### The reason for antiparticles

Why do we need antiparticles in a relativistic quantum theory? I will review the usual arguments based on spacetime symmetries and CPT invariance and show that antiparticles are not particles that travel backwards in time as Feynman claimed.

In a recent twitter thread , Martin Bauer states that antiparticles are needed in a relativistic quantum theory because if we swap space and time in a quantum scattering process, the particles would travel backward in time and this is a puzzle, forcing us to reinterpret those exotic particles that travel backward in time as antiparticles that travel forward in time.

His argument is not valid. First, because applied to classical scattering events it would lead to the conclusion that antiparticles are also necessary in the classical theory, which is not the case.

Secondly, because relativity does not establish that space and time are equivalent and can be freely interchanged. It is a common misunderstanding of relativity that space and time are equivalent, but they are not even on the geometric level. Consider the spacetime line element

$\mathrm{d}s^2 = -\mathrm{d}t^2 + \mathrm{d}x^2 .$

It is not invariant when swapping $$x$$ and $$t$$. Spacetime is not a four-dimensional space as Minkoswki claimed. Hawking wanted to transform spacetime in a true four-space by eliminating the special role of time. He did it for 'aesthetic' reasons because he believed geometry is a fundamental property of the physical world. He introduced an imaginary time $$T=it$$ to remove the minus sign on the line element to give it more 'symmetry'

$\mathrm{d}s^2 = \mathrm{d} T^2 + \mathrm{d}x^2$

Hawking even claimed that $$T$$ would be the real concept of time and $$t$$ would be only a fictitious time, but this is all wrong. He did not even eliminate the distinction between space and time. Of course, the line element now looks like the element of a proper four-dimensional space, but the distinction between space and time remains since three of the coordinates are real, while the fourth, $$T$$, is now imaginary.

Now, let us review the argument based on $$\mathcal{CPT}$$ invariance. Quantum field theory satisfies $$\mathcal{CPT}$$ symmetry, which implies that the theory is invariant to the combined symmetries of Charge, Parity, and Time. Consider quantum electrodynamics as an illustration. Electromagnetic interactions preserve parity $$\mathcal{P}$$, so the symmetry reduces to $$\mathcal{CT}$$. The equations are invariant if we reverse the signs of charge and time together. This is the argument used by Feynman in his treatise on quantum electrodynamics , in which he took the classical equation of motion for a charged particle

$m \frac{\mathrm{d}^2x^\mu}{\mathrm{d}\tau^2} = q F_\nu^\mu \frac{\mathrm{d}x^\nu}{\mathrm{d}\tau}$

and argued that its invariance by replacing $$\tau \to -\tau$$ and $$q \to -q$$ implies that "a particle moving backward in time looks precisely like an antiparticle with the opposite charge moving forward in time". Nothing is further from reality! First, Feynman is using a classical equation of motion, but there is no antiparticles in classical electrodynamics. Second, $$\tau$$ is proper time, not the time we measure at laboratory and use in the Schrödinger equation. Third, the replacement $$\tau \to -\tau$$ does not imply moving backward in (proper) time, because when we change the sign we also change the order of the temporal events. The generic process $$A \Rightarrow B$$ becomes $$B \Rightarrow A$$ when time is reversed, but both processes take place forward in time.

## THE REASON

The existence of antiparticles is due to the existence of negative-energy solutions. The relativistic energy is given by a square root

$E = \sqrt{m^2c^4 + \boldsymbol{p}^2c^2}$

and square roots have two solutions. We can ignore the negative-energy solutions in classical relativity, because they are separated by a $$2mc^2$$ gap from the positive-energy solutions and no classical process (which are continuous processes) can transform a positive-energy particle into a negative-energy particle (or vice versa). Therefore, we always take the positive-energy solutions and ignore the rest. The situation is radically different in a quantum context, because energy is quantized and quantum processes can provide the needed energy to link the positive- and negative-energy solutions. A possible process is drawn in the following figure

This process implies that matter would be unstable, since any positive-energy electron would fall into some of the negative-energy levels. Dirac was the first to address this problem by assuming that all the negative-energy levels are occupied. This is the concept of "Dirac sea". Sometimes a negative-energy electron could absorb enough energy and reach a positive-energy level, leaving a hole in the Dirac sea. Dirac then considered those holes to be antiparticles. This picture is inconsistent (it does not really explain the stability of matter) and, to quote Julian Schwinger, "the picture of an infinite sea of negative energy electrons is now best regarded as a historical curiosity, and forgotten".

Quantum field theory treats antiparticles the same way it treats particles. So, electrons and positrons only differ in that one has a negative charge and the other a positive one. Although this is omitted from most textbooks, the quantum field theory approach to fermions is really based on the hole interpretation of antiparticles. The theory initially assigns creation and destruction operators to both positive- and negative-energy particles, then interprets the destruction of a negative-energy particle as the creation of a positive-energy antiparticle in the Dirac sense. The remnant of this approach is the appearance of the so-called "energy of the vacuum", which is just the energy of the Dirac sea.

I have presented the three mainstream ways to think about antiparticles: holes in a Dirac sea, particles traveling backwards in time, and the quantum field theory approach. All three have problems, so I have developed a fourth way of thinking about antiparticles. My own approach shows that antiparticles are related to negative energy levels and not to spacetime symmetries or $$\mathcal{CPT}$$ invariance. I start with a rewrite of the Dirac Hamiltonian, then project this Hamiltonian into the Hilbert subspace of positive-energy solutions and, as a byproduct of the projection, new charges appear in the interaction part of the Hamiltonian, which correspond to antiparticles. This new approach does not use the unfounded Dirac sea, eliminates the infinite "vacuum energy" of quantum field theory, and avoids the nonsense of particles traveling backwards in time.