en.wikipedia.org website review
![](/include/images/menno/screenl1.png)
![](/include/images/loader.gif)
![](/include/images/pixel.png)
![](/include/images/menno/screenl2.png)
![](/include/images/loader.gif)
![](/include/images/pixel.png)
![](/include/images/loader.gif)
![](/include/images/pixel.png)
![](/include/images/menno/highlight.png)
Improve your SEO :: free trial!
en.wikipedia.org is 54% geoptimaliseerd!
SEO Keyword summary for en.wikipedia.org/wiki/dirac_equation
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be displaystyle
Focus keyword
Short and long tail
Short Tail Keywords displaystyle psi dirac |
long Tail Keywords (2 words) dirac equation partial mu mu nu bar psi mu psi |
long Tail Keywords (3 words) mu partial mu partial mu psi quantum field theory space and time u1displaystyle textu1 symmetry sigma mu nu omega mu nu |
en.wikipedia.org On-Page SEO Scan
Descriptive Elements
The <head> element of a en.wikipedia.org/wiki/dirac_equation page is used to inform the browser and visitors of the page about the general meta information. The head section of the page is where we place the page title, the definition of the HTML version used, the language of in which the page is written. In the head section we can also include JavaScript and CSS (markup) files for the page.
Page title
Title length
dirac equation wikipedia
Meta description
Meta description legth
Meta description SEO
No meta relevance in the description detected !
Content SEO
Number of Words
Spam detected?
Headings
Heading distribution
Heading normalisation
Heading SEO impact
Emphasis (bold and italic)
Emphasis SEO impact
Images
Number of images
Images dimensions
Image alt descriptions
Images SEO impact
wikipedia free encyclopedia displaystyle ihbar frac ddtpsi rangle hat hpsi leftbeta csum alpha npnrightpsi xtihbar partial psi xtpartial beta beginalignedalpha ialpha jalpha quad ineq ibeta endaligned ipartial xbeginbmatrixpsi endbmatrixipartial ybeginbmatrixpsi zbeginbmatrixpsi endbmatrixmbeginbmatrixpsi tbeginbmatrixpsi endbmatrix hbar mnabla phi tphi leftfrac nabla rightphi rho jfrac mphi cdot leftpsi tpsi right jmu mleftpsi epsi vec pbeta mpsi ecsqrt leftapartial xbpartial ycpartial zfrac icdpartial trightleftapartial tright abba ldots dots trightpsi kappa leftnabla rightpsi tfrac mchbar aibeta bibeta cibeta dbeta beginaligneddgamma aigamma bigamma cigamma gamma mcpsi beginpmatrixi endpmatrixquad beginpmatrix sigma xsigma ysigma zsigma endpmatrix leftgamma eta ababba ptextoppsi mathrel stackrel mathrm def big prime udagger beginalignedleftiudagger upartial mrightpsi leftxprime tprime igamma mupsi upsi leftigamma vfrac epsilon vigamma amu cmathbf hfrac mleftboldsymbol leftmathbf emathbf rightright ephi mleftmathbf ehbar mboldsymbol mathbf eamu mcrightpsi beginpmatrixmc eephi cboldsymbol rightcboldsymbol rightmc endpmatrixbeginpmatrixpsi endpmatrixbeginpmatrix beginalignedeephi approx mmathbf mcboldsymbol leftemc textu hgamma leftmc cgamma kleftpkqakrightrightcqa hcsqrt qmathbf mathbb rightarrow times textmat ibeginpmatrix isigma agamma bar xpsi xdagger dagger xigamma overleftarrow xgamma mbar mathcal lbar beginalignedpsi mapsto eialpha oalpha lmapsto ldelta delta lpartial xialpha ibar xumathbf eipcdot pemathbf textstyle sqrt umathbf pmu mrightumathbf beginpmatrixsqrt lihbar coverline overline sint xbar textso lambda slambda mmu mrho mnu smu exp omega xmpsi xmslambda islambda srho rightfrac leftsrho jrho tmu xmapsto xspsi sexp ixmu xnu xexp xlambda xmu xprime leftlambda rightnu rightmu gmu gnu galpha ifrac oleftlambda xleftifrac xomega leftifrac xendaligned beginpmatrixpsi lpsi rendpmatrix rendpmatrixbeginpmatrixsigma roverline lendpmatrix mbeginpmatrixpsi ibeginpmatrixsigma lendpmatrixmbeginpmatrixpsi rmpsi ioverline lmpsi dmu ieamu xfrac epartial xdmu idbig int stextmaxwellint xleftfrac fmu stextqedint ejmu xbeginpmatrixpsi xendpmatrix eibeta expibeta gtextsun cdots ijkcdots otimes umathbb textsun uxpsi xudagger uxamu xux gpartial uxux igamu igbar uxdmu xuxdagger stextymint texttrfmu anu igleftamu stextqcdint gmathbb gxpsi mathfrak gtextlg edit wikidata wikimedia foundation powered mediawiki
Mobile SEO en.wikipedia.org/wiki/dirac_equation
Mobile rendering
![](/include/images/menno/screenl1.png)
![](/include/images/menno/screenl2.png)
![](/include/images/menno/highlight.png)
Mobile optimizations
Responsive design detected (mobile css)
No flash detected !
Mobile improvement
![](/include/images/icons/loader.gif)
![](/include/images/icons/loader.gif)
Marketing / lead generation for en.wikipedia.org/wiki/dirac_equation
Social Media
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Conversion form
Search form
Analytics
Online presence
SERP Preview
SERP Title
SERP Link
SERP Description
Domain Level SEO
Domain name
16 characters long
Domain name SEO Impact
Path name
dirac found in path !
equation found in path !
Structured data
Publisher Markup
Other Structured data
Website configuration
Correct processing of non-existing pages?
![](http://www.webcijfers.nl/include/images/icons/loader.gif)
Favicon icon found?
![](http://www.webcijfers.nl/include/images/icons/loader.gif)
Robots.txt found?
![](http://www.webcijfers.nl/include/images/icons/loader.gif)
Sitemap found?
![](http://www.webcijfers.nl/include/images/icons/loader.gif)
Navigation and internal links
Navigation
Url seperator
Human readable urls
Number of links
Link SEO Impact
statistics
|
en.m.wikipedia.org |
en.wikipedia.org wikimedia foundation inc
|
w view history
httpsenwikipediaorgwindexphptitlediracequationoldid1230182800
page information
printable version
|
wiki read
dirac delta function
quantum mechanics
schrdinger
introduction
history
classical mechanics
old quantum theory
braket notation
hamiltonian
interference
complementarity
decoherence
entanglement
energy level
measurement
nonlocality
quantum number
quantum state
superposition
symmetry in quantum mechanics
tunnelling
uncertainty
wave function
collapse
bell test
davissongermer
doubleslit
elitzurvaidman
franckhertz
leggettgarg inequality
machzehnder interferometer
popper
quantum eraser
delayedchoice quantum eraser
schrdingers cat
sterngerlach
wheelers delayed choice
formulations
heisenberg
interaction
matrix mechanics
phase space
schrdinger
path integral formulation
kleingordon
pauli
rydberg
interpretations
bayesian
consistent histories
copenhagen
de brogliebohm
ensemble
hiddenvariable
superdeterminism
manyworlds
objective collapse
quantum logic
relational
transactional
von neumannwigner
relativistic quantum mechanics
quantum field theory
quantum information science
quantum computing
quantum chaos
epr paradox
density matrix
scattering theory
quantum statistical mechanics
quantum machine learning
aharonov
bell
bethe
blackett
bloch
bohm
bohr
born
bose
de broglie
compton
dirac
davisson
debye
ehrenfest
einstein
everett
fock
fermi
feynman
glauber
gutzwiller
heisenberg
hilbert
jordan
kramers
lamb
landau
laue
moseley
millikan
onnes
pauli
planck
rabi
raman
rydberg
schrdinger
simmons
sommerfeld
von neumann
hermann weyl
wien
wigner
zeeman
zeilinger
particle physics
relativistic wave equation
spinup
massive particles
electron
quark
parity
symmetry
special relativity
fine structure
hydrogen spectrum
standard model
antimatter
phenomenological
intrinsic angular momentum
complex numbers
bispinor
weyl equation
positron
theoretical physics
newton
maxwell
westminster abbey
rest mass
spacetime
momentum
momentum operator
speed of light
reduced planck constant
physical constants
atomic spectra
angular momentum
atomic nucleus
matrices
hermitian
involutory
anticommute
chiral representation
w k clifford
hermann grassmann
partial differential equations
planck units
free particle
maxwell equations
length of this fourvector
probability density
positive definite
conservation law
probability 4current density
pi meson
higgs boson
abuse of notation
relativistic expression for the energy
infinite series
half derivative
relativistic invariance
4gradient
implied summation
pauli matrices
identity matrix
minkowski metric
anticommutator
clifford relations
metric signature
dirac algebra
geometric algebra
eigenvalue
4momentum operator
proportionality constant
natural units
similarity transformation
unitary
unitary matrix
lorentz transformations
manifest relativistic invariance
oriented surface
epsilon symbol
tensor
determinant
metric tensor
lvyleblond equation
inhomogeneous
magnetic field
silver
ground state
integer
hamiltonian
si units
euclidean vectors
dimensions
minkowski space
fourvector
electromagnetic fourpotential
classical hamiltonian of a charged particle
minimal coupling
gyromagnetic ratio
rest energy
diffusion equation
creation
annihilation
observable
hermitian operators
hilbert space
quantum measurement problem
photons
vacuum
dirac sea
pauli exclusion principle
carl anderson
jellium
bogoliubov transformation
creation and annihilation operators
condensed matter physics
conduction electrons
electrical conductor
fermi sea
chemical potential
fermionic field
quantum electrodynamics
second quantization
dirac spinor
feynman slash notation
conserved current
noethers theorem
squareintegrable functions
lie algebra
here
lorentz covariant
majorana spinor
manifold
coordinate systems
local frames of reference
frame of reference
fiber
fiber bundle
frame bundle
rotations
lorentz boosts
section
spinor bundle
associated bundle
principal bundle
generators
total angular momentum
intrinsic angular momentum
general relativistic
frame field
affine
spin bundle
dirac equation in curved spacetime
dirac equation in the algebra of physical space
weyl spinors
helicity
zitterbewegung
poisson process
scalar electrodynamics
gauge field
connection
anomaly
quantum chromodynamics
gauge group
color symmetries
special unitary group
yangmills
standard model
strong interactions
gluon
electroweak
representation
contractibility
dirac field
gordon decomposition
klein paradox
nonlinear dirac equation
breit equation
dirackhler equation
raritaschwinger equation
twobody dirac equations
majorana equation
feynman checkerboard
foldywouthuysen transformation
isbn
world scientific publishing
zichichi antonino
ohlsson tommy
bibcode
arxiv
s2cid
jstor
halzen francis
martin alan
timeline
born rule
excited state
degenerate levels
zeropoint energy
waveparticle duality
science
quantum biology
quantum chemistry
quantum cosmology
quantum differential calculus
quantum dynamics
quantum geometry
quantum mind
quantum stochastic calculus
quantum spacetime
technology
quantum algorithms
quantum amplifier
quantum bus
quantum cellular automata
quantum finite automata
quantum channel
quantum circuit
quantum complexity theory
quantum cryptography
quantum electronics
quantum error correction
quantum imaging
quantum image processing
quantum information
quantum key distribution
quantum logic gates
quantum machine
quantum metamaterial
quantum metrology
quantum network
quantum neural network
quantum optics
quantum programming
quantum sensing
quantum simulator
quantum teleportation
quantum fluctuation
casimir effect
history
quantum gravity
in popular culture
wigners friend
quantum mysticism
|
Links to external pages
Outloing links
als.wikipedia.org
ar.wikipedia.org
cs.wikipedia.org
da.wikipedia.org
de.wikipedia.org
et.wikipedia.org
el.wikipedia.org
eo.wikipedia.org
fa.wikipedia.org
ga.wikipedia.org
ko.wikipedia.org
hy.wikipedia.org
it.wikipedia.org
he.wikipedia.org
hu.wikipedia.org
mr.wikipedia.org
ja.wikipedia.org
uz.wikipedia.org
pa.wikipedia.org
pt.wikipedia.org
ro.wikipedia.org
ru.wikipedia.org
sl.wikipedia.org
fi.wikipedia.org
sv.wikipedia.org
tt.wikipedia.org
uk.wikipedia.org
zh-yue.wikipedia.org
zh.wikipedia.org
www.wikidata.org
www.wikidata.org
en.wikiquote.org
books.google.com
books.google.com
www.dirac.ch
books.google.com
www.worldscientific.com
www.doi.org
www2.ph.ed.ac.uk
www.ghostarchive.org
books.google.com
www.doi.org
ui.adsabs.harvard.edu
www.doi.org
www.arxiv.org
ui.adsabs.harvard.edu
www.doi.org
api.semanticscholar.org
www.math.ucsd.edu
ui.adsabs.harvard.edu
www.doi.org
www.jstor.org
web.archive.org
gallica.bnf.fr
ui.adsabs.harvard.edu
www.doi.org
www.jstor.org
ui.adsabs.harvard.edu
www.doi.org
api.semanticscholar.org
www.archive.org
www.archive.org
en.wikiquote.org
lamediateca.infn.it
www.mathpages.com
electron6.phys.utk.edu
www.wikidata.org
catalogue.bnf.fr
data.bnf.fr
www.d-nb.info
olduli.nli.org.il
foundation.wikimedia.org
foundation.wikimedia.org
foundation.wikimedia.org
www.wikimediafoundation.org
SEO Advice for en.wikipedia.org
In this section we provide pointers on how you can to optimize your web page so it can be found more easily by search engines and how to make it rank higher by optimizing the content of the page itself. For each of the individual criteria the maximum score is 100%. A score below 70% is considered to be indication that the page is not complying with general SEO standards and should be evaluated and/or fixed. Not every factor is weighted the same and some are not as important as others. Relatively unimportant factors like meta keywords are not included in the overall score.
Item | Factor | Pointers | |
---|---|---|---|
PageTitle | 100% | Far too many sites lack a page title. A page title is the first thing that shows in the search results so always use the title element. | |
Title relevance | 87% | A title should reflect the contents of a site. This site has a 67 % match | |
Title Length | 30% | Limit your title to anywhere between 40 and 70 characters. Your title was 27 characters long | |
Meta Description | 0% | A meta description is the second element that shows in the search results so always use the meta description. | |
Meta description length | 0% | The meta description should be between 145 and 160 characters. This meta description is 1 characters long. | |
Meta description relevance | 0% | Meta Description should reflect the contents of a site. This site has a 0 % match | |
Number of internal links | 30% | Linking to internal pages makes pages easier to find for search engines. Try to keep the number of links on your page roughly below 100. There are 594 internal links on this page. | |
Folder structure | 100% | We found a folder structure in the links on your page. A good folder structure makes a site easier to navigate. We found 3 level 1 folders and 7 folders above or in the first level of navigation. | |
Headings | 21% | Headers should reflect the contents of a site. This site has a 9 % match | |
Links | 6% | Link anchors should to some degree reflect the contents of a site. This site has a 3 % match | |
Image alt tags | 14% | Image alt tags should to some degree reflect the contents of a site. This site has a 5 % match | |
Bold and italic | 24% | Bold and italic tags should reflect the contents of a site to some degree. This site has a 8 % match | |
Html ratio | 30% | Try to keep the html / text ratio as low as possible. More html means longer loading times. Layout should be handled in a serpate css file | |
Image descriptions | 99% | 98.826979472141 % of all images have been described via the "alt" attribute. Describing images with relevant text may lead to better results in the search engines. | |
Page errors | 100% | Pages with no errors display significantly faster on most browsers. We detected 0 errors and warnings | |
WordCount | 20% | An ideal page contains between 400 and 600 words.This page contains 12283 words | |
Server response time | 100% | A fast server speeds up a website. This server responds 92.59% faster then average | |
Gzip compression | 30% | This site does not use Gzip compression. Pages may not display as fast as they could | |
Keywords in Domainname | 30% | There are no important keywords in your domain name | |
Keywords in domain path | 100% | There are important keywords in the domain path | |
Structured Data | 100% | Structured data makes it easier for search engines to index your website | |
Inline css | 0% | Do not use inline css declarations. Inline css will slow down the rendering of the website. We detected 750 inline style declarations ( <a style="color:green">) with a size of 26608 bytes | |
Excessive use of the same words | 100% | There is no indication that there are one or more keywords that are used excessively. | |
Frames or iframes | 100% | Perfect, detected not (i)frames on your webpagina | |
Flash | 100% | Perfect, we detected no flash objects on your page | |
Css | 30% | We detected too much (2) CSS files on your page. Css files block the loading of a webpage. | |
Javascript | 100% | Perfect, we did not detect too many blocking JavaScript files | |
Mobile Website | 100% | Perfect, we found a responsive design for mobile users | |
Most important heading | 100% | Perfect, we detected a correct use of the most important (h1) heading! | |
Normalized headings | 40% | We dit not font a normalized heading structure. A heading 2 (h2) for example should be followed by a heading of an equal level (h2), a child heading (h3) or even a aprent heading (h1). |
How would you like to have SEO advice for all your pages ?? Start your SEO Dashboard and optimize your website!
en.wikipedia.org images and descriptions
264 images found at en.wikipedia.org Images can improve the user experience for a website by making a pag visually appealing Images can also add extra keyword relevance to a webpage by using alt tags. Images can also slow down a website. If the width and height for a picture is not specified for a browser know in advance how large the image is. A browser must first load the picture and see before it knows how much space should be on the page. Upon reservation In the meantime, the browser can do little but wait. When the height and width for the plate are given in the HTML code, a browser just continues to build for a page while the images load in the background.
http://en.wikipedia.org/static/images/icons/wikipedia.png height: 50 width: 50 description: no alt description found |
|
http://en.wikipedia.org/static/images/mobile/copyright/wikipedia-wordmark-en.svg height: height attribute not set width: width attribute not set description: wikipedia |
|
http://en.wikipedia.org/static/images/mobile/copyright/wikipedia-tagline-en.svg height: 13 width: 117 description: the free encyclopedia |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c height: height attribute not set width: width attribute not set description: {\displaystyle i\hbar {\frac {d}{dt}}|\psi \rangle ={\hat {h}}|\psi \rangle } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/462b6baf980241a20627f784d6b1c360e908ced3 height: height attribute not set width: width attribute not set description: {\displaystyle \left(\beta mc^{2}+c\sum _{n=1}^{3}\alpha _{n}p_{n}\right)\psi (x,t)=i\hbar {\frac {\partial \psi (x,t)}{\partial t}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6d57f7d7b73cb365bd40f655214fca1e16388275 height: height attribute not set width: width attribute not set description: {\displaystyle \alpha _{i}^{2}=\beta ^{2}=i_{4}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/28a458b64a8d23784df25a07ae50d07cc14e054e height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\alpha _{i}\alpha _{j}+\alpha _{j}\alpha _{i}&=0\quad (i\neq j)\\\alpha _{i}\beta +\beta \alpha _{i}&=0\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/82b54fdc7eadb7060aed15821f689854fb2ec482 height: height attribute not set width: width attribute not set description: {\displaystyle i\partial _{x}{\begin{bmatrix}+\psi _{4}\\+\psi _{3}\\-\psi _{2}\\-\psi _{1}\end{bmatrix}}+i\partial _{y}{\begin{bmatrix}+\psi _{4}\\-\psi _{3}\\-\psi _{2}\\+\psi _{1}\end{bmatrix}}+i\partial _{z}{\begin{bmatrix}+\psi _{3}\\-\psi _{4}\\-\psi _{1}\\+\psi _{2}\end{bmatrix}}-m{\begin{bmatrix}+\psi _{1}\\+\psi _{2}\\+\psi _{3}\\+\psi _{4}\end{bmatrix}}=i\partial _{t}{\begin{bmatrix}-\psi _{1}\\-\psi _{2}\\+\psi _{3}\\+\psi _{4}\end{bmatrix}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f06e53e9275e76e36565d145b8ae0bf0a5af425b height: height attribute not set width: width attribute not set description: {\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\phi =i\hbar {\frac {\partial }{\partial t}}\phi ~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/01ef6bd7664976b50a3ba51f56615d7f331e72da height: height attribute not set width: width attribute not set description: {\displaystyle e^{2}=m^{2}c^{4}+p^{2}c^{2}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9742ab48dcab62f89bdd2ad27ee2d66bfdd807 height: height attribute not set width: width attribute not set description: {\displaystyle \left(-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}+\nabla ^{2}\right)\phi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\phi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/472282fbc5e443b715ed2ca2361544f5f5e5eef4 height: height attribute not set width: width attribute not set description: {\displaystyle \rho =\phi ^{*}\phi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/03ceb1b39c1afeb3c61ba6beb01fa6f4c2d9b5c9 height: height attribute not set width: width attribute not set description: {\displaystyle j=-{\frac {i\hbar }{2m}}(\phi ^{*}\nabla \phi -\phi \nabla \phi ^{*})} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/bac713ddbbeb8d432b914c0c87ebcd2049cee0be height: height attribute not set width: width attribute not set description: {\displaystyle \nabla \cdot j+{\frac {\partial \rho }{\partial t}}=0~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/fce48a787a6442bb682c37783704b43535337975 height: height attribute not set width: width attribute not set description: {\displaystyle \rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}\partial _{t}\psi -\psi \partial _{t}\psi ^{*}\right).} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d49fea709dc99dad5702ef8fbda8f08dcbe58740 height: height attribute not set width: width attribute not set description: {\displaystyle j^{\mu }={\frac {i\hbar }{2m}}\left(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*}\right).} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/acdc3c339cb4ba45a865931981be3a4887ea4364 height: height attribute not set width: width attribute not set description: {\displaystyle e\psi =({\vec {\alpha }}\cdot {\vec {p}}+\beta m)\psi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/33cde84bc34e55a5613379a8c6db05e49c308991 height: height attribute not set width: width attribute not set description: {\displaystyle ({\vec {\alpha }},\beta )} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ab754a7526dc207c6e1810fe527e54cae06f8bb4 height: height attribute not set width: width attribute not set description: {\displaystyle ({\vec {p}},t)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/75b5a668b21420131ed56f44130efcf94b87a6db height: height attribute not set width: width attribute not set description: {\displaystyle ({\vec {x}},t)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/102c4ccff20e577a7d956237743c7bf048c6ee75 height: height attribute not set width: width attribute not set description: {\displaystyle e=c{\sqrt {p^{2}+m^{2}c^{2}}}~,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3a346c5da0315d2feab4e3d651d39aad8f9d4f height: height attribute not set width: width attribute not set description: {\displaystyle \nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}=\left(a\partial _{x}+b\partial _{y}+c\partial _{z}+{\frac {i}{c}}d\partial _{t}\right)\left(a\partial _{x}+b\partial _{y}+c\partial _{z}+{\frac {i}{c}}d\partial _{t}\right)~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f9196807d4de65df5fcd380fb1cc55f5847ed1c9 height: height attribute not set width: width attribute not set description: {\displaystyle ab+ba=0,~\ldots ~} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/61dd42452b962e89203ce3e322726ee32c4ae1c9 height: height attribute not set width: width attribute not set description: {\displaystyle a^{2}=b^{2}=\dots =1~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/82ca67277ef4b9c11449c2eac63d9e1ea2b5b4d3 height: height attribute not set width: width attribute not set description: {\displaystyle \left(a\partial _{x}+b\partial _{y}+c\partial _{z}+{\frac {i}{c}}d\partial _{t}\right)\psi =\kappa \psi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d height: height attribute not set width: width attribute not set description: {\displaystyle \kappa } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0c902aea5d09469ef8f8b2edf5a80a3ac6d7a918 height: height attribute not set width: width attribute not set description: {\displaystyle \left(\nabla ^{2}-{\frac {1}{c^{2}}}\partial _{t}^{2}\right)\psi =\kappa ^{2}\psi ~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3a0aa9a303f155614439c17d7fed99435b054fa3 height: height attribute not set width: width attribute not set description: {\displaystyle \kappa ={\tfrac {mc}{\hbar }}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6b10520d0846873a01c1298f0afa4a861fd876a8 height: height attribute not set width: width attribute not set description: {\displaystyle \left(a\partial _{x}+b\partial _{y}+c\partial _{z}+{\frac {i}{c}}d\partial _{t}-{\frac {mc}{\hbar }}\right)\psi =0~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/412f1d0602b47b3afa66f4cac9a96a1bea2095e7 height: height attribute not set width: width attribute not set description: {\displaystyle a=i\beta \alpha _{1}\,,\,b=i\beta \alpha _{2}\,,\,c=i\beta \alpha _{3}\,,\,d=\beta ~,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf0d5a37333e867c6c704c8f48140c80ef00a62 height: height attribute not set width: width attribute not set description: {\displaystyle d^{2}=\beta ^{2}=i_{4}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/867bd8e48e0564fa853dacadae6a49851c3dd723 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}d&=\gamma ^{0},\\a&=i\gamma ^{1},\quad b=i\gamma ^{2},\quad c=i\gamma ^{3},\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ac38859ceaaaaffca56a77087d640db903af72 height: height attribute not set width: width attribute not set description: {\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0b9cf9918eb959f7aa8f2382ccd8457c597f1220 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{0}={\begin{pmatrix}i_{2}&0\\0&-i_{2}\end{pmatrix}},\quad \gamma ^{1}={\begin{pmatrix}0&\sigma _{x}\\-\sigma _{x}&0\end{pmatrix}},\quad \gamma ^{2}={\begin{pmatrix}0&\sigma _{y}\\-\sigma _{y}&0\end{pmatrix}},\quad \gamma ^{3}={\begin{pmatrix}0&\sigma _{z}\\-\sigma _{z}&0\end{pmatrix}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1f70ea26c9f892780e0bc946685e111f7063af4d height: height attribute not set width: width attribute not set description: {\displaystyle \left\{\gamma ^{\mu },\gamma ^{\nu }\right\}=2\eta ^{\mu \nu }i_{4}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/26b1381a9421792076cb3dff8671fde43af78bd4 height: height attribute not set width: width attribute not set description: {\displaystyle \{a,b\}=ab+ba} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/161e5cad12e0ce8a87cf6d027a7c904c52daa9cd height: height attribute not set width: width attribute not set description: {\displaystyle p_{\text{op}}\psi =mc\psi \,.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c410c9a4eee038b52a8b0022277f72e32516eb6d height: height attribute not set width: width attribute not set description: {\displaystyle {\partial \!\!\!/}\mathrel {\stackrel {\mathrm {def} }{=}} \gamma ^{\mu }\partial _{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fd2df37b2e7392c51f956efd692ace7b36cd34 height: height attribute not set width: width attribute not set description: {\displaystyle {\partial \!\!\!{\big /}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2a306e5fd9bdd4ce43ac178fa618a2da2ef7cf height: height attribute not set width: width attribute not set description: {\displaystyle i\hbar {\partial \!\!\!{\big /}}\psi -mc\psi =0\,.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0638363e10a430afd04e74fd9e5b70048d844c height: height attribute not set width: width attribute not set description: {\displaystyle (i{\partial \!\!\!{\big /}}-m)\psi =0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ca6b0e10c41431b14a796987c61a013e4db913d1 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{\mu \prime }=s^{-1}\gamma ^{\mu }s\,.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/dee09524ff43d92cc506f7d873e4c5a15bebf060 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{\mu \prime }=u^{\dagger }\gamma ^{\mu }u\,.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/175799c2b301cc7ec7031d12721746ec344e1670 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\left(iu^{\dagger }\gamma ^{\mu }u\partial _{\mu }^{\prime }-m\right)\psi \left(x^{\prime },t^{\prime }\right)&=0\\u^{\dagger }(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m)u\psi \left(x^{\prime },t^{\prime }\right)&=0\,.\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ae25f7003897abb9601642ddc01e42c1fe9c0e71 height: height attribute not set width: width attribute not set description: {\displaystyle \psi ^{\prime }=u\psi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e0826ee1b97d9eb4dd9a5220d315bfeab33647 height: height attribute not set width: width attribute not set description: {\displaystyle \left(i\gamma ^{\mu }\partial _{\mu }^{\prime }-m\right)\psi ^{\prime }\left(x^{\prime },t^{\prime }\right)=0\,.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/294dfd80fad532201bc1e2629e65c1777f57ef81 height: height attribute not set width: width attribute not set description: {\displaystyle v={\frac {1}{4!}}\epsilon _{\mu \nu \alpha \beta }\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\alpha }\gamma ^{\beta }.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9e2d73f8d2f07d11ccab24c010ad41f833582e height: height attribute not set width: width attribute not set description: {\displaystyle v=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/faa9123929e25f20509fd5da9321df64f498f091 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma _{5}={\begin{pmatrix}0&i_{2}\\i_{2}&0\end{pmatrix}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/31a96750a8a6270832e01c3147c16e0307502883 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{5}\gamma ^{\mu }+\gamma ^{\mu }\gamma ^{5}=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d1e2b5d14e111debb84f4af7d3a93a67e34fadac height: height attribute not set width: width attribute not set description: {\displaystyle a_{\mu }=(\phi /c,-\mathbf {a} )} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b60422fa589288412f75468dbd9d8d4d6c1cc560 height: height attribute not set width: width attribute not set description: {\displaystyle h={\frac {1}{2m}}\left({\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {a} \right)\right)^{2}+e\phi ~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4 height: height attribute not set width: width attribute not set description: {\displaystyle \phi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2fffd6239d0b2769c9d6f82220dc542bf78e9560 height: height attribute not set width: width attribute not set description: {\displaystyle h={\frac {1}{2m}}\left(\mathbf {p} -e\mathbf {a} \right)^{2}+e\phi -{\frac {e\hbar }{2m}}{\boldsymbol {\sigma }}\cdot \mathbf {b} ~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9e49a210bcc73134899ff60b36229db974c3a179 height: height attribute not set width: width attribute not set description: {\displaystyle \left(\gamma ^{\mu }(i\hbar \partial _{\mu }-ea_{\mu })-mc\right)\psi =0~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/17864d1dfcb0df71fb224e5a21de032ed89c5cc6 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{pmatrix}mc^{2}-e+e\phi &c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {a} \right)\\-c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {a} \right)&mc^{2}+e-e\phi \end{pmatrix}}{\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e30d485b7874fb6f4cbda7c2fcb338ef2e602fa8 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}(e-e\phi )\psi _{+}-c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {a} \right)\psi _{-}&=mc^{2}\psi _{+}\\c{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {a} \right)\psi _{+}-(e-e\phi )\psi _{-}&=mc^{2}\psi _{-}\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/07bd226234d65fb92dffcb0b8dee2f0044eebda2 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}e-e\phi &\approx mc^{2}\\\mathbf {p} &\approx m\mathbf {v} \end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e645308354f14e46f69f3cd7a22ed91bcfcfd426 height: height attribute not set width: width attribute not set description: {\displaystyle \psi _{-}\approx {\frac {1}{2mc}}{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {a} \right)\psi _{+}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2daeec18f62db4508349d3392ceb0d0bd20f5d72 height: height attribute not set width: width attribute not set description: {\displaystyle \left(e-mc^{2}\right)\psi _{+}={\frac {1}{2m}}\left[{\boldsymbol {\sigma }}\cdot \left(\mathbf {p} -e\mathbf {a} \right)\right]^{2}\psi _{+}+e\phi \psi _{+}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e57f21007575fd03e3be0da20af34d25829cc9a7 height: height attribute not set width: width attribute not set description: {\displaystyle m=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/799f83909c3169d94893468b6aa2e83b40fee010 height: height attribute not set width: width attribute not set description: {\displaystyle {\text{u}}(1)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/188deadb17fc5e3a33fa216263e10137a29faf80 height: height attribute not set width: width attribute not set description: {\displaystyle h=\gamma ^{0}\left[mc^{2}+c\gamma ^{k}\left(p_{k}-qa_{k}\right)\right]+cqa^{0}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/14fe6b8c15bd318b62cad9db611fb3c7e1eedf28 height: height attribute not set width: width attribute not set description: {\displaystyle h=c{\sqrt {\left(\mathbf {p} -q\mathbf {a} \right)^{2}+m^{2}c^{2}}}+qa^{0}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a height: height attribute not set width: width attribute not set description: {\displaystyle \psi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/990797533f68706456ef046fb4e390c6d064f8a8 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {c} ^{4}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/97bbc96ea97da9adfb87ac79a78c8e604c97cd7b height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} ^{1,3}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/501173910e6da8425b4e9d44a4e8643620bc2464 height: height attribute not set width: width attribute not set description: {\displaystyle m>0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ad8f395a20f33e3ce8c723ee28b41d0a08d89aef height: height attribute not set width: width attribute not set description: {\displaystyle \psi :\mathbb {r} ^{1,3}\rightarrow \mathbb {c} ^{4}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bb6f75019e72bfa972786424c22af07d6145ee height: height attribute not set width: width attribute not set description: {\displaystyle (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi (x)=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b6942692584f37d53ac71686a5f630f22d6772 height: height attribute not set width: width attribute not set description: {\displaystyle (i\partial \!\!\!/-m)\psi (x)=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/89eb2e0f4ddfe5f30c8016a0f2aa1fb5ecedfe20 height: height attribute not set width: width attribute not set description: {\displaystyle 4\times 4} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e1dc68dea6f50632bf33e79661283eeef36bde height: height attribute not set width: width attribute not set description: {\displaystyle {\text{mat}}_{4\times 4}(\mathbb {c} )} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/39ad14fc012327beace837daa6cf8541bf0336d5 height: height attribute not set width: width attribute not set description: {\displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=2\eta ^{\mu \nu }i_{4}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/027ec2b822550a7f099c1c739a05738ed6be7831 height: height attribute not set width: width attribute not set description: {\displaystyle \eta ^{\mu \nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/489a4ac1b5d76663fa5328e6f3c1d382655cb496 height: height attribute not set width: width attribute not set description: {\displaystyle \mu ,\nu } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/5aad32fe98cb37bf38bd468b97e03635de094d0b height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{0}={\begin{pmatrix}i_{2}&0\\0&-i_{2}\end{pmatrix}},\quad \gamma ^{i}={\begin{pmatrix}0&\sigma ^{i}\\-\sigma ^{i}&0\end{pmatrix}},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1d0a098aad3c723834e6044b830ce3ddc83407 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/607365b9067701be0de14bc586f165284479fdb3 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a73874cb68bf4084bc738c53dca9911364c1cf2e height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{0}={\begin{pmatrix}0&i_{2}\\i_{2}&0\end{pmatrix}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/731586a289bf5d0e78edf1df6c825ae091d5688f height: height attribute not set width: width attribute not set description: {\displaystyle a\!\!\!/:=\gamma ^{\mu }a_{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3 height: height attribute not set width: width attribute not set description: {\displaystyle a} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1be3e5cf12940b60f8569a9927ad28823277177a height: height attribute not set width: width attribute not set description: {\displaystyle \partial _{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161 height: height attribute not set width: width attribute not set description: {\displaystyle \mu } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a596a1fb4130a47f6b88c66150497338bd6cbccc height: height attribute not set width: width attribute not set description: {\displaystyle \psi (x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9bbd281b090adb3e753d2ef06febbc33a37002 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {\psi }}(x)=\psi (x)^{\dagger }\gamma ^{0}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9ad28fd0224d333e10919fe473cf40e52ac6c9 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/7996ee9fba5571cef10030a55221e81eafeb079f height: height attribute not set width: width attribute not set description: {\displaystyle (\gamma ^{\mu })^{\dagger }=\gamma ^{0}\gamma ^{\mu }\gamma ^{0},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e188650006a8d530e7086eeb8b4c1b9c6ba33ef9 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{0}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1057aea898b8a6b14df249a67dac475eacb5de04 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {\psi }}(x)(-i\gamma ^{\mu }{\overleftarrow {\partial }}_{\mu }-m)=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/cc283c58d4607fd966570167ac8fbeb6b31e20c3 height: height attribute not set width: width attribute not set description: {\displaystyle {\overleftarrow {\partial }}_{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd7a9de2742ff615699c9b81ae837b1a9cce540 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {\psi }}(x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/eef0f33d67b0f5783ec6f4bab6389663322eec3b height: height attribute not set width: width attribute not set description: {\displaystyle -i\partial _{\mu }{\bar {\psi }}(x)\gamma ^{\mu }-m{\bar {\psi }}(x)=0.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a545d3c9a1d698023333b9f2415768c79d52efc8 height: height attribute not set width: width attribute not set description: {\displaystyle i\partial \!\!\!/+m} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/478fba8c1020fbf3b20417d1bf8b83cebbd6a315 height: height attribute not set width: width attribute not set description: {\displaystyle (\partial _{\mu }\partial ^{\mu }+m^{2})\psi (x)=0.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4a561380bac5b6337ce09eba3ddc53a82b536ed6 height: height attribute not set width: width attribute not set description: {\displaystyle j^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/881f845c41f32280aa81fd47f99906553429d52d height: height attribute not set width: width attribute not set description: {\displaystyle i((\partial _{\mu }{\bar {\psi }})\gamma ^{\mu }\psi +{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi )=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1126a7ee196733f04f5d9ccb65f7ee25a4d9b8ab height: height attribute not set width: width attribute not set description: {\displaystyle i\partial _{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/feaf36c5731316bf031ab47a4d597a6405410d93 height: height attribute not set width: width attribute not set description: {\displaystyle j^{\mu }.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bca71b7c4732324eb522b0f6b0a3395a3d4e08 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {l}}={\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi .} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e62b00d74ee0cefb86cc052365625abff56d43e6 height: height attribute not set width: width attribute not set description: {\displaystyle u(1)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb9d5cb140da658c797a5fddd4b24ffda12dc12 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\psi &\mapsto e^{i\alpha }\psi ,\\{\bar {\psi }}&\mapsto e^{-i\alpha }{\bar {\psi }},\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3 height: height attribute not set width: width attribute not set description: {\displaystyle \alpha } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/042c7c36b56cddae4e89e2e97f9e643f5fd44b71 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {o}}{\alpha ^{2}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/5a9465fae3508ee68fedb4b7131878a20fa60017 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {l}}\mapsto {\mathcal {l}}+\delta {\mathcal {l}},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/630fe89b387c45ced85724ee86186efa9ec01439 height: height attribute not set width: width attribute not set description: {\displaystyle \delta {\mathcal {l}}=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/35de2a319364dc3f6a5802d134230effb394f8b0 height: height attribute not set width: width attribute not set description: {\displaystyle \psi ,{\bar {\psi }}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/94bdb3ac2f71eaa99d3b79798a1818b5cfe8895e height: height attribute not set width: width attribute not set description: {\displaystyle \delta {\mathcal {l}}=\partial _{\mu }\left({\frac {\partial {\mathcal {l}}}{\partial (\partial _{\mu }\psi )}}\delta \psi +{\frac {\partial {\mathcal {l}}}{\partial (\partial _{\mu }{\bar {\psi }})}}\delta {\bar {\psi }}\right)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/890adebe2730294079a81b7bf08b2fe0f2c59909 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {\psi }}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2003c0fcb8fbf980507d6c7064a31a0d3182681a height: height attribute not set width: width attribute not set description: {\displaystyle \delta \psi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd58a1cdd48c2c81ec572b145e9a44590e10367 height: height attribute not set width: width attribute not set description: {\displaystyle \delta \psi (x)=i\alpha \psi (x).} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcb89991b05da3f3419cb295786e9080d21bb99 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\partial {\mathcal {l}}}{\partial (\partial _{\mu }\psi )}}=i{\bar {\psi }}\gamma ^{\mu }.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/75171b8c0b9400cfac5c37d6523453bf5a1c1e72 height: height attribute not set width: width attribute not set description: {\displaystyle 0=-\alpha \partial _{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/32e78db25b7b1459f19aeaf1b60742c451cfcbfa height: height attribute not set width: width attribute not set description: {\displaystyle \psi (x)=u(\mathbf {p} )e^{-ip\cdot x}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8665a66bf73fb744cb5c8e548fdacef6ef07d96a height: height attribute not set width: width attribute not set description: {\displaystyle p=(e_{\mathbf {p} },\mathbf {p} )} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/686ff49beaac5e46e2692cd6757a36ab1852b4d8 height: height attribute not set width: width attribute not set description: {\textstyle e_{\mathbf {p} }={\sqrt {m^{2}+|\mathbf {p} |^{2}}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/eb05a27f493c89513b75f2ff44b5427e55cfef3b height: height attribute not set width: width attribute not set description: {\displaystyle u(\mathbf {p} )} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea8f7827274b073057b2c2e48568bdd88784259 height: height attribute not set width: width attribute not set description: {\displaystyle \left(\gamma ^{\mu }p_{\mu }-m\right)u(\mathbf {p} )=0.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {c} ^{2}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db height: height attribute not set width: width attribute not set description: {\displaystyle \xi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb5f8207683548dbadef5e820908ad99740d912 height: height attribute not set width: width attribute not set description: {\displaystyle u(\mathbf {p} )={\begin{pmatrix}{\sqrt {\sigma ^{\mu }p_{\mu }}}\xi \\{\sqrt {{\bar {\sigma }}^{\mu }p_{\mu }}}\xi \end{pmatrix}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ac26217f42dcdf3966b0e6d59900cd9ebaed88 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{\mu }=(i_{2},\sigma ^{i}),{\bar {\sigma }}^{\mu }=(i_{2},-\sigma ^{i})} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/023052aea0c7332f3b5a8a581e03a6b4b1119caa height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\cdot }}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/060bf077a7803b33e65610824acaeb12aeee4844 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {l}}=i\hbar c{\overline {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -mc^{2}{\overline {\psi }}\psi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a50de1deae3d811bf18962611436ca05990b5d18 height: height attribute not set width: width attribute not set description: {\displaystyle s=\int d^{4}x\,{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/74fcaa978430bd2a8c9fb6957192b62fa191d9bd height: height attribute not set width: width attribute not set description: {\displaystyle j^{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/dac29ed3b32e940a2746f4b549fe0742b1d20628 height: height attribute not set width: width attribute not set description: {\displaystyle {\text{so}}(1,3)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/78496dc40739f5959f7b0745e927af29bf6c5e6e height: height attribute not set width: width attribute not set description: {\displaystyle {\text{so}}(1,3)^{+}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733 height: height attribute not set width: width attribute not set description: {\displaystyle \lambda } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/077a74b520c1b15b7b8fb3ab6bfd5e42186a8798 height: height attribute not set width: width attribute not set description: {\displaystyle s[\lambda ]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1eb3dba5f5aafb09220b1ec9d9560a2485d590 height: height attribute not set width: width attribute not set description: {\displaystyle \{m^{\mu \nu }\}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a94422f6b8296881ef47ac24c1a8bbec802fa3 height: height attribute not set width: width attribute not set description: {\displaystyle (m^{\mu \nu })^{\rho }{}_{\sigma }=\eta ^{\mu \rho }\delta ^{\nu }{}_{\sigma }-\eta ^{\nu \rho }\delta ^{\mu }{}_{\sigma }.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4bdc929543888431897809cdd2df86180ff5ed09 height: height attribute not set width: width attribute not set description: {\displaystyle \rho ,\sigma } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/94a9022837b52dcbbcf58bf59339dd5754657dfe height: height attribute not set width: width attribute not set description: {\displaystyle [m^{\mu \nu },m^{\rho \sigma }]=m^{\mu \sigma }\eta ^{\nu \rho }-m^{\nu \sigma }\eta ^{\mu \rho }+m^{\nu \rho }\eta ^{\mu \sigma }-m^{\mu \rho }\eta ^{\nu \sigma }.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa1d46ce1c9924b2baf13f7d47275d6fba96795 height: height attribute not set width: width attribute not set description: {\displaystyle s^{\mu \nu }={\frac {1}{4}}[\gamma ^{\mu },\gamma ^{\nu }]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2efd2386e6af720ba38a7ab0d22f685d9036eae0 height: height attribute not set width: width attribute not set description: {\displaystyle \lambda =\exp \left({\frac {1}{2}}\omega _{\mu \nu }m^{\mu \nu }\right)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/938bfe327ff95bcb3b6a76bd7cc181159baf86f2 height: height attribute not set width: width attribute not set description: {\displaystyle \omega _{\mu \nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/10722092977f8485627304963fca0083fc6ef228 height: height attribute not set width: width attribute not set description: {\displaystyle s[\lambda ]=\exp \left({\frac {1}{2}}\omega _{\mu \nu }s^{\mu \nu }\right).} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c8224ceefae69c6e46f08d195b1b50eee7374491 height: height attribute not set width: width attribute not set description: {\displaystyle \omega _{\mu \nu }.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/5e923264bc0f1f03e2a84fd8ac528e5f923deca1 height: height attribute not set width: width attribute not set description: {\displaystyle i\gamma ^{\mu }\partial _{\mu }\psi (x)-m\psi (x)=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/406788da24426d18a98234247f994455acded045 height: height attribute not set width: width attribute not set description: {\displaystyle i\gamma ^{\mu }((\lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })s[\lambda ]\psi (\lambda ^{-1}x)-ms[\lambda ]\psi (\lambda ^{-1}x)=0.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/599451b0fac5eb9ffd9fcff587ba85f8189ab397 height: height attribute not set width: width attribute not set description: {\displaystyle s^{-1}[\lambda ]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4 height: height attribute not set width: width attribute not set description: {\displaystyle x} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/7020f3da1d164c0669eb5b53d5155f08d42e2bbc height: height attribute not set width: width attribute not set description: {\displaystyle is[\lambda ]^{-1}\gamma ^{\mu }s[\lambda ]((\lambda ^{-1})_{\mu }{}^{\nu }\partial _{\nu })\psi (x)-m\psi (x)=0.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/41bc1b06fd934caccc9008ed8ca291a33432eb05 height: height attribute not set width: width attribute not set description: {\displaystyle s[\lambda ]^{-1}\gamma ^{\mu }s[\lambda ](\lambda ^{-1})^{\nu }{}_{\mu }=\gamma ^{\nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d493d98586cc5011c2eeb929b4e26ea6c8a985b1 height: height attribute not set width: width attribute not set description: {\displaystyle s[\lambda ]^{-1}\gamma ^{\mu }s[\lambda ]=\lambda ^{\mu }{}_{\nu }\gamma ^{\nu }.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8 height: height attribute not set width: width attribute not set description: {\displaystyle \omega } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ea731e151719d4d00a5273e65ae3ac6e04e6c0f2 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{2}}\omega _{\rho \sigma }(m^{\rho \sigma })^{\mu }{}_{\nu }\gamma ^{\nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/927a272a97d477df7b517379cf939abfa81fd2ab height: height attribute not set width: width attribute not set description: {\displaystyle \left[{\frac {1}{2}}\omega _{\rho \sigma }s^{\rho \sigma },\gamma ^{\mu }\right]={\frac {1}{2}}\omega _{\rho \sigma }\left[s^{\rho \sigma },\gamma ^{\mu }\right]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/09f53e3ecb985af28069f8f19f6e13ffc857ed28 height: height attribute not set width: width attribute not set description: {\displaystyle m^{\rho \sigma }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/7793f3e718f73b103cc55a8720ddb9a1e778c7fe height: height attribute not set width: width attribute not set description: {\displaystyle ({\mathcal {j}}^{\rho \sigma })^{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6d36857e2c75bdfbd5403a04c617a73b65171a9b height: height attribute not set width: width attribute not set description: {\displaystyle t^{\mu \nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc height: height attribute not set width: width attribute not set description: {\displaystyle a} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2 height: height attribute not set width: width attribute not set description: {\displaystyle x'} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c24315cfafa681aec77b353aab9d2e85a63b6e76 height: height attribute not set width: width attribute not set description: {\displaystyle x\mapsto x',} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3f04e27e1870051b7fabbd51b9c2364072209e height: height attribute not set width: width attribute not set description: {\displaystyle \psi '(x')=s\psi (x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2 height: height attribute not set width: width attribute not set description: {\displaystyle s} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b97313a2e0df03f5965f515ea104bc5edb7b6b4b height: height attribute not set width: width attribute not set description: {\displaystyle s=\exp \left({\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1dcfeeb5699089c110d8704c8fed3ed4f1de38d6 height: height attribute not set width: width attribute not set description: {\displaystyle \omega ^{\mu \nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2a5225e32ad6a094f47f8bb48dbb742e303797 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma _{\mu \nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e2271999698158d61e356297a43e9821fab684 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{\mu \nu }={\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/db78d877d8430a7d8189010063c2db973726547b height: height attribute not set width: width attribute not set description: {\displaystyle j_{\mu \nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8fdb9ea5f9813eaff6845d395fc42fc0e9f6edd7 height: height attribute not set width: width attribute not set description: {\displaystyle j_{\mu \nu }={\frac {1}{2}}\sigma _{\mu \nu }+i(x_{\mu }\partial _{\nu }-x_{\nu }\partial _{\mu })} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d2692887e350c15a4847193887e479ab812190b2 height: height attribute not set width: width attribute not set description: {\displaystyle \psi ^{\prime }(x)=\exp \left({\frac {-i}{2}}\omega ^{\mu \nu }j_{\mu \nu }\right)\psi (x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e2f7f304f1bfe40c8e148a344684b6daac2593 height: height attribute not set width: width attribute not set description: {\displaystyle x\mapsto x'} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9f8d2f80b11e6492301be671f38b4a5b044355 height: height attribute not set width: width attribute not set description: {\displaystyle \psi (x)\mapsto \psi '(x')} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c8eb773682695f914a4a65ca86e2f34631381e25 height: height attribute not set width: width attribute not set description: {\displaystyle x'\mapsto x} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b78e1e0799faa8b859224c9a33e01d4abde274 height: height attribute not set width: width attribute not set description: {\displaystyle x~.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6b803da9c45c1186883bde55107e9ccb102c92c6 height: height attribute not set width: width attribute not set description: {\displaystyle a.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/270ddf1e50b881963d5960e52d173b61802fc822 height: height attribute not set width: width attribute not set description: {\displaystyle x'=\lambda x} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8074af204563c8e1d08a38ac6739986d9a677220 height: height attribute not set width: width attribute not set description: {\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi (x)-m\psi (x)=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f90b9184495662c4fb060ae80da59262dfa385 height: height attribute not set width: width attribute not set description: {\displaystyle i\gamma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi ^{\prime }(x^{\prime })-m\psi ^{\prime }(x^{\prime })=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb5ad10b735f22aa3306f320b4617db77474f1b height: height attribute not set width: width attribute not set description: {\displaystyle \psi ^{\prime }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6ea51dd7e4f3e87885e097502a0b1f63826617 height: height attribute not set width: width attribute not set description: {\displaystyle \psi ^{\prime }(x^{\prime })=s(\lambda )\psi (x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3646c0b5192ce60f1f927f71de4a1325b9ace24d height: height attribute not set width: width attribute not set description: {\displaystyle i\gamma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}s(\lambda )\psi (x)-ms(\lambda )\psi (x)=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/453d3de7ade40b89154be71e0198d8b2ae516b25 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}={\left(\lambda ^{-1}\right)^{\nu }}_{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1bba7be07b54f6b2d7511ab29a9bea2dc92e359f height: height attribute not set width: width attribute not set description: {\displaystyle s(\lambda )\gamma ^{\mu }s^{-1}(\lambda )={\left(\lambda ^{-1}\right)^{\mu }}_{\nu }\gamma ^{\nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd25e13e1ac3334ba077d85e80f25fa5e2b9988 height: height attribute not set width: width attribute not set description: {\displaystyle s(\lambda )} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1ceaebf7fbf6ad8b43b08ca39f0c55172febed3c height: height attribute not set width: width attribute not set description: {\displaystyle {\lambda ^{\mu }}_{\nu }={g^{\mu }}_{\nu }+{\omega ^{\mu }}_{\nu }\ ,\ {(\lambda ^{-1})^{\mu }}_{\nu }={g^{\mu }}_{\nu }-{\omega ^{\mu }}_{\nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/411bc4b28ba6ae8d9f8d0af96b9946d7c114e102 height: height attribute not set width: width attribute not set description: {\displaystyle {g^{\mu }}_{\nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc5efff8a35053c42c56f88f67df9fda8bb1dcd height: height attribute not set width: width attribute not set description: {\displaystyle {g^{\mu }}_{\nu }=g^{\mu \nu '}g_{\nu '\nu }={\delta ^{\mu }}_{\nu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/de9410a0b681d1061c95b89d7fb2a935d2f20d2a height: height attribute not set width: width attribute not set description: {\displaystyle \omega _{\mu \nu }={\omega ^{\alpha }}_{\nu }g_{\alpha \mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4b2b110fccc62c9fac41b45a31a91b4d74b544 height: height attribute not set width: width attribute not set description: {\displaystyle s(\lambda )=i+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }+{\mathcal {o}}\left(\lambda ^{2}\right)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0ba5e0514cff9eae8b231e09932002b04cb89d height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{\mu \nu }={\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8412b657c05849fbcacfbf6f3c69d2cdcfced7dd height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\psi '(x')&=\left(i+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x)\\&=\left(i+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }\right)\psi (x'+{\omega ^{\mu }}_{\nu }\,x^{\prime \,\nu })\\&=\left(i+{\frac {-i}{4}}\omega ^{\mu \nu }\sigma _{\mu \nu }-x_{\mu }^{\prime }\omega ^{\mu \nu }\partial _{\nu }\right)\psi (x')\\&=\left(i+{\frac {-i}{2}}\omega ^{\mu \nu }j_{\mu \nu }\right)\psi (x')\\\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f5cf42cf86b6815a3d06f3b1a64bb09c2c76dec1 height: height attribute not set width: width attribute not set description: {\displaystyle \psi \mapsto \psi '} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/55b291e6cb29a961cc71c3619e1604eaf0d848ca height: height attribute not set width: width attribute not set description: {\displaystyle \psi ={\begin{pmatrix}\psi _{l}\\\psi _{r}\end{pmatrix}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/577ea88983dfe6dd88e20ff7402dce2c7fd981a0 height: height attribute not set width: width attribute not set description: {\displaystyle \psi _{l}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/eacf8c5f2587de7c59bbfd798e11bdbeaa01707b height: height attribute not set width: width attribute not set description: {\displaystyle \psi _{r}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9fa0456cacd6574be141fca221982cb450c860 height: height attribute not set width: width attribute not set description: {\displaystyle \gamma ^{\mu }{\begin{pmatrix}\psi _{l}\\\psi _{r}\end{pmatrix}}={\begin{pmatrix}\sigma ^{\mu }\psi _{r}\\{\overline {\sigma }}^{\mu }\psi _{l}\end{pmatrix}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c871ccb3f7fbaa7461220d9a7223d423314e27da height: height attribute not set width: width attribute not set description: {\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m){\begin{pmatrix}\psi _{l}\\\psi _{r}\end{pmatrix}}=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d31dfcbe2165e8491972b90aae48315168f4a900 height: height attribute not set width: width attribute not set description: {\displaystyle i{\begin{pmatrix}\sigma ^{\mu }\partial _{\mu }\psi _{r}\\{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{l}\end{pmatrix}}=m{\begin{pmatrix}\psi _{l}\\\psi _{r}\end{pmatrix}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/83fbf0fec605dfa9a62c63a5b585a17ac6f524f4 height: height attribute not set width: width attribute not set description: {\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{r}=m\psi _{l}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ddbe4690b7c358f1b354244d6d2f07fcadb79586 height: height attribute not set width: width attribute not set description: {\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{l}=m\psi _{r}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc height: height attribute not set width: width attribute not set description: {\displaystyle m} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467 height: height attribute not set width: width attribute not set description: {\displaystyle e} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c67b39358369d132b9d3e219719a265abfe2d177 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\psi (x)&\mapsto e^{i\alpha }\psi (x),\\{\bar {\psi }}(x)&\mapsto e^{-i\alpha }{\bar {\psi }}(x).\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/76b1b2a4ddc63a82a0cce9221723ac6aa1762910 height: height attribute not set width: width attribute not set description: {\displaystyle j^{\mu }(x)={\bar {\psi }}(x)\gamma ^{\mu }\psi (x).} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/92b728b20518ede0c488d78b1605202c63c36869 height: height attribute not set width: width attribute not set description: {\displaystyle \alpha :\mathbb {r} ^{1,3}\to \mathbb {r} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/14d01c1713b9327c11e9067e5076fdffef20375f height: height attribute not set width: width attribute not set description: {\displaystyle e^{i\alpha }:\mathbb {r} ^{1,3}\to {\text{u}}(1),} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/43941ef40b4eb751a8daea118d0012ea998f5610 height: height attribute not set width: width attribute not set description: {\displaystyle \alpha (x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/267983ed233945e70f42935df28e623cfce15c12 height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9d83306aba55cd6fce0194e89352c46210ce91 height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mu }\psi =\partial _{\mu }\psi +iea_{\mu }\psi ,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cc41cae7722be7263f30b4491f696aed878db height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-iea_{\mu }{\bar {\psi }}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9277f5286335ab99c040c9c9151ab752d3bedc49 height: height attribute not set width: width attribute not set description: {\displaystyle a_{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/431d0e43965aaea6c69303cd88c6bbbc5051dd53 height: height attribute not set width: width attribute not set description: {\displaystyle a_{\mu }(x)\mapsto a_{\mu }(x)+{\frac {1}{e}}\partial _{\mu }\alpha (x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5ad9b0987256e70699f11b9b2f1cf45ec71c2d height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mu }\psi (x)\mapsto e^{i\alpha (x)}d_{\mu }\psi (x),} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1175112f75abcfcc926e12850b1752bdd507c57f height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mu }{\bar {\psi }}(x)\mapsto e^{-i\alpha (x)}d_{\mu }{\bar {\psi }}(x).} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b47191d8910a3d1ec1ab3019ce918600bb10a908 height: height attribute not set width: width attribute not set description: {\displaystyle s=\int d^{4}x\,{\bar {\psi }}\,(id\!\!\!\!{\big /}-m)\,\psi =\int d^{4}x\,{\bar {\psi }}\,(i\gamma ^{\mu }d_{\mu }-m)\,\psi .} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d97665309b41d549b1aba23d483d599f8031af5b height: height attribute not set width: width attribute not set description: {\displaystyle s_{\text{maxwell}}=\int d^{4}x\,\left[-{\frac {1}{4}}f^{\mu \nu }f_{\mu \nu }\right].} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a025c51e08f1173b33252a5ccc3e5357151d4c2e height: height attribute not set width: width attribute not set description: {\displaystyle s_{\text{qed}}=\int d^{4}x\,\left[-{\frac {1}{4}}f^{\mu \nu }f_{\mu \nu }+{\bar {\psi }}\,(id\!\!\!\!{\big /}-m)\,\psi \right]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/384edf181c86f4fafcb6a2e4115d8be55f4c404e height: height attribute not set width: width attribute not set description: {\displaystyle s_{\text{qed}}=\int d^{4}x\,\left[-{\frac {1}{4}}f^{\mu \nu }f_{\mu \nu }+{\bar {\psi }}\,(i\partial \!\!\!{\big /}-m)\,\psi -ej^{\mu }a_{\mu }\right]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6f40b8a17530679b2f9478d78c0c6472de65efbe height: height attribute not set width: width attribute not set description: {\displaystyle \psi (x)={\begin{pmatrix}\psi _{1}(x)\\\psi _{2}(x)\end{pmatrix}},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9f332f8c7bddf80d44c5f5220f879ae3497aa65c height: height attribute not set width: width attribute not set description: {\displaystyle i\gamma ^{\mu }\partial _{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8637b048fbb584161c831a8d0b2d45e24a4330fc height: height attribute not set width: width attribute not set description: {\displaystyle i\gamma ^{\mu }\partial _{\mu }={\begin{pmatrix}0&i\sigma ^{\mu }\partial _{\mu }\\i{\bar {\sigma }}^{\mu }\partial _{\mu }\ &0\end{pmatrix}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9e3f88bdd05d08345945334cf491d972c2132493 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma ^{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/47b285a0b97e645ae64e69402acfb428e044cf24 height: height attribute not set width: width attribute not set description: {\displaystyle (i_{2},\sigma ^{i})} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b25596cca3a356fada0453caf72e45a51c1b6d height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {\sigma }}^{\mu }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1a998f721ea4f9c8b85a4c2fffa1ee749b08b0 height: height attribute not set width: width attribute not set description: {\displaystyle (i_{2},-\sigma ^{i})} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/427fdb0111a2a5ad80c80014660aab414c833803 height: height attribute not set width: width attribute not set description: {\displaystyle s=\int d^{4}x\,\psi _{1}^{\dagger }(i\sigma ^{\mu }\partial _{\mu })\psi _{1}+\psi _{2}^{\dagger }(i{\bar {\sigma }}^{\mu }\partial _{\mu })\psi _{2}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfdde1da54e02a016fe2a230c58b25dfcc014d6 height: height attribute not set width: width attribute not set description: {\displaystyle \psi _{1}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c5083a526766f85c4f39ab695791b0b739f06897 height: height attribute not set width: width attribute not set description: {\displaystyle \psi _{2}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ac69e529d4a5118d297c03ec8249e6746a268d5b height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\psi _{1}(x)&\mapsto e^{i\beta }\psi _{1}(x),\\\psi _{2}(x)&\mapsto e^{-i\beta }\psi _{2}(x).\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9881423825e2d5c9980c5cf72a9a51f1a1bd25 height: height attribute not set width: width attribute not set description: {\displaystyle \psi (x)\mapsto \exp(i\beta \gamma ^{5})\psi (x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1185b570f67b4221307626254f64f9e619e769 height: height attribute not set width: width attribute not set description: {\displaystyle \exp } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b height: height attribute not set width: width attribute not set description: {\displaystyle g} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4e0013caf8d63a1716475799928774c2b8f2e8cc height: height attribute not set width: width attribute not set description: {\displaystyle g={\text{su}}(n)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a093c818bd6ea93f8a94d0aed556d907c4ba46 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {c} ^{n}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/22d8a53594ee60b9679484c9b88fbdc8e21e02c6 height: height attribute not set width: width attribute not set description: {\displaystyle \psi :\mathbb {r} ^{1,3}\mapsto \mathbb {c} ^{4}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/47c39d15ec4d15fddac17715e6d1b118a539b399 height: height attribute not set width: width attribute not set description: {\displaystyle \alpha ,\beta ,\gamma ,\cdots } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d1befdb28ed15d91de6ddb9b5303fe6e5683d61f height: height attribute not set width: width attribute not set description: {\displaystyle i,j,k,\cdots } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc080b972918855f349631dd1d838c538162366 height: height attribute not set width: width attribute not set description: {\displaystyle 4n} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/47aa8c6ec88565764006096bacf8910c3cd586da height: height attribute not set width: width attribute not set description: {\displaystyle \psi ^{i,\alpha }(x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4b30b2abf34553558590b1d89d06d90a57fc4925 height: height attribute not set width: width attribute not set description: {\displaystyle \psi :\mathbb {r} ^{1,3}\to \mathbb {c} ^{4}\otimes \mathbb {c} ^{n}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9ffe9941b895ff3f70af568a0614106d45ac8c height: height attribute not set width: width attribute not set description: {\displaystyle u:\mathbb {r} ^{1,3}\rightarrow {\text{su}}(n),} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/55c68a1dbbabe7cc4855bd5f42a1719ab19e257b height: height attribute not set width: width attribute not set description: {\displaystyle \psi (x)\mapsto u(x)\psi (x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a60098a25029643e08e5417007535ae0c549a432 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {\psi }}(x)\mapsto {\bar {\psi }}(x)u^{\dagger }(x).} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d40b4f913e99b50055f5d7e7c4a8c9b9e05623cd height: height attribute not set width: width attribute not set description: {\displaystyle {\text{su}}(n)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3180f124063429bad52144d0c4d2f9789bf8803e height: height attribute not set width: width attribute not set description: {\displaystyle a_{\mu }(x)\mapsto u(x)a_{\mu }(x)u(x)^{-1}+{\frac {1}{g}}(\partial _{\mu }u(x))u(x)^{-1},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/edb18e9db6699bbe7fb3249eac5c9930beff5ea4 height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mu }\psi =\partial _{\mu }\psi +iga_{\mu }\psi ,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6ccaff66b84ceca5a9a5aff9142f60e7eebc53ce height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mu }{\bar {\psi }}=\partial _{\mu }{\bar {\psi }}-ig{\bar {\psi }}a_{\mu }^{\dagger }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4440edfa9d5807820447d171d8eaa6fa5dccd8eb height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mu }\psi (x)\mapsto u(x)d_{\mu }\psi (x),} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/37fe0f81726501dc34203832d20711624ce0e6c5 height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mu }{\bar {\psi }}(x)\mapsto (d_{\mu }{\bar {\psi }}(x))u(x)^{\dagger }.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/62c1638a4f48db7494393fba54a524598360ce3f height: height attribute not set width: width attribute not set description: {\displaystyle s_{\text{y-m}}=\int d^{4}x\,-{\frac {1}{4}}{\text{tr}}(f^{\mu \nu }f_{\mu \nu })} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3adb7fe181bfb570ee03cade284fb340313060b6 height: height attribute not set width: width attribute not set description: {\displaystyle f_{\mu \nu }=\partial _{\mu }a_{\nu }-\partial _{\nu }a_{\mu }-ig\left[a_{\mu },a_{\nu }\right]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/28dd4c22d60192519c1c12cf645b040f368db9e9 height: height attribute not set width: width attribute not set description: {\displaystyle [\cdot ,\cdot ]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/13c8641fc62950244e108adcff4e051466e9c49c height: height attribute not set width: width attribute not set description: {\displaystyle s_{\text{qcd}}=\int d^{4}x\,-{\frac {1}{4}}{\text{tr}}(f^{\mu \nu }f_{\mu \nu })+{\bar {\psi }}\,(id\!\!\!\!{\big /}-m)\,\psi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ab951d779fcdeea3ec188bb5c73518c46b19de height: height attribute not set width: width attribute not set description: {\displaystyle n=3} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/405d64b14536deffc3465f1e81b1b7fe9358ad2a height: height attribute not set width: width attribute not set description: {\displaystyle n=2} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7 height: height attribute not set width: width attribute not set description: {\displaystyle w} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/77254725424c2b3b6c4aa7dcf10d2e8e6806a7d6 height: height attribute not set width: width attribute not set description: {\displaystyle (\rho ,g,v)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845 height: height attribute not set width: width attribute not set description: {\displaystyle v} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/322826e026fff506994bd9aaa209cd774fbe41d3 height: height attribute not set width: width attribute not set description: {\displaystyle \psi :\mathbb {r} ^{1,3}\to \mathbb {c} ^{4}\otimes v.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9b081eedfce3d3fa6bb154cb965d36c96cae96b7 height: height attribute not set width: width attribute not set description: {\displaystyle g:\mathbb {r} ^{1,3}\to g} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3940e596dfb0a769ee6001efd8e5842096ac8133 height: height attribute not set width: width attribute not set description: {\displaystyle \psi (x)\mapsto \rho (g(x))\psi (x)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d79e66d4b5c6c2ae4f4ebd96b2c5664b3d380fdf height: height attribute not set width: width attribute not set description: {\displaystyle d_{\mu }\psi =\partial _{\mu }\psi +\rho (a_{\mu })\psi } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64 height: height attribute not set width: width attribute not set description: {\displaystyle \rho } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/64bc89999b2bd542faac4d103e2b839dcfc1fe58 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathfrak {g}}={\text{l}}(g)} |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/f/fa/wikiquote-logo.svg/34px-wikiquote-logo.svg.png height: 40 width: 34 description: no alt description found |
|
https://upload.wikimedia.org/wikipedia/en/thumb/9/96/symbol_category_class.svg/16px-symbol_category_class.svg.png height: 16 width: 16 description: no alt description found |
|
https://upload.wikimedia.org/wikipedia/en/thumb/8/8a/oojs_ui_icon_edit-ltr-progressive.svg/10px-oojs_ui_icon_edit-ltr-progressive.svg.png height: 10 width: 10 description: edit this at wikidata |
|
https://login.wikimedia.org/wiki/special:centralautologin/start?type=1x1 height: 1 width: 1 description: no alt description found |
|
http://en.wikipedia.org/static/images/footer/wikimedia-button.svg height: 29 width: 84 description: wikimedia foundation |
|
http://en.wikipedia.org/static/images/footer/poweredby_mediawiki.svg height: 29 width: 84 description: powered by mediawiki |
How are images contributing to your SEO site-wise ? Your leading content tool has the awnsers!