Quantum Measures and Integrals

Yongyun Hu

On the basis of quantum measure and integral theory,the complex-value quantum integral and its properties are discussed.At the same time,the relationship between the integral of the complex-value measurable function and absolute integral is considered,and then a bounded,monotone convergence theorem for this complex-value quantum integral is attained.

Дмитрий Петрович Санкович

The first steps in the application of methods for integrating functions defined on abstract sets were taken by Wiener. Most widely, the ideas of functional integration were developed in Feynman's works. The Feynman continual integral is well known to a wide community of physicists. Along with this, there is another approach to the construction of a functional integral in quantum physics. This approach was proposed by Bogolyubov. Bogolyubov's methods are relevant in quantum statistical physics, and have natural ties with probability theory. We review some mathematical results of integration with respect to a special Gaussian measure that arises in the statistical theory for quantum systems. It is shown that the Gibbs equilibrium averages of the chronological products of Bose operators can be represented as functional integrals with respect to this measure (the Bogolyubov measure). Some properties of this measure are studied. We rewrite partition function of many particle Bose systems in terms of Bogolyubov functional integral.

Gudrun Kalmbach

Carlton M. Caves

Consider measurements that provide information about the position of a nonrelativistic, one-dimensional, quantum-mechanical system. An outstanding question in quantum mechanics asks how to analyze measurements distributed in time---i.e., measurements that provide information about the position at more than one time. I develop a formulation in terms of a path integral and show that it applies to a large class of measurements distributed in time. For measurements in this class, the path-integral formulation provides the joint statistics of a sequence of measurements. Specialized to the case of instantaneous position measurements, the path-integral formulation breaks down into the conventional machinery of nonrelativistic quantum mechanics: a system quantum state evolving in time according to two rules---between measurements, unitary evolution, and at each measurement, ``collapse of the wave function'' (``reduction of the state vector''). For measurements distributed in time, the path-integral formulation has no similar decomposition; the notion of a system quantum state evolving in time has no place.

Jan Hamhalter