Editing Uncertainty principle (section)

===Phase space===
In the [[phase space formulation]] of quantum mechanics, the Robertson–Schrödinger relation follows from a positivity condition on a real star-square function. Given a [[Wigner quasi-probability distribution|Wigner function]] <math>W(x,p)</math> with [[Moyal product|star product]] ★ and a function ''f'', the following is generally true:<ref>{{Cite journal | last1 = Curtright | first1 = T. |last2= Zachos | first2= C. | title = Negative Probability and Uncertainty Relations| journal = Modern Physics Letters A | volume = 16 | issue = 37 | pages = 2381–2385 | doi = 10.1142/S021773230100576X | year = 2001 |arxiv = hep-th/0105226 |bibcode = 2001MPLA...16.2381C | s2cid = 119669313 }}</ref>
<math display="block">\langle f^* \star f \rangle =\int (f^* \star f) \, W(x,p) \, dx \, dp \ge 0 ~.</math>

Choosing <math>f = a + bx + cp</math>, we arrive at
<math display="block">\langle f^* \star f \rangle =\begin{bmatrix}a^* & b^* & c^* \end{bmatrix}\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x \star x \rangle & \langle x \star p \rangle \\ \langle p \rangle & \langle p \star x \rangle & \langle p \star p \rangle \end{bmatrix}\begin{bmatrix}a \\ b \\ c\end{bmatrix} \ge 0 ~.</math>

Since this positivity condition is true for ''all'' ''a'', ''b'', and ''c'', it follows that all the eigenvalues of the matrix are non-negative.

The non-negative eigenvalues then imply a corresponding non-negativity condition on the [[determinant]],
<math display="block">\det\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x \star x \rangle & \langle x \star p \rangle \\ \langle p \rangle & \langle p \star x \rangle & \langle p \star p \rangle \end{bmatrix}
= \det\begin{bmatrix}1 & \langle x \rangle & \langle p \rangle \\ \langle x \rangle & \langle x^2 \rangle & \left\langle xp + \frac{i\hbar}{2} \right\rangle \\ \langle p \rangle & \left\langle xp - \frac{i\hbar}{2} \right\rangle & \langle p^2 \rangle \end{bmatrix}
\ge 0~,</math>
or, explicitly, after algebraic manipulation,
<math display="block">\sigma_x^2 \sigma_p^2 = \left( \langle x^2 \rangle - \langle x \rangle^2 \right)\left( \langle p^2 \rangle - \langle p \rangle^2 \right)\ge \left( \langle xp \rangle - \langle x \rangle \langle p \rangle \right)^2 + \frac{\hbar^2}{4} ~.</math>