But What Actually Is A Particle How Quantum Fields Shape Reality
read summary →TITLE: But What Actually Is a Particle? How Quantum Fields Shape Reality CHANNEL: Physics Explained DATE: 2025-06-14 URL: https://youtu.be/KLIS4lq1mBE
Have you ever wondered what a particle really is at the deepest possible level? When we talk about electrons, quarks, and neutrinos, what are we actually talking about? In this video, I want to answer a deceptively simple question. According to our best theory of physics, what is a particle? Now, the answer is surprisingly easy to state. A particle is the smallest possible vibration of a quantum field. So, if all you want is a convenient sound bite, there you have it. But if you’d like to truly understand what this means and in the process discover one of the most astonishing and successful theories in all of physics, quantum field theory, then stick around. This is going to be a wild ride, so buckle up and enjoy the show.
To answer this question, we’re going to need a strategy. First, we’ll learn how to mathematically describe vibrations and waves. Then, we’ll explore the concept of a field, what it is, and how it can ripple and vibrate. Next, we’ll see what happens when we bring quantum mechanics into the picture. And finally, we’ll be ready to understand what a particle truly is according to quantum field theory.
We’ll begin our journey by considering the simplest kind of vibration first studied by Robert Hooke, a mass on a spring. Consider a spring fixed at one end with a mass m attached to the other end. If you then pull on the mass and extend the spring by some amount, which we’re going to call X, then the spring exerts a restoring force F in the opposite direction to the extension. And Hooke’s law tells us that the force is proportional to the displacement but in the opposite direction. And we can turn this into an equation by writing F = -Kx where K is the spring constant. Now if we combine this with Newton’s second law f = ma we can write ma = -kx and if we then rearrange for the acceleration we find that a = -k/m * x and we see that the acceleration is proportional to the negative of the displacement which is characteristic of a type of motion known as simple harmonic motion.
Okay. So what is this equation telling us? Well, the first step to answering this question is to recall that the acceleration is defined as the rate of change of the rate of change of displacement with respect to time, which we can write as d2x/dt^2. And if we then combine this with our simple harmonic motion equation, we find the following second order differential equation. This equation is telling us that the displacement of our mass changes with time in a very specific way such that if we take the displacement function and differentiate twice with respect to time, we get back the same function with a minus sign.
To answer that question, we can plot a graph with displacement on the y-axis and time on the x-axis. And if we then allow the mass to oscillate, we see that the displacement follows a characteristic periodic curve, which you hopefully recognize as a cosine function. More specifically, we can write the displacement as x(t) = a cos(omega t), where a is the amplitude or maximum displacement of the spring and omega is known as the angular frequency, which we can relate to the time period and oscillation frequency in the usual way.
Okay, so let’s now double check that this displacement function is indeed a solution to the simple harmonic motion differential equation we derived earlier. To do that, we simply need to substitute it into our equation. And to do that, we need to differentiate the function twice with respect to time. And we note that this is simply equal to -omega^2 x. If we then sub this result into our equation and rearrange and then take the square root, we see that this function is indeed a solution provided that omega is equal to the square root of k/m. So what is this telling us? Well, if we recall the relation between the angular frequency omega and the oscillation frequency f, we can then combine these relations and derive an expression for the oscillation frequency as well as the time period of our harmonic oscillator. And so we see that once we specify the spring constant and mass of our system, we can predict the frequency and time period of our harmonic oscillator.
Okay. But what about the energy of the harmonic oscillator? How might we determine that? Well, the total energy of our oscillator could be written as the sum of the kinetic and potential energy. And we can write this as 1/2 m (dx/dt)^2 + 1/2 kx^2 where the first term is the usual 1/2 mv^2 kinetic term and the second is the standard quadratic potential energy term for a harmonic oscillator. We observe something interesting if we then sub in the displacement function and the derivative of the displacement function into the energy equation and we find the following expression involving a sin squared and cosine squared term. If we then rewrite the coefficient of the cosine squared term by using the omega equation we derived earlier and rearranging this in terms of k then we can factorize the energy equation and we note that the term inside the bracket is the well-known trig identity which is simply equal to 1. And so our energy term drastically simplifies and becomes 1/2 m omega^2 a^2.
We notice a few things about this relation. Firstly, all time dependence has dropped out and this is simply telling us that energy is conserved. In other words, the total energy remains constant in time. We also note that for a given mass and spring constant, the total energy is proportional to the square of the amplitude. And since in classical physics, the amplitude can take any value, so too can the energy. As the amplitude increases, the energy does in a smooth and continuous fashion.
Now, a natural extension of a simple harmonic oscillator is to consider a classical wave which consists of a series of coupled harmonic oscillators. An example of a classical mechanical wave is provided by a rope held by two people under tension. If one of the people then rapidly moves their hands up and down, you will see a wave pulse moving along the rope. Alternatively, if you were to wiggle your hand up and down in a periodic fashion, then you can send a train of waves along the rope. And if we then highlight just one section of the wave, we notice that this point is simply moving up and down in a vertical direction rather than side to side. More specifically, if we zoom in and focus on this point, we actually see that it’s undergoing simple harmonic motion, much like a mass attached to a spring.
Now we can describe the displacement of a wave in the y direction as a function of position x and time t according to the relation y(x,t) = a cos(kx - omega t) where a is the amplitude omega is the angular frequency which if you recall is related to the oscillation frequency f and k is the wave number which defines the spatial frequency of the wave and is related to the wavelength by the relation k = 2 pi / lambda. We see from the displacement equation that for a given value of x, say for example x = 0, the function reduces to the displacement function of a simple harmonic oscillator. And it’s in this sense that we can think of a wave as a series of coupled harmonic oscillators.
Now, in much the same way that the displacement function for a harmonic oscillator was a solution to a differential equation, the displacement function for a propagating wave is also the solution to a differential equation. This equation is known as the wave equation and it’s a characteristic equation that describes many types of waves within the universe. We note that the velocity V of the wave appears explicitly in this equation. Now we can check that the displacement function is indeed a solution by subbing it into the wave equation. To do this we note that the second derivative of displacement with respect to time is equal to -omega^2 y and the second derivative with respect to x is equal to -k^2 y. If we then sub these results into the wave equation and simplify, we find that the displacement function is a solution provided that omega^2 = v^2 k^2. This relation is known as a dispersion relation and it will play an important role in what follows.
A key point to emphasize at this stage is the relation between the wave number K and the angular frequency omega. We see that as the wave number K reduces to zero, the angular frequency omega also reduces to zero. In other words, for very long wavelength waves, the frequency tends to zero.
To see this explicitly, let’s take the square root and rearrange our dispersion relation in terms of v. And then if we sub in the definitions for omega and k, we see that we recover the familiar relation v = f lambda. Now here is the key point. For a given physical system such as waves propagating down a particular rope, the velocity of the waves is constant for that system. And so we see from the equation v = f lambda that for a wave with a fixed velocity as the wavelength tends to infinity the frequency must tend to zero. And this is a feature of any wave that obeys this type of wave equation.
Okay. So to summarize, we’ve seen that waves such as those that move along a stretched rope obey a wave equation. And the displacement of such a wave is described by a wave displacement function. And if we sub this function into the wave equation, we find that it is a solution provided that v = f lambda. And then we saw that for a given medium, all waves traveling through that medium have the same speed. And this then implies that the frequency and wavelength of these waves can take any values. And importantly, in the limit of very long wavelengths, the frequency reduces to zero with no limit. Finally, because we’re dealing with classical physics, the amplitude of these waves can in principle take any value and can vary continuously. And hence the energy which is proportional to the square of the amplitude can also vary continuously and take any value.
Now the next question is can we modify this wave equation and if so what effect will that have to answer these questions let’s consider a slight modification of our wiggling rope example. Let’s imagine that along our rope there is a series of springs that connect the rope to the ground such that when the rope is vertically displaced there is a restoring force back towards the equilibrium position. To simplify matters, we’ll also assume that there is no friction or air resistance in our system.
So, a natural question arises. What is the wave equation for this new system? Well, if you recall, we previously had the following expression for our wave equation without the spring restoring force. And it turns out that the effect of anchoring our rope to the ground with springs is to introduce a new linear term in our wave equation equal to omega^2 y where for the case of the rope system omega^2 is equal to k/mu where k is the spring constant per unit length of the rope due to these restoring springs and mu is the mass per unit length of the rope. This additional term represents the fact that at every point along the rope, there is now an extra restoring force pulling the rope back towards the equilibrium position, even if the rope is perfectly flat all along its length. And this has very important consequences for the behavior of this system.
To see how this system differs from the simple wave equation we were previously dealing with, we can make use of the displacement wave function we used earlier and sub this into our modified wave equation. As before, the second derivative of y with respect to t is equal to -omega^2 y and the second derivative of y with respect to x is equal to -k^2 y. If we sub these results into the modified wave equation, we find the following result. And if we then cancel off the y’s and rearrange for omega^2, we find the following modified dispersion relation. And if we then compare this with our unmodified wave equation dispersion relation, we see that there is a new omega^2 term.
So the natural question is what is the effect of this new term? Well, to answer that question, we see that as the wave number reduces to zero, the angular frequency does not reduce to zero as before, but rather reduces to omega. Now, if you recall, since the wave number k = 2 pi / lambda, we see that k approaching zero is equivalent to the wavelength becoming very long. And we see that in this limit there exists a minimum angular frequency which is non zero. And this in turn means that there is a minimum oscillation frequency equal to omega / 2 pi. We can see this even more clearly if we plot k on the x-axis and omega on the y-axis. And if we then plot the unmodified dispersion relation corresponding to the rope without springs, we see that as k goes to zero, so does the angular frequency. Whereas if we plot the modified dispersion relation as k goes to zero the angular frequency tends to omega n at k equals zero.
We can visualize this by sending wave down our two ropes one with and one without the springs. And we see that as we steadily increase the wavelength then in the limit of infinite wavelength the rope without the springs stops oscillating as its frequency tends to zero. Whereas the rope with springs ends up oscillating with a frequency equal to omega n.
This makes perfect sense from a physical perspective. If we uniformly displace both ropes across their entire length and then release, then the rope with springs attached is pulled back causing an oscillation. Whereas there is no restoring force in the rope without springs and therefore no oscillation.
Okay. So to summarize we have seen that the unmodified wave equation takes the following form and in this case the dispersion relation implies that as the wavelength becomes very long the frequency reduces to zero. On the other hand we have seen that the modified wave equation with restoring term has a modified dispersion relation that implies that there is a minimum possible frequency of vibration for these types of waves. Now, you might be wondering why we’re spending so much time on this particular modification of the wave equation. The reason is that this modified wave equation with its built-in minimum frequency is the simplest example of a structure that will soon become very important to us. In fact, when we transition from ropes to fields, the exact same mathematical structure reappears. And the physical meaning of this minimum frequency will become central to our understanding of mass in quantum field theory.
But before we get there, we need to take the next step in our journey and ask a new question. What exactly is a field? And how do we describe waves in a field?
Well, the good news is a field is easy to define. A field is simply a function of space and time. Perhaps the simplest type of field is known as a scalar field. This is just a single number that we assign to each point in space. For example, think of the temperature in your room. At every point in the room, we can assign a number representing the temperature at that location. We can of course represent the field in a number of ways. For example, we could use a heat map to show the distribution of temperature throughout the given room. It’s also possible for our temperature field to change as a function of time. For example, if a local heat source enters our space, then we see that the field updates as the heat source moves. And so we see here that our field is changing as a function of both space and time.
But not all fields are scalar fields. Some fields assign not just a number, but a vector to each point in space and time. These are called vector fields. A vector field tells you both a direction and a magnitude at every point in space. For example, the electric field is a vector field. At each point in space, the electric field specifies both the strength and the direction of the force that would act on a positive test charge if placed at that location.
Now, whether we’re dealing with scalar fields or vector fields, one remarkable fact about nature is that the equations of motion describing waves in these fields often look very similar. Even though the underlying physics can be quite different.
For example, the height of the water surface in a tank is an example of a scalar field. And as waves pass through the water, the corresponding height field oscillates. Likewise, the variations in air pressure as a sound wave passes through air defines a pressure field. And we see that ripples exist in this field. And finally, consider an electromagnetic wave. Here the electric field oscillates, but there is no underlying medium. The field is a fundamental entity that exists throughout space and time. Its oscillations are not the motion of something deeper. The field itself is what is waving.
And here is something truly profound. Not only can the EM field support waves, it does so in a way that is consistent with the principles of special relativity. In fact, this is exactly how Einstein’s theory of relativity was born. In the 19th century, James Clerk Maxwell showed that the equations of electricity and magnetism could be combined to form a characteristic wave equation, one that predicted a wave speed of 300 million m/s.
This remarkable result perfectly matched the recently measured speed of light, leading Maxwell to conclude that light itself is an electromagnetic wave. But this raised a profound question. If light travels at this fixed speed relative to what frame of reference is that speed measured? This was precisely the question that Einstein considered when developing his special theory of relativity at the beginning of the 20th century. One of Einstein’s key insights was to postulate that all observers, regardless of their state of motion, must measure the same speed of light. This required a radical new understanding of space and time. observers moving relative to one another would measure time and distance differently in order to preserve the constancy of the speed of light. And crucially, Einstein’s theory showed that the idea of an underlying medium was unnecessary. Light propagates as a ripple in the underlying relativistic electromagnetic field itself. It is the relativistic field that is waving with no corresponding underlying medium.
Okay, but what actually is a relativistic field? Well, a relativistic field is any field whose equations of motion respect the principles of special relativity. That the laws of physics take the same form for all inertial observers.
Now, here is the key point. Everything we’ve done so far, waves on ropes, ripples in water, oscillations of air pressure, has been to build our intuition. But from this point forward, we’re going to focus specifically on relativistic fields and waves within these fields.
Now, the good news is that the equations that describe waves in relativistic fields have the exact same form as the unmodified and modified wave equations we’ve already seen. The only difference is that we replace y(x,t) which represented for example the displacement of a rope with the field value phi(x,t) which could represent a scalar field or a component of a vector field. And second and this is absolutely critical for a relativistic field we replace the wave speed v with the speed of light c the universal speed limit. When we make these replacements, we end up with the following two relativistic field equations.
And just as before, each of these wave equations has a corresponding dispersion relation. And it’s still the case that waves described by the first equation can in principle have zero frequency at long wavelengths. Whereas waves described by the modified field equation have a minimum vibration frequency characterized by omega.
So, let’s pause for a moment and take stock. We’ve seen how the physics of simple harmonic motion led us to wave motion, then to fields, and now to equations that describe waves in relativistic fields. But what about quantum mechanics? After all, we live in a quantum mechanical universe. So, how does this picture change when we bring quantum mechanics into the story?
So far we’ve only discussed the classical harmonic oscillator such as the mass on a spring. And we’ve already seen in such a classical system the amplitude can vary continuously and take any value. And since the total energy of such an oscillator is proportional to the amplitude squared. It follows that the energy can also vary smoothly and take any value.
In classical physics the simple harmonic oscillator is an example of a continuous system. Meaning you can give the oscillator as little or as much energy as you like and it varies smoothly. But according to quantum mechanics, this is no longer true. When we apply the rules of quantum mechanics to a harmonic oscillator, the energy of the system becomes quantized. It can only take on specific discrete values.
More specifically, if you solve the Schrodinger equation for a harmonic oscillator potential, then you find that the energy is quantized according to the following relation, where n is an integer that labels the different energy states, h bar is Planck’s constant divided by 2 pi, and omega is the angular frequency of the oscillator. We note that the separation between each of the energy levels of the quantum harmonic oscillator is always equal to h bar omega. In other words, the oscillator can only gain or lose energy in chunks of size h bar omega. This is what it means to say that energy is quantized. And we refer to a chunk of energy equal to h bar omega as one quantum of energy. And the energy of the quantum oscillator then depends on the number of quanta of energy that it possesses.
We also note something very interesting in the case where n equals zero corresponding to the ground state of the quantum harmonic oscillator. We see that even in this lowest energy ground state, the oscillator still has nonzero energy equal to half h bar omega. And this energy is known as zero point energy. This is essentially telling us that even in its lowest energy state, the quantum oscillator is not perfectly still. It still fluctuates with some small amount of energy.
Now, here is where things get really interesting. Let’s momentarily return to our wave on a rope example. We saw that a wave on a rope is essentially a large number of coupled harmonic oscillators. And we’ve just seen how quantum mechanics causes the energy of the oscillator to become discrete. So can we apply these rules to our entire wave? For simplicity, let’s imagine that we fix both ends of the rope and then allow the rope to oscillate. In this case, only certain standing wave patterns are allowed. The condition for a standing wave is that only an integer number of half wavelengths can fit between the fixed ends. And each unique wavelength is referred to as a vibrational mode of the system.
If we label each of the allowed modes with an integer m and if we imagine that the length of the rope is given by L, then the wavelength of the nth mode will be equal to 2L / M. And then given the relation between wavelength and wave number, the allowed wave numbers will be given by M pi / L. The allowed angular frequencies can then be calculated using the dispersion relation for the rope.
It then follows that a general wave here pictured in yellow can be built up by forming a superposition of different wave modes and by adjusting the amplitude of each of the different modes.
Okay, so let’s now ask what happens when we apply quantum mechanics to this system. Well, the amazing thing is that each wave mode behaves like an independent quantum harmonic oscillator. In other words, each oscillation mode m has discrete quantized energy levels given by the following relation where n counts the number of quanta in that particular mode. This is telling us that the energy in any given mode of oscillation is quantized. Meaning the energy can only be increased or decreased in chunks of h bar omega m. And we say that adding or removing this amount of energy corresponds to adding or removing one quantum of energy.
Okay, so we’re now ready to ask the big question. What happens if we apply these same ideas to relativistic fields? Now, it turns out that the answer to this question completely revolutionized our understanding of the universe and provided an entirely new framework, one we now call quantum field theory.
To show quantum field theory in action, we’re going to consider the modified relativistic field equation that we encountered earlier, which if you recall gave rise to the following dispersion relation. Now, here’s the key point. When we quantize this field, in other words, when we apply the rules of quantum mechanics, exactly the same thing happens as with the rope. Namely, each allowed wave mode of the field behaves like a quantum harmonic oscillator. Meaning that the energy of each mode is quantized and the field can only oscillate with discrete amounts of energy and the energy needed to create one quantum of oscillation of the field is equal to h bar omega.
To see what this means within the quantum field theory framework, let’s take our dispersion relation and multiply both sides by h bar^2. In that case, we end up with the following expression. Next we observe that the left hand side is simply equal to the square of the energy of one quantum of the field. And so we can write our equation in the following form. And this equation now relates the energy of one quantum of oscillation of our field with the wave number K and the minimum angular frequency omega n.
Now, if you stare at this expression long enough, and if you’re familiar with Einstein’s special theory of relativity, then you may notice that this equation bears a striking similarity to Einstein’s energy momentum relation E^2 = P^2 C^2 + M^2 C^4. This equation tells us how the total energy E of a particle depends on its momentum P and its rest mass M. It’s a foundational result in relativistic physics. Now step back and look at both expressions side by side. They’re structurally identical. Both have a total energy squared on the left hand side and both have a term on the right hand side that represents some kind of minimum. In the case of Einstein’s energy momentum relation, the rest mass energy represents the smallest amount of energy a particle can possess just by virtue of existing. Whereas in the dispersion relation we have a term representing the minimum frequency with which the field can oscillate. This structural match is not a coincidence.
To see why, let’s match terms. If we equate the first two terms and then divide both sides by C^2 and then take the square root, we obtain the relation P = h K. And if we then use the definitions of h bar and k, we recover the famous de Broglie relation linking a particle’s wavelength and momentum. But the magic doesn’t stop here. If we compare the second terms on the right hand side, we find the relation m^2 c^4 = h^2 omega^2. And if we then rearrange for m, we find the following remarkable relation. M = h omega / c^2.
This is an astonishing result. This equation is telling us that within a quantum field theory context, the rest mass of a particle is directly determined by omega n, which if you recall represents the minimum frequency of oscillation of the corresponding field.
This is telling us that each wave mode of the field behaves like a quantum harmonic oscillator. And each quantum of oscillation, each chunk of energy h bar omega corresponds to one particle with total energy equal to h bar omega and momentum equal to h bar k with rest mass energy equal to h bar omega / c^2.
And so we’ve discovered something truly remarkable. Particles are the quanta of relativistic fields. The field is the fundamental entity and particles are the smallest possible excitations of this field. This is what we stated right at the start of the video and now you know what it means. And this identification that one quantum of the field equals one particle is one of the most profound and beautiful ideas in all of modern physics. It connects waves, fields, and particles in a single unified framework.
And we can take this idea even further and consider how to describe massless particles within this framework. If we take the equation we’ve just derived for the mass of a particle in terms of the minimum oscillation frequency and set this equal to zero, then we see that this implies that omega must be zero. Meaning that the waves corresponding to massless particles do not have a minimum frequency and are therefore described by the unmodified field equation. Furthermore, we see that if omega is zero, then the energy equation resulting from the dispersion relation simplifies and after a bit of rearranging and simplifying, we end up with the famous result E = HF, which as you know is the energy of a massless photon, which Einstein famously used to explain the photoelectric effect. And because there is no lower limit on the frequency of this type of field, a photon can have any arbitrarily small amount of energy.
And so now we’re finally in a position to answer the question that was posed at the beginning of this video. What is a particle? Hopefully, if you followed along this far, you’re now ready to understand that a fundamental particle is the smallest possible vibration of a quantum field, a single quantum of the underlying relativistic quantum field.
And so to summarize and wrap up this fantastical journey, we have seen that according to quantum field theory, particles with mass are described by the modified relativistic wave equation where the field has a minimum oscillation frequency omega which determines the particle’s rest mass. On the other hand, massless particles are described by the unmodified relativistic wave equation with no minimum frequency.
And in the modern framework of quantum field theory, this is the deep unifying picture that underpins our understanding of nature itself. Particles are quanta of relativistic fields. Within this framework, quantum fields are regarded as fundamental and the familiar particles of the standard model emerge as the quanta of these fields. For example, a quantum of the electron field is an electron. A quantum of the electromagnetic field is a photon. A quantum of the quark field is a quark and a quantum of the Higgs field is the famous Higgs Boson. Every particle in the universe is in essence a tiny quantum mechanical ripple of an underlying relativistic field.
And this is where our journey ends for now. Of course, there are many questions that have been left unanswered in this video. For example, what determines the minimum vibration frequency of a fundamental quantum field? Why do some particles have mass and others do not? And how do particles interact with each other within this quantum field theory framework? These are deep questions and in many ways they take us right to the cutting edge of modern physics. But for now, I hope that this journey has given you a deeper intuition for one of the most beautiful ideas in all of physics.