equations:dirac_equation
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equations:dirac_equation [2018/03/26 17:15] jakobadmin |
equations:dirac_equation [2023/04/02 03:11] (current) edi [Concrete] |
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| - | ====== Dirac Equation: $ \quad (i\gamma_\mu \partial^\mu - m ) \Psi =0 $ ====== | + | <WRAP lag>$ (i\gamma_\mu \partial^\mu - m ) \Psi =0 $</WRAP> |
| + | ====== Dirac Equation ====== | ||
| - | </note> | ||
| + | <tabbox Intuitive> | ||
| + | The Dirac equation describes how the state of a relativistic (= fast moving) quantum system with half-integer spin changes in time. | ||
| - | <tabbox Why is it interesting?> | + | The analogous equation for systems without spin is the [[equations:klein-gordon_equation|Klein-Gordon equation]]. |
| - | The Dirac equation is the correct equation of motion that describes free [[basic_notions:spin|spin]] $1/2$ particles. | + | If the system only moves slowly, the Dirac equation becomes the [[equations:pauli_equation|Pauli equation]]. |
| - | + | <tabbox Concrete> | |
| - | + | ||
| - | <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 Layman> | + | |
| - | + | ||
| - | <note tip> | + | |
| - | 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]] | * A nice discussion can be found in [[http://www.quantumfieldtheory.info/website_Chap04.pdf |chapter 4 of Klauber's Student Friendly QFT book]] | ||
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| * For a nice description, see section 7.4.3 "Diracology" in the book The Conceptual Framework of Quantum Field Theory by Duncan | * For a nice description, see section 7.4.3 "Diracology" in the book The Conceptual Framework of Quantum Field Theory by Duncan | ||
| - | <tabbox Researcher> | + | |
| + | ---- | ||
| + | |||
| + | **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> | ||
| The motto in this section is: //the higher the level of abstraction, the better//. | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| </note> | </note> | ||
| + | |||
| + | <tabbox Why is it interesting?> | ||
| + | |||
| + | The Dirac equation is the correct equation of motion that describes free [[basic_notions:spin|spin]] $1/2$ particles. | ||
| + | |||
| + | |||
| + | |||
| + | <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> | <tabbox Definitions> | ||
| Line 43: | Line 53: | ||
| * $\partial _{\mu} $ denotes the partial derivative and $ \gamma_{\mu} \partial^{\mu}$ stands for a sum using the Einstein sum convention, i.e. $\gamma_{\mu} \partial ^{\mu} = \gamma_0 \partial^0 - \gamma_1 \partial^1 -\gamma_2 \partial^2 -\gamma_3 \partial^3$, | * $\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, | * $m$ 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, | + | * $\Psi$ 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. | * $\gamma_\mu$ are the Dirac gamma matrices. | ||
| | | ||
equations/dirac_equation.1522077354.txt.gz · Last modified: 2018/03/26 15:15 (external edit)
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