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arXiv math-ph/0312065math-phmath.MP
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用代数变形连接的经典和量子费米子

Classical and Quantum Fermions Linked by an Algebraic Deformation

Ali Mostafazadeh

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7 pages
发表信息：
Int. J. Geom. Meth. Mod. Phys. 2, 777-782 (2005)
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摘要：

研究了单费米子代数${\cal A}_ζ$的正则表示$ρ_ζ$，即，$c^2=c^{+2}=0$、$cc^++c^+c=ζ~1$，用于$ζ\in [0,1]$。我们证明了$ρ_0$是${\cal A}_0$的四维非酉表示，它是忠实不可约的（它不允许一个适当的忠实子表示）。此外，$ρ_0$是${\cal A}_0$的最小忠实不可约表示，因为${\cal A}_0$的每个忠实表示都有一个等价于$ρ_0$的子表示。因此，我们确定一个经典的费米子与$ρ_0$和查看其量子化的变形：$ζ:0\to 1$的$ρ_ζ$。后者具有将$ρ_0$映射到${\cal A}_1$的四维酉（忠实）可约表示$ρ_1$的效果，$ρ_1$正是与狄拉克费米子相关的表示。

英文摘要：

We study the regular representation $ρ_ζ$ of the single-fermion algebra ${\cal A}_ζ$, i.e., $c^2=c^{+2}=0$, $cc^++c^+c=ζ~1$, for $ζ\in [0,1]$. We show that $ρ_0$ is a four-dimensional nonunitary representation of ${\cal A}_0$ which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, $ρ_0$ is the minimal faithfully irreducible representation of ${\cal A}_0$ in the sense that every faithful representation of ${\cal A}_0$ has a subrepresentation that is equivalent to $ρ_0$. We therefore identify a classical fermion with $ρ_0$ and view its quantization as the deformation: $ζ:0\to 1$ of $ρ_ζ$. The latter has the effect of mapping $ρ_0$ into the four-dimensional, unitary, (faithfully) reducible representation $ρ_1$ of ${\cal A}_1$ that is precisely the representation associated with a Dirac fermion.

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