Linear positivity and virtual probability

We investigate the quantum theory of closed systems based on the linear positivity decoherence condition of Goldstein and Page. The objective of any quantum theory of a closed system, most generally the universe, is the prediction of probabilities for the individual members of sets of alternative coarse-grained histories of the system. Quantum interference between members of a set of alternative histories is an obstacle to assigning probabilities that are consistent with the rules of probability theory. A quantum theory of closed systems therefore requires two elements: (1) a condition specifying which sets of histories may be assigned probabilities and (2) a rule for those probabilities. The linear positivity condition of Goldstein and Page is the weakest of the general conditions proposed so far. Its general properties relating to exact probability sum rules, time neutrality, and conservation laws are explored. Its inconsistency with the usual notion of independent subsystems in quantum mechanics is reviewed. Its relation to the stronger condition of medium decoherence necessary for classicality is discussed. The linear positivity of histories in a number of simple model systems is investigated with the aim of exhibiting linearly positive sets of histories that are not decoherent. The utility of extending the notion of probability to include values outside the range of 0–1 is described. Alternatives with such virtual probabilities cannot be measured or recorded, but can be used in the intermediate steps of calculations of real probabilities. Extended probabilities give a simple and general way of formulating quantum theory. The various decoherence conditions are compared in terms of their utility for characterizing classicality and the role they might play in further generalizations of quantum mechanics.

Figure 1

The geometry of the two-slit experiment. An electron gun at left emits an electron traveling toward a screen with two slits, its progress in space recapitulating its evolution in time. The electron is detected on the screen at right at a position $y$ with a probability density that exhibits an interference pattern. A coarse-grained set of histories for the electron can be defined by specifying which slit ($U$ or $L$) it went through and ranges of detected positions $y$.

Figure 2

Candidate probability densities for the two-slit experiment. The candidate probability density $\wp (y,U)$ to go through the upper slit and arrive at $U$ is shown at top for $kd=kD=60$ where $k$ is the wave number of the electron [cf. Fig. 1]. The corresponding candidate probability density $\wp (y,L)$ for going through the lower slit is just below. The sum ${\wp }_{\mathrm{tot}}$ is the probability density to arrive at $y$ irrespective of which slit is passed through is shown at bottom. The amplitude $a$ has been assigned arbitrarily which is why the scale of the vertical axis not indicated. When these probability densities are integrated over ranges of $y$, they give candidate probabilities for the histories. The size of well-chosen ranges that yield positive probabilities is smaller than the size that would wash out the interference pattern.

Figure 3

Candidate probabilities for two different spin directions of a spin-1∕2 particle. States of a spin-1∕2 system can be labeled by two angles $(\theta ,\varphi )$ as in Eq. (4.13). These plots are concerned with the candidate probabilities for histories specified by values of the spin along a direction ${\vec{n}}_1$ followed (or preceded) immediately by a value of the spin along a second direction ${\vec{n}}_2$ making an angle $\delta$ with respect to the first. The plots show the states for which this set of histories is linearly positive (white) and those for which some candidate probabilities are negative (black) for given $\delta$. Only the partial range $0<\varphi <\pi$ is shown because the rest can be determined from the symmetries of Eq. (4.14). From top to bottom the values of $\delta$ are $\pi ∕2$, $\pi ∕4$, and $\pi ∕8$. For example, states with $\varphi =\pi ∕2$ imply these sets are linear positive for any value of $\delta$ but none is medium decoherent.

Figure 4

Candidate probabilities for a single free particle in a stationary Gaussian wave packet to remain in a position interval $\Delta$ after a time short compared to the wave-packet spreading time. The plot shows the candidate probability $p_L\left(\Delta \right)$ defined by Eq. (4.38a) for $\lambda ∕\sigma ={10}^{-2}$. This is positive over the whole of the range $\Delta$, indicating that the set consisting of the history in which the particle remains localized and the history where it does not so remain is linearly positive.

Figure 5

Candidate probabilities for a space-time coarse graining. A free particle moving in one dimension is initially in a Gaussian wave packet of width $\sigma$ traveling toward negative $x$ with a momentum $K_0=-20∕\sigma$ in $\hslash =1$ units. The solid curve shows the candidate probability $p_R$ for the particle to remain always at positive $x$ over a time $T$ short compared to the wave-packet spreading time. This is plotted as a function of the final position of the packet’s center $X$ that is connected to the initial position $X_0$ and $T$ by Eq. (4.54). The probability is positive over the entire range computed, showing that the set consisting of the history $\left(R\right)$, where the particle remains at positive $x$, and the history $\left(\overline{R}\right)$, where it does not, is linearly positive. The dashed curve shows the real part of the overlap $⟨{\Psi }_R\mid {\Psi }_{\overline{R}}⟩$ which must vanish for this set of histories to be medium decoherent. Evidently there is a significant range of linearly positive sets which are not medium decoherent.