Path Integrals

Because of the statistical nature of quantum physics, there is a degree of choice in how a particle moves, as a result there are a infinite number of possible paths that could reach a destination.

Today I’m going to talk about what we call Path Integrals in quantum physics. The last two articles on the principle of least action and propagators have provided some conceptual background that will help us along. Continuing on those concepts, I’ll introduce an important approach that extends the principle of least action to the quantum world and gives us a way of using it to calculate propagators.

Quantum Paths

With the details from last week details under our belt, we can start looking at quantum propagators from a different angle. In particular, the Hamiltonian approach that we discussed last time was a little troubling.  The problem stems from the fact that our formula includes specific references to time and the hamiltonian (a quantity that can be loosely interpreted as the total energy).

time_evolution.png

The Hamiltonian propagator suffers from being incompatible with relativity. Both time (t) and the hamiltonian (H) change when we look at different reference frames

Time can appear differently in different reference frames due to relativistic effects, and as a result, the hamiltonian also varies based on our perspective. We want a formula that is lorentz invariant, meaning that when relativistic effects come into play, this formula is still valid.  While this form of the propagator isn’t lorentz invariant,  it does come in handy for deriving a relativistic propagator that uses the Lagrangian instead. With the Lagrangian approach, we won’t have to worry about about relativity because we can work with lorentz invariant lagrangian functions instead of hamiltonian functions.

energy_relative.png

While energy is always conserved, it is different in different reference frames, and thus is not lorentz-invariant.  A ball moving in one reference frame would appear to have a different velocity in a another reference frame, leading to different observations of energy.

The main thing that we need to do is to somehow express the transition between our two points in spacetime. We can call them point A and B. The hamiltonian method just needed the difference in time to work properly, however the lagrangian is associated with the path. We can no longer just get by with the difference, in time, we need to account for different possible paths to a destination now.

Here is another way of looking at it.  In classical mechanics, any particle with a given energy and position will follow a single path to move to its destination. Because of the statistical nature of quantum physics, there is a degree of choice in how a particle moves, as a result there are a number of possible paths that could reach a destination. As it turns out, there is actually an infinite number.  This means that we need to consider the contribution of each possible path to calculate the propagator. This approach is referred to as the sum-over-histories.

paths.png

In classical physics, a particle has a single possible path that it can take from one spacetime point to another, this is considered to be a deterministic system. In quantum physics, there an infinite number of paths that a particle can take, and the outcome appears to be randomly determined. This is considered to be a stochastic system.

You can break the hamiltonian propagator down through a method called time-slicing. By splitting the overall time into segments of size delta-t, you get a ton of small time evolutions each moving the particle slightly further forward in time.

time_slice1.png

time_slice2.png

By splitting it up into a number of tiny points in time, you can use rewrite the equation as a series of smaller propagators on a variety of paths. The overall integral indicates that the propagator is comprised the the sum of all possible paths between the two points.

time_slice3.png

Note that I’m deliberately omitting the more complex calculations in this description, in order to focus on the overall picture. Skipping past a few steps, the action emerges from the hamiltonian. With the action, we just need to find some way of combining all these paths into the propagator. It turns out that you can perform a special kind of integral across all the possible paths. This is called the path integral and it looks something like this:

path_integral.png

This is the general formula for the path integral. The D[q(t)] means that this is a functional integral, that integrates over a changing function instead of a variable. In this case it is integrating over all the possible functions for the action (which are related to each individual path).  The overall picture here is that this formula is adding the contributions of an infinite number of ‘mini-propagators’ for each individual path. Remember that because the particle oscillates as it travels, the principle of superposition applies and it can either add constructively or destructively and some paths will cancel each other out.

The method of solving this particular equation is fairly complex and can differ depending on the form of the lagrangian, so we’ll leave the mathematical formulation for now.

What does it mean?

Now that we have a look at the general mathematical description, we can start looking at the implications of this formulation.

One particular feature or working with path integrals is that some of the paths can end up moving backwards in time for a portion of the path. This may seem pretty nonsensical but particles moving backwards in time are interpreted as anti-particles moving forwards in time. So these time switching paths would be observed as particles and antiparticles pairs being created and annihilating. This paints an interesting picture of the way particles can move on the quantum scale.

Another consequence of those infinite paths is that some of these paths can end up in extremely unlikely places.  In classical physics, we can devise some kind of impassable barrier that a particle could never cross, however in quantum physics we have to account for paths that go around the barrier, or even travel through time to circumvent this barrier. This leads to a non-zero propagator for passing through the barrier, so some particles will actually cross the barrier. This phenomena is called quantum tunnelling because in experiments it appears as if the particle as crossed the barrier through some kind of invisible tunnel.

As far as the sum-over-histories approach itself, the physical interpretation is divided. One possibility is that the particle only takes one possible path from one point to another, however there are some people who argue that the particle takes every single path to its destination. It is also equally possible that all of this is just a clever mathematical abstraction that can’t be interpreted in human terms, but where is the fun in that?

These are only a couple of the perspectives that open up through our formulation of path integrals. Here are a couple resources for those of you who want to learn more.

VIDEO: PSI 2016/2017 Quantum Field Theory II – Lecture 1,  Francois David
This provides a introduction to the general method, and provides a thermodynamic analogy that shows how this method can apply to more than just quantum physics.
http://pirsa.org/displayFlash.php?id=16110001

BOOK: Quantum Field Theory for the Gifted Amateur, Tom Lancaster,  Steven J. Blundell
I’ve drawn heavily from the derivation in Chapter 23 of this great book. I must admit that some of the equations in this post are right out of this chapter. Overall this book is a great introduction to Quantum Field Theory and I’d recommend it to anyone looking to get started in this kind of material.
https://www.amazon.ca/Quantum-Field-Theory-Gifted-Amateur/dp/019969933X

Learning Resources For Physics – Part 1: Video

Everyone learns in different ways and takes different paths to nurture their understanding

Learning is the bread and butter of science, we could be studying some esoteric equations, attending class, or even just looking at the myriad patterns in your everyday life.  The tricky part is that everyone learns in different ways and takes different paths to nurture their understanding.

It helps to have lots of different tools in our search of knowledge so that we can explore are these avenues of learning. I’ve found many people are interested in theoretical physics, but find it inaccessible and daunting. With that in mind, I’ve begun to compile a few helpful resources so that people can have more avenues of learning.  In the spirit of inclusivity, I will only include resources that are 100% free.

Today, I’ve gathered a few video learning resources to help people along at every step of their learning.

Khan Academy
This is a staple of online learning sites. It covers all the STEM fields from high-school up to at least second-year university level. It provides a reliable and accessible entry point into the foundation material of any scientific study.
https://www.khanacademy.org/

MIT OpenCourseWare
MIT has hosted course content from hundreds of their past courses. This includes a few pretty solid courses on physics. You can even filter the courses to see the ones that include video content.
https://ocw.mit.edu/courses/find-by-topic/#cat=science&subcat=physics

Perimeter Institute Recorded Seminar Archive
The Perimeter Institute has made recordings of all their lectures openly accessible for anyone to watch. If you have a solid grasp of undergraduate physics, these provide a doorway into graduate level material. The lectures are well laid out and cover a huge variety of topic material. Make sure to check out their seminars too.

  • 2014/2015 Course lectures:

https://www.perimeterinstitute.ca/training/perimeter-scholars-international/lectures/2014/2015-psi-lectures

  • Main video archive:

http://pirsa.org/

 
CERN Document Server
This site includes not only video lectures on cutting edge particle physics, but also foundational topics taught by some of the leading physicists and engineers working there. This is a treasure trove for anyone interested in particle physics.
https://cds.cern.ch/collection/Videos?ln=en

These are by no means the only resources out there. Just the ones that I tend to use. You can find even more resources on other lists like these:
https://futurism.com/ultimate-collection-free-physics-videos/
http://www.infocobuild.com/education/audio-video-courses/physics/physics.html
https://glenmartin.wordpress.com/home/leonard-susskinds-online-lectures/