A Children’s Picture-book Introduction to Quantum Field Theory

First of all, don’t panic.

I’m going to try in this post to introduce you to quantum field theory, which is probably the deepest and most intimidating set of ideas in graduate-level theoretical physics.  But I’ll try to make this introduction in the gentlest and most palatable way I can think of: with simple-minded pictures and essentially no math.

To set the stage for this first lesson in quantum field theory, let’s imagine, for a moment, that you are a five-year-old child.  You, the child, are talking to an adult, who is giving you one of your first lessons in science.  Science, says the adult, is mostly a process of figuring out what things are made of.  Everything in the world is made from smaller pieces, and it can be exciting to find out what those pieces are and how they work.  A car, for example, is made from metal pieces that fit together in specially-designed ways.  A mountain is made from layers of rocks that were pushed up from inside the earth.  The earth itself is made from layers of rock and liquid metal surrounded by water and air.

This is an intoxicating idea: everything is made from something.

So you, the five-year-old, start asking audacious and annoying questions. For example:

What are people made of?
People are made of muscles, bones, and organs.
Then what are the organs made of?
Organs are made of cells.
What are cells made of?
Cells are made of organelles.
What are organelles made of?
Organelles are made of proteins.
What are proteins made of?
Proteins are made of amino acids.
What are amino acids made of?
Amino acids are made of atoms.
What are atoms made of?
Atoms are made of protons, neutron, and electrons.
What are electrons made of?
Electrons are made from the electron field.
What is the electron field made of?

And, sadly, here the game must come to an end, eight levels down.  This is the hard limit of our scientific understanding.  To the best of our present ability to perceive and to reason, the universe is made from fields and nothing else, and these fields are not made from any smaller components.

But it’s not quite right to say that fields are the most fundamental thing that we know of in nature.  Because we know something that is in some sense even more basic: we know the rules that these fields have to obey.  Our understanding of how to codify these rules came from a series of truly great triumphs in modern physics.  And the greatest of these triumphs, as I see it, was quantum mechanics.

In this post I want to try and paint a picture of what it means to have a field that respects the laws of quantum mechanics.  In a previous post, I introduced the idea of fields (and, in particular, the all-important electric field) by making an analogy with ripples on a pond or water spraying out from a hose.  These images go surprisingly far in allowing one to understand how fields work, but they are ultimately limited in their correctness because the implied rules that govern them are completely classical.  In order to really understand how nature works at its most basic level, one has to think about a field with quantum rules.


The first step in creating a picture of a field is deciding how to imagine what the field is made of.  Keep in mind, of course, that the following picture is mostly just an artistic device.  The real fundamental fields of nature aren’t really made of physical things (as far as we can tell); physical things are made of them.  But, as is common in science, the analogy is surprisingly instructive.

So let’s imagine, to start with, a ball at the end of a spring.  Like so:


This is the object from which our quantum field will be constructed.  Specifically, the field will be composed of an infinite, space-filling array of these ball-and-springs.


To keep things simple, let’s suppose that, for some reason, all the springs are constrained to bob only up and down, without twisting or bending side-to-side.  In this case the array of springs can be called, using the jargon of physics, a scalar field.  The word “scalar” just means a single number, as opposed to a set or an array of multiple numbers.  So a scalar field is a field whose value at a particular point in space and time is characterized only by a single number.  In this case, that number is the height of the ball at the point in question.  (You may notice that what I described in the previous post was a vector field, since the field at any given point was characterized by a velocity, which has both a magnitude and a direction.)

In the picture above, the array of balls-and-springs is pretty uninteresting: each ball is either stationary or bobs up and down independently of all others.  In order to make this array into a bona fide field, one needs to introduce some kind of coupling between the balls.  So, let’s imagine adding little elastic bands between them:


Now we have something that we can legitimately call a field.  (My quantum field theory book calls it a “mattress”.)  If you disturb this field – say, by tapping on it at a particular location – then it will set off a wave of ball-and-spring oscillations that propagates across the field.  These waves are, in fact, the particles of field theory.  In other words, when we say that there is a particle in the field, we mean that there is a wave of oscillations propagating across it.

These particles (the oscillations of the field) have a number of properties that are probably familiar from the days when you just thought of particles as little points whizzing through empty space.  For example, they have a well-defined propagation velocity, which is related to the weight of each of the balls and the tightness of the springs and elastic bands.  This characteristic velocity is our analog of the “speed of light”.  (More generally, the properties of the springs and masses define the relationship between the particle’s kinetic energy and its propagation velocity, like the \( KE = \frac{1}{2}mv^2 \) of your high school physics class.)  The properties of the springs also define the way in which particles interact with each other.  If two particle-waves run into each other, they can scatter off each other in the same way that normal particles do.

(A technical note: the degree to which the particles in our field scatter upon colliding depends on how “ideal” the springs are.  If the springs are perfectly described by Hooke’s law, which says that the restoring force acting on a given ball is linearly proportional to the spring’s displacement from equilibrium, then there will be no interaction whatsoever.  For a field made of such perfectly Hookean springs, two particle-waves that run into each other will just go right through each other.  But if there is any deviation from Hooke’s law, such that the springs get stiffer as they are stretched or compressed, then the particles will scatter off each other when they encounter one another.)

Finally, the particles of our field clearly exhibit “wave-particle duality” in a way that is easy to see without any philosophical hand-wringing.  That is, our particles by definition are waves, and they can do things like interfere destructively with each other or diffract through a double slit.

All of this is very encouraging, but at this point our fictitious field lacks one very important feature of the real universe: the discreteness of matter.  In the real world, all matter comes in discrete units: single electrons, single photons, single quarks, etc.  But you may notice that for the spring field drawn above, one can make an excitation with completely arbitrary magnitude, by tapping on the field as gently or as violently as one wants.  As a consequence, our (classical) field has no concept of a minimal piece of matter, or a smallest particle, and as such it cannot be a very good analogy to the actual fields of nature.


To fix this problem, we need to consider that the individual constituents of the field – the balls mounted on springs – are themselves subject to the laws of quantum mechanics.

A full accounting of the laws of quantum mechanics can take some time, but for the present pictorial discussion, all you really need to know is that a quantum ball on a spring has two rules that it must follow.  1) It can never stop moving, but instead must be in a constant state of bobbing up and down.  2) The amplitude of the bobbing motion can only take certain discrete values.

oscillator_quantaThis quantization of the ball’s oscillation has two important consequences.  The first consequence is that, if you want to put energy into the field, you must put in at least one quantum.  That is, you must give the field enough energy to kick at least one ball-and-spring into a higher oscillation state.  Arbitrarily light disturbances of the field are no longer allowed. Unlike in the classical case, an extremely light tap on the field will produce literally zero propagating waves.  The field will simply not accept energies below a certain threshold.  Once you tap the field hard enough, however, a particle is created, and this particle can propagate stably through the field.

This discrete unit of energy that the field can accept is what we call the rest mass energy of particles in a field.  It is the fundamental amount of energy that must be added to the field in order to create a particle.  This is, in fact, how to think about Einstein’s famous equation \( E = mc^2 \) in a field theory context.  When we say that a fundamental particle is heavy (large mass \( m \)), it means that a lot of energy has to be put into the field in order to create it.  A light particle, on the other hand, requires only a little bit of energy.

(By the way, this why physicists build huge particle accelerators whenever they want to study exotic heavy particles.  If you want to create something heavy like the Higgs boson, you have to hit the Higgs field with a sufficiently large (and sufficiently concentrated) burst of energy to give the field the necessary one quantum of energy.)

The other big implication of imposing quantum rules on the ball-and-spring motion is that it changes pretty dramatically the meaning of empty space.  Normally, empty space, or vacuum, is defined as the state where no particles are around.  For a classical field, that would be the state where all the ball-and-springs are stationary and the field is flat.  Something like this:

flat_fieldBut in a quantum field, the ball-and-springs can never be stationary: they are always moving, even when no one has added enough energy to the field to create a particle.  This means that what we call vacuum is really a noisy and densely energetic surface:


This random motion (called vacuum fluctuations) has a number of fascinating and eminently noticeable influences on the particles that propagate through the vacuum.  To name a few, it gives rise to the Casimir effect (an attraction between parallel surfaces, caused by vacuum fluctuations pushing them together) and the Lamb shift (a shift in the energy of atomic orbits, caused by the electron getting buffeted by the vacuum).

In the jargon of field theory, physicists often say that “virtual particles” can briefly and spontaneously appear from the vacuum and then disappear again, even when no one has put enough energy into the field to create a real particle.  But what they really mean is that the vacuum itself has random and indelible fluctuations, and sometimes their influence can be felt by the way they kick around real particles.

That, in essence, is a quantum field: the stuff out of which everything is made.  It’s a boiling sea of random fluctuations, on top of which you can create quantized propagating waves that we call particles.

I only wish, as a primarily visual thinker, that the usual introduction to quantum field theory didn’t look quite so much like this.  Because behind the equations of QFT there really is a tremendous amount of imagination, and a great deal of wonder.

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About Brian Skinner

Brian Skinner is a physicist who specializes in the theory of strongly-interacting many-body systems. He is currently an assistant professor at Ohio State, and he blogs at Gravity and Levity


  1. What does “propagate” mean in this analogy? I’m sitting here at me desk, in what sense are the waves that constitute the particles that make up “me” propagating?

    • For simplicity’s sake, I mostly focused in this article on describing single, free particles and where they come from. Such free particles “propagate” through the vacuum in just the same way the ripples propagate across a pond — hence my use of the word.

      But the reason that large, complicated things like yourself can exist is that the particles (waves) that comprise you are not free. They exist in bound states of multiple particles that are held together by inter-particle forces. These forces bind the particles, which would otherwise freely propagate in straight lines, into stable orbits around each other. The middle school picture of these stable orbits is something like a planet orbiting around a sun. But the grown-up picture looks much more like a standing wave (e.g., http://www.acs.psu.edu/drussell/Demos/MembraneCircle/Circle.html ). Such standing waves are made of the same stuff as propagating waves, but they are just pulled into confined spaces by external forces.

      But still, it is definitely correct to think that all the particles that make up you are continually in motion. They have just found a complicated way of orbiting around each other that makes it look, to someone who can only see at a vastly different scale, like you are standing still.

      • Thanks for the reply. I have many more questions but it’s time for the waves that constitute the particles that make up me to interact with the waves the constitute the particles that make up my bed. But I’ll be back for more.

  2. Shari Taylor says

    Thank you for that clear and concise explanation. If you have time I would love a short explanation on the double-slit experiment. I’m not a physicist, and the children’s picture-book version is so helpful for lay people interested in this stuff.

    • I wrote something about the double-slit experiment a while back, which you might find helpful: https://gravityandlevity.wordpress.com/2009/05/10/surfing-and-the-double-slit-experiment/

      The basic problem is like this: you fire an electron toward a screen with two small slits in it, and you put a sensitive photographic film behind the screen to record where the electron lands after it passes through the slits. What you see after developing the film is a complicated wave-like pattern of places that the electron landed. This would be impossible to imagine if the electron were a bullet-like point flying through the air: the developed film would just show the “shadow” of the two slits projected onto the film. But if the electron is, in a fundamental way, a wave, then the wavefront that is the electron can easily pass through, and diffract around, both slits at once, in the same way that a water wave would. In doing so it produces the necessary wavy pattern on the screen.

  3. Fantastic explanations / visualizations!

    When mass gets turned in to energy do you have then the energy of Higgs field being transferred to other fields?

    Or to get more basic- how do the various fields interact with one another?

    • Thank you!

      Yes, energy can be transferred between fields. Whenever a particle is created or destroyed, what that means is that its energy is transferred from one field to another one.

      I’m not sure I have a good answer for the question “how do the various fields interact with one another”. For a physicist, the answer to that question is usually “in the mathematically simplest way that doesn’t violate any of the fundamental symmetries of nature.” But if you want at least one visual example, you can read my previous post: https://www.ribbonfarm.com/2015/06/23/where-do-electric-forces-come-from/ . That post is precisely describing how one field (the electron field) interacts with another (the photon/electromagnetic field). In that particular case the interaction consists of the electron deforming the photon field around it in a way that makes the electron look like a “hose” that is continually spraying “virtual photons” all around it.

  4. I still don’t understand the whole of reality,
    but your article gave me my daily recommended dosage of knowledge towards that goal.

    When you were using the “mattress” analogy, I also kept on thinking of those PinPression toys, albeit my analogy is not as good. But it makes me think of a hidden hand/energy pushing/creating everything.

    Hey, could you in any way elaborate on the topic of your article and how it would relate (if at all) to the collapsing of the wave function, or how the observer plays a role in measurements? Does it have something to do with light?

    Thanks, enjoyed your article very much.

    • You’re asking hard questions, but I think that the best (short) answer I can give is: no, I can’t use QFT to explain to you where wave function collapse comes from. In QFT, the basic laws of quantum mechanics, including something like “wave function collapse”, are imposed from the beginning. The fact that particles in the quantum field can exist in superpositions and undergo collapse is purely a consequence of my assumption that the individual ball-and-springs can exist in superpositions and undergo collapse.

      In that sense, QFT does not help to explain where the basic laws of quantum mechanics come from. It only uses them as inputs to explain how quantum particles behave. It is, as Freeman Dyson said, “descriptive and not explanatory”.

      I should also say that the question of “what does wave function collapse mean and where does it come from?” is a messy one. If you ask it to 100 physicists, 90 of them will tell you it’s a silly question that they don’t want to talk about, and the other 10 will say things that are either confusing or conflicting. I personally don’t feel confident taking a stand with either the 90 or (any of) the 10. The one thing that all 100 will agree on, though, is that wave function collapse has nothing to do with human consciousness or human observers. (Okay, at least 99 of them will agree.)

  5. You lost me at field, sorry, my eyes just sort of glazed over.

    • That’s too bad, although I would have thought that the word “quantum” immediately before it would have been much worse!

  6. Leonard Cuff says

    Superb! Thanks. Here is my question: Although it doesn’t explain the wave function collapse, is there a way to describe what a wave function collapse would look like in the context of your analogies? Even if at some moment in time you have to say “The system magically goes from being like ‘this’ to like ‘other’, without us knowing anything about how that happens?

    • In the language of the analogy in this post, a particle’s wave function is the pattern of spring oscillations that comprise it: how big the “ripple” is, which way it’s traveling, etc. A quantum superposition is when two or more such patterns, together, have one particle’s worth of energy. Think, for example, about some piece of the field having two simultaneous, superimposed standing waves (perhaps of the sort that I linked to above: http://www.acs.psu.edu/drussell/Demos/MembraneCircle/Circle.html ).

      A wavefunction collapse would be when some interaction with a large external thing (such as the measurement apparatus of a human observer) causes all but one of those oscillation modes to disappear.

  7. Aaron Thornton says

    Hi Brian,
    Thankyou, I love your work and writing.
    For me it is Modern science slowly understanding the science of old. Yoga.

    For your next blog, if enough work has been done, could you extend this concept of the Higgs field to include the observer effect? Could it be that our thoughts could be influencing this field too? And that vacuum fluctuations are not just some inherent property of this field, but are the total sum of our busy minds interacting with this field? Or is that getting too far ahead for now.

    • I’m afraid that you’re likely to be disappointed. Firstly, because I am a lowly condensed matter physicist (as opposed to a lofty high-energy physicist), and I am probably not sufficiently well educated about the Higgs field to write something adequate. But also because there is really no viable interpretation of quantum mechanics (or, at least, no useful interpretation of quantum mechanics) in which the human mind or consciousness plays any significant role. The language of “observers” and “observations” in quantum mechanics is probably a bit antiquated, and these days we tend to view the evolution of quantum systems in much less human-centric terms.

      To belabor the point a little bit (in a way that I hope doesn’t sound harsh): the best scientific description we have at the moment is our “thoughts” are really just a sequence of electrical impulses in a complex network of neurons. These impulses are themselves made of an intricate set of oscillations in various quantum fields. In that sense, our thoughts (like everything else) are made of quantum fields; they are not in any way outside of them.

      • Aaron Thornton says

        No disappointment, only gratitude to be able to converse at such an abstract level and to someone with an open enough mind to at least entertain others viewpoints without offence. I really love reading these comments and replies. I will take this opportunity though to expand on my previous questions. It does really involve the ability to look through the other end of the telescope though, I hope you don’t mind. What I was hinting at actually agrees, that thoughts are within a field. In fact they would also be energy, just a patterns of it. And what we call ‘our’ thoughts or ideas would be patterns of energy in a localised field. This slight adjustment in thinking gives rise to a paradigm type construct. That is that mind, or consciousness, is the field and matter(the body) is a condensed form of the energy in that localised field. My personal analogy of the how our brain functions it that is like a TV antenna: It has the ability to receive the subtle fluctuations in our localised field(mind) and transform that into electrical impulses that our body can make use of. Thus neurological activity in the brain can be called a thought, or idea, but its actual origin and form are slightly different. If your still with me in this topsy turvy world here are a few thing to ponder:
        – A field can be infinitely expanded, just like our mind.
        – If you close your eyes and move your awareness through your body, what is actually happening? Is something moving through our physical bodies or is it traveling through the field that surrounds it.
        – If, at the most fundamental level, the quantum field is mind and a whole universe can be born out of fluctuations in this field. It does beg the question… do we apply the three letter name we attribute to the creator of the universe?
        Maybe this last question is the reason we don’t want to turn the telescope around?

  8. Thank you for this and your other article on fields. Like you say, for visual learners this treatment of the material is a tremendous help. I hope you write more!

    • Thanks for the kind words. The plan is for me to write three more posts for ribbonfarm on the same theme. These should appear over the course of the next 3-4 months.

  9. Hi Brian, a couple things in this analogy are unclear to me.

    How is the particle represented as ripples when ripples expand outward in concentric circles? An electron doesn’t expand. How is the movement of the particle (let’s say across this mattress) represented? I could see the oscillation of the ball generating waves, but here I am viewing the oscillation as the particle and the wave as the field of force, when you stated a couple times that the wave was a particle. The wave resulting from the disturbance on the mattress is expanding in a concentric circle.

    Also, how is a ripple supposed to propagate on this quantum mattress at all? Let’s say I disturb a ball with 1 packet of energy. I can’t manufacture energy out of nowhere, so that disturbance fans out to the 4 neighboring balls in 1/4 parts. Their own oscillations would only be affected by at least 1 packet, so my initial disturbance dies on the vine?

    I don’t want to overextend a metaphor, but think if the metaphor is to be of use to me it would have to answer these questions. Thanks.

    • Hi Erik,

      My use of the word “ripple” was probably a little unclear. I really used that word synonymously with “wave”: any sort of collective oscillation of the field, and not necessarily the kind that expands outward in concentric circles as when you drop a rock in a pond. Of course, that concentric circle kind of wave certainly exists, and an electron can occupy such a state — it just means that the region over which the electron is confined gets increasingly large. But in general, an electron can be described by many different-looking kinds of waves.

      The second part of your question is harder, and it comes down to the difference between classical systems and quantum ones. In a quantum system, two or more springs can share one quantum of energy between them, even though it isn’t possible for a single isolated spring to have a fraction of one quantum. In the language of physics, we say that the two springs are in a quantum superposition of the first spring oscillating and the second spring oscillating.

      What really happens when you disturb some isolated part of the field is that the quantum superposition expands to include an ever-larger number of springs. It’s not quite right to say that each individual spring has, by itself, some fraction of the total energy (as it would for a classical field). It’s only that, together, they share the quantum of energy that is the electron.

      It seems likely to me that this explanation won’t feel terribly satisfying to you. If that’s the case, then all I can say is that quantum mechanics rarely feels very satisfying, and it’s never completely clear whether this is a failing of our understanding or a failure of our feeling.

      • I had a little realization here, not only does that wave propagate across the surface of the pond, and as it’s the most prominent manifestation of the imact, we stop at that visual. The wave also propagates down into the water, and also into the air in 3 dimensions – not only in 3 dimensions, but in 4, if you want to deal with the time it takes, not just the space.

        So as all models fall short, we sometimes get stuck at insisting the model is the reality.

        We are just poor 4th dimensional beings, after all.

        • This visualization is meant to be for two dimensions, but thats the case in most visualizations since it’s simpler to visualize 2 dimensions without losing generality. There’s no reason *that I see* that this couldn’t work in 4 dimensions as well, the problem being that one would need to visualize 4 dimensions with ideal springs attached, and the assortment of strings and the propogation of the wave. If you really want to imagine it, its something like a 4d sphere expanding outwards *assuming a spherical wave*.

          • I meant no criticism that _you_ see_ it that way. Our preferred perceptual systems often limit our understanding. Our models are only models.

            I just had that little enlightening vision, that’s all.

            And I imagined that single event moving into and affecting every dimension around it, through it, every dimension that it pervades and that pervades it. So the action affects all dimensions that it connects to. As many dimensions as we want to imagine. Perhaps that action expresses, travelling through some other manifold of space-time, as some particle manifest in a dimension we perceive.

            The whole plate of jelly wiggles, doesn’t it. And please don’t extrapolate some kind of NewAgeyWoo. I know it sounds a bit goofy.

  10. Dear Brian:

    Diving deep at a deeper level, it is true that we are made out of the quantum fluctuation arising from the vacuum field.

    As you have explained that in the jargon of field theory, the virtual particles can briefly and spontaneously appear from the vacuum and then disappear again.

    However, some of these particles can grow to give rise to universes and other disappear.

    As I perceive the following:
    1. Vacuum Field fluctuations occur only because the field itself is stretching. Which means that at a point they break and hence create a virtual particle. In other words these breakage in the vacuum field gives rise to causes for big-bang and hence the reason for origin of a new Universe.

    2. A Universe itself comes from the fluctuation of the Vacuum Field so it stays in the Vacuum Field.

    3. As we know about the acceleration of the Universe from Hubble’s Experiments and Super Nova / Redshift observations. A red-shift and acceleration of Universe means that the vacuum field itself is stretching within our Universe.

    4. This further implies that a Universe will give rise to another universe and this is an infinite process, however, with an end at the end…

    5. Also quantum field gives rise to different fluctuations which in turn gives rise to different elementary parts (matter field and force field). This means that each such variation and combination produces a different universe all together but all arising from the same vacuum field fluctuation.

    6. As we say that to have an effect comes from a cause. And it is true that cause lives in the effect. Simple example, a Au is a cause and ring is the effect. But Au lives in the ring being its cause.

    7. As the subject matter becomes more abstract, however, to add further, the origin of all the forces of the universe, including vacuum field, Nuclear, Gravity, EM, etc. must reside in the Universe itself….The fun part is to continue the voyage of ever evolving mind ( a collective human mind)…


    • Hi Mohan,

      Thanks for the comment. There’s a lot going on in it, but let me respond to just one or two points within it.

      First, you’re right that, in principle, all sorts of particles can arise from fluctuations in the quantum field. As I understand, it is a very real research question whether something like our universe could arise just from a random (and extremely rare) vacuum fluctuation. The problems with this idea are that: 1) the probabilities involved are probably smaller than any number you have tried to imagine before, and 2) particles arising from random vacuum fluctuations always appear together with antiparticles. So in order to explain the observed universe as a random fluctuation, one would also have to explain where all the antiparticles have gone.

      Of course, I should acknowledge that I am not really trained in cosmology, so I’m not really the right person to answer questions like “where do universes come from?”.

      I should point out, though, that it’s not correct to say that “vacuum fluctuations occur only because the field itself is stretching”. Vacuum fluctuations are an unavoidable consequence of quantum mechanics. Whether the field is expanding, contracting, or remaining static, they will always be there.

      • Mohan Bhan says

        Dear Brian:
        Thank you for the reply.
        1. Let me rephrase and as correctly pointed out by you that the vacuum fluctuations are an unavoidable consequence of QM. Irrespective of the direction of the field, the vacuum fluctuations will always be there. An accelerating field is stretching, a decelerating field is contracting (collapsing) or a balanced field is static and every fluctuation will yield into a different type or variety of particles.
        2. However, the origin of the vacuum field itself is the vacuum field (birth of vacuum field comes from the vacuum field itself), and it accelerates in a multi-directional way.. I can only correlate this to the observable expanding universe, which intern satisfies this condition..
        3. To support Point 2 above, the concept of big band further explains the argument that ever since its (universe) incubation, the ever expanding universe is the result of the ever expanding vacuum field (or accelerating dark energy)..
        4. The mystery of matter (particle) and antimatter (antiparticle), and their annihilation and somehow resulting into one matter, which consequently giving rise to the reason for the birth of all visible matter is an interesting fact of our life. Surely, it is further a thesis topic for many many young minds.
        5. Our Universe is an ever evolving universe, it goes from one state to another state or from one type of universe to another type, Our conscious is also an evolution from the same Universe. Weather our conscious is there are not, the universe will keep on evolving.
        6. In order to explain Point#4, as I correlate to the observable fact that when a particle is generated, it also creates an antiparticle and they both travel in the opposite directions (like electron and hole in a semiconductor material). So at the time of collision when all matter and antimatter cancel each other then where does the last matter come from.
        7. Further, it appears that a positive vacuum fluctuation will also generate a negative vacuum fluctuation. So as we know that the universe is expanding (anti-gravity or dark energy force) then there must exist a secondary force (+ve gravity or anti-dark energy).
        8. Since equal amount of particle and antiparticle are generated and upon meeting they annihilate and cancel each other so this reaction should involve attracting another matter from the nearest event.
        9. Yes the probability involved for such occurrences can be said 1 in billion… Our Universe seems also such a probability…

        Happy weekend…

  11. Hello, I would like to ask your opinion on the increasing expansion rate of the universe. Could this expansion be causing the vibrations needed to put the energy in the springs causing them to bounce. Sort of like pulling on a rubber band would transfer the energy needed to stretch it to the resistance created.
    I know this is a simplistic way of looking at the situation but the energy has to come from somewhere. Ex nihilo?

    • The responsible thing for me to do here is to acknowledge my ignorance about the question of dark energy, and politely decline to answer.

      But my irresponsible reply is this. My rudimentary understanding is that one of the primary candidate explanations for dark energy (which is responsible for the accelerating expansion of the universe) is that it comes from the vacuum fluctuations of some kind of scalar field. If this explanation is correct, then it would be similar to what I am describing in this post: random fluctuations of the quantum “mattress” endow empty space with a huge energy density, and this energy density leads to a universe with accelerating expansion.

  12. Amy Charles says

    Brian, thank you very much for this lucid and (I hope) helpful explanation. As I read the bit about vacuum fluctuations, I’m reminded of the 19th-c perplexity over the ether, and think — well, maybe they didn’t get it wrong, but for reasons and in ways they couldn’t possibly have been thinking of. But even so, the idea of “really nothing” seems to be gone — am I getting that right?

    • You’re definitely right that we have come back to the idea of “empty space isn’t really empty”. But now we know not to insist that the “stuff” in empty space is some kind of mechanical object. For a working physicist, it’s best to think of the vacuum, and all its fields, as a mathematical object that obeys the various fundamental laws that we have figured out: quantum mechanics and relatively, in particular. It just happens that this mathematical object behaves a lot like the “mattress” that I outlined in this post.

      • Amy Charles says

        Thank you! (I bet this post is giving you a lot of followup work and explainy-practice.) Funny, though, my first reaction to your reply is dismay at how it knocks a hole in high-modernist existential terror, which depends pretty heavily on the existence of nothing. I mean you put Beckett in a corner right there. Well, you can’t have everything. Or nothing. I’ll stop now.

      • Can you please shed some light on how an antimatter particle would be represented in the mattress analogy.
        Also, how a matter anti-matter particle annihilation would appear?
        And what the creation of energy be in this case, a propagation of another wave in another field? Or is “pure” energy somehow different?

        Thank you.

        • Unfortunately, the field that I drew here doesn’t really have antiparticles. Or, to say it another way, there is only one kind of excitation, and thus no distinction between particles and antiparticles. Two waves, properly constructed, can cancel each other out by interfering destructively, in the same way that a particle and antiparticle can. So particles in this field are their own antiparticles.

          To have a field with particles and antiparticles that are separate things, you need to make a field that has two different kinds of “defects”. For example, in my previous post (https://www.ribbonfarm.com/2015/06/23/where-do-electric-forces-come-from/) there were “sources” and “sinks”, and these have opposite effects on the field that can cancel each other out when they come together. I’ll give a better example of this in a future post.

          As you alluded to, when a particle and antiparticle collide, the corresponding excitations are removed from the field, and all that energy has to go somewhere. In general, it is released into one of the other fields in nature. (Recall that nature has many fields, coexisting in all of space.) When that energy is absorbed by some other field, new particles are created in it.

  13. At the risk of inappropriately extending the analogy:
    My thought when you talked about coupling the balls to allow wave propagation is that this is the place that quantum constraints would apply. In other words, rather than an elastic bond, some sort of ratchet that only permits fixed quanta of energy to pass from one ball to another. In that case, each ball could have any vibration (including zero), representing vacuum fluctuations, but only fixed quanta would actually propagate. Smaller amounts would dissipate or add with further stimulus from whatever invisible hand (leakage to/from other fields?) may be involved.
    This is only a minor variation from your model and my physics is far enough in history that I don’t know if it better represents reality – is this how virtual particles actually work? Or is just pushing the image a little too far.

    • There are sort of three problems with your alternate picture of a quantum field. The first is that it would allow any arbitrarily small amount of energy to be added to the field, as long as it remained in only one spring.

      The second problem is that, in your field, if you added one quantum of energy (one particle) it would always be localized at only one spring at a time. It could perhaps hop from one spring to another, but it wouldn’t be possible to build some big, wave-like collective excitation. What we know, on the other hand, is that particles can occupy large wave functions that extend over large regions of space. These large regions together collectively share one quantum of energy in a quantum superposition. (See my comment above to Erik).

      Finally, your field would be capable of complete quiet: in the absence of any particles, it would be placid and not fluctuating. What we know, on the other hand, is that there are always intense quantum fluctuations in a field. The statistics of these fluctuations are very accurately understood and measured, so it’s pretty clear that they are always present and always with a particular amplitude.

  14. You describe an infinite array of balls on springs connected with elastic bands, a “mattress”, as an analogy to a “quantum field”. The mattress is not relativistic since the wave speeds are relative to the mattress. Is the quantum field envisioned by physicists also non-relativistic?

    • No, any actual field (at least, any fundamental field) must respect the laws of relativity. And you’re right that the field that I drew definitely has a preferred reference frame. The problem is that I have no idea how to draw a picture of a field that is Lorentz-invariant!

  15. Mohan Bhan says

    Dear Brian:
    Thank you for the reply.
    1. Let me rephrase and as correctly pointed out by you that the vacuum fluctuations are an unavoidable consequence of QM. Irrespective of the direction of the field, the vacuum fluctuations will always be there. An accelerating field is stretching, a decelerating field is contracting (collapsing) or a balanced field is static and every fluctuation will yield into a different type or variety of particles.
    2. However, the origin of the vacuum field itself is the vacuum field (birth of vacuum field comes from the vacuum field itself), and it accelerates in a multi-directional way.. I can only correlate this to the observable expanding universe, which intern satisfies this condition..
    3. To support Point 2 above, the concept of big band further explains the argument that ever since its (universe) incubation, the ever expanding universe is the result of the ever expanding vacuum field (or accelerating dark energy)..
    4. The mystery of matter (particle) and antimatter (antiparticle), and their annihilation and somehow resulting into one matter, which consequently giving rise to the reason for the birth of all visible matter is an interesting fact of our life. Surely, it is further a thesis topic for many many young minds.
    5. Our Universe is an ever evolving universe, it goes from one state to another state or from one type of universe to another type, Our conscious is also an evolution from the same Universe. Weather our conscious is there are not, the universe will keep on evolving.
    6. In order to explain Point#4, as I correlate to the observable fact that when a particle is generated, it also creates an antiparticle and they both travel in the opposite directions (like electron and hole in a semiconductor material). So at the time of collision when all matter and antimatter cancel each other then where does the last matter come from.
    7. Further, it appears that a positive vacuum fluctuation will also generate a negative vacuum fluctuation. So as we know that the universe is expanding (anti-gravity or dark energy force) then there must exist a secondary force (+ve gravity or anti-dark energy).
    8. Since equal amount of particle and antiparticle are generated and upon meeting they annihilate and cancel each other so this reaction should involve attracting another matter from the nearest event.
    9. Yes the probability involved for such occurrences can be said 1 in billion… Our Universe seems also such a probability…


  16. I get eerie sense reading this article. Because of this https://en.wikipedia.org/wiki/Indra%27s_net

  17. “First of all, don’t panic. I’m going to try in this post to introduce you to quantum field theory, which is probably the deepest and most intimidating set of ideas in graduate-level theoretical physics.”

    I guess most people will panic, reading that. It’s a great post! But how about not starting with “this is incredibly difficult, scary stuff.”

    • Thanks! I guess I assumed that most people would be intimidated just by the “quantum field theory” in the title, as I’m sure I was even during graduate school. So I felt a need to address the elephant in the room before beginning. Perhaps this was unnecessarily anxiety-inducing.

      But, given the number of comments here, I guess I was lucky enough to find some intrepid readers who weren’t scared away.

      • Amy Charles says

        Oh, I always read things that start with “don’t panic”; I assume a towel is involved somewhere.

  18. Ken Abbott says

    Very nice! But where’s spin?
    Here a piece of topology that gives spin and perhaps you can blend into your model..

    • Nice analogy!

      You’re right; what I described is a scalar field, and scalar fields don’t have any concept of spin. In the future I’ll write some more posts about quantum fields, and we’ll see if I can come up with a nice visual analogy for the vector fields that do support spin.

  19. This just crossed my radar: Hawking Tries to Find Black Hole’s Emergency Exit.

    A relevant quote:

    “The complication that black holes can evaporate is a consequence of the weirdly wild quantum world. In a vacuum, quantum physicists predict that pairs of “virtual particles” are constantly popping in and out of existence. Therefore, a “true vacuum” is never truly empty; it’s actually buzzing with pairs of quantum particles that pop into existence and then annihilate within a minuscule time frame. Usually this continues without a hitch — particles appear out of the vacuum, they annihilate with their partner and the universe goes about its business.

    But if you add a black hole to the mix, things get a bit crazy.

    As we’ve already discussed, the black hole’s event horizon is the point of no escape. So what happens when a pair of virtual particles pop into existence at the horizon and only one falls in? Well, the second “virtual” particle, which has lost its annihilation partner, is ejected away from the black hole and becomes “real.” In the process of becoming a real particle, it steals a tiny amount of mass from the black hole (quantum mechanics never pretended to make logical sense). With this process happening all the time at the event horizon, the black hole starts to physically lose mass — it is evaporating via Hawking radiation.”

  20. Heather Folsom says

    Fields are my favorite concept in physics. I am also a visual person, and while I find mathematics wonderfully explanatory, I am not satisfied until I have a picture or a little video in my mind, often one that requires continual refining.

    Do you have an idea for a model of entanglement?

    By the way, condensed matter physics and physicists–not lowly at all!

    • I have sort of a way of thinking about entanglement, but I really haven’t subjected it to enough critical scrutiny to be confident writing about it publicly. One of these days, though.

  21. A few questions;

    1) Is it apt to think of anti-matter on this construction by thinking of a parallel “mattress” that is plucked at a time T’ after a particle P is created such that T’ is when the particle P has propagated 1/2 wavelength. Then the “annihilation” of P occurs when you touch/combine these two parallel matrices?

    2) To make an analogy to your analogy.. consider each individual spring as a pixel on a screen. By having distinct springs or pixels, wouldn’t that imply that there couldn’t be a true circle, or any sort of curve? Like how if you zoom in enough on a curve on a computer, you see it’s really an approximation of squares that look like a curve. The distribution for your springs is square shaped, so the wave propagation would be based off that, but I don’t see how any non-dense set of springs that form a mattress wouldn’t have an analogous (albeit different shaped) lack of curvature? I don’t know much about fields, so I’m essentially wondering if they’re considered everywhere dense, or at least having no distinct points.
    As a bit of a follow up, or at least along a similar vein, doesn’t the finite number of connections between any point and the surrounding points necessitate this same sort of pixilization?
    Just briefly pondering those two scenario’s, I don’t think that with one, and not the other, you could have all possible distinct waves.

    3) If you consider the propagation of a touch to be a particle, I’m confused by what happens when you have two of these waves that originate in different locations interact. For simplicity, assuming they’re both circular waves. Neither wave would be completely destroyed or amplified along its entire length at a single instant in time. The wave’s can still be called distinct, and individual pieces of matter, but I’m confused not so much about the wave itself, but about the points of constructive or destructive interference.

    • Hi Evan,

      1) Matter and anti-matter are properly thought of as different kinds of excitations in the same field. The example of a field that I gave here doesn’t have both matter and anti-matter. It only has one kind of excitation: the “ripple”.

      In the future I’ll try to give a different picture of a field that has both matter and antimatter excitations. But here’s a simple image that comes to mind at the moment. Imagine a one-dimensional “field” that looks like a zipped-up zipper. You can make a “particle” in this field by having a defect in the zipping. For example, you could accidentally zip two teeth from the left side in between two adjacent teeth on the right side. This defect would cause a lot of stress in the zipper, but it could still move up and down the chain. Notice, though, that you could also make an opposite kind of defect, where you skip a tooth on the left side, thereby having two adjacent teeth on the right side with nothing between them. This defect would also create a lot of stress, and could move up and down the chain. When the two defects came together, though, they would “annihilate” each other. The zipper would go back to looking normal, and all of that stress would be released.

      The point, though, is that in order to have particles and antiparticles, your field needs to have opposite kinds of defects. The mattress doesn’t.

      2) No one knows if there is a fundamental “pixel size” in the universe. In my analogy there would be, since the springs have a certain spacing between them, but we have no evidence for any nonzero spacing in the real universe. There is sort of a candidate possibility, which you can read about here if you like: https://gravityandlevity.wordpress.com/2015/04/11/how-big-is-an-electron/ (under “option #5”)

      3) Two waves do not, in general, destroy each other just by running through each other. But they do create points of constructive and destructive interference, which correspond to places in space where the particles have a larger or smaller influence. (You can say that these are places where an experiment would be more or less likely to detect particles.)

  22. Brian,

    First of all, thanks for taking the time to write this post and respond to comments. I’m a visual thinker, which I’ve concluded is not a benefit when thinking of all things quantum, but your post helps me to some degree.

    I have a huge amount of questions, but I’ll start with the simplest, starting with the assumption that there is a ball-and-spring mattress that is entirely at rest.

    1) How, physically, is energy put into the system? You mention that a ball gets “tapped” — what does that mean, physically?

    2) Imagine that a single ball is tapped with the energy of 1 quanta. What will happen next? The surrounding balls, attached by our elastic, will be pulled down with <1 quanta — correct? It seems that the energy of 1 quanta is "trapped" in one ball (and it's surrounding elastic) and cannot propagate.

    3) You mentioned the properties of the field are largely dictated by how Hookean the springs and, I assume, the elastic are — but do not clarify what our assumptions for this model actually are.

    4) What information most succinctly defines a particle? Is it the point origin and amplitude? Is an "electron" simply an oscillation of (making this up) amplitude=5 at point X,Y in the field? How would you define an "electron"?

    I realize these questions may be completely misguided, and if that is the case, I'd certainly appreciate your best attempt that offering me some clarity.

    Thanks again for your time,

  23. Hi Sam,

    Thanks; I’m glad that you found the post at least slightly clarifying.

    1) In the real universe, energy is neither created nor destroyed. It simply moves from one kind of field to another. You have to remember that the universe is filled with many different kinds of fields, all coexisting with each other and all having different kinds of properties. They can interact with each other, and when they do some ripples in one field are transferred to another.

    2) This is pretty similar to the question asked by Erik, above. You might want to check out my answer to him.

    3) I’m not really describing a specific field here. To do that rigorously, I would have to insert a few more equations. But the short answer is that there are all kinds of different fields in nature (essentially, one for each fundamental particle). To continue the imagery of the post, some of these fields have springs/elastics that are more Hookean than others. Usually, the approach of physicists is to write some equations that could describe a field, starting with the simplest possible equation that one can think of. If that equation describes nature well, then you stop. If it doesn’t, then you add the next simplest term and see how that equation looks, and so on. Hooke’s law is the mathematically simplest description of a spring, and you can make a decent field with it. But some of the fields in nature require something more complicated.

    4) This again comes down to a difference between different types of fields. One answer is that if you put in a single quantum of energy, you will get one particle unambiguously. But if your field has many quanta of energy in the field, then sometimes it is less clear. In the picture I drew here, you could imagine making a big disturbance of the field, and this could either have one ripple with large amplitude, or many ripples with small amplitude. (Keeping in mind that no individual wave can have less than one quantum of energy.) In technical language, one could say that the field created here does not conserve particle number (it is a bosonic field).

    But there are some alternate fields where the definition of particles is much clearer, because the “defects” in the field come naturally in discrete units. I’ll explain this better in a future post, but for the moment maybe you can check out my “zipper” example in the reply above to Evan.

  24. Speculative if not confused question: what does the stability of the element Gold ‘AU’ indicate about its Quantam field conditions for it to emerge?

    • A single gold atom is already an intricate composite object, made of many separate excitations of different fields. What we call a gold atom is really 79 excitations of the electron field, 237 excitations of the quark fields, and countless excitations of the photon (electromagnetic) and gluon (strong force) fields. These excitations are doing sort of a dance where they all stick together. And the fact that a gold atom is stable means precisely that such a dance of many excitations is possible.

  25. Hi Brian,

    First of all, thank you for this brilliant article, I’m amazed at how well you are able to get this idea across to people like me, who have only done high school physics. Also the fact that you take the time to reply to everyone’s questions is just fantastic.
    Now for my question, in this theory, can gravity be explained as the distortion of these fundamental fields, or is it something else? (The interaction of a “gravity field” with other fields?).

    Thank you.

    • Thanks for the kind words.

      I should first say that I am no expert in gravity. For the great majority of physical problems (on the smaller-than-a-planet scale) gravity is too weak to have any relevance, and thus I never had to learn about it very deeply.

      That said, my understanding is that the gravitational field is indeed a quantum field just like any of the other fundamental fields. Its (as yet undetected) particle is called the “graviton”, and it is a quantized ripple in the gravitational field in just the same way as I was describing. But it is also true that so far there is no consistent quantum field theory of gravity (despite decades of attempts). I’m not really expert enough to say why this has been so difficult, and whether this comes from some truly fundamental difference between the gravitational fields and the other fundamental fields.

      Sorry for the less-than-satisfying answer!

  26. Just someone interested in science. I loved this post, and it has allowed me to answer my son’s question, “What are electrons made of.”

  27. That’s very profound.They don’t come from aeynhwre. That is; you don’t need some kind of source for them from somewhere else . When new particles are created they have to obey a (surprisingly large) set of conservation laws. Things like (but not limited to);-If the new particle has charge, then you also have to create another particle with the opposite charge.-If you create a particle made of matter, you have to create an anti-matter particle.-Energy is conserved, so you can’t just declare particles into being, you need to spend some energy.What is not conserved is particle number. There are no physical laws preventing new particles from forming, or old particles being destroyed. In fact, it’s an integral part of how the universe works!Some particles are randomly generated more often than others. The probability of a new particle being of some type A is proportional to the number of states that A can be in, which is exactly how you go about maximizing entropy. So, you can write off the selection of new particles as just another attempt by the universe to increase entropy.

  28. first of all, thank you brian for your writeup and q&a. an absolute layman interested in physics, I have the following questions, appreciate your comments –

    1. taking electrical field as an example. a stationary positive charge creates a field with certain field energy, and a stationary negative equal charge creates a field with the same field energy. if we put two equal charges together and cancel each other’s charge, then field goes away. but where do the original energy of both fields go?

    2. where does the field energy come from when all of a sudden we drop an electron in the picture which gives rise from nothing to this electric field? similarly, looking at a stationary charge starting to create a field, as the field is excited from near distance to far away space (ie field is not instantly created in the whole universe by this charge), so the field energy starts to grow. nothing changes here (charged particle stay unmoved), so where the increasing field energy comes from?

    3. when field oscillates, it creates particles – but how are particles making up matter (say an electron) different from particles effecting force (say a photon), using field theory explanation?

    4. how many different fields do we have – is it right to say for each different “force” we have a field? since there are 4 known forces, so we have 4 types of fields?

    5. how do fields of different types interact with each other – eg electrons are products of electro field, but two electrons have gravitational force between themselves too through gravity fields. how to understand the mechanism of such cross-field interactions?

    6. on one hand, we are saying particles are created by field disturbance. on the other, we say fields are excited by the presence particle. so it’s a bit chicken and egg. which comes to being first then?