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Boson-Fermion Algebraic Mapping in Second Quantization.

Fabio Lingua1, Diego Molina Peñafiel2, Lucrezia Ravera3,4,5

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Summary
This summary is machine-generated.

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.

Keywords:
Grassmann variablesbosons and fermionsgauge invariancesecond quantization

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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.