Richard Arens

Algebraic Difficulties of Preserving Dynamical Relations When Forming Quantum‐Mechanical Operators

Proofs are presented showing impossibility of assigning differential operators (quantum observables) to classical mechanical observables in such a way as to preserve the usual bracket formalism. Difficulty is shown to arise even if we limit ourselves to preserving brackets between the Hamiltonian and a rather limited set of observables. Some other algebraic difficulties inherent in the operator assignment problem are also discussed.

Multiplicativity factors for seminorms. II

Let S be a seminorm on an algebra A. In this paper we study multiplicativity and quadrativity factors for S, i.e., constants μ > 0 and λ > 0 for which S(xy) ⩽ μS(x)S(y) and S(x2) ⩽ λS(x)2 for all x, y ∈ A. We begin by investigating quadrativity factors in terms of the kernel of S. We then turn to the question, under what conditions does S have multiplicativity factors if it has quadrativity factors? We show that if A is commutative then quadrativity factors imply multiplicativity factors. We further show that in the noncommutative case there exist both proper seminorms and norms that have quadrativity factors but no multiplicativity factors.

Classical Lorentz Invariant Particles

A classical Lorentz invariant completely Hamiltonian elementary one‐particle system is defined as having a state space K in which the Poincare group acts transitively, its infinitesimal actions having generating functions relative to some Poisson bracket, such that there can be associated with each state k a world line Γ(k) in Cartesian 4‐space. It is determined that there are nine families of such particles. Two have their speed in the usual range, three travel at the speed of eight, and four always faster. In each family the members are distinguished by one or two parameters such as mass and spin.

On the concept of Einstein–Podolsky–Rosen states and their structure

In this paper the notion of an EPR state for the composite S of two quantum systems S1,S2, relative to S2 and a set O of bounded observables of S2, is introduced in the spirit of the classical examples of Einstein–Podolsky–Rosen and Bohm. We restrict ourselves mostly to EPR states of finite norm. The main results are contained in Theorems 3–6 and imply that if EPR states of finite norm relative to (S2,O) exist, then the elements of O have discrete probability distributions and the Von Neumann algebra generated by O is essentially imbeddable inside S1 by an antiunitary map. The EPR states then correspond to the different imbeddings and certain additional parameters, and are explicitly given by formulas which generalize the famous example of Bohm. If O generates all bounded observables, S2 must be of finite dimension and can be imbedded inside S1 by an antiunitary map, and the EPR states relative to S2 are then in canonical bijection with the different imbeddings of S2 inside S1; moreover they are then give...

CLASSICAL RELATIVISTIC PARTICLES.

The concept of Lorentz-invariant classical elementary particle is made precise and it is found that there are exactly two families: (0) the well-known free point-particles distinguished only by their masses and (+) a family with eight-dimensional phase space whose members are distinguished either by their mass or their positive spin.

An axiomatic basis for classical thermodynamics

In 1908, Caratheodory [I, 21 proposed a deductive theory of therrnodynamics. The special features of his theory are less important than the fact that he set standards for rigorous thinking in the subject. Hence it is not necessary to emphasize this aspect of the present approach. Physically speaking, the present objective is to build the theory on the concepts of thermometry and calorimetry, taking temperature and heatgeneration as the primary concepts, and to show how entropy and absolute temperature arise as derivative concepts. By not having work production as a primary concept, we obtain a purely thermal theory to which the first law and internal energy need not be attached, unless desired. This more traditional approach is in contrast with Caratheodory’s (see also refs. 3,4, and 5). We do not use an axiom about inaccessible states, as Carat&odor-y does, because such an axiom tends to obscure the fact that the “thermodynamic relations” which are the practical objectives of the subject are statements about reversible processes. (Irreversible processes take one out of the space of equilibrium states.) Our form of the second law is based on a semi-Carnot cycle which consists of heat generation (or absorption) at one temperature only, plus an adiabatic return to the initial state. Our axiom says that in this case, the total heat generated is zero. The compelling inductive proof of the second law based on the impossibility of perpetual motion of the second kind applies directly to support this axiom, because if positive total heat were generated in such a cycle then that heat, all absorbed at one temperature, could be converted into work indefinitely by reversing the cycle. Here, because we want a deductive theory, we cannot wait until, say, the concept of absolute temperature arises, and then inject the physical observation that this cannot be zero. We have somehow to make such specifications on our primary concepts (which are: the space of states, the temperature

Hamiltonian structures for homogeneous spaces

The definition and classification of classical relativistic particles requires the classification of certain invariant tensor fields on the inhomogeneous Lorentz group. The entire 10-parameter set is exhibited. At the same time, a much larger class of Lie groups is treated. The connection with particles will be presented in the succeeding article.

Weighted l∞ norms for matrices

Abstract Let W = ( w ij ) be a fixed n × n matrix of positive entries, and consider the W -weighted l ∞ norm defined on C n×n by ‖A‖w, ∞ = max| i, j | w ij α ij |, A =( α ij ).The main purpose of this note is to prove that for this norm, multiplicativity, strong stability, and quadrativity are each equivalent to the condition ( W -1 ) 2 ⩽ W -1 , where W -1 = ( ω -1 ij ) is the Hadamard inverse of W . Among other things we also show that if ‖ · ‖ ∞ is k -bounded for some k ⩾ 2, then it is stable.

A class of seminorms on function algebras

Abstract Let A be a function algebra on a set T. In this paper we study seminorms on A of the form Sc(x)=‖cx‖ where c, 0 ≠ c ∈ A , is a fixed element and ‖·‖ is the sup norm on T. We begin by proving that under suitable assumptions, elements c, d ∈ A satisfy c ⩽ d on T, if and only if for some p, 0 B of A . These results are then used in order to study multiplicativity and quadrativity factors for Sc on B , i.e., constants μ > 0 and λ > 0 for which Sc(xy) ⩽ μSc(x) Sc(y) and Sc(x2) ⩽ λSc(x)2 for all x, y ∈ B . Finally, for a family T of functions in A , we define the seminorm S F (x)=sup{Sf(x):f ∈ F }, and provide conditions under which S F has multiplicativity and quadrativity factors by exhibiting an element c ∈ A such that S F =S c on A .

Weighted l1 norms for matrices

Let W =(Wij ) be a fixed m × n weight matrix, and let the W-weighied l1 , norm on Cm×n be defined by Given weight matrices U,V,W, of orders m × r r × n and m × n, respectively, we begin by proving that a constant μ > 0 satisfies In the second part of this note we restrict attention to a single weighted l1 norm on C n×n ,and show that We conclude that may be quadrative without being multiplicative on C n×n .