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.     

Perturbative quantum field theory at positive temperatures: An axiomatic approach

It is shown that the perturbative expansions of the correlation functions of a relativistic quantum field theory at finite temperature are uniquely determined by the equations of motion and standard axiomatic requirements, including the KMS condition. An explicit expression as a sum over generalized Feynman graphs is derived. The canonical formalism is not used, and the derivation proceeds from the beginning in the thermodynamic limit. No doubling of fields is invoked. An unsolved problem concerning existence of these perturbative expressions is pointed out.     

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.     

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.     

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.     

An axiomatic deduction of the pauli spinor theory

Pauli's spinor theory is deduced entirely from a postulate relating to the two-valuedness of the spectrum of the Pauli spin operators, without explicit use of the theory of group representations, or any assumption concerning the angular momentum properties of these operators.     

Between general relativity and quantum theory

Some possibilities of reconciling general relativity with quantum theory are discussed. The procedure of quantization is certainly not unique, but depends upon the choice of the coordinate conditions. Most versions of quantization predict the existence of gravitons, but it is also possible to formulate a quantum theory with a classical gravity whereby the expectation values ofT _µv constitute the sources of the classical metric field.     

A general framework for cosmological quantum theory

A self-contained structure for interpreting quantum theory in a cosmological context is presented which is free from the “unique predecessor” restriction previously imposed, and which allows mixed states involving only a finite part of the universe.     

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.     

An alternative to quantum theory

In a recent paper we have shown that continuous sets of resonances (as expressed by the nonvanishing of the kinetic collision operator) result in divergences in the traditional unitary transformation theory in addition to the usual ultraviolet divergences. Therefore, relaxation processes and lifetimes cannot be eliminated by unitary transformations diagonalizing the Hamiltonian. For this reason, we introduce a more general transformation theory based on nonfactorizable superoperators which “block diagonalize” the Hamiltonian superoperator and eliminate the divergence of the unitary transformation. This leads to a new concept of “observables” which are represented in general by operators which are both noncommuting and nondistributive. For example, to a single energy level we now associate a set of numbers corresponding to a probability distribution whose width is determined by the lifetime of the state. This new approach incorporates dissipation into the frame of quantum mechanics. It leads directly to a number of predictions such as the existence of a new anomalous Lamb shift dependent on lifetime as well as the appearance of a broken “time symmetry” in the structure of the energy spectrum. As this symmetry breaking depends on the arrow of time (thermodynamic equilibrium is approached in our future and not in our past) which is a property of our universe as a whole, we may call this new effect the “cosmological” Lamb shift. Of course subsequent experiments will have to explore the existence of this effect. Other consequences of this approach are briefly mentioned and will be developed in subsequent papers.     

Some remarks on general covariance of quantum theory

If one accepts Einstein's general principle of relativity (covariance principle) also for the sphere of microphysics (quantum mechanics, quantum field theory, theory of elementary particles), one has to ask how far the fundamental laws of traditional quantum physics fulfil this principle. The reason for presenting this short paper is to draw attention to a series of papers that have appeared during the last years, in which the author criticized the usual scheme of quantum theory (Heisenberg picture, Schrödinger picture, etc.) and presented a new foundation of the basic laws of quantum physics, obeying the principle of fundamental covariance (Einstein's covariance principle in space-time and covariance principle in Hilbert space of quantum operators and states) [1].Dedicated to Achille Papapetrou on the occasion of his retirement.     

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.