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 
Indistinguishable Particles
The question now arises as to whether swapping the coins makes any difference. In other words does
|ht> = |th> ?
I’m going to assume that it doesn’t make a difference and that the above equality is true. If we could distinguish between the coins, say one was silver and one was bronze, then we might want to distinguish between the two states. But I’ll assume that the two coins are indistinguishable. So we’ll only have the state |ht> and will assume that |th> is the same thing.
In particle physics, all particles of the same kind are identical. You cannot distinguish one electron from another. They all have identical mass, spin and charge. Despite this, when you swap two electrons, their PD does NOT remain the same. One probability distribution is the negative of the other. This is one of those cases where the fact that we’re really dealing with the square roots of probability distributions becomes really important. A negative PD makes no sense, but a negative square root is perfectly respectable. I’ll still call the states “probability distributions”.
It turns out that, for all particles with integer spin, swapping two particles leaves their PD the same. Such particles are called Bosons. For all particles with half integer spin, swapping two particles introduces a minus sign. These particles are called Fermions.
This may all sound a bit esoteric, but it has enormous ramifications. The minus sign means that no two half integer particles can ever occupy the same state. If they did occupy the same state then swapping them would make their PD the negative of itself. Since these must be equal for indistinguishable particles, that means the PD must be zero. The particles can’t be anywhere.
In particular (no pun intended), no two electrons in an atom can occupy the same state. They are forced into different energy levels, resulting in some electrons being tightly bound, close to the nucleus, while others orbit more remotely and are free to interact with other atoms. The entire periodic table, and all of chemistry and biology comes from that minus sign!
Conversely, integer spin particles CAN exist in the same state. In fact they positely relish it. Any number of particles can exist in the same state. Spin-1 photons build up in the same state to such an extent that they build classical waves that we perceive as light.
QFT explains why we need the minus sign, sort of. There’s a formula for calculating the total number of particles in an arbitrary state. Unfortunately, for spin 1/2 particles, the formula gives opposite signs for the number of particles and antiparticles. So the number of particles could be negative. That’s nonsense, even in QFT. The only way to make the counts of both spin 1/2 particles and antiparticles come out positive, is to introduce the swap particle minus sign. This result goes by the rather grand sounding title of “the spin-statistics theorem”.
Infinities
The infinite number of states isn’t going to be a problem for us. However it does illustrate where some of the difficulties in QFT come from. QFT often assumes that we can have an infinity of states, just like we’ve done for the coins. But in reality, there’s no such thing as infinity. If we counted all the coins in the universe it might be a very, very large number, but it’s almost certainly not infinite.
Quite often in QFT, calculations sum over all possible states, going all the way to infinity. Most times the sum converges. But occasionally it doesn’t and the calculation gives an infinite result.
The techniques for dealing with this are called “renormalization”. Sometimes there’s a clearly identifiable bit of the sum that creates the infinite part, in which case it just gets thrown away and the calculation proceeds without it. In other cases we can simply limit the summation to some arbitrarily large number. Other cases are more difficult still. I have to confess that this is one area of QFT that I haven’t really mastered yet.
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<blockquote>The entire periodic table, and all of chemistry and biology comes from that minus sign!</blockquote>
This is where I start to struggle with some of these theories. In much of classical physics, and in SR & GR, the mathematical model offers an explanation of both what is happening and why. In QFT it all seems to be just the maths.
I know that as models go, QFT is astonishingly accurate. But I could, for example, create a model of gravity simply by taking a million measurements, noting their initial conditions and outcomes, and then mapping the former to the latter as just a huge “if this, then that” algorithm. It would also be very accurate, but it would hold no explanatory power. Newton’s (and then Einstein’s) equations demonstrate the driver behind all this. They equally say what is happening, but they also say why.
There’s a wonderful article in Nature that suggests that it was the Manhattan Project that started all this, that the desperate wartime need for results over explanations turned particle physics into particle engineering. I say this as an engineer myself – nothing wrong with a bit pf practical engineering – but the map is not the territory.
Michio Kaku famously said, “In the beginning, God said let the four dimensional divergence of an antisymmetric second rank tensor equal zero, and there was light.” This is very funny and very T-shirtable. But it is of course the wrong way round. The tensor measures and describes light, but it isn’t light.
The utility of all this is unquestioned. It’s only by having the maths that the experiments to find and study new particles can be designed and interpreted. And QM is not without its deep explanatory theorems – the Eightfold Way (and the Standard Model) sorting out the particle zoo of the 50s and 60s, for example. But it feels in need of another one – a theory like GR that will get rid of renormalization and other tricks and will use instead a physical, not mathematical, model.
Please don’t think I am not enjoying your explanations immensely, Peter. I understand the structure of the maths better now than I ever did. And I do enjoy a good discussion, so feel free to tell me exactly how wrong I am. It’s just that in maths one sphere can be made into two identical copies of the original just by cutting and rotating, and in engineering pi is absolutely equal to 3.14. Physics should be neither of these. It should be superior to both.
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I think I agree with you. In fact, next week was going to be the “Big Reveal” where I point out something very similar. One of my aims in this series is to demonstrate that QFT is a purely descriptive theory. I’ll be interested to hear what you think about next week’s topic.
“This is very funny and very T-shirtable. But it is of course the wrong way round. The tensor measures and describes light, but it isn’t light.”
Again, I agree. The electromagnetic field tensor has to come from something deeper. And it does. It comes from Noether’s theorem, that every continuous symmetry of nature creates a conserved current. In the case of EM, it’s the symmetry of the wavefunction with respect to phase changes. This generates the conservation of electric charge. But, due to relativity, the conservation must be local, rather than global. (Electric charge doesn’t just disappear and reappear miles away. It has to move.) The requirement to make it local forces the existence of the EM field tensor.
Einstein’s field equations arise in exactly the same way. This time the symmetry is with respect to spacetime displacements, giving rise to conservation of energy (time displacement) and momentum (space displacement). Again, these must be locally conserved, giving rise to gravity.
But none of these say what matter or EM fields are. What they do, do, is constrain them to say how they must behave. QFT is along the same lines. It creates a model of extremely tight constraints on the behaviour of particles. So tight that we haven’t been able to find any experiment that allows us to probe deeper.
You’ve kind of stolen my thunder!
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“What they do, do, is constrain them to say how they must behave“
Yes, and this is of crucial importance. The huge amount of extraordinarily accurate data generated by the experiments and the model make creating a physical picture extremely challenging. It has to conform, or else it is wrong.
“You’ve kind of stolen my thunder!”
Sorry.
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