Yesterday, a reader voiced his dissatisfaction with the principle known as wave-particle duality. This Wikipedia article seems good to me, by the way, and I will refer to it a few times.

This opposition is widespread and it looks crazy to me. The principle is not even a precise principle of modern quantum mechanics. It's something more qualitative and older than quantum mechanics. The principle is obviously true and I simply can't imagine how someone may understand much about modern physics without agreeing that the wave-particle duality captures an important part of the defining insights of modern physics.

What is this duality? Wikipedia says:

Wave–particle duality is the concept that every elementary particle or quantic entity may be partly described in terms not only of particles, but also of waves. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the behavior of quantum-scale objects.You may find lots of similar but differently articulated assertions, e.g.

It’s a popular conception that in quantum mechanics microscopic objects, such as electrons or photons, are neither pure particles nor pure waves—they are both waves and particles. In some conditions they behave as waves, in some conditions, they behave as particles.Well, it's not a popular conception. It's the principle as actually stated by the pioneers of modern physics. What is

*popular*is to invent bogus complaints against this important principle.

And you see these bogus complaints almost everywhere. Here, Hrvoje Nikolic says that the duality is a myth, refers to his paper making the same claim, and opines that "serious books only talk about waves". What he forgets to say that he is not a serious physicist himself – just a 4th class Bohmian would-be physicist.

But the questions of the sort "please tell me that the principle is a myth" may be found everywhere, e.g. here on the Physics Forums. Why do the people act in this self-evidently stupid way?

The duality is a qualitative answer to the question whether the experiments in spaces with low particle numbers are manifestations of waves or particles. And the answer is that the correct entity that captures all the observed behavior is neither a classical wave, nor a classical particle, but a hybrid or interpolating entity that reconciles both of these characters of behavior but can't be identified with either. For a physicist to disagree with this statement is just crazy.

In the second sentence of the Wikipedia article, we may see a quote attributed to Einstein:

It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do.It's totally agreeable. The quote correctly captures the situation before the modern theory of quantum mechanics was understood. Well, the quote isn't really from "Einstein". It's a sequence of sentences from the 1938 book "The Evolution of Physics" by Albert Einstein and Leopold Infeld and some sources suggest that the whole book was written by Infeld. Nevertheless, Einstein agreed to put his name on the cover – and make the book much more famous as a result. I don't think that Einstein would ever disagree with the quote above.

The principle is often associated with the following slogans:

...all matter exhibits both wave and particle properties...Again, I just can't imagine how someone could justify his disagreement with these somewhat vague but innocent statements. The duality was recognized as one of the key observations that led to the birth of quantum mechanics. Werner Heisenberg said:

...an electron is a particle and a wave at the same time...

The problem of quantum theory centers on the fact that the particle picture and the wave picture are merely two different aspects of one and the same physical reality.That's just another, more refined formulation of the wave-particle duality. For Niels Bohr, the wave-particle duality was just the most specific and well-known example of his more general principle of complementarity.

In fact this new feature of natural philosophy [i.e., complementarity] means a radical revision of our attitude as regards physical reality....The point of the "new philosophy" is that objects display many kinds of behavior depending on the way how they're probed or measured. These ways of probing can't be done simultaneously and they couldn't be simultaneously respected by any classical theory. But the correct theory, quantum mechanics, is capable of reconciling all these features.

OK. We could talk about the history of the people's answers to the question "is light waves or particles?". For Newton, light was a stream of particles, the corpuscles, and this paradigm is therefore known as the corpuscular theory. Christian Huygens conjectured and the 19th century polymath Thomas Young clearly experimentally showed the interference properties of light – although Newton's rings were already known to Newton (he wasn't able to draw the right lessons) – and light became a manifestation of waves, electromagnetic waves that followed from Maxwell's equations of electrodynamics. Everything seemed clear – it's just the waves – almost until the end of the 19th century.

However, in the last months of 1900 (and the century), Max Planck partly reversed this "trend towards waves" when he showed that one needed to consider light quanta to cure the ultraviolet catastrophe and explain the black body radiation, especially its finiteness at high frequencies. In 1905, Albert Einstein took this idea of photons even more seriously and explained the relationships one may extract in the photoelectric effect. In 1923, Arthur Compton removed the remaining doubts when he observed the Compton scattering. In that process, photons behave totally as particles. The calculation of the final angles and frequencies is virtually isomorphic to the calculation of the collisions of billiard balls.

On the theoretical front, every particle was said to "carry" some de Broglie wave in 1924. Schrödinger generalized the wave to his wave function and told us additional details about the equations that this wave function obeyed. The wave function became a part of quantum mechanics. The wave function needed to be interpreted probabilistically and since that moment, it could have been used to predict the probabilities of any observations, whether they're observations showing the particle-like or wave-like properties of electrons, photons, or other particles.

In quantum field theory, the particles may also be derived as "quanta of energy" that a quantum field, basically an infinite-dimensional harmonic oscillator, carries. The correct picture is clearly some kind of a hybrid between particles and waves. The correct picture replaces the naive classical particles by using waves – wave functions – to calculate the probabilities that the particles will do one thing or another. On the other hand, the correct picture replaces the naive classical fields by quantum fields whose energy is no longer continuous but (the allowed values i.e. eigenvalues are) quantized in the units of \(E=\hbar\omega\), and that's why these excited fields unavoidably behave as sets of particles whose number is integer just like in any particle-like theory.

There are lots of people who try to deny or misinterpret quantum mechanics and they're silly. But to deny the wave-particle duality seems even nuttier to me. To me, it sounds like the complete denial of all the observations that were made even

*before*quantum mechanics was formulated and that unavoidably led to the quantum mechanical theories.

I am sure that a majority of professional particle and other quantum physicists would endorse my statements, especially the verdict that modern physics has proven that the "truth is almost exactly in between" the old pictures of particles and waves.

But it's not just the laymen and professional anti-quantum crackpots similar to Hrvoje Nikolic who are spreading bizarre criticisms against the duality. You may often hear totally serious physicists – who do excellent detailed technical work – saying that "it's just the particles" or "it's just the waves" or a similar denial of the fact that the correct modern description really has to interpolate between the two extreme pictures pretty much democratically.

Let me start with the less embarrassing one. Nima Arkani-Hamed likes to frame all the wisdom of quantum field theory by saying that it's all about "particles". There are just particles, not waves. His lectures of quantum field theory have always been brilliant and they didn't have any serious technical problems, as far as I can say. All the imaginable "physics of the fields" is just a collection of methods to calculate the probabilities and cross sections relating initial and final states with \(N_i,N_f\) particles with some momenta and polarizations etc.

It's a possible

*convention*to decide what is real, primary, and fundamental, and what is just auxiliary. But I would still argue that his asymmetric presentation is completely artificial (and moreover, it is only convincing if you fool yourself into believing that all the physics is about the scattering and the S-matrix). The quantum field such as \(\hat \Phi(x,y,z,t)\) at some point is at least

*as well-defined an operator*on the Hilbert space as the momentum \(\vec p_j\) of the \(j\)-th particle. In fact, the latter is much more problematic because such operators are only well-defined on the subspace of the Hilbert space with particular numbers of particles of each species, and after a convention is adopted how we identify the \(j\)-th particle. On the other hand, \(\hat \Phi(x,y,z,t)\) is well-defined on the whole Hilbert space. This is the reason why I tend to agree that the fields are

*more fundamental*than the properties of particles. They just work on the whole Hilbert space of a quantum field theory. But the

*observations*often end up with the same or similar results as observations would proceed according to classical physics if they were showing particle-like or wave-like properties of matter. According to the

*observations*, Nature and the correct quantum mechanical theories describing Her are capable of emulating the exact old concepts of particles and waves.

More embarrassingly, a European physicist I know – but there are others – often like to say that "everything is just waves" and pretty much explicitly deny that there are any particles or particle-like behavior. These folks typically buy some naive versions of the Many Worlds Interpretation where the wave function is "just a wave" – they really imagine it's conceptually a classical wave – and the observations of particular results, e.g. a position of a particle, are just some illusions from the "splitting of the wave function".

Needless to say, the reason why this attitude to the wave-particle duality is irrational is basically the same reason why it's silly to try to talk about quantum mechanics in terms of the "many worlds". There are no consistent laws – and there can't be any consistent laws – that would allow you to say how the wave function is "split" to the individual worlds. As far as fundamental laws of physics go, any way of writing one state vector as a sum of two or several other state vectors is

*equally good*. There is no universal "metric" on the Hilbert space that would allow you to write a state vector as a sum of two or a greater number of terms from "different worlds", and so on. You can't even say how many of these worlds there are.

But even if there were such rules, the split of the wave function would still not imply any viable relationship with the observations. It couldn't explain the Born rule – why the squared absolute values of the probability amplitudes give the probabilities of one outcome or another. And so on.

And even if all these self-evidently incurable defects of the many worlds paradigm were cured in some way, it would still be utterly irrational to deny that the particle-like and wave-like descriptions are basically equally fundamental, related by a duality. These two descriptions, particle-like and wave-like, may really be identified with bases on the Hilbert space of one particle. The particle-like description is composed of the position eigenstates\[

\psi_{x_0} (x) = \delta (x-x_0)

\] while the wave-like description has the natural basis of the momentum eigenstates\[

\psi_{p_0} (x) = \frac{ \exp(i p_0 \cdot x / \hbar) }{\sqrt{2\pi\hbar}}.

\] These two bases are exactly equally good. The wave functions expressed in these two representations are related by a simple Fourier transform. The Fourier transform is a classic "toggle on/off" button, a \(\ZZ_2\) switch. It's self-evident that it relates two perspectives that

*must*be considered on equal footing. Any asymmetry is clearly a bias. You may find it useful – or a preferred convention – to only work with one representation among the two. But this preference of yours doesn't invalidate the statement that the two representations are equally good.

The Wikipedia article also talks about the alternative views. Some people say that it's just particles; or it's just waves; or there are both; or it's neither; or it's "relational" which means that new content-free philosophical buzzwords are coined that pretend to disagree with the old picture – except that any disagreement obviously means that the new picture is just wrong.

Some of these "alternative views" are due to some people's inability to look at the system from different perspectives, or even to study and describe different kinds of experiments (those that mostly show the particle-like or wave-like behavior, respectively). Some of these "alternative views" are associated with the dysfunctional "alternative interpretations" of quantum mechanics. Some of the "alternative views" are spread by people like Rovelli who probably mostly understand that the founders of quantum mechanics were right but who want to sound interesting by pretending that they have discovered something new "again" even though they clearly couldn't.

The Bohmists may prefer to say that "there are both particles and waves" and they exist in the classical sense and on top of each other. This "doubling of the number of entities that exist" is a mistake for all the reasons why the Bohmian mechanics can't work as a complete theory of Nature. But even without the detailed analyses, we may see that the photon or the electron in an experiment is

*one object*with different features (which show up depending on how we probe the particles/waves), not several mutually interacting objects. If there were several objects, we could separate them but we can't.

At the end, all these "alternative views" are absolutely irrational. It is spectacularly clear that neither classical particles nor classical waves are enough to describe all the observations we have made and we can make. No classical theory may take care of everything. It is spectacularly clear that the correct picture is a "hybrid" or an "interpolation" in between the two extreme classical pictures. It is spectacularly clear that the two classical aspects (particle-like and wave-like) are two limits of the newer structures and these two limits should be admitted to have the same "status". In some cases, the switching between the two is literally a mirror reflection, like the Fourier transform.

We may say that the people's efforts to suppress or even deny the wave-particle duality is another major part of the jihad against quantum mechanics. What is unusually extreme about this part of the jihad is that it is no longer just quantum mechanics and the views of the founders that is being unjustifiably criticized. The wave-particle duality is a weaker, less specific observation about Nature. It's so weak that not only all the founders of quantum mechanics agreed with that principle (and with its importance). Even people like Albert Einstein did.

Efforts to deny the wave-particle duality are an activity meant to deny almost all of modern physics. A century ago, physicists weren't doing such things because they knew that they had to explain the observations – from all experiments that someone had done – and the wave-like and particle-like properties of matter were self-evident in these experiments. The critics of the wave-particle duality may easily ignore the experiments or one-half of them – because many other people are unfamiliar with them, even the basic ones – and repeat misconceptions that everyone would immediately see to be ludicrous a century ago. All the "alternative views" are ultimately struggling to return the mankind to the philosophy of classical physics and deny that there have been any experiments (about 100 years ago) that made that philosophy indefensible.

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