# Naive Uncertainty principle for string theory

Is it possible, in some sense, that a naive uncertainly principle for string theory could be expressed as :

$$\Delta x_i \Delta p_j \Delta \sigma ~=~ \delta_{ij} \hbar \ell_s$$

where $\ell_s$ is the string length scale, and $\sigma$ a proper distance?

-
Why did you combine exactly these three things into a product? – Luboš Motl Oct 31 '12 at 19:30
Naively, I think that, in String Theory, there is one (spatial) more dimension. So there should be one more term in uncertainty principle. But it could be a very naive and false idea. But I like it. – Trimok Oct 31 '12 at 19:35
Dear Trimok, first, the three objects in the product are called factors, not terms. Terms are things that are added, not multiplied. Second, string replacing point-like particles adds one more dimension to the dimensions along the elementary objects from which matter is composed. But $x,p$ are dimensions in something completely different, namely the phase space, and they have deep reasons, the nonzero commutators, why they enter the uncertainty principle. So you're mixing phase spaces with world volumes, apples with oranges. – Luboš Motl Oct 31 '12 at 19:43
Various new types of a stringy uncertainty principle have been proposed over the years. One of them is perturbative stringly $\Delta x \Delta t\geq l_{string}^2$. I think it's due to Yoneya and holds in many respect, even in the presence of D-branes which were unknown when he wrote it for the first time. A more general, strongly coupled, inequality may be "derived" from holography, namely that $D-2$-dimensional areas are never smaller than $G_N$, Newton's constant - because the horizon entropy can't be smaller than one (but positive). There are various ways to use this inequality. – Luboš Motl Oct 31 '12 at 19:46
@LubošMotl: OK, but you did not speak about a uncertainty relation including momenta. So a factor (thanks...) is missing, and there is no $\hbar$, and this is surprising because string theory is a quantum theory. – Trimok Oct 31 '12 at 19:55