The Quantum Thomist

Musings about quantum physics, classical philosophy, and the connection between the two.
On Incompetent Bishops and the measure of Success.


Building blocks
Last modified on Sun Mar 11 18:19:40 2018


I have (sporadically) writing a series of posts outlining the basis of quantum field theory. So far I have discussed how uncertainty is parametrised in quantum physics (including a notion that the same being can exist in numerous different states or potentia, with change being movement from one potentia to another); that particles are created and destroyed; that physics is ultimately indeterminate (meaning that even given complete knowledge of the universe at one moment in time, and complete knowledge of the laws of physics, it is impossible to predict what the universe will be like at a future moment in time -- the best we can do is calculate probabilities or likelihoods for different possible outcomes); and the crucial role that symmetry plays in contemporary physics.

I think that it is about time that I started applying all of this. But I'm naturally lazy. So what I will do instead is show how it might be applied. I'm interested in the principles of QFT, not so much (in this series) the practical applications. It is the principles which link to metaphysics and the religious applications.

I will focus on the question Given that the universe has a certain likelihood of being in a state A at one moment of time, and given an understanding of physics consistent with the ideas listed above, what is the likelihood that the universe will be in a state Z at a future moment in time?

This is obviously not the only question that physicists are interested in answering, but it is an important question. Aristotle's theory of natural motion (expressed in terms of final causality) was one attempt to answer this question; Newton's second law another; Einstein's theory of general relativity another; the principle of least action and related ideas provide a prescription to construct such theories. So while this question isn't everything a physicist is interested in, it is a good place to start. And the solution to this problem will give us inspiration about how other questions of physics, such as those concerning the properties and structures of materials, might be answered.

I should make one caveat before I start. I am going to limit my discussion to quantum electrodynamics (QED), the theory of electricity and magnetism. This is not because the other forces of nature are unimportant, or that I can't discuss them in this framework (I discuss them in some detail in my book). My reason is that it is both the simplest theory with real world applicability, and because it contains everything I need to make the points that I want to make. If you understand the philosophy behind quantum electrodynamics (which you won't: nobody understands quantum physics -- as soon as we step back from the equations and start thinking about what it means our brains start hurting), then the rest of the standard model isn't that big a step. The mathematical details are far more …, well I would use the word exciting, but that might not be everyone's cup of tea. But the underlying principles are sufficiently similar that the philosopher of physics can skip over the non-Abelian theories in his first reading to get a vague understanding of what is going on. Nor do I want to discuss quantum gravity at the moment (although again, I mention it in my book). I would rather stick with what is known than speculate about what is unknown.

This restriction to QED means one thing: likelihoods are represented by circles, or, with a little more mathematical formalism, by complex numbers. There is no reason why likelihoods have to be represented in this way. That's just what QED is: the quantum field theory subject to the various symmetries of special relativity and local gauge invariance where likelihoods have the same topology as a circle. [Don't worry about what is meant by gauge invariance, I will get round to explaining about it at some point. Or, if you are impatient, you can look it up.]

I also need to be clear before I start about what I mean by a particle. I will just define it for now as a basic unit of matter; something from which more complex objects can be constructed. What I have in mind in particular are things such as electrons or quarks. Note that I did not say that the particle is irreducible, or that it is the most fundamental building block of matter. The particles I consider might be the most fundamental layer of material reality, or they might not. That doesn't affect what I have to say. However, what I would claim is that if there is something more fundamental than the electron, it must obey the same basic principles of QED (and remember, I am interested in the principles). Maybe the symmetries would be different; maybe we would have to introduce extra dimensions of space, or something like that, but it would still be a quantum field theory. You can't generate an indeterminate quantum theory from a deterministic classical theory (Bell proved that much). A field theory where particle number isn't conserved can't emerge from a mechanical theory where it is.

But, although I am using the same word, these particles are not the same as the particles we encounter in classical physics. I want to leave open the possibility that they might have very different properties than classical particles. For the moment, "particle" is just a word I'm using to represent the concept I described above. We can refine the definition and make it more precise as we move along; it would be stupid to do so before we know where we are going.

My discussion in this post might all seem a bit abstract, and like I am not really going anywhere, but spending all my time creating a notation. Not even turning my attention to real world problems, but just writing down a abstract notation which I claim might be useful. Even though I skip over a lot of the details and proofs of the intermediate steps (this is intended as an non-technical introduction to the concepts behind quantum field theory -- of course this is a strange usage of the word non-technical with which most people were not previously aware), you will probably ask why I put so much effort into doing this. Is any of this relevant to the question? But we are not going to be able to understand the answer unless we first understand the question. Our intellect is great at making abstractions; and we can only understand things if we first make these abstractions. Even language is an example of such an abstraction. Words such as dog and pig are used to mentally and vocally represent various types of object. Indeed, I am already deep into abstraction by the time I pose the question. Likelihood is an abstract way of thinking about probability; probability is an abstract way of thinking about the prediction of frequency; frequency is an abstract way of counting; even counting is an abstraction, albeit one that is closer to the real physical world. I am also describing states of matter; I haven't yet formally defined what I mean by the term, but that there are such states is obvious, since the same man could be either standing or sitting. The same collection of water molecules could be either ice or water. In each case, there is both something the same and something distinct. The way to avoid it is to say that there a single substance that can exist in multiple states. So the states exist in reality; but our understanding of these states has to again be an abstract representation of that reality.

We cannot avoid abstraction if we are to try to think about things intellectually, because our intellect can only grasp abstract objects. Abstractions are necessary; everybody does them to a certain degree, even people like Hume who denied that abstract thinking was reasonable. Without realising what he was doing, he abstracted from the physical world, and his own experience, and reasoned from that abstraction to his conclusion that the only things we could be sure of were what we directly observed or which arose from simple geometrical proof. Thus he contradicted himself with his very first move.

But we must get the abstraction right. Abstractions are only helpful if they bear a close relationship to whatever it is they are meant to represent. We need to be able to get from the reality to the abstraction, and back again from the abstraction to the reality. If this is not the case, then the abstraction can be dangerous and misleading. Hume's main problem, and Descartes, and Kant, and all the others, was that they abstracted from the real world, but did so badly. Thus their premises bore no resemblance to reality, and neither did their conclusions. We must be meticulous in constructing our abstractions.

Now, I am not being meticulous here, because this is just a blog post which is meant to be vaguely interesting to the reader who hasn't encountered this way of thinking about physics before. I am skipping over too many details for my presentation to be rigorous. But what I am trying to do is present how one might be meticulous in constructing a way of answering the question. All you have to do is fill in the blanks in my argument; prove the statements I leave unproved. It can be done; this just isn't the place to do it.

So, if we look at the question, it is clear that first of all we need some symbol to represent the likelihood that the universe is in a particular state at a given time. Before that, we need some symbol to represent the state of the universe at a given time. OK, here is a symbol: |A(t)>, and it is as good as any other for the purpose. The symbol mentions a state, A, and a time, t, and that's what we need.

Note that the symbol represents the state of the entire universe. Isn't this a bit of an overkill? Do we need that much? Unfortunately, we do. The reason is related to the fact that particles are not indestructible. For example, if |a(t)> represented merely a single particle (quantum mechanics), then that would be fine until that particle decided to decay and split into two particles. Yet those two particles should somehow be contained within |a(t)>, which is by definition a single particle state. That doesn't make sense, so we can't restrict ourselves to single particle states. We can then think about a symbol that could represent either a one or two particle state, but then the same objection would apply. The only way to avoid this is to have our symbol able to represent any possible combination of particles anywhere; i.e. it represents the state of the universe as a whole.

So |A> represents one possible state of the universe, and |B> represents another possible state of the universe, and so on. To have a complete description, we need an infinite number of such symbols, which is a bit unfortunate, since there are only a finite number of letters in the alphabet to represent each state. We will have to make do. |A> and |B> are meant to represent different states, so if the universe is in state |A> then it is not in state |B> and so on.

Now this suggests that we need some way of comparing states; some way of saying that |A> is the same as |A>, but not the same as |B>. We do this by introducing another symbol, <A|, which basically asks the question Is this in state A?. Taking a cue from computer programmers, we can say that 1 means "yes" and 0 means "no". Thus <A| |A> = 1 means that |A> is the same as itself, while <A| |B> = 0, means that |A> and |B> are different. That looks a bit ugly with those two lines next to each other, so I will write <A|A> = 1 instead, meaning the same. Of course, we will need this comparison operator to answer the question, since at the end of the day we will want to compare what A evolves into against the state Z.

The next thing we want to do is associate a likelihood with each state. As I mentioned, a likelihood is associated with a complex number, c, which pictorially can be represented by a point that lies within a unit circle. We need some way of denoting that state |A> has the likelihood c1 while the state |B> has the likelihood c2. I will express this as c1|A>. c1|A> is a symbol that means that there is a likelihood c1 that the universe is in state |A>, nothing more and nothing less. There is a subtle shift here. The symbol |A> represents an actual possible state of the universe. The symbol c1|A> represents our knowledge of the state of the universe. Given that the universe is indeterminate, and we can't simultaneously measure everything, our knowledge is going to be incomplete. We therefore can't say (in general) that the universe is in state |A>; we can only say that there is a certain likelihood that the universe is in that state. Once again, if we are going to think about this problem intellectually (our ultimate goal), we need some abstraction to represent that, which is what c1|A> does for us.

We can now use the comparison object <A| to ask the question: what is the likelihood that the universe is in the state |A> given that our knowledge of the universe is represented by c1|A>. We get the answer <A|c1|A> = c1<A|A> = c1, which is a relief because that's how we defined c1.

Of course, this isn't a real life situation. We are not just interested in state |A>. Our knowledge is going to be more like There is a certain likelihood c1 that the universe is in state A, and a certain likelihood c2 that the universe is in state B, and so on. That's a closer representation of what we might know. Once again, we need to represent this knowledge somehow, and I'll do so by writing it as c1|A> + c2|B> + c3|C> + ….

That "+" symbol doesn't mean the same thing as when we add three integers together. It is used analogously. The reason this analogy is useful is because the operation that combines these knowledge states satisfies the same properties as standard addition, for example: commutativity, associativity, identity, inverse, and distribution under "multiplication" with a likelihood. I'm not going to prove that here (such a proof would be too technical for my present purposes). But it means that physicists can comfortably use all their favourite algebraic tricks they are familiar with when adding things while working with these objects. And because physicists are lazy, we just use the same symbol rather than going to the trouble of inventing a new one.

Again, we can use the comparison operator to extract likelihoods from the representation of our knowledge. So we want to write, < A| (c1|A> + c2|B> + c3|C> + …) = c1< A|A> + c1< A|A> + c1< A|A> + … = c1. This is just the result we expect from the distributions.

We can also construct more complicated comparison operators. We don't just want to compare our knowledge with an actual state, but one representation of our knowledge against another. This, after all, is what the question asks of us. It is comparing one likelihood against another, and each likelihood represents a part of our knowledge of the true state of the universe.

So we can construct an operator < A| c1 + < B| c2. I will write this as <W| to be concise. This plays the same role as the previous comparison operators we have encountered. We apply it to a state |X>, which represents our knowledge of some system X. <W|X> is the likelihood that the systems represented by W and X are in the same state, given our (incomplete) knowledge of both of them. Mathematically, c1 represents the complex conjugate of c1. There are reasons why this is the case, but again I will skip over those details. All you need to know is that it is not the same as c1, but is nonetheless related to it in some well defined way.

One of my premises was that the fundamental building blocks of matter can be created and annihilated. They are not (as the mechanical world view supposes) indestructible. This is confirmed by numerous particle physics experiments. So there is a process by which the universe exists in a state |A>, which lacks a certain particle a, and it moves into a state |Aa>, where that particle exists. |Aa> is a symbol representing a universe which contains a single instance of particle a. We need some symbol to describe that process of creation. This process acts upon one state, and spews out another. In mathematics an object which acts on one or more sources and produces one or more results is called an operator, so I will use the same terminology here. In particular, I am interested in a creation operator, denoted by â. This acts on any state and results in the same state but with an additional particle a. So it eats the state with no particles of type a, and vomits out the state with one particle of type a.
â |A> = |Aa>
Similarly, it acts on the state with one particle of type a and produces the state containing two particles of that type. And so on.

Equally, we need an annihilation operator, â, which performs the reverse operation.
â |Aa> = |A>
I'll again skip the proof of why the creation and annihilation operators share the same conjugate relationship to each other as the likelihoods c1 in the state vector and c1 in the comparison operator. It can be proven from my fundamental premises, but that will be too much of a digression here.

The creation operator not only describes the particle itself, but also which state it is in. So â1 creates a particle of type a in the state we choose to list first, â2 creates a particle in the second state, and so on.

Each particle is created or annihilated at a particular time, so that's another piece of information we need include in our notation. Thus the complete description we need is that we create a particle of type a in the first of our list of possible states for that particle at time t1. This operation is represented by the symbol â1(t1).

Change from one state to another can be described as a combination of creation and annihilation operators. For example, change from state 1 to state 2 (in Aristotelian terms, the process of actualisation of a potentia) can be written as the annihilation of the particle in the first state and the simultaneous creation of the particle in the second state: â2(t1) â1(t1).

But now, we are finally in a position where we can begin to express the question formally. For we now have the means to define any state we encounter. We start with a state with no particles in it, |0>. We then need to populate it with particles to get our required initial state.
|A(t1)> = â1(t1) â3(t1)… |0>
We do the same thing to get our required final state (also constructed, say, from particles of type a).
|Z(t2)> = â2(t2) â4(t2)… |0>
We need an operator, S(t1,t2), which describes the evolution of the universe from time 1 (the time of our initial state) to time 2 (when we will observe the final state).

Then the likelihood we want is just given by.
< 0 | a2(t2) a4(t2)… S(t1,t2) â1(t13(t1)… |0>

So this is great progress. We have now written the question in a abstract language we can process. We haven't yet answered the question, but we have at least now asked it in the right way, which is a good start.

But equally clearly, this isn't yet useful. We have a bunch of symbols, but what we need is a number for the likelihood, which we can then convert into a probability. We require two more things.

Firstly, we need some means of understanding how the creation and annihilation operator algebra works. How do we manipulate those symbols, and move them around the equation? In particular, what is the relationship between â â and ââ? In both cases we are creating a particle and destroying the same particle at the same moment of time. This might seem like a pretty pointless thing to do, and maybe it is, but thinking about this problem allows us to learn how to manipulate the equations. In particular, creating the particle and then destroying it is not the same as destroying the particle and then creating it. Think about what happens when we apply the operators to an empty state. You can't destroy something that isn't there.

Secondly, we need to pin down the operator S. This describes the rule governing how the universe evolves in time. We need to know what that rule is if we are to make any progress.

But speaking of time, I am out of it myself for the moment, so I will continue in the next post.



Order matters


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