arXiv 2203.030632022-03hep-th

海森堡代数与弦论

Heisenberg Algebra and String Theory

Norbert Dragon, Florian Oppermann

摘要：

如果庞加莱生成元的代数被时空位置算符$X=(X_0,\dots, X_{D-1})$扩大，那么动量$X=(X_0,\dots, X_{D-1})$和质量$P^2$的谱是无界的和连续的。特别地，协变弦的约束$(P^2 - m^2)Ψ_{\text{phys}}=0$在允许$X$的空间中没有解：所有物理状态消失，$Ψ_{\text{phys}}=0$。反之亦然，由质量本征态所跨越的空间不允许在$D$维中的位置算符$X$。无质量粒子不允许空间位置算子$\vec X$。海森堡域对$X^i$和$P^j$，$i,j\in \{1,\dots D-2\}$，$D > 2$，它们与$P^+=(P^0 + P_z)/\sqrt{2}$，$[P^+,X^i] = 0$交换，不允许有无质量或快子态的空间，这是通过旋转映射到自身的，更不用说洛伦兹变换了。这在所有维度上都是正确的，使得玻色弦的临界维度$D=26$的代数计算变得毫无意义：光锥弦不是洛伦兹不变的。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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