interacting with each other. For example, in a process like $A+(BC)\to A+B+C$, the initial channel Hamiltonian includes the potential $V_{BC}$ that forms the bound state $(BC)$. The final channel Hamiltonian is just the kinetic energy of the three free particles.
• Channel Subspace (${\mathcal{S}}_{\alpha }$): This is the Hilbert space containing the asymptotic states for channel $\alpha$.
• Channel Møller Operators (${\Omega }_{\alpha }^{\pm }$): Just as in the single-channel case, we define Møller operators for each channel. The operator ${\Omega }_{\alpha }^{+}$ maps an "in" state from the channel subspace ${\mathcal{S}}_{\alpha }$ (evolving under $H_{\alpha }$) to the corresponding true interacting state in the full Hilbert space (evolving under the full Hamiltonian $H$).
• The S-Matrix Element ($S_{\beta \alpha }$): The amplitude for a transition from an initial state $|{\varphi }_i,\alpha ⟩$ in channel $\alpha$ to a final state $|{\varphi }_f,\beta ⟩$ in channel $\beta$ is given by the matrix element of the S-operator: $$S_{\beta \alpha }=⟨{\varphi }_f,\beta |{\Omega }_{\beta }^{(-)†}{\Omega }_{\alpha }^{(+)}|{\varphi }_i,\alpha ⟩$$

The full S-matrix is a large matrix whose blocks $S_{\beta \alpha }$ connect the different channel subspaces. The unitarity of the full S-matrix, $S^{†}S=1$, now expresses the conservation of probability across all possible outcomes. For an initial state in channel $\alpha$, the sum of probabilities to end up in any of the possible final channels $\beta$ must be 1:

3. The Transition Matrix (T-matrix)

The S-matrix contains parts that correspond to "no scattering" or scattering within the same channel. It's often more convenient to isolate the part that is purely due to the interaction that causes transitions. This is the T-matrix.

The S-matrix is typically written in terms of the T-matrix as:

Let's dissect this important equation:

• The first term, ${\delta }_{\beta \alpha }{\delta }_{fi}$, is the "no interaction" part. It's only non-zero if the channel ($\alpha =\beta$) and the quantum numbers ($i=f$) remain unchanged. It represents the particles passing through without interacting.
• The second term contains all the interesting physics.
  • The Dirac delta function, $\delta (E_f-E_i)$, enforces energy conservation for the transition.
  • The matrix element $⟨f,\beta |T|i,\alpha ⟩$, or simply $T_{\beta \alpha }$, is the transition amplitude. It is directly related to the physically measurable cross-section for the process $\alpha \to \beta$.

To see how the T-operator is formally defined and connected to the S-matrix, you can study the following material.

The T-operator is defined in terms of the full Green's operator $G(z)=(z-H{)}^{-1}$ and the interaction potential $V$. A key result is the Lippmann-Schwinger equation for T:

where $G_0(z)=(z-H_0{)}^{-1}$ is the free Green's operator. This equation is fundamental because it provides a way to calculate the T-matrix, often iteratively through an expansion known as the Born series. Your familiarity with quantum field theory will make you appreciate that calculating S-matrix (and thus T-matrix) elements, often using Feynman diagrams, is the central task in predicting experimental outcomes.

4. The Analytical Power of the S-Matrix

The S-matrix is more than just a tool for calculating cross-sections. Its mathematical properties, when viewed as a function of energy, encode profound information about the system's structure, including its bound states.

The following video shows a remarkable result: the energies of a potential's bound states (which are negative-energy, non-scattering states) correspond to poles of the S-matrix when it is analytically continued into the complex energy plane.

This connection between scattering properties (like phase shifts and cross sections at $E>0$) and the bound-state spectrum ($E<0$) is a testament to the power of the S-matrix formalism. It unifies different physical regimes into a single analytic structure. This idea is central to many areas of modern physics, including the bootstrap program in quantum field theory.

Conclusion

In this lesson, we have extended the S-matrix formalism to handle multi-channel scattering, which is essential for describing most real-world interactions.

Key Takeaways:

• A channel represents a specific collection of particles and their internal states in the asymptotic t -> ±∞ region.
• The multi-channel S-matrix, $S_{\beta \alpha }$, gives the transition amplitude from an initial channel $\alpha$ to a final channel $\beta$. Its unitarity ensures probability is conserved across all possible outcomes.
• The T-matrix, defined by $S_{\beta \alpha }={\delta }_{\beta \alpha }{\delta }_{fi}-2\pi i\delta (E_f-E_i)T_{\beta \alpha }$, isolates the dynamics of the interaction. Its matrix elements are directly related to experimental cross-sections.
• The S-matrix, as an analytic function of energy, is a profoundly powerful tool, containing information not only about scattering processes but also about the bound-state spectrum of the system.

Preview of the Next Lesson:
We will continue our exploration of scattering theory by looking at how to approximate the T-matrix in the important case of low-energy scattering. We will introduce key parameters like the scattering length and effective range, which characterize the interaction potential without needing to know its detailed shape. This will be a practical application of the formalism we've built today.