This section of the article, authored by P. Jordan and E. Wigner, builds upon their previous work on the quantum mechanics of gas mixing. The authors aim to describe an ideal or non-ideal gas under the Pauli Exclusion Principle using concepts that do not rely on the abstract coordinate space of the gas's atomic ensemble but instead use the conventional three-dimensional space. This is achieved by representing the gas through a quantized three-dimensional wave field, where the non-commutative multiplication properties of the wave amplitude are responsible for both the existence of corpuscular gas atoms and the validity of the Pauli Exclusion Principle. The theory shares close analogies with the corresponding theory for Einstein's ideal or non-ideal gases, as developed by Dirac, Klein, and Jordan. The authors discuss the application of quantum mechanical methods to both the electromagnetic field and the description of material particles in the ordinary three-dimensional space, avoiding the use of abstract coordinate spaces. They introduce a decomposition of the number operator \( N_r \) into two factors \( b_r \) and \( b_r^{\dagger} \), which are defined such that \( N_r \) and \( \theta_r \) are canonically conjugate. This setup allows for the representation of both Einstein's and Pauli's statistics, where the eigenvalues of \( N_r \) can be either continuous or discrete. The formulas derived support the belief that this representation of the Pauli Exclusion Principle aligns with the essence of the problem and is likely to yield accurate results. These formulas are closely related to the issues of collisional interactions of particles and density fluctuations in quantum gases.This section of the article, authored by P. Jordan and E. Wigner, builds upon their previous work on the quantum mechanics of gas mixing. The authors aim to describe an ideal or non-ideal gas under the Pauli Exclusion Principle using concepts that do not rely on the abstract coordinate space of the gas's atomic ensemble but instead use the conventional three-dimensional space. This is achieved by representing the gas through a quantized three-dimensional wave field, where the non-commutative multiplication properties of the wave amplitude are responsible for both the existence of corpuscular gas atoms and the validity of the Pauli Exclusion Principle. The theory shares close analogies with the corresponding theory for Einstein's ideal or non-ideal gases, as developed by Dirac, Klein, and Jordan. The authors discuss the application of quantum mechanical methods to both the electromagnetic field and the description of material particles in the ordinary three-dimensional space, avoiding the use of abstract coordinate spaces. They introduce a decomposition of the number operator \( N_r \) into two factors \( b_r \) and \( b_r^{\dagger} \), which are defined such that \( N_r \) and \( \theta_r \) are canonically conjugate. This setup allows for the representation of both Einstein's and Pauli's statistics, where the eigenvalues of \( N_r \) can be either continuous or discrete. The formulas derived support the belief that this representation of the Pauli Exclusion Principle aligns with the essence of the problem and is likely to yield accurate results. These formulas are closely related to the issues of collisional interactions of particles and density fluctuations in quantum gases.