Complementarity in Physics and Biology

[dx]

As many of you know, Niels Bohr found an understanding of quantum theory in his idea of 'complementarity.' To him, this was not an 'interpretation' of quantum theory, in the sense that it is something added on top of quantum theory whose validity is not on the same footing as the theory itself, but an essential part of quantum theory that is inseperable from the formalism. I first encountered Bohr several years ago, at a time when I was so frustrated with quantum theory almost to the point of madness because of the fact that I couldn't get an insight into quantum theory and what it means. Ever since I encountered Bohr, I had an almost messianic desire to expose other people to it, because it seemed so important. However I did not talk to anyone about it because it took a long time to read all his papers on this subject (several times) before I felt that I had a reasonable idea about what he was trying to say.

I am starting this thread to hear what people think about complementarity. I found it convenient to choose an example that is not from physics, but from biology (described below). This is because the scope of the complementarity argument is far wider than physics, and is more appropriately viewed as a general lesson about the underlying conditions for the explanation of nature.

To set the stage, I will briefly describe complementarity in physics. From the usual simplest examples of quantum phenomena, like the double slit experiment and the photoelectric effect, we see that the experimental data seem to have different aspects which are seemingly contradictory. For example, in the double slit experiment, the observable fact is a set of dots on the photographic plate. The individuality of the dots finds its expression in the concept of a particle, while the distribution of the dots finds its logical representation in the concept of wave. The trouble is that a particle picture cannot lead to a distribution of the form observed, so evidently we cannot use either picture without restriction if we are to avoid contradiction. The reciprocal restriction of the applicability of the particle or wave picture finds its expression in the uncertainty relations.

Now I would like to describe an example from biology that Bohr gives, which as far as I know is not known to many people, other than people who have spent time with Bohr's original writings.

Many people have the intuition that the characteristic features of living organisms seem to be outside the scope of a purely mechanical or physical description. The description of living organisms requires the use of the word 'purpose' which belongs to a frame of concepts which may be termed vitalistic. On the other hand, a system specified in a mechanical sense leaves no room for such a concept, and there does not seem to be any restriction on our ability to carry out such a description for life forms using microscopes etc. The apparent contradiction is resolved in the following way: A situation in which a life form exhibits its characteristic vitalistic behavior is not compatible with a simultaneous specification of its state in a mechanistic sense, and a situation in which such a mechanistic specification is carried out is incompatible with vitality, because such a situation will kill the life form. Thus the vitalistic and mechanistic descriptions are complementary and apply to mutually exclusive experimental situations.

 

[atyy]

That's just Copenhagen, isn't it, with its Heisenberg cut somewhere along the von Neumann chain?

Via sean Carroll's blog, I came across this video of Witten agreeing that there is a hard problem in quantum mechanics and in neuroscience: http://www.preposterousuniverse.com/blog/2015/03/05/the-big-questions/.

Of course, Bohmians think the one in neuroscience is much harder ...

 

[bhobba]

I am starting this thread to hear what people think about complementarity.

Bohr was deep and profound. To go toe to toe with Einstein and actually win was no easy task. It was basically Bohr that made Einstein realize QM was correct - but for him still incomplete. Einsteins final challenge to Bohr was a lu-lu. It exercised Bohr well into the night - but eventually he cracked it. When he explained its error Einstein literally tipped his hat to Bohr and from that point on never challenged it on the grounds of being incorrect again.

Complementarity is of course correct - in QM there are things like momentum and position that are complementary. If you know position to a high degree of accuracy then momentum is unknown and conversely. I am however not that keen on the wave-particle thing as an example of complementarity because wave-particle duality is basically a myth:
http://arxiv.org/pdf/quant-ph/0609163.pdf

That said my issue is taking it as a central principle of QM. For me its central issue and mystery is the modern version of the measurement problem which has morphed quite a bit with the modern understanding of decoherence - it now - colloquially - is why do we get any outcomes at all - or more technically - how does a an improper mixed state become a proper one:
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

In my opinion a lot of water has passed under the bridge since Bohr's time and QM is now understood much better. Its now understood to be the simplest generalised probability model that allows continuous transformations between pure states - which if you think about it is pretty much required by time being continuous - if a system changes to something in a second it goes through half a second:
http://arxiv.org/pdf/quant-ph/0101012.pdf

Bottom line IMHO - yes Bohr is deep and complementarity true but outdated as a central principle.

Thanks
Bill

 

[dx]

Complementarity is of course correct - in QM there are things like momentum and position that are complementary. If you know position to a high degree of accuracy then momentum is unknown and conversely. I am however not that keen on the wave-particle thing as an example of complementarity because wave-particle duality is basically a myth:
http://arxiv.org/pdf/quant-ph/0609163.pdf

However, in more advanced and technical textbooks on QM, the wave-particle duality is rarely mentioned. Instead, such serious textbooks talk only about waves, i.e., wave functions ψ(x, t). The waves do not need to be plane waves of the form ψ(x, t) = e i(kx−ωt) , but, in general, may have an arbitrary dependence on x and t. At time t, the wave can be said to behave as a particle if, at that time, the wave is localized around a single value of x.

The wave ψ(x), even if it is not localized at a single value of x, still represents the probability of finding a particle at x. The particle interpretation is present even if the wave function is not localized. In fact, this probability ψ*ψ can only be calculated by a combined use of both the wave and particle pictures (as evident from the fact that the particle formula

H = P²/2m + V(X)

is used in the schrodinger equation for ψ which satisfies the wave-picture concept of the superposition principle). The experimental evidence, being a set of dots on the photographic plate, obviously needs the particle picture to explain the fact that you see individual dots. The distribution of the dots however requires the superposition principle which refers to the wave picture ψ(x), and does not apply to the particle picture.

 

[dx]

That's just Copenhagen, isn't it, with its Heisenberg cut somewhere along the von Neumann chain?

Yes. The idea of a Heisenberg cut, or a division between the objects under investigation and measuring bodies, also has analogs in other situations in which complementarity appears. For instance Bohr talks about the description of our mental activity, where we have to separate the objective content and the perceiving subject in any unambiguous communication of this activity, but this 'cut' is not necessarily fixed because in a different situation, the previous 'subject' can now become the object of communication. Whenever we try to communicate to other human beings our mental activity, i.e. in any use of language to talk about our mental activity, there must be this 'heisenberg cut', which can move along the 'von neumann chain' when we focus our attention on a different aspect of consciousness.

 

[bhobba]

In fact, this probability ψ*ψ can only be calculated by a combined use of both the wave and particle pictures (as evident from the fact that the particle formula

A couple of points here.

First, although its an advanced issue, position is not always an observable - you will find a number of threads about this if you do a search.

Secondly think about the wave-function of two entangled particles - what dimension is it? What about the amplitude - what's its physical significance? Multiply it by a phase factor exp*ix - again what's its physical significance? Taking all these point into account would you still consider it wave-like?

Finally if its wave-like exactly what is it waves of?

Basically QM is a theory about observations - saying anything about it when not observed is fraught with danger.

Thanks
Bill

 

[dx]

Even for two entangled particles, you still have to use the particle picture formula

H = P₁²/2m₁ + P₂²/2m₂ + V(x₁, x₂)

You cannot use the Schrodinger equation without defining the Hamiltonian using the particle picture.

By wave picture, I only mean the use of the superposition principle. psi(x1, x2) is not a wave in space, but the point is that it satisfies the superpositoin principle

 

[bhobba]

Even for two entangled particles, you still have to use the particle picture formula

The primary entity is the state - in can be expanded in all sorts of basis - precisely why do you think the expansion in the position basis fundamental?

You cannot use the Schrodinger equation without defining the Hamiltonian using the particle picture.

The Hamiltonian does not have to be expressed in terms of the wavefunction eg the Hamiltonian of a free particle is p^2/2m - that is most naturally expressed in terms of the momentum representation.

Thanks
Bill

 

[dx]

I'm not sure what you mean by that. Whether you use the position representation or the momentum representation, the Hamiltonian is the same, as a function H(X,P)

You can substitute X = -id/dp or P = -id/dx in that, to turn it into a differential operator. But the function H(X,P) comes before that, which is determined by the particle picture.

 

[bhobba]

I'm not sure what you mean by that

What I mean is its entirely arbitrary what you expand a state in - you can expand it in terms of momentum if you wish. Take the free particle - expanding it in terms of momentum is the most natural - there is nothing about position. What exactly is wave-like there?

But we also have the other issues I raised - its amplitude makes no difference - although its usually normalised and its invariant to a phase change - that is not the usual behaviour of waves.

I also forgot to add - there is nothing wrong when position is an observable thinking in terms of particles when it has an exact position - its thinking of it in terms of waves otherwise.

Thanks
Bill

 

[dx]

The wave function Ψ can be written either as a function of P or a function of X. Isn't that what you mean by position and momentum representation? You can also turn the function H(X, P) into a differential operator by either substituting X = -id/dP or by substituting P = -id/dX, so you can write the operator H in either representation.

However the form of the function H(X,P) is determined by the particle picture, before you turn it into a differential operator.

The 'wave picture' which I refer to is simply the superposition principle, which applies in both representations. This (the superposition principle) is what is needed to describe the distribution of particles on the photographic plate, by interference.

 

[bhobba]

The wave function Ψ can be written either as a function of P or a function of X. Isn't that what you mean by position and momentum representation? You can also turn the function H(X, P) into a differential operator by either substituting X = -id/dP or by substituting P = -id/dX, so you can write the operator H in either representation.

Of course

However the form of the function H(X,P) is determined by the particle picture, before you turn it into a differential operator.The 'wave picture' which I refer to is simply the superposition principle, which applies in both representations. This (the superposition principle) is what is needed to describe the distribution of particles on the photographic plate, by interference.

No its not. H is a function of the operators X and P - what that has to do with particle nature has me beat - other than of course X must be an observable for it to make sense. You could equally as well say its determined by its momentum nature - but such semantics doesn't really make sense to me.

I am however not that worried about saying if position is an observable it has a particle nature - that's almost the definition of what a particle, in the usual sense, means.

My concern is the wave part of the wave-particle duality - the wave-function does not behave like waves in any kind of usual sense.

Thanks
Bill

 

[dx]

When I say 'wave', I am merely referring to the wave-function, so I guess we agree about that. I also agree that you can write the wave-function in either the position or momentum representation. I am not claiming that either representation is better than the other. I am also not claiming that the wave-function is a wave in space. The only thing about the wave function in my argument above is that it satisfies the superposition principle, whereas the particle picture does not have a superposition principle.

But what I said about the Hamiltonian is that you only get the operator H by making subsitutions like X = -id/dP or P = -id/dX in the classical Hamiltonian

H = P²/2m + V(X)

This is the expression for the sum of the kinetic and potential energies of a particle.

 

[bhobba]

When I say 'wave', I am merely referring to the wave-function,

Then I agree entirely with you.

That link I gave about myths also did the same - although it starts out saying that's what it means by wave.

Its a bit of an issue here on the QM sub-forum we get a lot of beginners who don't know the difference.

Thanks very much for the discussion that may be helpful to beginners in understanding the difference between wave and wave-function.

Thanks
Bill

 

[TrickyDicky]

Certainly complementarity is just a consequence of the Heisenberg cut of Copenhagen as atyy said, so it is as artificial as the cut concept, in the sense that it is needed in QM to make sense of the divide between the SE deterministic evolution vs stochastic evolution of measurements with a minimum of coherence but It is more than likely that the cut is not physical as makes little sense to have a classical reality separated from a quantum reality.
I agree with Bhobba that wave-particle duality is obsolete, misleading and is not related to complementarity.

 

[dx]

The cut is not 'physical' in the sense that it is not inherent in the world, but the cut is a necessary component of the formalism. The well defined application of the Schrodinger equation for example can only be done once a cut has been introduced. This is exactly analogous to the example I gave before of consciousness. When one speaks about the content of consciousness, there is a cut between the subject and the object. This cut is not inherent in consciousness, but it is inherent or necessary whenever we speak about a particular aspect of consciousness.

About wave-particle duality, this is completely apparent in the Schrodinger equation. The wave-picture super position principle is involved in |Ψ> and the particle picture is involved in the definition of the Hamiltonian H = P²/2m + V(X)

The probabilities cannot be calculated unless both pictures are used.

 

[bhobba]

The cut is not 'physical' in the sense that it is not inherent in the world, but the cut is a necessary component of the formalism.

Indeed it is not physical, and it is of course necessary as was shown by Von-Neumann.

These days however the tendency is to put it just after decoherence. Of course it doesn't have to be put there, but things are simpler if you do eg Schroedinger's Cat is trivial.

Thanks
Bill

 

[dx]

So it looks like not many people are interested in the Biology part of my original post, but I just want to add a few comments here for those who may find this thread in the future from a search, because really, these are not more or less vague analogies but exact logical relationships which are found in many different fields.

Ultimately, the deepest and most profound thing that we find in quantum mechanics is the fact that the quantum of action h is "whole" and "indivisible". The corresponding situation in my biology example is that "life" is not something that can be subdivided and analyzed as a property of the collective behavior of its parts. The "vitality" of the whole is not a property of the collective "mechanistic" behavior of the atoms. These "vitalistic" and "mechanistic" pictures are complementary, and cannot be combined into a single picture.

It is not that we cannot subdivide the organism into atoms. That is clearly possible, just like it is possible to write h as h/2 + h/2. But the point is that the "whole" has a behavior that cannot be understood in terms of its parts. In that sense, h is indivisible.

It is exactly for this reason that things like life and consciousness can never be "explained" in terms of the atoms and so on. They are properties of the whole, which neither require nor are capable of rational foundation.

 

[bhobba]

It is exactly for this reason that things like life and consciousness can never be "explained" in terms of the atoms and so on. They are properties of the whole, which neither require nor are capable of rational foundation.

Suppose some ultra sophisticated neural net was put together that passed the Turing test. The designers obviously knows how it works - would that be enough to explain its behaviour? Indeed driver-less cars is now considered a solved problem - does its design explain how it works considering it has to be trained?

Thanks
Bill

 

[dx]

It is of course possible to design machines that modify themselves, like cellular automata, and very soon they may become so complex that the designer can no longer understand its behavior. But that is only because of the complexity of the object, not because its behavior cannot be understood in terms of its design. The simple rules that define a cellular automaton are ultimately responsible for its behavior, and how the automaton behaves can always be traced back to these rules. A machine can always be understood in terms of its parts.

The difference between this and life is that the very pictures/concepts that we use to describe its various aspects exclude each other. For example, in a cellular automaton, if you describe some aspect of its behavior using some word/concept, that word/concept does not contradict the defining rules of its evolution. The elementary rules that describe the evolution of the automaton can always be combined with any other pictures that one may use to describe its complex behavior. In our description of life, the pictures that describe its holistic behavior cannot be combined with the pictures that describe its parts. These pictures are contradictory when you try to apply them at the same time. So they must be regarded as complementary pictures.

 

[TrickyDicky]

So it looks like not many people are interested in the Biology part of my original post, but I just want to add a few comments here for those who may find this thread in the future from a search, because really, these are not more or less vague analogies but exact logical relationships which are found in many different fields.

Ultimately, the deepest and most profound thing that we find in quantum mechanics is the fact that the quantum of action h is "whole" and "indivisible". The corresponding situation in my biology example is that "life" is not something that can be subdivided and analyzed as a property of the collective behavior of its parts. The "vitality" of the whole is not a property of the collective "mechanistic" behavior of the atoms. These "vitalistic" and "mechanistic" pictures are complementary, and cannot be combined into a single picture.

It is not that we cannot subdivide the organism into atoms. That is clearly possible, just like it is possible to write h as h/2 + h/2. But the point is that the "whole" has a behavior that cannot be understood in terms of its parts. In that sense, h is indivisible.

It is exactly for this reason that things like life and consciousness can never be "explained" in terms of the atoms and so on. They are properties of the whole, which neither require nor are capable of rational foundation.

What you are talking about is usually referred to as "emergent properties" and is a philosophical concept that opposes simplistic reductionism. I'm not denying that emergentism might be somewhat related to the principle of complementarity in some way as both are general principles but I can't see the direct relation between the quantum of action h and emergentism, either in physics or biology.
Complementarity is basically a rationalization of empirically non-commuting quantities and the Heisenberg indeterminacy. In as much as this is a general feature of nature it obviously affects biological organisms too. But I don't think this necessarily implies some discrete indivisable h. 'Emergent/complementarity/non commuting properties' can be implemented some other way.

 

[dx]

I can give some examples of this kind of indivisibility in quantum mechanics, but it is a very involved problem (both mathematically and dialectically) to directly connect these to the indivisibility of the quantum.

In the double slit experiment, we have particles which are shot out of a gun onto a photographic plate. The phenomenon is a distribution of dots on the photographic plate. This phenomenon is 'indivisible' in the sense that if you try to understand it in terms of the trajectories of the particles, it is irrational. If you try to subdivide the phenomenon by observing the trajectories of the particles, the 'emergent' behavior of the interference is destroyed. The circumstance that any attempt at the subdivision of the phenomenon demands an experimental situation where the phenomenon is destroyed is what makes room for the quantum of action. So the phenomenon cannot be subdivided.

Similar comments would apply to the stability of atoms. This stability cannot be understood in terms of the orbits of the electrons in the atom. In situations where we apply conservation laws to the transitions between stationary states, we must on principle renounce a space-time description of the electrons in the atom.