## Saturday, February 27, 2010

### Symmetry for the "I don't want a PhD in Physics to understand that" People

Because there has been some talk of symmetry on this blog lately I thought that I would give an explanation of symmetry for those who really don't know (or want to know) what a Lie Group is. So this is an explanation of symmetry for the reasonably well informed, yet still baffled.

First let's start with a basic definition of symmetry. Symmetry implies a "sense of harmonious or aesthetically pleasing proportionality and balance". The key here is proportionality and balance. In other words, if you have two opposites then the amount of stuff on each side is equal. This also implies that if you turn the system around 180 degrees and look at it from the opposite side then it will still look the same. If you look at the "tree" on the left you can see that it is symmetric and if you look at it from the opposite side then it will still look the same. The other tree is asymmetric and has a definite and discernible left and right side to it. In this case we say that symmetry is broken. Again the key here in understanding symmetry is that there should be NO preference for one side over the other.

If we think of a coin toss, a "fair" coin would be perfectly symmetric and several tosses with that coin would produce even numbers of heads and tails. But with an "unfair" coin or a weighted coin, it would produce asymmetric results of heads and tails. So if we look at a large number of randomly chosen events and we detect an asymmetry then we can say that the system was predetermined to produce more of a certain result, and we claim that symmetry is broken.

So now how does this all relate to fundamental particle physics? Well, in the standard model there are three basic symmetries that determine a lot of the physics that we see. They are Charge symmetry (C-symmetry), Parity (P-symmetry) and Time symmetry (T-symmetry). I will attempt to give a simple (and comprehensible) explanation of these three symmetries.

First C-symmetry deals with the way electric charge fundamentally works. You might say that it is the basis for why opposite charges attract each other and why atoms are structured the way they are (it's actually much more complex than that but for our purposes it is adequate). The key here is that if the positive and negative charges are swapped (protons are now negatively charged and electrons are positively charged) then we would still have the same universe. That is, everything would still work the same and we would still have the same laws of physics.

But with C-symmetry there is a catch, it would seem that at some point C-symmetry must have been broken. If everything was truly symmetric for charge in our universe then why is it that, at least for 99.9999999999999999999% of known matter, the negative charge is associated with a light (not very massive) point like particle and the positive charge is associated with heavy particle of measurable size? From the looks of it, when the universe was being formed there were two possibilities for how charge could be associated with matter, and as it turned out the current configuration won out over the other. If you remember the coin toss analogy, this would seem to indicate a predisposition to one particular configuration. Which means that somehow, fundamentally, the negative charge prefers a particle like the electron and the positive charge prefers a particle like the proton. That realization can be both puzzling and insightful at the same time. Unfortunately the state of modern physics is more puzzled by it than enlightened.

The second symmetry, Parity, is also steeped in mathematical and theoretical complexities, but it too can be reduced to a simple concept. Parity is the basis for the law of conservation (i.e. conservation of energy, momentum, charge, baryon number, lepton number etc.) This particular symmetry demands that if a particle is created (such as a proton) then there must be a corresponding opposite particle with all the opposite properties (charge, spin etc.). (Note: it says more than this and there is much, much more to parity than I am describing here) But again like what happened with C-symmetry, P-symmetry also has some problems when we consider the early universe. If we look at the type of matter present in the universe we see predominately one type of matter, which would indicate that when the universe was created P-symmetry was broken.

It is interesting to note that one way to fix a break in the symmetry of P or C is to have a corresponding break in the symmetry of C or P (i.e. if we observe a break in C-symmetry then there must be a corresponding break in P-symmetry to compensate). What this means is that even if there is a C or P symmetry violation then the opposite violation of P or C preserves symmetry and thus over all CP-symmetry is preserved.

This is all fine and dandy until we find that in all cases CP-symmetry is not preserved, and this brings us to the third symmetry in the standard model, T-symmetry. T-symmetry stands for time symmetry. What T-symmetry implies is that if you reverse the flow of time then you will still have the same physics (and hence results). This symmetry shows up in two major places. First as a way of "fixing" CP-symmetry violations, much in the same way that CP-symmetry was supposed to fix C and P violations, which results in CPT-symmetry. Thus even if individual symmetries are broken, they are preserved overall through CPT-symmetry.

The second place we see a break in T-symmetry is the flow of time in our universe. On a small scale this symmetry is preserved as it makes no difference which direction time is flowing the physics is still the same (things may move "backwards" but everything still works). The problem is that on a large scale (i.e. where the number of particles is large, >10^23) then things start looking different. We find that on large scales there is a specific direction to time and it does matter if time is going forward or backwards, and the physics is different. This symmetry breaking is the source of entropy (or it may be argued that entropy is the source of the symmetry breaking).

The purpose of large scale collider experiments is to probe the high energies similar to those that were present in the early universe to observe the symmetry breakings, and to find out their origin and effect on us now. At these scales some physicists assume that there is no symmetry, others assume that there is more symmetry (supersymmetry), or that it is impossible to observe the symmetry breakings that occured in the early universe (i.e. something about our universe prevents us from probing that high of energies).

In the end what particle physicists (and theoretical physicists) are attempting to do is to figure out why our universe is the way it is, and why is it that we observe perfect symmetry for most physical processes, but when we get to extreme or special cases symmetry is broken. By answering these questions we can figure out why the universe is and works the way it does.

1. Wow, pretty extensive article on CPT. Well done. I like the picture as it is very helpful.

For those who don't know, there are more symmetries than C, P and T but these three are very important.

Great post Quantumleap42!

2. Nice piece. Have been thinking about the nature of charge while writing about the structure of atoms. Nicely summed up!

3. I'm not aware of any proposed mechanism linking particle physics CPT-violations (e.g. Kaon decay) with the thermodynamic arrow of time (related to entropy). Thoughts?

4. As for linking CPT to the arrow of time, I don't have anything intelligent to say.

However, a complete breaking of CPT is related to a violation of Lorentz invariance.

5. But how can one define precisely what it means to "reverse the flow of time"? More generally, how can we possibly motivate any of our definitions of T, C and P? We could start by just assuming that the world is reversible. If fundamental physics is both deterministic and time-translation invariant, then there is always *some* involution under which the equations are time-reversal invariant as well. In the standard model, that involution is CPT symmetry. But that would just beg the question -- we want to *know* if the world is reversible, without just assuming it from the start!

6. Hi Bryan,

"More generally, how can we possibly motivate any of our definitions of T, C and P? We could start by just assuming that the world is reversible."

Good question.

I may not be able to answer your concern, but saying the world has CPT invariance translates to only allowing terms in your Lagrangian with CPT invariance. Then, we calculate the equations of motion, cross sections, decays rates, etc... from this Lagrangian and compare with experiment.

If what we observe doesn't match, we need to have either more or less terms. If we need more, this means the restriction of only allowing CPT terms is too harsh. This translates to CPT being violated. This would let us "know" the world is not CPT invariant.

However, a CPT invariant Lagrangian does match observation so now we need to ask: Is this because the universe really has these symmetries or is it
just coincidence.?

7. Thanks -- I'm willing to grant that physical theories are more than instrumental models. They describe the world. And it's an empirical fact whether or not the world can be modeled by a given Lagrangian. Moreover, it's just a mathematical fact whether or not a Lagrangian exhibits CPT invariance -- given a particular conventional definition of C, P and T. But why choose any particular definition of C, P or T? Answering your question of whether or not the universe "really has these symmetries" requires us to come up with a well-motivated (not just conventional) definition of what these symmetries mean "in the world." The difficulty is that there's no real operation that corresponds to "implementing T" or "implementing P" -- so it seems we must seek some other justification for their definition. Only then can we be justified in saying the *world* is CPT-invariant, and not just our physical theories.

8. To put the point in terms of your picture: It's an empirical fact whether or not something in the world can be described by the tree-shape on the left. And it's a mathematical fact that the tree-shape on the left is invariant under a certain reversal transformation defined on the tree-shape. What is missing is a well-motivated account of what that reversal transformation means in the world.

9. Bryan, interesting food for thought. I have something new to dwell on today.

10. Has anyone else seen the colour of air or sat atop of and looked down into the enlightened gravity of the "UnBeheld"? {otherwise known as the "Akashic Records","The Universal Pool Of Conciousness"}and be almost pulled into its sence of belonging that we all want and need.(its easy to ignore what we truly need when were blinded by lobbyists and complacency lol)

11. I think that time reversal symmetry is a key element in the creation of the universe. If nothing exists before the start of the Universe, there is no time. We imagine this state to be a quantum supposition of all states with time reversal symmetry. These are considered to be the "no time" states since time flowing one way is balanced by time flowing the other way. The supposition state spontaneously decays - no cause - and one of the supposition states becomes real. We have a real Universe in which time reversal symmetry holds. If the time reversal symmetry is spontaneously broken - these things happen - we get a universe with one way time. There is no way of knowing which Universe will be the final result - it is a random selection - except by observing the current Universe.

12. What if what we call recorded history has had a series of forward flows in time followed by backward flows in time? Also postulate that each individual flow of time at the beginning of the Big Bang almost equally cancelled one another out. With the staggering number of cycles of forward and backward flow since the Big Bang, would not the minute difference in these flows accumulate to give the net forward flow of time we observe on the macrosopic level today.  Is there any mathematical model in support of this hypothesis?