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equations:dirac_equation [2017/10/21 15:22] jakobadmin [Why is it interesting?] |
equations:dirac_equation [2018/04/16 09:11] jakobadmin [Intuitive] |
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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 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. |
- | <tabbox Layman> | + | The analogous equation for systems without spin is the [[equations:klein-gordon_equation|Klein-Gordon equation]]. |
- | <note tip> | + | If the system only moves slowly, the Dirac equation becomes the [[equations:pauli_equation|Pauli equation]]. |
- | 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 Concrete> |
- | | + | |
- | <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 | ||
* A great discussion can be found in [[https://books.google.de/books?id=HbdEAgAAQBAJ&lpg=PA157&ots=mShBas00sp&dq=We%20could%20also%20have%20written%20down%20a%20rotationally%20invariant%20equation%20of%20motion%20for&hl=de&pg=PA158#v=onepage&q&f=false|chapter 10 of the quantum field theory book by Matthew Schwartz]]. | * A great discussion can be found in [[https://books.google.de/books?id=HbdEAgAAQBAJ&lpg=PA157&ots=mShBas00sp&dq=We%20could%20also%20have%20written%20down%20a%20rotationally%20invariant%20equation%20of%20motion%20for&hl=de&pg=PA158#v=onepage&q&f=false|chapter 10 of the quantum field theory book by Matthew Schwartz]]. | ||
* 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 | ||
+ | <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> | ||