Symmetry

 Symmetry

What is symmetry in the contexts of physics?

There are four types of symmetry: charge, parity, time and gauge.

Symmetry is easily recognizable everywhere around us. Look at buildings, mathematical formula, even biology, but the concept of symmetry in physics is more precise than that. It is in physics that symmetry has played one of its most important roles by unlocking the secrets of the forces in nature and of the fundamental quantum particles.

Mathematical symmetry has helped scientists to discover the quarks that make up the protons and neutrons in atoms, the gluons that bind them, and of course the Higgs boson which explains how particles get their mass. It has allowed researchers to unify some of the forces in nature for instance uniting electricity and magnetism into electromagnetism and later adding the weak force to make the electroweak interaction.

Symmetry, to a physicist generally means something different than it does for members of the public. It means that an object or a theory does not change when you inflict a transformation on this object’s mathematical model. The simple kinds of transformations we are referring too are rotating or moving it or doing something like that to the equations. The most symmetrical object one could think of would be a sphere, it looks identical no matter what you do to it, however you rotate it in any given direction.

Arvin Ash does a great description of this in his video.

Symmetries are used in the physical world and works to determine the rules of the game. The symmetries of nature determine for us things that remain constant, that can’t be changed. Those are the guideposts in physics, the quantities like energy and momentum. For instance, energy is conserved, we now understand, because there is a symmetry of nature that tells us the laws of physics don’t change over time.

Time Symmetry

With time symmetry things work the same way backwards or forwards in time. Time in most equations is reversible so therefore symmetric. If I run a movie film backwards then the characters in the movie undo what they just did and if you play music in reverse the waveforms are like a mirror image as the forward ones. When it comes to things like objects thrown upwards against the force of gravity they fall back again, short of looking at what threw the object one could not tell the difference between a video of the object’s motion if it was played backwards or forwards.

Light waves demonstrate time symmetry. When an electron moves to a lower band then a packet of energy called a photon is released in the electromagnetic domain. When that atom absorbs the packet of energy via an incoming photon then the packet of energy is absorbed.

Charge Symmetry

Charge symmetry means that there is no difference if all the charges are swapped. Yes, if our universe came with more positrons than electrons and if the nucleus of an atom were negatively charged then there would be no noticeable difference.

Parity Symmetry

Parity symetry is left and right handedness. With quantum particles parity is referred to as spin.

Gauge Symmetry

Gauge symmetry starts simply with a symmetry that most people are familiar with. That is the conservation of electric charge, the fact that whenever a positive charge is produced, there must be a negative charge to balance it if the system is neutral. Electrons have what is called negative charge, but negative charge is just a human term it doesn’t mean anything physically. We might have called electrons the positive charges. So, the sign of charge is an arbitrary thing. We could change the sign of every electric charge in nature and the world would look identical. That’s symmetry

But that’s not gauge symetry.

Again, symmetries imply laws that don’t change—like everything staying the same if every positive charge turned into a negative charge and vice versa. Well, gauge symmetry expresses a deeper, really weird symmetry, in which you can show equivalence between two things—“positive” and “negative,” for instance—by defining what is “positive” and “negative” locally, by setting certain conditions that map the two together. It’s a very subtle idea that has required us to stretch our minds to their limits, but it’s important to know about, because it helps constrain theorists and ensure that our mathematics are consistent and make sense.

So, imagine the universe as a big chessboard. I could change every white square on a chessboard to a black square and every black square to a white square and the game would be the same. That’s the simple kind of symmetry. Now I can turn it into a gauge symmetry by making it much trickier. I can say, “Let me just change locally, whenever I want, a white square to a black square or a black square to a white square. Not everywhere but place to place." Now the chessboard doesn’t look the same at all, so the game can’t be the same unless I also have a rule book—a coordinate system for what happens at every point—containing rules for the pieces of the chessboard to follow to keep the game the same, rules that account for everywhere I have changed the color of a square. That becomes a very weird symmetry.

This sort of symmetry tells you how to go from the conservation of charge to the theory of electromagnetism. It says, “I could change the sign of each electric charge in nature locally. But I have to have a rule book.” What's the rule book? In this case, it’s the electromagnetic field. Even though gauge symmetry is something that most people find obscure, it’s the most visible thing in the world—and if you don’t have it, things fall apart in surprising ways. Whenever you look at a lightbulb, you're able to see light because nature has this weird symmetry.

When did physicists realize how important gauge symmetry was?

All of this was discovered after the fact. Maxwell developed his equations of electromagnetism in the 1800s. But people began to recognize this symmetry once again when it came to Einstein’s general relativity, which was guided by another kind of gauge symmetry. Now this became really interesting because the two most fundamental theories that we knew of in nature at the time—electromagnetism and gravity—are suddenly determined by this weird mathematical symmetry. That leads to the question: If there are other forces in nature, are they determined by this same kind of symmetry?

But if you’d asked these questions in the 1930s through the early ‘50s, you still wouldn’t have heard physicists talking about gauge symmetry as the guiding principle. It was only after the fact they began to realize how useful it was. One of the reasons I wanted to write this book, and the reason it’s called The Greatest Story Ever Told, is if it were easy, it wouldn’t be the greatest story ever told. I wanted to show the long and convoluted path that people took to enlightenment. It’s a tremendous story because it involves great leaps of the imagination, great failures, great blind alleys and red herrings. In any case, the real heart of the idea is to understand that the world we live in and the other forces in nature was a search to ultimately apply this idea of gauge symmetry to these other forces.

How did the idea of “broken symmetry” fit into this search?

Some symmetries exist in the fundamental world, but when we look around we don’t see them. What’s happened is that as an accident of our circumstances, the symmetry isn’t manifested. We say it’s broken. An example is sitting down at a round banquet table and not knowing which water glass is mine. Say there are eight chairs at the table and it looks identical. There’s a symmetry because the glasses to the left and the right look exactly the same. But the first person to sit at the table and pick up a glass of water determines for everyone else which glass they have to pick up. That means they break the symmetry.

This idea of broken symmetry turns out to be the fundamental key that solves a problem with the other forces in nature—the weak and the strong forces—which don’t behave like electromagnetism. The weak force and the strong force both only operate on subatomic scales and they’re very, very different. Physicists eventually realized that in an underlying way those theories might actually look identical to electromagnetism and have a gauge symmetry that determines their nature, but some accident of our circumstances hides that fact from us.

An example is ice on a window on a cold day. Ice crystals are pointing in every different direction on the window. But if you were a civilization that lived on that ice crystal, then the direction along which the ice crystal pointed would be a very special direction. It would be a fundamental aspect of your universe. What would be hidden to you is the fact that the direction of the ice crystal is irrelevant, that the laws of physics are the same independent of this direction.

And this realization helped explain why these other forces behave differently than electromagnetism?

Let’s say we’re living in something very much like an ice crystal, and due to an accident of our circumstances some weird field froze into existence that fills empty space. If that’s the case, the particles that interact with that field act like they’re very heavy even if in reality they are weightless. Then you could understand the weak force because you could say that at a fundamental level, the particles that convey the weak force really are massless just like the photons that convey electromagnetism. In the world in which we live, those particles act like they’re massive, so the force we see is very different than it would be if we didn’t live in this accidental universe. The world of our experience is an illusion in which the forces are very different, but that’s only because there’s this invisible field that exists everywhere that affects the properties of certain particles and not the properties of other particles.

And this invisible field you’re talking about is the Higgs field, which is linked to the famous Higgs boson and which we think explains mass?

This is what physicist Steven Weinberg did, was say, “Maybe it’s not just the mass of these weird particles that convey the weak force, but maybe it’s the mass of all particles we see,” that certain particles interact more strongly with that background field and they behave heavier. Other particles interact less strongly with that background field and behave lighter. Some particles like the photon don’t interact at all. If that’s the case, then the universe of our existence is really a total illusion, because mass itself is not fundamental. If it weren’t for this weird accident that we live in this universe with this background field that’s frozen into existence, there would be no massive objects. There would be no stars, no galaxies, no planets, no people, no Scientific American, no Scientific American reporters. It would be a much less interesting world.

What kind of accident could have created this field?

We think symmetry has been broken in the past as the universe has cooled down in the aftermath of the big bang, causing the Higgs field to freeze in the state it did. It’s like the ice crystals on the window—and when the sun comes out, those crystals melt. Suddenly, the universe that those people thought they lived in disappears. The question then becomes, why did it settle into the configuration it did in the early universe? Why does it have the properties it has? None of that is explained within the theory.

Temperature of the energy soup prior to the big bang would be 10³² K. At this temperature nothing is stable. Quantum particles, the basic building block of our universe, only form for very short periods of time. Entropy is at its smallest value, practically 0. Entropy, conceived in thermodynamics, is the measurement of disorder stated in energy per unit of temperature so the units of entropy are in Joules per Kelvin. So with a very high temperature it is conceivable to have entropy of almost zero.

During inflation the temperature in the universe dropped to about 10²⁷K. This is the approximate temperature barrier that the electroweak crystalized out of the energy soup. During this time inflation runs rampant as the other forces have not separated out, there is no gravity yet, little attractive forces, no strong force, so inflation is huge.

At 10¹⁵K is when all of the four forces appeared separately and stable. The universe as we know is has begun.

And what's going to happen to that field in the future? It’s not fundamental. It’s the preferred state for that field to be in now. But maybe as the universe evolves, that field will either melt or another field may freeze. That would change the fundamental nature of the forces in the universe. If the Higgs field were to melt, then all particles become massless and everything we see would disappear. Or it could be that the Higgs field will freeze into a different configuration where the masses of particles will change completely and again, neutral stable matter will disappear. In some sense we’re at the hairy edge of a universe that might be unstable. The Higgs field probably won’t melt tomorrow—but in the far future, it might. We're lucky to be here to be able to ask these questions. We should enjoy our moment in the sun.

Bose–Einstein condensate

A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of atoms cooled to temperatures very close to absolute zero. Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum phenomena, particularly wavefunction interference, become apparent. A large quantity of atoms will act as one. A BEC is formed by cooling a gas of extremely low density, about one-hundred-thousandth the density of normal air, to ultra-low temperatures. Albert Einstein and Satyendra Nath Bose predicted that quantum mechanics could force a large number of particles to behave in concert as if they were only a single particle. This form of matter was called a Bose-Einstein condensation, and it wasn't until 1995 that researchers created the first such condensate of a gas of alkali atoms.

This state was first predicted, generally, in 1924–25 by Satyendra Nath Bose and Albert Einstein.

What good are they? Sensors. Particles in this state easily demonstrate the superposition nature of quantum particles, this is where they are in two states at once. This is useful for sensing in the electromagnetic spectrum, very sensitive sensing. The way I understand it is when an electron is in the superposition state and it is subject to a electromagnetic disturbance if you measure it, its collapse will tend towards certain levels and phase angels of spin. This can be used to measure very small magnetic fields or even perhaps pick up data.

Euler

This equation is amazing and is worth mention in this context. Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers and discovered its relation to the trigonometric functions. For any real number φ (taken to be radians), Euler's formula states that the complex exponential function satisfies…

e^iφ=cosφ+i*sinφ

..shown here in the following graph both sides of the above equation taken separately are shown to be equal. The symbol i in this case is taken to mean √(-1).

A special case of the above formula is known as Euler's identity…

e^iπ=-1

…is simply found by setting Ψ=π.

It was called "the most remarkable formula in mathematics" by Richard P. Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, e, i and π.