保形对称与手性费米子的半经典实现：螺旋度1/2粒子研究

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本文探讨了保形对称性、手性费米子以及半经典近似在物理学中的重要进展。研究者们在《物理 letters B》杂志2020年第800期的文章中，发现了保形发生器的显式形式，这是对庞加莱对称性的拓展，特别适用于无质量且具有任意螺旋度（或称为自旋）的粒子，其中螺旋度1/2的粒子被作为一个具体的例子进行讨论。这一发现对于理解高能物理中的手性现象具有重要意义。手性费米子是量子场论中的基本对象，它们在自然界中占据着关键角色，因为自然界的基本粒子如电子和夸克就是手性费米子，其自旋与运动方向的相对关系定义了它们的“手性”。在标准模型中，手性对称性是基本的对称性之一，但实际观测到的粒子似乎仅保留了一部分这种对称性，这就是著名的马约拉纳不相等原理。作者们通过求解Weyl方程的半经典状态，探讨了保形对称性在这一物理情境下的实现。Weyl方程是一种描述无质量粒子行为的方程，特别适用于描述低能量的光子和某些高能物理中的基本粒子。半经典近似是一种将量子力学与经典力学相结合的方法，它在处理强相互作用或者在极高温、极高密度条件下粒子行为时非常有效。在文章中，他们揭示了如何通过保形生成元的具体形式，使得在半经典状态下，手性费米子的行为可以展现出保形对称性，这可能有助于解释某些非平凡的物理现象，比如在凝聚态物理中可能存在的拓扑相变，或者是高能粒子碰撞过程中的新效应。此外，保形对称性的研究也为量子场论和弦理论中的对称性破缺问题提供了新的视角。这篇研究不仅深化了我们对手性费米子动力学的理解，还推动了量子场论的边界探索，特别是结合了保形对称性和半经典近似后的物理模型构建。它为理论物理学家提供了一个强大的工具，以更深入地探索宇宙的基本结构和自然法则。由于该论文是开放访问的，并遵循Creative Commons Attribution (CC BY) 许可证，读者可以无障碍地获取和研究这些前沿研究成果。

Physics Letters B 800 (2020) 135111

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Conformal symmetry, chiral fermions and semiclassical approximation

Joanna Gonera, Piotr Kosi

nski, Paweł Ma

slanka

∗

Department of Computer Science, Faculty of Physics and Applied Informatics, University of Lodz, Poland

a r t i c l e i n f o a b s t r a c t

Article history:

Received

30 October 2019

Accepted

20 November 2019

Available

online 25 November 2019

Editor:

M. Cveti

The explicit form of conformal generators is found which provides the extension of Poincare symmetry

for massless particles of arbitrary helicity. The helicity

particles are considered as the particular

example. The realization of conformal symmetry in the semiclassical regime of Weyl equation is obtained.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

In recent years the dynamics of chiral fermions became a ﬁeld

of intense research [1–20]. This topic is interesting due to its ap-

plications,

for example in exploring anomaly-related phenomena in

kinetic theory, but also because its complete theoretical description

requires to combine the diverse ideas concerning topology, sym-

metries,

anomalies, Berry’s phases etc.

In the most important case of helicity ±

massless fermions

the starting point is the Weyl equation in electromagnetic ﬁeld;

the latter is typically viewed as classical external one, at least in

the context we are concerned here. For many applications the elec-

tromagnetic

ﬁeld is assumed to be weak and slowly varying which

allows for the use of semiclassical approximation. The semiclassical

dynamics can be derived either from path-integral representation

of transition amplitudes [11]or using time-dependent variational

principle [20]popular among condensed matter physicists [21–23].

The result is summarized in semiclassical action functional involv-

ing

canonical variables describing chiral fermion (see eqs. (29), (30)

below).

One of the important problems inherent in semiclassical ap-

proach

is that of Poincare covariance. Weyl equation enjoys stan-

dard

Poincare covariance. However, the semiclassical action func-

tional

lacks manifest Lorentz covariance (which becomes Lorentz

symmetry in the absence of external ﬁelds), in spite of the fact

that it is derived from explicitly covariant Weyl equation. The way

to cure this situation was suggested in Ref. [11] where a modi-

ﬁed

transformation rule under Lorentz boosts has been proposed.

It involves O () corrections which are helicity-dependent and pro-

vide

on-shell implementation of Poincare covariance on semiclas-

Corresponding author.

E-mail

addresses: joanna.gonera@uni.lodz.pl (J. Gonera),

piotr.kosinski@uni.lodz.pl (P. Kosi

nski), pawel.maslanka@uni.lodz.pl (P. Ma

slanka).

sical level. The resulting situation attracted much attention [9–11],

[16–18], [19], [20]. Duval, Horvathy et al. [16–18]derived the mod-

iﬁed

transformation rules from Souriau model [24]of spinning par-

ticle

using the procedure called spin enslaving. Similar results were

obtained by using coadjoint orbit method [19]. On the other hand,

the time-dependent variational principle provides particularly clear

interpretation of canonical variables entering semiclassical action

which allows for derivation of the relevant transformation rules

from ﬁrst principles [20].

It is well known (see, for example, Ref. [25]) that the mass-

less

representations of Poincare group can be extended to those of

the conformal one. On the helicity ±

level this is equivalent to

the conformal covariance of Weyl equation (which persists in the

presence of external electromagnetic ﬁeld) [26].

The natural question arises if the conformal symmetry survives

the semiclassical approximation and what is the actual form of rel-

evant

transformations.

The present paper is addressed to this problem. In Sec. 2 we

present a simple method of constructing classical and quantum

conformally invariant Hamiltonian dynamics of massless particles.

The particular example of helicity −

fermion is discussed in

Sec. 3. The results obtained are then used in Sec. 4 to ﬁnd semi-

classical

form of conformal symmetry. Some conclusions are pre-

sented

in ﬁnal Sec. 5.

2. Conformally invariant dynamics

It is well known that the Hamiltonian systems admitting transi-

tive

action of some symmetry group can be classiﬁed using coad-

joint

orbit method [24], [29]. The resulting Hamiltonian dynam-

ics

may be then quantized leading in many cases to consistent

quantum theory. The procedure outlined is quite effective. Given a

symmetry group G it yields, in principle, all “elementary” systems,

classical and quantum, enjoying the symmetry described by G.

https://doi.org/10.1016/j.physletb.2019.135111

0370-2693/

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