Klein-Gordon
Fields| Klein-Gordon
Fields |
In General > s.a. klein-gordon fields
in curved spacetime; scalar
field theory.
* Hamiltonian:
H = 
Sigma (
2 + 
·
+ m22)
d3x , =
d/dt .
* Field equation: The usual
form is (if the field is complex, replace one
of the 's
by *, to
get a real 
)
– m2 =
0 , from
the Lagrangian =
– |g|1/2
(gab a b + m22)
.
* Interpretation: 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 
;
There is no problem if it is treated as a (classical or quantum) field equation.
* Solutions: A complete
set is uk(x)
= [2
(2)n–1]–1/2
exp{i k · x}, ka =
(,
k), k2
= –m2;
They are eigenfunctions of 
/t,
with eigenvalue –i,
and orthonormal wrt the Klein-Gordon inner product below.
@ General references: Wald 84, p461;
Oshima et al ht/05 [real
vs complex].
@ Derivation from classical field theory: Lehr & Park JMP(77);
Santamato JMP(84)
[Weyl curvature]; Morato PLA(91).
@ And pilot-wave theory: Horton et al JPA(00)qp/01,
comm Tumulka JPA(02)qp, qp/02;
Horton & Dewdney
qp/01.
@ Solutions: Hinterleitner JMP(96)
[separation of variables in 2+1]; Fodor & Rácz PRD(03)ht [expanding
shells]; Gönül CPL(06)qp [bound
states and non-relativistic limit]; Mosley a0707 [wave
packets].
@ Related topics: Kyprianidis PLA(85)
[and particle trajectories]; Grössing
PLA(02)
[sub-quantum Brownian movement]; Borghardt et al PLA(03)qp [superluminal
waves]; Comay Ap(04)qp/03,
Ap(05)qp/04 [difficulties];
Semenov et al PLA(08)ht/07 [states
with positive norm]; Wharton AIP(07)-a0706
[new probabilistic interpretation]; Hall PLA(07)-a0707 [1D
with non-singular Coulomb-like potential].
Space of Klein-Gordon Fields > s.a. complex
structures.
* Inner product: For ,
:
→ C,
a spacelike hypersurface
in spacetime, the Klein-Gordon inner product is:
|
KG:=
i
Sigma (* m – m *)
dsm = i Sigma (*,t –
*,t)
dn–1v .
* Properties: Independent
of
, because j a:= –i
(* a – a *)
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:
@ Hájícek & 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.
@ General references: Man'ko et al PLA(95)
[q-deformed, and other non-linearities];
Adler & Santiago ht/99 [
=
(k)]; Jacobson & Mattingly
PRD(01)ht/00 [with
high-f dispersion]; Santos & Silva JMP(05)mp
[variable mass, Wigner-Moyal]; Arminjon in(07)-a0706 [from
quantum mechanics]; Das a0811 [covariant
discrete phase space]; Jana & Roy PLA(09)-a0902 [with
minimal length].
@ Non-linear: Perel & Fialkovsky a0712 [arbitrary
dimensionality]; Smolyakov a0910 [Klein-Gordon-Maxwell,
no-go
result].
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send feedback and suggestions to bombelli at olemiss.edu – modified 27
oct
2009