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 July 22, 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 July 22, 2014).

Share This




More Matter & Energy News

Tuesday, July 22, 2014

Featured Research

from universities, journals, and other organizations


Featured Videos

from AP, Reuters, AFP, and other news services

Government Approves East Coast Oil Exploration

Government Approves East Coast Oil Exploration

AP (July 18, 2014) — The Obama administration approved the use of sonic cannons to discover deposits under the ocean floor by shooting sound waves 100 times louder than a jet engine through waters shared by endangered whales and turtles. (July 18) Video provided by AP
Powered by NewsLook.com
Sunken German U-Boat Clearly Visible For First Time

Sunken German U-Boat Clearly Visible For First Time

Newsy (July 18, 2014) — The wreckage of the German submarine U-166 has become clearly visible for the first time since it was discovered in 2001. Video provided by Newsy
Powered by NewsLook.com
Obama: U.S. Must Have "smartest Airports, Best Power Grid"

Obama: U.S. Must Have "smartest Airports, Best Power Grid"

Reuters - US Online Video (July 17, 2014) — President Barak Obama stopped by at a lunch counter in Delaware before making remarks about boosting the nation's infrastructure. Mana Rabiee reports. Video provided by Reuters
Powered by NewsLook.com
Crude Oil Prices Bounce Back After Falling Below $100 a Barrel

Crude Oil Prices Bounce Back After Falling Below $100 a Barrel

TheStreet (July 16, 2014) — Oil Futures are bouncing back after tumbling below $100 a barrel for the first time since May yesterday. Jeff Grossman is the president of BRG Brokerage and trades at the NYMEX. Grossman tells TheStreet the Middle East is always a concern for oil traders. Oil prices were pushed down in recent weeks on Libya increasing its production. Supply disruptions in Iraq fading also contributed to prices falling. News from China's economic front showing a growth for the second quarter also calmed fears on its slowdown. Jeff Grossman talks to TheStreet's Susannah Lee on this and more on the Energy Department's Energy Information Administration (EIA) report. Video provided by TheStreet
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