论文标题
海森伯格代数和弦理论
Heisenberg Algebra and String Theory
论文作者
论文摘要
如果庞加莱发电机的代数被时空位置运算符$ x =（x_0，\ dots，x_ {d-1}）$放大，则动量$ p $和质量$ p^2 $的光谱是无界和连续的。特别是，约束$（p^2 -m^2）ψ_ {\ text {phys}} = 0 $的协变量字符串在空间中没有解决方案，该空间允许$ x $：所有物理状态消失，$ψ_{\ text {phys}}} = 0 $。反之亦然，由大众特征态跨越的空间不承认$ d $ dimensions中的位置运营商$ x $。 无质量的粒子不允许空间位置操作员$ \ vec x $。 The domain of Heisenberg pairs $X^i$ and $P^j$, $i,j\in \{1,\dots D-2\}$, $D > 2$, which commute with $P^+=(P^0 + P_z)/\sqrt{2}$, $[P^+,X^i] = 0$, does not allow for a space with massless or tachyonic states, which is通过旋转映射到自己，不用洛伦兹的变换。这在所有维度上都是如此，并且使临界维度的代数计算（$ d = 26 $）的骨弦字符串无意义：光锥字符串不是lorentz不变的。
If the algebra of the Poincaré generators is enlarged by the spacetime position operator $X=(X_0,\dots, X_{D-1})$ then the spectra of the momentum $P$ and the mass $P^2$ are unbounded and continuous. In particular, the constraint $(P^2 - m^2)Ψ_{\text{phys}}=0$ of the covariant string has no solution in the space which admits $X$: All physical states vanish, $Ψ_{\text{phys}}=0$. Vice versa, a space spanned by mass eigenstates does not admit the position operator $X$ in $D$ dimensions. A massless particle does not allow a spatial position operator $\vec X$. The domain of Heisenberg pairs $X^i$ and $P^j$, $i,j\in \{1,\dots D-2\}$, $D > 2$, which commute with $P^+=(P^0 + P_z)/\sqrt{2}$, $[P^+,X^i] = 0$, does not allow for a space with massless or tachyonic states, which is mapped to itself by rotations, leave alone Lorentz transformations. This is true in all dimensions and makes the algebraic calculation of the critical dimension, $D=26$, of the bosonic string meaningless: the light cone string is not Lorentz invariant.
