I’ve had this idea since undergraduate school.  I’ve shared it with many people, whole audiences of people who I’m sure wish could forget it (and now you’re the lucky one).

From the first day we were taught about particle-wave duality in modern physics I wondered why we believe that there is this duality in nature.  What I mean by this is the following.  We are taught that matter is made of particles, point-like particles that are “hard”.  We are also fed a paradigm in which the elasticity and plasticity of bulk matter is due to the ability to “squeeze” or “pull” these hard point like objects to new relative positions.  We are taught that things like light are fields that extend through all of space supporting oscillations, i.e. waves.  Then comes quantum mechanics and we are fed the idea that sometimes a particle can act like a wave and sometimes a wave can act like a particle and that this is “sexy”.  I feel sorry for the nerd who really thinks that is sexy but hey it’s a free country.  Furthermore as we learn more advanced forms of QM and QFT we learn that the apparent waviness of the “particles” is explained by the wave function and its interpretation as a complex amplitude of a probability distribution.  Hey, it works and you can’t really knock that until you try it yourself.  So in short, we believe in this duality because “particles” stopped behaving nice and started behaving badly.

But there is more to story than what we are taught in school.  First of all the greatest minds in history have been contemplating the true nature of matter and light for more than about 1600 years, back to Aristotle and Democritus.  Even Newton, in a version of his text on Optics, stated that he believed that light was made of corpuscles and as such should be influenced by the pull of gravity, more than 200 years before Einstein’s theory of general relativity.

The idea that the macro world is made of tiny irreducible elements is the idea behind the ancient Greek concept of the Atom and later eloquently described by Leibniz in Modanology.  At the time we may not have known what the “Atoms” or the “Monads” of nature were but we’re convinced, by empirical evidence, that we can reduce complex systems to basic elements with concrete properties.  This is the foundation of reductionism and is a guiding philosophical principle for many scientists.  We were on the fence until the results of the Gold Foil experiment and Rutherford’s interpretation were available in c1908-1909.  We now had a picture of the atom as being mostly empty space, consisting of a very small positively charged “hard-core” called the nucleus surrounded by tiny (point-like) negatively charged satellites we call electrons.

The truly fascinating thing is that the entire periodic table of the elements (all of Chemistry) is built from three elementary objects, the electron, the proton and the neutron.  Protons and neutrons couple together to make the nucleus and the electrons fly around that nucleus.  It’s beautiful and a triumph for reductionism and the idea that there are irreducible elementary building blocks that make up the universe.  More than 100 distinct chemicals with unique electromagnetic properties are described by three (actually at the level of chemistry only the electron and proton are important) simple objects.  It turns out that not all of these elementary building blocks are irreducible.  Based on all evidence to date the electron is irreducible but the neutron and proton are made of quarks which, to the best of our ability to probe, are irreducible.  In the world of high energy particle physics we believe we’ve seen it all, quarks and leptons are irreducible and the properties of nuclear physics, massive particle decay, and all of chemistry can be explained by this simple model.

So let’s take a break for a second, there is a lot of history here, a lot of ideas being thrown around, and conclusions being drawn.  First, the idea behind an irreducible element of nature is an abstraction that doesn’t necessarily mandate a particle view of nature.  The notion of a particle is independent of the notion of an irreducible element.  When we think of particles we naturally think of something that can be confined to a finite volume of space, e.g. our hand.  This is a natural idea based on our experience of the macro world, our experience of our own bodies and the boundary defined by our skin.  Everything in our world seems to be capable of being confined to a space, rocks, trees, plates, food, small amounts of water (or scotch).  So why wouldn’t we believe that when we cut any of these objects into smaller and smaller pieces that the pieces wouldn’t be confinable to smaller volumes of space.

It is not necessary for the elementary irreducible components of nature to be confinable  to small volumes but based on our experience why would we think otherwise.  We can, and frequently do, consider these elements or “particles” as a collection of properties that cannot be described in terms of other properties.  When we get to that point you could say we have extracted a quantum of thought.

So what do our experiences tell us (or how do we interpret our experiences) and what do we really experience when we observe a particle?

The results of what we call the Rutherford experiment seem to imply that the elementary irreducible constituents of matter have properties that we attribute to a particle, that is they can be confined to a small or infinitesimal volume of space.  Of course this is relative to the resolution of the probes and/or processes used at the time.  Waves, on the other hand seem to fill space traveling in all directions, deforming around boundaries and obstacles.  A perfect example of this, one that most people have experienced, is ripples on the surface of a body of water created by skipping a stone or other object hitting the surface.

Things got weird during the development of quantum mechanics when we began to see behavior of electrons and nuclei that seemed to imply that there was a wave-like nature to them, e.g. diffraction and tunneling for example.  Keep in mind that this is only weird because we had convinced ourselves that once and for all matter was made of particles.  When a particle diffracts, that is weird.  But one could have taken the position that we were simply wrong to have jumped to the conclusion that matter was made of particles.  In other words, instead of living with this dual nature of matter we could have changed our position on what really constitutes matter.  This is a position I have taken since undergraduate school.  Though I do not offer a set of mathematical equations and/or procedures for making predictions based on this paradigm I offer an alternate description of the nature or matter.  One day I hope to have a description in terms of postulates and equations but QFT may be enough to satisfy this.  The only change required may be a paradigm shift.

My view of all the observations during the golden age of physics is that they imply that the irreducible constituents of matter are fields.  What type of field exactly, I am not sure.  Rather than view a particle as an entity I view a particle as a specific behavior or state, and similarly for waves.  Waves and particles are behaviors that are both supported by (or can be described as) disturbances in a field.

A field is a mapping of some mathematical object into space-time, x.  The object, F, could be a scalar, vector, spinor, group elements, etc, pretty much anything.  We have a field configuration or section of the mapping when we specify F(x).  For example a real scalar field, like acoustic pressure in a fluid, would be described by {p: R⁴ –> R}, taking values from the real number line or some subset of thereof.  The particular behavior of the possible field configurations are governed by a partial differential equation (PDE), linear or non-linear, typically second order but there are examples of higher order PDEs, e.g. stiffness vibrations are governed by a 4-th order linear PDE in the spatial variables.  To describe all possible configurations requires a complete set of basis functions.  When we have a self-adjoint PDE it’s eigen solutions furnish us with such a basis.  However this is not necessary for describing states in a field configuration space in the abstract sense.

Given a field mapped into space-time we can describe waves that fill all of space or some region of space and we can describe spikes in the field which are confined to a finite volume and travel in space-time like a particle.  In other words a field theory supports both paradigms, particle-like behavior and wave-like behavior.  The reader may be thinking that this is already what physicists believe since our description of matter is based on a field theory, QM or QFT.  This is not an accurate view of these theories.  In QM one states a priori that the system being quantized is a classical particle system.  What I am proposing in this essay is the view that the classical system under study is a field theory, not a quantized particle theory.  I am proposing that the interpretation of Rutherford be stated as the components of the atom are amplitude distributions in a field with a very small covariance, or spread, in space.  This type of distribution, if probed non-destructively, would appear to be a particle.  Even though it’s a configuration state of a field this state is confinable to a finite volume of space and so behaves as we expect a particle to behave based on our experiences.  If probed in a destructive manner this field configuration may start to break apart into wave-like disturbances.  Rather than accept a dual nature to the meaning of what matter is we accept a singular meaning for the description of matter and accept that this single description supports dual behavior.  The apparent duality came about in early QM, as I previously stated, when we (perhaps falsely) convinced ourselves that matter was made of dimensionless particles.

I don’t offer any results based on this paradigm, I am primarily trying to point out that some of the decision points in the history of physics could have branched into different views of reality that are just as valid as those we currently hold to be true.  In fact I would state that Rutherford’s conclusion is not iron clad, the gold foil experiment did not prove that no other paradigm of matter was valid, merely that the nucleus was not a uniformly distributed smear of jelly (or bread).  There were two extremes being considered and the middle road was lost.

To consider the irreducible constituents of matter to be something other than a particle is not new in modern physics, or in ancient Greek philosophy.  We are taught that Aristotle promoted a paradigm that matter was made of a continuum and that Democritus promoted a paradigm that matter was made of irreducible atoms.  This statement does not do justice to either philosopher, I merely want the reader to know that this is a very old competition.  In the 20th century when we were trying to understand the behavior of the nucleon the string paradigm was born.  Though the string view of a nucleon died out theoretical physicists quantized free moving strings and the result was interesting.  String theory takes the position that the irreducible elementary constituents of matter, the classical system we are quantizing, are strings.  The specific particle states we experience in our world are different vibration modes of these strings, which move about in a space-time of more than 4 dimensions.  Again, I’m not doing justice to string theory but the point of bringing this up is to illustrate that the particle paradigm has been challenged in modern times.  My view is different in that I don’t restrict these irreducible elements to be 1-dimensional objects but fields which could be n-dimensional, or even have non-integer dimensionality.

There is another important difference between string theory and other theories.  When we quantize a system we are taught to start with the classical system, that which our senses tell us exists and is supported by macroscopic evidence, i.e. we are taught that quantum mechanics is a procedure you apply to a classical system, a system you already know.  Then the quantum version of that system accounts for the strange and spooky behavior that we have all been taught to think of as “sexy”.  Then, miracle of miracles, take the large N limit of the quantum results and you recover the very classical system you started with, along with small quantum corrections which will approach zero in the limit as N goes to infinity.  String theory does not seem to contain this circular logic, which quite frankly is disturbing in QM and refreshing in string theory.  What we are basically saying in string theory is that we don’t know what the elementary constituents of matter are based on our senses, we can’t since we are classical entities!  So why not just quantize anything and see what we get.  If it looks like it matches data it is a viable theory, if not scrap it.  Although it may seem a bit mystical to those of us who follow reductionism and empiricism this is the legacy of quantum mechanics.  We do not have direct experience of the quantum world, we experience tables and chairs and cathode ray tubes.  But we don’t experience the electron as an individual entity.  In fact I would go so far as to say the whole particle paradigm was a kind of trick played on humanity.  We extrapolated a macro paradigm to the small scale world.  We didn’t really have a reason to do this other than it was a simple thing to do, and there is some virtue to that.  But once we did it we were reluctant or unwilling to undo it.  When nature gave us hints that our decision could have been wrong we would have rather believed that nature was spooky than that our previous decision was wrong.  String theory sets an important and impressive precedent (even though I personally didn’t like string theory in graduate school the approach is very important to the development of physics and the philosophy of science) in that it eliminated the need to start with a system that was available to our senses as a mathematical model of nature.  If quantum mechanics is a flawless theory then all that matters at the end of the day is what the large N limit looks like.  This is where we live and the large N limit is the foundation of all our collective experience.

There is one criticism I get from well educated scientists.  “What about dispersion?”  To which I have to say, how is that connected to this discussion?  The comment is a red herring in a sense and I’ll explain why.  First of all these critics will not state how dispersion hurts the paradigm, so I am left to guess.  If the issue is that dispersion would cause the confined distribution (like a Delta function) to spread, I have a couple of things to say about this.  First our experience tells us that things fall apart, particles (oops, I used that dirty word, I mean particle-like states) decay into smaller particles so if the existence of dispersion leads to an eventual spreading of the particle like state this is not necessarily a bad feature of the theory.  Second, dispersion is caused by the specific frequency-wave number relation which is determined by the linear PDE which describes the evolution of the field state.  It is well-known in the standard model that many “particles” do not exhibit dispersion, some do and some do not.  The mass of elementary particles is not an intrinsic property of said particle any more but due to a coupling of spinors and gauge fields to the Higgs field.  Furthermore, the correct description of these particle-like states within the field may be governed by a non-linear PDE which supports solitons that do not change shape as they evolve.  The idea being presented here is not that particle like states are eternal, they don’t need to be.  The idea is that given a field theoretic picture of classical mechanics one is able to construct states which have all the properties of a particle, including decay.  The converse is never true.  Given a single particle one cannot produce a model of a wave or extended field configuration.  As stated in the first paragraph, from a reductionist point of view, one can make continuum models by sewing together a large number of “particles” with mathematical relationships and take the limit as that number goes to infinity.  In fact we get fairly accurate descriptions of elasticity, classical fluid mechanics, etc by this technique.  Assuming that the elementary constituents of matter are particles we can construct models of extended bodies and this is a very successful feature of that paradigm.  But at the end of the day what we know to be true is that these elementary constituents are not truly particles.

As a final note I would challenge any reader, especially physicists, to form a convincing argument that they have actually “seen” an electron or any other elementary particle.  What I would say is that through the course of many decades, or centuries, of experimental work we have amassed a collection of behaviors and attributes that we have placed into the simplest collection of categories possible.  With these categories and some simple laws governing how one category affects another, e.g. force fields that comprise more categories, we are able to predict new phenomenon and explain old phenomenon in a self consistent manner.  Perhaps this is all science can or should do, and it’s impressive enough.  One could argue that to get wrapped up in whether or not the elementary constituents of matter are particles, strings, or something else is a problem for someone else.  But we need to care about this problem because what we believe to be true about these monads will drive how we model them and how we interpret data.  I am perfectly happy accepting that an “electron” is a collection of experiences and that the equations governing electron motion and interactions reflect the patterns present in those collected experiences.  I cannot say that one of those experiences is confinement of the electron to an infinitesimal region of space.

copyright 2014 David R Bergman