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With contributions by many of today's leading quantum physicists, philosophers and PDF · Algebraic Quantum Mechanics. Nicolaas P. Landsman. Pages The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for. Germany, the. following 30 years the concepts of quantum mechanics were e.g., http://pdg. dancindonna.info
Skip to main content. Log In Sign Up. Paul Busch. Pekka Lahti. Measurement Theory Compendium entry.
We mentioned already that Birkhoff and von Neumann had noticed that the set of closed subspaces L H of a complex Hilbert space H is not a Boolean algebra, because distributivity between conjunction and disjunction, like expressed in 1 , is not satisfied.
The set of closed subspaces of a complex Hilbert space forms however a lattice, which is a more general mathematical structure than a Boolean algebra, moreover, a lattice where the distributivity rule 1 is satisfied is a Boolean algebra, which indicates that the lattice structure is the one to consider for the quantum mechanical situation.
To make again a reference to general relativity, the lattice structure is indeed to a Boolean algebra what general Riemann geometry is to Euclidean geometry. And moreover, meanwhile it has been understood why the structure of operational propositions of the world is not a Boolean algebra but a lattice.
This is due to the fact that measurements can have an uncontrollable influence on the state of the physical entity under consideration . Hence the intuition of Birkhoff and von Neumann, and later Mackey, Piron and others, although only mathematical intuition at that time, was correct. Axiomatic quantum mechanics is more than just an axiomatization of quantum me- chanics. In this sense, we believe that it is one of the candidates to generate the framework for the new theory to be developed generalizing quantum mechanics and rela- tivity theory .
Let us explain why we believe that operational quantum axiomatics has the potential to deliver such a generalization of relativity theory and quantum mechan- ics.
General relativity is a theory that brings part of the world that in earlier Newtonian mechanics was classified within dynamics to the geometrical realm of reality, and more specifically confronting us with the pre-scientific and naive realistic vision on space, time, matter and gravitation. But there is one deep lack in relativity theory: it does not take into account the influence of the observer, the effect that the measuring apparatus has on the thing observed.
It does not confront the subject-object problem and its influence on how reality is. It cannot do this because its mathematical apparatus is based on the Riemann geometry of time-space, hence prejudicing that time-space is there, filled up with fields and matter, that are also there, independent of the observer.
That is also the reason why general relativity can easily be interpreted as delivering a model for the whole universe, whatever this would mean. We know that quantum mechanics takes into account in an essential way the effect of the observer through the measuring apparatus on the state of the physical entity under study.
In a theory generalizing quantum mechanics and relativity, such that both appear as special cases, this effect should certainly also appear in a fundamental way. Of course Riemann is not equivalent to general relativity, a lot of detailed physics had to be known to apply Riemann resulting in general relativity. This is the same with operational quantum axiomatics, it has the potential to deliver the framework for the theory generalizing quantum mechanics and relativity theory.
We want to remark that in principle a theory that describes the possible actions in the world, and a theory that delivers a model for the whole universe, should not be incompatible. It should even be so that the theory that delivers a model of the whole universe should incorporate the theory of actions in the world, which would mean for the situation that exists now, general relativity should contain quantum mechanics, if it really delivers a model for the whole universe.
Non-locality means non-spatiality, which means that the reality of the micro-world, and hence the reality of the universe as a whole, is not time-space like.
Time-space is not the global theatre of reality, but rather a crystallization and structuration of the macro-world. This fact is the fundamental reason why general relativity, built on the mathematical geometrical Riemannian structure of time-space, cannot be the canvas for the new theory to be developed. A way to express this technically would be to say that the set of events cannot be identified with the set of time-space points as is done in relativity theory.
Recourse will have to be taken to a theory that describes reality as a kind of pre-geometry, and where the geometrical structure arises as a consequence of interactions that collapse into the time-space context. We believe that operational quantum axiomatics can deliver the framework as well as the methodology to construct and elaborate such a theory. In later work Piron made a stronger attempt to found operationally part of the axioms , and 4 this attempt was worked out further in , to arrive at a full operational foundation only recently .
Also mathematically the circle was closed only recently. There do exist a lot of finite dimensional generalized Hilbert spaces that are different from the three standard exam- ples, real, complex and quaternionic Hilbert space. But since a physical entity has to have at least a position observable, it follows that the generalized Hilbert space must be infinite dimensional. At the time when Piron gave his five axioms that lead to the representa- tion within a generalized Hilbert space, there only existed three examples of generalized Hilbert spaces that fitted all the axioms, namely real, complex and quaternionic Hilbert space.
Years later Hans Keller constructed the first counterexample, more specifically an example of an infinite dimensional generalized Hilbert space that is not isomorphic to one of the three standard Hilbert spaces . She proved that an infinite dimensional generalized Hilbert space that contains an orthonormal base is isomorphic with one of the three standard Hilbert spaces .
Cluster States. Coherent States. Complementarity Principle. Complex-Conjugate Number. Compton Experiment or Compton Effect.
Consistent Histories. Copenhagen Interpretation. Correlations in Quantum Mechanics. Correspondence Principle. Counterfactuals in Quantum Mechanics.
Creation and Annihilation Operators. Creation and Detection of Entanglement. Davisson—Germer Experiment. About this book Introduction With contributions by many of today's leading quantum physicists, philosophers and historians, including three Nobel laureates, this comprehensive A to Z of quantum physics provides a lucid understanding of the key concepts of quantum theory and experiment. History of quantum mechanics Philosophy of physics Quantum Compendium Quantum Lexicon Quantum mechanics Quantum theory linear optimization science.
Editors and affiliations. Bibliographic information DOI https: In the second case, there is no consistent quantum measurement theory, unless an appropriate reinterpretation of what it means for an observable to have a definite value can be found. There is an enormous amount of literature dealing with the quan- tum measurement problem, and as yet there is no generally accepted resolution. Rigorous technical presentations of the problem and the spectrum of interpretational options are found, for example, in  and , whereas philosophical aspects are discussed in .
A valuable cross-section of the older literature until is reprinted in the vol- ume . Interestingly, the founders of quantum mechanics e.
Adopting the collapse postulate has since been taken by many as a pragmatic way of suspending the measurement problem.
Following this route, there remains the task for quantum measurement theory to show that quantum mechanics entails the possibility in principle of measuring any of its observables.
However, whenever that condition does apply, it implies the reproduction of probabilities for the object observable in terms of the pointer statistics. This latter condition, called probabil- ity reproducibility condition , can always be taken as the defining criterion for a measurement scheme to constitute a measurement of a given observable.
The formal implementation of these ideas, which constitute the mathematical framework of quantum measurement theory, are briefly summarized in the text box below. If the pointer observable Z is represented as a POVM on the Borel sets of R say , then for each state T of the object system, the following defines a probability measure on the real line X denotes any Borel subset of R and I is the identity operator: It is a fundamental theorem of the quantum theory of measurement that for every observable there are measurement schemes in fact, infinitely many such that 3 is fulfilled for all object states T .
With the existence of premeasurements for any observable thus secured, another task of quantum measurement theory is the description of the effect of a measurement on the object system.
Given a measurement scheme for an observable E, one can ask for the probabilities of the out- comes of any subsequent measurement. Dividing the joint probability in 4 by the latter probability gives the conditional probability for the occurrence of an outcome in Y given that the first measurement led to an outcome in X.
Thus IX T can be taken to play the role of the final object state in accordance with the collapse postulate. Any instrument arising from a measurement scheme has the property of complete positivity: It is another funda- mental theorem of quantum measurement theory that every completely positive instrument can be realized by some in fact, infinitely many measurement schemes .
With the conceptual tools of measurement theory outlined in the above box, it has become possible to eliminate some long-standing myths and corroborate a number of equally long-standing folk truths.
Measurements with that property are called repeatable. Moreover, according to a theorem due to Ozawa , in order for an observable to admit a repeatable measurement, this observable must be discrete, that is, have a countable set of values.
The realization that measurements necessarily disturb the object sys- tem was made early on in the history of quantum mechanics. Yet another fundamental theorem of quantum measurement theory is given by the statement that there is no measurement which does not change at least some of the states of the system under investigation: Thus, there is no information gain in quantum measurements without some disturbance. This is one example for the importance of quantum measurement theory as an applied discipline besides its foundational role.