Informational Approach to Identical Particles in Quantum Theory

Abstract

A remarkable feature of quantum theory is that particles with identical intrinsic properties must be treated as indistinguishable if the theory is to give valid predictions. In the quantum formalism, indistinguishability is expressed via the symmetrization postulate, which restricts a system of identical particles to the set of symmetric states (`bosons') or the set of antisymmetric states (`fermions'). However, the physical basis and range of validity of the symmetrization postulate has not been established. A well-known topological derivation of the postulate implies that its validity depends on the dimensionality of the space in which the particles move. However, this derivation relies on the labeling of indistinguishable particles, a notion that cannot be justified on an informational basis. Here we show that the symmetrization postulate can be derived by strictly adhering to the informational requirement that particles which cannot be experimentally distinguished from one another are not labeled. Our key novel postulate is the operational indistinguishability postulate, which posits that the amplitude of a process involving several indistinguishable particles is determined by the amplitudes of all possible transitions of these particles when treated as distinguishable. The symmetrization postulate follows by requiring consistency with the rest of the quantum formalism. The derivation implies that the symmetrization postulate admits no natural variants. In particular, the possibility that identical particles generically exhibit anyonic behavior in two dimensions is excluded.

Figures (7)

• Figure 1: Measurements on two particles, distinguishable and indistinguishable. Two particles are each subject to measurements at times, $t_1$ and $t_2$, yielding outcomes $\ell_1, \ell_2$ at $t_1$ and $m_1, m_2$ at $t_2$. Left: If the particles are distinguishable, one can experimentally determine which transition actually occurs---the 'direct' one (top), or the 'indirect' one (bottom). Right: If the particles are indistinguishable, one cannot say what happened in the intermediate time on the basis of the information experimentally obtained.
• Figure 2: Amplitude for one- and two-stage experiments on two indistinguishable particles. Two indistinguishable particles are each subject to a measurement at successive times. (a) Measurements at time $t_1$ and $t_2$ yield outcomes $\ell_1, \ell_2$ and $m_1, m_2$, respectively. The figures on the left show the transitions of two distinguishable particles compatible with these outcomes: the 'direct' transition of amplitude $\alpha_{12}$, and the 'indirect' transition of amplitude $\alpha_{21}$. We postulate that the amplitude of the indistinguishable-particle process is $H(\alpha_{12}, \alpha_{21})$, where $H$ is a continuous function to be determined. (b) Measurements are performed at three successive times, yielding the indicated outcomes. On the left are shown the four possible transitions of two distinguishable particles, with respective amplitudes $\gamma_{11}, \gamma_{12}, \gamma_{21}, \gamma_{22}$, compatible with the observed outcomes. We postulate that the amplitude of the indistinguishable-particle process is $G(\gamma_{11},\gamma_{12}, \gamma_{21}, \gamma_{22})$.
• Figure 3: Isolation condition. In the center, two indistinguishable particles are each subject to measurements at times $t_1$ and $t_2$, yielding the indicated outcomes. Each particle is bound to its own subexperiment, which are isolated from one another. Accordingly, the transition probability can be computed in two different ways. As on the left, one can compute the transition probabilities, $|u|^2$ and $|v|^2$, for each of the particles separately, and multiply these to obtain $|uv|^2$. Alternatively, as on the right, one can use the $H$ function to compute the amplitude $H(\alpha_{12}, \alpha_{21}) = H(uv, 0)$, which yields transition probability $|H(uv, 0)|^2$. Consistency requires that $|H(uv, 0)|^2 = |uv|^2$.
• Figure 4: Origin of the $G$-product equation. In the center, two indistinguishable particles are each subject to measurements at times $t_1$, $t_2$ and $t_3$, yielding the indicated outcomes. The amplitude for this process can be computed in two different ways. On the left are shown the four possible transitions of two distinguishable particles compatible with the observed sequence. These transitions can each be expressed as series $(\mathop{\bm{\cdot}})$ combinations of the indicated transitions, yielding amplitudes $\alpha_{12}\beta_{12}, \alpha_{12}\beta_{21}, \alpha_{21}\beta_{12}, \alpha_{21}\beta_{21}$. Hence, from Eq. \ref{['eqn:G-def']}, the indistinguishable-particle process amplitude is $G(\alpha_{12}\beta_{12},\alpha_{12}\beta_{21}, \alpha_{21}\beta_{12}, \alpha_{21}\beta_{21})$. On the right, the indistinguishable-particle process is first decomposed into two smaller indistinguishable-particle processes in series $(\mathop{\bm{\cdot}})$, with respective amplitudes, $H(\alpha_{12}, \alpha_{21})$ and $H(\beta_{12}, \beta_{21})$, yielding an overall indistinguishable-particle amplitude $H(\alpha_{12}, \alpha_{21}) \, H(\beta_{12}, \beta_{21})$.
• Figure 5: In the center, a system of two indistinguishable particles is subject to measurements at three successive times. One of the outcomes of the measurements at $t_2$ is a coarse-graining of atomic outcomes $m_2$ and $m_2'$, denoted $(m_2,m_2')$. The amplitude of this indistinguishable-particle process can be computed in two different ways: (i) on the right, the process is decomposed into two processes combined in parallel $(\lor)$, and each of these in turn is expressed as two processes combined in series $(\mathop{\bm{\cdot}})$. The amplitudes of each of the four component processes can then be determined using $H$ in terms of the amplitudes of distinguishable-particle transitions, yielding $[H(\alpha_{12}, \alpha_{21}) + H(\alpha_{12}', \alpha_{21}')]\,H(\beta_{12}, \beta_{21})$; (ii) on the left are shown the four compatible transitions of two distinguishable particles. The amplitudes of these transitions, $\gamma_{11} = (\alpha_{12} + \alpha_{12}')\beta_{12}, \gamma_{12} = (\alpha_{12} + \alpha_{12}')\beta_{21}, \gamma_{21} = (\alpha_{21} + \alpha_{21}')\beta_{12}$, and $\gamma_{22} = (\alpha_{21} + \alpha_{21}')\beta_{21}$, are computed using Feynman's sum and product rules as indicated. The amplitude of the indistinguishable-particle sequence is then obtained using the $G$-function to be $G(\gamma_{11},\gamma_{12}, \gamma_{21}, \gamma_{22})$.