Posts Tagged ‘symmetry’

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!

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.