Attempt of an axiomatic foundation of quantum mechanics and more general theories V

We continue here the series of papers treated byLudwig in [1–5]. Using some results ofDähn in [6], we point out that each irreducible solution of the axiomatic scheme set up in [5] is represented by a system of positive-semi-definite operator pairs of a finite-dimensional Hilbert-space over the real, complex or quaternionic numbers.This paper is an abridged version of the author's thesis presented to the Marburg University and written under the direction of Prof.G. Ludwig.     

Attempt of an axiomatic foundation of quantum mechanics and more general theories. III

Starting from axioms as physical as possible [1, 2, 3] about effects and ensembles, we shall investigate further consequences.Concerning part I and II [4, 5] the axioms can be so formulated as to be surveyed more easily.Besides, it is possible to prove some important theorems more simply.New structures of the lattice of decision effects are pointed out, leading in two subsequent papers at last to the final aim, the structure of Hilbert-space.     

Attempt of an axiomatic foundation of quantum mechanics and more general theories,II

The consequences of an axiomatic formulation of physical probability fields established in a first paper [1] are investigated in case of a finite dimensional ensemble-space.It will be shown that the stated number of axioms can be diminuished essentially. Further the structure of an ortho-complemented orthomodular lattice for the decision effects (also often called properties or still more misunderstandingly propositions) and the orthoadditivity of the probability measures upon this lattice, both, can be essentially inferred from the axioms 3 and 4,only. This seems to give a better comprehension of the lattice structure defined by the decision effects.Particularly, it is pointed out that no assumption (axiom) concerning the commensurability of two decision effectsE ₁ E ₂ withE ₁E ₂ must be made but that this commensurability is a theorem of the theory.     

Attempt of an axiomatic foundation of quantum mechanics and more general theories. IV

This contribution continues the series of papers on the same subject which has been treated byLudwig in [1–3]. Using the system of axioms as given in [3], we shall succeed in constructing an orthomodular lattice of linear operators on the real vector space generated by the physical decision effects. There results an isomorphism between the orthomodular lattice of all physical decision effects and the lattice to be constructed.     

An axiomatic scheme more general than quantum theory

An axiomatic scheme generalizing the operational approach to quantum theory is described. Only quite general axioms ensuring the existence of well-behaved probabilities are postulated. The space-time location of macroscopic apparatus interacting with the object is explicitly taken into consideration. The states and observables are defined and their time development is considered. The classification of physical processes with respect to their reversibility or irreversibility in time is given. The conditions of Lorentz and translational invariance are formulated. Linear transformations corresponding to operations on the object are introduced. In the case of reversible processes these transformations form an algebra and linear representations of the Poincaré group arise naturally. These results are, in general, invalid for irreversible processes. The position of quantum theory in the scheme described is clarified.     

State automorphisms in axiomatic quantum mechanics

A new metric which we call the intrinsic metric is introduced on the states of the generalized logic of quantum mechanics. It is shown that every automorphism on is an isometry. A norm can be defined on the linear spanE of which reduces to the intrinsic metric on. IfX is the completion ofE then every automorphism, on has a unique extension to a linear isometry onX. A comparison is made between these results and those of Kroniff.     

Uncertainty and complementarity in axiomatic quantum mechanics

In this work an investigation of the uncertainty principle and the complementarity principle is carried through. A study of the physical content of these principles and their representation in the conventional Hilbert space formulation of quantum mechanics forms a natural starting point for this analysis. Thereafter is presented more general axiomatic framework for quantum mechanics, namely, a probability function formulation of the theory. In this general framework two extra axioms are stated, reflecting the ideas of the uncertainty principle and the complementarity principle, respectively. The quantal features of these axioms are explicated. The sufficiency of the state system guarantees that the observables satisfying the uncertainty principle are unbounded and noncompatible. The complementarity principle implies a non-Boolean proposition structure for the theory. Moreover, nonconstant complementary observables are always noncompatible. The uncertainty principle and the complementarity principle, as formulated in this work, are mutually independent. Some order is thus brought into the confused discussion about the interrelations of these two important principles. A comparison of the present formulations of the uncertainty principle and the complementarity principle with the Jauch formulation of the superposition principle is also given. The mutual independence of the three fundamental principles of the quantum theory is hereby revealed.     

Events and observables in axiomatic quantum mechanics

It is shown that in fairly general circumstances the event and observable frameworks for axiomatic quantum mechanics are equivalent.     

The orthogonality postulate in axiomatic quantum mechanics

Letp(A,,E) be the probability that a measurement of an observableA for the system in a state will lead to a value in a Borel setE. An experimental function is a function f from the set of all statesI into [0,1] for which there are an observableA and a Borel setE such thatf()=p(A, , E) for all I. A sequencef ₁,f ₂,... of experimental functions is said to be orthogonal if there is an experimental functiong such thatg+f ₁+f ₂+...=1, and it is said to be pairwise orthogonal iff _i+f _j 1 forij. It is shown that if we assume both notions to be equivalent then the setL of all experimental functions is an orthocomplemented partially ordered set with respect to the natural order of real functions with the complementationf=1–f, each observableA can be identified with anL-valued measure _A, each state can be identified with a probability measurem onL and we havep(A,,E)=m oA(E). Thus we obtain the abstract setting of axiomatic quantum mechanics as a consequence of a single postulate.     

Lattice operations between observables in axiomatic quantum mechanics

Observables are treated as-homomorphisms of the Borel sets of the real line into an orthomodular-latticeL. By means of corresponding spectral-resolutions operations meet and join are defined between observables which endow the set of all observables with a lattice structure in caseL is-continuous and which give rise to lattices of observables in caseL is chosen arbitrarily and the observables commute.     

Logical foundation of quantum mechanics

The subject of this article is the reconstruction of quantum mechanics on the basis of a formal language of quantum mechanical propositions. During recent years, research in the foundations of the language of science has given rise to adialogic semantics that is adequate in the case of a formal language for quantum physics. The system ofsequential logic which is comprised by the language is more general than classical logic; it includes the classical system as a special case. Although the system of sequential logic can be founded without reference to the empirical content of quantum physical propositions, it establishes an essential part of the structure of the mathematical formalism used in quantum mechanics. It is the purpose of this paper to demonstrate the connection between the formal language of quantum physics and its representation by mathematical structures in a self-contained way.     

An improved formulation of some theorems and axioms in the axiomatic foundation of the Hilbert space structure of quantum mechanics

In Ref. [1] the axiomatic foundation of the Hilbert space structure of quantum mechanics was outlined. Apart from a set of physically plausible axioms, the (mathematical) assumption (V 1) of the minimal-decomposition property of the basenorm spaceB was incorporated into the axiomatic scheme of the theory.It is shown in the present paper that the assumption (V 1) is superfluous. In the first part of the paper we give a short summary of the axioms; in the second part the main theorems are proved without using assumption (V 1).     

Experimental tests of general quantum theories

Bell-type inequalities can be used to test the general framework not only of classical theories but also of quantum theories. A common algebraic origin of such inequalities is discussed.     

Semiclassical quantum mechanics. III. The large order asymptotics and more general states

We solve the time dependent Schrödinger Equation $\text{i?(}\text{?ψ}\text{?t}\text{) = ?(?}{}^2\text{/2m) Δψ + Vψ}$ modulo errors which have L² norms on the order of $\text{?}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ for arbitrarily large l. The initial conditions we consider are fairly general states whose position and momentum uncertainties are proportional to $\text{?}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.     

Minimum length from quantum mechanics and classical general relativity

We derive fundamental limits on measurements of position, arising from quantum mechanics and classical general relativity. First, we show that any primitive probe or target used in an experiment must be larger than the Planck length lP. This suggests a Planck-size minimum ball of uncertainty in any measurement. Next, we study interferometers (such as LIGO) whose precision is much finer than the size of any individual components and hence are not obviously limited by the minimum ball. Nevertheless, we deduce a fundamental limit on their accuracy of order lP. Our results imply a device independent limit on possible position measurements.     

Hypothesis on the foundation of stochastic interpretations of quantum mechanics

The hypothesis that a particle exists in a cavity space in thermal equilibrium with photons and executes a Brownian motion is due to collisions with the photons leads to a realistic stochastic interpretation of quantum mechanics.