Mathematical Construction of Hilbert Space of Infinitely Many Particles in Quantized Dirac Field

Metias Apul Abraham 

In 1928, motivated by the difficulties arose in Klein-Gordon equation, Dirac discovered a relativistic equation that provides the description of first order relativistic differential equation which allowed a quantum mechanical interpretation. He succeeded in developing the equation but there was difficulty with the presence of arbitrarily negative energy which also appeared in the Klein-Gordon equation. Then, He came up with the idea to resolve these problems by assuming that the particles were fermion and the negative energy states were filled which led to the resulting sea of particles known as Dirac Sea.
Although, a satisfactory mathematical framework of Dirac Sea has not been developed in detail but in my thesis I am taking this seriously based on the framework constructed by Dimock (2018) and Tumulka (2021). So, the goal is to give us a brief idea of the mathematical construction by taking n-fermion state that is modeled by an n-fold wedge Hilbert space then the Dirac Sea will be described as an infinite wedge product of Hilbert space. Finally, I will present you some mathematical statements which are part of my thesis project. Those statements are still conjecture and have not been fully proven.