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首页> 外文期刊>Journal of Physics Communications >Teaching renormalization, scaling, and universality with an example from quantum mechanics
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Teaching renormalization, scaling, and universality with an example from quantum mechanics

机译：用量子力学中的一个例子教授重归一化，缩放和通用性

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Wediscuss the quantum mechanics of a particle restricted to the half-line x 0 with potential energy V = α/x~2 for -1/4 α 0. It is known that two scale-invariant theories may be defined. By regularizing the near-origin behavior of the potential by a finite square well with variable width b and depth g, it is shown how these two scale-invariant theories occupy fixed points in the resulting (b, g)-space of Hamiltonians. A renormalization group (RG) flow exists in this space and scaling variables are shown to exist in a neighborhood of the fixed points. Consequently, the propagator of the regulated theory enjoys homogeneous scaling laws close to the fixed points. UsingRGarguments it is possible to discern the functional form of the propagator for long distances and long imaginary times, thus demonstrating the extent to which fixed points control the behavior of the cut-off theory. By keeping the width fixed and varying only the well depth, we show how the mean position of a bound state diverges as g approaches a critical value. It is proven that the exponent characterizing the divergence is universal in the sense that its value is independent of the choice of regulator. Two classical interpretations of the results are discussed: standard Brownian motion on the real line, and the free energy of a certain one-dimensional chain of particles with prescribed boundary conditions. In the former example, V appears as part of an expectation value in the Feynman-Kac formula. In the latter example, V appears as the background potential for the chain, and the loss of extensivity is dictated by a universal power law.
机译：我们讨论了在-1/4 <α<0时，势能为V =α/ x〜2时，限于半线x> 0的粒子的量子力学。已知可以定义两个尺度不变理论。通过用宽度为b和深度为g的有限方井对势的近原行为进行正则化，表明了这两个尺度不变理论如何在哈密顿量的结果（b，g）空间中占据固定点。在该空间中存在一个归一化组（RG）流，并且缩放变量显示为存在于固定点附近。因此，规范理论的传播者享有接近固定点的均匀缩放定律。使用RG参数，可以辨别传播器的功能形式，适用于长距离和较长的假想时间，从而说明了固定点控制截止理论行为的程度。通过保持固定的宽度并仅改变阱深度，我们展示了当g接近临界值时束缚态的平均位置如何变化。从其独立于调节器选择的意义上说，证明了差异的指数是普遍的。讨论了两种经典的结果解释：实线上的标准布朗运动，以及具有规定边界条件的特定一维粒子链的自由能。在前一个示例中，V在Feynman-Kac公式中显示为期望值的一部分。在后一个示例中，V出现为链的背景电势，而扩展性的损失由通用幂定律规定。

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