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 helped solve problems that would have been very difficult 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.

Representation of physical reality

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

These diagrams are important because they connect theory with real experiments. Even though they use special math, they can help explain many 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.

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. They are not the same as pictures taken in real experiments, but they give us a way to understand how 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, added up over all possible field paths, tells us the chance to go from one field setup to another. For this to work, the field theory must have a clear ground state, and the adding up must be done in a special way called a Wick rotation. The path integral way is the same as the usual operator way.

Scalar field Lagrangian

A simple example is the free relativistic scalar field in d sizes. The chance for a process is given by an adding up, where A and B are flat areas that set the limits. The group of all the field numbers on the start area gives the field's start number, and the field numbers at each point of the end area set the end field number.

The path adding up gives the hope value of workers between the start and end state. When A and B go far back and far forward, the only thing that counts is from the ground state.

On a lattice

On a small grid, the field can be split into Fourier modes. The action must be made into small parts, and this should be seen as saying what the slope means.

Monte Carlo

The path adding up sets a chance way to make a European scalar field setup. Pick at random the real and imaginary parts of each Fourier mode at wave number k to be a Gaussian chance number with a certain width. This makes a setup at random, and the Fourier change gives the field.

Scalar propagator

Each mode is separately Gaussian given out. The hope of field modes is easy to work out. For two different k-values, the hope is zero. When the two k-values are the same, the hope is given by a certain rule.

Equation of motion

The way of the hope can be more simply found by using the way of motion for the field. From the Lagrangian, the way of motion is given, and in a hope value, this says something about the field.

Wick theorem

Because each field mode is an alone Gaussian, the hope values for the product of many field modes follow Wick's theorem. This means it is zero for an odd number of fields, and for an even number of fields, it is the same as a giving out from each pair by itself.

Interaction

Mixing up is shown by more order bits. The simplest mixing up is a certain type, with an action that has a part with four fields. Writing the action in terms of Fourier modes gives a free action and a mixing up part.

Feynman diagrams

The spreading out of the action in steps of the mixing up gives a line of parts with more and more mixing up bits. The giving out from the part with just n mixing up bits is called nth order.

The nth order parts have inside half-lines and outside half-lines. By Wick's theorem, each pair of half-lines must be put together to make a line, and this line gives a bit that times the giving out.

Loop order

A forest picture is one where all the inside lines have push that is totally set by the outside lines. A tree picture is a joined forest picture. A picture that is not a forest picture is called a loop picture.

Symmetry factors

The number of ways to make a given Feynman picture by joining half-lines is big, and by Wick's theorem, each way of putting the half-lines together gives out the same. The not-canceled bottom part is called the picture's symmetry factor.

Connected diagrams: linked-cluster theorem

A Feynman picture is called joined if all points and passing lines are linked by a line of points and passings. Joined Feynman pictures decide something important.

Vacuum bubbles

A fast bit is that all empty bubbles, pictures with no outside lines, clear out when working out linking bits. A linking bit is given by a share of path-adding ups. The top has the same bits of empty bubbles as the bottom. Dividing gets rid of the second bit.