The Greek philosopher Plato is well known for his allegories and metaphors. Of particular interest is his Allegory of the Cave, which appeared in The Republic, written around 380BCE. In it, Plato describes a group of prisoners which are chained to a wall within a cave, and have been all of their lives. They have no direct interaction with the world outside of the cave. They only know of the world via shadows that are cast on the wall opposite of them. For the prisoners, the shadows are their reality. Though you and I know the shadows are only a very low-resolution representation of that reality.
Theoretical physicist Steven Weinberg, a Nobel Prize winner who works out of the University of Texas at Austin, once likened himself to a prisoner in Plato’s cave. We are forever chained to this cave by the limitations in measurements we can make and experiments we can perform. All that we can know are shadows of the reality that exists in the sub-atomic world. We can see the shadowy figures lurking in our math and as wisps of misty vapor trails in our cloud chambers. We attempt to pierce the veil with the power of our imagination and draw nifty looking charts and animations depicting what our mind’s eye thinks it can see. But in the end, we are all trapped in a cave… staring at shadows. Reflections of a reality we can never truly know.
In our last Quantum Mechanics article, we introduced you to the idea of quantum electrodynamics, or to put it more simply — quantum field theory. In this article, we’re going to explore how QED lead to the prediction and eventual confirmation of something known as the Higgs Boson, also known as the God Particle. As usual, we’ll aim to keep things as simple as possible, allowing anyone with a curious mind to know what this God particle talk is all about. Like so many things in the quantum world, it all started with an unexpected outcome…
The start of World War II threw quantum theory research into disarray. Many of the European physicists left Europe all together, and research moved across the ocean to the shores of the United States. The advent of the atomic bomb thrust American physicists into the spotlight, and physicists began to meet on Shelter Island to discuss the future of quantum theory. By this time one thing was certain: the Copenhagen interpretation of quantum theory had triumphed and challenges to it had mostly died off.
This allowed physicists to focus on a different kind of problem. At this point in time quantum theory was not able to deal with transitional states of particles when they are created and destroyed. It was well known that when an electron came into contact with a positron, the two particles were destroyed and formed at least two photons with a very high energy, known as gamma rays. On the flip side, gamma ray photons could spontaneously turn into positron-electron pairs.
No one could explain why this occurred. It had become obvious to the physicists of the day that a quantum version of Maxwell’s electromagnetic field theory was needed to explain the phenomenon. This would eventually give rise to QED, short for quantum electrodynamics. This is a severely condensed story of how that happened.
Einstein referred to her as the most important woman in the history of mathematics. Her theorem has been recognized as “one of the most important mathematical theorems ever proved in guiding the development of modern physics.” Yet many people haven’t the slightest clue of who this woman was, or what she did that was so significant to our understanding of how our world works. If you count yourself as one of those who have never heard of Emmy Noether and wish to enlighten yourself, please read on. I can only hope I do her memory justice. Not just by telling you who she was, but by also giving you an understanding of how her insight led to the coming together of symmetry and quantum theory, pointing academia’s arrow toward quantum electrodynamics.
Being a female in Germany in the late 1800s was not easy. She wasn’t allowed to register for math classes. Fortunately, her father happened to be a math professor, which allowed her to sit in on many of his classes. She took one of his final exams in 1904 and did so well that she was granted a bachelors degree. This allowed her to “officially” register in a math graduate program. Three years later, she earned one of the first PhD’s given to a woman in Germany. She was just 25 years old.
1907 was a very exciting time in theoretical physics, as scientists were hot on the heels of figuring out how light and atoms interact with each other. Emmy wanted in on the fun, but being a woman made this difficult. She wasn’t allowed to hold a teaching position, so she worked as an unpaid assistant, surviving on a small inheritance and under-the-table money that she earned sitting in for male professors when they were unable to teach. She was still able to do what professors are supposed to do, however – write papers. In 1916, she would pen the theorem that would have her rubbing shoulders with the other physics and mathematical giants of the era.
Noether’s Theorem – The Basics
Emmy Noether’s Theorem seems simple on the onset, but holds a fundamental truth that explains the fabric of our reality. It goes something like this:
For every symmetry, there is a corresponding conservation law.
We all have heard of laws such as Newton’s first law of motion, which is about the conservation of momentum. And the first law of thermodynamics, which is about the conservation of energy. Noether’s theorem tells us that there must be some type of symmetry that is related to these conservation laws. Before we get into the meaning, we must first understand a little known subject called The Principle of Least Action.
The Universe is Lazy
I would wager a few Raspberry Pi Zeros that many of you already have an intuitive grasp of this principle, even if you’ve never heard of it before now. The principle of least action basically says that the universe has figured out the easiest way possible to get something done. Mathematically, it’s the sum over time of kinetic energy minus potential energy as the action occurs. Let us imagine that you’re trying to program an STM32 Discovery eval board in GCC. After about the 6,000th try, you toss the POS across the room and grab your trusty Uno. The graph depicts the STM32 moving through time and space.
The green points represent particular points of how how high the STM32 is at a given point in time. Note that there are no values for height and time – this example is meant to explain a principle. We can say that at these points (and all points along the curve), the SMT32 has both kinetic and potential energies. Let us call the kinetic energy (kt) and the potential energy (pt). The ‘t‘ subscript is for time, as both the energies are functions of time. The action for each point will be called s, and can be calculated as:
However, action is the total sum of the difference of energies at each point between t1 and t2. If you’ve read my integral post, you will know that we need to integrate in order to calculate the total action.
Now before you get your jumper wires in a bunch, all that is saying is that we’re taking the difference in potential (p) and kinetic (k) energies at each point along the curve between t1 and t2, and we’re adding them together. The elongated S symbol means a sum, and the (dt) means as it changes over time. The path that the STM32 will take will be the path where the action S is at its minimum value. Check out the video in the source section below if you’re confused. It’s only 10 minutes and goes into this concept in easy to follow details.
Noether’s Theorem – The Details
Noether’s theorem is based upon a mathematical proof. It’s not a theory. Her proof can be applied to physics to develop theories, however. Now that we know what the principle of least action is, we can do just this.
Any law of nature can be traced back to a symmetry and the least action principle. Let’s consider two very simple examples – Newton’s first law of motion and the first law of thermodynamics.
Conservation of Momentum
Space has what is known as translational symmetry. That’s just fancy-pants talk for saying that what you do in one point in space is the same as what you do in another point in space. It doesn’t matter what hacker space you throw your STM32, it will act the same at all hacker spaces on earth. Space itself provides the symmetry. And because the principle of least action applies, you have a natural law – the first law of motion.
Conservation of Energy
Time has the same translational symmetry as space does. If I toss the STM32 now, and toss it tomorrow, it will act the same. It doesn’t matter what point in time I toss it, the results will always be the same. Thus energy is conserved between different points in time. Time is our symmetry, and the 1st law of thermodynamics is the result.
Now, I realize these examples might seem a bit useless. But when you dig a bit deeper, things get interesting. Electrical charge is also conserved. Noether says there must then be some type of symmetry involved. What do you suppose that symmetry might be? Keep following that rabbit hole, and you’ll end up face to face with QED. We’ll get there in a future article, so for now just keep Noether’s Theorem in mind.
Physics Helps, The principle of least action, video link.
Ransom Stephens, Ph.D., Emmy Noether and The Fabric of Reality, video link
Symmetry is everywhere in our natural world. Just take a look at your hands, a butterfly, or a sunflower. It’s easy to pass off the idea of symmetry and symmetric structures as a simple quirk of existence, and to pay it little mind. If this is your view, I can assure you it will no longer be by the end of this series. If we force ourselves to look beyond the grade school applications of symmetry, we find a world rich in connections via many different types of symmetric identities. One of the most interesting is Gauge Symmetry, which lies at the heart of Quantum Electrodynamics, or QED (we’ll get into this a bit later in the series). Several branches of higher level mathematics study symmetry in detail, allowing a host of sciences, from physics to chemistry, to view difficult problems and theories from a different perspective.
The subject matter of the ideas explored in symmetry is complicated, and not well known outside of academia and the theoretical sciences. It is the goal of this series of articles to simplify some of the concepts that underpin the study of symmetry, so that the average hacker can gain a basic (and I mean basic) understanding of this fascinating body of knowledge, and put it to use in future projects. We’ll start things off by taking a look at a machine that has crossed the Hackaday server many times – those nifty Rubik’s Cube solvers. Just how do those things work anyway?