# How can energy be negative for antiparticles in the solutions of Klein Gordon equation?

Although similar questions have been asked before I'm still confused.

This is from Greiner, Relativistic Quantum Mechanics

$$E^2=c^2\sqrt{\vec{p}^2+m_0^2c^2}$$

Consequently, there exist two possible solutions for a given momentum $$\vec{p}$$: one with positive, the other with negative energy. $$E_p=\pm\sqrt{\vec{p}^2+m_0^2c^2}, \psi_{(\pm)}=A_{(\pm)}exp[\frac{i}{\hbar}(\vec{p}\cdot\vec{x}\mp|E_p|t)]$$. $$A_{(\pm)}$$ are normalization constants.

Charge density $$\rho_{(\pm)}=\pm\frac{e|E_p|}{m_0c^2}\psi\ast_{(\pm)}\psi_{(\pm)}$$

This suggests the following interpretation: $$\psi_{(+)}$$ specifies particles with charge $$+e$$; $$\psi_{(-)}$$ specifies particles with charge $$-e$$.

I have just started studying Klein Gordon Equation so my question might seem silly. The wave has energy $$E_p$$ for both particles and antiparticles. Also, both have positive energy given by $$E=\sqrt{p^2c^2+m^2c^4}$$. Then what is $$E_p$$ and $$-E_p$$? Both wave and antiparticles don't have negative energy. Then what is the significance of negative energy i.e. $$-E_p$$?

• Tbh, I don't get the point. What do you mean Ep has energy for both particles and antiparticles? The term for Ep is either a particle or, if negative, an antiparticle. ? – Ben Aug 19 '19 at 21:58
• physics.stackexchange.com/a/242554/197399 This link says that particles and antiparticles have equal rest mass energy i.e. E (as given). My question is: Since $-E_p$ is not the energy of antiparticle as it is E and positive, what is its significance? – Asit Srivastava Aug 19 '19 at 22:23
• Asit, to be sure, the question you link to above doesn't involve the KG equation, i.e., the question there is about matter and antimatter. This section of the KG Wiki article seems relevant. – Alfred Centauri Aug 19 '19 at 23:01
• Yeah, I misread it. en.wikipedia.org/wiki/Annihilation#Examples This link says that electron and positron annihilate each other to give photons. Their total energy is equal to the summation of their rest mass energies neglecting their kinetic energies. My question is: Antiparticles have positive rest mass energy. Then $-E_p$ cannot be their energy. So why is it associated with them as given in the paragraph of Greiner? – Asit Srivastava Aug 20 '19 at 8:02
• Asit, at the KG Wiki article section I linked to above, there is this: "By integration of the time–time component $T_{00}$ over all space, one may show that both the positive- and negative-frequency plane-wave solutions can be physically associated with particles with positive energy. This is not the case for the Dirac equation and its energy–momentum tensor" - the citation for this is Greiner's "Relativistic Quantum Mechanics. Wave Equations" – Alfred Centauri Aug 20 '19 at 10:27