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.

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

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.

Propagators

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.

propagator

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.

Time-Evolution

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:

time_evolution

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.

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: https://waves.neocities.org/superposition.html

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.
https://www.amazon.ca/Modern-Quantum-Mechanics-Revised-Sakurai/dp/0201539292

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.
http://www.lecture-notes.co.uk/susskind/quantum-entanglements/