Nonlinear Bogolyubov–Valatin transformations and quaternions

@article{Holten2005NonlinearBT,
  title={Nonlinear Bogolyubov–Valatin transformations and quaternions},
  author={Jan Willem van Holten and Klaus Scharnhorst},
  journal={Journal of Physics A},
  year={2005},
  volume={38},
  pages={10245-10252}
}
In introducing second quantization for fermions, Jordan and Wigner (1927, 1928) observed that the algebra of a single pair of fermion creation and annihilation operators in quantum mechanics is closely related to the algebra of quaternions H. For the first time, here we exploit this fact to study nonlinear Bogolyubov–Valatin transformations (canonical transformations for fermions) for a single fermionic mode. By means of these transformations, a class of fermionic Hamiltonians in an external… 
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References

SHOWING 1-10 OF 97 REFERENCES
Nonlinear Transformation Group of CAR Fermion Algebra
AbstractBased on our previous work on the recursive fermion system in the Cuntz algebra, it is shown that a nonlinear transformation group of the CAR fermion algebra is induced from a U(2p) action on
Diagonalisation of the quadratic fermion Hamiltonian with a linear part
It is well known that the spectrum of a homogeneous quadratic Hamiltonian in m fermion construction-operator pairs is characterised by m mode energies. In this paper it is proved that the spectrum of
Quantum transformation theory in fermion Fock space
In this article a general linear quantum transformation U for the following fermion system with n modes, (b’+,b’)=U(b+,b)U−1=(b+,b)(B, A, CD), is studied. All above transformations in Fock space are
A geometric-algebra treatment of the Feynman-Vernon-Hellwarth space in the two-state problem
Pauli algebra is reviewed and then related to Dirac’s braket algebra and a generalization of the Wigner–Jordan operators in the two‐state problem. A method similar to the Feynman–Vernon–Hellwarth
Introduction to quantum statistical mechanics
Problem Non Ideal Bose Gas, Superfluidity and Fundamental Aspects of Quasiaverages Application Quasiaverages in the Theory of Superconductivity Correlations Weakening and Theorems on Singularities of
Local canonical transformations of fermions.
TLDR
The systematic method of generating all ocal canonical transformations enables us to discover a ``nonlinear'' local U(1) gauge symmetry of the Heisenberg-Hubbard model that remains a local symmetry away from half filling.
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