The Quantum Thomist

Musings about quantum physics, classical philosophy, and the connection between the two.
The failure of nominalism

Aquinas' first way and modern physics?
Last modified on Mon Jul 23 23:35:24 2018

The first and more manifest way is the argument from motion. It is certain, and evident to our senses, that in the world some things are in motion. Now whatever is in motion is put in motion by another, for nothing can be in motion except it is in potentiality to that towards which it is in motion; whereas a thing moves inasmuch as it is in act. For motion is nothing else than the reduction of something from potentiality to actuality. But nothing can be reduced from potentiality to actuality, except by something in a state of actuality. Thus that which is actually hot, as fire, makes wood, which is potentially hot, to be actually hot, and thereby moves and changes it. Now it is not possible that the same thing should be at once in actuality and potentiality in the same respect, but only in different respects. For what is actually hot cannot simultaneously be potentially hot; but it is simultaneously potentially cold. It is therefore impossible that in the same respect and in the same way a thing should be both mover and moved, i.e. that it should move itself. Therefore, whatever is in motion must be put in motion by another. If that by which it is put in motion be itself put in motion, then this also must needs be put in motion by another, and that by another again. But this cannot go on to infinity, because then there would be no first mover, and, consequently, no other mover; seeing that subsequent movers move only inasmuch as they are put in motion by the first mover; as the staff moves only because it is put in motion by the hand. Therefore it is necessary to arrive at a first mover, put in motion by no other; and this everyone understands to be God.


This argument is a variation of the cosmological argument, one of the most important philosophical arguments for the existence of God. I am writing this post in response to a criticism that was brought to my attention on the Thomas Aquinas facebook group that claimed that this argument was effectively disproved by modern physics.

I'll get to the objections in a moment, but first of all I want to outline the main steps of Aquinas' argument.

  1. Some things are in motion.
  2. Everything in motion must have been put into motion by something else.
  3. This chain of movers and moved is a hierarchical rather than an accidental series.
  4. Every hierarchical series must have a primary member; in this case something which moves others but is not moved itself.
  5. Therefore there must be at least one unmoved mover.
  6. Therefore there must be at least one unmovable mover.
  7. This being is God.

Many critics pounce on the last two steps, and ask how we get to God from an unmoved mover. In part, it is down to definition. The classical definition of God is simply a being of pure actuality, that is to say something devoid of potentiality, and therefore incapable of change. The second part is that God is act, i.e. interacts with the world, so moves others. So, by definition, God is an unmovable mover. To anyone who complains that the definition of God must include attributes such as omnipotence, omniscience and so on, that's your choice. We can quibble over definitions if you like. It's no obstacle to the theist, because Aquinas (and just about everyone else who has wielded the cosmological argument in any more than a brief article such as this) has shown that such attributes follow directly from the definition. I devote chapter 13 of my own book here to the task, and others have done it in more detail and rigour than I did. An important part of that argument is to overcome the other obvious objection to this part of the argument, namely why must there be only one unmoveable mover (how do we get from at least one at the end of the cosmological argument to a singular God).

But that's not the topic I want to discuss here. All the cosmological argument does is show that there is at least one unmovable mover; to go beyond that requires further argumentation. So let us keep that as our immediate goal. I am here more concerned with the first few steps of the argument. And before I get onto them, I need to unpack them and explain them to those not familiar with the philosophy.

I'll start with the distinction between a hierarchical series compared to an accidental series. Each of these represent chains, in the simplest and easiest to visualise form, where one member is linked to another which is linked to another and so on. The only direct interaction between members of the chain is with their neighbours (in the language of physics, the direct interactions are local). But what about indirect influence? So let us say that A is a member of a chain and Z is another member, and they are not neighbours. Does A influence the state of Z, with everything between them acting as instruments? In the specific example of the series of movers, can the motion of Z ultimately be attributed to the power of A to move? There are three possible answers to this question: yes, no, and in part. A hierarchical series is one where the answer is either yes or in part.

An analogy that I like to use is that of a series of train carriages. Suppose that the carriages are moving. The carriage we are in is pulled by the carriage next to it. That carriage is pulled by the carriage next to that. And so on. Eventually, you get to an engine, that is to say something that is capable of pushing the carriage next to it without being pushed itself. Now, if the carriages are in motion, then there must be an engine. So if there is one carriage in motion, it must be coupled to an engine. If there are two carriages, at least one of them must be coupled to an engine. If there are N carriages, then at least one of them must be coupled to an engine. And this remains true even when we take N to infinity. The engine is what I have called the primary mover. (This translation of Aquinas calls it the first mover, but that translation is slightly misleading to the modern ear, since the first merely implies the end of the chain, and not this special property of being able to move without being moved itself. The chain could be infinite, but it would still need a primary mover.) So this is a hierarchical series.

One of the examples that Aristotle used to illustrate the series was a hand moving a stick moving a stone, which perhaps moves another stone and that another stone and so on. The point is that the stones themselves cannot initiate the motion, no matter how many stones you have. There must be a hand at the start of the series. Aquinas refers to this analogy in the first. The students who first read his work, who, at this stage of their education would have been experts on all the Socratic philosophers including Aristotle, would have immediately picked up on the reference and understood his meaning. In our degenerate days with a different educational emphasis, the concept needs explaining. Which is why I have just explained it.

An accidental series would be like a group of train carriages coupled together without an engine. Here one could (in principle, if not in practice) have an infinite number of carriages without an engine. Perhaps the carriages are stationary. Perhaps they each have their own in-built engines. In either case, there doesn't need to be a primary mover pushing the system along.

So what distinguishes a hierarchical series from an accidental one? There are various possibilities.

Hopefully, it is clear from my examples that a hierarchical series needs to have at least one primary member (while that is not true for an accidental series).

Every cosmological argument sets up a sequence of some sort, claims that that sequence cannot continue indefinitely, concludes that the sequence must have a first member, and identifies that first member with God. This business with the hierarchical series is how Aquinas (and Aristotle before him) argued the sequence could not continue indefinitely. Now, I should hasten to add that this is not the only argument one can use to justify that step. But it is the argument used in the first way, so it is what I will focus on here.

The second concept I need to discuss is that of motion. This word is confusing, because Aristotle and Aquinas meant something very different by the word than we do today. Our understanding of the word is a legacy of the early mechanists and the enlightenment, who assumed that the only type of motion (in the sense that Aristotle meant the word) of physical relevance was local or locomotion, i.e. movement from place to place. We have inherited that, and when we think of motion, we think of locomotion. But when we come to the first way, we then get confused. I will put the source of the confusion clearly: locomotion is not motion in the sense used by Aristotle and Aquinas.

That statement would come as a big surprise to the Aristotelian philosophers. But that's because their physics was wrong. They had a version of Newton's second law, which was that force is proportional to velocity. Newton, of course, proposed that force is proportional to mass times acceleration (equal to, if one chooses convenient unity to measure things in). The Aristotelian physicists (not, I think Aristotle himself -- I am being anachronistic in my language; but I am talking about his late medieval successors) would define force as something which induces a motion. So in Aristotle's physics, motion can be equated to velocity. Newtonian physics removes that link. Something has got to change somewhere. We either have to no longer equate motion with velocity, or no longer define force as something that induces a motion. The early modern philosophers chose the second option. I think that was the wrong decision. In doing so, they undermined the deeper classical concept of motion, and made it impossible for their followers (which includes people of our generation, since English and other modern languages developed to follow their lead) to understand arguments such as the first way.

Of course, Aristotle's physics is not as bad as it might seem to someone who has had too many classes on classical mechanics. Force is proportional to velocity is how things seem to work in the world around us. That's what induction from a basic experiment will tell you (and, presumably why Aristotle came up with the idea that I expressed in that formula). This is because of friction. Friction is a force acting on a body travelling through a gas or a liquid that is, to a good approximation, proportional to the velocity of that body. To keep the body moving at constant velocity, one needs to apply an equal force in the opposite direction. If one only measures the force one directly applies, and neglects the friction, it will indeed appear as though force is proportional to the terminal velocity of the being (with the constant of proportionality dependent on the density of the medium, the mass of the body, and also its coefficient of friction determined by its shape and texture, among other things).

But, of course, we live in a post-Newtonian world, and know that force is instead associated with acceleration. (Indeed, we live in a post-quantum world, and know that even this is too simple.) So now it is a matter of translating Aristotle's terminology into that reality. Which means delving into what the concept of motion really meant to the Aristotelian philosopher.

The concept, as Aquinas stated in the first way, is derived from the concept of actuality and potentiality. In addition to actual existence and non-existence, something can exist potentially. That is that it exists within the being as a possible state it could change into. Every being could exist in a multitude of stable or meta-stable states. One of those states is actualised at any given time. Motion is when a being stops being in one state and moves to another. One of the potential states is actualised, and what was the actual state ceases to be actual and becomes potential.

Each meta-stable state has a natural motion towards a stable state. In Aristotle's physics, this stable state is to be at rest in relation to the centre of the universe (which is the centre of the earth). Now, obviously this is wrong. Newtonian physics, on the other hand, posits that bodies have a natural tendency to continue at the same velocity or momentum (mass times velocity). The state is defined by location and momentum. Thus a body will naturally maintain a constant momentum, and change its location at a constant rate.

In addition to this natural motion, we have artificial motion. This is when something is forcibly moved from one state to another by an external agent, against its natural tendency. In Aristotle's terms, an example would be when we lift a stone off the floor. In Newtonian physics, this might be when we divert the course of an Imperial Star Destroyer by ramming our corvette into it.

The first way was originally conceived in a world-view that accepted Aristotelian physics. Indeed, it was originally conceived by Aristotle himself. Defenders of the first way, however, state that the philosophical basis stands apart from that physics, and can be adapted to modern physics. In that case, it should be possible to translate the first way into a language more amenable to modern science. In particular, we are going to have to adapt what is meant by the concepts of motion and thing to remain as true as possible to the spirit of what Aquinas meant by those terms in the light of subsequent developments in physics.

There are two main questions which need answering:

  1. Is it true that whatever is in motion is put in motion by another.
  2. Is it true that the chain on movers and moved is a hierarchical rather than accidental series?

The first way and Newtonian Physics

The objection.

I was asked to write this post in answer to a comment by an internet user, who claimed to have found a way to refute the first way. (Actually, I wasn't asked to write this post, but only to comment on the objection, but this is my way of commenting). The objection is the following:

The First Way also hinges on the assertion of the impossibility of an infinite regress.

The erroneous world-view of Aquinas is what leads him to a hierarchical regress analysis in his first three ways.

If an object in uniform linear motion required "another" to act upon it to sustain its motion then a hierarchical regress would be called for.

If an object needed a cause to persist in existence then a hierarchical regress would be called for. But as someone pointed out "Whoa, where did that come from?"

An object in uniform linear motion persists in motion because there is no change in its kinetic energy to do so, and no change means no changer is necessary to account for observed uniform linear motion.

An existent object persists in existence because there is no change in its mass/energy to do so, and no change means no changer is necessary to account for observed existential inertia.

If an object in uniform linear motion had to be externally acted upon to maintain its uniform linear motion the First Way would make sense and be a very powerful argument for a hierarchical first mover in the present moment.

In that case the uniform linear motion of X1 would require an X2 to act upon it. Then X2 would require an X3 to act upon it, and so forth, calling for a first mover that acts upon Xn without itself being acted upon.

However, it is manifest and evident to the senses that uniform linear motion persists without any external actor.

To act upon an object in order to move it is to apply a force to it. The application of a force to an object that imparts motion transfers kinetic energy to that object and accelerates that object approximately by F=ma.

An object in uniform linear motion does not change in its mass/energy. Since it does not change in its mass/energy there is no necessity of an external actor upon that object, because no change calls for no changer.

Thus, the First Way is false as an argument for the necessity of a first mover to account for observed uniform linear motion.

Thus, I have utterly destroyed the First Way. I have utterly destroyed the Second Way. As arguments for the necessity of a hierarchical first mover both fail under my clear and unrefuted arguments.

This objection defines motion as linear motion in a straight line and at constant velocity. It then states that this motion continues interrupted. Changes in this motion are thus disjoint. It is not necessary that the thing moving the object is at that time being pushed by something else, as in the example of the stick pushed by the hand and pushing the stone. Indeed, it is not necessary that there is any outside mover at all. If there is any series of movers, it is disjointed and thus not hierarchical. Therefore there is not the sort of hierarchical series of movers needed by the first way. Therefore the first way is false.

The response.

The first way and Newtonian Physics has been discussed by Edward Feser in an article The medieval principle of motion and the modern principle of inertia , and also an essay Motion in Aristotle, Newton and Einstein contained within his Neo-scholastic Essays. One of his suggestions was that we should not consider a being in an initial state as being in motion. By definition, something in a stable state is not in motion. Instead, we should be thinking of changes in state, which implies acceleration.

Now I am here discussing Newtonian physics (as in the physics of Newton's day), where forces are transferred by physical contact. I will discuss electromagnetic and gravitational forces in later sections. So acceleration is caused when something pushes against it. Now that thing doing the pushing must also be accelerating. Either it is a rigid body, or perhaps a body that was compressed and is now expanding back to its original state -- but even there, the part that is expanding is being pushed by the released energy, which is itself the result of something else changing, and so on (although we soon get down to the molecular and atomic level and thus quantum physics, which I am not discussing at this point). Now this is precisely the sort of series that Aquinas and Aristotle were referring to in the first way.

Thus as long as as we define motion to be acceleration rather than loco-motion, which is truer to Aristotle's definition of motion as being from one meta-stable state to another, then we run straight into the requirements of the first way. Newton's first law guarantees that everything that is moved (i.e. accelerated) is moved by something else, and the series we get is completely analogous to Aquinas' example.

The primary error of the objection was to assume that because motion and locomotion can be equated in Aristotle's physics, they should also be equated in Newtonian physics. That assumption was not justified, and I have argued that it is, in fact, false.

Further objection

In Newtonian physics, a state is defined by two, and only two, parameters: location and momentum. However, the argument above defines states in terms of a single parameter, momentum. Therefore locomotion is still a change from one state to another, and thus counts as a type of motion. The first way can't be used to describe this change of state.


Aristotle, of course, knew nothing of the importance of momentum. He only described location and linear velocity. Thus states (for Aristotle) were determined by location -- not only by location (he considered other types of change, such as change in temperature), but not including velocity. Aristotle also had this distinction between natural and artificial motion. But the two concepts in Aristotle's thought are somewhat confused. They act on the same parameter space. Newtonian mechanics places a clear distinction between them. Natural motion affects the location of a particle. Artificial motion results in a change in its momentum.

The existence of natural motion doesn't invalidate the first way. If it did, one wouldn't need to appeal to Newtonian physics. Both Aristotle and Aquinas acknowledged it. The first way only requires that artificial motion forms a hierarchical series. That there is also natural motion does not preclude that the sort of motion needed by the first way also exists. Indeed, it could be thought that Newtonian physics in this respect strengthens Aquinas' hand by making the distinction between the two types of motion clear.

This objection is fully resolved in quantum physics, because there the states are solely described by momentum, as discussed below. We are left with a system entirely analogous to Aristotle's physics.

Further objection

Natural motion in Aristotle's thought is characterised by motion towards some end. Inertial motion continues indefinitely. Therefore the two can't be equated.


As already seen with the concept of motion, when adapting Aristotle to modern science, we have to divorce some of his categories with the physical consequences that he applied them to, and reattach them to different concepts in physics. In Aristotle's physics, natural motion has a tendency towards an end. For example, the natural motion of a brick (which is predominately made of the element of earth) is towards the centre of the universe. In Newtonian physics, we keep the concept of natural motion, but change its end.

But doesn't the concept of an end indicate a degree of finality? A body undergoing linear motion in Newton's physics continues to infinity, and infinity does not have an end. But Aristotle was comfortable with the idea that natural motion could continue forever. Quintessence, the material which (in Aristotle's view) the stars and planets were made of, had a natural motion of circling the centre of the universe. It would continue indefinitely, rather than reaching some final destination. Again, fire had a natural motion away from the centre of the universe. No end there. So it is not true that natural motion necessarily implies an end.

Nor is it true that there are no ends in classical physics. It is not true for complex systems of particles; thermodynamics states that these tend towards a state of statistical equilibrium or maximum entropy. General Relativity changes the concept of inertia. It is no longer that things naturally travel at constant velocity unless an external force is applied, but that things naturally travel along geodesics of the metric unless an external force is applied. Geodesics need not be straight lines with constant velocity. That is only true in flat geometries such as the Euclidean geometry that underlies Newtonian physics, or the hyperbolic geometry that underlies special relativity. Geodesics in the more complex geometries of general relativity can be circles, or lines heading into a black hole. When something falls under the force of gravity, it is undergoing natural motion along a geodesic. It seems that Aristotle's and Einstein's theories of gravity are far closer than many would give them credit for. The difference, of course, is that (were the earth not there, and there was no friction) Aristotle's falling body would come to a rest at the centre of the earth, while Einstein's will oscillate around that centre. But both are following natural motion, as Aristotle defined it.

The first way and Maxwell-Faraday Physics


My internet friend didn't reach this far, but if we go a couple of centuries beyond Newton, we have a more serious difficulty for the first way than the principle of inertia.

The argument above assumed that acceleration was only caused by contact. However, we know that that is not true. The electromagnetic force is a clear example. Here we have force at a distance. Each charged particle generates by itself an electric force which draws other charged particles into it or away from it. Thus a charged particle can move others based on its own inherent power.

This means that while there might be a series of movers and moved required by the first way, it is not the hierarchical series needed by Aquinas in which most of the members are inert intermediaries between God and the object being moved.

The basic problem are these. Firstly, the force between charged particles is mutual and mechanistic. There is no asymmetry; no direction of causality. Indeed, it is not clear that the concept of causality is useful here. Secondly, the particles generate the force themselves. Every charged particle generates an electric force. It does not need to be moved itself in order to move others. It just does it automatically.


As discussed, there are two principles in the first way which need defending, with everything else basically following from these or the definitions, or being self-evident. The first is the principle that every motion needs a mover. I have argued that this remains true in classical physics; indeed, it is demanded by Newton's first law once we have defined our terms correctly. The second notion is that the sequence of movers and moved is hierarchical rather than accidental. While a lot of responses to the first way concentrate on the first premise, it is this second one which in my view is the harder of the two to justify. This objection targets the second premise. It acknowledges the existence of motion (correctly defined as acceleration); acknowledges the sequence of movers and moved, but denies that that chain is a hierarchical series and instead asserts that it is an accidental series.

In my answer here, I will be using Maxwell and Faraday's conception of electromagnetism, and bring in elements from special relativity.

Certainly, Aquinas' example of the stick pushing the stone no longer seems relevant in the context of the electromagnetic force. It is not the case of one thing being moved itself and on account of that motion pushing something else. So is there some other way of determining whether the series is hierarchical or accidental?

First of all, we need to be more specific in describing the electromagnetic force. Maxwell and Faraday established that the early mechanists got something wrong. They believed that the only things that influenced physics were corpuscles, location and motion. Corpuscles are discrete and localised. Localised means that they only exist at one point in space. Discrete means that the number of particles has to be an integer. It is not a generic real number. Faraday and Maxwell introduced the second concept of fields. Classical fields are continuous and de-localised. De-localised means that that they are spread across all space. Continuous means that the intensity of the field (the closest analogy to particle number) can take any real value.

Note that the physical field is not the electromagnetic force, usually denoted by E or B in the physics textbooks, but the electromagnetic potential. The potential energy of the electric force is usually denoted by φ, and the magnetic potential as A. The relationship is that the force is the variation of the potential field in space (E = φ; B = × A).

The electromagnetic force is an interaction between corpuscles and fields. Variations in field strength generate forces on corpuscles. The fields themselves, in the absence of corpuscles have a natural motion described by Maxwell's equations. The motion of particles induces changes in the fields. Each interaction is localised, so it comes through contact. So there is no action at a distance. So one could say that the particles are pushed by the fields, which in turn push the particles.

Now we have to think about how this translates into Aristotelian terminology. It seems clear to me that we have to treat both fields and corpuscles as different types of matter. Fields move further away from Aristotle's vision of matter; but that vision is a physical application of the underlying philosophical principle, and we are trying to ditch the original physical application to preserve the philosophy as much as we can. Fields exist (this can be demonstrated experimentally), they are not supernatural, a mathematical artefact or pure form, therefore in Aristotle's terminology they must be a kind of matter. The fields exist in various states; these correspond to the different possible values of the potential. Natural motion for electromagnetic fields can be towards a flat potential or in the form of oscillating waves. Artificial motion is movement from one field strength to another, induced by an external source, a corpuscle.

So once again, we have a sequence. A particle induces motion (change of state) in a field in its vicinity. The field at that location moves the field at the point next to it to a different value, which again moves the next bit of the field, and so on until we reach the next particle. That particle is then moved (accelerated) by the field, and in turn induces a further change of state in the field. All of this happens 'simultaneously'. Although we have to redefine a little what is meant by 'simultaneous. The motion propagates along the field at the speed of light, and in that perspective it might not seem to be simultaneous. However, the notion of time is dependent on one's movement through space. The faster one moves, the slower one's perception of time. If we are sitting on the wave of motion in the electromagnetic field, it will seem as though time does not pass; everything is instantaneous. So simultaneity depends on the reference frame. But whether a series is hierarchical or accidental is not frame dependent.

One of the determining factors of whether a series was hierarchical or accidental was simultaneity of cause and effect. However, in special relativity (and Maxwell's theory of electromagnetism is taking us to special relativity), the concept of two events at different places being at the same time is a difficult one to define. The naive interpretation, inherited from Galileo (and Aristotle) posits an absolute time, which is the same across all space. However, measurement of duration depends on choice of coordinate system, and this is not absolute. Each observer can choose their own system. Given that the speed of light is absolute (the same in all inertial reference frames), observers travelling at different speeds experience time differently. This contradicts the notion of an absolute time; and without that, we lose the naive notion of things being simultaneous at different locations. However, we can define simultaneous as meaning being on the same light cone (this was Einstein's definition when developing special relativity). That two things are on the same light cone is not observer dependent. In this sense, we can say that the movement of the field by the first particle is simultaneous with the movement of the second particle by the field. The movement of the third particle by the field will be simultaneous with the movement of the field by the second particle. However, the interaction between the third particle and the field need not be simultaneous with the interaction between the first particle and the field. That concept is not intuitive, but when you turn to non-Euclidean geometry you get a lot of concepts which are not intuitive but happen to be right. We changed the definition of simultaneous; some of the logic associated with the old way of thinking about it is no longer valid. Thus we have local simultaneity between neighbours of the chain, but not global simultaneity between every member of the chain. But I would say that that is enough to maintain the nature of a hierarchical series.

Equally, there is a direction in the sequence of causality. The corpuscle moves the field which moves a corpuscle which moves the field which moves a corpuscle and so on. This direction is not necessarily temporal; that depends on the reference frame. But there is a definite sequence. That sequence is not symmetrical. The way particles interact with fields is different from the way that fields interact with particles. The series is not static. There is motion.

So with regard to the criteria I listed above for identifying a hierarchical, the first, fourth, and sixth conditions are satisfied.

But now we come to the more difficult issues. Firstly, the series is not as obviously a linear chain of mover to moved. Particle A influences particle B through the changes in the field along the line connecting them. The field hits particle B, and then progresses out in every direction, with the field strength getting a boost from particle B, but still with indirect influence from A. That field then interacts with particle C. So there is some indirect influence from A to C, but also direct influence. It is sort of like a stick with two prongs, pushing both the first stone and the second stone, while the second stone is simultaneously pushed by both the first stone and the stick. I don't think that this amendment is enough to affect the underlying metaphysics. All the motion is directly or indirectly dependent on the first member, so that's still consistent with the series being hierarchical.

Then we have the worry that at the same time as A influences B, B is influencing A. If it were truly the same time, then that would put a huge dent in the idea that this is a hierarchical sequence of movers and moved. It is no longer asymmetric.

However, although it is true that A's movement of the field can be viewed as being simultaneous with B's movement by the field, and that B's movement of the field is viewed as being simultaneous with A's movement by the field, this does not mean that A's movement of the field is simultaneous with A movement by the field as induced by B in the same moment that B is moved by the field coming from A. The two events of A moving the field and its being moved are not on the same light cone. The chain of interaction has to move from A to B and then back again from B to A. There is a change of direction in the middle there, which preserves the order of interaction. It's non-Euclidean geometry playing havoc with our intuitive notions again. And, of course, we now have the notion that two members of the chain are the same being but at different moments in time. But that doesn't stop the series being hierarchical; we would not complain if two of Aristotle's stones were identical, and that's analogous to the situation here. So the order of interaction is still preserved, and we still maintain the asymmetry.

The next point of discussion is the requirement that the members of the chain lack the power to induce movement of themselves. It seems that this is violated in the case of classical electromagnetism. Charged particles just have the natural power to move electric fields. Or so it seems.

But this doesn't violate the principle that is needed for a hierarchical series. The stick has the natural power to push a stone; but power isn't activated unless the stick is put into a certain motion between various states. Similarly, the power of the charged particle to move electric fields does not mean that those fields will be moved. A static or constant velocity particle generates a constant field (i.e. one unchanging in time, in which there is no motion). It is only when the particle accelerates that the electric field is changed. A static field (this time meaning unchanging in space) does not generate an electric force on a particle, but only one which is different from one place to the next or changing in time.

Energy in physics is basically just a label of states. For each being, there are numerous possible states it can be in. These states are distinguishable, and so we need to attach a name to each state so we can tell one from the other. Since we are trying to map these states to an algebraic system, we use a set of numbers as the label. Those numbers are known as the energy and momentum. Of course, there is no single unique way of doing this. The choice that physicists use is advantageous because total energy and momentum are conserved in every local interaction. The principles of causality and special relativity guarantee that every interaction is local, and this is true in practice for all the interactions we know of.

In classical mechanics, the situation is complicated because the states are also labelled by location. In quantum physics, this isn't the case. Only energy, momentum, and a few other qualities (which don't really influence this discussion) such as gauge, polarisation and what physicists unfortunately call spin and colour colour (nothing to do with the spin and colour we observe in the everyday world) are required to label the states. But in either classical or quantum physics, motion is movement from one energy state to another. In classical physics, that is one of only two types of motion; in quantum physics it is the only type of motion.

Each particle has a spectrum of possible energy states. In classical physics, this spectrum is continuous; in quantum physics it is discrete in certain bound systems such as the nucleon or the atom. This energy is always positive (this is linked to the direction of time in quantum physics), which means that the energy spectrum of a particle has a minimum value. For a free massive particle, that minimum value is given by E0 = mc2, where m is the particle's mass and c is the speed of light, and I assume that we have chosen our units so that there is no constant of proportionality (this equation works if energy is measured in Joules, mass in kilograms, and the speed of light in meters per second; it fails if energy is measured in calories, mass in grams, and the speed of light in furlongs per Jovian year). So you start with this value, and build up from that a tower of possible energy states.

Note another interesting point of overlap with Aristotle's physics. Aristotle stated that objects had a natural motion towards one particular rest state, for the elements of earth and water the centre of the earth, and could only be raised from that state by artificial motion. Well, he got his theory of gravity and the elements wrong. But now we have another tower of states, and one at the bottom, known as the ground state. Particles have a natural motion towards the ground state, and will drop there unless they are prevented from doing so (perhaps because it is already occupied by other fermions). They can only rise up the tower of states by artificial motion, an external interaction such as the absorption of an incoming photon.

[Also, can we construct a first way argument defining motion as being in a non-ground state. I don't think that this definition is true to the spirit of Aristotle, but it certainly makes the everything can only be raised from the ground state by something that isn't in the ground state part of the argument clear. But we still need to show that it is a hierarchical series.]

Now energy is always conserved. That means that if there is an interaction between two particles (and here I include both corpuscles and fields), then if the energy of the second of them increases, then the energy of the first must decrease. But this is only possible if the energy of the first particle had previously been raised above its minimum value. So the first particle can only move the second if it had previously been moved itself. In the case of electromagnetism, we can start with two static particles and an electric field. The electric field will be in a state of high energy, the particles (being static in our laboratory frame) their minimum energy. Then energy is gradually transferred from the electric field to the particles as they approach each other.

That energy had to come to the electric field from somewhere (perhaps a previous motion of the particles, as we put them into place), which had to come from somewhere else, and so on. So just as the stone always has the power to move another stone, but that power has to activated by the stone itself being pushed (moved out of its minimum energy state), so the electric field has the power to move the corpuscles, but needs that power to be activated by it initially being moved into a higher energy state.

So it is not true that the particles solely act on their own power. They act because they had previously been moved.

The only question which remains is that word previously? Does that rule out the simultaneity needed in a hierarchical series? No. Firstly because the starting position of this experiment is artificial. We are holding the two corpuscles in place; applying a force to them. By doing so, we set up another hierarchical series of movers and moved, only this time involving ourselves. Secondly, previously here means previously in the chain of causality rather than previously in time.

So we seem to have satisfied the conditions listed above for a hierarchical series. However, this question is one which I would like to see a professional philosopher give more attention to. I don't tend to see it, either by detractors of the first way (who don't tend to understand the argument well enough to get to this point), or defenders of it (who don't tend to give a strong defence of this point in terms of the actual interactions described by modern physics; mostly they just use the basic contact pushing model of early classical physics rather than the electromagnetic picture that classical physics adopted when it reached its maturity).

Existential inertia

The objection.

Back to my internet adversary.

Existential inertia of material is manifest and evident to the senses. Modern science calls this the conservation of mass/energy, described with E=mc2. We never observe new mass/energy persistently coming into existence, nor do we ever observe mass/energy persistently ceasing to exist. The amount of mass/energy or material remains constant, thus, material does not change in its existential respect, only in its structure, or shape, or organization, or form.

Since the existential respect of material does not change no changer is necessary. One can speculate that an unseen changer is actually changing material in just the right way so that it appears to our senses that no change in the existential respect of material is occurring, but such a speculation is not necessary, and thus the Second Way fails as an argument for the necessity of a hierarchical first mover acting upon matter to sustain it in existence in the present moment.


Edward Feser again discussed existential inertia in his Neo-Scholastic essays, but this time I will go my own way. The questioner badly misunderstands the physics (and advanced physics is one of Feser's weaknesses as well. He needs to read my book). I will say it again: Energy is just a label used to distinguish different stable or metastable states. Nothing more. Nothing less. This definition can be used in classical physics, but isn't usually, because classical physics is usually taught without bringing in notions of states. And that's fine, for the physicists. One can use the language of states to classical physics, but it adds to the work that physicists have to do, without really aiding the calculation. The point is that classical physics is consistent with both the (stateless) mechanistic and the (requiring states) Aristotelian philosophy. But it was developed by mechanists who were trying to find a way to avoid Aristotle. So their notation and philosophy is usually used to teach it. Of course, leaving states out had huge philosophical consequences, but most classical physicists didn't care too much about that because they didn't subscribe to Aristotelian philosophy. So in the traditional way classical physics is taught (and the way it historically developed), energy is just some mysterious number that happens to be useful in the equations.

But when we come to quantum physics, particularly the more advanced quantum field theory (QFT), the notion of states is unavoidable. The energy operator (the Hamiltonian) is the time evolution operator. Thus stable states are states of constant energy.

Now if energy is just a label, then it is not a thing. The things are the particles themselves; the electrons, the photons, the quarks and so on. When we discuss motion, we discuss one particular thing moving from one energy state to another. The only requirement that we need when two or more particles interact is that the sum of the numbers used to label the energy of each particle remains constant.

Now, it can be that the energy of the initial particle is larger than the minimum energy required to create new ones. For example, suppose we have a photon which has the energy a little over twice the electron mass. Among the rules we have to describe possible interactions are conservation laws: of spin, energy/momentum, electric charge and so on. The decay of the photon into an electron and anti-electron is consistent with these rules. Thus it can happen. And it does happen. All the time. In fact, every interaction in quantum field theory -- thus every interaction in physics except (possibly) gravity -- is of this type. At least one particle is created or annihilated in each interaction; the remaining particles (if not annihilated themselves) move into a different state (i.e. have a different energy).

The objector is thus quite wrong. Firstly, he identifies mass (a property essential to each type of particle, ultimately linked to the strength of its interaction with the Higgs Boson) with energy (a label describing the accident of which particle state is actualised). Secondly, he states that because energy is conserved, there is no change in the universe. But change is precisely when energy is transferred from one particle to another. That the total amount of energy is conserved is irrelevant. Thirdly, he states that mass/energy doesn't change in its existential aspect. But it does. An electron hurries along in a state with an energy more than three times its rest mass. It emits a photon. The photon decays into an electron/anti-electron pair. You started with one particle. You now have three. New beings have come into existence. If that's not a change in the existential aspect of the universe, I don't know what is.

This notion of particles being emitted or absorbed is precisely what Aristotle meant by efficient causality. Efficient causality (rather than the alternative versions of causality introduced during the enlightenment which focussed on events) links one particle, or more precisely one particle state, with another. Take the example I gave in the previous paragraph. The efficient cause of each of the the particles in the electron/anti-electron pair is the photon. The efficient cause of the photon is the initial electron. The efficient cause of the initial electron in its final state is that electron in its initial state. [A particle can, of course, have more than one efficient cause, when defined in this way; or rather the efficient cause can be a set of rather than a single particle]. Event causality is dead because of the spontaneous nature of particle creation. But the notion of efficient causality, at least in this form, is in a stronger evidential position now than it has ever been in human history.

One of the problems that the person I am responding to had is that he was still living in a mechanistic framework, where the fundamental building blocks of matter could not be created or destroyed, only rearranged. I don't think that his objection stands even in that framework. Particles still move from one state to another. But his knowledge of physics is a hundred years out of date. The mechanical philosophy is dead, buried, mourned for, had its obituary published in the Times, dug up by archaeologists, and had its rotting remains displayed in the museum of ancient artefacts as a prime example of philosophical dead ends. Classical physics was its midwife; quantum physics its undertaker.


[This is one of mine.] But is the chain of efficient causes a hierarchical rather than an accidental series? It seems that it is only an accidental series, because the steps in the chain are not simultaneous with each other. It is not like the stick simultaneously moving the stone which simultaneously moves another stone which simultaneously moves another stone, and so on. Instead, we have the electron gaining energy and then at a later time emitting the photon, which then a little while later decays and so on. There is a gap in time, so the series is more like a father begetting a son, who begets another son, and so on, a series which is usually described as being accidental.


I raise this objection, because I would like to see a professional philosopher address it. This is something of a weak point in my book (one which I hope to address in a future revision). I say that the series of efficient causes can't be infinite by falling back to an argument from the second law of thermodynamics. It has the weakness that it is an argument drawn solely from physics (i.e. the natural), and there is a jump needed to get to the supernatural. I use a variation of the Kalaam argument to get to a supernatural cause, then jump back to Aquinas and his third way to get from there to God. Now I think that this combination of arguments works, but I still find it unsatisfying. The thing that stopped me from using the second way (or argument from efficient causality) all the way to God was this doubt that I could prove that the chain of efficient causes is a hierarchical rather than an accidental series, by using examples from QFT rather than the baby physics that Aristotelian philosophers tend to illustrate their arguments with.

So if anyone has any ideas, I would be very grateful for the opportunity to plagiarise them.

However, there is one thing I can say which is of importance. In quantum physics, the energy and time operators don't commute. That means that time is undefined for a being in a definite energy state (possibly one can connect this to the Aristotelian notion that time is a measure of change, so something in a stable state doesn't change so doesn't have a notion of time). The sequence I drew up of an electron emitting a photon which in turn decays was expressed in the energy/momentum description of reality. In that description, time is undefined. We can't say that the interactions are instantaneous, simultaneous, or at different times simply because there is no concept of time in the illustration (at least for free, bare, unrenormalised particles -- which are rather unphysical in themselves. For bound and complex states, it is a little different). To get back to a space/time formulation, we need to integrate over all possible energy states (and thus in this case the energy of a particle is undefined). There is an order of causality, but not an order in time. I don't know if this would help or hinder the argument about whether this is a hierarchical or accidental series, but it is something to bear in mind. [It's another place where quantum physics defies any attempt we have to imagine or illustrate it with macroscopic examples].

Indeed, since we are discussing special relativity, we face the same problems in defining simultaneity as we did in classical electromagnetism. We could fall back on the same definition (two events on the same light cone are at the same time), and then we could say that the creation of the electron/anti-electron pair is simultaneous with the photon being emitted. But that doesn't address the issue of the efficient cause of the original electron. Is that in some sense inconsistent with it being an accidental series of causes simultaneous with the photon decay later in the order of causality? Does that matter?

Secondly, energy is still conserved. This is of the same importance as it was in classical physics. Although every electron has the power to emit a photon, that power can only be activated if it has enough energy. It had to get that energy from something else, and that thing from something else, and so on. So even though the emission of the photon is spontaneous, the electron by itself cannot do so; it needs to have been moved into the right state.

Note also the initial comment from the person I was replying to:

If an object needed a cause to persist in existence then a hierarchical regress would be called for.

This statement, if it can be proved, might be a different way forward. Obviously, it was originally stated as an argument against the hierarchical regress; the claim was that things don't need a cause to persist. But is that claim justified in the light of QFT?

One thing is clear: things don't naturally persist in existence. That's the whole point of QFT. There is a continual process of creation and annihilation going on. That means that there is a genuine question of why things persist, or at least seem to persist. Indeed, the propagator of an object (which is the part of the Hamiltonian which leads to the amplitude staying in the same state) is made of an annihilation operator and a creation operator for the same state. Whether that means that in reality the particle is going in and out of existence, or that is just an artefact of the mathematical prescription is harder to say. But there is nothing in the QFT toolbox which states that things must persist. Spontaneous destruction is quite possible. Now, physicists will shout at me here and say, "What about physics? What about locality? The Hamiltonian contains both the creation and annihilation operators, and there are various good reasons why it would be inconsistent if it were otherwise." I would agree in everything said here except that it answers the question. Physics just provides a description that things do seem to persist; it doesn't provide an explanation of why; nor show that no explanation is needed. We have to turn to the philosophy of physics. QFT shows us that we can't just turn to some Newtonian concept of existential inertia, because the annihilation operator exists, and represents a genuine physical event. But, in any case, the idea of existential inertia is not something we should blindly accept, because the process of spontaneous emission shows that things don't have existential inertia. They don't naturally remain in the same state if they are left alone.

So it seems likely to me that the same arguments that apply in classical or Aristotelian physics also apply here, and this series is a hierarchical series, but I haven't yet been able to prove it to my satisfaction (let alone anyone else's). Any suggestions would be welcome.


Does the conservation of energy preclude God's creation out of nothing.


This is another objection I have put in, this time not because it is something I doubt, but because this response is something I want to say.

For the theist, God is continually active upholding and sustaining the universe (that's the difference between theism and deism). Physics provides a mathematical description of how the universe is upheld and sustained. It is relatively easy for the theist to connect the two together. This, of course, is anathema to the atheist or the deist. They regard as a first principle that the laws of physics operate independently of God. But I have argued that it is easier to deduce the the laws of physics from the theist perspective than the atheist or deist ones. The symmetry requirements follow from God's nature outside space and time, assuming that God is indifferent to the universe. The indeterminacy of Quantum Physics is a reflection of God's free will. The physical constants are constrained by the anthropic principle. Occasionalism is ruled out because God respects the powers and natures of matter.

The principle of conservation of energy arises from the locality of the theory, which in turn arises from the Lorentz symmetry of special relativity combined with the principle of causality. The Lorentz symmetry arises in part from the assumption that God is indifferent to the universe, so there is no preferred inertial frame, so all observers in all inertial frames experience God in the same way, so all observers in all inertial frames experience the same physical Lagrangian. But, of course, most religions and most especially Christianity deny that God is wholly indifferent to the universe. God calls the Church to be a holy and blameless people, and we can't achieve that without His help. Thus we expect signs and events that provide evidence that the assumption of God's indifference is at best only approximate (these events are miracles). There is no reason in these cases for energy to be conserved, since we have taken out one of the premises from which the conservation of energy is deduced. Thus the philosophy on which both physics and theology are built is not inconsistent with miracles in general, and the creation from nothing in particular. The law of efficient causality is still maintained; only this time God would be the direct efficient cause rather than an indirect one.

Remaining objections

The rest of the person I was asked to answer's objection was largely built his initial misunderstanding of physics, so doesn't really go anywhere. But there are a couple of points to make.

All these formulations suffer a major defect in that they posit material changing itself at base in its existential respect. A-T typically argues with naked eye examples, such as a cup of hot water, which is observed to go from actually hot to potentially hot. This is a change. To transition from actual to potential in a particular respect is a change in that respect. In the case of the hot water that appears to change itself but in truth the cup of hot water is not a single material entity; rather it is composed of moving molecules. As those molecules collide with the molecules of the air kinetic energy is transferred in a vast number of mutually causal temporal interactions with no change in the existential respect of the material of the water-air system.

I actually agree with this point. I don't think that it helps the objection, because as I have argued, once we get beyond the molecule "collisions" (not that they really collide, but they transfer energy and momentum through the exchange of photons), the microscopic interactions are not inconsistent with the sequence of efficient causes. But I agree with him that I wish that the philosophers would use examples from quantum field theory. It is at the level of fundamental physics that it interacts with philosophy, and the thermodynamics of coffee cups only indirectly.

When faced with a coffee cup or a banana, people's natural inclination is to say "Well, yes, but mechanistic science has explained all that without recourse to all this nonsense about act and potentiality and final causes and so on." And then they switch off, and don't listen to anything else you, even if you are making a good and sound point that goes beyond the illustration. You need to get to the point where science demands that we introduce notions of potentiality and finality, without any alternative mechanistic explanations. Of course, that's a lot harder to explain. You need twelve chapters describing the physics before you get to the good bit.

In the continual change speculation it may be asserted that material is being continually changed, that is, moved from potentially existent to actually existent. But non-existence has no potential to exist. Non-existence has no potentialities at all.

But, just supposing non-existence could have the potential to exist how would actually existent material be continually actualized to exist? Actually existent material is already actualized in its existential respect. A thing cannot be both potential and actual in the same respect at the same time. So material would need to change itself to not existing, so the unseen changer could change it back to existing, or alternatively simply bring new material into being . Thus, a new universe is continually being created, while the old universe is continually being destroyed. This leads to severe problems with the notion of temporal continuity of self, besides the rather fantastic notion that the universe really was created a moment ago with the appearance of age and all things in just the right state and just the right motion so as to have the appearance of temporal continuity.

This looks closest of his arguments to the description we get in physics. So I will address this one.

Firstly, his statement Non-existence has no potentialities at all. That is just gibberish. Potentiality does not belong to concepts such as existence or non-existence, but to form and substance. So we should say, Can a non-existent being come into existence? Or rather, (since this obviously happens) can the framework of potentiality and actuality cope with this situation? Yes it can. The final causes of an electron include the creation of a photon. The notion of actuality and potentiality is invoked to describe the notion of change of state in a being that continues to exist. But there is no difficultly in tacking on the idea that as well as beings moving from one state to another, there is also creation of new particles.

Are these called out of non-existence? No. Because non-existence isn't a thing. They have the efficient cause of the particle that emitted them. It is not a case of continual creation and destruction of the universe. Firstly, each interaction is local. It doesn't involve the whole universe, but only the particles affected. Secondly, energy eigenstates aren't temporal eigenstates. You don't have destruction then re-creation in the next moment of time. (You might have destruction and then re-creation, or might not, but it isn't in time.) Discussion of moments of time doesn't make sense in this context. Thirdly, particles of the same type are identical. One can't tell if the electron at the start of an experiment is the same one as the electron at the end of the experiment. An electron could continue as it were, or it could emit a photon, which decays into an electron/anti-electron pair, where the anti-electron annihilates with the original electron into a photon which is absorbed into the second electron. We are left with an electron of the same energy and momentum, but it's not the same one we started with. But why does that matter? It is identical to it in every respect, and there is still the sense of continuity contained within the sequence of efficient causes. The original state is still the efficient cause of the final state in either case, and this allows us to maintain a temporal continuity of ourselves.

Thus, the Second Way, the argument from efficient cause, fails as an argument for the necessity of a hierarchical first changer in the present moment to account for the observation of existential inertia that is manifest and evident to our senses.

No the second way doesn't fail (at least, not for this reason). Because there is no such thing as existential inertia.

Is it Hatred or Love?

Reader Comments:

1. Jack Napier
Posted at 05:43:20 Wednesday July 25 2018

What can I say when "thank you" is not enough?

I hope writing this out didn't inconvenience you too much. The good news is that the objection that I brought to your attention was probably the best objection that i've seen. The vast majority of the other one's don't even try to understand any of it. Once, I heard someone ask why the Prime Mover couldn't be a blueberry muffin. (Yes i'm serious. The video is still on youtube)

I'm still shocked that QFT is actually vindicating Aristotle. Interestingly, in the Thomism group, I came across a thesis that "Makes a case for the revival of physics as it was originally conceived, as the philosophical investigation of the meaning of motion"

Would you be interested in reading it?

2. Nigel Cundy
Posted at 21:39:43 Wednesday July 25 2018

Thanks for your comment. I also saw that thesis appear a few days OK, and saved it as something for me to read when I get the time. Looks to be somebody else thinking along the same lines as me, only he got in thirty years earlier ...

3. Callum Savage
Posted at 17:47:54 Thursday July 26 2018

All I can say is brilliant post.

4. Scott Lynch
Posted at 00:12:47 Tuesday August 14 2018

Suggestions for further reflection

Mr. Cundy,

Excellent post! I really enjoy your blog, although it is a bit technical for a lowly engineer like myself. I am also very familiar with Edward Feser’s work and A-T metaphysics, although I am certainly not a professional philosopher.

I have a few questions and comments.

First, regarding your rejection of locomotion as motion in the Aristotelian sense, that certainly could work (and was proposed by Dr. Feser among others). However, I wonder if the traditional Aristotelian sense, which includes locomotion, can be salvaged from QFT and Special Relativity. If I’m not mistaken, the only reason any object does not travel at the speed of light is because the particles are interacting with the Higgs Field which gives it mass. Therefore, couldn’t you say that the object’s change in location is being actualized by the Higgs Field? Where there is no Higgs Field, there is no change (from the perspective of the object, since due to SR, objects traveling at light speed to not experience change). I think that this would make velocity (change in location) much more intelligible. Why is it that an object’s change in location is never discontinuous? The answer is because the Higgs Field is not discontinuous. If there was nothing actualizing the object’s change in position (it was just a brute fact) then continuous motion would seem to be a miracle. I feel like it may be a bit rash to suggest that location is not a real feature of an object and that an object does not change when its location changes.


Taking my previous comment into consideration, one could say that the Higgs Field (as well as any other relevant fields or even spacetime itself) is one of the links in the hierarchical chain. Since momentum and energy are proportional to change in location with respect to time, and the Higgs Field is directly responsible for that change in location with respect to time, it seems that the Higgs Field fills in the gaps between interactions by constantly actualizing the particles change in position. The other component of the energy is the particles rest mass, but that can be considered to be the form of the particle in question. I think it’s also important to note from a philosophical perspective, that eventually one will get to a bottom level of physics that must be explained either by a primary mover or a brute fact (which is not an explanation at all). So my appeal to the Higgs Field is not a true solution; it merely pushes the problem back one level. In order to explain why the Higgs Field is always giving particles mass, you have to appeal to the nature of the Higgs Field, a more fundamental theorem, or a brute fact. We really want to avoid brute facts because they make science a miracle. A more fundamental theorem is fine, but that too pushes the problem back a step and so is not ultimately explanatory (it is simply more useful for making calculations and predictions). Upon analyzing the Higgs Field, it does not take much effort to realize that it is not infinite in the relevant sense (it does not act on all particles, it depends on spacetime, etc.) and is therefore not Pure Act. So eventually, we will either bottom out and need to appeal to God, or we will have an infinite regress without ultimate explanation in which case we will still need to appeal to God.

Third, you made a comment about God being “indifferent” to creation. It is a doctrine of Classical Theism that God is not really related to creation, although creation is really related to God. I recommend you read the Stanford Encyclopedia of Philosophy article on Medieval Theories of Relations. This is considered a non-paradigmatic relationship. It is a great read. This does not mean that God does not intervene in creation (he sustains it after all), it merely means that creation does not emanate from God out of necessity. It is a free act of creation. Per Divine concurrentism, there is a constant “creation out of nothing”. This does not mean that the universe is constantly being annihilated and re-created, rather it means God is preventing the universe from going out of existence. Rather than saying God merely creates the matter, one could say that God creates the Law of Conservation of energy as well. And if that is the case, a supernatural suspension of that law (by a miracle) should not be a problem for God.

5. Nigel Cundy
Posted at 00:00:08 Wednesday August 15 2018

Thanks very much for your comment. You have certainly given me some things to think about.

I'll respond to your comments in reverse order.

1) I've added the Stanford Encyclopedia article to my list of things to read. I agree that my use of the word "indifferent" is not optimal (for one thing, I don't believe it to true even in the sense in which I use the word -- I merely use it as an initial approximation in view of man's rejection of God's providence at Eden). But were I a better writer, I would almost certainly be able to find a better word for the concept.

2) I am not so keen on saying that God creates the law of conservation of energy as well as matter. Firstly, the law of conservation of energy is very different in nature to matter, so while I agree that God is ultimately responsible for it, that is not in the same way as God is responsible for the continued existence of matter. So I think that using the same word in both cases is misleading. Secondly, for the physicist, matter is basically fundamental. The physicist can get as far as matter, but can't go any further without getting into philosophy. The law of conservation energy, however, is derived from locality and symmetry principles. It is not fundamental in the sense that it comes from something both more fundamental and physical. I would rather say that the symmetry and locality principles that give rise to conservation of energy are a (partial) description of the relationship between God and matter, while matter is the object of that relationship.

3) But I would agree fully that God's (continuous) acts of creation are free and not necessary.

4) The question of whether matter is continually annihilated and recreated is an interesting one for me. I personally don't like the idea, but it is the most natural reading of the form of the Hamiltonian (the operator that describes time evolution) of QFT. Of course, that might just be an artefact of the mathematical representation. In any case, there is no observational difference between a universe that is constantly annihilated and recreated by God (although locality ensures that there is still some sense of continuity over that process -- the destruction and recreation would be simultaneous with each other and in the same location), and one where there is continuity between each emission or absorption event. So its not a question which science can answer. Without evidence, I would rather keep an open mind unless I see a clear philosophical argument.

5) Now to your main point. I think that I see where you are coming from, but there are a few things to bear in mind. Don't forget that quantum physics is different from classical physics, and in particular the notion of velocity as understood in classical physics doesn't really exist in the same sense as in classical physics. But one can take the ratio between the momentum and the Energy eigenvalues and relate that to the velocity. After taking the classical limit, we wind up with E^2 = p^2 + m^2 (using units where the speed of light is one), or v^2 = (p/E)^2 = 1- (m/E)^2. From that we conclude that massless particles travel at the speed of light.

The second part of the argument is that in the standard model, the mass of the fundamental particles (the fermions and the particles which carry the weak interaction force) is proportional to their coupling with the standard model scalar Boson (note not the coupling with the Higgs Boson; the Higgs particle is one of the four fields contained within the scalar Boson and emerges after symmetry breaking. So it is related to but different from the scalar field which generates mass). If the coupling constant between (say) the electron and the scalar field were zero then the electron would be massless. And a massless particle would travel at the speed of light. So you are saying that if all these couplings were zero then all particles would be massless and travel at the speed of light and there would be no motion in the Aristotelian sense.

But the Higgs Boson is not the only source of mass in nature. It is not even the most important. The interaction with the Higgs gives mass to the bare, unrenormalised, particles. These bare particles are the ones described in the way that people tend to write down the standard model. These are the particles that physicists start with, because calculations with them are easier. But they are not the particles seen in nature. The point is that they are constantly interacting with each other. The electron can't be separated from its surrounding electromagnetic field. What we observe as the electron is in reality a horrible mix of bare electrons, bare photons and other bare particles. That doesn't change the philosophy -- we can in principle write down creation operators for the renormalised particles -- but it is much easier to calculate the Feynman diagrams and then renormalise rather than the other way round.

There are four basic sources of mass that I can think of off the top of my head:

a) The coupling with the scalar field and electroweak symmetry breaking (i.e. the Higgs mechanism).

b) Binding energy for compound particles. For example, the mass of the proton is about 940MeV. (MeV is a unit of mass which physicists like to use, mainly because it is much easier to write down the number 940 than mess around with factors of 10^{-27}). The proton is made of three quarks, with masses of less than 10MeV, and a bunch of massless gluons. Clearly the bulk of its mass comes from something other than the masses of its constituent particles. It comes from the binding energy primarily due to the strong interaction (electromagnetism and the weak interaction are also involved, but to a much smaller extent). Even if the quarks were massless, the proton would still have a mass of around 920-930MeV (I can't remember the exact figure, but I recall that it is somewhere around there). There are, of course, no free quarks at standard temperatures and densities -- so we, being mostly made of protons and neutrons, would only be a little bit lighter if there was no scalar field interaction. We would still travel at our usual mundane speeds. (Of course, this doesn't apply to the electron which is a fundamental particle. However, I'm not sure that the Hydrogen atom would be stable if the electron were massless. I've only ever calculated its energy levels in the non relativistic limit of quantum mechanics and field theory, and those calculations wouldn't be valid if the mass of the electron were zero and it were fully relativistic. I'm not sure what would happen in relativistic QM and QFT, but I would expect that it wouldn't be pretty.)

c) Self energy. This is basically when a particle emits a photon, and then reabsorbs the same photon, and more complicated processes of the same nature. The net result of this is to add a term to the propagators (which describe the amplitude for a particle to travel from A to B) which plays the same role as a mass. Since you can't avoid this sort of process, you won't get a massless fermion.

d) The process of Renormalisation also affects the mass of particles. Admittedly this tends to be multiplicative rather than additive (so if you start with a zero mass, you might keep it), but it could also play a role in conjunction with the other things I mentioned.

The photon is protected from acquiring a mass in this way through gauge symmetry. But that doesn't apply to the fermions. So we can only sensibly speak about the speed of particles after taking the classical limit of the theory. However, we have to incorporate the binding energy, self-energy and renormalisation before taking that limit (since they are all quantum effects). So even if you tuned the scalar field coupling to zero, you would still have mass, and things would still travel at much smaller than the speed of light.

So there is more to the question of motion than just the Higgs field.

6) Why is it that an object’s change in location is never discontinuous? Primarily because of the locality of the Hamiltonian, which arises as a consequence of the symmetries behind special relativity.

7) I feel like it may be a bit rash to suggest that location is not a real feature of an object and that an object does not change when its location changes. I guess that it does sound like I am saying that location is not a real feature of an object. If so, that is slightly misleading. There are two basic principles: 1) the meta-stable states of free bare particles are expressed in terms of momentum states. 2) A state in a definite momentum state is in a superposition of location states. Those arise directly from quantum physics.

We have freedom in quantum physics to express the basis states in numerous different representations. One of those is the energy/momentum representation. The other is the space/time representation. They are linked together by the Fourier transform (so the creation operator for a particle at a particular location is the Fourier transform of the momentum creation operators). So I was not saying (or if I did it was inadvertent) that location is not a real feature of the object. What I said was that the states are parametrised by momentum alone. We can still extract the location (or rather a distribution of amplitudes of possible locations) by rotating the basis states. For a non-interacting massless particle, this location is equally likely to be anywhere in the universe. Once we switch on the interactions and start binding the particle, the distribution of the possible locations becomes more localised -- which is why particles seem to us to have definite location; we can't directly see at a small enough scale to see the fuzziness.

The idea that something is in a superposition of states is a difficult concept to get your head around. It is completely alien to both classical physics and Aristotle's physics. Aristotle fares better -- we can talk about a particle that is actually in one momentum state being potentially in this location state or potentially that location state. But that still gives some headache to Aristotle, since he would have held that it had to be at actually at one location at any given time, and this notation bypasses that. Another possible way round this (and one I favour) is to say that this description refers to our knowledge of reality only. The particle is really at one location, we just can't know which one until we take a measurement (at which point our knowledge of the particle will shift to it being in a definite location state, and therefore undefined momentum state). Location is a real feature of the world, just not in the way that Aristotle or Newton would have understood it.

However, superposition is probably the hardest thing in the philosophy of quantum mechanics to get your head around. (Different representations of the same data, some in superposition and one not, is the next hardest.) It is where quantum physics gets really weird, and impossible for our imaginations, trained as they are in classical or Aristotle's physics, to grasp. As Feynman might have said, if you think you understand it, then you don't. It is something we can only really appreciate by laying our imagination to one side and focussing on the mathematics. Location is a real feature of quantum physics. One can do the calculation in terms of creation operators for particles in definite location states. However, those states are not stable. Particles constantly shift from one location state to another on their own. The momentum states are less obvious to us (since energy/momentum is a conjugate space to space/time -- it is not something we directly observe -- I am confusingly using the word space in two different senses here; primarily the mathematical notion of a Hilbert space, but also referring in a different sense to the three physical dimensions we observe. Sorry about that.), but they are stable, at least until the next absorption or emission event. It is therefore much easier to calculate in energy/momentum space. It is possible to do the Feynman diagram calculation in space/time. Rules exist for space/time Feynman diagrams. But when it is calculated, one invariably converts to the energy/momentum representation, does the calculation, and then converts back.

Equally, Aristotle's dictum that a potentia can only be actualised by something already actual is clear in momentum space, where states do only change when there is a new particle emitted or destroyed, but not in location space where there is constant motion from one location state to another. That motion can be characterised by Aristotle's idea of natural motion, but it does hide the principle of causality, while in momentum space it is obvious to everyone. That's why I tend to focus on the momentum representation -- it makes the link to the philosophy clearer. Even though it seems to suggest that location isn't real or important (which is a misleading suggestion).

8) From what I have previously said, we can't treat the Higgs field alone as the source of mass and therefore the source of change.

Sorry, that is all I have time for for tonight. I didn't get a chance to proof-read and revise, so I hope that makes some sort of sense, explains things a bit better, and there weren't too many typos.

6. Scott Lynch
Posted at 04:26:21 Wednesday August 15 2018

Mr. Cundy

Thank you for your insightful response. I understand you are busy (aren’t we all?), so don’t worry about nitpicking or minor typos. I would just like to briefly comment on your points.

1. I think you would benefit from it! There is also a blog called Reading the Summa (which is where I first saw the reference) which is invaluable to me and gives a more accessible synopsis of the article. (Right before his article on Question 27).

2. I agree that “create” is an imprecise word regarding Energy Conservation. Make intelligible is a better word (after all, energy conservation isn’t anything over and above the intrinsic matter/energy being conserved).

4. The only argument I could give against creation/annihilation philosophically is that this could essentially entail Occasionalism. If objects do not maintain their existence ever during an interaction, it is not clear how they could communicate their existence to another object. Both objects would be created instantaneously by God with properties that merely look like they interacted. Then I would proceed to argue why occasionalism is not a good philosophical hypothesis (see Edward Feser for that).

5. I appreciate the insightful technical analysis. I think it shows that mass arises from the nature of physics (binding energy, Higgs mechanism, etc.). For the binding energy, what causes the protons to be bound? It is the strong force which is mediated by gluon exchange (and possibly other factors). No matter how deep you go, the point is, if you admit brute facts into the picture, there is no reason that the location of a particle (or change in location) should be intelligible at all. As you know, Aristotle did not require something to be completely determinate to be intelligible. As long as it has some determinacy (for example it can be predicted with statistical methods, be given boundaries, etc.), it will be intelligible in the relevant sense. This intelligibility needs to be explained either by recourse to a brute fact or by a primary mover. But even a statistically predictable location range would be miraculous if it was not intelligible. If there are a thousand winning lottery tickets out of a bag of infinite lottery tickets, it is just as much a miracle to get a winning lottery ticket as if there is only one ticket in the bag.

6. I presume the locality of the Hamiltonian following from symmetries basically means that it is due to the nature of spacetime and particles’ interactions with the quantum field. In this case, it is still something that needs to be made intelligible by recourse to a prime mover (or more fundamental physics).

7. My physics is definitely rusty. How do particles move from one location state to another without a cause? From what I understand, a particle’s location is not exact but is determined by its wave function. The confusion comes from thinking of particles as billiard balls that have an exact location. When they are instead conceived of as wave-packets, the “multiple locations at once” seems to simply follow from the form of the particle in much the same way as I am in “multiple locations at once” by being extended in space. Anytime a particle assumes an exact location due to wave-function collapse is in my opinion a direct observation of an actualization of a potential, so I do not see any problem there.

7. Nigel Cundy
Posted at 23:38:56 Wednesday August 15 2018

Thanks again for your response.

4) I agree. That's why I said that I personally don't like the idea. However, in my view, Occasionalism (which I take as the denial that particles have final causes according to their nature which God acts through in upholding the universe and efficient causes which are other particles) is undermined because particles do have by their nature a list of possible decays and other decays which are ruled out, and we can track the energy and momentum through a sequence of efficient causes.

5) The binding of the protons is mostly caused by the strong force (i.e. gluon exchange). The short answer is that it comes down to E=mc^2. The energy that holds a proton together acts in the same way that a mass would. The long answer is, as you might expect, longer and more technical.

There are numerous different ways in which we can represent nature. We can write physics in terms of creation operators for quarks and gluons, or we can equivalently write down a theory constructed from creation operators for protons, neutrons, pions and so on. If you are familiar with calculus, then think of it as similar to a change of variables in integration (particularly a multidimensional integration). Or it is a bit like switching from Cartesian coordinates to polar coordinates. Going through that calculation (which is mainly done in practice through respecting the symmetries of the theory -- the calculation is very difficult to do properly, so we take shortcuts) leaves us with time evolution operator for protons, neutrons, pions and so on, and generates a mass term for them. That mass ultimately arises from the self-interactions of the particles that constitute the proton. The Hamiltonian for a particle has three types of terms, those where the particle interacts with other particles, and those where it doesn't. There are two ways to parametrise terms where it doesn't interact with other particles: kinetic terms proportional to the momentum (or the square of the momentum for some types of particles) (and which, in the classical non-relativistic limit, lead to Newton's kinetic energy), and mass terms which are independent of the particle's momentum. When we go through the process of converting from the Hamiltonian for quarks and gluons to the effective Hamiltonian for protons and neutrons, we still need to keep track of the interactions between the constituent particles of the proton. These interactions are going to have to affect some part of the effective Hamiltonian (we can't just sweep them under the rug, because they happen). They won't appear in the interaction terms, since they don't involve external particles. They aren't proportional to the momentum of the particle, since they happen even when the proton has zero momentum. Thus the only option left is that they give rise to something which resembles a particle mass. We can calculate this mass in computer simulations of the strong interaction, and, as I said, almost all of the mass of the proton arises from this effect (with the remainder being from the binding energy of the electromagnetic and weak interactions, and the small masses of the up and down quarks).

The rest of your comment I agree with.

6) Yes. Indeed, I do make that case (or something similar to it) elsewhere. At least, it is part of the argument to get from God to physics.

7) It is more movement without an external cause; I would characterise this as natural motion, just as a Newtonian particle with fixed momentum moves from place to place. But, of course, there is more to it than this.

An Eigenstate of an operator is a state that remains unchanged when we apply the operator to it. So an operator acting on an eigenstate gives the same eigenstate (usually multiplied by a number). Think about matrix times vector = vector. The matrix is an operator; the vectors represent states. For every matrix, there are some vectors where if you multiply that vector by the matrix you get the same vector times a constant. It is the same principle. Location states -- describing particles of fixed location -- are eigenstates of the location operator. Stable states -- describing particles which don't change -- are eigenstates of the time evolution operator (the Hamiltonian), which describes the rate of change -- these correspond to states with constant energy/momentum. As you know, I find it easier to think in terms of energy/momentum states because it makes the pattern of efficient and final causes easier to recognise. But let's look at it in terms of location states instead.

Location states are not eigenstates of the Hamiltonian. That means when the Hamiltonian acts on a particle in a fixed location state, it doesn't stay in the same state. Instead, it moves into a superposition of location states. [This superposition is basically just the wave-packet you mentioned described in a different notation. Schrodinger expressed quantum mechanics in terms of wavefunctions or wave-packets, and that's how most physics undergraduates are introduced to the subject. Dirac came up with an alternative state notation, which tends to be more useful at a more advanced level and is how I naturally think about things -- I was fortunate enough to have a mentor who taught me using Dirac's formulation from the start so my mental processes weren't polluted by wavefunctions. The two notations are mathematically equivalent; the wavefunction is a function over space; its value at a particular location is the amplitude for a particle to be in that location state. The state notation, however, works better because it can deal with both continuous states (such as location) and discrete states (such as the energy levels in an atom, or spin states, or most importantly states with different particle numbers) in the same way.] Then we apply the Hamiltonian operator again to move onto the next moment of time, and we get a different superposition of states. Each superposition is a list of all possible states combined by a number describing the amplitude that the particle is in that state.

Of course, when we next measure the location of the particle, we will find it in a fixed location. The probability that it is found in that location is given by the square of the amplitude. This is, of course, where the indeterminacy of physics comes in, which is why we need to express things in terms of probability. We can't say which path the particle took to get from A (the initial location) to B (the final location), but we can calculate the amplitude for the particle to travel from A to the next point, and the amplitude from the next point to the point after that, and so on until we reach B. The total amplitude for that path is all the individual amplitudes for each infinitesimal movement multiplied together. We can then sum up the amplitudes for every possible path to get from <i>A</i> to <i>B</i>, and square it to get the probability.

The amplitude for each possible outcome for a particle, starting in a given location state, as it goes to the next moment in time, is determined by the Hamiltonian operator. As I said, this has three terms: the mass term, the kinetic term, and the interaction term. Location is an eigenstate of the mass term, so this part of the Hamiltonian increases the likelihood that the particle will stay in the same place. The kinetic term links one location state with its neighbour, so this describes the likelihood that the particle will undergo natural motion to the next point in space. The interaction term describes the likelihood that the particle will interact with another particle (such as emitting or absorbing a photon). The constant moving around is thus a fight between three "forces" [I don't mean that word in the Newtonian sense], one of keeping it where it is, one of moving the particle in a particular direction (how likely each direction is is determined by the gradient of the amplitudes in the superposition -- how rapidly the amplitudes change in space in that direction), and one of interacting with an external particle. What happens in practice is determined by the winner of that fight, and like all good sporting events, you can't predict with certainty what the outcome is, only assign odds to each possibility. So each instance of motion does have a specific efficient cause -- either the particle itself (through natural motion), or the particle that is interacting with it.

There are then two ways to think about this (well, more than two, but these are the ones I will discuss here). The first is, as you suggested, to use the Copenhagen interpretation. To say that each location state remains potential until it is actualised by the measurement. This is, I think, roughly Heisenberg's approach in his Philosophical work.

It is also what is generally taught to physics undergraduates. It was favoured by a number of the founders of QM, and has generally stuck. The second approach, which I favour, is to say that the superposition refers to our knowledge of the situation rather than reality. At each moment, one of the potentia is actualised (one of the three forces wins the fight), but we don't know which one until we next make a measurement, so we mentally have to keep track of every possibility.

Neither approach is entirely intuitive. The Copenhagen interpretation runs into the measurement problem -- why does the act of measurement cause the actualisation process to occur? What is it about measurement that has that magical effect? Is the particle pure potentiality (i.e. a superposition) between measurements but actual at the moment of measurement? And it is that last thought I find particularly troubling. The knowledge approach resolves this problem -- there is no difficulty in accepting that our knowledge of the situation collapses when we make a measurement. But it instead runs the risks of collapsing into a hidden variables theory, which have been disproved by Bell's inequalities. The natural way of thinking about the particle actually being at a specific location but saying we don't know what that location is is to say there is a certain probability that it is there. But as soon as you do that, you violate Bell's inequalities and thus experiment. To avoid this, I say that we have to parametrise our uncertainty using amplitudes. This is unintuitive, because it means interference effects between different potential paths affect the physics despite there being an actual path of the particle. [Don't worry if you don't understand that last sentence -- I don't either.] But this second approach, despite being impossible for us to imagine and wholly counter-intuitive, still strikes me as being more rationally coherent than the alternatives. There is no contradiction in the mathematics; the contradiction is with our common-sense thinking about uncertainty.

Quantum physics is unnatural to us, and there is some point where we will sit back and think that it doesn't make sense. The use of amplitudes rather than probabilities to denote uncertainties is not incoherent for the intellect, just impossible for us to get our imaginations to grasp. [Not too different from the doctrine of the Trinity.]

But who said that the philosophy of quantum physics was going to be easy?

8. Scott Lynch
Posted at 04:40:57 Thursday August 16 2018

Mr. Cundy,

Thank you for your additional remarks. Perhaps a blog on ”observation” as described by quantum mechanics would be of benefit to your readers. I never liked the term “observation” because, from what I have understood, an “observation” is really just an interaction with a measuring device. It seems much less mysterious that an interaction should affect the behavior of a particle than an “observation”. Even if the interaction doesn’t apply any force to the particle, it is still interacting in some way (or else it could not be a measuring device). The way many scientists throw the word observation around when explaining to laypersons, you would think that an electron’s behavior changes based on which direction your head is turned.

I have one last quick question for now. Do you have any suggestions for undergraduate level mathematics books that could help prepare for a more in depth (but still accessible) study of QFT? I plan to read your book as well as QFT for the Gifted Amateur, which I have heard good things about. However, I am thinking a math refresher course could be very beneficial for me (it’s been a long time since I’ve had to use anything more advanced than algebra, although I did linear algebra and differential equations in college).

9. Nigel Cundy
Posted at 22:02:11 Thursday August 16 2018

Mathematics books

The books I used as an undergraduate were Mary Boas "Mathematical Methods in the Physical Sciences" and G. Stephenson "Mathematical Methods for Science Students", and I still refer to them and tend to use them today. Boas' book is a bit old now, but I still don't think that it has been bettered. One weakness of those two books is that they don't contain much group theory, which isn't used much in undergraduate physics, but more important at graduate level. Riley, Hobson, and Bence "Mathematical Methods for Physics and Engineering" is also recommended. The important topics you need to understand (beyond basic algebra, trigonometry, arithmetic and the theory of functions) are calculus, matrix and vectors (especially eigenvectors), complex numbers and complex analysis, and probability. You need to at least understand the basic principles of these topics.

If you can pick up a good book on set and group theory as well, that would help, although I am not sure it is necessary. You don't need that large an understanding of group theory to understand the basic outline of QFT, but you need to know some, (and you need to know quite a bit to do research). Geoffrey Stephenson's "An Introduction to Matrices, Sets and Groups for Science Students" is a reasonable starting point. Elliot and Dawber "Symmetry in Physics" was the book I used myself as a student, and it was OK but I wasn't the biggest fan of it. You can also find introductions to group theory online.

As far as quantum mechanics is concerned, I would personally favour Townsend's "A Modern Approach to Quantum Mechanics" or Binney and Skinner's "The physics of quantum mechanics." These are perhaps controversial choices, since they focus on the state (Dirac) notation rather than the standard wavefunction notation, but that's why I like them. A more conventional choice is Griffith's "Introduction to Quantum Mechanics," but in my view Townsend's gives you a much better understanding of the underlying structure of the theory, without getting bogged down in numerous partial differential equations and so on (QM is full of PDEs, but QFT not nearly so much). And it is a much smaller leap from the approach taken by those books to QFT.

Of course, like most academic books, these are seriously overpriced. But that goes with the territory.

10. Christopher
Posted at 10:11:09 Saturday October 6 2018

Hi Dr. Cundy,

I love your blog and hoped you could answer a question. I really like argument from motion (change) but have a question about it. The objection is basically that on a B-theory of time (which is frequently claimed to be the only theory of time compatible with modern physics) then the universe itself exists tenselessly, with all times being on equal ontological footing. Basically, the idea is that, for instance, the universe doesn't really come into being at the big-bang anymore than a ruler comes into being at the front edge. So, because there is no objective "present" moment, then all instances of change are in some sense already actual. Because past, present and future are all on equal ontological footing, then the times at which any given potential is actualized in some sense "already" is actual. Feser discusses this objection in "Motion in Aristotle Newton and Einstein" in his Neo-Scholastic essays book, but I wanted to know what you think. Basically, can we affirm the reality of change if there is no objective present moment in which things are changing? If my question is not already clear enough, let me quote from Feser's book, "To take the spacetime view seriously us indeed to regard everything that ever exists, or ever happens, at any time or place, as being just as real as the here and now. And this rules out any conception of free will that pictures human agents, through their choices, as selectively conferring actuality on what are initially only potentialities. Contrary to this common-sense conception, the world according to Minkowski is, at all times and places, actuality through and through: a four dimensional block universe." Or as Hermann Weyl says, "The objective world simply is, it does not happen." Feser sums up the views as follows," So, in short, I have two questions. First, is that on the B-theory of time can we get a robust enough sense of "change" to get these arguments working? Second, are you aware of any way to make the A-theory of time work with modern physics? I have seen some models from George Ellis and others (Lee Smolin etc) but weren't sure if something came to your mind that might be better than theirs. Forgive me if my question is too long, I have a hard time condensing such a complex subject matter.

11. Christopher
Posted at 10:13:03 Saturday October 6 2018

Hi again,

Sorry for the formatting error in my prior comment; I can't seem to get quote marks to work right. I apologize.

12. Nigel Cundy
Posted at 19:38:17 Saturday October 6 2018


Dear Christopher,

Thanks for your question. Firstly, fixing your formatting is much easier than answering your question. Actually, you found a bug in my script for the comments. I substitute certain characters with the html codes to prevent people from inserting malicious commands into the website. I got the code for the " character wrong. Easy enough to fix the bug, and easy enough to fix the comment.

With regards to your question, it is, I recognise, somewhat controversial among philosophers, and please don't take my thoughts as any more than a few stabs in the dark which are likely to miss the target. I can't say in this case that I understand all the philosophical issues involved, and there might be something I have missed. This is also just a few brief thoughts. If I have made an obvious mistake, then please correct me.

1) There are actually two distinct but equivalent formulations of quantum physics. The first is the Hamiltonian/Schroedinger picture, where the states evolve in time, and the second is the Lagrangian/Heisenberg picture, which offers a four dimensional view and the states remain constant in time. The states are the description of the particle. In quantum mechanics, the state of the particle is described by the wavefunction, which provides all the information about the particle. It is a mathematical function that describes at one glance the amplitude that the particle is in any given potentia. In QFT, the corresponding item is the Fock state. I'll discuss QFT because that's what is consistent with special relativity, which is where the problems come in.

In quantum field theory, the Hamiltonian (three dimensional plus time or A-theory) approach is called canonical quantisation, while the Lagrangian (four dimensional or B-theory) approach is called the path integral approach. Canonical quantisation is based around the creation/annihilation operators which I reference frequently here. It explicitly takes each slice of time separately and has the states evolving in time. The commutation relations are non-temporal, and there is no mention of time until we ask how the given state will change in the future. Creation operators and Fock states are three dimensional objects. In other words, it gives objects at one particular moment of time (the initial state) a special status, and then tries to predict what will happen in the future. This is fully consistent with the A theory.

The path integral approach basically involves a sum over all possible entire trajectories; in other words you have to calculate bear in mind what happens at each moment in the past and the future. It is explicitly four dimensional. The path integral approach is often (in my view at least) easier to calculate in. The symmetries tend to be more manifest, while you have to fiddle around to get them right in the canonical approach. But canonical quantisation is just as valid.

The idea that B theory is more natural in physics comes from the theory of relativity, but we only need special relativity to get to it. QM is, of course, inconsistent with special relativity (Schroedinger equation) or not self-consistent (Dirac/Klein-Gordon equations). But canonical quantisation is consistent with special relativity. Since the canonical formulation of QFT is also consistent with some forms of the A theory of time, I don't think that it is entirely fair to say that relativistic physics implies a B theory. The canonical quantisation approach takes each time slice separately (note that this means that we first have to select some particular coordinate system, and work in that).

Special relativity is based around a symmetry transformation, which relates how we measure distances with how we measure durations in time. It basically describes a relationship between the speed your clocks are running at and the ticks on your ruler. These are different for different observers depending on their relative velocity, and it allows us to compare the clock speed of one observer with that of another. There is nothing explicitly in this statement which describes a B theory of time. General relativity elevates this symmetry from a global linear symmetry (i.e. the relationship between different observers is the same at each point in space) to a local symmetry (the relationship between different observers could be different at different points in space), but adds little to this fundamental picture. The four dimensional block universe is a convenient mathematical way of thinking about this. It is easier to do calculations in that framework. But it is not directly implied by special relativity. Don't forget: Einstein formulated his theory of relativity, and then Minkowski introduced the block universe. The block universe came after relativity, not alongside it.

2) However, in physics, the Hamiltonian and Lagrangian approaches are equivalent. They lead to the same predictions, and they are convertible into each other. I think that there is likely to be philosophical significance to this. In other words, A and B theories of time need not be as mutually inconsistent with each other as most philosophers like to claim. In other words, there is some form of A theory which is convertible with some form of B theory. What we have to do is think of what those forms are.

3) The B theory does allow for change. Particles are in one state at one moment of time. Particles are in a different state at the next moment of time. This is as good a definition of change as any, and it still allows for motion in the Aristotelian sense. Now we can debate about what it means for these states to on an equal ontological footing. In the B (Lagrangian) theory, we say "At this time, these states are actual, and at that time those states are actual." There is still a relative sense of past and future, and there is still a sense of things being simultaneous with each other (although given special relativity to have an unambiguous definition of simultaneous we also need locality, otherwise it is coordinate system and observer dependent). What we can't do in the B theory is point to one particular time and make it special and declare it to be the present. But we don't need to. We still get act, potency, efficient causality, final causality, motion and so on. I think that the only change needed to the philosophy is that instead of discussing things as actual in an absolute sense, we have to discuss them being actual at a given moment in time.

This means that something can't be both actual and potential in the same respect and the same sense even in a B theory. Take the example of something, X which is is potentially hot becoming actually hot (as expressed in the A theory). In the B theory, we say that at time t1 it is actually cold and potentially hot, while at time t2 it is actually hot and potentially cold. Can we say from this that X is both actually cold and actually hot? No, because such a statement as "X is actually hot" is meaningless in the B theory without the additional context. In this case, we can't use the word actual without specifying what time we are referring to (just as in physics, we can't describe something as being a particular colour without specifying our relative velocity to it: it might be blue in one inertial frame and red in another, but there is no contradiction as long as we always specify the frame in which we take the measurement). What has meaning is the statement "X is actually hot at time t2". That doesn't contradict the statement "X is actually cold at time t1."

This is the same basic principle as taking a piece of string, and putting one end in a flask of boiling water and the other end in a bucket of ice. Is the string actually hot or actually cold? The answer is both, but in different respects. We can only say "This end of the string is actually hot, and that end of the string is actually cold". For composite objects, we have to state which part of the object we are referring to as actually in a given state even in the A theory. We are happy to add qualifications to something's ontological status when different parts of it are in different states. What I am suggesting for the B theory is similar to this.

4) I think that a equivalence between the A and B theories is useful from a theological perspective. Traditionally, God is viewed as timeless, in the sense that He views each moment of time together rather than in succession. So, for God, which moment of time is actual? The answer has to be either all of them or none of them or that the question is meaningless. In any case, God looks at the universe from the perspective of the B theory. On the other hand, we are temporal creatures. We live in the universe, and thus take one given moment of time as special. We observe things in the A perspective. There would be an issue here unless we can say that the A and B theories of time are in some ways convertible to each other, as the Hamiltonian and Lagrangian formulations of physics are convertible.

Note that in my book, I start of with the canonical (temporal) approach in order to construct the basic premises behind classical philosophy (asking what sort of God can we expect given physics); and then later (when asking the question what sort of physics can we expect given God) I develop the path integral (block) approach. In the first case, I am starting from the human perspective and getting to God, in the second case I start from the divine perspective and get to the universe around us.

5) I think that the main issue is what we understand by ontology. As I understand it (and here my training as a physicist rather than a philosopher limits me) the issue concerns the nature of being. My feeling is that different formulations of ontology are valid in the A and B pictures (same underlying theory, but different ways of expressing it). So when we say in the B theory "past, present and future are all equally real" and in the A theory "Only the present is real", we mean different things by the word real in each case. In the A theory, we are equating reality with a particular time. We can get the same effect in the B theory. We can introduce some particular coordinate system, and some particular origin of that coordinate system. This gives every point in space time a particular label saying what time, t it is at. We can then define the word real by saying that only those things at time t=0 are real. This would then be equivalent to what we mean by real in the A theory. The same set of objects are regarded as real. But, of course, this is different to how advocates of the B theory would usually define the word real. Which suggests that the two statements use different definitions of that word, and thus are not truly in contradiction.

6) Another way of thinking about it is that the A theory of time refers to physics after we have imposed a coordinate system, and the B theory a theory of physics without a coordinate system. Both are valid ways of looking at the world.

So, in conclusion, I would say that I am neither an A theorist nor a B theorist but perhaps an A+B theorist. I believe that there are expressions of each picture which are equivalent to each other, and I will use whichever one is most convenient for whichever problem I want to solve.

Of course, if that doesn't make sense, or I have made an obvious error or omission, then please tell me. I make no claims to be an expert on the philosophy of time, and it is one area where I am not particularly well read. It is an important topic, though, and I would like to learn more.

13. Me
Posted at 21:56:31 Saturday October 6 2018


Ignore this message

14. Christopher
Posted at 10:28:24 Sunday October 7 2018

Thank you so much! That reply was really helpful and incredibly well explained. I wish my physics teachers at my university had been as clear as you. I will go over the articles I have been reading and post further comments when I have something of greater substance to ask about. Enjoying your book btw.

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