Posts Tagged ‘group theory’

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!


The Poincaré Conjecture Explained

March 13, 2010

The Poincaré Conjecture is first and only of the  Clay Millennium problems to be solved.  It was proved by Grigori Perelman who subsequently turned down the $1 million prize money, left mathematics, and moved in with his mother in Russia.  Most explanations of the problem are overly-simplistic or overly-technical, but on this blog I try to hit a nice middle-ground.  Here is the statement of the conjecture from wikipedia:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

This is a statement about topological spaces.  Let’s define each of the terms in the conjecture:

simply connected space – This means the space has no “holes.”  A football is simply connected, but a donut is not.  Technically we can say \pi_1(X) = 0 but I’ll explain this notation further on.

closed space – The space is finite and has no boundaries.  A sphere  (more technically a 2-sphere or S^2) is closed, but the plane (R^2) is not because it is infinite.  A disk is also not because even though it is finite, it has a boundary.

manifold – At every small neighborhood on the space, it approximates Euclidean space.  A standard sphere is called a 2-sphere because it is actually a 2-manifold.  Its surface resembles the 2d plane if you zoom into it so that the curvature approaches 0.  Continuing this logic the 1-sphere is a circle.  A 3-sphere is very difficult to visualize because it has a 3d surface and exists in 4d space.

homeomorphic – If one space is homeomorphic to another, it means you can continuously deform the one space into the other.  The 2-sphere and a football are homeomorphic.  The 2-sphere and a donut are not; no matter how much you deform a sphere, you can’t get that pesky hole in the donut, and vice-versa.

A donut and a coffee mug are homeomorphic.

What the conjecture is basically saying then is this:

Any finite 3-dimensional space, which doesn’t have any “holes” in it, can be continuously deformed into a 3-sphere.

This was proven for all dimensions > 3, but 3-manifolds proved to be the trickiest.

Now, how do you show that 2 spaces are homeomorphic to each other?  One way is to show that the spaces share the same fundamental group.  Hopefully you remember what a group is, but basically it’s a collection of elements which includes an identity element, and a  product.  Also each element has an inverse.  The fundamental group basically describes how many ways you can draw a path in a space.  It is organized as follows, where each element is a path which starts and ends at the same point.  The identity fundamental group is the trivial path of length 0, i.e. a single point.  The product in the fundamental group is just combining 2 paths end-to-end.  Since all paths start and end and the same point you can always take their product and get a new element in the group.

The fundamental group of the n-sphere is equivalent to the group with one element (the identity element), because any path can always “shrink” down into a single point due to the fact the n-sphere has no hole.  If a space’s fundamental group is trivial, then that means it is simply connected, as described above.  To say the fundamental group for any n-sphere is trivial you use the following notation:

\pi_1(S^n) = 0

Where \pi_1 means this group contains elements of 1-dimensional closed paths.   A little bit off topic but even more fascinating, is that you can also examine higher dimensional paths and these non-fundamental groups develop an interesting and complex structure.

Anyway, the Poincaré conjecture says that if for any 3-manifold, M^3, if \pi_1(M^3) = 0, then M^3 can be continuously deformed into S^3.

Perelman finally solved this for 3-manifolds by using Ricci flow techniques to show that these manifolds are homeomorphic.

Update: 6 years after his proof of the Poincare Conjecture and 5 days after this post, Clay Mathematics officially awarded the Millennium prize to Grigori Perelman.  If he declines, this could be the first time in modern history where someone living at home refuses $1 million.

Group Theory

December 14, 2008

I already wrote about Set theory and I eventually want to write about symmetry, so this seems like the perfect opportunity to discuss Group theory.  Let’s start where we left off with Set theory.  We had a very abstract concept of a Set: a collection of “things”.  Then we added some basic structure to that collection (the operation of set concatenation), and before we knew it we were counting and adding.

Well, to formalize it a bit, when you take a set and add a binary operation, and an identity element, then what you get is called a monoid.  Monoids are extremely abstract yet very powerful.  (In a way they even form the underpinning of all of computer science, where the set is a collection of functions, and the product is simply chaining these functions together, called function composition, and the identity is simply a “noop“.)  I’ll talk more about this in a future post because it is so cool how these ideas tie together.

A Group is just like a Monoid, except every element in the set must also contain its inverse.  When an element operates on its inverse, it results in the identity!  Some basic examples, are the integers\mathbb{Z}, where every element also has its addtive inverse, called its “negative”.  Another example are the rationals, \mathbb{Q}, where every element has its multiplicative inverse, called its “reciprocal”.  In the former example, the operation is addition and addtive identity is 0, while in the latter the product is multiplication and the multiplicative identity is 1.  Both of these groups also commute (i.e. the order of operation does not matter), and we call these groups “abelian“.  They are named after the mathematician Niels Henrik Abel, but strangely the word is not capitalized.

These groups may seem pretty abstract at first, but here are a few more tangible examples.  First, is that to physicists, a set with elements and their inverses will immediately remind one of a set of particles and their antiparticles.  But we can actually get much more tangible than that if we pull the same trick we pulled above with our monoid:  instead of thinking about objects, get Categorical, and think about processes.  It’s easy to imagine a set of transformations and their inverses: raise and lower, shrink and grow, cut and paste.  All of these examples have the identity of  “do nothing”.

Groups are strongly related to symmetry because an object which is equivalent to the original after transformations is at the heart of symmetry!  In future posts we will talk about the groups of rotations and reflections to explore these concepts.