非线性Bogolyubov-Valtin变换：2种模式
Nonlinear Bogolyubov-Valatin transformations: 2 modes
摘要：
扩展我们以前的研究非线性Bogolyubov-Valatin变换（费米子的正则变换）的一个费米子模式，在本文中，我们进行了深入的研究一般（非线性）正则变换的两个费米子模式。我们发现可以用酉SU（2^n = 4）变换实现的n=2费米子模的Bogolyubov-Valatin群同构于SO（6;R）/Z_2。这项调查涉及许多问题。作为一个新颖的从数学的角度来看，我们研究的结构的非线性基变换的Clifford代数[具体地说，在Clifford代数C（0，4）]，需要（超对称）变换之间的不同等级的多向量。一个突出的代数作用在这方面正在发挥biparavectors（线性组合的产品狄拉克矩阵，quadriquaternions，sedenions）和自旋bivectors（反对称复杂的矩阵）。所研究的双仿矢等价于爱丁顿的E-数，并且可以用四元数H的除代数的两个可交换副本的张量积来理解。从物理的角度，我们提出了一种对角化任意两费米子哈密顿量的方法。基于两自旋-1/2和单自旋-3/2系统的Jordan-Wigner变换，我们还研究了非线性自旋变换和对角化任意两自旋-1/2和单自旋-3/2哈密顿量的相关问题。最后，从计算的角度来看，我们适当注意SU（4）和SO（6;R）矩阵（分别大小为4x 4和6x6）的显式参数化及其相互关系。
英文摘要：
Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, in the present paper we perform a thorough study of general (nonlinear) canonical transformations for two fermionic modes. We find that the Bogolyubov-Valatin group for n=2 fermionic modes which can be implemented by means of unitary SU(2^n = 4) transformations is isomorphic to SO(6;R)/Z_2. The investigation touches on a number of subjects. As a novelty from a mathematical point of view, we study the structure of nonlinear basis transformations in a Clifford algebra [specifically, in the Clifford algebra C(0,4)] entailing (supersymmetric) transformations among multivectors of different grades. A prominent algebraic role in this context is being played by biparavectors (linear combinations of products of Dirac matrices, quadriquaternions, sedenions) and spin bivectors (antisymmetric complex matrices). The studied biparavectors are equivalent to Eddington's E-numbers and can be understood in terms of the tensor product of two commuting copies of the division algebra of quaternions H. From a physical point of view, we present a method to diagonalize any arbitrary two-fermion Hamiltonian. Relying on Jordan-Wigner transformations for two-spin-1/2 and single-spin-3/2 systems, we also study nonlinear spin transformations and the related problem of diagonalizing arbitrary two-spin-1/2 and single-spin-3/2 Hamiltonians. Finally, from a calculational point of view, we pay due attention to explicit parametrizations of SU(4) and SO(6;R) matrices (of respective sizes 4x4 and 6x6) and their mutual relation.
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