Dirac Equation / Fields| 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 \(j^0\):= ψ†ψ.
* Applications: It yields
the Sommerfeld fine structure for the hydrogen atom energy levels.
* Properties: It has
negative-energy states, interpreted as positrons when quantized.
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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