For all the news talking about quantum theory, most people don’t know or understand what the theory actually says. In fact, it takes years to understand it because it is not only very complex, but it rests on centuries of classical physics for its vocabulary. You have to understand all of classical mechanics to understand quantum mechanics. You have to understand classical field theory and statistical mechanics of fields to understand quantum field theory. Not to mention operator theory and Hilbert spaces.
There are quite a few descriptions of the Standard Model of quantum physics. This model includes all the particles that we know exist, all the forces, and how they all fit together. This is not one of those descriptions. What I want to talk about is more fundamental than that.
Ultimately, quantum theory is concerned with fields. There are spinor fields, fields of gauge bosons, Higgs field, and so on. All these fields combine in various ways to create the Standard Model.
There are two general approaches to quantum theory, one is via field operators, which are kind of like little mathematical machines that act on infinite dimensional vectors which represent particles and forces.
The other way is what Richard Feynman did his Ph.D. thesis on back in the 1940s, called the Feynman Path Integral.
You can prove that the operator method and his method are equivalent. You can think of the operator method as a kind of black box where you put your starting setup for your experiment in and get your results out. The path integral does the same but it explicitly models all the stuff in between, the path that a particle takes, for example, from emission to detection.
Feynman path integrals are based on the principle of least action. An action is a mathematical expression for any system in classical physics such that that system will always take the path that is minimal.
The action is just a sum or integral of the Lagrangian, sometimes called action density. The Lagrangian is usually the kinetic energy, which comes from the velocity of an object, minus the potential energy, which represents any forces. As an example, suppose I have a ball of mass m and it has velocity v(t) where z(t) is its height above the ground and g is acceleration due to gravity:
I can put this into an integral over time and have an action. Then I can compute the least action and that will give me the equations of motion of the ball.
Feynman took the principle of least action and incorporated it into quantum mechanics. In doing so he showed that the least action principle is only an approximation of what happens. Particles do not take a single, least action path from emission to detection. Instead all paths contribute to the final result. The path that has the least action just contributes the most. To get your measurement prediction, you have to sum over all these random paths. This is called Feynman’s sum over histories method. One bizarre result of this is that no matter where I place my detector I might detect the particle there.
I first encountered path integrals in graduate school and ultimately did my Ph.D. thesis on them. They are probably one of the most fundamental discoveries about the universe because they show how action describes quantum physics.
The way I learned it was to imagine the famous double slit experiment. You have a particle fired at a barrier with two slits in it. Traditional quantum mechanics says that the particle travels through both slits to create a diffraction pattern on a detector. Now imagine you have not two but many many slits cut into an infinitely long barrier. Quantum mechanics says that the particle travels through all the slits but with lower probability for the slits that are further from the straight line path.
Now imagine that you have many copies of that barrier and you place them at regular intervals between the emitter and detector. Quantum mechanics says that the particle passes through all the slits of the first barrier, second barrier, and so on but with differing probabilities. And that once it has passed through a slit in the first barrier it can travel through any of the slits in the second one with different probabilities depending on how long the path is. Now, if we take the number of slits to infinity and make them infinitesimally close to one another and take the number of barriers to infinity, we end up with empty space, but all the previous principles still apply. The particle travels through every path. That is the path integral according to Feynman and Hibbs (1965).
With this tool you can turn any classical theory into a quantum theory.
Feynman originally presented the path integral as a non-relativistic evolution operation. In other words it described how a quantum wavefunction changed in time. Thus it did as a path integral what Schroedinger’s equation did as an operator equation. But to understand quantum fields it was quickly generalized to describe fields in four dimensions.
What does a four dimensional “path” look like? If you can imagine a one dimensional path as a piece of string, a two dimensional path is like a mattress of springs that is flexed in some way, a three dimensional path is a block of them, and a four dimensional one is like that in one more dimension.
It’s hard to visualize like that but you can at least visualize the mattress version and imagine how you can have many different configurations of a mattress depending on what is sitting on it. Now imagine that the mattress is constantly being shaken and vibrated even with nothing there. Then you get the idea of what quantum field theory is like. The least action would be for the mattress to be flat but quantum field theory instead flexes it away from flatness randomly. All the different configuration of the mattress contribute (Zee, 2010).
And that, my friends, is quantum theory in a nutshell.
Feynman, Richard Phillips, and Laurie M. Brown. Feynman’s thesis: a new approach to quantum theory. World Scientific, 2005.
Feynman, R. P., and A. R. Hibbs. “Path integrals and quantum mechanics.” McGraw, New York (1965).
Taylor, Edwin F., et al. “Teaching Feynman’s sum-over-paths quantum theory.” Computers in Physics 12.2 (1998): 190–199.
Dirac, Paul Adrien Maurice. The principles of quantum mechanics. №27. Oxford university press, 1981.
Zee, Anthony. Quantum field theory in a nutshell. Vol. 7. Princeton university press, 2010.