The Observation Problem of Many-Particles System in the Quantum Theory

Tadashi Nakajima

Abstract Nearly a century has passed, since the birth of quantum mechanics, and yet the measurement problem has not been solved. We investigate the measurement problem from two aspects. First we scrutinize the basic postulates adopted by existing theories and identify the postulate of classicality of apparatus (PCA) to be the origin of the trouble. Second, we analyze the simplest possible experimental setup, a single photon particle as an observed system S and a detector as an apparatus A, and we find that a quantum jump occurs as a microscopic interaction between S and a single particle in A. We call this a microscopic quantum jump (MIJ). The MIJ selects system eigenvalues (SEVs) such as a two-dimensional position and arrival time for an incident photon. The MIJ outputs a microscopic particle (MIP), which carries the information of the SEVs potentially. In the apparatus A, the MIP triggers amplification cascade of secondary particles, which we call the intermediate particles (IMPs). The IMPs are initially a few, but become plenty after the amplification. The output of the amplification is a macroscopic observable (MAO) such as a current pulse, which carries the information of the SEVs in actuality. The measurement is complete when the MAO is obtained. By adopting the postulate of the MIJ and by discarding the PCA, we have constructed a measurement theory, which is consistent with standard quantum mechanics.

Leo Van Hove, Nicolaas Marinus Hugenholtz, L. P. Howland

Léon Van Hove, Nicolaas Marinus Hugenholtz, L. P. Howland

Gray, Andrew

A new formulation of quantum mechanics is developed which does not require the concept of the wave-particle duality. Rather than assigning probabilities to outcomes, probabilities are instead assigned to entire fine-grained histories. The formulation is fully relativistic and applicable to multi-particle systems. It shall be shown that this new formulation makes the same experimental predictions as quantum field theory, but without having to rely upon the notion of a system evolving in a superposition of quantum states until collapsed by an observation. It is thus free from the problem of deciding what exactly constitutes an observation (the measurement problem) and may therefore be applied just as readily to the macroscopic world as to the microscopic.

H. Kummer

In this paper a rigorous description of a system consisting of identical particles is given for which the particle number is a superselection rule. A state of such a system is described by a sequence of density operators D = (D0, D1, …, Dn, …), where Dn acts in n-particle space and the asymptotic behavior is determined by the requirement ∑ n=0∞kn|Dn|<∞ for all k=1,2,…,where | | denotes the trace norm. The (bounded) observables in turn are described by sequences of bounded self-adjoint operators: B = (B0, B1, …, Bn, …) such that sup lim nk−n ‖Bn‖<∞ for some k=1,2,…. The expectation value of the observable B in the state D can be expressed as 〈B〉D=(Γ(B)|D)=(B|L(D)),where (B|D)= ∑ n=0∞Tr(BnDn);Γ denotes the expansion operator, whereas L stands for its adjoint, the contraction operator. The analog to the n-representability problem, i.e., the so-called representability problem, is put forward and its solution is connected with the solutions of the n-representability problem for different n. Finally, a possible mathematical foundation of the BCS theory is given.

Henry Mehlberg