# Is there an equivalent of Heisenberg's uncertainty principle in the decision sciences ?

From memories of a quantum mechanics class and Wikipedia:

In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can be known simultaneously. For instance, the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.

In practical terms the product of the uncertainty in the position and the uncertainty in the momentum is at least of the order of magnitude of the Planck constant.

A reason why an analogous of this principle could be found in social sciences is that (in my understanding) its origins are purely mathematical. It is a Cauchy-Schwarz-like inequality applied to the scalar product Bra-ket under the dynamic of a particular pde. However, the nanoscopic nature of quantic measurements and the ~10^-34 Planck constant are so small that the uncertainty principle is irrelevant in physics of larger particles, not to mention the other sciences.

A common operation in decision sciences is < belief || outcome > (or < price vector || vector of goods > at a larger scale) and the solution concepts in game theory, for example, allow tremblings as small as possible (epsilon > 0) with fixed outcomes! Therefore, Is there (yet) an equivalent of Heisenberg's uncertainty principle in the game theory, decision theory etc. ?

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One possibility is the Cramer-Rao inequality in information theory. –  Deane Yang May 8 '13 at 18:45

Let me answer in terms of operator theory. The uncertainty principle can be interpreted as some particular inequality (as you say) such as $$\frac{\hbar}{2}\Vert{u}\Vert^2\le \Vert{D_xu}\Vert\Vert{xu}\Vert,$$ inequality due to the identity $2\Re\langle\hbar\frac{1}{i}\partial_x u, ixu\rangle=\hbar\Vert{u}\Vert^2,$ and in fact to the non-commutation of the unbounded operators $D_x=\frac{\hbar}{i}\partial_x$ with $x$ (multiplication by $x$). Incidentally you can note that the non-commutation of bounded operators $A,B$ (say finite matrices) could never lead to $$[A,B]=I\tag{NC}$$ as it is the case for $A=\frac{1}{i}\partial_x$, $B=ix$. In fact (NC) is not possible for matrices since the trace of a commutator is 0. This is one reason for which the quantization requires (necessarily infinite dimensional) unbounded operators.

Now this non-commutation property is at the source of quantum mechanics and produces the uncertainty principle inequalities. Each time you will have to deal with a non-commutative algebra of operators, you will have some type of uncertainty principle. In particular if your unknown quantities do not belong to a commutative algebra, you will have an uncertainty principle: the paradigmatic example remains the transfer from classical mechanics where the unknown quantities belong to a commutative algebra (functions on the phase space, such as position $x$ or momentum $\xi$), transfer to quantum mechanics where the unknown quantities are quantization of the previous quantities and are operators, such as $D_x,x$.

The uncertainty principle is simply the signature of non-commutativity and is not limited to quantum mechanics.

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It is worth noting that the Cramer-Rao inequality can be viewed as the real analogue of the Heisenberg principle, because it derives an inequality similar to the one given in this answer from the real version of (NC): $[x, \partial_x] = -1$. –  Deane Yang May 16 '13 at 22:03