费米子谐振子

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定义 · 1.5 费米粒子的路径积分量子化

这称为费米子谐振子(fermionic harmonic oscillator)，它可以看作描述相对论性费米子的狄拉克哈密顿量的一个傅里叶分量。如果算符 $c$ 与 $c^{†}$ 满足与玻色子相同的对易关系，那么哈密顿量将是一个常数 $H=-\omega /2$。反之，假设它们满足反对易关系

$$\{c,c^{†}\}\equiv c{c}^{†}+c^{†}c=1\{c,c\}=\left\{c^{†},c^{†}\right\}=0.\text{\hspace{1em}(1.167)}$$

英文原文

This is called the fermionic harmonic oscillator, which may be regarded as a Fourier component of the Dirac Hamiltonian, which describes relativistic fermions. If the operators $c$ and $c^{†}$ should satisfy the same commutation relations as those satisfied by bosons, the Hamiltonian would be a constant $H=-\omega /2$ . Suppose, in contrast, they satisfy the anti-commutation relations

$$\{c,c^{†}\}\equiv c{c}^{†}+c^{†}c=1\{c,c\}=\left\{c^{†},c^{†}\right\}=0.\text{\hspace{1em}(1.167)}$$

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