Space-time trajectories of quantum particles

Hrvoje Nikolić

States in quantum field theory (QFT) are represented by many-particle wave functions, such that a state describing n particles depends on n space–time positions. Since a general state is a superposition of states with different numbers of particles, the wave function lives in the configuration space identified with a product of an infinite number of four-dimensional Minkowski space–times. The squared absolute value of the wave function is interpreted as the probability density in the configuration space, from which the standard probabilistic predictions of QFT can be recovered. Such a formulation and probabilistic interpretation of QFT allows one to interpret the wave function as a pilot wave that describes deterministic particle trajectories, which automatically includes a deterministic and continuous description of particle creation and destruction. In particular, when the conditional wave function associated with a quantum measurement ceases to depend on one of the space–time coordinates, then the 4-velocity of the corresponding particle vanishes, describing a trajectory that stops at a particular point in space–time. In a more general situation a dependence on this space–time coordinate is negligibly small but not strictly zero, in which case the trajectory does not stop but the measuring apparatus still behaves as if this particle has been destroyed.

Shinil Cho

Yakir Aharonov, M. Vardi

In this article we give an operational meaning to an individual "Feynman path." In other words, we describe a process of dense measurements, made in temporal sequence, which check whether the particle moves along any given trajectory in space-time. We show that in this process the two assumptions of the space-time formulation of quantum mechanics, are realized: (a) The weight that the particle moves along a trajectory that has been checked by this process is the same for all trajectories, and in fact, we show that the particle follows, with probability 1, the trajectory that is being checked. (b) A phase is systematically accumulated, so that, at the end of this process, the state is multiplied by the familiar factor $\mathrm{exp}[(\frac{i}{\ensuremath{\hbar}})\ensuremath{\int}L\mathrm{dt}]$. As an immediate extension of the above formalism, we suggest a setup that measures the relative phase between any two trajectories. Finally, our approach points toward the possibility of extending the Feynman formalism in order to cover more general Hamiltonians.

V. Frolov

The hypothesis that the Universe in its evolution exhibits, to one degree or another, strongly violated scale invarianceis discussed. The fundamental group of space-time now is not the Poincaré group, but the Poincaré-Weyl group. In case with a very high probability a particle is found near its classical trajectory, but there is an extremely small (but nonetheless nonzero!) probability of detecting this particle far from its classical trajectory, providing quantum mechanical nonlocality both in space and in time. The considerations presented here, in their further study, may lead to the formulation of a new interpretation of quantum mechanics. А possible change in Planck’s constant over time is discussed.