Chapter 3 – Elementary particles

Quantum Field Theory

What we have been describing so far is wave mechanics:

• Particles are fundamental entities that have positions with probabilities given by the wave function
• Schrödinger equation tells us how wave functions change over time

It is common in physics to have two apparently completely different paths to the same solution. For example, the effects of gravity can be calculated by treating gravity as a force or by treating it as a potential energy field. Both approaches give the same result so use the one that is easiest for the problem being solved.

What is a field?

A field in mathematics is structure that has a value at every point in space. Here are three common examples of fields:

Each of these fields has a value at every point but the types of values depend on the field. The Atmospheric pressure field just has a single value. the magnetic and wind fields have a direction and strength (i.e. a vector) at every point

What is Quantum Field theory (QFT)?

Quantum Field Theory is an alternative to wave mechanics where particles are replaced by a field for each particle type:

• Photons are ripples in the electromagnetic
• Electrons are ripples in the electron field
• etc.

Each particle type has its own field and the type of value given by each field affects how the particle operates.

How do waves represent particles?

The hurricane in the wind field above is a good example. We are used to talking about hurricanes having a position, speed, size and energy and yet there is no physical thing that represents the hurricane. Similarly we can give a wave in the ocean attributes expected for physical objects.

The electromagnetic field gives two values for every point – the electric and magnetic influence at the point. A photon is a local oscillating ripple of these values that is moving

Other fields have different values at each point and local oscillations produce each field’s particles

How do fields interact?

Visualize two separate fields occupying the same space:

• ripples in the surface of a trampoline
• ripples in a layer of water on top of the trampoline

The ripples in one field affect ripples in the other – this is fields interacting. Interactions between fields appear as interactions between particles

Quantum Field Theory is…

QFT is a precise mathematical model that explains particle behavior

QFT adds support for Special Relativity

QFT gives accurate calculations for:

• How many virtual particles will be created and their effect
• Exact properties of all particles in the standard model
• Which particles will be created in collisions
• Decay rates in radiation

Most important: QFT is the mathematical language behind the standard model