On the Sharpness of Localization of Individual Events in Space and Time

Abstract

The concept of event provides the essential bridge from the realm of virtuality of the quantum state to real phenomena in space and time. We ask how much we can gather from existing theory about the localization of an event and point out that decoherence and coarse graining—though important—do not suffice for a consistent interpretation without the additional principle of random realization.

Appendix: Non-uniqueness of Decomposition of a General State—Effective Coherence Length

Let us start from a pure state with almost sharp momentum with mean value $\bar{p}$, mean position $\bar{x}$ and momentum uncertainty γ ^−1/2 described by the wave function in momentum space (we omit normalization factors)

$$ \psi_1(p)= \mathrm{e}^{-\frac{\gamma}{2}(p-\bar{p})^2+\mathrm {i}p\bar{x}}, $$

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or in x space

$$ \psi_1(x)= \mathrm{e}^{-\frac{1}{2\gamma}(x-\bar{x})^2+\mathrm {i}\bar{p} x}. $$

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We consider a mixture of such states corresponding to an ignorance of the precise values $\bar{p}$ and $\bar{x}$ expressed by the weight function $\mathrm{e}^{-\frac{\beta}{2}(\bar{p} -\hat{p})^{2}-\frac{1}{2\alpha }{\bar{x}}^{2}}$. The statistical matrix in x space is $\langle x'|\rho_{1} |x\rangle = \int\mathrm{d}\bar{x} \mathrm{d}\bar{p} \mathrm{e}^{-K_{x}}$ with

$$ K_x= \frac{\bar{x}^2}{2\alpha} + \frac{\beta(\bar{p} -\hat{p})^2}{2} +\frac{1}{2\gamma}{ \bigl[(x-\bar{x})^2+\bigl(x'-\bar{x}\bigr)^2 \bigr]} - \mathrm{i}\bar{p} \bigl(x-x'\bigr). $$

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Integration over $\bar{p}$ gives $\langle x'|\rho|x\rangle= \int\mathrm{d}\bar{x} \mathrm {e}^{-K_{1}}$ with

$$ K_1 = \frac{\bar{x}^2}{2\alpha}+\frac{(x-x')^2}{2\beta'}+ \frac{1}{\gamma} \biggl(\bar{x}-\frac{x+x'}{2} \biggr)^2- \mathrm {i}\hat{p} \bigl(x-x'\bigr) $$

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with

$$ \frac{1}{\beta'}=\frac{1}{\beta}+\frac{1}{2\gamma}. $$

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The same statistical matrix is obtained if we start from a pure state given by the wave function in x space

$$ \psi_2(x)=\mathrm{e}^{-\frac{(x-\bar{x})^2}{\beta'}+\mathrm{i}\hat {p} x} $$

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and consider the mixture given with a weight factor $\exp(-\frac{\bar{x}^{2}}{2\alpha'})$. It leads at first sight to the following expression for the statistical matrix

$$ \langle x'|\rho_2 |x \rangle = \int\mathrm{d} \bar{x} \mathrm{e}^{-K_2} $$

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with

$$ K_2=\frac{\bar{x}^2}{2\alpha'}+\frac{1}{2\beta'}\bigl(x-x' \bigr)^2 +\frac{1}{2\beta'} \biggl(\bar{x}- \frac{x+x'}{2} \biggr)^2 -\mathrm {i}\hat{p}\bigl(x-x'\bigr). $$

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After integration over $\bar{x}$ we see that ρ ₁=ρ ₂ provided

$$ 2\alpha+\gamma= 2\alpha'+\frac{\beta'}{2}. $$

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If γ≫β, in the first version we then have a very large coherent extension γ ^1/2 of the pure components. In the second version the effective coherence length β′^1/2 is much smaller corresponding to the much larger subjective ignorance of the momentum.