The definition of the complementarity principle

Question

I am looking for a precise definition of the complementarity principle. It is rather briefly mentioned in my textbook, and I feel that authors have deliberately avoided defining it precisely. I'm a math major. Perhaps I didn't get the point. The most understandable description so far is one I found in Wikipedia:

The complementarity principle holds that objects have certain pairs of complementary properties which cannot all be observed or measured simultaneously.

It is at best intuitive content. From the examples, it is not clear what properties are complementary. How can we consider the pair of complementary properties of a system? Would it be enough to consider non-commuting Hermitian operators?

I found another definition in the book, "Niels Bohr and Complementarity" by Plotnitsky:

Complementarity, then, is defined by

(a) a mutual exclusivity of certain phenomena, entities, or conceptions; and yet

(b) the possibility of applying each one of them separately at any given point; and

(c) the necessity of using all of them at different moments for a comprehensive account of the totality of phenomena that we must consider.

Firstly, it is perhaps Bohr's definition that Plotnitsky formulated as a comprehensive definition. Secondly, is it how physicists understand the complementarity principle? Thirdly, I just read somewhere that Bohr's correspondence principle implies the complementarity principle but that the correspondence principle was rejected during the '80s. How about then fate of complementarity principle in the contemporary physics?

Answer 1

Oh, the complementarity principle. One of those things that you read about and get confused mess mostly talked by philosophers. Then you read the original guy and find something completely different from what all those people were talking!

If you go to Bohr's "Discussions with Einstein on Epistemological Problems in Atomic Physics" you will find the following

The new progress in atomic physics was commented upon from various sides at the International Physical Congress held in September 1927, at Como in commemoration of Volta. In a lecture on that occasion, I advocated a point of view conveniently termed "complementarity," suited to embrace the characteristic features of individuality of quantum phenomena, and at the same time to clarify the peculiar aspects of the observational problem in this field of experience. For this purpose, it is decisive to recognise that, however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. The argument is simply that by the word "experiment" we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangement and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.

I.e. 1). The quantum physics is fundamental and contains classical physics as its limit 2). But we are macroscopic beings and all our experiments are formulated in classical terms 3). All theories in physics have to be formulated as statements about experiments

So here's the complementarity principle. We have to formulate the more fundamental quantum physics using its classical limit to make statements about the experimental setup. In this way the quantum and classical terms complement each other.

Answer 2

I am looking for a precise definition of the complementarity principle. It is rather briefly mentioned in my textbook, and I feel that authors have deliberately avoided defining it precisely. I'm a math major.

I won't start off by addressing this question directly. Rather, I'm going to ask this question: What would be required to understand what quantum theory implies about how the world works.

To do that you need to have a specific, clearly stated theory with specific, clearly stated means of making predictions. The clearest way of doing that with quantum theory is to use the equations of motion such as the Schrodinger equation or equations of motion of relativistic quantum fields and work out what those equations imply. And you would also have to look at actual experiments and work out accounts of what is happening in reality to produce the outcomes: explanations.

If you take the equation of motion of quantum theory and apply them then you can start to discuss a theory of measurement. For example, if you're measuring a physical quantity that acts as a record: a piece of information that can be copied indefinitely often, then the evolution of that piece of information is described by a Hermitian operator: an observable, see Section II of:

https://arxiv.org/abs/0707.2832

In many treatments of quantum theory in textbooks the authors state something along the lines that when you do a measurement the state is projected onto the corresponding eigenstate. However, measurements are just physical interactions that transfer or copy information and as such they are constrained by the equations of motion of quantum systems and those equations aren't compatible with this postulate. A lot of measurements don't fit into the neat box of copying a record, e.g. - destructive measurements of photons and the quantum Zeno effect and so the projection approach is inadequate to deal with many real measurements:

https://arxiv.org/abs/1604.05973

The reason complementarity sounds vague and woolly in comparison with the account above is that it was originally invented by Bohr as a broader philosophical idea. For a relatively sympathetic account see:

https://arxiv.org/abs/quant-ph/0412195

People keep mentioning it partly out of tradition and partly because openly describing it as vague, bad and obsolete philosophy is regarded as impolite.

If you want to understand quantum theory look at its actual implications, which have been worked out in the literature on decoherence not at bad philosophy.

Answer 3

This is only a partial answer to the question. A quote from Wiki in the question states: "The complementarity principle holds that objects have certain pairs of complementary properties which cannot all be observed or measured simultaneously."

While that may be a true concept as it applies to the specific term "complementarity", it is very deceptive. It was known even early on (by 1935 for sure, really by 1927) that there is actually a trade off between one extreme (particle-like) and the other (wave-like).

Of course: you can simultaneously know both the momentum (wave-like property) and position (particle-like property) of a quantum object. But there are strict limits on the precision of that knowledge. That of course is embedded in the Heisenberg Uncertainty Principle (HUP). This trade-off applies to non-commuting observables, but does not apply to pairs of commuting observables (spin and momentum commute, for example).

The Complementarity Principle is often mentioned in lay or conversational terms. But it is understood only to represent cases where focus is on a specific property or a specific quantum effect. The HUP is the more general rule.