# Particle-Antiparticle of Klein-Gordon equation

When we solve Klein-Gordon equation, it gives both positive energy solution and negative energy solution. These two solutions are responsible for positive and negative value of $$\rho$$ which was previously known as probability density in Born's interpretation. But this freedom of $$\rho$$ of being both positive and negative makes us leave the probability density interpretation. But if we multiply $$q$$ (value of charge irrespective of sign) with $$\rho$$ and define a new quantity $$\rho'=q\rho$$, then the positive and negative value of $$\rho'$$ can be interpretated as the positive charge density and negative charge density. The sign of the charge density comes from the sign of the energy. If energy is positive, then $$\rho'$$ is positive. If energy is negative then $$\rho'$$ is negative. For spinless particles, positive charge is associated with "particle" and negative charge is associated with "antiparticle". e.g. Pions.

On the other hand, Feynman-Stueckelberg interpretation says that a negative energy particle running backwards in time is mathematically equivalent to a positive energy antiparticle running forwards in time. Can we use this interpretation in the above case without any difficulties? I didn't understand the part where they are saying "negative energy particle" in the FSI. Because for spinless particles, the sign of the energy determines the sign of the charge density $$\rho'$$, and the sign of the charge density determines whether the entity is either particle or antiparticle. Then how can we say negative energy particle or positive energy particle? Should not the positive energy entity be positively charged, hence identified as particle? and negative energy entity be negatively charged, hence identified as antiparticle?

Perhaps a good starting point: Its a mathematical model for a stable, massive, charged, spinless elementary, point particle.

All assumptions have been proved to be wrong. Nevertheless, the electron with mass, charge and spin 1/2 obeys the Dirac equation, that has the same problem, because

Diracs operator squared results in the Klein-Gordon operator for each of its four components

The problem is inhaerent in wave equations of the second order everywhere. For

$$-\partial_{t,t} \psi +\partial_{x,x} \psi = m^2 \psi$$

to have solutions in one dimension

$$\psi= f( i (\pm \omega t \pm k x) ) \ : \ \omega^2 -k^2 = m^2$$

To our surprise, there are no continuity demands. Any function goes, and indeed, very steep waves exist (tsunami, bore).

It follows that, mathematically without alternative, the space of functions used in Fourier synthesis in space at any time, eg in order to describe a hard wave front, must be a dense subspace in the space of all complex waves in x, so that all physical possible solutions can be represented by approximations.

The elementary waves in $$x \to e^{i (\pm \sqrt{m^2-k2} t \pm k x)}$$

can be categorized accoding to the four signs. For $$m=0$$ this has been accomplished for masseless waves, light, and the result has been called, advanced and retarded solutions.

For massive charged particles, the situation is more complex.

The frequency $$\omega$$ can always be taken to be positive, with a factor $$\pm t$$. The spatial wave vector has to be smaller than $$\omega$$ in order to yield a positive mass term. It follows that the wave vector is confined either to the future or to the past inner of the light cone, made up by $$k$$-vectors of light of the retarded and advanced potential of the em-field.

There are particles and antiparticles in nature. They all move time forward in nature. They can emerge from $$t,x=0$$ and move time forward. Or they come in from past to interact at $$t,x,=0$$. Incoming spherical waves are not known to exist. There is wave expansion, but not coherent wave concentration onto a point.

So its sure, that the time inversion symmetry of the wave equations is broken by the condition of start form preparing. Waves simply cannot prepared in the past or future at in certain forms.

Particles and antiparticles can annihilate producing two photons of positive frqency on the forward light cone, by energy-momentum conservation in the center of gravity system, two massive vectors from the past, inside the backward light cone, meet at the cones vertex with equal $$\omega$$ and opposite $$k$$, and are converted into two light wave vectors on the forward light cone.

So in the many particle theory the antiparticle comes from future, is anihilated, particle is created and two photons spend the missing energy momentum, one anuhilated from future and the same created to future. Photons are their own antparticles, so there are no problems witch charge or mass.

Now all physics is in the future cone, nothing has to be assumed to exist in the past. And, trivially, the total energy momentum sum is zero.

Thats the great point: From Feynman until the Nobel price for the Higgs model as a mass creation mechanism, today all spirals around one question: How to get a common energy zero point for all kind of particles.