Last week we took our first steps into the world of Quantum Field Theory (QFT).
We defined a set of Probability Distributions (PDs) that included zero or more coins, each of which could be heads or tails.
|0> = (1,0,0,0,…) no coins, the vacuum state
|h> = (0,1,0,0,…) one coin, heads up
|t> = (0,0,1,0,…) one coin, tails up
|hh> = (0,0,0,1,…) two coins, heads up
etc.
Just as we did with Quantum Mechanics (QM), we can define operators that act on the coin states and turn them into other coin states. So, for example, we could define a “heads creation operator”, H+, that creates a coin in the heads up state. This acts as follows.
H+ |0> = |h>
H+ |h> = |hh>
H+ |t> = |ht>
and so on.
One thing that I think is quite important to point out. H+ is just a bunch of numbers that specifies a set of arithmetical operations that transforms the bunch of zeros and ones in one state into a different bunch of zeros and ones in another state. Despite its name, the “heads creation operator” has absolutely no idea what a coin is, how to make one, what the difference between heads or tails is. If that seems obvious to you, then congratulations. You won’t believe how many years it took me to figure that out. I’ll give a more significant example of this in a moment.
The behaviour of the tails creation operator, T+, is similar.
T+ |0> = |t>
T+ |t> = |tt>
T+ |h> = |ht>
and so on.
Just as we have coin creation operators, we also have coin annihiliation operators, H- and T-. Some of their actions are obvious.
T- |tt> = |t>
T- |ht> = |h>
T- |t> = |0>
and similarly for H-. However, some of their actions are not quite so obvious.
T- |0> = 0.
The result of acting on the vacuum with an annihilation operator, any annihilation operator, is the number zero. Note that zero (“0”) and the vacuum (“|0>”) are quite distinct things in QFT. The vacuum is a state that can be acted upon by creation operators to create new states. Once something gets annihilated to zero though, there is no coming back.
Perhaps even less obvious is the following.
T- |h> = 0
If the tails annihilation operator acts on a state without any tails, then that result is also zero.
The order that we apply creation and annihilation operators also matters. Two different creation or annihilation operators can be applied in any order.
T+ H+ |0> = |ht>
H+ T+ |0> = |ht> (remember, we’re treating |ht> = |th>)
However, we have to be careful with creation and annihilation operators of the same type.
H- H+ |0> = H- |h> = |0>
But
H+ H- |0> = H+.0 = 0
So H-H+ has a very different result from H+H-. QFT expends a significant amount of effort to get all the annhilation operators to the right. The reason for this will become apparent when we start to explore particle physics, but basically it allows us to delete terms because they annihilate the vacuum state. When all the creation operators are on the left and all the annihilation operators are to the right, the product is called “normal ordered”.
Some of these rules may seem a little arbitrary, but they’re based on analogy with the quantum harmonic oscillator (QHO), one of the few quantum systems for which there is an exact, analytic solution. In the QHO there is a ground state and a set of equally spaced energy levels that can be reached via creation and annihilation operators, just like the ones here. These rules are absolutely central to the way QFT works.
And that completes the rules that we need for our creation and annihilation operators. We’re now ready to define our coin quantum field, C.
C = H+ + H- + T+ + T-
i.e. The quantum field is the sum of all possible creation and annihilation operators. That’s what a “field” is in QFT – a thing that can create or annihilate stuff. I’ll give a more realistic example in the comments.
Next week, we’ll move onto something a bit more like particle physics.
Platitude of the Day 