谐振子与基本粒子

摘要：

具有相同对称性的两个动力系统应该具有共同的特征，并且就它们的共享对称性而言，一个可以代表另一个。强子的三种轻夸克成分，a）具有近似的味SU（3）对称性，b）具有精确的色SU（3）对称性，c）作为自旋1/2的粒子，具有洛伦兹SO（3，1）对称性。3D谐振子也是如此。a）它的哈密顿量具有SU（3）对称性，如果3个基本振荡模式不相同，则是可破的。b）3个振荡方向具有置换对称性。这使得我们能够为每种振荡模式创建三个不可破坏的SU（3）对称性副本，并模仿基本粒子的颜色。三维振子的拉格朗日量具有SO（3，1）对称性。这可以用来适应粒子的自旋。本文建立了三维振子Poisson括号算符的本征模与粒子味多重态、振子置换模与粒子色多重态和反色多重态的一一对应关系。胶子由振子的色SU（3）对称性的生成元表示。谐振子是常见的位置对象，无论在哪里遇到，都是解析可解的。另一方面，基本粒子则是一个遥不可及的抽象实体。了解其中一个可能有助于更好地欣赏另一个。

英文摘要：

Two dynamical systems with same symmetry should have features in common, and as far as their shared symmetry is concerned, one may represent the other. The three light quark constituents of the hadrons, a) have an approximate flavor SU(3) symmetry, b) have an exact color SU(3) symmetry, and c) as spin 1/2 particles, have a Lorentz SO(3,1) symmetry. So does a 3D harmonic oscillator. a) Its Hamiltonian has the SU(3) symmetry, breakable if the 3 fundamental modes of oscillation are not identical. b) The 3 directions of oscillation have the permutation symmetry. This enables one to create three copies of unbreakable SU(3) symmetry for each mode of the oscillation, and mimic the color of the elementary particles. And c) The Lagrangian of the 3D oscillator has the SO(3,1) symmetry. This can be employed to accommodate the spin of the particles. In this paper we draw up a one-to-one correspondence between the eigen modes of the Poisson bracket operator of the 3D oscillator and the flavor multiplets of the particles, and between the permuted modes of the oscillator and the color and anticolor multiplets of the particles. Gluons are represented by the generators of the color SU(3) symmetry of the oscillator. Harmonic oscillators are common place objects and, wherever encountered, are analytically solvable. Elementary particles, on the other hand, are abstract entities far from one's reach. Understanding of one may help a better appreciation of the other.

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