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equations:dirac_equation [2017/10/21 15:35]
jakobadmin [Why is it interesting?]
equations:dirac_equation [2023/04/02 03:11] (current)
edi [Concrete]
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 +<WRAP lag>​$ ​ (i\gamma_\mu \partial^\mu - m ) \Psi =0 $</​WRAP>​
 +
 ====== Dirac Equation ====== ====== Dirac Equation ======
  
-<tabbox Why is it interesting?> ​ 
  
-<note tip>It'​s ​the correct equation ​of motion that describes free [[basic_notions:​spin|spin]] $1/2$ particles. +<tabbox Intuitive 
-</​note>​+The Dirac equation describes how the state of a relativistic (= fast moving) quantum system with half-integer ​spin changes in time.
  
 +The analogous equation for systems without spin is the [[equations:​klein-gordon_equation|Klein-Gordon equation]].
  
-<​blockquote>​In fact, Dirac'​s ​equation ​for the electron must be rated, alongside the Maxwell and Einstein ​equations, as one of the Great Field Equations of physics.<​cite>​page 289 in "The Emperors new Mind" by Penrose</​cite></​blockquote>​ +If the system only moves slowlythe Dirac equation ​becomes ​the [[equations:​pauli_equation|Pauli equation]]
-<tabbox Layman> ​+
  
-<note tip> +<​tabbox ​Concrete
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. +
-</​note>​ +
-   +
-<​tabbox ​Student+
  
 +  * A nice discussion can be found in [[http://​www.quantumfieldtheory.info/​website_Chap04.pdf |chapter 4 of Klauber'​s Student Friendly QFT book]]
   * For an elementary derivation and a discussion of what the solution of the Dirac equation mean, see Physics from Symmetry by Schwichtenberg   * For an elementary derivation and a discussion of what the solution of the Dirac equation mean, see Physics from Symmetry by Schwichtenberg
   * Alternative,​ two possible derivations can be found at page 100ff in Relativistic Quantum Mechanics by Paul Strange   * Alternative,​ two possible derivations can be found at page 100ff in Relativistic Quantum Mechanics by Paul Strange
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   * Another great discussion, especially of question how many degrees of freedom a Dirac spinor has can be found in http://​www.damtp.cam.ac.uk/​user/​tong/​qft/​four.pdf   * Another great discussion, especially of question how many degrees of freedom a Dirac spinor has can be found in http://​www.damtp.cam.ac.uk/​user/​tong/​qft/​four.pdf
    
-<​tabbox ​Researcher+ 
 +** Gamma Gymnastics:​** 
 + 
 +There are many important rules for the $\gamma$ matrices that appear in the Dirac equation. These rules are important for many practical calculations. 
 + 
 +  * For a nice description,​ see section 7.4.3 "​Diracology"​ in the book The Conceptual Framework of Quantum Field Theory by Duncan 
 + 
 +---- 
 + 
 +**Graphical Summary** 
 + 
 +The diagram below shows the Dirac equation and its Lagrangian in various forms. For a more detailed explanation see [[https://​esackinger.wordpress.com/​blog/​lie-groups-and-their-representations/#​dirac|Fun with Symmetry]].  
 + 
 +{{:​equations:​dirac.jpg?​nolink}} 
 + 
 +<​tabbox ​Abstract
  
 <note tip> <note tip>
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 </​note>​ </​note>​
  
---Common Question 1#+<tabbox Why is it interesting?​
  
-  +The Dirac equation is the correct equation of motion that describes free [[basic_notions:​spin|spin]] $1/2$ particles.
-<--+
  
---> Common Question 2# 
  
-  
-<-- 
-  ​ 
-<tabbox Examples> ​ 
  
---Example1#+<​blockquote>​In fact, Dirac'​s equation for the electron must be rated, alongside the [[equations:​maxwell_equations|Maxwell]] and [[equations:​einstein_equation|Einstein equations]],​ as one of the Great Field Equations of physics.<​cite>​page 289 in "The Emperors new Mind" by Penrose</​cite></​blockquote>
  
-  +<tabbox Definitions>​
-<--+
  
---> Example2:# 
  
-  +  * $\partial _{\mu} $ denotes the partial derivative and $ \gamma_{\mu} \partial^{\mu}$ stands for a sum using the Einstein sum convention, i.e. $\gamma_{\mu} \partial ^{\mu} =  \gamma_0 \partial^0 ​\gamma_1 \partial^1 ​-\gamma_2 \partial^2 -\gamma_3 \partial^3$,​ 
-<--+  * $m$ denotes the mass of the particle, 
 +  * $\Psi$ is either the wave function of the spin $1/2$ particle if we use the Dirac equation in a particle theory, or describes the spin $1/2$ field if we work in a field theory. In any case, $\Psi$ is not a vector but a [[advanced_tools:​spinors|spinor]].  
 +  * $\gamma_\mu$ are the Dirac gamma matrices.
   ​   ​
 <tabbox History> ​ <tabbox History> ​
 +<​blockquote>​“A great deal more was hidden in the Dirac equation than the author had
 +expected when he wrote it down in 1928. Dirac himself remarked in one of
 +his talks that his equation was more intelligent than its author. It should
 +be added, however, that it was Dirac who found most of the additional
 +insights.” <​cite>​Weisskopf on Dirac</​cite></​blockquote>​
 +
 +<​blockquote>​Niels Bohr: “What are you working on Mr. Dirac?”
 +Paul Dirac: “I’m trying to take the square root of something” </​blockquote>​
  
 </​tabbox>​ </​tabbox>​
  
  
equations/dirac_equation.1508592916.txt.gz · Last modified: 2017/12/04 08:01 (external edit)