Order matters

I have (sporadically) been 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.

In my last two posts, I have started to go into the details. Two posts ago I introduced a notation to describe the different states that matter can exist in.

The states roughly (not precisely, or at least not precisely at this level of discussion) correspond to Aristotelian potentia. These states cover the entire universe, i.e. every possible particle. We use the notation to represent a state where the universe is in a particular configuration . is applied to that state to ask the question: what is the likelihood that the states represented by and correspond to the same, given our knowledge of both of them.

We can construct a complete basis out of these states, i.e. there is one state vector to represent every possible configuration of the universe. However, we also have to deal with our knowledge of the universe being incomplete. Because of the indeterminacy of nature (although there is more to it than that), our knowledge can't be anything but incomplete. All we can do is to assign likelihoods that the universe is in one state or another. If represents our knowledge of the universe, the possible states the universe could be in, and the corresponding likelihoods (calculated from what we know, perhaps from observation or perhaps from reasoning) that the universe is in those states, then we write this as . These likelihoods are mathematically represented by complex numbers of unit modulus or less; objects with the same topology as the points within a circle of radius .

Then we need a representation of change, and for this end I introduced the idea of operators, which take in one state as input and produce a state as output. The output state could be the same as the input state, but usually isn't. The operator represents the creation of a particle of type at location , while the operator represents the destruction of a particle of type at location . So if is a the representation of a state without a particle at then represents the state which is exactly the same as in every detail but one: there is now a particle at location . Movement of a particle from to is represented by the operator . Going from right to left in the notation, we destroy the particle at and simultaneously create a new particle at location . This doesn't mean that in reality that particles are continually created and destroyed; this is merely our representation of one aspect of reality, namely change.

I am asking myself the question: Given that the universe has a certain likelihood of being in a state at one moment of time, what is the likelihood that the universe will be in a state at a future moment in time? By the end of that post, I came up with an expression for this likelihood. represents the empty state, with nothing in it. The state represents a universe with particles at locations , , , and any other odd number index you can care to think of. represents a universe with particles at locations , , , and the other even indices. We take an initial time and a final time , and the likelihood is given by

represents the time evolution operator, and our big problem is that we don't know what that is. The goal of this post is to find .

But finding is not enough; we need to know how to calculate with it. In my last post, I showed how we can do that. An annihilation operator acting on the empty state gives us zero. So what we want to do is swap the order of the operators in the expression above so that all the annihilation operators are on the right of the expression and all the creation operators on the left. But when we swap the order of two operators, we often have a consequence to pay. In particular, it can be rigorously shown that there are two different types of particle in the universe. The first type of particle has

where is the identity operator when and the zero operator when the particles aren't at the same location. This type of particle is known as a Fermion.

The second type of particle (I will call this type ) are called Bosons, and when we swap the order of creation and annihilation operators of this type we replace

In this way, when we swap the order of the operators in our expression for the likelihood, we are left with a bunch of identity operators. Identity operators are easy to deal with, so this allows us to turn this expression of creation and annihilation operators into a number.

The other point I made in that last post is going to be important in this one. So far, I have written the states as though each particle is located at a specific place in the universe. However, we don't know that particles are localised. Indeed, the state notation I have been using has all the mathematical properties of a vector space; and one thing we know about vector spaces is that there is no unique basis. For example, in a universe with one particle and two possible locations, we could treat and , the states of definite location, as fundamental, and perform our calculations using those states. But the states and will do the job just as well. This freedom to rotate the basis is difficult to get our heads around. It is a completely alien concept to both Newton and Aristotle. It is one of the things that makes quantum physics (both quantum mechanics and quantum field theory) different.

But don't forget, that we are just talking about our representation of reality here rather than reality itself. One particular basis will correspond to reality. We don't know, at this stage of the argument, what that basis is. But equally, at this stage of the argument, we don't really care. We know how to rotate from one basis of our representation to another. So the plan is going to be to start in the basis that represents reality, rotate our representation to a basis where it is easy to calculate things, perform the calculation, and then go back to the representation of reality at the end. Rotating the basis is relatively easy; performing the actual calculations is much harder. So why not make the calculating step the least challenging that we can?

So what basis will be least challenging for calculations? Surely the one in which a particle in a given state remains in that state always. The basis where particles are stable. It might be thought that this basis is really boring, and one where nothing happens, but that is not quite true. The likelihood that a particle is in a given state has the same topology as a circle. Everywhere on the radius of the circle represents the same probability. Thus we can have the points representing the likelihoods rotating around the circle at constant radius. This doesn't have any physical consequences (as long as we are just discussing the single particle). We can, however, distinguish between states based on the speed at which this rotation occurs for particles in those states. We can form a mathematically consistent basis where the rotations are at constant speed. I will call this speed of rotation the Energy of the particle, and denote it by the symbol . The symbol , which I will call momentum, is used to distinguish between states. is a numerical constant that is there for historical reasons. States of particle type with a given energy are denoted as . At any given time, the state of the particle is then , where .

As I said, we can use this to create a completely self consistent mathematical basis to represent reality. In practice, the physical particles are going to be a superposition of these states -- but we will worry about that at the end of the calculation.

So let us take one of these states at one moment in time, and ask what it is going to be like at the next moment in time. We will call the time difference . The operator needed to represent this change contains four parts. Firstly, we need the annihilation operator . This returns if the particle is not present in the state, i.e. it ensures that we only implement the change for particles that are actually there. The second thing we need is to actually perform the change, by multiplying by . Thirdly, we want the final result to contain the particle, so we need the creation operator . And finally, we need to apply this to every possible particle in every possible state. Thus our first, slightly naive, attempt to construct a change operator is given by

is just a notation saying that we are summing over all the states. The precise details of why it has been written in this way are lost in the mathematical details which I have been omitting.

Of course, we don't just want a change operator that acts on this particular basis, but one that will work on any basis. In particular, we are interested in the basis where particles are located at particular locations. This might not be (and indeed isn't) what we will want at the end of the day, but it is, at least, more familiar to us and easier to visualise. The first step is to replace with an operator which gives multiplied by the state when acting on any of these stationary states. That operator is well known from the mathematics of calculus: it is the differential with respect to time, denoted as . This is a standard mathematical measurement of how fast something is changing. We can replace with this operator in the equation above, and it gives exactly the same result. The only difference is that this makes general: we are no longer dependent on being in this particular basis. So we can happily rotate to the spatial basis:

is again a way of saying that we are summing over all possible states.

Again, this is interesting, but doesn't quite get us as far as we want to go. All we have done so far is to say that the particles change in time in whatever way they change in time. What we need is to replace that differential with respect to time with something that depends on how the particles are scattered around in space, since that is (ultimately) what we measure.

This is where symmetry comes into play to help us. Einstein's theory of special relativity states that space and time are linked together. At the heart of Einstein's theory is a symmetry linking differentials with respect to time and differentials with respect to space. If this symmetry holds (and there are good experimental and philosophical reasons to say that it does), then we have what we need: we can substitute in the for an expression involving .

It is not, however, quite as simple as this because time is a scalar quantity, while location is a vector quantity; it basically represents three numbers. It is not quite so obvious how these three numbers are put together. We seemingly need to take the spatial differential in some direction, but any direction we choose would be arbitrary. The solution was provided by the Paul Dirac, one of the great physicists of the early twentieth century. He realised that if instead of treating particles as a single scalar operator, we can suppose that each of the operators actually describes the possible creation of at least one of at least four different particles. There could be more than four particles, but four is the minimum number for this trick to work. thus represents not one but four related operators. The mathematics is the same as the mathematics of vectors, but we call these numbers a spinor because these numbers aren't related to space and time. We then suppose that when we transform from one inertial frame to another (having the centre of our coordinate system move with a different velocity), the transformation at the heart of special relativity, we simultaneously perform another rotation between the different directions" contained within the spinor. This rotation is parametrised by four operators, labelled , , and (represented, in the simplest case, by four by four matrices). is the time coordinate, , and the three spatial coordinates. These matrices then allow us to put together the spatial derivatives in a way that preserves the symmetry of special relativity. The formula is (again, I skip over the mathematical details explaining why it has to be of this precise form, but it does).

is a parameter allowed by special relativity. Since it is allowed, we have to put it in the equation (we can always set it to zero if we later find out that we don't need it). Since we have put it in the equation, we should give it a name: mass. Again, I take the name from classical mechanics, since (as it turns out) this is a good part of the origin of the mass of classical objects (the rest of classical mass arises from certain interactions between particles, which leads to another term in the final result which just adds onto the mass). Just as the , just introduced as a parameter in this theory, ultimately gives rise to the Energy of classical mechanics and the parameter to classical mechanical momentum. [I should add as an aside that in the full standard model of particle physics, this term is forbidden by other symmetries I am not considering here, but the mass ultimately arises from interactions between these Fermions and a Boson.]

That there are four strands of each particle has been confirmed experimentally. By the time that Dirac wrote down his theory, two of them were already known: the spin-up electron and the spin-down electron (the word "spin" is an unfortunate historical anomaly. People first thought what we call spin was related to the rotation of particles. That hypothesis turned out to be false, but we are now stuck with the name). Dirac predicted the existence of two more: the spin-up anti-electron and the spin-down anti-electron. Shortly afterwards, these particles were observed in experiment, with precisely the properties predicted by Dirac's theory.

Now we are not quite there, because there is one further symmetry we need to consider. As I keep emphasising, each state vector is multiplied by a likelihood and each likelihood is characterised by an angle or phase. We can only measure angles between two lines, so in our notation we have to arbitrarily declare one direction to be angle zero. However, in nature there is no such preferred angle: it is only a consequence of our representation of nature. The difference between two phases does not depend on where we put the zero-line. Thus the only variables of physical consequence are relative phases, rather than absolute phases.

That is all very well when we only compare phases for particles at one fixed location. We have already seen that the phase can change in time; it can also be that the phase changes in space. And if so, so can orientation of the zero phase can change from one place to another. We can transform the point of zero phase at each location in space (and, indeed, time) independently, and this transformation cannot affect any of the physics. Thus this transformation, known as a gauge transformation, is a symmetry of the theory.

This is a problem when we differentiate with respect to space. The differential operator compares the phase at one location with the phase at another location, and this seemingly depends on our choice of origin for each phase. When we perform a gauge transformation, the physical consequences of the operator , as written down above, are different from one gauge to another. This is inconsistent with our conclusion that gauge transformations are a symmetry of the theory.

The solution to this apparent inconsistency is to say that is incomplete. We need to introduce another particle into the theory, in such a way that when we perform a gauge transformation we simultaneously transform the creation and annihilation operators for this new particle in such away that the gauge symmetry is preserved. This new particle needs to be a Boson field, and I will denote it by the symbol , so it has a creation operator and an annihilation operator . also has spinor indices; in this case three of them, and these are combined using a polarisation vector . There is one for each direction. We add in a constant to describe how strong this field interacts with the fermion fields. Adding this into gives

with meaning that we how the system changes in each instant between the starting time and finishing time , and then add up the result as we pass through each successive moment, and

This is becoming a bit more complicated, and unfortunately we are still not done yet. We still have to figure out how the field changes in time. The reasoning here is similar to the case of the fermions. We need to figure out an expression for the time differential acting on the field which is consistent with gauge and Lorentz invariance. The final result for the time evolution operator is not unique, because the operator directly depends on the choice of gauge (although no likelihood we calculate will depend on the which of the forms of the operator we choose, as long as we are consistent throughout the calculation), but one possibility for the time evolution operator for the Boson field is

So now we have almost everything we need to calculate the likelihood.

The expression we started with was

We know how to manipulate the operators to give a numerical result. We know what is. We are seemingly well set. All we have to do is to sit down and actually calculate.

Of course, the calculation itself is not going to be easy or trivial. In fact, it is a lot of blooming hard work. But it can be done, almost. We know how to get approximations to the answer. This approximations are to arbitrarily good precision. If you want a more precise answer, you just have to put in a couple more hours (for the first steps) or a couple more decades (for the precision we have reached in the calculations now) work into it. And, after all, our final aim is to compare against experiment, and the experimental results are also imprecise. So as long as the theorists are more precise than the experimentalists, they have done their job.

And there is one step still missing in my calculation: we have used states distinguished by location. I mentioned at the start that we rotated from the physical basis to a convenient basis to perform the calculation. The basis where the state represent localised particles is convenient. The basis where the states represent particles of fixed energy and momentum is also convenient; and these are the two I have presented above. However, neither of these bases are actually physical. So we still have to go back to the physical basis before we can compare against experiment. If you aren't calculating with the real physical particles, you get nonsense answers (infinite probabilities and the like). This is as we would expect; if we are in an unphysical basis, we should expect unphysical answers.

This process is known as re-normalisation. It is actually easy (in principle, not so much in practice) to find the right basis, if we adopt one more axiom. These is where we need to impose scale invariance. This is the statement that there is no preferred length scale, or energy scale, in the universe. You can measure things in meters, or in inches, or in furlongs. The universe doesn't care, as long as you do things consistently.

There is one basis which is independent of any scale artificially entered into the system. As a slightly naive (but more easily understood than the methods used in practice) example, if you artificially put a large energy cut-off into the equations, all the likelihoods in the scale invariant theory can't depend on the size of that cut-off, as long as it is considerably larger than the largest mass in the system. We can, however, keep track of all the ways this energy scale enters the calculation, and keep making adjustments to the basis as we go through each step of the reasoning to ensure that any scale dependence cancels out of the final result.

I should say that what I have done is not quite rigorous. The steps I have taken have all been following the right ideas, but because this is a blog post rather than an academic textbook, I have taken short-cuts in one place or another. For example, I have been a bit lax in the finer details of my discussion of Lorentz invariance (there are various factors of the square of the energy which have to appear somewhere, and which could have been put in the equations I wrote in this post). If you want a more rigorous presentation of all this, then please see chapter 10 of my book (or, for even more rigour, one of the standard textbooks on quantum field theory, albeit that I use a non-standard notation in places).

But all I have done so far is provide an equation and suggested a means to calculate it. What does it mean? That is the topic for next time.

Putting it all together

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What is the next highest prime number after 11?