# Is “Field” a more fundamental quantity or “Force”(in classical mechancis)?

Consider an isolated system consisting of two particles. We can say the two particles are exerting gravitational forces to each other due to their masses. Also we can say each particle has a gravitational field around itself again due to its mass. But if the system happens to have just one particle in itself, there would be no force in the system but the field is still present. It seems that a "Field" might be defined regardless of respective "Force" being present or not. What bothers me is in classical mechanics "Fields" are always defined by "Force". For example:

$\vec{g}=\frac{\vec{F}}{m}$ (Gravitational field)

$\vec{E}=\frac{\vec{F}}{q}$ (Electric field)

Also in quantum mechanics I've never seen an equation including "Force" and there we tend to use the notion of "Field" more.

I want to know which concept is more fundamental/primitive and which one is a derived concept? Field or Force?

EDIT(A second question):

If we consider "Field" a more fundamental quantity, can we define a quantity named "Field" independent of the quantity "Force" and also independent of the "Nature of the field/force" weather being gravitational, electric, etc and then deduce these specific fields as a special case of the first quantity? And if this is true, how can we define it?(consider I want a definition in classical mechanics)

## 4 Answers

I think the way to conceptualize this is to first imagine your introductory physics class, where you learn to calculate the trajectories of objects given certain forces. This leaves unanswered the origins of those forces. In a sense this is perfectly "fundamental", being totally agnostic to those origins. In another sense it is "derived", since describing nature in practice will require a theory of why forces behave in the way they actually do.

One next moves to (e.g.) electromagnetic theory, which attempts to explain why the forces have the values they do. First, you learn that bodies exert a certain force upon one another depending on their charge and position. They act upon one another directly, without necessarily bearing any influence on the intervening space. Next, you reformulate the theory in terms of fields. Bodies no longer act upon one another directly: they instead act locally upon the field, and that action propagates outwards, eventually influencing distant objects.

At least classically these are perfectly equivalent, and thus equally "fundamental", points of view. The major advantage of the field viewpoint is that it provides a vastly simpler description of how things work out in spacetime. As charges change values and move around, their distant counterparts feel difference forces not instantaneously, but only after some time delay. You can describe this in two ways: a) the moving charges disturb the fields, and those disturbances propagate outward with a finite velocities; b) there are no fields, and the force laws just have time-delays built into them. Either approach will work, but (a) is much simpler to handle.

Once relativity and quantum theory get simultaneously thrown into the mix, however, you get much more stuck with the field viewpoint. Indeed one of the major successes of QED (for example) is that it unifies mechanics (what forces do) and dynamics (where forces come from) into a single framework. The theory requires there to be an electron and a photon field, while simultaneously telling you how those fields affect one another dynamically.

• So can we define a quantity named "Field" independent of the quantity "Force" and also independent of the "Nature of the field/force" weather being gravitational, electric, etc and then deduce these specific fields as a special case of the first quantity? And if this is true, how can we define it?(consider I want a definition in classical mechanics) – Hamed Begloo Nov 7 '16 at 15:02

This question is rather philosophical, or at least, cannot be rigorously posed, but I will interpret the term fundamental as whichever is of greater importance, and is foundational.

Experimentally, we know forces acting instantaneously, or action at a distance does not occur, that is, say, that the effect of the motion of a distant star immediately impacts a planet. This is an issue of locality, and it is remedied by a field theory approach, and I would argue given this, it is often more appropriate to be speaking of fields in both classical and quantum systems.

If we take for example, general relativity, it is described by an action (in the absence of matter) as,

$$S = \frac{1}{16\pi G}\int d^4x \, \sqrt{|g|} \, R$$

which describes gravitation in terms of a field $g$, rather than directly in terms of forces. However, it can be said the analogous expression for the Lorentz force for a charge particle is,

$$\frac{d^2 X^\mu}{ds^2} = \frac{q}{m} F^\mu_\nu \frac{dX^\mu}{ds}$$

but this is stated in terms of the field strength tensor $F$ which in turn depends on the field $A_\mu$, or 4-potential of electromagnetism.

In quantum field theory, if we consider a scalar field periodic in a particular direction with two reflecting plates, this gives rise to the Casimir force which can be derived by considering the ground state energy of the field, for which we find,

$$\frac{F}{A} = -\frac{\pi^2}{240 d^4}$$

where $d$ is the plate separation. Thus, the notion of a force can arise from field theory, which further bolsters the viewpoint of the field as fundamental.

• So can we define a quantity named "Field" independent of the quantity "Force" and also independent of the "Nature of the field/force" weather being gravitational, electric, etc and then deduce these specific fields as a special case of the first quantity? And if this is true, how can we define it?(consider I want a definition in classical mechanics) – Hamed Begloo Nov 7 '16 at 15:01

## None is more fundamental, they are concepts for describing phenomena, but they cannot be tested at a fundamental level. However, some are more general than others...

The concept of force is historically older, originated by ancient philosophers, but in Newton we see the most useful conception of force, who understood that in its absence, motion can occur. This conception changed drastically the previous idea that no motion can occur without force, which seems a natural notion from human experience. Newton was also the first that introduced the notion of non-contact force, by explaining gravity as a universal action at a distance force. This however does not make force a fundamental concept, but rather a very powerful because it allows explaining and quantifying so many phenomena.

The concept of field is introduced much later, and was necessary because the notion of force was not sufficient to describe electromagnetic phenomena. When using the idea of force in explaining electromagnetic phenomena, we had to accept electrical and magnetic forces as different. In this conception, we couldn't have understood magnetic induction of current, what is light, etc. Only when Maxwell unified the previous knowledge, mainly from Faraday, by using the concept of lines of force whose variations can produce other other types of interaction (electric to magnetic and vice-versa). Still this is not more fundamental, it is a better description for the mentioned phenomena, but is not a better description for classical gravitation or statics laws.

Finally, the concept of force cannot be used in General Relativity and Quantum Mechanics.

GR considers gravitation not as a force, but as a phenomenon resulting from the simple movement in a 4D space (3D space plus time).

QM cannot describe particles as points following trajectories defined by forces, because all of those concepts give answers not observed in reality, and therefore wrong.

Still non is more fundamental, but yes maybe more inclusive, as GR can contain classical gravitation, and QED can contain ED.

Answer to comment: So can we define a quantity named "Field" independent of the quantity "Force" and also independent of the "Nature of the field/force" weather being gravitational, electric, etc and then deduce these specific fields as a special case of the first quantity? And if this is true, how can we define it?(consider I want a definition in classical mechanics)

No, this is not possible. The definition of field is based on the definition of force and interaction. As I mention above, the phenomena in Gravitation, Electrostatics and Magnetostatics have all a similar formulation of force and do not require field necessarily. Example of this are that laws like Gauss Law apply to all of them because the form of the force is equivalent. But when it comes to Electrodynamics and Gravitodynamics, the analogy does not amount to be an equivalency (see Gravitoelectromagnetism). As for Quantum theories, the "field" while still being used formally, the concept is very different, because the classical idea of force is inapplicable (as trajectories and positions are meaningless, rendering the concept of inertia useless). Hence a quantum field is completely different than the classical field, the only common thing is the name* (see QFT)

*Note: Yes the have Hamiltonian and Lagrangian formulations, but the mathematical framework and objects are different and have no more connection with the concepts of classical mechanics, than the relation between classical and quantum momentum or position.

• So can we define a quantity named "Field" independent of the quantity "Force" and also independent of the "Nature of the field/force" weather being gravitational, electric, etc and then deduce these specific fields as a special case of the first quantity? And if this is true, how can we define it?(consider I want a definition in classical mechanics) – Hamed Begloo Nov 7 '16 at 15:00
• @HamedBegloo: Answer to comment too long for a comment. Appended it to answer. – rmhleo Nov 7 '16 at 15:29
• Since you provided the most descriptive answer, I gave the 50 reputation to you. But still I want to see others' opinions. So I prefer to let this question be open to others' answers. Thank you. – Hamed Begloo Nov 10 '16 at 17:23

I would have thought that you couldn't say which is more fundemental and which is more of a concept. As long as you have two particles or any bodies of mass, you'll have both force and fields. The individual fields of each particle and the force acting between the two. If you only had one fundamental particle and nothing else for it to act on as far as forces go, then perhaps you could say a field is more primative. It reminds be of the 'what came first the chicken or egg' question.

Hope this helps and sorry if none of the above made sense!