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.     

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.     

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

This contribution continues the series of papers [2, 4, 5, 12] treated by Ludwig and collaborators. It is based on the generalized frame given in [6]; there Ludwig has set up an infinite axiomatic scheme as extension of the finite system [4, 5]. The results of [12] are then proved for a locally finite case; they lead to an extended representation theorem.This paper was supported by the Deutsche Forschungsgemeinschaft.     

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,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. 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. 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.     

Atomicity and maximality in axiomatic quantum theory

The “weak-maximality” condition is proved to be equivalent to atomicity of the lattice of “propositions” (“decision effects”) in quantum axiomatics, satisfying certain simple conditions. In particular, it is shown that these conditions are fulfilled in Ludwig's axiomatic formulation of quantum mechanics. It is further proved that atomicity of the lattice of propositions follows from the condition of “strong maximality”. The maximality conditions have a clear physical interpretation. They are also fulfilled in the Hilbert space formulation of quantum mechanics. Since the atomicity property is used in theories based on Type I factors, the connection between atomicity and maximality seems of general interest. Useful theorems are proved.     

Adiabatic theorem in axiomatic quantum field theory

It is shown that Møller matricesS _± and scattering matrixS in axiomatic field theory can be expressed through their adiabatic analogs. In particular, it is proved under certain conditions that \(S_ - = \mathop {s\lim }\limits_{\alpha \to 0} S_\alpha (0,\infty )W_\alpha \) whereW _α is a trivial phase factor [i.e. a unitary operator of the form exp i / α ∝r(k)a ⁺ (k)a(k)dk]. Corresponding results in Hamiltonian approach are discussed.     

The transitions among classical mechanics,quantum mechanics,and stochastic quantum mechanics

Various formalisms for recasting quantum mechanics in the framework of classical mechanics on phase space are reviewed and compared. Recent results in stochastic quantum mechanics are shown to avoid the difficulties encountered by the earlier approach of Wigner, as well as to avoid the well-known incompatibilities of relativity and ordinary quantum theory. Specific mappings among the various formalisms are given.     

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).     

An axiomatic measure of one-way quantum information

I introduce an algorithm to detect one-way quantum information between two interacting quantum systems, i.e. the direction and orientation of the information transfer in arbitrary quantum dynamics. I then build an information-theoretic quantifier of one-way information which satisfies a set of desirable axioms. In particular, it correctly evaluates whether correlation implies one-way quantum information, and when the latter is transferred between uncorrelated systems. In the classical scenario, the quantity measures information transfer between random variables. I also generalize the method to identify and rank concurrent sources of quantum information flow in many-body dynamics, enabling to reconstruct causal patterns in complex networks.     

Quaternionic quantum mechanics is consistent with complex quantum mechanics

Quaternionic quantum mechanics is investigated in the light of the great success of complex quantum mechanics. It is shown that to reproduce the results of complex quantum mechanics, quaternionic quantum mechanics must contain complex quantum mechanics.     

Decoherence in quantum mechanics

We study decoherence in a simple quantum mechanical model using two approaches. Firstly, we follow the conventional approach to decoherence where one is interested in solving the reduced density matrix from the perturbative master equation. Secondly, we consider our novel correlator approach to decoherence where entropy is generated by neglecting observationally inaccessible correlators. We show that both methods can accurately predict decoherence time scales. However, the perturbative master equation generically suffers from instabilities which prevents us to reliably calculate the system’s total entropy increase. We also discuss the relevance of the results in our quantum mechanical model for interacting field theories.     

Modalities and quantum mechanics

Three approaches concerning the usage of modalities in the language of quantum mechanics were considered; Mittelstaedt and I built up a dialog semantics for modalities on a metalinguistic level, and a calculus of quantum modal logic is known that is complete and sound with respect to this dialogic semantics. Van Fraassen replaced the usual interpretation of quantum mechanics (with the projection postulate) by his modal interpretation based on a modal object language. Dalla Chiara translated a nonmodal object language for quantum mechanics and the appropriate quantum logic into a modal language. Specifically we are interested in the similarities and the differences of these three approaches.