In the framework of quantum field theory, one finds multiple load-bearing locality and causality conditions. One of the most important is the cluster decomposition principle, which requires that scattering experiments conducted at large spatial separation have statistically independent results. The principle grounds a number of features of quantum field theory, especially the structure of scattering theory. However, the statistical independence required by cluster decomposition is in tension with the long-range correlations characteristic of entangled states. In this paper, we argue that cluster decomposition is best stated as a condition on the dynamics of a quantum field theory, not directly as a statistical independence condition. This redefinition avoids the tension with entanglement while better capturing the physical significance of cluster decomposition and the role it plays in the structure of quantum field theory.
In ordinary, non-relativistic, quantum physics, time enters only as a parameter and not as an observable: a state of a physical system is specified at a given time and then evolved according to the prescribed dynamics. While the state can, and usually does, extend across all space, it is only defined at one instant of time, in conflict with special relativity where space and time are treated on an equal footing. Here we ask what would happen if we defined the notion of the quantum density matrix for multiple spatial and temporal measurements. We introduce the concept of a pseudo-density matrix which treats space and time indiscriminately. This matrix in general fails to be positive for timelike separated measurements, motivating us to define a measure of causality that discriminates between spacelike and timelike correlations. Important properties of this measure, such as monotonicity under local operations, are proved. Two qubit NMR experiments are presented that illustrate how a temporal pseudo-density matrix approaches a genuinely allowed density matrix as the amount of decoherence is increased between two consecutive measurements.
In time-dependent scattering theory, the collision problem is described by assuming that a particle was localized outside the interaction domain where V = 0. After collision, there exists a definite probability that the particle is deflected from its impact direction at a certain scattering angle. After collision, there exists a definite probability that the particle is deflected from its impact direction at a certain scattering angle. This probability cannot be directly observed in experiments but is indirectly accessible to measurements through differential cross sections. The unitarity of the operator S is one of the most important properties of the S-matrix theory of scattering. Of course, the study of the S-matrix will greatly be facilitated by the knowledge that the S-operator belongs to a narrower class of the unitary operators.
Recently David Albert presented an argument that relativistic quantum theories are non-narratable. That is, specifying the state on every space-like hypersurface in a given foliation of space-time is not in general sufficient to determine the states on other hypersurfaces, so the history of the universe cannot be told as a narration of states at successive times. We show that the system Albert examined to arrive at this conclusion violates cluster decomposition of the S-matrix, a locality requirement satisfied by relativistic quantum field theories. We formulate the general requirements for a system to display non-narratability, and argue that a large class of systems satisfying them violate the cluster decomposition principle.
The application of the "causality condition" to the
S matrix for nonrelativistic particles encounters several difficulties: (a) there is no maximum velocity; (b) the interference of ingoing and outgoing waves has to be taken into account; (c) wave packets with a sharp front do not exist. The condition is therefore reformulated as follows: At any time the total probability of finding the particle outside the scattering center shall not be greater than 1, for every form of the incident wave packet. From this follows for spherical waves that
S, as a function of the momentum
p, is analytic and holomorphic in the first quadrant and that
e2iapS(p) (where
a is the radius of the scattering center) has an imaginary part \ensuremath{\leqslant} 1. That suffices to give an explicit integral representation and a product expansion for
S, but these permit a more general form for
S than is usually envisaged. If, however, the usual symmetry relation $S(\ensuremath{-}p)=S{(p)}^{*}$ is assumed in addition to the causality condition, more specific equations can be derived, which are direct generalizations of those in Part I. In particular, integral relations between the real and imaginary parts of
S, and the properties that Wigner found for the
R matrix can be deduced.