The paper by P. Jordan and E. Wigner presents an extension of their previous note on the quantum mechanics of gas degeneracy. It aims to describe an ideal or non-ideal gas subject to the Pauli exclusion principle using ordinary three-dimensional space rather than abstract coordinate space. This is achieved through a quantized three-dimensional wave field representation, where the non-commutative 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 has close analogies to the theory of Einstein's ideal or non-ideal gases as developed by Dirac, Klein, and Jordan. The authors discuss how the matrix theory of quantum mechanics suggests that difficulties in radiation theory might be overcome by applying quantum mechanical methods to both matter particles and the electromagnetic field. They introduce a decomposition of the number operator $ N_r $ into two factors $ b_r^\dagger b_r $, with specific forms for $ b_r $ and $ b_r^\dagger $, where $ N_r $ and $ \theta_r $ are canonically conjugate. This approach allows for the representation of both Einstein and Pauli statistics. In the Pauli case, the eigenvalues of $ N_r $ are restricted to 0 and 1, reflecting the exclusion principle. The derived formulas are closely related to problems of particle collisions and density fluctuations in quantum gases, supporting the idea that this representation accurately describes the Pauli exclusion principle and will lead to correct results.The paper by P. Jordan and E. Wigner presents an extension of their previous note on the quantum mechanics of gas degeneracy. It aims to describe an ideal or non-ideal gas subject to the Pauli exclusion principle using ordinary three-dimensional space rather than abstract coordinate space. This is achieved through a quantized three-dimensional wave field representation, where the non-commutative 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 has close analogies to the theory of Einstein's ideal or non-ideal gases as developed by Dirac, Klein, and Jordan. The authors discuss how the matrix theory of quantum mechanics suggests that difficulties in radiation theory might be overcome by applying quantum mechanical methods to both matter particles and the electromagnetic field. They introduce a decomposition of the number operator $ N_r $ into two factors $ b_r^\dagger b_r $, with specific forms for $ b_r $ and $ b_r^\dagger $, where $ N_r $ and $ \theta_r $ are canonically conjugate. This approach allows for the representation of both Einstein and Pauli statistics. In the Pauli case, the eigenvalues of $ N_r $ are restricted to 0 and 1, reflecting the exclusion principle. The derived formulas are closely related to problems of particle collisions and density fluctuations in quantum gases, supporting the idea that this representation accurately describes the Pauli exclusion principle and will lead to correct results.