![]() (6) It is a deeper quantum mechanical requirement than the condition about a minimal scale. Example 21 shows that it is not equivalent to it. Indeed, inequality ( 11) implies immediately the Robertson–Schrödinder uncertainty principle. ![]() Indeed, inequality ( 15) is an equality iff the state is pure. 1 (4) It includes a pure state condition. Here we use the squares | W ρ ~ | 2 and | F σ ( W ρ ~ ) | 2, and so we are treating W ρ as a wave function in ordinary quantum mechanics on a 2 n-dimensional configuration space. (3) It makes a direct connection with harmonic analysis, as it amounts to an inequality relating W ρ and its Fourier transform F σ ( W ρ ). (2) It is invariant under linear symplectic and anti-symplectic transformations (see Theorem 19). In fact, we only have to compute the covariance matrices of W ρ, | W ρ ~ | 2 and | F σ ( W ρ ~ ) | 2 and check inequalities ( 11). (1) It is parsimonious, in the sense that it is a computable test as the RSUP, but not a complicated one as sets of necessary and sufficient conditions such as the Kastler, Loupias, Miracle-Sole (KLM) conditions. Roughly speaking, it can be obtained from the ordinary Fourier transform F ( F ) by a symplectic rotation and a dilation ( F σ F ) ( z ) = 1 ( 2 π ħ ) n ( F F ) Jz 2 π ħ.įor a given measurable phase-space function F, satisfying ![]() In the sequel F σ ( F ) denotes the symplectic Fourier transform of the function F. In order to state our results precisely, let us fix some notation. ![]() This prompted us to look for an alternative uncertainty principle which goes beyond the RSUP. This means that being a quantum state is not only a question of scale but also of shape. More emphatically, we will show that any measurable function in phase space F with a positive definite covariance matrix Cov ( F ) > 0 satisfies ( 4) after a suitable dilation F ( z ) ↦ λ 2 n F ( λ z ), while most of them remain non quantum. We shall give an example of a function in phase space which saturates the RSUP, but which is manifestly not a Wigner function. This condition is not sufficient to ensure that the state is quantum mechanical (not even if saturated). In fact, ( 4) is only a requirement about a minimal scale related to ħ. Having said that, there is nothing about inequality ( 4) which is particularly quantum mechanical, with the exception of the presence of Planck’s constant. More specifically, the RSUP ( 4) is saturated, whenever all the Williamson invariants of Cov ( W ρ ) are minimal : In particular, we say that the RSUP is saturated if we can find n two-dimensional symplectic planes, where the uncertainty is minimal. By a suitable linear symplectic transformation, the RSUP makes it a simple task to determine directions in phase space of minimal uncertainty. It has a nice geometric interpretation in terms of Poincaré invariants, and it is intimately related with symplectic topology and Gromov’s non-squeezing theorem. It is invariant under linear symplectic transformations (unlike the more frequently used Heisenberg uncertainty relation). For a Gaussian measure G it is both a necessary and sufficient condition for G to be a Wigner distribution. Nevertheless it has many interesting features. It can be shown that condition ( 4) is a necessary but not sufficient condition for a phase space function to be a Wigner distribution. ![]()
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