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Boson-Fermion Algebraic Mapping in Second Quantization
Fabio Lingua1, Diego Molina Peñafiel2, Lucrezia Ravera3,4,5
1Department of Applied Physics, KTH Royal Institute of Technology-KTH, SE-10691 Stockholm, Sweden.
Entropy (Basel, Switzerland)
|January 8, 2025
Summary
We developed an algebraic method to map bosonic and fermionic operators, creating a deformed Grassmann-type algebra. This approach clarifies gauge invariance in second quantization and applies to harmonic oscillators.
Area of Science:
- Quantum mechanics
- Algebraic physics
- Theoretical physics
Background:
- Bosonic and fermionic algebras are fundamental in quantum mechanics.
- Mapping between these algebras is crucial for theoretical advancements.
- Understanding their relationship aids in developing new quantum theories.
Purpose of the Study:
- To present an algebraic method for mapping bosonic and fermionic algebras.
- To introduce a deformed Grassmann-type algebra using anticommuting variables.
- To discuss the implementation of gauge invariance in second quantization.
Main Methods:
- An algebraic method is employed to derive the structure of the mapping.
- A suitable identification between bosonic and fermionic generators is introduced.
- The resulting deformed Grassmann-type algebra is analyzed.
Main Results:
- A novel algebraic structure, a deformed Grassmann-type algebra, is derived.
- Anticommuting Grassmann-type variables are utilized within this new algebra.
- The mapping procedure is successfully applied to bosonic and fermionic harmonic oscillators.
Conclusions:
- The developed algebraic method provides a new framework for understanding bosonic-fermionic mappings.
- The introduced deformed Grassmann algebra offers insights into gauge invariance in second quantization.
- This work has potential applications in various areas of quantum physics, including quantum harmonic oscillators.
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