Fundamental Forces and Yukawa Theory
Yukawa proposed a model in which particles interact by exchanging virtual quanta (virtual packets of energy or
virtual particles) which mediate the force. This is known as
quantum field theory.

In physics a (force)
field is a phenomenon that has a value at all points in space, even if that value is zero. For
example, space is filled with an electric field, generated by charged particles, though at some positions in space
this field may be essentially zero at a given time. The
vacuum of space, however, is not empty, but instead
consists of a sea of particles (quanta) that phase in and out of existence – being spontaneously and randomly
created and then later absorbed and destroyed by specific particle interactions. These particles are transient
and can not be directly observed and are called virtual particles. These particles give even empty space a
certain energy and mass.

The law of
conservation of energy, states that energy can neither be created nor destroyed. The existence
of virtual particles ‘violates’ this law, however, it does not violate the laws of quantum mechanics. One such law
Heisenberg’s uncertainty principle, which states that both the momentum and position of a particle can
not, in principle, be measured to precision. It places a limit on the accuracy, which has nothing to do with the
limits of accuracy of the instruments used to measure things, but which is a fundamental limit. Particles are so
minute that simply measuring one changes it. To measure any particle we have to interact with it. Think of a
particle of soot under a microscope – a clump of many carbon atoms. This particle is large enough to be seen (it
is not a fundamental or elementary particle like an electron) and we see it by shining light (a beam of photons)
upon it. The photons in this light interact with the carbon atoms and change them slightly, the soot may become
warmer, but unless we use very bright light it will not change much. Similarly, elementary particles must interact
in some way with other particles or a force field in order to be measured. However, such particles are so minute
that measuring them can cause dramatic changes in their properties. What we actually measure is the state of
the particle after the interaction. If we try to pinpoint an electron by measuring its position to very high accuracy,
then it will react by spreading out the range of possible values of its momentum – its velocity becomes more
uncertain. Likewise, measuring the momentum very accurately means that we can no longer be sure where
exactly the electron is – it spreads out, at least until we measure its position more accurately (then its
momentum spreads out).

In essence, the particle is really behaving like a group of interacting waves (a
wave packet). The properties of
the particle are determined by its wave function which is a mathematical equation that is a solution of a quantum
mechanical wave equation (which could be approximated by Schrodinger’s wave equation, or for particles
moving at an appreciable fraction of the speed of light we would use either Dirac’s equation or the Klein-Gordon
equation). The wave function determines the possible values that each measurable property of the particle can
take, such as its momentum, position and energy. We say ‘possible values’ because there is inherent
uncertainty in quantum mechanics and we can never precisely predict the behaviour of a single particle, only
the probabilities or likelihoods of each possible type of behaviour, generating a
spectrum of possible values for
observable (measurable property), such as energy. Another way of looking at it is that for a very large
group or ensemble of particles, initially in the same state for any given observable, such as in the same energy
state, we can state approximately how many will behave in a certain way.

Another uncertainty relation of importance here is the
energy-time uncertainty relation. This tells us that to
measure the energy of a particle to higher and higher precision requires a longer period of time; or to put it
another way, a particle can have a very large uncertainty in its energy if it only exists for a very short period of
time. Virtual particles exist for such a small fraction of time that we can never directly measure their energy (they
are virtual) but their energy (or energy uncertainty) can be very large. This means that a massive or highly
energetic particle can exist for a very short period of time without essentially violating the law of conservation of
energy, since they do not exist for long enough for the violation to be measured! It’s as if they are cheating or
bending the laws of Nature (of course they are not, rather they obey more subtle rules than suggested by crude

This means that a virtual particle can pop into existence, creating energy from nothing, and disappear again
before anyone can notice this violation of energy conservation. However, the existence of such quantum fields
does lead to indirect changes which we can measure. this is why Yukawa’s theory is more than just a theory – it
gives reliable and accurate predictions about the behaviour of particles when they interact, so apparently
particles do interact by exchanging virtual quanta.

This concept may be strange to those used to thinking of subatomic particles as billiard balls – billiard balls
interact by physically colliding with one-another. However, on the atomic scale the atoms in the surfaces of two
colliding balls have electric charges and these charges repel one-another when the balls approach to minute
distances – the electrons in the atoms of one ball repel the electrons in the atoms of the second ball (since
electrons have negative electric charge and like charges repel) so that there is no real physical contact as such.
Indeed there is nothing physical to contact in the intuitive sense – atoms behave like wave packets (vibrating
energy) and the force of repulsion between the approaching electrons is mediated by virtual quanta. In the case
of this electric interaction, the virtual quanta are virtual photons.

Physics currently identifies four fundamental forces:

1. The electromagnetic force (including electric and magnetic interactions).
2. The weak nuclear force (governing certain particle interaction, such as those involving neutrinos).
3. The strong nuclear force (which binds quarks together inside a proton or neutron).
4. Gravity.
The more massive the particle, the higher its minimum energy (what used to be called the rest energy: E = mc^2)
and the shorter the time it can exist for without violating energy conservation (the energy-time uncertainty
principle) and so the shorter the distance it can travel in that time and the shorter the range (the smaller the
value of a, the range parameter).

Different fundamental/elementary forces are mediated by different virtual particles. These particles are all bosons
and are called exchange bosons or
gauge bosons. Bosons are particles with integral spin (spin = 0 or 1, spin 1
in the case of gauge bosons) in contrast to fermions which have half-integer spin (e.g. spin = ½). Electrons are
an example of fermions; whilst photons are an example of bosons with spin = 1. [Recall that spin is the quantum
mechanical equivalent to intrinsic angular momentum, which in our macroscopic world an object has by rotating
about its axis, but in the atomic and subatomic worlds is quantised, with only certain values possible.]

The force exchange or gauge bosons are as follows:
Yukawa originally applied his theory to study the nuclear force, an attractive force between nucleons (protons and
neutrons). Nucleons are composite particles made up principally of quarks (fundamental or elementary particles)
and when they interact with one-another via the strong nuclear force, it is the quarks in the nucleons that interact
and they do so by the strong force, by exchanging gluons. Since quarks also can not exist in isolation, the smallest
strong-force conveying particle that can leave a nucleon is a quark/antiquark pair. Such a pair is a meson and
mesons act as exchange particles in the nuclear force – they carry quarks between the nucleons and then these
quarks interact with quarks in each nucleus via gluon-exchange in the strong interaction. One should not confuse
the strong nuclear force, consisting of meson exchange, which is a composite interaction, with the
fundamental/elementary strong force due to gluon-exchange, though they are aspects of the same strong force.
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