论文标题
带有光滑锥弦的粗糙
The Rough with the Smooth of the Light Cone String
论文作者
论文摘要
庞加莱组统一表示的发电机中的多项式构成了一个代数，该代数将光滑的密集子空间映射到平滑的子空间中，迅速减少了波形。这一数学结果非常欢迎物理学家，他以前只是假设其对无限算子的代数处理是合理的。然而，平滑度具有副作用，即必须证明，粗糙的操作员r映射了S自身的密集子空间，以允许其他r和所有发电机都映射到自身的其他密集域。否则，他们的代数产品（它们的串联）将无法定义。光锥字符串的规范量化假设运算符$ -i x^1 $和$ p^ - =（p^0-p^z）/2 $，作为其换向器的乘法运算符r = p^1/（p^0 + p^z）。这不是光滑的，但在无质量动量的负z轴上很粗糙。仅使用p^m与发电机的换向关系$ -i m_iz $在$ p^i $ - $ - $ - $ p^z $ - 平面中我们表明，在无数状态下，操作员$ r $与SO（D-1）的单一表示不一致。这使得对临界尺寸的代数确定，$ d = 26 $，毫无意义：如果光锥字符串的无质量状态允许r，那么他们不承认庞加莱集团的亚组SO（d-1）的单一表示。通过类似的论点，我们表明：无质量的多重组与一个空间动量的翻译组不一致，该平均动量由自我接合空间位置操作员$ x $产生。
The polynomials in the generators of a unitary representation of the Poincaré group constitute an algebra which maps the dense subspace S of smooth, rapidly decreasing wavefunctions to itself. This mathematical result is highly welcome to physicists, who previously just assumed their algebraic treatment of unbounded operators be justified. The smoothness, however, has the side effect that a rough operator R, which does not map a dense subspace of S to itself, has to be shown to allow for some other dense domain which is mapped to itself both by R and all generators. Otherwise their algebraic product, their concatenation, is not defined. Canonical quantization of the light cone string postulates operators $-i X^1$ and $P^- = (P^0 - P^z)/2$ and as their commutator the multiplicative operator R = P^1/(P^0 + P^z). This is not smooth but rough on the negative z-axis of massless momentum. Using only the commutation relations of P^m with the generators $-i M_iz$ of rotations in the $P^i$-$P^z$-plane we show that on massless states the operator $R$ is inconsistent with a unitary representation of SO(D-1). This makes the algebraic determination of the critical dimension, $D=26$, of the bosonic string meaningless: if the massless states of the light cone string admit R then they do not admit a unitary representation of the subgroup SO(D-1) of the Poincaré group. With analogous arguments we show: Massless multiplets are inconsistent with a translation group of the spatial momentum which is generated by a self-adjoint spatial position operator $X$.
