Calculating Scattering Amplitudes in Quantum Field Theory

Scattering amplitudes are fundamental to quantum field theory (QFT), describing the probability of particles interacting and transforming into other particles. Calculating these amplitudes allows physicists to predict the outcomes of particle collisions, a cornerstone of experimental particle physics.

The Role of Feynman Diagrams

Feynman diagrams provide a powerful visual and calculational tool for determining scattering amplitudes. Each diagram represents a specific process or contribution to the overall amplitude, built from fundamental interactions.

Feynman diagrams are graphical representations of particle interactions.

These diagrams, invented by Richard Feynman, consist of lines representing particles and vertices representing interactions. They offer an intuitive way to visualize complex quantum processes.

Each line in a Feynman diagram corresponds to a particle propagating between interactions. Internal lines represent virtual particles that mediate forces, while external lines represent incoming and outgoing real particles. Vertices, where lines meet, signify the fundamental interaction points governed by the theory's Lagrangian. The complexity of a scattering process is often related to the number of diagrams contributing to it.

The S-Matrix and its Elements

The S-matrix (Scattering matrix) encapsulates all possible outcomes of a scattering process. Its elements, the scattering amplitudes, quantify the transition probability from an initial state to a final state.

The S-matrix relates initial and final states of a quantum system.

The S-matrix, denoted by S, is a unitary operator that transforms the initial state of a system at time $t = -\infty$ to its final state at time $t = +\infty$ . Its elements, $S_{fi} = \langle f | S | i \rangle$ , are the scattering amplitudes.

The probability of transitioning from an initial state $|i\rangle$ to a final state $|f\rangle$ is given by $|S_{fi}|^2$ . In QFT, the S-matrix is often expressed as $S = T \exp(-i \int H_I(t) dt)$ , where $H_I$ is the interaction Hamiltonian and $T$ denotes time ordering. Perturbative expansions of this expression lead to the Feynman diagrams.

Calculating Amplitudes: The Perturbative Approach

In most practical scenarios, QFT calculations are performed using perturbation theory. This involves expanding the scattering amplitude in a power series of a small coupling constant.

What is the primary method used to calculate scattering amplitudes in most QFT scenarios?

Perturbation theory.

Each term in the perturbative expansion corresponds to a specific Feynman diagram. The amplitude for a given process is the sum of the amplitudes associated with all relevant diagrams up to a certain order in the coupling constant.

Feynman Rules and Calculation Steps

To translate Feynman diagrams into mathematical expressions, we use Feynman rules. These rules assign specific mathematical factors (propagators, vertices, external leg factors) to each element of a diagram.

Feynman rules convert diagrams into calculable mathematical expressions.

Each component of a Feynman diagram (propagators for internal lines, vertex factors for interactions, and external leg factors for incoming/outgoing particles) is assigned a specific mathematical term according to the Feynman rules of the particular QFT.

For example, in Quantum Electrodynamics (QED), the propagator for an electron is proportional to $1/(p^2 - m^2)$ , where $p$ is the momentum and $m$ is the mass. The vertex factor for electron-photon interaction is proportional to the electric charge $e$ . After assigning these factors to all elements of a diagram, one integrates over all undetermined internal momenta (loop momenta) to obtain the amplitude for that diagram.

A typical Feynman diagram for electron-electron scattering (Møller scattering) in QED involves two incoming electrons, two outgoing electrons, and a photon exchanged between them. The amplitude calculation involves assigning propagators for the electron lines, a vertex factor at each electron-photon interaction point, and a photon propagator for the exchanged photon. The calculation requires integrating over the momentum of the exchanged photon.

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Loop Integrals and Renormalization

Calculations involving internal loops in Feynman diagrams often lead to divergent integrals. Renormalization is a crucial procedure to handle these divergences and extract physically meaningful finite results.

Renormalization is the process of absorbing infinities arising from loop diagrams into redefinitions of physical parameters like mass and charge.

The process involves introducing a cutoff or a regularization scheme (e.g., dimensional regularization) to make the integrals finite, then absorbing the divergent parts into redefinitions of the fundamental parameters of the theory. This ensures that the calculated scattering amplitudes are independent of the arbitrary cutoff and depend only on experimentally measured quantities.

Beyond Perturbation Theory

While perturbation theory is widely used, it is an approximation. For strongly coupled theories or phenomena where the coupling constant is not small, non-perturbative methods are necessary. These include lattice gauge theory, Dyson-Schwinger equations, and other advanced techniques.

Learning Resources

Feynman Diagrams - Wikipedia(wikipedia)

Provides a comprehensive overview of Feynman diagrams, their history, construction, and applications in quantum field theory.

Introduction to Quantum Field Theory - Cambridge University Press(documentation)

A foundational textbook that covers the principles of QFT, including the calculation of scattering amplitudes and the use of Feynman diagrams.

Quantum Field Theory Lecture Notes - David Tong(documentation)

A widely respected set of lecture notes that delves deeply into QFT, with detailed explanations of scattering amplitudes and their calculation.

Introduction to Quantum Field Theory - MIT OpenCourseware(video)

Video lectures from MIT covering the core concepts of QFT, including the calculation of scattering amplitudes and Feynman rules.

QFT for the Gifted Amateur - Cambridge University Press(documentation)

An accessible yet rigorous introduction to QFT, explaining complex topics like scattering amplitudes in a more digestible manner.

Feynman Rules for QED - Physics Stack Exchange(blog)

A community discussion and explanation of the specific Feynman rules used in Quantum Electrodynamics for calculating amplitudes.

Renormalization - Wikipedia(wikipedia)

Explains the concept of renormalization, its necessity in QFT, and how it's used to handle divergences in calculations.

Introduction to Quantum Field Theory - Lecture 10: S-matrix and Feynman Diagrams - YouTube(video)

A focused video lecture explaining the S-matrix and its connection to Feynman diagrams for calculating scattering amplitudes.

Quantum Field Theory and the Standard Model - Cambridge University Press(documentation)

A comprehensive text that covers QFT in the context of the Standard Model, including detailed methods for calculating scattering amplitudes.

The S-Matrix Approach to Quantum Field Theory - arXiv(paper)

A research paper discussing the S-matrix approach in QFT, providing a more theoretical perspective on scattering amplitude calculations.