Klein-Gordon Fields| Klein-Gordon Fields |
In General > s.a. klein-gordon fields in curved
spacetime [and different media]; scalar field theory.
* Hamiltonian: For a scalar field
φ(x) of mass m (and conjugate momentum
π(x)), the Hamiltonian on a spacelike hypersurface Σ is
H = \(1\over2\)∫Σ (π2 + ∇φ · ∇φ + m2φ2) d3x , π = dφ/dt .
* Field equation: The usual form is (if the field is complex, replace one of the φs by φ*, to get a real \(\cal L\))
\(\square\)φ − m2φ = 0 , from the Lagrangian \(\cal L\) = −\(1\over2\)|g|1/2 (gab ∇aφ ∇bφ + m2φ2) .
* Interpretation: It cannot be
interpreted as a single-particle equation because (1) It has negative-energy
solutions; (2) The current j a
does not give a positive-definite probability density ρ; There is no
problem however if it is treated as a (classical or quantum) field equation.
* Solutions: A complete set is
uk(x) = [2ω
(2π)n−1]−1/2
exp{i k · x}, ka
= (ω, k), k2
= −m2; They are eigenfunctions of
∂/∂t, with eigenvalue −iω, and orthonormal
with respect to the Klein-Gordon inner product below.
@ General references: Wald 84, p461;
Oshima et al ht/05 [real vs complex];
Gravel & Gauthier AJP(11)may [classical applications];
Dai a1106 [Hamiltonian with appropriate surface terms].
@ Derivation from classical theory: Lehr & Park JMP(77);
Santamato JMP(84) [Weyl curvature];
Morato PLA(91);
Alonso-Blanco a1201
[as a compatibility condition for Maxwell-Lorentz dynamics in Newtonian mechanics];
Andriambololona et al a1401
[derivation of field equations using dispersion-codispersion operators];
Donker et al AP(16)-a1604 [logical inference approach].
@ And pilot-wave theory: Horton et al JPA(00)qp/01,
comment Tumulka JPA(02)qp,
qp/02;
Horton & Dewdney qp/01.
@ In a box: Koehn EPL(12)-a1301 [infinite square-well potential with a moving wall];
Alberto et al EJP(18)-a1711 [Klein-Gordon vs Dirac equations].
@ Other solutions: Hinterleitner JMP(96) [separation of variables in 2+1 dimensions];
Fodor & Rácz PRD(03)ht [expanding shells];
Gönül ChPL(06)qp [bound states and non-relativistic limit];
Mosley a0707 [wave packets];
Tolish & Wald PRD(14)-a1401 [particle on a null geodesic, retarded solution].
@ Superluminal waves: Borghardt et al PLA(03)qp;
> s.a. klein-gordon fields in curved spacetime.
@ Interpretation: Wharton AIP(07)-a0706,
AP(10) [new probabilistic interpretation];
Heaney FP(13) [Symmetrical Interpretation];
Sutherland a1509 [and retrocausal influences];
Kazemi et al a1802 [new probability current density].
@ (1+1)-dimensional: Hall PLA(07)-a0707 [with non-singular Coulomb-like potential];
Opanasenko & Popovych a1810 [generalized symmetries and conservation laws].
@ Related topics: Kyprianidis PLA(85) [and particle trajectories];
Grössing PLA(02) [sub-quantum Brownian movement];
Comay Ap(04)qp/03,
Ap(05)qp/04 [difficulties];
Semenov et al PLA(08)ht/07 [states with positive norm];
Hall PRA(10) [comparison theorem for energy eigenvalues];
Wong JMP(10) [in hydrodynamical form];
> s.a. thermodynamic systems.
Space of Klein-Gordon Fields > s.a. complex structures.
* Inner product: For φ,
ψ: Σ → \(\mathbb C\), with Σ a spacelike hypersurface
in spacetime, the Klein-Gordon inner product is
\(\langle\)φ | ψ\(\rangle\)KG:= i ∫Σ (φ* ∇m ψ − ψ ∇m φ*) dsm = i ∫Σ (φ*ψ,t − ψ φ*,t) dn−1v .
* Properties: Independent of
Σ, because j a:= −i
(φ* ∇a ψ −
ψ ∇a φ*) is a
conserved current; Positive-definite only if restricted to (combinations of) positive-frequency
solutions of the Klein-Gordon equation (according to the timelike vector field t).
* Symplectic structure:
@ Hájíček & Isham JMP(96)gq/95 [in curved spacetime].
* Observables:
@ Inner product:
Mostafazadeh gq/02 [positive-definite],
CQG(03)mp/02 [Hilbert space],
& Zamani AP(06)qp [covariant];
Kleefeld CzJP(06)qp.
Modifications and Quantization
> s.a. dispersion; klein-gordon quantum field
theory; scalar fields; Sine-Gordon Equation.
@ General references: Adler & Santiago ht/99 [\(\hbar\) = \(\hbar\)(k)];
Santos & Silva JMP(05)mp [variable mass, Wigner-Moyal];
Arminjon in(07)-a0706 [from quantum mechanics];
Das CJP(10)-a0811 [covariant discrete phase space];
Thibes a2011 [higher-order generalization];
Giardino a2105 [quaternionic].
@ Fractional Klein-Gordon fields:
Lim & Teo a1103 [and Casimir effect];
Garra et al JSP(14)-a1308 [and related stochastic processes].
@ Quantum-gravity corrections: Cheon IJTP(78) [with fundamental length, Bopp equation];
Jacobson & Mattingly PRD(01)ht/00 [with high-f dispersion];
Moayedi et al IJTP(10)-a1004,
Jana & Roy PLA(09)-a0902 [with minimal length].
@ Non-linear:
Man'ko et al PLA(95) [q-deformed, and other non-linearities];
Perel & Fialkovsky a0712 [arbitrary dimensionality];
Smolyakov JPA(10)-a0910 [Klein-Gordon-Maxwell, no-go result].
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