how does Klein-Gordon equation admit "spontaneous transitions" ?

[Widdekind]

http://wiki.physics.fsu.edu/wiki/index.php/Klein-Gordon_equation

 

The above website says the KG equation admits "spontaneous transitions from positive-energy particle solutions, to negative-energy anti-particle solutions"...

 

Q1: could someone succinctly show, such a "spontaneous transition" sort of solution ?

 

Q2: i personally perceive, a parallel, between the opposite spins, of particles (left-handed) and antiparticles (right-handed)... and the opposite signs, of their frequencies, from the KG equation, which signs cause the phase-factors of particles to spin left-handed [math]e^{- \imath \omega t}[/math], and the phase-factors of antiparticles to spin right-handed [math]e^{+ \imath \omega t}[/math], around on the complex-plane... is it correct, to interpret the positive and negative energies, of particles and antiparticles, as actually merely positive (LH) and negative (RH) phase-factor frequencies ?

 

Q3: the "density" of antiparticles is conserved, but equals -1 (whereas particles normalize to +1)... so the KGE seemingly states, that antiparticles are "absent missing (but whole) holes" in the Dirac sea... is that not essentially the same, as what the Dirac equation demonstrates ? What does the Dirac equation actually demonstrate, that the KGE does not ?

[TrappedLight]

http://wiki.physics.fsu.edu/wiki/index.php/Klein-Gordon_equation

 

The above website says the KG equation admits "spontaneous transitions from positive-energy particle solutions, to negative-energy anti-particle solutions"...

 

Q1: could someone succinctly show, such a "spontaneous transition" sort of solution ?

 

Q2: i personally perceive, a parallel, between the opposite spins, of particles (left-handed) and antiparticles (right-handed)... and the opposite signs, of their frequencies, from the KG equation, which signs cause the phase-factors of particles to spin left-handed [math]e^{- \imath \omega t}[/math], and the phase-factors of antiparticles to spin right-handed [math]e^{+ \imath \omega t}[/math], around on the complex-plane... is it correct, to interpret the positive and negative energies, of particles and antiparticles, as actually merely positive (LH) and negative (RH) phase-factor frequencies ?

 

Q3: the "density" of antiparticles is conserved, but equals -1 (whereas particles normalize to +1)... so the KGE seemingly states, that antiparticles are "absent missing (but whole) holes" in the Dirac sea... is that not essentially the same, as what the Dirac equation demonstrates ? What does the Dirac equation actually demonstrate, that the KGE does not ?

 

 

 

Well, to show the negative and positive values of the KG equation, is second order in time and admits the solutions

 

[math]\psi(r,t) = e^{\frac{i(pr - Et)}{\hbar}}[/math]

 

where the sign of the energy is

 

[math]E = \pm c \sqrt{p^2 - Mc^2}[/math]

 

The difference between the Dirac equation and the KG equation, is that the Dirac equation describes heavy fermion particles with spin 1/2 states. The KG equation is an equation used to describe spinless particles. Another large difference is that the Dirac equation is first order in time.

Edited September 20, 2013 by TrappedLight

[Widdekind]

do you not mean "sign of the frequency" (not "energy") ? anti-Fermions' phase frequencies are negative (not necessarily their energies). anti-Fermions also have opposite electric (and weak) charge... is there some sort of close connection, between charge <----> energy / frequency ?