The Quantum Thomist

Musings about quantum physics, classical philosophy, and the connection between the two.
On the time evolution of fields.

Putting it all together
Last modified on Sat Aug 22 19:57:56 2020

I have 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 three posts, I have gone into more details. As I have said, my aim in this series is not to give a complete and rigorous description of Quantum Field theory as it is used in today's cutting edge physics. There are plenty of good books which do that. Instead, my goal is to present an outline to the simplest quantum gauge theory to give the reader enough of a background to understand what it is all about and the underlying theoretical ideas. There is far more to the standard model of particle physics than I am discussing here. But the ideas I discuss in these posts underlie everything else. If you understand this, then you will understand the basic principles behind everything else. And it is the principles above all which I want to convey.

Three posts ago I introduced a notation to describe the different states that matter can exist in. The idea is that we could use a mathematical notation to represent states (or potentia). This forms a vector space. We then introduce a notation to tell us which of these states are occupied at any given moment of time (or more precisely, given that our knowledge is incomplete, the likelihood that the states are occupied). If such a knowledge state accurately describes the universe, then we can use it to specify which particles exist in which states at any given moment of time. I also introduced operators, which change the configuration of the state vectors, and represent change (movement from one state to another) in the physical universe. Central to this process are the creation and annihilation operators, which describe particles being created from nothing or annihilated into nothing. This "from nothing" is simply a mathematical convenience; in practice we pair creation and annihilation operators together to describe motion from one state to another.

Two posts ago I looked at the commutation relations between creation and annihilation operators. One would naively think that if one instantaneously applied a creation operator to a system (creating a particle in that state) and then an annihilation operator corresponding to the same state (removing a particle in that state), one would not really do much at all. Equally, if one annihilated and then created the same state, one might expect that there would be no effect on the system. But one can't annihilate something that isn't there. This means that the operator that creates and then annihilates is not equal to the operator that annihilates and then creates. This simple observation allows us to simplify expressions for the mathematical likelihood that the system would move from an initial state to a final state. It allows us to take a bunch of creation and annihilation operators acting on an initial state and compared against a final state, and turn it into a number.

A consequence of that discussion was that there are and can only be two different types of matter, Fermions and Bosons. These differ in how the creation and annihilation operators for those particles relate to each other. One important physical consequence of this is the Pauli exclusion principle, which states that you can't have two Fermions in the same state at the same time.

Finally, in the previous post, I looked at how matter evolves in time. I started off by assuming a system with one type of fermion field (reasonable enough that I should start with Fermions, since most of the matter we are made of, such as electrons and quarks, are observed to be Fermions), and I considered how it might evolve in time. If we suppose that the universe is constrained by a number of symmetries, most importantly the Lorentz symmetry linking space and time that is central to the theory of special relativity, and gauge invariance, which states that only the relative rather than the absolute phase of a likelihood can be observed, and discovered that there was only really one possible way in which the system could evolve. There are stable, or metastable states, known as the energy and momentum eigenstates. That momentum states are stable means that one cannot change momentum except under interaction with an external field. In other words, we have discovered Newton's first law and the principle of inertia just by thinking about symmetry and change. There is, however, an important difference from Newton mechanics, which I will come to later.

Two further consequences followed from that discussion. Firstly, the only way we could get it to work meant that there couldn't be a single electron field; but we needed (at least) four different types of electron (and, indeed every other type of fermion). These are known as the spin up electron, the spin down electron, the spin up positron, and the spin down positron. There are unfortunate historical reasons for these names: the distinction in spin was observed experimentally before Dirac wrote down his equation and understood it; the name positron was given before the theory was fully understood and people realised that anti-electron would be better. As with almost everything that was named before it was understood, we can regret the nomenclature established in those early days, but we are stuck with it.

The second conclusion drawn was that the theory was only possible if there was also a second, Bosonic, field in the system, called the photon. Thus the smallest, simplest, self-consistent possible arrangement of matter contains two fields: one Fermion field (the electron) and one Boson field (the photon). These fields can interact with each other, and these interactions allow states to change.

I should now mention an important difference between quantum physics and classical physics. In classical physics, a physical state is described by two parameters, the location and the momentum. In quantum physics, we only use one (slightly naively, but good enough for this level of discussion) the momentum: the location is not used to identify the state. This has an important philosophical consequence. Key to Aristotelian philosophy is the principle of causality: that things can only be in motion (i.e. change their state) if there is something to move them. Aristotle's physics, of course, assumed that states were distinguished by their location as the only dynamical parameter (there are also things such as colour, humidity, temperature and so on, but they are not dynamical parameters); he knew nothing of momentum.

Then Newton came along (actually, this was first proposed well before Newton, by the fourteenth century early mathematical physicists, but I won't let facts such as that spoil a good myth), and said that force was related to acceleration rather than velocity. This meant that a state had to be parametrised by two variables: momentum and location. Aristotle's principle of causality was directly applicable to momentum: there was no movement from one momentum state to another without a mover. But the principle is seemingly violated by location: particles do travel from one place to another without something pushing them. The Aristotelians had some counters to this. Aristotle's philosophy allows for natural motion, where something, in effect, moves itself (or rather its inherent final causes leave it with a natural tendency to move from one state to another). But these answers were brushed aside by the proponents of the new mechanical philosophy as ad-hoc explanations after the fact. The discovery of the principle of inertia was one of the main reasons why Aristotle's philosophy was abandoned alongside his physics. There were many reasons why people historically rejected Aristotle; most of these were built on misunderstandings or a refusal to even try to understand Aristotle's system. The numerous straw men presented by the early modern writers against Aristotle are obviously invalid to anyone who does understand Aristotle's system -- that is anyone who learnt Aristotle's philosophy from his classical and medieval commentators rather than his renaissance and early modern critics. However, a few of the objections do need to be taken seriously; and the objection from inertia is, in my view, prominent among them.

However, now we come to quantum physics, and the situation changes again. Now states are distinguished by momentum alone. This is a bit weird: we think about particles being localised in space, and that's no longer true, and it is difficult for us to get our head around it. Localisation makes a bit more sense when we consider compounds of elementary particles rather than the elementary particles themselves. One can, by rotating the basis of the creation and annihilation operators of the fundamental particles, derive creation and annihilation operators that describe changes between states of compound particles. From these, one can develop a quantum field theory of more complex states. For these states, the momentum eigenstates are not the simple waves that characterise free elementary particles, so they are not spread out across the universe but strongly peaked around a certain location. Looking at it from a distance, it looks as though these compound states are localised; only when we get up very close do we realise that the location of the particles is still fuzzy and undetermined. Everything we observe is in one of these compounds; and that you need a very powerful microscope to see the fuzziness explains why the classical prejudice that particles are localised works so well. But really, it's not so.

But what this means is that the objection to Aristotle's philosophy disappears. States are distinguished by a single dynamical parameter; which can only change in an interaction between particles. Aristotle got it wrong that that parameter was location; but his metaphysics doesn't depend on this claim, and after we change every reference to "location" and replace it with "momentum", this part of his philosophy continues undaunted. Every movement requires a mover. Of course, we have to redefine what we mean by "move." Aristotle defined movement as a transition from one potentia (what I am calling a state) to another; and he defined states in terms of their location. Thus movement became synonymous with changing from one place to another. However, now we know that this synonym is mistaken. Sticking to Aristotle's definition of motion, we now have to say that things move when they change their momentum. Location has nothing to do with it. With that change in terminology, Aristotle's principle of causality works again, seemingly without any fudges involving final causality. Again, this is no more than a direct consequence of our natural axioms concerning the nature of change and principles of symmetry.

So let us now turn to the end point of the last post, the operator that describes how matter evolves in time. This expression was in two equations and three parts. The first equation is

     Rt1   0
^S = eit0 dtH^

This describes two things, the natural motion of the fermion fields (the part that only depends on the fermion creation operators ^ax1   ), and the part that describes the interaction between the fermion fields and the photon fields described by the creation operator  ^
A  . The second equation describes the self-evolution of the photon field,

Z        Z    d4p   y         2             ip (x  y)
   dtH^g =    (2 -)4E2-^Ax;  (   k  + p p )A ^y; e      :

Self-evolution terms have one creation operator and one annihilation operator of the same type of particle. The annihilation operator checks to see if the particle is present; we then get some number which adds to the likelihood, and then the creation operator returns to the original state. In short, this term tells us the likelihood that a particle remains in the same state.

An interaction term contains more than two operators. Either two particles are destroyed and one created, or two created and one destroyed, or more than that. In this theory, the only possible interaction has three terms, corresponding to three particles involved in the interaction: a Fermion creation operator, a Fermion annihilation operator, and either a photon creation or annihilation operator.

The self-evolution of the fields, if they are in a momentum state, just means that the likelihood changes its phase, with no change in the state. Remember, the likelihood can be represented by a circle, with the probability, what we observe, only depending on the distance from the centre of the circle. A change in the angle, or phase, does not change the probability that that state is occupied; thus these terms in the time evolution operator don't lead to a change in state, but only in our parametrisation of the likelihood. These contributions to the time evolution equation describe particles that are left alone and don't interact with anything, and states that a particle in a momentum state remains in that momentum state (unless it interacts with something else). As I said, Newton's first law of motion.

The remaining term is the interaction term, and it involves a fermion creation operator, a fermion annihilation operator, and either a Boson creation or annihilation operator. Now, the annihilation operator ^ax1   describes either the annihilation of a particle or the creation of an anti-particle. The underlying mathematics of this term guarantees the conservation of energy and momentum. Momentum can be transferred from the fermion to the Boson, but cannot be created or destroyed in a purely physical process. The following interactions are possible:

  1. A spin-up particle state is destroyed and a photon and spin-down particle state created.
  2. A spin-down particle state is destroyed and a photon and spin-up particle state created.
  3. A spin-up anti-particle state is destroyed and a photon and spin-down anti-particle state created.
  4. A spin-down anti-particle state is destroyed and a photon and spin-up anti-particle state created.
  5. A spin-up particle state and photon state are destroyed and a spin-down particle state created.
  6. A spin-down particle state and photon state are destroyed and a spin-up particle state created.
  7. A spin-up anti-particle state and photon state are destroyed and a spin-down anti-particle state created.
  8. A spin-down anti-particle state and photon state are destroyed destroyed and a spin-up anti-particle state created.
  9. A photon state is destroyed and a spin-up particle state and spin-up anti-particle state are created
  10. A photon state is destroyed and a spin-down particle state and spin-down anti-particle state are created
  11. A photon state is created and a spin-up particle state and spin-up anti-particle state are destroyed
  12. A photon state is created and a spin-down particle state and spin-down anti-particle state are destroyed

Any other possibilities you might think of are forbidden by either the matrix structure of this equation or the conservation of energy and momentum.

Recall that the time evolution operator acts on one likelihood and spits out another likelihood. None of these interactions are certain to occur. But each of these interactions could occur, and, moreover, we can compute the likelihood that each of them occurs. We don't know what will happen in practice. But we do know what all the options are, and how likely each of those options are to happen.

For example, suppose that we have a single spin-up electron sitting in our laboratory. That electron could continue as it is without anything happening to it. Or it could emit a photon, and move to become a spin-down electron with a different momentum, and then nothing happens to that spin-down electron. Or the spin-down electron could emit a photon, so we would be left with a spin-up electron (of a different momentum to our original one) and two photons. One of those photons could decay into an electron and anti-electron. Then we would have two electrons, one anti-electron and one photon. Or the spin down electron could absorb a photon. It could even absorb the same photon that was emitted by the spin up electron in the first place; in that case we would be left with a single spin up electron in exactly the same state that it started in. If we look at the electron some time later, it might seem as though it has had a very boring existence, but behind our back it might have been continually emitting and then re-absorbing various photons.

Some people find pictures easier to visualise than words or equations, so the great and greatly lamented Richard Feynman invented a way to describe this in diagrams. A straight line represents an electron. A wiggly line represents a fermion. The direction of efficient causality goes from left to right, so the things on the left are the causes, and the things on the right the effects, and everything in the middle is an instrumental cause. These diagrams represent some possible ways in which particles can interact with each other.

Feynman diagrams More Feynman diagrams

Interactions between fermions are mediated by exchanging photons. For example, the following diagrams represent possible routes where there are two electrons in the initial state and two electrons in the final state. The first diagram indicates the electrons proceeding with nothing happening, but there are also many other ways in which they could interact. These are just a very small sample.

Feynman diagrams More Feynman diagrams

Remember, in Aristotelian philosophy, an efficient or final cause always refers to a physical state. This is often confusing for us, because those idiot philosophers of the enlightenment decided to redefine causality to include events as well. We have inherited this confusion from them. So when we think of a cause of a state, we think of an event; or we can think of causes of events or processes. Put that thought away; it is useless and contrary to physics. The event of an emission of a photon by an electron has no cause, not because the principle of causality is invalid, but because the word "cause" only applies to objects and not events, and any philosopher who tries to convince you otherwise doesn't know anything about anything and is not worth listening to (except maybe to have a good laugh at their pompous stupidity before we go back to studying philosophers who do know what they are talking about). However, the results of the decay do have an efficient cause, namely the original electron. As usual, Aristotle got it right and the enlightenment philosophers laughably wrong.

Clearly, there are many different routes by which we can get from an initial state A to a possible final state Z. We can't say which of these routes will occur. In fact, even having observed Z, we can't say how the system got from A to Z. The only thing we know is that it must have taken one of the paths that lead to Z, but we can't say which one. So we have to consider all of them; each of the infinite number of possibilities.

Now we want to calculate the probability that, given with a specified initial state at a particular time, we will end with a specified state at a later time. Nearly all we have to do is write down every possible route from A to Z, calculate the likelihood that each route would have occurred, sum up all the likelihoods, and square it to get the probability. Job done; as easy as that. Well, as I said, nearly. We still have to renormalise (rotate the basis of creation and annihilation operators to reflect the physical particles rather than the idealised states used so far: the physical particles are still momentum eigenstates, but they are compounds of the bare fermion and photon operators we have used so far). But that is one of the technical details behind my description; the basic principle still stands.

You will note that in my list of possible physical interactions, there were none that lead to all the particles being destroyed, or all the particles being created. There is always something at the beginning of the process and something at the end. For every particle, we can always trace back a history of causes; a history which must remain unbroken either for eternity or until it reaches something which does not have a cause but exists for eternity (and thus something which could not, given its nature, have a cause; i.e. a being which could only exist in one possible state; i.e. a being of pure actuality, an uncausable cause or unmovable mover). Of course, this conclusion is a bit hasty: we have only been considering one of the non-gravitational forces of nature, and (more importantly) in special relativity rather than general relativity. It might be that gravity has something to say on the matter (or it might not; it's not that there is no answer, it's just that I haven't yet discussed the subject). But everything we see around us, and everything that can have a caused given its nature, is caused.

But what about the philosophy of all this? What we see is that we are lead in a direction which is uncomfortably Aristotelian. The following basic principles of Aristotle's philosophy just drop out immediately:

  1. The principle of potentiality and actuality, and motion as movement from one state to another, is analogous to the state notation of quantum physics. I've pointed this out already, and I am by no means the first person to note this similarity. But it is key, because Aristotle's metaphysics can largely be derived by taking this principle as an axiom.
  2. The principle of causality, that every change in state needed substance to exist beforehand. A spin-up electron has to exist to decay into a photon and spin-down electron; the movement is not possible without a mover (namely the electron in its initial state, which in this case is destroyed in the process).
  3. The principle of proportionate causality, namely that an effect is implicit in its cause (or causes). This basically means that given a cause, there are only a certain number of possible effects, limited by the potential of the original cause. Only a limited number of decays or absorptions are allowed in each interaction, determined by the initial state vector.
  4. Efficient causality. For every effect, there is a specific cause; for every substance except an unmoveable mover there is something that caused it. Of course, we can't know what all the links in the chain of causality are; but they are still there. We don't know what route the particle took from A to Z, but we do know that it had to have taken some route.
  5. Final causality: for every cause there is a list of possible effects, dependent only on the type of being the cause is. The being has the inherent tendency towards these effects, and only these effects. Indeed, we can define a type of object in terms of its final causes or inherent tendencies. [Note that for Aristotle, final causality was not the same as purpose. That confusion between final causality and purpose was introduced in the Renaissance period, and we ought not to blindly repeat it.] I gave such a list above.
  6. Formal causality. The form is an abstract description of an object, describing its possible states and distinguishing between different types of object. In the simple example I am discussing here, we just have a small number of different objects; perhaps seven at the most generous (the four types of electron, and the three different polarisations of photons). It is not so easy to make compound objects out of these building blocks. Nonetheless the abstract state notation developed here provides at least a partial mathematical representation of the Aristotelian form. The representation can only be physically useful if there is something in reality that it is a representation of. And thus the form must exist in reality. I'll expand on this point a bit more below.
  7. Material causality. However, form cannot be everything. Form is an abstract principle, as indeed is the representation of states in mathematical physics, yet physical substances are not abstract but concrete. There must be some non-abstract principle that unites with the form to give an actual substance. This principle is matter.

These principles, to my mind, define an Aristotelian philosophy. Accept them (or something sufficiently close to them; there is room for quibbling over the precise definitions) and you are a disciple of Aristotle, in one school or another (there has, of course, been a lot of variation in Aristotelian thought beyond these points). Deny at least one of them, and you are not an Aristotelian. And yet we have seen that all of them are pretty much a consequence of quantum field theory. This suggests that if quantum field theory is true (and all the evidence we have thus far suggests that it is), then so is some form of Aristotelian philosophy. Of course, that still leaves the question of which form of Aristotelian philosophy (and it will not be exactly the same as Aristotle's or any of his major medieval disciples once we go down to the finer details), but nonetheless it would still be recognisably Aristotelian.

If true, this means that the Aristotelian interpretation of quantum physics is not one interpretation among many (Copenhagen, many worlds, De Broglie-Bohm, Quantum Idealism and so on). It seems to be forced on us by the physics. Of course, it might be that some of those other interpretations have an Aristotelian version, in which case I haven't said anything against them. And, of course, I have made a few philosophical assumptions which can be questioned (as opposed to the physical assumptions I have made, which are validated by experiment), particularly involving the way in which I claim that physical states are represented mathematically.

Of course, what I have described here is just the simplest possible quantum gauge theory, quantum electrodynamics. There is more to the real world than just that. But we can add in the other known forces (except gravity), and it doesn't change much. They are described by quantum field theories, and the basic philosophy doesn't change. Likelihood no longer maps to the circular U(1) group, but a more complicated group, and we have more particles: the neutrinos, quarks, weak interaction gauge Bosons, gluons, and scalar Boson. But the basic picture of how they interact with each other and evolve in time doesn't change, and so neither does the philosophy.

One thing that does change is that we have more particles to play around with, which means that we can easily make compounds of them. Each compound substance has its own creation and annihilation operators, which can be constructed using a suitable transformation of the basis of the underlying quark, lepton and gauge Boson operators. These transformations are not linear; a pion is not simply the sum of two quarks; its creation operator does not simply reduce to the two quark creation operators. If that were the case, we could question the utility of form: everything would, in principle, reduce to its constituent parts. Describing those parts would be enough. But that is not what happens. A proton is not simply three quarks and a couple of gluons. It has its own creation operator, which is in complementary basis to that of the quarks and gluons. Since it is in a new basis, and operators in different bases don't commute with each other, it cannot be reduced to its constituent parts. One cannot look at a proton and say "There is a quark." The quarks are absorbed into it in such a way that they loose their own individual identity. The proton is a new substance in its own right. We can use the creation operators of the pion (in its various different states) to construct state vectors that represent pions. From these state vectors, we can identify the form of the pion as being in part the set of all the pion states. This form is distinct from (for example) the proton, because it cannot be connected to the proton via a continuous deformation of the state indices. Thus we can identify representations of the form of the proton and form of the pion and distinguish between them.

The same principle that allows us to construct creation operators for simple compounds such as pions and protons has been extended to other sub atomic particles, simple atoms, and large crystalline substances such as metals. It could in principle be done for things like water molecules, butane molecules, elks, whelks and trees (the only reason it hasn't been isn't that it is impossible but that it is too difficult, and the classical and quantum mechanical approximations are good enough for these objects of study). Thus incorporating the other field theories only enhances the power of this approach, and makes the philosophy of physics more Aristotelian.

So that just leaves gravity. The embarrassing step-child of theoretical physics, which doesn't quite fit in with the others. But I have written enough for the moment, so won't discuss gravity here.

The mechanical philosophy against Quantum Physics

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