As a curious reader of theoretical physics, it can feel frustrating to repeatedly come upon the "foreign language" of complex mathematics, but it is here where the concepts and ideas written out for us stand in their naked just-born form and I suspect that those who can read it see nuances, elegant machinery and connections unavailable to most of us.

I don't have the credentials to take us into a proper mathematical discussion of this theory, nor do I have the years of training required to look at gauge theory from all different angles and thus have a complete working understanding of the concept. Here, I am a beginner learning to appreciate gauge theory. I want to try to build for myself an accurate idea as possible of what gauge theory is and find the secret of why it pops up over and over in modern theories.

As complex and, frankly, intimidating as gauge theory is, we haven't escaped it. In fact, we've already been introduced to it and we just didn't know it. The fundamental forces are all treated as gauge theories, and all the symmetry-breaking I mentioned over past articles - that mind-blowing idea of how forces originate - is rooted in the equations of gauge theory.

The Standard Model Is a Gauge Theory

The Standard Model of particle physics is a relativistic quantum theory of particles of matter, such as leptons and quarks. Particles of matter are shown as purple and green squares in the Standard Model below.

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All particles of matter are treated as quantum wave functions, rather than discrete points.

Atoms and subatomic particles behave according to the rules of quantum mechanics, where probability clouds rather than exact values tell us where a particle is and how fast it's going, and it restricts what energies particles can have. For example, the figure below compares a particle moving according to classical mechanics versus a particle moving according to quantum mechanics. It compares possible trajectories for a theoretical particle that is restricted to movement in one direction only where velocity is constant. In A it moves according to classical mechanics, but in B through F it moves as one of five possible wave functions, where blue is the real part of the wave function and red is the imaginary part of the wave function.

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Special relativity means that all massless particles travel at the speed of light, regardless of which frame of reference you use to measure their velocity. This tells us that space must be more complex than the static three-dimensional Cartesian coordinate system it appears to be. An additional dimension of time must be incorporated into how space works. Physicists must use something called Lorentz transformation to measure distances in four-dimensional space-time. For example, the diagram below shows how special relativity works with respect to time. The dots represent sets of events that are identical on the left and the right. The left diagram represents the events occurring for an observer at rest. On the right, the same events are observed by someone in relative motion, traveling close to the speed of light. The two red events are simultaneous for the left observer but for the right observer the right red event occurs before the left red event. This is an example of a Lorentz transformation of space-time.

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The Standard Model revolutionized how we understand atomic behaviour, such as how atoms bond chemically with each other, how electromagnetic radiation works and how magnetism works.

The Standard Model is also mathematically a quantum field theory for three of the four fundamental forces - the electromagnetic force, the weak force and the strong force. Virtual particles called gauge bosons act as carriers of force. They arise from the quantization of, or the application of quantum mechanics to, force fields. For example, the powerful attractive force that holds quarks together inside a proton is carried out by the exchange of virtual gluons, quantized units of the strong force. Below right, gluon exchange is shown by yellow wavy lines between the three quarks of a proton.

Arpad Horvath;Wikipedia |

This mathematical framework also makes the Standard Model technically a quantum gauge field theory. Gauge theory gives the model various kinds of symmetry. This has great implications for how particles of force and matter interact and it even offers a deeper glimpse into what they really are and where they come from.

The recently discovered Higgs Boson is a gauge boson. In fact, it's a particle that was accurately described and predicted by gauge theory before it was ever "seen" in CERN's supercollider. This confirmation bolstered electroweak theory (I'll mention more about this later on) and furthered physicists' acceptance of gauge theory as a fundamental concept that, if we can get our heads around it, will greatly deepen our understanding of the nature of matter and energy in our universe.

General Relativity is not part of the Standard Model. Physicists do not yet understand how gravity works, though it is described geometrically through Einstein's field equations for general relativity. Mass, energy and momentum bend and stretch the fabric of space-time itself. The image below offers a simplified representation of how Earth's mass warps four-dimensional space-time.

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What does a gauge transformation look like? They are very complex, but we can get a rough idea. Below, a Cartesian coordinate grid is distorted by a coordinate transformation. As the distortion takes place, point (x,y) on this grid will move. The relationship between old (x,y) and new (x,y) is a non-linear one, meaning that the new coordinates are not directly proportional to the original ones, that the relationship is complex. Likewise, Einstein's equations for general relativity are valid before and after distortion. They are invariant under the transformation. Changes in a coordinate system like this one represent the gauge transformations in general relativity.

A Definition

What do physicists mean when they say "gauge" theory?

A. The Idea of a Transformation

A gauge in physics is not terribly complicated. It is a coordinate system that varies depending on where you are looking at it from with respect to some reference location. A gauge can transform by changing the reference location. Transformation is a core concept in physics. Phase changes, such as water freezing to ice, are physical transformations, shown in the graph below.

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^{1},t

^{1}) is a frame moving close to the speed of light. The change in shape from a rectangle to a rhomboid illustrates length contraction. As a simple example, a spaceship moving past you at close to light speed (you are at rest relative to it) will appear shortened or horizontally squished as you watch it go by.

Transformations are not necessarily geometric. They can be mathematical functions or variables that are translated into different functions or variables, while preserving the structure of the formula. Gauge transformations fall into this category.

Gauge theory is a mathematical model of a system in which you can apply gauge transformations, and where all physically measurable or observable quantities are left unchanged. You can't "see" a gauge transformation. Adding quantum mechanics and special relativity to gauge theory, as in the Standard Model, not surprisingly complicates the mathematical formulation.

B. The Idea of a Field: Scalar, Vector and Tensor

Gauge theory is also a type of field theory. In the Standard Model, this field theory assigns a value to every point in four-dimensional space-time. We'll use electric and magnetic fields as examples of fields, in simpler two-dimensional space, as shown below. The electric fields (E) around positive and negative charges as well as the magnetic fields (B) around north and south magnetic poles (can be thought of as magnetic charges here) are shown below left. These two fields induce each other. A moving electric charge induces a magnetic field and a moving magnetic charge induces an electric field (shown to the right), illustrating how these two fields are connected as part of the fundamental electromagnetic field. Vector arrows (in black) indicate the strength and direction of each field's force.

A positive charge placed near the positive charge above left will accelerate away because the field created by the original positive charge will act as a repulsive force on the new charge. If this positive charge is placed near the negative charge below left, it will accelerate toward it because the negatively charged field will exert an attractive force on it. These fields are not just fields of force that act on the motion of particles; they have an independent reality because they carry energy.

Fields are used to describe how forces, such as gravitational, electrical, and magnetic forces, act on objects that are not in direct physical contact with each other. Notice in the example above that you cannot see or measure the field itself but you can measure charge, acceleration, momentum, etc., which indicate that the field is a reality and it has energy. You can measure the force of the field and how that force is changing, at any reference point in two or three spatial dimensions or even within the four-dimensional matrix of space-time. You can also measure one or more different interacting and non-interacting forces acting in various directions on a particular reference point within a particular space. For this hefty job you need to pull out the tensor.

A field can be a scalar, vector or a tensor field, depending on whether the value of each point in the field is a vector, scalar or tensor value. You many have heard of scalar and vector values already. Scalar fields give you a single value of some variable for every point on a two-dimensional grid or in space or space-time. An example would be temperatures across Alberta at 3 pm today. You get one number (one value) in degrees Celsius for each point on a two-dimensional field. A vector field, on the other hand, assigns two values to each point - magnitude and direction - such as those of the magnetic and electric fields above. A scalar or vector field can be two, three or even four-dimensional.

With a simple vector, I am a bit limited because I can only describe magnitude and a single direction in two or more dimensions.

A tensor field opens up a whole new toolkit. It is a little more complex and much more useful. Rather than just one or two values assigned to every point, here an array of values can be assigned. A tensor field also allows physicists to describe a point in three-dimensional space or four-dimensional space-time. To help you understand the concept of a tensor, try this excellent 12-minute video:

Scalar and vector values can be thought of as very simple (or low rank) tensor values. A scalar value (just magnitude) is a 0-dimensional, or 0th order tensor value. A scalar value doesn't transform. It's the same no matter where you look at it from.

A single vector value (magnitude + direction; usually drawn as a arrow of specific length) is a 1-dimensional, or 1st order, tensor value. An electric field around a point charge can be described using a rank -1 tensor. Any reference point in a 1st order tensor field can undergo a transformation. Imagine a vector arrow plotted in a three-dimensional Cartesian matrix. It can look shorter or longer, for example, depending on which direction you are looking at it from. Likewise, an electric field around a point charge can undergo a transformation into a magnetic field and vice versa.

Higher rank tensor fields are indispensible because they can provide concise descriptions of realistic phenomena, which are often far too complex to visualize. Here it may seem that we are just going to add spatial dimensions to the tensor but the rank of a tensor may be independent of the number of dimensions of a space. A rank-2 tensor is magnitude and two directions, or 2nd order. It can be described as a 3 x 3 matrix, giving it 9 values for each reference point. An example is a stress tensor, shown below.

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_{1}, θ

_{2}and θ

_{3}, giving you nine values.

An electromagnetic field, such as the example above, is described as a rank-2 tensor field, and, like all rank-2 tensors, it can undergo transformations. A rank-3 tensor is magnitude and three directions, or a set of 27 scalar values. Most tensors you find in physics are 2nd order tensors.

Special relativity was formulated as a four-dimensional (called four-vector) rank-2 tensor called the Minkowski tensor or Minkowski metric, which is subject to several rank- 0, 1 and 2 tensor transformation laws and which rotates in four dimensions using formulas called Lorentz transformations, given by the Lorentz tensor. The Minkowski diagram of a Lorentz transformation is greatly simplified and reduced to two dimensions so that we can visualize it. General relativity describes space-time as a four-dimensional metric tensor, shown below.

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An example of a higher rank tensor is elasticity in a material such as concrete. You can define stress on any single point in the material as a function of a differential equation in two directions. A differential equation allows the stress measurement to be a dynamic rather than static value. You need two directions because you need the direction of the applied force and the direction of the area to which it's applied. This function adds up these vectors to get a overall stress tensor of the material. When you apply stress to a solid object, that object will experience strain, and this, like stress, is a function and a rank-2 tensor. The cement is now described by a 2nd rank stress input tensor and a 2nd rank strain output tensor. Added together, you can describe the cement's elasticity, which is a 4rth rank tensor.

Tensors, like the elasticity tensor, where one or more point values are functions of (dependent on) other values, can offer mathematical descriptions of highly complex smoothly changing systems. The changing elasticity of concrete as it cures, or the fluid mechanics of plasma in the Sun or even complex special relativity problems become accessible. Tensors offer tremendous power to physicists. A tensor, for example, can tell you how much momentum and energy exists at a particular location, what direction it is moving in, and if a function is used for example, how the momentum is changing. A tensor can also be used as a frame of reference, which can experience rotation or a transformation in many other ways to reflect a transformation in a field. General relativity was formulated entirely in the language of tensors.

Gauge theory is about field transformations, and tensors are used to describe them mathematically.

C. Gauge Theory is About Symmetry

The symmetry of a system is a physical or mathematical feature of the system that is preserved under some transformation, and there are two basic kinds - discrete and continuous. A very simple example of a discrete transformation is your reflection in a mirror. All the laws of classical mechanics are also symmetric under mirror inversion. Take any motion, for example and imagine viewing it in a mirror. That mirror image, that reflected motion, would still satisfy the laws of classical mechanics. This is called a parity transformation, so we can say classical mechanics is invariant under a parity transformation.

The simplest example of a continuous transformation is a circle rotating about an axis. Here, we are dealing with a continuous transformation. These can be described by continuous or smooth mathematical functions. A function is simply a relation that associates one set of values with another set of values. These functions are described mathematically by using derivatives. Derivatives represent an infinitesimal change in a function with respect to some variable. Nature is filled with smoothly changing phenomena, so derivatives are used to get values for instantaneous rates of change at any particular point in space or time, or space-time.

Different kinds of continuous transformations gives rise to different kinds of symmetries. Gauge symmetries are mathematically described using Lie groups. Lie groups are smoothly varying families of symmetries. The rotational symmetry of a circle is easy to visualize. A circle can be rotated by any any angle and remain unchanged.This series of smooth or continuous transformations about the circle's axis makes up the simplest possible one-dimensional Lie group, called the circle group or U(1) symmetry.

Values that change in a continuous transformation equation are called variables. In this case, they can be described as transforming smoothly over infinitely small degrees, using a differential equation. Gauge theory specifies this change and tells us which kinds of change are permissible and which aren't, because any changes that are made must cancel out in terms of observable quantities. What does this mean?

Gauge theory allows variables to undergo a group of local transformations, or local changes, called gauge transformations, which leave the physics of the system unchanged, or invariant. For example, none of the fundamental fields can be directly measured, but observable quantities associated with these fields - such as charge, particle energy and particle velocity - can be measured. These observable qualities don't change under a gauge transformation, even though they are associated with fields that do change under the transformation.

This basic unchanging backdrop is called the Lagrangian, which means that the overall dynamics of a physical system is invariant under a continuous group of local transformations. This concept where the system as a whole is invariant is called gauge invariance. The Lagrangian for the Standard Model is quantum field theory (QFT). QFT provides the overall mathematical framework that controls the kinematics and dynamics (how particles move and behave) of the field as a whole, in space-time, while allowing smaller (local) internal transformations to take place.

Gauge theory, treated mathematically, gives internal symmetries to the Standard Model. These local symmetries define or pose some limits on the way a field can interact with other fields or the way it interacts with particles. The local SU(3)xSU(2)xU(1) gauge symmetry group is an internal symmetry Lie group that lies at the heart of the Standard Model. It gives rise to three of the four fundamental forces - the strong force (SU(3)), the weak force (SU(2)) and electromagnetism (U(1)).

Another internal gauge symmetry in the Standard Model is colour symmetry, which defines or restricts how quarks behave inside protons and neutrons. It's an internal symmetry of the strong fundamental force. How quarks, each one possessing one of three possible colour charges, interact and behave is the study of Quantum Chromodynamics (QCD) For example, force fields due to colour charges in quarks are shown below using black vector arrows. Colour charges are mediated by gluons. They create tiny but powerful fields that are mathematically described by the gluon tensor, G. These fields tightly bind quarks together inside all hadrons such as protons and neutrons (top middle), antiprotons and antineutrons (top right) as well as inside (unstable) 2-quark hadrons called mesons (three bottom images).

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The difference is that you do not see any colour charge field lines arc outward to infinity as you do with electric and magnetic fields. The strong force is so powerful that field lines are pulled together tightly by the gluons. This is why the range of the strong force is so short (about 1 fm) and why it is confined inside hadrons, unlike magnetic, electric and electromagnetic force fields that have infinite range (and is why electromagnetic radiation, such as visible light, can travel right across the universe).

When you see Standard Model particles with different flavours, you see other examples of internal symmetry. Flavour quantum numbers (isospin, charm, strangeness, topness and bottomness) are all described mathematically using gauge theory. Some of these flavor symmetries can be broken and some can't, depending on the particle or the interaction. For example, neutrinos can change flavours spontaneously, which means they undergo spontaneous flavour symmetry transformations. In QCD however, flavour symmetry is global. This symmetry, SU(3), isn't broken so it is called a global symmetry of QCD. However, this symmetry is broken in the electroweak theory, allowing neutrino flavour oscillations and quark decay. Leptons, such as electrons, have six flavor numbers, which are all conserved in electromagnetic interactions but are violated in weak interactions. In this case, parity and charge-parity symmetries are broken.

The simplest local transformation in a Lagrangian can be described as a circle rotating about an axis, as mentioned above. It represents the simplest symmetry Lie group, called U(1). The Standard Model is built on three symmetry Lie groups - U(1), SU(2), and SU(3). (SU stands for special unitary group, a complex mathematical term)

The electromagnetic force can be described mathematically as a U(1) symmetry group. The weak nuclear force is described as an SU(2) symmetry and the strong nuclear force is an SU(3) symmetry. The Standard model cannot mathematically describe gravity.

Gauge theory also ascribes a force particle to each of these fundamental forces. It treats these particles as generators of these symmetry groups. U(1) symmetry has one particle generator - the photon. The photon mediates or carries out the electromagnetic force. SU(2) symmetry has 3 generator particles - the Z, W

^{+}and W

^{-}particles (collectively called gauge bosons of the weak force (nuclear decay and fusion). SU(3) has 8 generators. This symmetry is linked to the strong force; these particles are the eight different gluons (Why eight when gluons come in three possible colour charges? This again is a question for Quantum Chromodynamics (QCD), an article to come). Gauge theory isn't just about symmetry; it's about breaking it as well.

Symmetry-Breaking in Gauge Theory: The Electroweak Force as An Example

Some of my earlier Universe articles mentioned forces and particles coming into existence for the first time in the universe through the process of symmetry-breaking, and I used an example of a glass of water (one homogenous system) breaking into two parts - water and ice - as the system's energy decreased and the water underwent a phase transition. This is a vastly simplified notion of what gauge theory does through the process of gauge transformation.

The diagram below helps us visualize how symmetry-breaking works. At high enough energy, a ball rests at the lowest energy central point. This system is symmetrical. However, as energy decreases, the center becomes unstable and the ball rolls down to a new (arbitrary) lower energy position, shown right. Now the system is no longer symmetrical. The symmetry is broken.

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Gauge theory allows physicists to perform theoretical transformations on the fields - change water to ice and back to water so to speak - changes that you can think of as local changes within an overall invariant system. In the analogy above, the glass is still full of water molecules; you are not changing the Langranian in other words. In gauge theory the Langrangian is the kinematics and dynamics of the particles as dictated by quantum field theory.

The fundamental forces (and their mediator particles) come from constraints placed on local gauge symmetries. As mentioned in examples above, these symmetries can be broken under certain circumstances.

To see how symmetry-breaking works, let's go far back in time to the very early universe. It is unimaginably energetic and dense. The strong force exists and gravity exists but the electromagnetic force and its mediator particle, the photon, are not part of this universe. At this energy this force is identical with the weak force and its mediator particles, the W and Z particles. It is a single force called the electroweak force, and the theory predicts that, right now, these four very different kinds of particles are all identical electroweak bosons.

Now we wait for just a minute fraction of a second. The universe has significantly cooled in just this time. Three particles with mass (W and Z particles - the very first mass in the universe) and one massless particle, the photon, are arising from just one kind of particle, and two very different fundamental forces now exist (along with the strong force and gravity already in existence). What happened here is a process called symmetry-breaking and it is roughly analogous to the water freezing into ice, a phase change in other words. The particles that arise from a gauge transformation, such as the weak and electromagnetic forces breaking from the electroweak force above, are not part of the transformation itself. They arise instead from the underlying fields that change, or put even more accurately, the new fields arise from them.

All this change occurred while the backdrop remained invariant; the physics of the universe didn't change. Quantum field theory was intact throughout and space-time didn't change. It is only because the universe is in its current low-energy state that electromagnetism and the weak force appear to be so different from each other and the symmetry between the two is not apparent (except mathematically). Described mathematically, this is a gauge transformation and it is part of the captivating story behind the Higgs Boson discovery.

Under Gauge, Forces Can Unify

Many physicists believe that ultimately the gauge theories for the electroweak force and the strong force can be combined in a similar way into a single gauge theory, called the Grand Unification Theory (GUT), unifying all three gauge fundamental forces into one force called the electronuclear force.

This is a theory in progress. To unify the theories, physicists have to formulate a gauge transformation that couples quarks to leptons, a process that would violate the conservation of baryon number. Extensive study of particle interactions led to this basic conservation law in particle physics. Baryon number, like electric charge, is considered to be an absolutely conserved flavour number. Unlike neutrino flavor numbers, baryon number is considered a global symmetry in quantum field theory. Breaking it would permit the decay of the proton, and there is no evidence that it decays. Still, physicists speculate that a baryon, such as a proton or neutron, can transform into several leptons at high enough energy (around 10

^{16}GeV). This grand unified force operating under a single unified gauge group called SU(5), might be mediated by X and Y particles, analogous to, but much more massive than, the W and Z particles for the weak force. According to this model, leptons and quarks combine into irreducible representations while the proton is allowed to decay into leptons and pions, but it is given an extremely long half-life, perhaps longer than the universe itself.

Next, physicists wonder if the GUT force, assuming it can be worked out, could couple theoretically with gravity into an ultimate "Theory of Everything" (TOE). This is a very far reach. Quantum mechanics would additionally have to be coupled to general relativity.

How Gauge Theory Developed

The idea of the existence of symmetries within various fields began with the work of physicists such as Hendrik Lorentz, Albert Einstein and Henri Poincaré at the turn of the twentieth century, around the same time as quantum theory, a cornerstone theory of modern particle physics, was being hashed out. These physicists noticed something interesting in James Maxwell's equations for electromagnetism (these classical equations, published in 1864, describe how magnetism and electricity arise from one field of force, electromagnetism). They found symmetry in the equations, Lorentz symmetry to be specific. This is the same symmetry that underlies special relativity, another cornerstone theory that Einstein had recently developed. Other physicists looked for further symmetries within Maxwell's equations and lo and behold a researcher named Hermann Weyl found one. This symmetry is the gauge symmetry within electromagnetism. Weyl attempted unsuccessfully to use it to unify general relativity with electromagnetism in 1918. Later, with the development of quantum mechanics, Weyl and others modified their gauge choice and were able to apply their modified gauge theory to electromagnetism. Chen Ning Yang and Robert Mills created a mathematical framework for gauge symmetry in 1954, and later on it was used to unify electromagnetism with the weak force (this was the electroweak theory, developed in 1979) and to construct the Standard Model of particle physics.

Maxwell's equations can be thought of as prototype gauge symmetry equations. In 1916, Einstein published the general theory of relativity (special relativity actually came first). He was on to the importance of symmetry, so he wrote these field equations with symmetry in mind. Some physicists now look back at general relativity as yet another example of a gauge theory. Gauge theory lies at the heart of so many backbone theories in physics today, such as the Standard Model, the electroweak theory and quantum electrodynamics (QED). These theories have also been successfully described as quantum theories. In other words, we now have a quantum understanding of electromagnetism, of particles of energy (gauge bosons) and matter (fermions), the weak force and of the strong force (QCD). The exception is general relativity, which is a theory with gauge symmetries that have not yet been successfully quantized.

The Standard Model, though extraordinarily powerful and elegant, is not entirely satisfactory to many physicists. Gauge theory tells us how particles and forces relate to each other but it says nothing about why. If you look at the list of particles there, do they still not seem kind of arbitrary? Why are there three families of quarks and leptons, and why is the gauge group SU(3)xSU(2)xU(1), for example? Gauge symmetry does something beautiful to particle physics but it seems to ask its own new questions. Is the gauge symmetry we see part of a larger symmetry (filled with yet more - very-high energy - particles), such as supersymmetry, or is it part of a universe filled with extra dimensions that we never see but become significant at much higher energies such as when particles and energies are sorting themselves out, and perhaps where the graviton if it exists has its origin?

thank you for this. it's very well done!

ReplyDeletenice article, using great equipment to measure the thickness of your paint will produce you with good information about your stuffs.

ReplyDeleteVery good explanation. I will re-read several times to fully grasp. One question. You write: "Values that change in a continuous transformation equation are called variables." What is the variable that is changing for the rotation of the circle?

ReplyDelete