# Dirac equation and clifford algebra

• TimeRip496
In summary, it is not necessary to have knowledge of Clifford algebra in order to derive the Dirac equation. Some sources, such as the video by drphysics, use two waves moving in opposite directions to derive the equation without touching Clifford algebra. However, most introductory books do use Clifford algebra in the process of deriving the equation, as it is a natural representation of the proper orthochronous Lorentz group. The famous Handbook "Enzykloädie der mathematischen Wissenschaften" and the book "Atombau und Spektrallinien" also use alternative methods, such as quaternions, to derive the Dirac equation. Ultimately, all methods lead to the same result.

#### TimeRip496

Is it a must to know clifford algebra in order to derive the dirac equation?
I recently watch drphysics video on deriving dirac equation and he use two waves moving in opposite directions to derive it, without touching clifford algebra. If this possible, what is the intuition behind it?

No you don't "need to" know Clifford Algebra... you derive it in the process in order to get your equation working [and reproducing the relativistic relationship between energy and mass/momenta] . At least by the time one adds the gamma matrices..

vanhees71
ChrisVer said:
No you don't "need to" know Clifford Algebra... you derive it in the process in order to get your equation working [and reproducing the relativistic relationship between energy and mass/momenta] . At least by the time one adds the gamma matrices..
Do you have any sources that will allow me to derive the dirac equation without clifford algebra? Cause I have been searching on the Internet and all of it involves clifford algebra. Besides the one by drphysics he didnt really derive it fully such as the dirac matrix.

I haven't seen what you are talking about, I don't know which is that drphysics book... Maybe this can help? http://arxiv.org/vc/quant-ph/papers/0607/0607001v1.pdf (sec.4)

However, I don't understand this "clifford algebra" thing. In every introductory book, they never used the Clifford algebra to derive the Dirac equation.
Most of them are starting by a linear Hamiltonian operator, which they act to reproduce the result of the relativistic energy-momentum-mass relation $E^2 = m^2 +p^2$. By doing so, you see that the objects that you used in your operator were not just complex numbers, but they should satisfy certain relations, one of them is the Clifford Algebra. You don't "use" it, you "result" in it...

Well, you can call it as you like. The Clifford algebra is basically the algebra realized in the usual treatment by the Dirac matrices in the one or the other representation. At some time ago in history also quaternions and octonions were en vogue. In the 19th century at least; Maxwell formulated his equations first in terms of quaternions; the usual vector notation was introduced into physics by Heaviside and became widely used in the beginning of the 20th century. It was not the least spread by the famous Handbook "Enzykloädie der mathematischen Wissenschaften", edited by Felix Klein around 1900. You can find a derivation of the Dirac equation, using quaternions in the even more famous book by Sommerfeld "Atombau und Spektrallinien" (freely translated: Atomic Structure and Spectral Lines). As stressed before: No matter, how you derive it you end up with somehow with the Dirac formalism. This is no surprise, because it's a very natural representation of the proper orthochronous Lorentz group augmented by spatial reflections to be able to describe the parity-conserving interactions, especially the electromagnetic interaction, which was the first application of the Dirac formalism of course.

## 1. What is the Dirac equation?

The Dirac equation is a relativistic quantum mechanical wave equation that describes the behavior of fermions, such as electrons, in a relativistic quantum system. It was developed by physicist Paul Dirac in 1928.

## 2. What is clifford algebra?

Clifford algebra, also known as geometric algebra, is a mathematical structure that extends the concepts of complex numbers and vectors to higher dimensions. It is based on the work of mathematician William Kingdon Clifford and is used in a variety of fields, including physics, computer graphics, and robotics.

## 3. How is clifford algebra related to the Dirac equation?

Clifford algebra is used to represent the mathematical objects, known as spinors, that are necessary for the formulation of the Dirac equation. The properties of clifford algebra, such as its ability to handle rotations and reflections, make it a useful tool in understanding the behavior of particles described by the Dirac equation.

## 4. What are the applications of the Dirac equation and clifford algebra?

The Dirac equation and clifford algebra have a wide range of applications in various fields of physics, including quantum mechanics, particle physics, and cosmology. They are also used in engineering, particularly in the fields of computer graphics and robotics.

## 5. Are there any limitations to the Dirac equation and clifford algebra?

While the Dirac equation and clifford algebra have been successful in describing the behavior of particles at the quantum level, they are not able to fully explain all phenomena, such as gravity. Additionally, the mathematics involved in clifford algebra can be complex and difficult to work with, making it challenging to apply in some situations.