Propagators and Time Evolution

On the quantum scale, particles no longer travel in well-defined paths or states, new tools are needed to describe the uncertain states of reality.


This week I’m going to quickly introduce a couple more foundational concepts in quantum physics. This discussion is a little brief, and one of these days probably end up writing something in a little more detail on some of these topics. Regardless, this hopefully gives a  starting point in some of these concepts.


When developing our understanding of that Quantum world, it is easy to see that things in the classical world never quite work in the same way. On the quantum scale, instead of working with simple particles travelling in well-defined paths, you usually get particles with paths and properties that can best be described as ‘smeared’ across spacetime. Most of the time you can’t categorize them as either a discrete particle or a continuous wave.  Particles can be considered to simultaneously existing in different states and locations at the same time. Given these difficulties, physicists tend to work with probabilities and statistical descriptions instead of focusing on some absolute truth of some singular reality.

One important function in quantum physics is the propagator. The propagator is a function that takes an observed position and state, and tells us the likelihood of observing that same particle at a another position and state at a different time. Propagators help us predict how a particle will behave on quantum scales, so calculating them is an important part of any quantum theory.


If a particle exists at spacetime point A, the propagator can tell us how likely we will observe it at another spacetime point B. Note that only one of many possible paths is shown,

The tricky part is actually calculating the propagator. You may be recalling my last post on the principle of least action.  ‘Hey!’ You might say, ‘Can’t we just find use the action again and find a minimum/maximum path to predict how the particle moves?’ That would be a good approach, however there is no absolute certainty in the quantum scale so the best we can do is find a probability. Luckily, we can tweak our approach of using the action just a little and it’ll be able to find a probability instead.

Our solution to this problem hinges on our understanding on how a particle moves through time on a quantum scale. Since particles are ambiguously wave-like (see wave-particle duality), they also travel like waves.  This means that they oscillate in a way that is most conveniently described by imaginary numbers. The full derivation involves exploring the Schrodinger equation and something called the Hamiltonian, and we’ll touch on that a little.


So without dipping too much into the details, I’ll lay out some of the technical formulation of how this works. We can represent the transition along a path between two spacetime points like this:


In this equation, |x-A> represents our initial spacetime state.  <x-B| represents our final spacetime state, and e^-(iH(tB-tA))  is our time evolution operator. tB-tA represents the change in time between the two spacetime points. G is our propagator.

For those unfamiliar with this kind of exponential with an imaginary number, it means that the state is going through a complex (or imaginary) rotation. It isn’t easy to describe in relatable physical terms, but it is heavily related to the concept of superposition. When two different things are in sync with one another, they tend to add up and cause constructive interference. When they are oppositely aligned, they cancel out and cause destructive interference.


Waves in sync can combine to larger waves through constructive interference. Waves out of sync with produce smaller waves through destructive interference. Image courtesy of Joe’s waves revision page:

As mentioned earlier, the value of H is the Hamiltonian, a quantity that is closely related to the Lagrangian. The Hamiltonian probably deserves its own article so I’ll be pretty brief here. In most cases the hamiltonian can be interpreted as the total energy of the system. So this time evolution operator tells us that the particle undergoes a complex rotation through time, and that rotation is related to the total energy of the system (ie: the particle in this case).

What does it mean?

The time-evolution operator and the propagator are both used to describe and understand the nature of quantum physics.  While these are both human tools used to describe a physical realm the defies our intuitive understanding, examining them can reveal a lot about the nature of quantum reality.

The imaginary rotation through time evolution is more than just an abstract mathematical description, it is what tells us that particles can undergo constructive and destructive depending on a difference in some kind of phase. From this equation, we know that the progression of this wave-like behavior is related to energy, and from here we can begin to ask why? What would energy be connected to the passage through time in this way?

The propagator gives a basic tool to start working with the uncertainties of the quantum world. Through it we can start working with complex interrelations between particles in Feynman diagrams or we can used them to describe the odd phenomena of tunneling. Possessing these tools moves us away from the assumptions of the classical world and empowers us to tackle much more difficult problems in quantum physics.

Many of these tools are central to the field of quantum mechanics. Here are some resources for anyone who wants to learn more:

BOOK: Modern Quantum Mechanics, J. J. Sakurai
One of the most commonly cited textbooks on Quantum Mechanics. Develops all the concepts in a detailed approach. Chapter 2 tackles some of the topics here.

VIDEO AND READINGS: Quantum Entanglements, Leonard Susskind
These lectures provide a great introduction to Quantum Mechanics. Each lecture contains video and written material. Lectures 8 and 9 are related to what I have discussed in this post.


The Principle of Least Action and Lagrangian Mechanics

The principle of least action is a idea that helped transition physics away from Newtonian methods and toward a more general description of physical systems.

Lately I’ve been studying methods of calculating path integrals, and while I would very much like to write about that, I think I should start with a more foundational topic. The principle of least action is a idea that helped transition physics away from Newtonian methods and toward a more general description of physical systems. It has been so important that it is still used across many different areas of physics today.

Principle of least action

To start, we should discuss what the principle of least action actually is. In the 1600s, Pierre de Fermat discovered that light always takes a path that takes the least amount of time to traverse. Knowing the initial point, and the final point of light, you would know what path it took by finding that path that minimized the time taken by the light. This would become known as Fermat’s principle or the principle of least time.


The light ray moves slower when passing through a material. As per Fermat’s principle, the ray is deflected in such a way that minimizes the light takes to pass from beginning to end.

This was a novel idea, since older approaches of physics first considered the present state of a system and then apply laws to determine what further point it will transition to. In Fermat’s principle, you consider the endpoints of the system (beginning and end of the light path), and then you could derive the laws based on some kind of constraint (ie: minimizing the time that light travels).

Lagrangian Mechanics

As it turns out, light isn’t the only place you can apply this kind of approach. In Lagrangian mechanics, you could define the starting point of a system, and the end point of that system, then you’d try to minimize and maximize a particular quantity known as the action (S). The action is an abstract value that depends on the path the system takes; if find the path that minimizes or maximizes the action, that path corresponds to the real path that the system takes. This sounds pretty abstract but there is a well defined system for calculating the action for a path, and it comes from formula you can build for your system called the Lagrangian function or just the Lagrangian (L).


 The Lagrangian is defined at each point in time. Integrating along the path of the system provides the action for that path.

So by deriving the Lagrangian function and applying the principle of least action, you can describe almost any physical system. This has been a very successful approach and was later adapted to a similar technique in what we call Hamiltonian mechanics.

There are some conceptual oddities of Lagrangian mechanics. Particularly, the idea of both the past and the future influencing the behaviour of the present can be a little discordant with our idea of causality. Regardless, it has been proven as an effective approach to understanding the behaviour of a large variety of scenarios in physics.

What does it mean?

The principle of least action is an extremely handy concept that lets us tackle some problems that would otherwise be extremely difficult in physics. A commonly used example is the problem of coupled oscillators. In this problem, a series of masses are connected by springs and strung between two walls. Deriving it using tradition newtonian methods can be extremely tedious for a large number of masses. On the flip-side using lagrangian mechanics can vastly simplify the problem. If you are interested in lagrangian methods for harmonic systems, this problem can illustrate can illustrate the power of lagrangian methods:

Coupled Harmonic Oscillators by Irina Yakimenko

Keep in mind that this approach can offer more than just mathematical power, it is conceptual change in how we look at interactions in the universe. Instead seeing the universe as of a series of sequential events, we can see a system of relationships across time and space that emerge from the optimization of a few constraints.

The principle of least action also provides interesting implications for our perspective of time. As mentioned earlier, since the lagrangian derives our equations of motion from past and future constraints, it violates our intuition about causality. Some have argued that it implies an eternalist universe where the past, present and future all exist simultaneously and are separated only by the dimension of time. This is largely philosophical and as such, is difficult to demonstrate with any certainty.


Here are a few resources for getting a handle on Lagrangian Mechanics and the principle of least action:

PDF: Introduction to Lagrangian and Hamiltonian Mechanics, Melanie Ganz

A short pdf that introduces the main concepts and equations of Lagrangian Mechanics

BOOK: Mechanics: Volume 1, Landau and Lifshitz

The go-to book on classical mechanics. Uses Lagrangian methods from the beginning and covers many different methods applying them to mechanical situations.

VIDEO: A Special Lecture: Principle of Least Action, Kenneth Young

A fantastic introduction to the material and great speaker.