Charles, indeed.

I have some remarks about particles...

Conceptually such an understanding can be formulized in terms of indivisibility, but in reality `particles' may only be realized in the context of measurement. Therefore one is forced to conclude that the reality of a particle is as much a part of the measurement process as it is a part of the measured subject. For example, the ability of the apparatus to locate.

Indeed, there is an intimate connection with language and what one may state about a situation. Belavkin's axioms of quantum probability are essentially those regarding the logic of statements about reality, and the non-trivial conditioning of some events by other inconsistent events.

However, the assertion that particles have a form of reality outside of measurement/observation is meaningful only as an idea.

I) a definition of `particle' is proposed.

ii) the definition of a `measurement apparatus' is proposed

iii) a measurement is performed

iv) a result is obtained

v) the result is interpreted in the framework of definitions i and ii

vi) conclusions are drawn

However, soon language becomes insufficient. If you sit down to eat that is not language, but you may use language as a way of understanding that. One might ask `what is the reality of anything if I am not turning it into words to understand it?' Well, there is certainly something...

I would say that language is necessary but insufficient.

back to particles..

To the average person a particle is a very specific thing. So, what I meant above is this: The existence of `a particle as a concrete physical thing, not just an idea, but a material thing that exists independent of consciousness', may only be considered true if the particle is actually detected (measured/observed). Otherwise it is nothing more than an idea, something that is potential (or possible) but not actual.

I'm trying to illustrate the realization of material reality as a consequence of measurement, and the realization of measurement/observation as the mechanism by which possible is transformed into actual.

Of course we can talk about where a particle might be found if we were to measure it, but the idea that the particle is `actually' there is very different from the possibility that the particle may `possibly' be found there. One is the concrete realization of a particle, the other is not.

It is incorrect, even from a linguistic point of view, to state that there is actually a particle, until the particle is actually observed.

I hope that what I am saying as at least a little clearer. Indeed the particles do have a form of reality outside of measurement in so far as their potential existence may be discussed in terms of probability; but as particles they do not exist.

I think it was William Lawrence Bragg who said `the future is a wave and the past is a particle'.

First of all, Belavkin's formulation of quantum mechanics has roots in von Neumann's formalism, but it is an extension of this. von Neumann's theory does not account for stochastic processes. Stochastic processes are characterised by Ito algebras, or more generally quantum Ito algebras, which were formulated by Hudson and Parthasarathy in the early 80's. If you study quantum Ito algebra you will see that these are not C*-algebras, which were central to von Neumann's axioms.

Secondly, the idea that the structure of matter is independent of human `thought' (sensation would be better here, as there is more to one's being than thought), is itself conceived as a human thought. There is not really any logical way to get around that. So such an 'independent of human thought' notion of a particle is a belief. Which is fine, but should not be confused with any kind of intrinsic truth.

Finally, the statement that 'but to assume particles do not exist we must assume reality does not exist' is false. Instead we simply make an assertion that particles are not the intrinsic structure of reality. However, as a notion of particles appears as the result of measurement, we may infer that particles have their place within the structure of reality: the output of a measurement process.

Of course we can distinguish a thought from the object described by the thought, because there are many ways that an object can be detected other than thought. If an object is undetectable then it is not there. If there is any means by which the existence of an object can be inferred then there is a means by which that object is being detected.

Belavkin's work is based in deductions from mathematics. He wasn't confused, but many a reader has been confused in trying to understand what it was he was saying. English wasn't his first language and a lot of his work was translated into English from Russian. Even in quantum probability many find it hard to understand his work.

Quantum Measurement Basics:

The state of a quantum system is inferred by virtue of its entanglement with a second system corresponding to a measurement apparatus. A measurement apparatus may be prepared in a known state and an observation of an apparatus observable may be used to infer a state of the quantum system.

Such is the role of measurement apparatus (the notion of which may be generalized). Generally speaking, observations are made and one wishes to understand why such observations are made. The observations serve as those of a genaralized apparatus, and this appartus is considered to be interacting with another system. This other system is inferred to exist simply as a means to explain observation. And this system may be regarded as real as an object that we are interacting with in order for our observations to occur.

Whether referring to a human's observations, or laboratory apparatus measurements, we have the notion of an apparatus as the thing which is collecting the data about observations. In the simplest example this is a sequance of measurements. The (quantum) stochastic calculus provides a general tool for representing the apparatus. In addition to the apparatus space we have the system space too. And the measurement process for the example of counting measurement is a sequential entangling dynamics.

We can also consider the other basic example of a time-continuous measurement process given by a Wiener process. Which is a nice example because Lindblad dynamics may be obtained as the `drift' of such a quantum stochastic process. This means that if we consider a very special case of the quantum stochastic dynamics (a geometric quantum Brownian motion) and then we take the expectation of the apparatus (i.e. `trace out' the apparatus) then we are left with Lindblad's equation describing the evolution of the quantum system. This also means that the Lindblad dynamics of the quantum system is a marginal state of a geometric quantum Brownian motion.

Indeed, in the `orthodox' quantum mechanics there is no such sequence of interactions corresponding to a measurement process. This is a failing of such a formulation of quantum mechanics. The measurement process itself cannot be neglected from the formalism.

If one is to speak of `empirical evidence' of the existence of anything, then one must have an explicit understanding of the measurement process. As, after all, empirical evidence is that which is based on observation.

If the state of a quantum system is defined by the way it is prepared then that quantum system is the apparatus. This is not the same as the system under measurement.

The differential structure of eventum mechanics (event enhanced quantum mechanics) is as follows: a state of the apparatus is prepared and then there is a single measurement. But eventum mechanics makes the effort to recognize that the apparatus is distinct from the that which is being measured.

By the way, entanglement is only meaningful in the context of a compound system. And yes, it does have a technical meaning.

Thank you, I will have a look. In return I strongly recommend that you have a thorough look at this paper http://arxiv.org/pdf/quant-ph/0208087.pdf it gives a good introduction to the general theory of quantum measurement without getting too mathematically heavy. It also gives a very nice historical account of the development of quantum mechanics and the rise of quantum measurement as a development of quantum mechanics. Belavkin taught me both maths and physics for many years, and he had some good teachers himself. You may also find it interesting to look at some of the work of Stratonovich on the development of quantum mechanics and quantum measurement, and I also recommend the quantum probabilistic approach, building on the classical axioms of Kolmogorov, this may even be necessary to understand fully some of the foundations laid down by Belavkin. All the best.

Where do I start... second quantization does not mean replacing waves with operators. Second quantization can be done purely on the level of operators. It is related to the construction of product systems.

"the algebra of the actual observables, or beables"

Charles, the word you've not considered the importance of here is `actual'. This wasn't just thrown in for fun. When Belavkin speaks of actual observables (as oppose to just `observables' which have a more general mathematical definition) he is being very specific. He is talking about that which is ACTUALLY observed. Real output from experiment.

Actual observables live in a diagonal subalgebra that forms the classical part of a system. The classical part is that part where the measurement results live.

no no no...

There is an algebra B that contains observables - i.e. the self-adjoint operators on the Hilbert space of the quantum system. Whilst there is a very particular subalgebra A contained in B that contains the particular set of observables called `actual' observables. These are considered as the only contenders for Bell's 'beables'.

In quantum mechanics we do not consider operators without considering linear functionals on these operators. This is were the states and the wave-functions come from. If an operator algebra is used to describe the observables of a quantum system, then this algebra may be second quantized. This is usually done by forming a continuous tensor product structure over space and/or time. However, as a quantum system there is the corresponding second quantization of the wave-functions. This is why generating a Fock space from a Hilbert space k say is a second quantization of a quantum system described by K.

However, in the modern mathematical quantum theory we tend to generalize to the setting of Hilbert modules. In which case we have, as a simple example, a vector (in a Hilbert space K) of operators (in an algebra B), these are vector operators used for describing decoherence of waveifunctions (rather than the decoherence of the density matrix).

That is, consider the decoherence of a pure quantum state P=|h> ). Then this decoherence may be given on the wave-function alone as (E,F)h = (Eh,Fh) where (E,F) is a vector operator in a Hilbert B module having the B-valued inner product =E*E+F*F=I

i.e. (E,F) is a vector operator. This may be understood as a generalization of wave-functions to the setting of Hilbert modules.

If the decoherence, as a statistical event, is second quantized (over an interval of time) then we arrive in the operator setting of Fock modules. Being even more specific, we often use a mathematical structure called the `interior tensor product' for the treatment of such a quantum system, so we sometimes have objects called `interior tensor-product Fock modules'.

"The terminology, be-able as against observ-able, is not designed to frighten with metaphysic those dedicated to realphysic. It is chosen rather to help in making explicit some notions already implicit in, and basic to, ordinary quantum theory. For, in the words of Bohr, "it is decisive to recognize that, however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms." It is the ambition of the theory of local beables to bring these "classical terms" into the mathematics, and not relegate them entirely to the surrounding talk.

"The concept of "observable" lends itself to very precise mathematics when identified with "self-adjoint operator". But physically, it is rather a wooly concept. It is not easy to identify precisely which physical processes are to be given the status of "observations" and which are to be relegated to the limbo between one observation and another. So it could be hoped that some increase in precision might be possible by concentration on the beables, which can be described in "classical terms", because they are there. The beables must include the settings of switches and knobs on experimental equipment, the currents in coils, and the readings of instruments. "Observables" must be made, somehow, out of beables. The theory of local beales should contain, and give precise physical meaning to, the algebra of local observables."

- J.S.Bell

*For anyone that should stumble upon this string of answers:

This was originally a back-and-forth discussion with another researcher/scientist. However, he has, for some reason unknown to me removed all of his contributions from the discussion.

I will not do the same as I think it is nice to see how one's own opinions change as time goes on.