In looking for symmetries, Einstein was following in the footsteps of James Clerk Maxwell, a physicist who died the same year Einstein was born, in Purely for mathematical balance, Maxwell added to equations describing electromagnetism an extra term that related electrical currents to a resulting magnetic field.
In doing so, he unified all electric and magnetic phenomena then known, as well as the laws of optics. The symmetries that give rise to the Higgs boson are different. They originate in the probabilistic world of quantum physics — 'internal symmetries' reflect the robustness of universal laws to changes in the identity of some elementary particles. Just as a particle's position may be uncertain according to quantum rules, so may its identity. Some types of particles are thus interchangeable within the equations.
Electrons and neutrinos can be swapped without altering the laws of nature. This symmetry is central to the 'electroweak theory', developed in the s by physicists Steven Weinberg, Sheldon Glashow and Abdus Salam. The theory unifies electromagnetism with the weak nuclear force, which is responsible for some radioactive decays such as a neutron to a proton and underlies nuclear reactions in stars. The Higgs boson has a role in breaking the symmetry of weak interactions. The strong nuclear force, which acts among quarks to make up protons and neutrons, and binds protons and neutrons to form atomic nuclei, is also subject to symmetries.
The laws of physics are blind, for instance, to exchanges of up quarks and down quarks protons are made of two up quarks and one down quark. The standard model of particle physics combines the electroweak theory with quantum chromodynamics, the theory of strong interactions within the nucleus. The model's great achievement is that it unifies three of the four known forces through which particles interact gravity stubbornly continues to resist unification.
Internal symmetries are local, or independent of the locations of particles. Quarks or electrons and neutrinos at different points in space and time may be exchanged without consequence. Fields, with their associated gauge bosons, mediate these remote interactions. The electromagnetic interaction is carried by the photon. Gluons shepherd the strong force between quarks. The symmetries associated with a given fundamental force act on all associated elementary particles, not just the force mediators.
The symmetry requirement dictates which particles and interactions are necessary for a given theory. Yet the cosmos does not always manifest perfect symmetry.
The equations describing the electroweak interaction, for example, are symmetrical. They do not change when a photon is swapped with a W or Z particle. But the solutions to these equations — the particles themselves — are not identical. The photon is massless, and the W and Z particles are about times heavier than a proton. The symmetry of the governing equations is somehow lost, or broken, in physical reality. The concept of spontaneous symmetry-breaking allows physicists to preserve a symmetric theory while confronting puzzling observations.
In the s, the Jewish—Russian physicist Lev Landau realized that phase transitions are accompanied by a loss of symmetry. Take magnetic materials. Its internal magnetic fields, pointing in random directions when hot, collectively settle on one orientation. The symmetry is broken. The equations describing the magnetic field within the iron are symmetric — they have no preferred orientation.
It is the physical state of the iron that changes. Higgs and his colleagues realized in that symmetry-breaking could be applied to particle physics. They proposed that a fraction of a second after the Big Bang, as the Universe expanded and cooled, it went through a dramatic phase transition see 'Fundamental forces'.
The internal symmetry of the weak interactions, which held true at very high energies, broke when the Universe's energy dropped below some threshold. The mechanism by which it did so the Higgs mechanism involves a quantum field the Higgs field , which has a non-zero value associated with every point in space. The Higgs particle is a ripple, a parcel of energy, in the Higgs field.
However, this idea of an 'invariant' or 'conserved quantity' - something which remains the same no matter how you look at it - is fundamental to symmetry.
We can tell that an object is symmetrical if it looks the same when we change our viewpoint: if you rotate your head by 90 degrees when looking at a square then it looks the same. This is caused by one of the symmetries of the square.
Similarly, if we're looking at a beam of light, and we decide to 'change our viewpoint' by moving in some direction and at some speed, then Einstein says that the speed of light must always stay the same.
This must then be caused by some kind of underlying symmetry. The type of symmetry which causes this is called a 'Lorentz Symmetry'. The exact details of this are horribly complex, but we can still appreciate Einstein's genius without them. Whereas many lesser minds would have seen the constancy of the speed of light as an error to be ironed out, Einstein's mind took it as an undisputable, fundamental fact of nature, as the equations which produced it were too beautiful not to be true.
He then had to rethink our understanding of space and time to incorporate it. The theories of relativity which sprung from this realisation explained how and why gravity works, as well as quantifying it more accurately than any previous theories. On a day to day basis, this is fundamental to the idea of GPS, as it allows us to measure time differences much more accurately. But to reduce such a ground-breaking theory to a relatively trivial application simply does not do it justice.
Einstein's way of thinking marked a change in the entire philosophy of Physics. Whereas previously, physicists would observe data and make theories to explain it, Einstein championed the formulation of theories on the basis of pure thought, with symmetry as a guiding light. These theories of Relativity work well on large scales - galaxies moving through the universe, planets orbiting the sun or even just humans standing on Earth.
If we zoom in further though, we enter a whole new world governed by yet more strange and counterintuitive laws. This is the world of quantum mechanics, which describes the fundamental building blocks of the universe. In Einstein's day, we knew about atoms, the nucleus at the centre of each atom and the electrons surrounding it. The protons and neutrons which make up the nucleus were discovered in the 20s and 30s respectively, but that was by no means the end of the story.
As technology developed after WWII, scientists built increasingly powerful particle accelerators, smashing things together at ever increasing speeds largely out of curiosity. As they did so, they produced more particles, which looked kind of like protons and neutrons.
More and more of these particles were being churned out, sometimes at a rate of one per week, each one being added to the fast-growing 'particle zoo'. To use a technical term, this created a mess: scientists believed that each particle found must also be a new fundamental one to sit alongside the proton and the neutron et al. Such disorder grated horribly - physicists wanted desperately to arrange and categorise their discoveries into a physics equivalent of the periodic table of chemistry.
For all the disorder, though, there were some hints at patterns, with different particles often sharing some properties but not others. What caused these seeming coincidences remained a mystery until , when two separate groups of physicists both postulated the existence of 'quarks': the building blocks of particles like protons and neutrons.
Once again, the link to symmetry is not explicitly obvious. This time, physicists had really exploited the idea of symmetry groups, like those which Galois had introduced. They studied the patterns amongst the 'zoo' of particles, linking these with certain symmetry groups as they went. This revealed the underlying structure of the particles, which physicists could then define in terms of fundamental particles and certain laws which determine how these particles interact.
The result of this is the Standard Model, which has very complex symmetries formed of a number of different groups combined. When this theory was proposed, we hadn't yet discovered all of the particles that it predicted would exist. Over the past 40 years, then, this has provided an excellent test of the Standard Model, checking to see that each and every predicted particle can be observed in nature, and that they all have the correct properties.
Fortunately, there were no contradictions. The development of Relativity and the formation of the Standard Model epitomise the philosophical effect that symmetry has had on physics. Einstein spotted a 'conserved quantity' in the speed of light and decided not to dismiss it, deducing the underlying symmetry of space and time as a result. The Standard Model on the other hand, was born from a search for symmetry - having already observed its power scientists hoped that it would restore some kind of order to their world.
Two different approaches at either end of the symmetrical revolution, but both fundamentally symmetric in nature.
If our physical theories are so symmetrical, though, then why when we look out of the window are we not living in an entirely symmetrical world? The answer to that question is simple - randomness. Ever since the introduction of quantum theories, the universe has had a decidedly unpredictable side to it.
Particles' properties are determined by probabilistic processes rather than definite, deterministic ones, spontaneously becoming certain when we choose to observe them. This causes what are incredibly symmetric physical laws to produce remarkably asymmetric objects. For example, consider a pencil standing on its pointy end on a piece of paper. If you rotate the piece of paper, the pencil still looks the same. Similarly, there's no 'bias' as to which direction the pencil wants to fall in ignoring any kinds of wind movement etc, the pencil would fall due to quantum fluctuations in the particles which form it, which are entirely random in nature.
As a physicist would say, the system has 'no preferred direction'. However after the pencil has inevitably fallen over, this symmetry breaks - if you then rotate the paper, the pencil changes how it looks, pointing in a certain 'preferred' direction which changes as the paper rotates. Here we have a symmetrical system producing asymmetrical results, but what would happen if we repeated this over and over again? As the pencil's direction of fall is random, it would end up falling in just about every direction possible, all of the way round the circle.
If you then take an average, the different preferred directions created by each fall all cancel out: each time the pencil falls in one direction, it will also fall in the exact opposite direction at some point in the future.
On average, then, we reclaim the original symmetry of the system, with no preferred direction at all. In a similar way, if you could take all of the Oak trees in existence and in some way 'average' them, then you would end up with something which looked very symmetrical indeed. Whilst this isn't something we can actually do, it reassures us that we haven't been barking up the wrong tree all along in our pursuit of symmetry. Students begin to use symmetry with commutativity and associativity in arithmetic, making more use of it in Euclidean geometry and plane geometry, and may eventually see it in terms of transformation groups.
Nevertheless, it is natural to want to teach these concepts in their full value from the very beginning. This paper will describe how I have been introducing students in a general education geometry course to the concept of symmetry in a way that I feel gives them a comprehensive understanding of the mathematical approach to symmetry. Symmetry is found everywhere in nature and is also one of the most prevalent themes in art, architecture, and design — in cultures all over the world and throughout human history.
Symmetry is certainly one of the most powerful and pervasive concepts in mathematics. In the Elements, Euclid exploited symmetry from the very first proposition to make his proofs clear and straightforward. Recognizing the symmetry that exists among the roots of an equation, Galois was able to solve a centuries-old problem.
The tool that he developed to understand symmetry, namely group theory, has been used by mathematicians ever since to define, study, and even create symmetry. Students are fascinated by concrete examples of symmetry in nature and in art. The study of symmetry can be as elementary or as advanced as one wishes; for example, one can simply locate the symmetries of designs and patterns, or one use symmetry groups as a comprehensible way to introduce students to the abstract approach of modern mathematics.
Furthermore, the ideas used by mathematicians in studying symmetry are not unique to mathematics and can be found in other areas of human thought.
By looking at symmetry in a broader context, students can see the interconnectedness of mathematics with other branches of knowledge. For these reasons, many mathematicians today feel that the mathematical study of symmetry is worthwhile for general education students to explore.
The central idea in the mathematical study of symmetry is a symmetry transformation, which we can view as an isomorphism that has some invariants.
For example, a symmetry transformation of a design in the plane is an isometry that leaves a certain set of points fixed as a set. I would like students to realize that this concept of symmetry transformation, as abstract as it may appear, can be connected to ideas that may seem more central to a view of life as a whole; for this, I introduce a verse from the Bhagavad-Gita.
In the Bhagavad-Gita, Lord Krishna lays out the complete knowledge of life to his pupil Arjuna, just as a great battle is about to begin. This work has long been appreciated for the great wisdom that is expounded in just a few short chapters. He who in action sees inaction and in inaction sees action is wise among men.
He is united, he has accomplished all action. How is this related to symmetry? A geometric figure that we wish to study is usually given as a set of points existing in some ambient space. For example, a tiling pattern may be given as a collection of line segments in the plane. Thus, in inaction, we see action.
But a symmetry transformation is not just any action; it must leave the pattern as a set of points invariant. Thus, what is important to us is that in this action the transformation , we are able to see inaction the invariance of the set of points making up the pattern.
This is the seed of all that I want students to know about symmetry: action and inaction, a transformation and its invariants, what changes and what stays the same.
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