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37-534: [REDACTED] Look up APS in Wiktionary, the free dictionary. APS or Aps or aps or similar may refer to: Businesses and organizations [ edit ] Businesses and public bodies [ edit ] Algeria Press Service , a news agency Arizona Public Service , an electric power company Australian Protective Service , an Australian Commonwealth law enforcement agency Australian Public Service ,

74-469: A United States Navy maritime safety initiative Stechkin automatic pistol ( Avtomaticheskiy Pistolet Stechkina ) APS, a classification symbol United States Navy amphibious warfare ships Other uses [ edit ] Adult Protective Services , social services in the United States Advanced planning and scheduling , a manufacturing management process Annual Population Survey ,

111-404: A United States Navy maritime safety initiative Stechkin automatic pistol ( Avtomaticheskiy Pistolet Stechkina ) APS, a classification symbol United States Navy amphibious warfare ships Other uses [ edit ] Adult Protective Services , social services in the United States Advanced planning and scheduling , a manufacturing management process Annual Population Survey ,

148-652: A compact, unified and geometrical formalism for both classical and quantum mechanics. APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cl 1,3 ( R ) of the four-dimensional Minkowski spacetime . In APS, the spacetime position is represented as the paravector x = x 0 + x 1 e 1 + x 2 e 2 + x 3 e 3 , {\displaystyle x=x^{0}+x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3},} where

185-690: A household survey in Great Britain Anpartsselskab (ApS), the Danish term for a limited liability company Average propensity to save , in Keynesian economics See also [ edit ] All pages with titles beginning with Aps All pages with titles beginning with APS All pages with titles beginning with ApS All pages with titles beginning with APs All pages with titles containing aps Apse (disambiguation) AP (disambiguation) Topics referred to by

222-518: A household survey in Great Britain Anpartsselskab (ApS), the Danish term for a limited liability company Average propensity to save , in Keynesian economics See also [ edit ] All pages with titles beginning with Aps All pages with titles beginning with APS All pages with titles beginning with ApS All pages with titles beginning with APs All pages with titles containing aps Apse (disambiguation) AP (disambiguation) Topics referred to by

259-506: A human rights group Alternative Press Syndicate , a network of countercultural newspapers and magazines American Philatelic Society , a stamp collecting foundation American Philosophical Society , an American scholarly organization American Physical Society , a not-for-profit membership organization American Physiological Society , a non-profit professional society American Phytopathological Society , an international scientific organization Armenian Professional Society ,

296-506: A human rights group Alternative Press Syndicate , a network of countercultural newspapers and magazines American Philatelic Society , a stamp collecting foundation American Philosophical Society , an American scholarly organization American Physical Society , a not-for-profit membership organization American Physiological Society , a non-profit professional society American Phytopathological Society , an international scientific organization Armenian Professional Society ,

333-686: A membership organization Association for Psychological Science , an international non-profit organization Association for Psychosocial Studies , a British learned society Australasian Proteomics Society , a learned society Australian Psychological Society , a professional association Places [ edit ] Alba-la-Romaine , Ardèche, France, known as Aps until 1904 Apsley railway station , England, station code APS Science and technology [ edit ] Adenosine 5'-phosphosulfate, metabolic precursor to 3'-phosphoadenosine-5'-phosphosulfate Antiphospholipid syndrome , an autoimmune state Autoimmune polyendocrine syndrome ,

370-686: A membership organization Association for Psychological Science , an international non-profit organization Association for Psychosocial Studies , a British learned society Australasian Proteomics Society , a learned society Australian Psychological Society , a professional association Places [ edit ] Alba-la-Romaine , Ardèche, France, known as Aps until 1904 Apsley railway station , England, station code APS Science and technology [ edit ] Adenosine 5'-phosphosulfate, metabolic precursor to 3'-phosphoadenosine-5'-phosphosulfate Antiphospholipid syndrome , an autoimmune state Autoimmune polyendocrine syndrome ,

407-409: A more compact form by defining the ordinary velocity as v = d d x 0 ( x 1 e 1 + x 2 e 2 + x 3 e 3 ) , {\displaystyle \mathbf {v} ={\frac {d}{dx^{0}}}(x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3}),} and recalling

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444-1314: A news agency Arizona Public Service , an electric power company Australian Protective Service , an Australian Commonwealth law enforcement agency Australian Public Service , the federal civil service of the Commonwealth of Australia ASTRA Platform Services, now SES Platform Services Education [ edit ] Abbottabad Public School , in Pakistan Adarsh Public School , in India Alamogordo Public Schools , in New Mexico, U.S. Albuquerque Public Schools , in New Mexico, U.S. Allendale Public Schools , in Michigan, U.S. Arlington Public Schools , in Virginia, U.S. Associated Public Schools of Victoria , in Australia Atlanta Public Schools , in Georgia, U.S. Indian Army Public Schools Other organisations [ edit ] Aborigines' Protection Society ,

481-453: A rare disease Advanced Photon Source , a synchrotron X-ray source at Argonne National Laboratory Algebra of physical space , in physics Ammonium persulfate , in chemistry Apus , abbreviation Aps, a constellation Application Packaging Standard , for integrating application software Active-pixel sensor , an image sensor Advanced Photo System , a film format in photography Asynchronous Protocol Specification , or X.445,

518-453: A rare disease Advanced Photon Source , a synchrotron X-ray source at Argonne National Laboratory Algebra of physical space , in physics Ammonium persulfate , in chemistry Apus , abbreviation Aps, a constellation Application Packaging Standard , for integrating application software Active-pixel sensor , an image sensor Advanced Photo System , a film format in photography Asynchronous Protocol Specification , or X.445,

555-556: A single equation: ∂ ¯ F = 1 ϵ j ¯ , {\displaystyle {\bar {\partial }}F={\frac {1}{\epsilon }}{\bar {j}}\,,} where the overbar represents the Clifford conjugation . The Lorentz force equation takes the form d p d τ = e ⟨ F u ⟩ R . {\displaystyle {\frac {dp}{d\tau }}=e\langle Fu\rangle _{R}\,.} The electromagnetic Lagrangian

592-606: A telecommunications standard Atmospheric Plasma Spray, a thermal spraying technique Application support sublayer of the Zigbee specification Accessible Pedestrian Signal , a pedestrian crossing Alstom APS , electrical ground-level power supply for trams Auxiliary Propulsion System of the S-IVB launch vehicle Military [ edit ] APS underwater rifle , a Soviet firearm Active protection system , for vehicles against missile attacks Africa Partnership Station ,

629-511: A telecommunications standard Atmospheric Plasma Spray, a thermal spraying technique Application support sublayer of the Zigbee specification Accessible Pedestrian Signal , a pedestrian crossing Alstom APS , electrical ground-level power supply for trams Auxiliary Propulsion System of the S-IVB launch vehicle Military [ edit ] APS underwater rifle , a Soviet firearm Active protection system , for vehicles against missile attacks Africa Partnership Station ,

666-416: Is L = 1 2 ⟨ F F ⟩ S − ⟨ A j ¯ ⟩ S , {\displaystyle L={\frac {1}{2}}\langle FF\rangle _{S}-\langle A{\bar {j}}\rangle _{S}\,,} which is a real scalar invariant. The Dirac equation , for an electrically charged particle of mass m and charge e , takes

703-465: Is x → ( x 0 + x 3 x 1 − i x 2 x 1 + i x 2 x 0 − x 3 ) {\displaystyle x\rightarrow {\begin{pmatrix}x^{0}+x^{3}&&x^{1}-ix^{2}\\x^{1}+ix^{2}&&x^{0}-x^{3}\end{pmatrix}}} The restricted Lorentz transformations that preserve

740-618: Is a positive unimodular paravector, which implies the following condition in terms of the Clifford conjugation u u ¯ = 1. {\displaystyle u{\bar {u}}=1.} The proper velocity transforms under the action of the Lorentz rotor L as u → u ′ = L u L † . {\displaystyle u\rightarrow u^{\prime }=LuL^{\dagger }.} The four-momentum in APS can be obtained by multiplying

777-408: Is different from Wikidata All article disambiguation pages All disambiguation pages APS [REDACTED] Look up APS in Wiktionary, the free dictionary. APS or Aps or aps or similar may refer to: Businesses and organizations [ edit ] Businesses and public bodies [ edit ] Algeria Press Service ,

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814-471: Is different from Wikidata All article disambiguation pages All disambiguation pages Algebra of physical space The Clifford algebra Cl 3,0 ( R ) has a faithful representation , generated by Pauli matrices , on the spin representation C ; further, Cl 3,0 ( R ) is isomorphic to the even subalgebra Cl 3,1 ( R ) of the Clifford algebra Cl 3,1 ( R ). APS can be used to construct

851-587: Is invariant under a gauge transformation of the form A → A + ∂ χ , {\displaystyle A\rightarrow A+\partial \chi \,,} where χ {\displaystyle \chi } is a scalar field . The electromagnetic field is covariant under Lorentz transformations according to the law F → F ′ = L F L ¯ . {\displaystyle F\rightarrow F^{\prime }=LF{\bar {L}}\,.} The Maxwell equations can be expressed in

888-496: Is the double cover of the Lorentz group . The unimodularity of the Lorentz rotor is translated in the following condition in terms of the product of the Lorentz rotor with its Clifford conjugation L L ¯ = L ¯ L = 1. {\displaystyle L{\bar {L}}={\bar {L}}L=1.} This Lorentz rotor can be always decomposed in two factors, one Hermitian B = B , and

925-902: The Hermitian part representing the electric field E and the anti-Hermitian part representing the magnetic field B . In the standard Pauli matrix representation, the electromagnetic field is: F → ( E 3 E 1 − i E 2 E 1 + i E 2 − E 3 ) + i ( B 3 B 1 − i B 2 B 1 + i B 2 − B 3 ) . {\displaystyle F\rightarrow {\begin{pmatrix}E_{3}&E_{1}-iE_{2}\\E_{1}+iE_{2}&-E_{3}\end{pmatrix}}+i{\begin{pmatrix}B_{3}&B_{1}-iB_{2}\\B_{1}+iB_{2}&-B_{3}\end{pmatrix}}\,.} The source of

962-528: The definition of the gamma factor : γ ( v ) = 1 1 − | v | 2 c 2 , {\displaystyle \gamma (\mathbf {v} )={\frac {1}{\sqrt {1-{\frac {|\mathbf {v} |^{2}}{c^{2}}}}}},} so that the proper velocity is more compactly: u = γ ( v ) ( 1 + v ) . {\displaystyle u=\gamma (\mathbf {v} )(1+\mathbf {v} ).} The proper velocity

999-455: The direction of time and include rotations and boosts can be performed by an exponentiation of the spacetime rotation biparavector W L = e W / 2 . {\displaystyle L=e^{W/2}.} In the matrix representation, the Lorentz rotor is seen to form an instance of the SL(2, C ) group ( special linear group of degree 2 over the complex numbers ), which

1036-646: The federal civil service of the Commonwealth of Australia ASTRA Platform Services, now SES Platform Services Education [ edit ] Abbottabad Public School , in Pakistan Adarsh Public School , in India Alamogordo Public Schools , in New Mexico, U.S. Albuquerque Public Schools , in New Mexico, U.S. Allendale Public Schools , in Michigan, U.S. Arlington Public Schools , in Virginia, U.S. Associated Public Schools of Victoria , in Australia Atlanta Public Schools , in Georgia, U.S. Indian Army Public Schools Other organisations [ edit ] Aborigines' Protection Society ,

1073-460: The field F is the electromagnetic four-current : j = ρ + j , {\displaystyle j=\rho +\mathbf {j} \,,} where the scalar part equals the electric charge density ρ , and the vector part the electric current density j . Introducing the electromagnetic potential paravector defined as: A = ϕ + A , {\displaystyle A=\phi +\mathbf {A} \,,} in which

1110-534: The form: i ∂ ¯ Ψ e 3 + e A ¯ Ψ = m Ψ ¯ † , {\displaystyle i{\bar {\partial }}\Psi \mathbf {e} _{3}+e{\bar {A}}\Psi =m{\bar {\Psi }}^{\dagger },} where e 3 is an arbitrary unitary vector, and A is the electromagnetic paravector potential as above. The electromagnetic interaction has been included via minimal coupling in terms of

1147-1301: The other unitary R = R , such that L = B R . {\displaystyle L=BR.} The unitary element R is called a rotor because this encodes rotations, and the Hermitian element B encodes boosts. The four-velocity , also called proper velocity , is defined as the derivative of the spacetime position paravector with respect to proper time τ : u = d x d τ = d x 0 d τ + d d τ ( x 1 e 1 + x 2 e 2 + x 3 e 3 ) = d x 0 d τ [ 1 + d d x 0 ( x 1 e 1 + x 2 e 2 + x 3 e 3 ) ] . {\displaystyle u={\frac {dx}{d\tau }}={\frac {dx^{0}}{d\tau }}+{\frac {d}{d\tau }}(x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3})={\frac {dx^{0}}{d\tau }}\left[1+{\frac {d}{dx^{0}}}(x^{1}\mathbf {e} _{1}+x^{2}\mathbf {e} _{2}+x^{3}\mathbf {e} _{3})\right].} This expression can be brought to

APS - Misplaced Pages Continue

1184-587: The potential A . The differential equation of the Lorentz rotor that is consistent with the Lorentz force is d Λ d τ = e 2 m c F Λ , {\displaystyle {\frac {d\Lambda }{d\tau }}={\frac {e}{2mc}}F\Lambda ,} such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest u = Λ Λ † , {\displaystyle u=\Lambda \Lambda ^{\dagger },} which can be integrated to find

1221-459: The proper velocity with the mass as p = m u , {\displaystyle p=mu,} with the mass shell condition translated into p ¯ p = m 2 . {\displaystyle {\bar {p}}p=m^{2}.} The electromagnetic field is represented as a bi-paravector F : F = E + i B , {\displaystyle F=\mathbf {E} +i\mathbf {B} ,} with

1258-403: The same term [REDACTED] This disambiguation page lists articles associated with the title APS . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=APS&oldid=1258043630 " Category : Disambiguation pages Hidden categories: Short description

1295-403: The same term [REDACTED] This disambiguation page lists articles associated with the title APS . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=APS&oldid=1258043630 " Category : Disambiguation pages Hidden categories: Short description

1332-1060: The scalar part equals the electric potential ϕ , and the vector part the magnetic potential A . The electromagnetic field is then also: F = ∂ A ¯ . {\displaystyle F=\partial {\bar {A}}.} The field can be split into electric E = ⟨ ∂ A ¯ ⟩ V {\displaystyle E=\langle \partial {\bar {A}}\rangle _{V}} and magnetic B = i ⟨ ∂ A ¯ ⟩ B V {\displaystyle B=i\langle \partial {\bar {A}}\rangle _{BV}} components. Here, ∂ = ∂ t + e 1 ∂ x + e 2 ∂ y + e 3 ∂ z {\displaystyle \partial =\partial _{t}+\mathbf {e} _{1}\,\partial _{x}+\mathbf {e} _{2}\,\partial _{y}+\mathbf {e} _{3}\,\partial _{z}} and F

1369-510: The time is given by the scalar part x = t , and e 1 , e 2 , e 3 are the standard basis for position space. Throughout, units such that c = 1 are used, called natural units . In the Pauli matrix representation, the unit basis vectors are replaced by the Pauli matrices and the scalar part by the identity matrix. This means that the Pauli matrix representation of the space-time position

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