[论文解读] The Principle of the Fermionic Projector II, Derivation of the Effective Gauge Group

Felix Finster|arXiv (Cornell University)|Feb 11, 2002

Quantum and Classical Electrodynamics参考文献 5被引用 2

一句话总结

本文从七种重费米子与一种无质量左手费米子场的模型出发，基于费米子投影原理，推导出有效规范群 SU(2)⊗SU(3)⊗U(1)。在一般假设及手征性/赝标量相互作用下，各场区自发形成块结构，产生一种有效理论，其手征性规范场与标准模型中的电弱力和强相互作用相匹配，包括 SU(2) 的自发对称性破缺，以及对右手中微子组分的零势耦合。

ABSTRACT

We study the principle of the fermionic projector for the two-point action corresponding to the Lagrangian L[A] = |A2|2 − μ |A|4 , μ ∈ R and a fermionic projector which in the vacuum is the direct sum of seven identical massive sectors and one massless left-handed sector, each of which is composed of three Dirac seas. It is shown under general assumptions and for an interaction via general chiral and (pseudo)scalar potentials that the sectors spontaneously form pairs, which are referred to as blocks. The resulting so-called effective interaction can be described by chiral potentials corresponding to the effective gauge group SU(2)⊗ SU(3)⊗ U(1) . This model has striking similarity to the standard model if the block containing the left-handed sector is identified with the leptons and the three other blocks with the quarks. Namely, the effective gauge fields have the following properties. • The SU(3) corresponds to an unbroken gauge symmetry. The SU(3) gauge fields couple to the quarks exactly as the strong gauge fields in the standard model. • The SU(2) potentials are left-handed and couple to the leptons and quarks exactly as the weak gauge potentials in the standard model. Similar to the CKM mixing in the standard model, the off-diagonal components of these potentials must involve a non-trivial mixing of the generations. The SU(2) gauge symmetry is spontaneously broken. • The U(1) of electrodynamics can be identified with an Abelian subgroup of the effective gauge group. The effective gauge group is larger than the gauge group of the standard model, but this is not inconsistent because a more detailed analysis of our variational principle should give further constraints for the Abelian gauge potentials. Moreover, there are the following differences to the standard model, which we derive mathematically without working out their physical implications in detail. • The SU(2) gauge field tensor F must be simple in the sense that F = Λ s for a real 2-form Λ and an su(2)-valued function s. • In the lepton block, the off-diagonal SU(2) gauge potentials are associated with a new type of potential, called nil potential, which couples to the right-handed component. These results give a strong indication that the principle of the fermionic projector is of physical significance.

研究动机与目标

• 从多场区费米子系统中的费米子投影原理出发，推导有效规范群。
• 证明手征性和赝标量相互作用可导致费米子场区自发配对形成块结构。
• 通过变分原理重现标准模型规范群 SU(2)⊗SU(3)⊗U(1) 的结构，而无需预先假设该群。
• 在费米子投影框架中，阐明弱同位spin、色荷与电磁力的起源。

提出的方法

• 使用拉格朗日量 L[A] = |A²|² − μ|A|⁴（实参数 μ）构造两点作用量。
• 将费米子投影算符定义为七个重狄拉克海与一个无质量左手费米子场区的直和。
• 施加一般的手征性和赝标量势，以诱导场区之间的相互作用。
• 应用费米子投影原理，通过变分极小化推导有效动力学。
• 识别在相互作用下自发形成的场区块结构。
• 证明有效规范场满足 SU(2)⊗SU(3)⊗U(1) 对称性，且具有特定耦合结构，包括对右手中微子组分的零势耦合。

实验结果

研究问题

• RQ1 是否可在不预先假设的前提下，从费米子投影原理中自然涌现出有效规范群 SU(2)⊗SU(3)⊗U(1)？
• RQ2 手征性和赝标量势如何导致费米子场区自发配对形成块结构？
• RQ3 在轻子块中，零势在耦合右手中微子组分时起到何种作用？
• RQ4 为何在此框架中，SU(2) 规范场张量必须为简单形式（即 F = Λs）？
• RQ5 有效 U(1) 对称性如何与电磁力相关联？为何有效规范群大于标准模型的规范群？

主要发现

• 在手征性和赝标量相互作用下，费米子场区自发形成块结构，导致有效规范结构为 SU(2)⊗SU(3)⊗U(1)。
• SU(3) 规范场与夸克的耦合方式与强相互作用完全一致，且规范对称性未被打破。
• SU(2) 规范场为左手性，其耦合方式与弱相互作用一致，非对角分量需引入世代混合。
• SU(2) 规范场张量必须为简单形式，即 F = Λs，其中 Λ 为实 2-形式，s 为取值于 su(2) 的函数。
• 在轻子块中，非对角 SU(2) 势通过一种新型相互作用——零势，耦合至右手中微子组分。
• 有效 U(1) 对称性对应电磁力，是更大有效规范群中的阿贝尔子群，与标准模型电弱规范结构一致。