Paul Adrien Maurice Dirac:And the Discovery of the Dirac Equation

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When young Paul with a wee ladPaul Adrien Maurice Dirac was born in Bristol,

England on August 8th, 1902. His father, Charles Adrien Ladislas Dirac, was originally from Switzerland. His mother, Florence Hannah Dirac, was born in Cornwall, England.

Including his parents, Paul’s family also consisted of his younger sister, Beatrice Marguerite, and his older brother, Reginald Charles Felix.

It appears that Paul got along well with his mother, but unfortunately struggled with his father.

Charles apparently forced his children to speak to him in French only. This was his way of teaching them the language.

Paul found that in the moments when he could not find a way to express something in French, he would remain silent instead. This lead him to be a very quiet person later on in life.

Disclaimer: speaking French; not related to french fries

Unfortunately, when he was in his early 20s, his older brother, Felix, committed suicide in March 1925.

Paul later on in life talks about his parents and how they were distraught about his brothers passing. Apparently he did not think parents loved their kids.

Paul went on to marry Margit Wigner, and adopt her daughters Mary Elizabeth and Florence Monica.

These particular events most likely contributed to shaping Paul into the detailed thinker he became later on in life.

I like thinking!

The Man Behind the MadnessPaul Dirac was a quiet and precise individual.

He was also a little socially awkward, sometimes described as a geek and not always picking up on human intentions.

He strongly believed that the equations of physics should be aesthetically pleasing and should be built upon elegant mathematics.

He often criticized Quantum Electrodynamics (QED), particularly, the renormalization procedure.

When asked what he thought of QED, he is quoted as saying,

“I might have thought that the new ideas were correct if they had not been so ugly.”

In his younger years, Paul also criticized God and religion, dismissing it as nonsensical, with no basis in reality and seeing no need for “the postulate of an Almighty God.” Although, in his later years he seemed less critical, attributing the reason to the elegance of physics to God constructing a mathematical universe.

Works and Activities in PhysicsPaul worked on a number of subjects in physics

and mathematics.He quantized the gravitational field.He independently worked on and discovered

Fermi-Dirac Statistics, which is used for multi-fermion systems or multi ½ spin particle systems i.e. electrons.

He introduced the idea of vacuum polarization.He worked on the theory of magnetic monopoles.He formed the basis for gauge field theory

techniques.

Paul Dirac’s greatest work is that of the Dirac equation.

It is the relativistic invariant version of the Schrodinger equation.

The description of electron spin naturally comes out of the mathematics.

The equation naturally describes the positron or anti-electron, theorizing its existence four years before its discovery.

It is extremely accurate and has been the corner stone to QED (which, oddly enough, Paul disagreed with, despite the fact that his theory lead to its construction.)

The Equation and its DerivationTo derive the Dirac equation, one must first

look for a relativistic version of the Schrodinger equation.

The Schrodinger equation is essentially a conservation of energy equation, with the non-relativistic form,

where the first term on the left hand side is the kinetic energy of the particle.

The kinetic energy term in the Schrodinger equation is the non-relativistic energy, and is only accurate at low speeds. In order to have an accurate model of the electron, we need to use the relativistic version of energy given to us by Einstein,

Notice this equation doesn’t have a U term. I’ll get back to this later (in the mean time, our equation will describe a free electron i.e. free from external fields.)

In order to get a wave equation describing the electron, we need to start how we started with deriving Schrodinger’s equation.

We will eventually get to the same point where we plug in our differential operators,

in place of our momentum and energy variables.

Using our differential operators, our new conservation of energy equation becomes,

Due to this equation having a second order time derivative, this causes a number of issues in the interpretation of the equation and use of it.

To try and fix this, rewrite the equation in the form of,

Paul Dirac considered this equation. In order to get a time derivative of order one, you would need the derivative with respect with time to be done once. In the above equation, it is done twice. If one could take a square root of the left hand side so that one ends up with an equation of the form,

where H is some differential operator, this would then solve that problem.

Taking a literal algebraic square root would not be helpful because solving a differential equation with a square rooted derivative is highly unorthodox and difficult.

Dirac came to the idea that if the hypothetical operator H could be expressed as linear combination of first order spatial derivatives and the square root of the constant term, then squaring that operator would be the same as distributing the linear derivatives and constant term with themselves. In this way, our operator H would look like,

where the a’s and beta are yet to be determined.

Squaring our H operator we get,

This equation looks pretty ugly, but all we did was just distribute the terms of H with itself like you would normally do with parentheses.

Now remember, we want the squared H to reduce to the second order differential equation we produced earlier. Because derivatives commute, we can see that in order for this to happen, we need,

and,

for i, j = x, y, z.

If we define our alphas and beta in this way, then our H squared and our differential equation agree.

In this way we found our much desired equation,

because,

which is what we were looking for. It is at this point an electromagnetic potential (the

U that we had earlier) could be added in order to describe an electron near electromagnetic fields (in an atom for example.) I won’t do that here for simplicity.

The Constants and Dirac’s Solution(s)So what are our a’s and our beta? There are a number

of answers, all of which are equivalent.These constants can be put together to form an algebra

between the constants known as the Dirac Algebra.They can also be put together to form a Clifford

Algebra known as Spacetime Algebra.The most widely used method of handling these

constants is to write them as 4 x 4 matrices which have the properties mentioned earlier.

We can always absorb an ‘i’ or a (-1) into the matrices, so their defined properties can vary, and there are a number of matrices which can satisfy each, so which equation and matrices are used will change from author to author.

Because our a’s and beta are matrices, our probability function can no longer be treated as a scalar function, because otherwise, in the energy conservation equation, you would get a matrix equal to a scalar, which makes no sense mathematically.

Instead, the probability function must be treated as a 4 x 1 matrix with functions for its indices. This way, when you solve the energy equation you can get a reasonable result.

Because of this, one ends up with 4 different ‘types’ of electrons that solve the equation because of the 4 components of the probability matrix.

The 4 different ‘types’ of electrons that the equation describe are:

1. An electron with positive energy with spin up2. An electron with positive energy with spin down3. An electron with negative energy with spin up4. An electron with negative energy with spin down

The equations naturally describe a particle with spin. It literally falls out of the equations!

The negative energy solutions caused a lot of concern for physicists. If electrons could have negative energy, then that means that positive energy electrons could continue to loose energy until they become negative, and then continue loosing energy, potentially releasing an infinite amount of energy over an infinite amount of time, which is not what we observe.

To solve this problem, Dirac came up with a theory known as the negative energy sea (no longer thought to be valid). In this theory, he predicted there would be a particle similar to the electron, but with positive charge, the opposite of the electron.

About 4 years later, the positron was discovered!

The common interpretation today is to regard the negative energy solutions as positrons going backwards in time.

Since then, the Dirac equation has a had number of abstractions built upon it that build the foundations of QED, the best model we have for the electron and electromagnetism.

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