In Week 1, we introduced Probability Distributions (PDs).
The Remarkable Theory of Quantum Fields
Week 2 looked at PDs in Quantum Mechanics (QM) and how to compare one against another.
Summary of Quantum Mechanics
Week 3 covered operators in QM and included a brief diversion to the Schrodinger Equation.
Operators in Quantum Mechanics
Now we finally have all the machinery that we need to move on from QM to Quantum Field Theory (QFT).
QM is an enormously successful tool when dealing with atomic and molecular problems. It is largely the low energy theory of electron behaviour. It’s high energy, relativistic, version is also extremely successful, although it introduces a number of conceptual problems, such as negative energies. Then there is the fact that radiation and matter get treated differently in QM. QFT addresses these problems.
I’m going to begin by introducing another simple probability model. This time I’ll use a spinning coin rather than a die, mainly because it only has two outcomes: heads or tails.
In QM we could model this with a set of two numbers to hold the Probability Distribution (PD). There would be two possible results of spinning the coin and watching it come to rest.
|h> = (1,0) the coin lands heads up
|t> = (0,1) the coin lands tails up
When the coin is spinning and there is equal probability of either outcome:
|e> = (1/2,1/2) (ignoring the “square root” rule)
In QFT we expand this set of possibilities to include zero or more coins. Instead of just two numbers, we need an infinite set of numbers to encode this. Let’s start to list the states.
|0> = (1,0,0,0,…)
|0> is a state with no coins in it. It has a special name in QFT, it’s called the vacuum state. The “…” above means a repeating set of zeros that go on forever.
The next state has a single coin, heads up.
|h> = (0,1,0,0,…)
In our old model this would have been the state (1,0).
A single coin, tails up, occupies the next state.
|t> = (0,0,1,0,…)
Note that we can still represent a single spinning coin as (0,1/2,1/2,0,…).
Now we add our first state with multiple coins in it.
|hh> = (0,0,0,1,0,0,0,…)
This state has two coins showing heads up. And the following state has two coins showing tails up.
|tt> = (0,0,0,0,1,0,0,…).
We can have a state with one coin heads up and one tails up.
|ht> = (0,0,0,0,0,1,0,…).
We can keep on going, defining three coin states, four coin states and so on. You can see that we need an infinite number of states since there is no limit to the number of coins we could model.
As is the case in QM, the overlap of each of these states with itself is one. The probability of being in a state, given that it’s known to be in that state, is 1, certainty.
<0|0> = 1
<h|h> = 1
<ht|ht> = 1
etc.
But the overlap between different states is zero.
<0|h> = 0
<h|t> = 0
<hh|h> = 0
etc.
Note that last equation. There is no overlap between the two heads and one head state. They are completely different states.
I’ll discuss a couple of problems with this QFT type of model in the comments.
Next week I’ll discuss creation and annihilation operators.
Platitude of the Day 