Sunday, 15 July 2012

Understanding the Higgs Boson

Unless you have been hiding under a rock somewhere you will almost certainly have seen the news that the Higgs boson has been 'discovered', I say this tentatively as it has not officially been confirmed just a 'Higgs-like' particle has been observed. But what even is the Higgs boson?

By the 1970s physicists had devised what is known as the "Standard Model" of particle physics - it is a theory of the elementary particles, how they interact with each other and what these interactions mean. The elementary particles can be broken down into two core categories: fermions and bosons. Fermions are matter particles, they are what make protons, atoms, stars and us, whereas bosons are what helps the fermions to "communicate", this communication is called a force.

The standard model predicted the existence of a lot of particles far before they were discovered: the bottom quark, top quark, tau neutrino and the Higgs boson. All of these particles were required for the standard model to work in it's current form. In 1964 Peter Higgs (as well as Robert Brout and Francois Englert) devised a theory (that was later proven) for how elementary particles are able to have mass, this was called the Higgs mechanism.

The Higgs mechanism requires a field pervading through the whole universe and depending on how particles interact with this field dictates what mass they have, this is called the Higgs field. In quantum theory when a field interacts with another object it acts as a particle. In the electromagnetic field this particle is a photon, in the Higgs field the particle is the Higgs boson. The discovery of this particle means that the equations that are currently in place are correct and able to describe every particle we know about and how these particles interact together via the forces incorporated in the standard model (all except gravity), so essentially we now certainly understand the quantum world to a great deal of precision!

The reason the Higgs boson has eluded us for so long is that it is very, very unstable and decays incredibly quickly - in fact it only exists for approximately one zeptosecond, a thousandth of a billionth of a billionth of a second. Because it exists for such a short amount of time you have to observe the after effects of a collision and map how much energy the produced particles have and compare this to what you would expect if a Higgs boson had decayed, the problem is that it can decay into a lot of things. So you have to perform a lot of high energy collisions between particles, observe the data and see if it fits the predictions about how the Higgs boson should behave.

Properties of Higgs Boson:

Mass: 126 GeV/c2 which is equivalent to 2.25 × 10-25kg or 134 protons.
It has no electric charge or spin
Read about how the Higgs interacts with other particles

But is this it? Do we understand everything now that we have discovered the long elusive 'God particle' (a terrible, terrible term coined by the media)? The answer is a wonderfully fanatic, no! Thankfully. In fact, the discovery really means that we can now raise more questions and further advance our understanding of everything. At the smallest scale gravity is still a total mystery to us, it is not incorporated into the current standard model at all - one of the most primitive every day forces is still a complete mystery to us! Quantum mechanics and general relativity need to be united before we can even begin to ponder that we understand everything, we are some way off from a theory of everything but the best hat in the ring currently is string theory.

Just as one final note, the Higgs boson has not been officially confirmed as to existing but that a new, previously unknown boson with a mass between 125-127 GeV/c2 that has behaviour "consistent with" a Higgs particle. Further analysis is required to fully confirm it's existence but a very cautious tick is currently next to it!

Monday, 9 July 2012

Fermat's Little Theorem

Before reading through this post it is useful if you have some knowledge of modular arithmetic.

Fermat's little theorem (
it's 'little' because of Fermat's comparatively more difficult to prove last theorem, not because it is less useful, in fact it is more useful) is a vital result from the field of number theory as it provides a method of checking (within reason) whether a number is prime or not. Also it is a very interesting result that utilises modular arithmetic.

Fermat's little theorem is that given a prime number, p, and any integer, a, ap - a will divide by p to give an integer value. Or, more mathematically, p|ap - a. So from the definitions in modular arithmetic this means that ap ≡ a (mod p); it can also be wrote as ap-1 ≡ 1 (mod p) - both are equally valid but the second is used more often mainly because it has a 1 on the right hand side and this is 'neater' and mathematicians love things to be neat!

There is a condition for the integer
a and that is that it is coprime to p. Coprime means that the highest common factor between two numbers is 1; so for example 3 and 46 are coprime as the only factor they have in common is 1. The reason for why a must be coprime to p will come later.

When Pierre de Fermat first stated this theorem in a letter to his friend (in 1640) he did not include a proof of why this is the case as it was "too long". To attempt to 'one up' one of the greatest mathematicians in the past 400 years I will include a proof of this fact.

There are three points of contention within this proof that I feel I should justify/explain further. (A) is that these numbers are chosen intelligently in order to prove Fermat's little theorem, this was not the original proof of the theorem as it is difficult to be able to choose these numbers initially but in hindsight it is clear that these work very well.

(B) requires further explanation. The reason that when a, 2a, 3a,..., (p - 1)a are divided by p will give remainders 1, 2, 3,..., (p - 1) in some order is because of 
Euclid's lemma, if a prime number divides the product of two numbers then the prime number must divide at least one of the factors. Clearly none of the numbers in list A can be divided by p (a is coprime to p), so some integer in the range of list B, n, is coprime to p too - this means that na can not be divided by p so this means that a, 2a, 3a,..., (p - 1)a when divided by p must have remainders in the region of 1, 2, 3,..., (p - 1). Read more on this.

For (C) why can we cancel out (p - 1)! in modular arithmetic? It isn't an equals sign after all. If we consider ax 
≡ ay (mod n), this means that ax - ay can be divided by n, factorising this we get a(x - y) can be divided by n implying that either a can be divided by n or x - y can be. In our case a is (p - 1)!, x is ap - 1 and y is 1; but (p - 1)! is clearly coprime to p so that means that ap - 1 - 1 can be divided by p so we can in fact cancel the (p - 1)!s.

And that is Fermat's little theorem fully proved! You will have almost certainly encountered Fermat's little theorem whether you knew it or not, almost all online security and encryption utilises prime numbers to stop data being intercepted and Fermat's little theorem is used to check the primality of numbers to be used.

If you do not understand anything in this post, feel I have made a mistake or need extra explanation on anything covered here leave a comment and I will get back to you as soon as I can.