How Do Forces Work?
To understand how the universe works, one must understand how the fundamental particles of matter interact with each other. Thus, one requires theories of the basic forces of nature, and how they affect particles. This blog post explains the concepts behind our best theories for describing forces: gauge theories.
The fundamental forces
Four fundamental forces have been identified and have typically required separate theories for each. Two of these four forces, gravitation, and electromagnetism, have an unlimited range. Consequently, they are known and felt by everyone as they push and pull. The remaining two forces, the weak nuclear and strong nuclear forces have a short range (shorter than the diameter of an atomic nucleus), and so are not felt in everyday life. The strong force is what binds together protons and neutrons in the nucleus of an atom. To be specific, the strong force is binding the so-called quarks that make up these nucleons. On the other hand, the weak force is responsible for the decay of particles. It has often been said that the overarching aim of physics is to construct a single, all-powerful theory that incorporates all the fundamental forces. This is known as the ‘Grand Unified Theory’ (GUT). Such a theory should reveal a deep connection between the forces, whilst allowing for their apparent diversity. This has not yet been discovered, but progress has certainly been made.
Two of the forces, weak nuclear and electromagnetism, can now be described by a single theory, known as the ‘electroweak’ theory. The forces are still distinct but become mathematically intertwined in the theory. However, what may ultimately be more important, is that all four forces are now described by theories that take the same general form. As Gerald ‘t Hooft puts it, “although we don’t have a single key for all the known locks, all the needed keys can be cut from the same blank”. The theories in this single class are known as ‘non-abelian gauge theories with local symmetry’. Now, I understand that’s a pretty scary title to give to something, so let’s break it down. In more simple terms, it means that the theories relate the properties of the forces to symmetries of nature.
Symmetries
Physical theories from as far back as the times of Galileo and Newton have been constructed upon the symmetries in the laws of nature. Most people are typically familiar with spatial and geometric symmetries. A geometric symmetry can be defined as the invariance of a pattern that is observed when some operation or transformation is applied to it. The most common example is a perfect snowflake. In this case, the transformation would be to rotate the snowflake by 60 degrees. If one does this, no change in the pattern will be perceived, so one can say the snowflake is invariant with respect to 60-degree rotations. A square has a 90-degree symmetry, and a circle is said to have continuous symmetry, as no matter what angle you turn it by it will appear the same.
Although the concept of symmetries originated in geometry, it can be applied throughout the universe. For example, the charge symmetry of electromagnetism. Imagine a large number of electrically charged particles are set out in a definite configuration, and all the forces acting between the particles have been measured. Now, reverse the polarity of all the changes, and one will find that the net forces remain unchanged.
Another example is that of isotopic spin, which is a property of protons and neutrons. The basis of the symmetry lies upon the fact that protons and neutrons are extremely similar particles in that they have practically the same mass, and all other properties are identical (except for charge). If the electromagnetic force were to be ‘turned off’, the isotopic spin symmetry, which determines whether a nucleon is a proton or neutron, would be exact. In that case, if all protons and neutrons in the universe were to be interchanged, the strong force (which acts between nucleons) would be unaltered. Exact symmetry with respect to isotopic spin states that no effect of this transformation can be detected.
The symmetries addressed thus far can be characterised as ‘global’ symmetries, which means the transformations happen everywhere at once. This was essential with the proton-neutron example, where the transformation had to occur all throughout the universe at the same time. In addition to these ‘global’ symmetries, there are also ‘local’ symmetries. The difference here is that these symmetries can be decided independently at every point in space and at every point in time. Local symmetries are much harder to construct, making them much more powerful. Global symmetries state that some law of physics does not change when a given transformation is applied everywhere at once. A local symmetry, however, states that some law of physics does not change even when a different transformation occurs at each point in space and time. Gauge theories constructed upon local symmetries have the greatest interest today. In order for a theory to be invariant with respect to a local transformation, a new factor must be introduced: a force.
Fields
Fields are a key component in how particle interactions are described. But what is a field? A field is simply a quantity that is defined at every point in space and time in a given region. A good example is temperature. The region for this may be the top of a hob. The field consists of temperature values for every point on the surface. Given that temperature is simply represented by a magnitude, it is called a scalar quantity, and thus, this field is a scalar field. After a scalar field, you have a vector field, where each point corresponds to a vector, represented by an arrow. A vector has a magnitude, which correlates to the length of the arrow, and a direction, which correlates to two angles (in two-dimensional space only one angle is required). A field is a very convenient concept when examining charged objects and describing how the electromagnetic force operates in the space around it. All charged particles emanate an electromagnetic field and interact with the sum of all the fields around them. In quantum mechanics, the particles themselves can be seen as fields. For example, an electron can be considered as a packet of waves (a quantum mechanical field can be represented as a particle too).
Field quanta
The interaction of two particles through their interacting fields can be summarised by saying that the two particles exchange a third particle, known as the excitation ‘quantum’ of the field (the plural is quanta). For example, when two electrons with electromagnetic fields around them repel each other, they are said to have exchanged a photon (the quantum of the electromagnetic field). The thing about the quantum is that one can’t actually detect it — it is ‘virtual’. After it has been emitted, it must be reabsorbed in a finite period. The larger the energy of these virtual particles, the briefer their existence. The range of the interaction/force is depending on the mass of the exchanged quantum. If it has a larger mass, it requires more energy to exist (E = mc 2), and this energy must be absorbed sooner for it to remain virtual, meaning it will have a shorter range. However, in the unique case where the quantum is massless, the force has an infinite range.
The number of components of a field depends on the number of quantum mechanical states of the quantum, which is dependent on the intrinsic spin angular momentum of this particle. This property can only take discrete values: an integer or a half integer. Like the magnitude, the orientation of the spin is quantised. The number of possible orientations is equal to double the magnitude plus one. For example, an electron has spin ½ meaning it has two possible spin states: it can be parallel or antiparallel to its direction of motion. A spin 1 particle has 3 spin states: parallel, antiparallel or transverse. A spin 0 particle has 1 spin state (all orientations are equivalent, so it has no spin axis). A scalar field (like with the temperature example earlier), has only one component (magnitude) must therefore be represented by a quantum with spin 0 — these are conveniently called scalar particles. A particle with spin 3 spin states (spin 1), would correspond to a field with three components: a vector field, thus making the particle a vector particle. The electromagnetic field is a vector field, meaning its exchange particle (the photon) has spin 1.
Summary
So that is basically a summary of how particles interact with each other everywhere in the universe. Fields are regions that can be finite or infinite in range, and have corresponding quanta which exist for very brief amounts of time. The intrinsic spin of these quantum excitations defines the characteristics/components of the fields. In their short life span, they transfer energy and momentum between particles within their field, giving rise to the appearance of a force. There are four fundamental forces in the universe, and all can be described by so-called ‘gauge theories’ which are based on the invariance of certain laws of nature at every point in space and time, known as symmetries.
Originally published at http://thephysicsfootprint.com on January 27, 2022.
