Dirac Equation / Fields  

Dirac Equation > s.a. lagrangian systems.
$ Def: The equation for a 4-component spinor \(\psi\) describing spin-1/2 particles, given by

\(({\rm i}\gamma^a\nabla_{\!a} - m)\, \psi = 0\;, \qquad \gamma^a\) = Dirac matrices .

* In 2D: With potentials \(v\) and \(w\), \((\alpha p + \beta v + \gamma_5 w)\, \psi = E\, \psi\), with α, β, and γ5 Pauli matrices, and p momentum.
* Probability density: The positive-definite quantity \(j^0\):= ψψ.
* Applications: It yields the Sommerfeld fine structure for the hydrogen atom energy levels.
* Properties: It has negative-energy states, interpreted as positrons when quantized.
blue bullet Related topics: see dirac fields in curved spacetime [eigenvalues]; generalized dirac fields; quantum dirac fields.

References > s.a. huygens' principle; lattice field theory; thermodynamic systems.
@ General: Dirac PRS(36); in Bethe & Jackiw 68; Laporte & Uhlenbeck PR(31); Fauser in(98)ht/96 [quantum field theory point of view]; Gsponer & Hurni in(98)phy/05 [history]; Wilczek IJMPA(04); Caban et al a1206, PRA(13) [and relativistic spin operators in terms of unitary representations of the Poincaré group]; Nikolić EJP(14)-a1309 [Lorentz covariance]; Nieto & Pereyra IJMPA(13) [Majorana representation, 3+1 and 2+2 dimensions].
@ Interpretation: Savasta et al a0905; Kissling & Tahvildar-Zadeh JPA(16)-a1411 [quantum mechanical]; Poluektov a1901.
@ Evolution: Adler & Jacob JMP(96) [initial-value formulation]; Bigazzi & Lusanna IJMPA(99) [Hamiltonian, observables]; Lukyanenko & Lukyanenko SPIE(06)gq [path-integral representation]; Demikhovskii et al PRA(10)-a1007 [Gaussian wave packets]; Castrigiano a2004 [free wavefunction evolution].
@ Numerical: Mocken & Keitel JCP(04).
@ Derivations: Rylov IJTP(80) [from random world-lines]; Ord AIP(02)qp [from classical statistical mechanics]; Ng & van Dam PLA(03)ht/02 [lattice]; Arminjon FPL(06) [classical Hamiltonian]; Santamato a0808 [and the relativistic top]; Sartor a0903; Santamato & De Martini FP(13) [from conformal differential geometry]; D'Ariano & Perinotti PRA(14)-a1306, D'Ariano et al PLA(14)-a1406 [from the dynamics of a quantum cellular automaton]; Simulik & Krivsky a1309; Deriglazov & Nersessian PLA(14) [from quantized relativistic particle model]; Diaz & Rossignoli PRD(19)-a1806 [history state formalism, from timeless Wheeler-DeWitt-like equation].
@ Related topics: Konopelchenko JPA(97) [and Gauss-Codazzi equations]; Solomon qp/99 [inconsistencies?]; Marchuk NCB(01)mp/00, mp/00 [as tensor equation]; Liu mp/01 [vector representation]; Solomon qp/02, qp/03 [and gauge invariance], ht/03 [Schwinger-term issues]; Mosna & Vaz PLA(03) [tomographic reconstruction of the internal degrees of freedom]; Chavoya-Aceves NCB(03), Hiley & Callaghan a1011 [de Broglie-Bohm type model]; Arminjon IJTP(14)-a1211 [spin-rotation coupling]; Henheik & Tumulka a2006 [interior-boundary conditions, point sources].

Systems and Solutions > s.a. aharonov-bohm effect; Dirac Matrices; particle models; solitons.
@ General references: de Castro & Franklin IJMPA(00)hp, hp/00-in; Hiller AJP(02)may-qp/01 [1+1 scalar potential]; Thaller qp/04 [wave packet visualization]; Voronov et al TMP(07)qp/06 [strong Coulomb field]; Nogami & Toyama AJP(10)feb [minimum-uncertainty wave packets]; Park PRA(12) [wave packets with small momentum spread]; Datta a1709 [variation problem, review of min-max approach]; Moaiery et al a2104 [2+1 dimensions].
@ V(r): Goodman & Ignjatović AJP(97)mar + comments; Esposito & Santorelli JPA(99)ht; Bose et al PLA(01); Alhaidari PRL(01)ht, IJMPA(03); Dong PS(03) [2+1 Coulomb]; de Castro AP(04) [2D, linear + Coulomb-like V]; Ciftci et al PRA(05)mp [D dimensions]; Hall PRL(08)-a0808 [comparison theorem for parameter dependence]; Caruso et al AP(15)-a1411 [Coulomb-like potential in higher dimensions].
@ Cylindrically symmetric: Booth & Radford JMP(97)ht/96; Skarzhinsky & Audretsch JPA(97)ht [in B field].
@ Scattering: Dombey & Kennedy JPA(02)ht/01; Murguia & Moreno JPA(03)qp/02 [in B]; Kennedy et al IJMPA(04)mp [phase shifts]; De Leo & Rotelli EPJC(09)-a0908.
@ Potential barrier: Dombey et al PRL(00)ht [transmission]; De Leo & Rotelli EPJC(06)ht [multiple scattering], PRA(06)ht [and Klein-like paradox].
@ Uniform magnetic field: Bhattacharya a0705; Pitschmann & Ivanov a1205; Pedram et al IJMPD(15)-a1412.
@ Other electromagnetic field: D'Ancona & Fanelli m.AP/05 [dispersion]; Bourouaine EPJC(05)mp [in plane electromagnetic wave]; Adler et al PRD(12)-a1108 [and a weak gravitational field]; > s.a. electromagnetism.
@ In a box: Alhaidari a0908-proc [square well and box]; Alberto et al EJP(18)-a1711 [Klein-Gordon vs Dirac equations].
@ Special situations: Benguria et al JPA(00) [δ-function potential]; Gulveren et al PS(01) [in Yukawa field]; Tenjinbayashi et al AP(07) [on a torus, zero-modes]; Rivas JPA(07)ht [two coupled particles]; Giachetti & Sorace PRL(08)-a0706 [confining potentials].
@ Semiclassical motion: Dayi & Kilinçarslan PLB(15)-a1508 [kinetic theory, Thomas precession]; Gutiérrez-Jáuregui et al PRA(17)-a1711 [spin effects].
@ Related topics: Chruściel & Bartnik m.DG/03 [boundary-value problem]; Dürr & Pickl JMP(03) [flux across surfaces theorem]; Kassandrov G&C(08)-a0907 [and solutions of Klein-Gordon equation]; Lepori et al EPL(10)-a1004 [massive, realization using ultracold atoms]; Hall PRA(10) [comparison theorem for energy eigenvalues]; Kechriniotis et al CTP(20)-a1208 [and electromagnetic 4-potentials].
> Related phenomena: see Dirac Sea; Induced Gravity; Klein Paradox; quantum effects [tunneling].

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