If one considers only linear equations without auxiliary conditions or redundant components, which are, moreover, derivable from an invariant Lagrangian, then aside from unitary infinite-dimensional equations the only finite-dimensional equation is the Dirac equation. This powerful result puts the Dirac equation in a unique position, such that someone (such as J. Schwinger, Phys. Rev. 130, 800 (1963)) strongly suggests that only a Dirac particle can be considered as elementary. A Bose particle can be constructed out of spin-1/2 particles but not vice versa. In group-theoretical language, the fundamental representation of the Lorentz group is a spin-1/2 representation, out of which other representations are constructed.

在相对论量子力学领域，唯独Dirac粒子可以定义合适的位置算符（见A. O. Barut and S. Malin, Rev. Mod. Phys. 40, 632-651 (1968)）

晚安！

PS: We need a position operator to tell us whether a particle is in a large enough box, although a Dirac particle cannot be localized to within less than its Compton wavelength (an attempt at strong localization leads to creation of particles and antiparticles and this takes us beyond the limit of a single-particle theory).

（以上所谈的，对于学物理的人而言，不过是ABC，因此可以把这里的帖子看作是英文写出来的科普:-))

在相对论量子力学领域，唯独Dirac粒子可以定义合适的位置算符（见A. O. Barut and S. Malin, Rev. Mod. Phys. 40, 632-651 (1968)）


Sounds weird. How about a scalar, Weyl fermion?

美女比文章更抢眼，星空兄，色诱也有副作用的

Sounds weird. How about a scalar, Weyl fermion?
－－－－－－－－－－－－－－－－－
先简要解释一下，以后想另开主题专门谈（顺便带有和大家请教与讨论的含义）。上面可能我用“自旋为1/2的粒子”代替“Dirac粒子”更合适，这样你就不会问及Weyl fermion了。在相对论量子理论框架下，标量粒子和矢量粒子不再有非相对论量子力学中那种意义上的位置算符，这也能从另一个角度上看到：在相对论量子理论框架下，标量粒子和矢量粒子没有概率幅意义上的波函数。关于位置算符，非相对论量子力学中，有如上图形文件中所论述的

紧接上面：
人们对于位置算符的合理定义问题，提出以下“一致性检验法则”（The consistency Test）：The description of a free particle in quantum theory is done along the following steps:
(a) A basis for the space of wave functions, which describe all the possible states of a particle, is defined by a wave equation.
(b) A scalar product is defined in the space of the wave functions.
(c) Expressions for the probability density and probability current are found. They should form a 4-vector whose divergence vanishes. The expression for the probability density should be positive definite.
(d) Operators which correspond to measurements are defined, in particular, momemtum and position operators.
(e) The eigenfunctions of the operators, normalized to 1 (in the case of discrete spectrum) or to a δ-function (in the case of continuous spectrum), are found.
(f) The position operator, defined in (d), and the scalar product, defined in (b), uniquely determine an expression for the probability density. The theory is consistent only if this uniquely determined expression is identical with the one defined in (c) to satisfy a continuity equation. This is a consistency test.

以上从(a)到(f)的步骤，还可以用群论的语言重新表述如下：
(a)’ The states of a free particle are described by the vectors of a representation of the Galilei or the Poincaré group (which may be reducible), extended in general by discrete operators such as parity.
(b)’ The representation is unitary and hence a scalar product is defined. (For example, for the Klein-Gordon equation this invariant scalar product can be defined in momentum space （因为存在合理的动量算符，它对应空间平移生成元）, but there is no such simple form in the configuration space, because the density is indefinite).
(c)’ The representation is so chosen that a four vector operator exists which is conserved.
(d)’ Position and momentum operators are identified in the usual way within the Galilei or the Poincaré group.
(e)’ The eigenfunctions are normalized within the unitary representations.
(f)’ The position operators and the scalar product determine the probability density which must coincide with the matrix elements of the temporal component of the four vector operator

唯有Dirac粒子的位置算符才能通过以上一致性检验。对于其他粒子，在相对论量子理论中尽管无法寻求非相对论量子力学中的那种位置算符，人们退而求其次，寻求推广的意义上的位置算符。比如给出不同的内积空间，在该空间上可以谈论位置算符。位置算符问题之所以重要，因为它关联到其他许多问题的研究。例如，凝聚态物理中，谈到介质中的电偶极子矢量算符时（电荷乘以位置矢量算符），位置算符是无法绕过去的问题。