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 \(\cal L\))

\(\square\)φm2φ = 0 ,   from the Lagrangian   \(\cal L\) = −\(1\over2\)|g|1/2 (gabaφbφ + 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 ψψ ∇m φ*) dsm = i Σ (φ*ψ,tψ φ*,t) dn−1v .

* Properties: Independent of Σ, because j a:= −i (φ* ∇a ψψa 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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