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Symmetries in physics

The principle of symmetry has emerged as a useful tool for scientists in guessing and formulating new laws of physics

By Hassaan Saleem |
PUBLISHED October 03, 2021

In the last century, physicists have learned to formulate physics laws and guess new laws of nature using a nontrivial tool. This tool is the use of symmetry principles. Physicists impose certain symmetries on the systems that they are studying and then, they study the consequences of these symmetries. Before delving into the topic of symmetry principles, let me define that what do I mean when I say that a certain system has a certain symmetry.

When we say that a certain system has a particular symmetry, we mean that there is a transformation that doesn't change the system or the conditions in which the system is residing. Well, in a certain sense, that is true. For a more complete description, we will have to ask the question that what property of the system will have to change for us to conclude that the system has changed. This question can take us into a nontrivial detour and thus, I will skip it for now. For now, we can just assume that a "symmetry" is a transformation that "doesn't change the system".

Spacetime symmetries

So far, so good. However, now we can ask the question that what kind of symmetries are possible? Well, let's start with a point particle. How can we "transform" it? Well, the only thing that we can change about this point particle is its position in space (and time) or its velocity in spacetime. These transformations are known as spacetime translations (if they change the position) or spacetime boosts (if they change the velocity). There is no other property of a point particle apart from its spacetime coordinates and its velocity that we can change (Off course, I am assuming that the particle doesn’t have any intrinsic properties i.e., the properties that don’t depend on its position in velocity). A transformation that changes the spacetime coordinates or velocity of a system is called a space-time transformation.

Now consider an extended system (i.e., a system that doesn't have zero size). We can still change the position and velocity of the canter of mass of this system. However, now, there is another transformation that we can do. We can rotate the system around the centre of mass while keeping the position and velocity of the centre of mass constant. Transformations that can achieve this effect are known as rotations. Rotations are spacetime transformations as well because they are also changing the space coordinates of points that make up the system (except the centre of mass, as its position is not changing).

Until now, we have seen three kinds of spacetime transformations i.e., translations, rotations, and boosts. Now, there is a subtle point that I need to address here. These three transformations are known as continuous transformations. What do I mean by that? Well, without being too technical, I will state a simple mnemonic to understand what I mean by this. For translations, rotations, and boosts, you can perform "just a little bit" of the concerned transformations. You can translate things "just a little bit", you can change the velocity of things "just a little bit" and you can rotate things "just a little bit" (more formally, you can write down the infinitesimal forms of these transformations). Until now, this realisation seems useless but, this realization is important. You can ask the question that, "Well, is there any spacetime transformation that isn't continuous? (i.e., for which we can't do "just a little bit" of it?)". Well, yes, we do have such transformations. These transformations are called parity transformation and time reversal. The effect of time reversal transformation is clear from its name while the effect of parity transformation is to reflect the universe into its mirror image (it is just like looking at the image of the universe in a mirror). It is easy to see that you can't reflect the space "just a little bit", nor can we reverse the time "just a little bit". These two "non-continuous" transformations are called discrete transformations.

So, we have gone through all mainstream continuous transformations (which include translations, rotations, and boosts) and the discrete transformations (that include time-reversal and parity). What do I mean by "mainstream" continuous spacetime transformations? Well, I used this word because there are speculative spacetime transformations called supersymmetric transformations but let's not go there right now.

Now, different sets of the above-mentioned transformations have specific names (for more involved readers, I can mention that these transformations form groups, but I won't go into the details here). The names of these sets are listed below.

* The set of all possible rotations and boosts is called the Lorentz group

* The set of all possible translations (in space and time), boosts, and rotations is called the Poincare group. Since all boosts and rotations are contained in the Poincare group, therefore the Lorentz group is contained in the Poincare group.

Physical systems are governed by equations. These equations are conventionally known as equations of motion (or EOM for short). If EOM for a system doesn't change its form, (or their forms, in case there are more than one EOM for a system) after we apply a transformation on that system, then this transformation is called a symmetry of that system (as I mentioned before, a symmetry is a transformation that "doesn't change the system"). For example, if the EOM of a system doesn't change under rotations and boosts, then the system is said to have Lorentz symmetry (or Lorentz invariance). This name is adopted since the set of all boosts and rotations is called the Lorentz group. If the EOM of this system doesn't change under translations as well (in addition to rotations and boosts), then this system is said to have Poincare symmetry (or Poincare invariance). A similar definition can be made for parity invariance and time- reversal invariance.

According to our very best experiments, (most of which are cosmological experiments) our universe does have Poincare invariance. Any theory of physics must make predictions that respect Poincare invariance. However, it was discovered in the 1950s that some processes that are related to beta decay (beta decay is a radioactive process in which a neutron decays to give a proton, an electron and a lesser-known particle called the anti-neutrino), don't respect parity invariance. Moreover, it was discovered in 1964 that some particles called kaons violate time-reversal invariance. Much stronger evidence for this violation of time-reversal invariance was found in 2001 in California and Japan. This stronger evidence involved another kind of particle known as beauty particles. (More formally, for people with some background knowledge, I can state that the stronger evidence was the evidence for direct CP violation).

Particle physicists and cosmologists have developed calculational methods such that Poincare invariance is respected in every step of the calculation. However, this is not necessary. We can have a calculation that doesn't respect the Poincare invariance at every step. Only the final physical prediction should respect Poincare invariance. Some calculations don't respect Poincare invariance at every step but give Poincare invariant results at the end (for readers with some background knowledge, I can state that one example of such calculations is the set of light-cone gauge calculations).

Gauge symmetries

In addition to the spacetime symmetries, particle physicists have also learned the importance of another kind of transformations. These transformations are called gauge transformations. Before trying to explain what gauge transformations are, I must admit that describing these transformations without any esoteric mathematics is a bit harder than describing the spacetime transformation. However, I will try my best to explain the meaning of gauge transformations using an example that can be understood by anyone.

Suppose that there is a competition going on between two players i.e., player A and player B (The details of the type of competition don’t matter). The rules are simple; the player who scores more points wins and the amount of money (in dollars) that the winner receives is equal to the difference in the points of two players. For example, if player A scores 1000 points and player B scores 1500 points, then player B gets $500 because player B scored 500 points more then player A. In short, it is only the difference in points that “matters”.

Now, here is the interesting realization. If we increase (or decrease) the score of both players by the same amount, the difference in the points doesn’t change. For example, if player A and B scored 1000 and 1500 points respectively (leading to a difference of 500 in favour of player B), then decreasing their scores by 100 will lead to a score of 900 and 1400 for player A and B respectively. However, the difference in their score is still 500 and thus, player B will still win $500. Therefore, the thing that “matters” (i.e., the difference in scores) hasn’t changed by this “transformation” of scores. However, please note that the individual scores of two players were required to calculate the difference in scores (i.e., the numbers that didn’t matter are required to calculate the number that matters). Roughly, this is the idea behind gauge transformations.

In physics, we have two kinds of quantities. One type of quantities is called observable (or physical) quantities (i.e., the quantities that we can measure). The other type of quantities is called unobservable (or unphysical) quantities. The physical quantities correspond to the difference in the scores of players while the unphysical quantities correspond to the individual scores of the players.

Just like the example above, these unphysical quantities are required to calculate the physical quantities. However, it turns out that if some values of unphysical quantities lead to a particular value of a physical quantity (like the scores of 1000 and 1500 from player A and player B respectively gave a difference of 500), then the values of unphysical quantities can be transformed in a very particular way, such that the transformed values of the unphysical quantities will give the same value for the physical quantity (just like in the example above, where increasing or decreasing the scores of both players by the same number leads to the same difference between the scores). These transformations of the unphysical quantities are called gauge transformations. In other words, values of unphysical quantities that are related by a gauge transformation give the same value for physical quantities. Values of unphysical quantities that are related by a gauge transformation are called gauge equivalent.

In our example, the gauge transformation was very simple i.e., just increase or decrease the scores by the same number. However, in particle physics, the gauge transformations have more subtle forms than that. Moreover, different gauge transformations form different sets that are called gauge groups (I am ignoring the technical issue of the existence of an operation for forming a group right now - as someone with relevant background knowledge might object-).

In particle physics, a particular theory of fundamental interactions is based upon different gauge groups. A particular theory is called gauge invariant under a particular gauge group if the gauge transformations from that gauge group don’t change the physical quantities in that theory. For example, the theory of electricity and magnetism at very small distances (called quantum electrodynamics) is based upon a gauge group whose name is 𝑈(1) (don’t worry, it is just the name of a gauge group). The theory of strong force (the force that keeps the nucleus intact) is called quantum chromodynamics and it is based upon a gauge group whose name is 𝑆𝑈(3). Lastly, the standard model of particle physics, that describes all the known forces (expect gravity) with very good precision is based upon a gauge group called 𝑆𝑈(3) × 𝑆𝑈(2) × 𝑈(1).

Many times, the search for new theories can be boiled down to guessing the correct gauge group for the theory (although there are other things to be worked out but the search for appropriate gauge groups is a major task). Describing such speculative theories require another article and thus, I won’t talk about them here.

(The writer is a PhD student studying theoretical physics at the University at Albany, State University of New York. He studies string theory, makes videos and hosts a podcast series on Physics)