Feynman diagram

In theoretical physics, a Feynman diagram is a simple picture that helps scientists understand how tiny particles, called subatomic particles, behave and interact with each other. These diagrams were created by an American scientist named Richard Feynman in 1948.

Instead of using very complex math, Feynman diagrams let scientists see these math problems as pictures. This makes hard ideas much easier to understand. Since the middle of the 20th century, these diagrams have become an important tool in physics.

Feynman diagrams are mostly used in a part of physics called quantum field theory, but they can also help in other areas, like solid-state theory. They made it possible to solve problems that would have been very difficult or even impossible to think about without them. For example, they helped win a Nobel Prize in Physics in 2004 and were important in discovering the Higgs particle.

In these diagrams, particles that are the opposite of normal particles, called antiparticles, are shown as moving backward in time. This idea came from another scientist named Ernst Stueckelberg.

Motivation and history

Feynman diagrams are simple drawings that help scientists understand how tiny particles, like those inside atoms, interact with each other. They were created by a scientist named Richard Feynman in 1948. These diagrams make complicated math easier to handle by turning it into pictures.

In particle physics, these diagrams show different ways particles can bounce off each other. They help scientists calculate the chances of different outcomes when particles collide. Even though the math can sometimes give impossible results, scientists have ways to fix these problems so the answers match what they see in experiments.

Representation of physical reality

In particle physics, Feynman diagrams are a helpful way to show how tiny particles interact. They were created by a scientist named Richard Feynman and are used to understand complex math that describes these interactions.

These diagrams are important because they connect theory with real experiments. Even though they use special math, they can still help explain many different kinds of particle behavior.

Particle-path interpretation

A Feynman diagram is a picture that helps us understand how tiny particles interact with each other. In these diagrams, particles are shown as lines that can be straight or wiggly, and they may have arrows depending on the type of particle. When lines meet at a point, this is called a vertex, and it shows where particles interact in different ways, like giving off or taking in other particles, changing direction, or turning into different types of particles.

There are three kinds of lines in these diagrams: internal lines that connect vertices, incoming lines that start from the past and lead to a vertex, and outgoing lines that start from a vertex and go toward the future. Usually, the bottom of the diagram shows the past and the top shows the future, but sometimes the past is on the left and the future is on the right. These diagrams help us see all the possible ways particles can interact, even if those paths seem unusual, like going backward in time. They are not the same as pictures taken in real experiments, but they give us a way to understand the many different ways particles can behave when they meet.

Description

A Feynman diagram is a picture that helps scientists understand how tiny particles change and interact with each other. It shows how particles move from a starting point to an ending point.

For example, when an electron meets a positron, they can create two particles called photons. In these pictures, the starting particles are shown on the left, and the ending particles are on the right. Different types of particles are drawn as different kinds of lines.

QED looks at two main types of particles: matter particles like electrons or positrons, and particles that carry forces. In Feynman diagrams:

• An electron at the start is a solid line with an arrow pointing toward a meeting point.
• An electron at the end is a solid line with an arrow pointing away from the meeting point.
• A positron at the start is a solid line with an arrow pointing away from the meeting point.
• A positron at the end is a solid line with an arrow pointing toward the meeting point.
• A particle that carries a force is shown as a wavy line.

Each meeting point in the diagram has three lines connected to it. These lines can be different types, showing how the particles change during their interaction.

The meeting of an electron and a positron to create two photons is an example. In the beginning, there is one electron and one positron. In the end, there are two photons.

Canonical quantization formulation

A Feynman diagram is a simple drawing that helps us understand how tiny particles, like those in atoms, interact with each other. These diagrams were created by a scientist named Richard Feynman.

When we want to calculate how likely it is for particles to interact, we need to use complex math. Feynman diagrams make this easier by turning parts of the math into pictures. Each line and point in the picture stands for a different part of the calculation.

The diagrams follow special rules, called "Feynman rules," which depend on the type of particles involved. For example, one rule says that a wiggly line stands for a particle called a photon, while a straight line stands for an electron. These rules help scientists predict how particles will behave when they meet.

Path integral formulation

In a path integral, the field Lagrangian, integrated over all possible field histories, defines the probability amplitude to go from one field configuration to another. For this to work, the field theory must have a well-defined ground state, and the integral must be performed a little bit rotated into imaginary time, called a Wick rotation. The path integral formalism is completely equivalent to the canonical operator formalism.

Scalar field Lagrangian

A simple example is the free relativistic scalar field in d dimensions. The probability amplitude for a process is given by an integral, where A and B are space-like hypersurfaces that define the boundary conditions. The collection of all the field values on the starting hypersurface give the field's initial value, and the field values at each point of the final hypersurface define the final field value.

The path integral gives the expectation value of operators between the initial and final state. In the limit that A and B recede to the infinite past and the infinite future, the only contribution that matters is from the ground state.

On a lattice

On a lattice, the field can be expanded in Fourier modes. The action needs to be discretized, and the discretization should be thought of as defining what the derivative means.

Monte Carlo

The path integral defines a probabilistic algorithm to generate a Euclidean scalar field configuration. Randomly pick the real and imaginary parts of each Fourier mode at wavenumber k to be a Gaussian random variable with a certain variance. This generates a configuration at random, and the Fourier transform gives the field.

Scalar propagator

Each mode is independently Gaussian distributed. The expectation of field modes is easy to calculate. For two different k-values, the expectation is zero. When the two k-values coincide, the expectation is given by a certain formula.

Equation of motion

The form of the propagator can be more easily found by using the equation of motion for the field. From the Lagrangian, the equation of motion is given, and in an expectation value, this says something about the field.

Wick theorem

Because each field mode is an independent Gaussian, the expectation values for the product of many field modes obey Wick's theorem. This means that it is zero for an odd number of fields, and for an even number of fields, it is equal to a contribution from each pair separately.

Interaction

Interactions are represented by higher order contributions. The simplest interaction is a certain type, with an action that includes a term with four fields. Writing the action in terms of Fourier modes gives a free action and an interaction term.

Feynman diagrams

The expansion of the action in powers of the interaction gives a series of terms with progressively higher number of interaction terms. The contribution from the term with exactly n interaction terms is called nth order.

The nth order terms have internal half-lines and external half-lines. By Wick's theorem, each pair of half-lines must be paired together to make a line, and this line gives a factor that multiplies the contribution.

Loop order

A forest diagram is one where all the internal lines have momentum that is completely determined by the external lines. A tree diagram is a connected forest diagram. A diagram that is not a forest diagram is called a loop diagram.

Symmetry factors

The number of ways to form a given Feynman diagram by joining half-lines is large, and by Wick's theorem, each way of pairing up the half-lines contributes equally. The uncancelled denominator is called the symmetry factor of the diagram.

Connected diagrams: linked-cluster theorem

A Feynman diagram is called connected if all vertices and propagator lines are linked by a sequence of vertices and propagators. Connected Feynman diagrams determine something important.

Vacuum bubbles

An immediate consequence is that all vacuum bubbles, diagrams without external lines, cancel when calculating correlation functions. A correlation function is given by a ratio of path-integrals. The top includes the same contributions of vacuum bubbles as the denominator. Dividing gets rid of the second factor.