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 theoryis 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: , 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.

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!

Tags: group theory, Higgs, Langrangian, LHC, symmetry

May 18, 2010 at 12:04 pm

When particles behave like groups of numbers and groups of numbers can be shown to behave like particles… is it still weird to ask if there might be a case for fundamental interactions to be emergent properties of number? More specifically, if it takes a minimum number of bits (literally) to describe a physical property of the early universe (or sub-atomic particle interactions) … where do the math and the physics separate in reality?

July 9, 2010 at 10:32 am

Interesting that there are people trying to explain interesting facts from science. you might want to consider the fact that we are living in a 4d world, but at subatomical level math works with more that 7 dimensions. how do we think in 7d?

July 18, 2010 at 7:06 am

how do we think ‘in’ the subatomic level? using maths… our eye-perceived 4-ds are ‘built’ (or at least what we can perceive) from/of the subatomic 7-ds… like perceivable approximations of how the 7ds are organised?

We can’t perceive individual electrons without mechanical help, we perceive their effects ‘en masse’ as it were with an electric shock.

(I dunno lol)

June 30, 2011 at 5:52 am

dannyburton posts and interesting point. Mathematics is the only true way to perceive the multi-dimensional aspects of quantum physics. The simple trick of looking at derivative equations can help abstract things into our level of understanding. We live in a 3d world by our perceptions. Time, of course is not a physical element but more a perception. That is why Einstein posted the idea of “time-space” which is far more accurate. From our perception reality is composed of trillions of snapshots (still photos) that are cognitively simplified and represented in our minds as a memory. At any given time we are looking at the past, not the present, in that this process takes time. In order to get around all the error both in our own machinery and the tools we use we must look at the relative relationships of objects indirectly. This is especially true at a quantum state. Again, mathematics is the only way to represent these relationships with any hope of accuracy. You can not draw a picture of multi-dimensional space-time as there are no biological tools (e.g. eyes, ears, etc.) to upload the data. Therefore one must keep the abstractions in the mind in order to perceive them.

… now if that does not feel like a “Jack Handy” moment I do not know what does …