Journal of Mathematical Physics 

Abstract: According to ’t Hooft the combination of quantum mechanics and gravity requires the three‐dimensional world to be an image of data that can be stored on a two‐dimensional projection much like a holographic image. The two‐dimensional description only requires one discrete degree of freedom per Planck area and yet it is rich enough to describe all three‐dimensional phenomena. After outlining ’t Hooft’s proposal we give a preliminary informal description of how it may be implemented. One finds a basic requirement that particles must grow in size as their momenta are increased far above the Planck scale. The consequences for high‐energy particle collisions are described. The phenomenon of particle growth with momentum was previously discussed in the context of string theory and was related to information spreading near black hole horizons. The considerations of this paper indicate that the effect is much more rapid at all but the earliest times. In fact the rate of spreading is found to saturate the bound fro...

Abstract: We establish the general form of the generator of a completely positive dynamical semigroup of an N‐level quantum system, and we apply the result to derive explicit inequalities among the physical parameters characterizing the Markovian evolution of a 2‐level system.

Abstract: The individual spins of the Ising model are assumed to interact with an external agency (e.g., a heat reservoir) which causes them to change their states randomly with time. Coupling between the spins is introduced through the assumption that the transition probabilities for any one spin depend on the values of the neighboring spins. This dependence is determined, in part, by the detailed balancing condition obeyed by the equilibrium state of the model. The Markoff process which describes the spin functions is analyzed in detail for the case of a closed N‐member chain. The expectation values of the individual spins and of the products of pairs of spins, each of the pair evaluated at a different time, are found explicitly. The influence of a uniform, time‐varying magnetic field upon the model is discussed, and the frequency‐dependent magnetic susceptibility is found in the weak‐field limit. Some fluctuation‐dissipation theorems are derived which relate the susceptibility to the Fourier transform of the time‐dependent correlation function of the magnetization at equilibrium.

Abstract: The Einstein tensorGij is symmetric, divergence free, and a concomitant of the metric tensorgab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.

Abstract: Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions. Generally this time is proportional to the number of lattice points.The number of distinct points visited after n steps on a k‐dimensional lattice (with k ≥ 3) when n is large is a1n + a2n½ + a3 + a4n−½ + …. The constants a1 − a4 have been obtained for walks on a simple cubic lattice when k = 3 and a1 and a2 are given for simple and face‐centered cubic lattices. Formulas have also been obtained for the number of points visited r times in n steps as well as the average number of times a given point has been visited.The probability F(c) that a walker on a one‐dimensional lattice returns to his starting point before being trapped on a lattice of trap concentration c is F(c) = 1 + [c/(1 − c)] log c.Most of the results in this paper have been derived by the method of Green's functions.