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.

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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.

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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

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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

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.

FermatsPrinciple.png

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).

Action.png

 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

https://people.ifm.liu.se/irina/teaching/sem6.pdf

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

http://image.diku.dk/ganz/Lectures/Lagrange.pdf

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.

https://www.amazon.ca/Mechanics-1-L-D-Landau/dp/0750628960

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

A fantastic introduction to the material and great speaker.

https://www.youtube.com/watch?v=IhlSqwZBW1M