Symmetry with respect to positive and negative energies of the electromagnetic field

It is well known that the Dirac equation (DE) is symmetric with respect to positive and negative energy solutions. Symmetry means that positive and negative eigenvalues have a one–one correspondence and have contrary numbers to each other. The positive and negative eigenvalues correspond to positive kinetic energy (PKE) and negative kinetic energy (NKE) states, respectively. This symmetry is retained when the non-relativistic approximations of the DE are made, which are the Schrödinger equation (SE) and the NKE SE. The relativistic quantum equations for spin-0 particles also have this symmetry. In all three cases, it is seen that the non-relativistic motion of particles, the relativistic motion of particles with spin-0 and spin-1/2, and quantum mechanics equations are symmetric with respect to the PKE and NKE.

The energy spectrum of the PKE (NKE) branch has a lower (an upper) limit and has no upper (lower) limit, so the system has a positive (negative) temperature. A PKE system absorbs or emits photons with positive energies when it transits between energy levels, while an NKE system has to absorb or emit photons with negative energies.

The energy–momentum relation of photons is . This means that a photon can carry either a positive energy E=cp or a negative energy . The electromagnetic field with negative energy is called negative radiation (NR). We take the second quantization of the NR. The spectrum of the radiation (NR) has a lower (upper) limit and no upper (lower) limit, so it has a positive (negative) temperature, which is called the symmetry between the radiation and NR.

In our universe, the PKE (NKE) systems absorb and emit the radiation (NR). We say that PKE (NKE) systems match with radiation (NR) and call this pair of matches a symmetry of the universe.