Featured Research

from universities, journals, and other organizations

A mathematical study of the famous Dirac equation that describes particles

Date:
January 7, 2013
Source:
Elhuyar Fundazioa
Summary:
In 1928 the British physicist Paul Dirac put forward one of the fundamental equations that we use today to mathematically describe a spin one-half particle from a relativistic point of view. The mathematical representation that Dirac came up with enables certain particles, including the electron, to be better understood. Nevertheless, much more remains to be discovered.

Naiara Arrizabalaga, PhD holder in mathematics, UPV/EHU.
Credit: Image courtesy of Elhuyar Fundazioa

In 1928 the British physicist Paul Dirac put forward one of the fundamental equations that we use today to mathematically describe a spin one-half particle from a relativistic point of view. The mathematical representation that Dirac came up with enables certain particles, including the electron, to be better understood. Nevertheless, much more remains to be discovered.

For the case of particles like electrons that move at great speed, it is very important that the equation that describes them should bear in mind the contribution of the theory of relativity, since at high speeds the effects of this theory become clear. Although Schrödinger had previously discovered an equation that describes the movement of the electron, his equation does not take the theory of relativity into consideration.

The complexity of the structure of Dirac's equation makes it very difficult indeed to study it. "There are fewer pieces of work on Dirac's equation than on other equations on partial derivatives like, for example, that of waves or that of Schrödinger," says the mathematician Naiara Arrizabalaga. "It has a very complicated structure. Just as the equations that describe heat or waves are written as a single equation in partial derivatives, the Dirac one is a system of four equations related to each other. This is because the operator associated with the Dirac equation is a differential matrix operator. "

Making the unresolvable resolvable

Arrizabalaga's PhD thesis has studied Dirac's relativistic equation for the precise reason that few pieces of work have been done on it. Specifically, the thesis has set out to study the self-adjoint extensions of the Dirac operator with different potentials, including the electromagnetic potentials with singularity at the origin, using inequalities of the Hardy-Dirac type for this purpose.

There is one condition in particular that must be met so that the Dirac equation has a solution and that this solution is the only one:the operator associated with the equation must be self-adjoint, in other words, it must be symmetrical and its domain must coincide with that of its adjoint. In the cases in which it is not possible to prove that the operator is self-adjoint in a certain domain, then it is interesting to build self-adjoint extensions.

Arrizabalaga has studied what these extensions have to be like when the Dirac equation is applied to different potentials. "The Dirac equation is based on a physical reality which is the movement of certain particles. But in the reality around us these particles are not alone, they interact with others and are under the influence of electromagnetic fields," says Arrizabalaga. And that is why she has studied the Dirac operator with electrical and magnetic potentials. The first part of the thesis deals with diagonal electrostatic potentials, and the second tackles more general electromagnetic potentials that have a Coulomb-type singularity.

The construction of the self-adjoint extensions for all the potentials studied are related to Hardy-Dirac type inequalities, which are proven in this same piece of work and which are of independent interest owing to the methods involved in the demonstrations and the different uses they have.

Another interesting aspect about the Dirac equation is that it can be understood as a dispersive equation, in other words, it describes a wave system that is dispersed in time and space. This is why the equation meets certain dispersive estimates. The thesis has concentrated specifically on Strichartz estimates. Counterexamples are builtin the last part of the thesis for the Strichartz estimates for the Diracmagnetic equation, and what is more, counterexamples have been found for the wave equation.

In short, the thesis has striven to further certain mathematical methods that allow progress to be made in resolving the Dirac equation. What is more, it is believed that the methods created in this piece of work will be of use in other equations.


Story Source:

The above story is based on materials provided by Elhuyar Fundazioa. Note: Materials may be edited for content and length.


Cite This Page:

Elhuyar Fundazioa. "A mathematical study of the famous Dirac equation that describes particles." ScienceDaily. ScienceDaily, 7 January 2013. <www.sciencedaily.com/releases/2013/01/130107082226.htm>.
Elhuyar Fundazioa. (2013, January 7). A mathematical study of the famous Dirac equation that describes particles. ScienceDaily. Retrieved April 23, 2014 from www.sciencedaily.com/releases/2013/01/130107082226.htm
Elhuyar Fundazioa. "A mathematical study of the famous Dirac equation that describes particles." ScienceDaily. www.sciencedaily.com/releases/2013/01/130107082226.htm (accessed April 23, 2014).

Share This



More Matter & Energy News

Wednesday, April 23, 2014

Featured Research

from universities, journals, and other organizations


Featured Videos

from AP, Reuters, AFP, and other news services

Is North Korea Planning Nuclear Test #4?

Is North Korea Planning Nuclear Test #4?

Newsy (Apr. 22, 2014) — South Korean officials say North Korea is preparing to conduct another nuclear test, but is Pyongyang just bluffing this time? Video provided by Newsy
Powered by NewsLook.com
China Falls for 4x4s at Beijing Auto Show

China Falls for 4x4s at Beijing Auto Show

AFP (Apr. 22, 2014) — The urban 4x4 is the latest must-have for Chinese drivers, whose conversion to the cult of the SUV is the talking point of this year's Beijing auto show. Duration: 00:40 Video provided by AFP
Powered by NewsLook.com
Hagel Gets Preview of New High-Tech Projects

Hagel Gets Preview of New High-Tech Projects

AP (Apr. 22, 2014) — Defense Secretary Chuck Hagel is given hands-on demonstrations Tuesday of some of the newest research from DARPA _ the military's Defense Advanced Research Projects Agency program. (April 22) Video provided by AP
Powered by NewsLook.com
Lytro Introduces 'Illum,' A Professional Light-Field Camera

Lytro Introduces 'Illum,' A Professional Light-Field Camera

Newsy (Apr. 22, 2014) — The light-field photography engineers at Lytro unveiled their next innovation: a professional DSLR-like camera called "Illum." Video provided by Newsy
Powered by NewsLook.com

Search ScienceDaily

Number of stories in archives: 140,361

Find with keyword(s):
 
Enter a keyword or phrase to search ScienceDaily for related topics and research stories.

Save/Print:
Share:  

Breaking News:
from the past week

In Other News

... from NewsDaily.com

Science News

Health News

Environment News

Technology News



Save/Print:
Share:  

Free Subscriptions


Get the latest science news with ScienceDaily's free email newsletters, updated daily and weekly. Or view hourly updated newsfeeds in your RSS reader:

Get Social & Mobile


Keep up to date with the latest news from ScienceDaily via social networks and mobile apps:

Have Feedback?


Tell us what you think of ScienceDaily -- we welcome both positive and negative comments. Have any problems using the site? Questions?
Mobile iPhone Android Web
Follow Facebook Twitter Google+
Subscribe RSS Feeds Email Newsletters
Latest Headlines Health & Medicine Mind & Brain Space & Time Matter & Energy Computers & Math Plants & Animals Earth & Climate Fossils & Ruins