1.2 Forces

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In this first module, we are introducing the objects studied in particle physics,

namely matter, forces and space-time.

In this video, we will take a quick tour of elementary forces,

and their action at the subatomic level.

After watching this video, you should be able to name these forces and

their associated charges, to know the particles which transmit these forces,

and know the difference between real and virtual particles.

Here is a table comparing elementary interactions.

Note the vast differences between their respective strengths and ranges.

The strong force acts between quarks and particles containing quarks.

It has a large strength but a very short range.

It is transmitted by gluons.

The electromagnetic force concerns all particles carrying an electric charge, and

it has a modest strength but an infinite range, and it is transmitted by photons.

The weak force concerns all matter particles.

It has a low strength at long distances but

becomes comparable to the electromagnetic force at short distances.

It is transmitted by the W and Z bosons.

The gravitational force is weaker than these three but acts on all particles.

Because of the large mass of astronomic objects,

the gravitational force dominates the evolution of the Universe at cosmological scales,

but is however, negligible at the subatomic scales.

We don't know how this force acts at these scales and

we don't know if it is quantized as the other three forces.

Note also that the strength of the forces depends on distance or

equivalently on momentum transfer.

We will discuss this fact for each forces separately because the distance

laws for these forces are quite different.

So forces are transmitted by exchange of bosons of spin 1.

The photon for electromagnetic interactions, the W± and

the Z bosons for the weak forces and gluons for the strong force.

These bosons are collectively called gauge bosons.

Particles can emit or absorb such a boson if they carry the required charge.

To emit a photon, an electric charge is required.

But the concept of charge is not limited to electric charge.

A particle must have a non-zero weak isospin to emit a W or a Z boson.

And a color charge to emit a gluon.

The Higgs boson is the only entry in this table, which has a spin 0.

Its role is to keep particles from moving at the speed of light, and

this applies to all particles, even to the Higgs boson itself.

The only exceptions are the photons, which always move at the speed of light,

and which are however confined to bound states

between quarks as we will discuss in module 5.

So, how does the transmission of forces via the exchange of particles work?

To understand this,

we need to recall how forces act in classical and quantum physics.

How does classical physics deal with the action of forces?

Particles are described as mass points.

Their position x and momentum p can be known with certainty.

Their motion is a smooth curve, in space a function of time,

which we call a trajectory.

It is determined by Newton's law, once we know the vector

sum of all forces acting, and the initial conditions on the position and velocity.

The action of forces is continuous, thus the trajectory is a smooth curve.

The number of particles is conserved,

mass points cannot disappear or appear.

Field and potential are notions which are subordinate to the central ones of force and energy.

We give here the example for the electric force and

energy which are related to the potential as shown in the last equation.

This description is not wrong.

After all, it allows you to drive your car.

It's just that its validity is limited to non-relativistic velocities and

distances, much larger than the size of an atom.

To talk about subatomic systems at high energies,

we need to go beyond classical physics.

Quantum mechanics radically changes the approach.

Particles are fields described by a probability amplitude psi, which is

a function of the position x and the time t, which is also called a wave function.

The square of this amplitude rho, which must be always positive definite

gives the probability density to find the particle at the time t in the position x.

This density is a probability per unit volume.

The trajectory does not exist at small scales.

Multiple measurements of the position of a particle do not form a smooth curve.

Probability amplitude and probability itself

evolve starting from initial state, described it by the Schrödinger equation.

Particles interact with the potential, the action the potential is continuous.

However, the origin of the potential remains unexplained.

Even though the action the potential is continuous it can be implemented

as a sequence of point-like interactions in a pertubative approach.

The number of particles is conserved, since probability amplitudes

follow a continuity equation as shown at the bottom of the page.

This law relates the local probability density rho to the flux density j.

If the probability density diminishes at a certain place,

there must be a divergent flux at the same place.

So the flux j is thus a flux of probability density.

The validity of Quantum Mechanics is limited to non relatively velocity,

since the Schrödinger equation is not covariant,

its forms depends on the reference frame.

To overcome this limitation, we have to use relativistic field theory.

Here, we are going just to introduce its language, its concepts and

its results, but we will not use it in a formal way.

The evolution of particles is described by a relativistic equation of motion,

the Klein-Gordon equation,

which is covariant because it contains only scalars under Lorentz transformations.

The number of particles is no longer conserved about

the electromagnetic current density is.

This current is analogous to the probability current, but

is proportional to the charge e of the particle.

This conservation means that it is the electrical charge,

which is locally conserved, and not the number of particles.

This allows to describe the creation of charged particles in pairs of

particle-antiparticle.

The potential no longer comes out of nowhere,

but is generated by a second current density j_2

according to Maxwell's laws.

The four-potential A^µ is generated by the current density j^µ

by means of the propagator 1/q^2.

This quantity describes the probability amplitude for

the exchange of a photon of invariant mass q between the two currents.

Those who follow carefully may have noticed that there are at least three

unfamiliar notions, in the formulae that come with relativistic field theory.

The dimensions of the quantities do not seem to fit.

In the first equation, there is

a relation between a quantity which is an energy, a momentum and

another quantity which is a mass, which have all different

units in the international system which we are familiar with.

Even worse in the second equation, since psi has no dimension,

the first time has dimension 1/s^2,

the second 1/m^2, and the third kg^2.

The solution to this apparent inconsistency is the use of natural units.

Secondly, what does the notation j^µ and p^µ mean?

This is the notation used for four-vectors,

with implicit summation over Greek indices in scalar products.

Finally, the last equation requires the the photon has non-zero mass.

The real photon of course has zero mass.

After all, it moves at the speed of light.

The solution is the notion of your virtual particles,

which is central to the action of forces in field theory.

Virtual particles have all the same properties as the real ones,

except that they can have a different mass,

which can even be negative or imaginary.

We will introduce these three notions in the videos 1.2a, b, and c, one by one.

In the next video, we will explain how the probability for

a reaction between particles to happen is expressed by the cross section.

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Particle Physics: an Introduction

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