Free Fermion Field vs Free Boson Field

Free Boson Field

• Mathematical Characterization: The free boson field is characterized by a complex Hilbert space $K$, a Weyl system $W$ with values in $K$, a continuous representation $\pi$ from the unitary group $U(H)$ into $U(K)$, and a cyclic unit vector $v$ in $K$​​.
• Weyl Relations: The Weyl system $W(z)$ must satisfy the Weyl relations, which are crucial for defining the structure of the boson field​​.
• Positive Operator: A key condition is the positivity of the operator $T$, which ensures that the free boson field is well-defined and corresponds to physical reality​​.
• Wave and Particle Duality: The free boson field has representations that emphasize either particle properties (Fock-Cook representation) or wave properties (functional integration representation)​​.
• Historical Context: The free boson field was introduced by Dirac in 1926 and later rigorously constructed by Cook in 1953​​.

Free Fermion Field

• Clifford Systems: The free fermion field is based on Clifford systems, which are algebraic structures used to define fermion fields​​.
• Existence: The existence of the free fermion field relies on certain algebraic conditions and the representation of the Clifford algebra​​.
• Real and Complex Wave Representations: The free fermion field can be represented in both real and complex wave representations, which provide different perspectives on the field’s properties​​.
• Quantization: The quantization of fermion fields involves different mathematical techniques compared to boson fields, reflecting the underlying anticommutation relations (as opposed to commutation relations for bosons)​​.

In summary, the free boson field is characterized by commutation relations and a focus on wave-particle duality, while the free fermion field is defined through Clifford systems and anticommutation relations. Both fields have distinct mathematical frameworks and representations that reflect their unique properties in quantum field theory.