Archive for the ‘Physics’ Category

The Higgs Particle For Semi-Dummies

May 16, 2010

With the LHC ramping up, the hype about the Higgs particle is at frenzy levels, but what exactly is it?  You may have heard some frivolous descriptions, for example “the God particle”, but at the very least you might have heard that it gives the elementary particles their mass.  But how does it do such a thing?  In this post I will try to explain what it’s all about in a way which you wont need a physics degree to understand.  Of course I’m not an expert so for any corrections or suggestions please comment!

The Standard  Model

The best description we have of all the elementary particles is called the Standard Model.  It’s very technical, and not intuitive at all (to me at least).  It’s pretty amazing that it describes the way Nature works with such amazing accuracy.  The model is a gauge theory, and before we get to the Higgs, we have to understand what that means

Wikipedia tells us:

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations.

Let’s define each of the important terms:

• field theory – A theory which assigns a value to every point in space and time.
• Lagrangian – Defines the complete dynamics of a physical system.  The langrangian may have terms which depend on mass, for example, a standard lagrangian for a classical object is: $\mathcal{L}= Kinetic Energy - Potential Energy = 1/2mv^2 - mgh$, where m is the mass of the object.
• invariant – independent of.  If Y is invariant under X, then the value of Y will not change if X changes (and all other inputs remain the same).
• continuous group – we’ve discussed Groups on this blog before because they’re so fundamental, but if you need a refresher it’s an algebraic object which has a set of elements and is closed under a product operation.  Each element has an inverse so that the product of an element with its inverse is the identity element.  A continuous group is one where the product is a continuous function: infinite and without any discontinuities.
• local transformations – these are the elements of the above group.  These types of groups are called Lie groups.  an example is rotations of the sphere.

Putting it all together, a gauge theory defines a system where the global state of the system is unchanged with specific local changes.

Symmetry Group

These “local changes” are the elements of symmetry groups.  An example of such a group is the Circle group, which is  the rotations of a circle about its axis, called U(1).  The rotations have a product  (adding angles together), an identity, (the rotation of 0 degrees), and an inverse, (a rotation in the opposite direction).  It’s called U(1) because this group of transformations is represented by the set of all unitary matrices of dimension 1.

The Standard Model has 3 such symmetry groups: U(1), SU(2), and SU(3).  They represent 3 of the fundamental forces of nature: electromagnetic, weak nuclear, and strong nuclear respectively.  SU stands for special unitary, and SU(n) is the group of special unitary matrices of n-dimensions.

Somewhat bizarrely, The Standard Model says that the generators of these symmetry groups actually represent particles!  For U(1) there is 1 generator, which is the photon.  SU(2) has 3 generators which are the Z, W+, and W- particles.  SU(3) has 8 generators which are the 8 different gluons.

Symmetry Breaking

The Lagrangians described by these symmetry groups describe the complete dynamics of these quantum systems, however, they do not allow for any of these particles to have mass.  While this is fine for the photon and the gluon, the Z, and W+- must have mass to explain their observed behavior: namely their very limited range.

Incredibly, one can add mass to the Lagrangian for the weak force, if the SU(2) symmetry is broken.  As a first example, let’s see what it means to break U(1) symmetry.  If someone hands you a perfect circle, it is impossible to differentiate any point on the circle from any other.  However, if one breaks the symmetry by marking a point, then all points can be differentiated by describing how far they are from the marked point.

Spontaneously broken symmetry with a Higgs Field

The Higg field breaks the symmetry of SU(2) and gives us mass terms in the Lagrangian.  This  field can be represented by a Mexican Hat Potential, at the very center of which is a local maxima which represents a meta-stable state.  The stable states are in a circle around this local maxima which are the minimums of the potential.

Fields in Quantum Field Theory have an associated particle, and the particle representing this Higgs field is our Higgs boson.

It is the only particle predicted by the Standard Model which has not been discovered, but the LHC hopes to change that.  What’s interesting is that there are other ways to get mass to appear in the lagrangians.  The Higgs field is the simplest and most intuitive, however, it is possible that there is no Higgs particle and that mass is derived in some other manner.  That would certainly shake up the field of physics!

UPDATE (7/22/2012): The LHC found the Higgs and it appears to be just as the Standard Model predicted!

Why Time Flows from Past to Future

April 6, 2010

The direction of time is something most of us take for granted because it is so central and consistent in all of our experiences, but why does time seem to flow from past to future?  Why doesn’t our universe allow experiencing everything at once?  Why can’t we reverse the flow of time and go into the past?

Well the answer to these riddles lies in a surprisingly simple statement known as the 2nd law of thermodynamics:

In a closed system, heat flows from hot to cold.

Yup, that’s it.  This statement is pretty clear in every day life, if you drop an ice cube in hot water, the ice cube will melt and cool the water.  What we never experience is the heat from the ice cube flowing into the water, which would make the ice cube grow and cause the rest of the water to get hotter.  Of course adding work to a system can cause the 2nd law to locally go in reverse.  For example a freezer causes an ice cube to grow, but it does so by heating the the air outside the freezer.  The system as a whole still heats up.

He’s a chart of some equivalent ways to describe the 2nd law in other domains:

 Domain Statement of the 2nd law Example “special” state Example equilibrium state Heat Heat flows from hot to cold Ice cube in hot water Water all the same temperature Entropy Disorder increases A person Ashes Information Information is lost Beethoven’s 5th Symphony Static noise

The first thing to notice, is that these “special” states can be considered start states, while the equilibrium states can be considered end states.   That is, if some how the system is put into a special state, it will after a series of random and natural processes end in the equilibrium state.

So why does the universe work this way, and what does this have to do with the flow of time?  Let’s look at the ice cube in hot water example in more detail because it is the most tangible of the above examples, but the other examples can be explained in a similar fashion.

Why heat flows from hot to cold

As heat flows from hot to cold, the system will eventually reach a equilibrium, where everything is the same temperature.  At that point there is no more heat flow.  Dropping an ice cube in a cup of warm water will eventually lead to a cup of water which is all the same temperature somewhere in between the 2 starting temperatures.

If you think about all of the water molecules in the cup, let’s say for this example there are 2 billions molecules (1 billion hot and 1 billion cold), we can count all the ways in which these molecules can be distributed.  There are $2^{2 billion} \approx 10^{10^8}$ total possible states in this system.   These state spaces are so large it’s impossible to even begin to imagine their enormity.  These make other “large” numbers like the number of elementary particles in the universe ($\sim 10^{90}$)  and 1 googol seem minuscule in comparison.  The vast majority of these states are equilibrium states, while the special starting states are insignificant.

If you pick a state at random from the total space of water molecules in a cup, the state where some of the molecules are ice and some are hot, is effectively 0.  (To do a quick calculation try a binomial distribution with something like Binomial(2 billion, 2 million) = ${2 billion \choose 1 million} / 2^{2 billion} \approx 0$.)

Maxwell’s Demon

Many physicists thought the 2nd law was purely statistical and could be easily be violated by intelligent manipulation of states.  One example of this is called Maxwell’s demon, where a theoretical “demon” reverses the flow of heat without using any work.  However, physicists have shown that no matter how hard you try, the system as a whole cannot violate the 2nd law.  For example Maxwell’s demon must either conduct work, which will causes heat in another part of the system, or lose information, which we learned is equivalent to the 2nd law.  So the 2nd law survives even with small state spaces, however, the huge numbers of states in everyday molecular interactions is what makes time so indomitable.

Time

So the answer to why heat flows from hot to cold is essential a question of probability.  The equilibrium states dominate the special states, so unless you start in one of the special states you will never reach it.   The same can be said about our universe.  The Big Bang is the ultimate start state and Heat Death is the ultimate end state.  The big bang represents time = 0 while heat death represents the end of time.

The flow of time from past to future can therefore be defined as special states of a system transitioning to their equilibrium.  The sheer magnitude of the state space is what makes time so dominant in the Universe.

3D Movies and Quantum Mechanics

February 3, 2010

As 3D media becomes more popular, it’s interesting look at the technology which allows light to be targeted to the left and right eyes separately.  If you saw Avatar in the theaters you donned glasses which have polarized lenses.  We can call the filter over the left eye “L-polarized” and the right filter “R-polarized”.  The movie projector emits unpolarized light, but that light can be passed through a polarization filter which is synchronized to each frame of the movie.  Frames meant for the left-eye are L-polarized so that they match the left-eye of the glasses, while frames meant for the right-eye are R-polarized.  In the RealD technology, the filter switches between L and R polarization 240 times per second, allowing 120 Hz frame rate for each eye.

Something interesting about polarization is that it works even with a single photon.  If you send an unpolarized photon through a L-polarization filter there is a 50% chance it will be blocked and a 50% chance it will pass through.  If you then this L-polarized light through another L-polarization filter it will pass through 100% of the time, but with an R-polarization filter it will pass through 0% of the time.  This is a purely quantum mechanical (QM) effect that cannot be explained through classic means.

A basic rule of QM is that if you want to observe the state of a system, you must make an observation by operating on the system.  In this case our observable is the polarity of the photon and our operator is the polarization filter.

Incredibly, QM says that the eigenvalues of a quantum operator are the observables, where the states of the quantum system are the eigenvectors!  If we can remember our eigenvectors and eignvalues from linear algebra we have the following:

$H\, \psi = \lambda\, \psi$

where, $H$ is the operator, $\psi$ is the eigenvector and $\lambda$ is the eigenvalue.

Now for our observables to be real our eigenvalue solutions to this equation must be real, since QM tells us they are equivalent. (We assume our observables are real because we can only measure real values in nature, not imaginary one.)  In linear algebra, to assure that we have real eigenvalue solutions, the operator should be a Hermitian matrix.  This is a square matrix where the entries on opposite sides of the main diaganol are complex conjugates of each other.

Here is a table which sums up the relations in this equation:

Symbol Math QM 3D Movies
$H$ Hermitian matrix quantum operator L or R polarization filter
$\psi$ eigenvector quantum state L or R polarized light
$\lambda$ eigenvalue observable L or R measurement

What’s bizarre is that when unpolarized light passes through the filter, it must “decohere” into the eigenvectors of L or R polarized light so that we get a real observable. The process of decoherence is one of the great mysteries of QM: particles only exist as probabilities until they are measured, at that point, Nature appears to “reset” them into a well-defined state.

Hello World!

November 10, 2008

I’m putting together a blog of things I find interesting.  Hopefully I will update it at least once a day, since I’ve always felt that is the sign of a good blog.

What a better topic to start with than the exciting results from the CDF experiment at Fermilab running on a particle super-collider called the Tevatron in Illinois.  All high-energy physics experiments have confirmed the Standard Model (SM), which was discovered in the 60s,  but some new results from CDF do not fit into this model!  There could be many reasons for this, but it could be the signature of an unexpected particle!  Considering that the SM has been perfect for the past 40 years, it seems quite likely that there is a more mundane explanation for the results (e.g. unexpected background noise), but that will not keep the physicists from theorizing about new potential particles.

Here is a basic explanation of the results:CDF Ghost Muons

Here is more detail from one of the experimenters: CDF publishes multi-muons!!!!

And here is the actual paper: Study of multi-muon events produced in p-pbar collisions at sqrt(s)=1.96 TeV

Another interesting phenomena is not only the physics involved, but also the physicists themselves and their communication.  Many world-class physicists regularly post comments on other physics blog.  (Is there another academic subject which has this same level of open communication??)  And when physicists communicate sometimes there is fireworks.  In the case about the CDF, one physicist accused another of getting leaked results of the CDF experiment and used it his paper which happens to coincide with the unexpected CDF results.

The author of the paper stated: “I can tell you officially we had no word on this [the CDF reuslts]. This blog is, in fact, the first I’d heard of it”.

To which a CDF experimenter replies: “That is pretty hard to digest. Lepton jets with lifetimes. Come on. I think you owe it to the physics community to let us know where the leak came from.”

Which then received a several paragraph diatribe scorning the skeptics and the blogging community: “I find your cynicism remarkable . . . at least most of us don’t think of physics as a soap opera rife with rumor and innuendo, or spend the precious time we have cynically tossing around completely baseless and deeply offensive accusations.”