Is the concept of field the proper way to describe physical processes?

The drawback of the "interaction from a distance view" is that the source must know that there is a particle in its neighborhood. The best way is by the radar technique, that is, by a field.

Manouchehr Amiri

Dear Demetris, your notion of field and source, as has been appeared in your question details, is a classical approach. In contemporary view, to any physical existence attributed a "field" as has been described in QFT language or a "wave amplitude" in traditional quantum physics, whereby all the interactions, Lagrangian, Hamiltonian and equation of motions could be derived. Therefore "field" can not be a separated concept of "source" or "test particles", Nevertheless these concepts being used in classical approaches to facilitate the calculations in the context of electromagnetic fields etc. The general notion of fields started from the very beginning of the quantum theory and asserting that the all governing forces are mediated by Bosons as carriers of fundamental forces, i.e. photons, gluon, gravitons etc. On the other hand space-time and fields are interconnected physical concepts, that could not be existed without each other. Respect to the term" interaction from a distance" it is wisely to mention that this concept has been ruled out by the aforementioned concepts of bosons and scattering amplitudes calculation based on QFT, QED and agreement with experimental results, a very distinguished evidence is the discovery of W+ ,W- and Z vector bosons as the carriers of weak interactions of elementary particles, However we have not to equate the " entanglement" with "action form a distance". For a brief answer to your questions, I recall the debate on the "non-locality" in quantum explanation of Aharnov- Bohm effect, which may illustrate some ambiguities mentioned in your comment.

Juan Casado

The problem of obtaining an infinite value for the total energy is not critical from the physical point of view. Furthermore, it is natural since you are integrating a non-vanishing function all over an infinite volume. The concept of field is really useful when you consider that the source of the field does not need to know the presence of a test particle to act on it. From this point of view, it is easier to interpret the fact that, for instance, the acceleration of gravity at certain point of the field is the same regardless of the mass of the test particle "feeling" the gravitational field.

Mihai Prunescu

"Field" is only a concept, and is useful as such. By the way, no simplified model helps. The electromagnetic field of the universe is the composite "field" of all "sources". If we think like this, it is no more surprising that it has infinite energy and (by E = mc2) infinite mass, dissipated in a (probably) infinite space. Moreover, the reality is not only an electromagnetic field, but a unified field of all interactions - some of them being unknown. I propose to replace the notion of field with something like "material space". I also agree with all comments from the direction of Quantum Field Theory.

Krishnan Umachandran

Fields exists in Physical processes such as follows,

Magnetic fields
Velocity fields
Radiation field

leading to surface inhomogeneities. http://lesia.obspm.fr/MagMas/cartography2014/MRieutord.pdf

1. Physics uses twomain concepts : material bodies (matter) and fields (force fields). The characteristic of material bodies is that they have a trajectory : they have a definite location at each time. Force fields represent of actions without contact, so they must exist everywhere, and they propagate by interaction (with themselves whre there is no particle), and with particles (which the source of the field and are acted upon by the filed).

2. The physics is best expressed in the fiber bundle formalism. Force fields are connection on a principal bundle (thus there is a specific group attached to each kind of field). Chargef particles are then represented as sections of an associated vector bundle. Such a section can be assimilated to a virtual particle : it is defined everywhere, but its materialization depends upon the initial conditions (or in another way, a section corresponds to a general solution of differential equations).

3. The interactions are represented with the principle of last action through a lagrangian. The "point particle issue" appears only when one tries to find a solution (we have multiple integrals for fields and single integrals for particles), but this is a mathematical problem, for which there are rigorous solutions without the need of Dirac's functions and others. The key physical quantity is the energy-momenum tensor. As Juan said above, there is no issue of diverging integrals, anyway whenever one considers an infinite region, one should also consider what is in this region, so it is purely hypothetic.

4. Almost all the theory can be done inside classic physics, including gravitation, which is a field. QM is required only to explain that some kind particles and charges appear (quantization per se) and the spin. Actually in the usual QTF using Feynman diagram, creation and absorbtion of particles is a way to reintroduce the action at a distance.

I think Demetris (again) has asked a good question and hinted at where the difficulties might lurk. For instance the total energy and the associated problem of infinities are problematic.

On a cosmological level I would advocate the zero-energy universe scenario as the most realistic one. This would be formulated in terms of energy-momentum operators and their space-time conjugates. Using operators instead of variables will free the model from the unrealistic point concept. Another fix would be to incorporate the force-momentum-energy law inconsistency that lies at the bottom of Einstein's theory of General Relativity.

The problem is not really the concept of point, it is in quantum mechanics itself. In QM, a particle is described with the dynamics of a point that is quantized by introducing a wave function. But already with the Dirac equation, there is no longer a point particle classical counterpart.

Theoretically, a particle can be localized with an arbitrary precision. But in practice, a higher and higher energy is needed, and that doesn't work when it is greater than the mass of the particle, as particle-antiparticle pairs can be created. That is the gist of renormalization.

It is thought that this limitation is not fundamental, and experiments are made to test whether a particle is a point by simply verifying the predictions of quantum field theory. Though, it is but one conception of a point particle, the one corresponding to the hypothesis.

Renormalization is a recipe that works very well, yet is but a recipe without firm mathematical foundation, and which can't be made to play along with general relativity. Perhaps what it says is that this limitation in precision is fundamental, and the Dirac equation describes something else than the quantized motion of a point particle.

There is no trouble with the concept of point-particle. Physics use location so the concept of points. What matters is the physical quantity which is located at some point, and represents either a field or a particle. For fields this is a connection, and for particles a spinor. The issue of "infinite-energy" which would be stored in a point particle has no physical meaning : if you condider a particle alone, there is no field. If you consider a field you have to take a non null region. In fluid mechanics or plasma physics one deals with this issue by considering "densities", that is continuous distributions, as an approximation. It works well.

Do particles really can be observed, or are really limited to a point ? I do not know, and this is not critical when one uses the representation of the concept of particle.

Claude Pierre: The mathematical point, e.g. the transit time at a point or its image in a natural trajectory, has no physical meaning whatsoever. True that we can define an appropriate neighbourhood and fine grain the result towards a limit. Already Zeno taught us that such a reduction cannot be indefinite. In QM the link of conjugate variables prohibits the location of "the point".

Horst: I agree! Fundamental theories should avoid concepts like point particles, points in trajectories or in transit time situations etc.

There is no issue with the "physical meaning" of a point. It is in the theory who departs from the mathematical concept of point particle, which generates infinities. We have no theories that satisfy the Zeno requirement, because the idea of "quantizing" space and time never worked. It is true that in quantum mechanics, the probability of presence of a particle is a density. Yet, the same quantum mechanics describes a point, that with the appropriate experimental setup can be located with an arbitrary precision. In other words, the wave function in the position representation gives the probability density of finding the particle at a mathematical point.

In QM, there are two different evolutions. There is the unitary evolution, described by the Schrödinger, Dirac equations etc. which present no difficulty. But there is also the projection postulate, without which QM would be a mere classical field theory. And according to this postulate, a particle can (theoretically) be projected onto a state with a well defined position, although it is still described by a wave function, which in this case takes the form of a Dirac function. It is then not true that we can make without these functions.

To Claude,

In fluid mechanics and plasma physics one uses densities as an approximation, without QM. No problem.

Perhaps when using the traditional QM one needs Dirac's functions (put everywhere without great concern about mathematics) but this is the weakness of QM, not the requirement of a physical theory. QTF requires many patches, but physics is not guilty, and even less the "real world" for these shortings.

Claude Pierre,

Repeating, there is no physical meaning to a point, yet you and others seem to make it an issue. Quantum mechanics in hilbert space according to von Neumann does not yield any inconsistencies or problems. The Dirac "functions" can be mathematically incorporated via Gelfand Triplets or Rigged Hilbert Spaces. However, this is mathematics and not physics!

Narendra / N. Nath

Concepts of field has been created while particles exist as fundamental masses on creation of the Universe itself. Mathematics does not lead Physics. Concepts and postulates precede before and mathematical theorem is worked out.

Erkki, if you think QM yields no problem, then you don't have understood it. The very formalism of QM, with any mathematical sophistication you like, is logically inconsistent. Neumann didn't spot it in his time, but many others after him did.

Dear Narendra, the existed particles have to fulfil a specific and finite fraction of space, do we agree on this? If we agree, then, with whatever way we can define a concept like energy, this must be also finite, OK till now? So, if we accept finite entities as fundamental bricks of our world, then how can we associate with them infinite amount of 'energies'?

Claude,

Who said, quote, unquote: If you say that you understand quantum mechanics you have not understod it?

The mathematics is consistent, but not the interpretations.

Dear Demetris,

We do not need to see a particle like some kind of small solid. All we need to know about a particle is a tensor representing its state, located at a point. For instance the momentum of a particle in classic physics is a vector, located at a point. The physics is in the vector, not in the point. And the kinetic energy is a scalar linked to the vector.

Erkki, Bohr said that, Feynman said that, everybody. The mathematical axioms of QM are logically inconsistent, meaning both mathematically, conceptually, and interpretationally. "Shut up and calculate" doesn't mean that all the issues are now solved, it means that until now, nobody succeded in solving them, and the only remaining solution is be be great pretenders. So I leave you lonely pretending that the solution is around.

Oscar Chavoya-Aceves

@Horst If there is a problem with the concept of point then, considering that a field is, by definition, a function of position, or a function whose domain is a set of points, we are in trouble.

Dear jean, I worry about the problem of 'infinite locality' in Physics: we are quietly accepting that some features can be extended to infinity and this physically speaking is not true, since, as you better know, there exist a rapid vanishing after a small distance from any kind of source. The problem is not Mathematics, there you can define what you desire to, the problem is the connection with reality.

Niels G. Gresnigt

I think this is a very interesting question.

To challenge Jean Claude Dutailly’s comment above that if you consider a particle alone then it has no field: what if the particle is the field?

Coulomb’s law describes the electrostatic force between two charged particles well. In addition it treats both charged particles symmetrically in the sense that the force law contains both q_1 and q_2. The problem is that the law is contrary to the precepts of special relativity. To remedy this problem with Coulomb’s law and to visualize the mechanics of the electric and magnetic forces, Faraday and Maxwell introduces the idea of a field to carry the force. The concept of a field acting as intermediaries allows an object to act on a distant object. The force law is now that of Lorentz and (in the absence of a magnetic field) we may write F=qE. In the process we have lost the symmetry that was apparent in Coulombs law. The force now results from the local interaction of one charge with the electric field of the second charge.

What if instead we only consider the fields only? The energy density of the superposition of the waves is a quadratic force that contains the self-energy of both the charges but also a cross term involving both fields. Integrating this over all space and applying Hamilton’s principle, the force law is recovered and is symmetric with respect to the fields. In this view it is the fields themselves that interact through the non-linear energy density. The charges themselves do not enter the theory except incidentally, via Gauss’s law.

Jean Claude Dutailly also mentions that we do not need to see a particle as some small solid, and all we need to know is a tensor representing its state. But is it fair to say that a physical particle if something more than a tensor? Sure we have the mathematical tools to describe the behavior and properties of particles but what of the question about what the particle actually is?

Given my comments above about the possibility of considering the force resulting from the global interactions of the fields of two charges, is it then perhaps possible that on small scales the fields lines do not terminate at a point (which is problematic as the author of this question notes), but rather converge to some other configuration of finite extent? For example, Ranada has shown that there exist solutions to Maxwell’s equations in vacuum where each of the field lines are linked, i.e. an electromagnetic knot. Perhaps instead of considering point particles, we could consider particles to correspond to such knotted solutions to Maxwell’s equation?

http://arxiv.org/abs/1211.6072

Claude Pierre,

I think there is a misunderstanding here. I am not defending QM as such, but it is important to put the blame where it belongs.

Mathematics is a succinct version of a language that we use in order to communicate, like, I hope, we are doing here. Thus we have the levels of syntax, semantics and pragmatics. We know from Gödel’s theorem(s) that any deductive theory based on axiomatic systems would be incomplete (or if assumed complete, inconsistent). However this is really another discussion, see also other RG questions.

The current problems of QM rests in the postulates, e.g. the projection axiom and the collapse of the wavefunction, leading to the various attempts to find “consistent” interpretations.

The problem of QM is not a mathematical problem but an interpretational one. I am not as pessimistic as you are. To me the fix would be to extend QM to open systems leading to new exciting possibilities, see also other RG queries. In this world the “particle” or a “wave” can, without contradictions, be represented by its energy momentum tensor (as Jean Claude prefers) or by its operator representations together with their conjugate variables (operators).

To Niels,

1. I think that we cannot play indefinitely with the words : field and particle have different characteristics, they represent different objects or phenomena, they are linked to measures, and this is because of this link that we are in science and not metaphysic. Particles are the concept of material body, which are located at definite position at a given time, so they have a trajectory, and they carry a momentum. A field is defined everywhere. In the usual QM interpretation a particle can behave as a field and conversely: this is unacceptable if one cannot tell before hand if it is a particle or if it is a field. With some God's will one can prove the existence of miracles.

2. An issue which is usually bypassed is the propagation of fields. It is said that they propagate at the speed of light, but in the general case,notably in GR, it is not easy to define the speed of propagation of a field (I have never seen any equation on this subject). This is easy when considering signals (discountinuous fields) or plane waves in SR context, but the wave equation, which supports the idea of a speed of propagation, is not as simple in a curved space time.

3. Replace particle with fiield is nice, why not ? but we have to account for the obvious discontinuities that we see ata macroscopic scale. Or to go back to the metaphysic idea of collapse of a wave function (which has no physical meaning according to its inventors).

4. "Sure we have the mathematical tools to describe the behavior and properties of particles but what of the question about what the particle actually is?" I do not know, and the purpose of science is not to tell what reality is, it is to provide a satisfying and efficient representation of reality. Take an example: in chemistry we have the atomic theory, which provides an efficient and simple representation, and one can deal with it for almost all the practical work. Or DNA in genetics. Of course it would be useful to know more, but to do that it is necessay at least to give up the usual "nobody can understand QM". What would be the purpose of pretending to know reality, if this is by using a theory that one cannot understand ?

5. To Demetris. The concept of infinity is a great invention of mathematicians , but it has no physical correspondant. This is what distinguishes mathematics from the other science, one can invent concepts which have no physical realisation. Speak about fields vanishing at infinity is right formally, and useful,, but of course at infinity there is no physical, isolated, system. You cannot conceive an experiment at infinity.

Horst Reinhard Beyer

In reply to Qscar's question: In QFT, there is no (densely-defined, linear and selfadjoint) field operator in Fockspace corresponding to a space-time time point. On the other hand there is such "smeared out" field operator corresponding to a, e,g., rapidly decreasing, test function. QFT is a little more realistic than classical theory in this respect.

To Errki,

The problem with the usual view of QM is that it starts wiith axioms related to concepts whcich have no physical precise meaning : wave function, observables,.. This is indeed a very strange way to do physics : start with axioms. The right way is to define concepts, closely related to phenomena (as in a empiricist way, beloved by anglo-saxons), with characeristics, linked to measure, and then to state falsifiable laws governing the behavior of the objects. Why physics should start with Hilbert spaces, eigen vectors and commuting operators ??? Clearly there is something fundamentally wrong with this approach, and try to reinvent reality in order that it complies to the axioms is totally unscientific. Actually I have proven that a in model (such as commonly used in physics) one can retrieve all the axioms, which are mathematical artefacts, linked to the way one proceeds in formulating laws. So these axioms have an explanation, but it is not in nature. And one can use almost all the usual computational tricks of QM, without the need to invent interpretations. This is safer and scientific.

Fields are certainly the better mathematical tool for describing physical processes. In nature forces and energy exist, forces have vector characteristic while energy has scalar characteristic. The field has vector nature and it represents the mathematical tool for describing the force. For every field a scalar potential function is defined and it is connected with the field. The potential function is the mathematical tool for defining the energy, in particular the potential energy.

Whether the field or the potential function are defined in every point of space and they are always characterized by the property that at infinite distance they have null value. I don't see problems with the geometrical definition of point, that has null volume, because in a reference frame the point is identified by space coordinates. It is manifest that any body has non-null volume, but in that case coordinates relate to the barycentre of the body.

When the barycentre of any test particle is placed in a point of the field, the test particle doesn't feel the force comes from the remote source, but it instantly feels the force produced by both the potential function and the vector field generated in that point by the source on the initial formation of field. The concept of force derived by the concept of field isn't based on the action at distance theorized by Newton who didn't know the concept of field. It is manifest that the potential energy produced by a force field is alwyas related to the presence of a second body or particle and this energy is always null at infinite distance. From the physical viewpoint the integration of energy on infinite volumes doesn't have meaning because this energy physically doesn't exist.

Jean Claude:

I am not saying that physics should start with wave functions and Hilbert Spaces etc. I can even support your assertion “The right way is to define concepts, closely related to phenomena”.

As I have stated many times on RG our foremost respected and valued philosopher Immanuel Kant did already stress such a direction by his declaration that “Space and Time are the two essential forms of human sensibility.” Hence any energy entity (particle or wave) evolves (momentum, velocity etc.), as we perceive in space-time, prompting the most natural starting point! These variables (operators) are conjugate quantities irrespective of whether their formulation being “classical” or quantum.

Depending on the physical situation we formulate classical “point particles” or waves or quantum mechanical wave functions etc. In the second case we might e.g. use the mathematical apparatus of Hilbert Space Theory. Note that the latter is not introduced as a “prephysical assumption”, it is rather a particularly efficient way or language used to communicate and document the most accurate and relevant portrait of what we hope to be an objective outer reality.

I do agree with Horst that “There is more a problem with the ancient concept of points in space”. The concept of a mathematical point is unphysical and starting any fundamental description of physical reality with points and trajectories (although classically very successful) would nonetheless be severely limiting and not fundamental.

What about the concept of the field? First there is the question of mathematical apparatus that most practitioners believe are rigorous enough to make sensible limits and tricks to avoid infinities. Second follows the physical interpretation, which is in the eye of the beholder.

To Daniele,

1. In their most general definition, fields are represented by connection, and the precise definition of their potential makes that it can be vectorial (as in the magnetic field)

2. What is interesting in the definition of a material body, is that its rotation matters : the action of the magnetic field depends on the rotation. In the vision of point particle it is not clear what the rotation means. Actually one can see that the common definion of the motion of a material body is not consistent in the relativist context (the 4 dimensional rotation mixes the spatial rotation and the translational motion). So this leads to represent the motion (both translational and rotational) of a particle (any material body where the internal structure is neglected) by a different quantity, and one gets spinors and a sensible explanation of the spin with its bizarre properties. Spin is not a quantum effect, it is the unavoidable consequence of relativist geometry.

Moreover the motion of a body, in this background, is physically linked to its momentum, so it seems logical to take spinors as the main dynamical characteristic of material bodies, that is their momentum. For instance the rotation of a spinning ball cannot be ovserved through its movement (it is symmetric) but by its inertial rotational momentum.

Definition. Field of a force X is called a part of the space where, if we put a proper test particle, then a force X will be applied to it.

This definition has inside it the potential placement of a test particle and this is a serious problem in theoretical Physics.

Why? We have not proven the existence of a filed without the use of a 'test particle', so it is similar to a Gedankenexperiment: If you like the math convenience of accepting a scalar function (potential), which is the force generator, then it is OK.

But if you want an experimental proof, then you have troubles.

To Demetris

The problem is what is a force ? The usual way to proceed is through the change in the trajectory or the rotation that is its momentum observed on a test particle. And from there fields are measured through a network of experimenters, using similar procedures, measuring the changes on known particles. Which leads to how compare the results of different observers, and unvoidably to the definition of connections. One says that the value of the field is by how two observers have to adjust their instruments to get the same result. This is similar to the measure of a gravitational field. And why this is necessary to define fields (force fields) through connections.

We can associate the concept force as the cause of a spatial change. Why is the concept of field-force irreplaceable?

To Demetris,

because a force exercised by a field depends also of a characteristic of the particle, through its charge.

Jean Claude:

Quoting Lévy-Leblond see Charles Forgy at http://quantumchymist.blogspot.se/2014/04/is-spin-relativistic-effect-l-and-first.html.

Relativity itself does not necessitate the existence of spin. Spin is a purely quantum mechanical effect that arises from enforcing first order spatial dependence on a quantum wave equation, and can equally well be described by the Galilei group. It will emerge naturally from a quantum wave equation without any relativistic considerations whatsoever.

To Jean Claude,

1. I agree that fields can be represented also by lines of connection, but I think all fields are always vectors unless you are thinking about tensor fields that are scalar. My viewpoint nevertheless is that vector fields are the best tool for describing physical processes.

2. Any particle, massive or energetic, is a point for convenience of representation but we know they have always a non-null finite volume. In particular a massive particle has also an axis of rotation that is a symmetry axis for particle. I think the spin of the massive particle is the angular momentum with respect to this axis of symmetry. I have demonstrated in the Non-Standard Model the spin is connected with the electric charge, and because the charge is quantum also the value of spin is quantum.

To Demetris,

you report a physical definition of field, but we have a mathematical definition of field through the vector field and the scalar potential that is concordant with the previous physical definition, that nevertheless is a little vague, because it doesn' t define the part of space and doesn't define the modality of application of the force on the test particle.

I don't see problems you report. The source of the field (mass, charge, etc..) generates the field and the scalar potential in every point of space that you don' t see. Then if you want to prove the presence of the field you have to use a test particle or a measurement instrument of the field that will confirm the theoretical concept of field. Troubles that you report are related to the accurate measurement of the field in every point of space, like jean claude has specified.

To Errki,

I fully disagree with Lévy-Leblond, sorry. Explain the spin through a wave equation that nobody understands is just b...

SU(2) is the two-fold covering group of the SO(3) subgroup of the Galilei group. Spin can exist in a non relativistic theory, but not because of quantum mechanics, because of the symmetry group. In relativity, the SU(2)xSU(2) group plays the same rôle.

To Claude,

The groups SU(2), SU(3), U(1) are involved only for force fields. Does that mean that the spin depends only on the action of these fields, or that it has a deeper meaning ?

Anyway galilean Geometry being a proxy for Relativist geometry, there is no problem.

No, SU(N) are matrix groups, just like SO(3) and SO(3,1) which are subgroups of respectively the Galilei and the Poincaré group. The spin is associated to SU(2) only, and depends on space-time symmetries. To the weak force field corresponds the isospin.

To Claude

I know my group theory. If the spin was associated to SU(2) only it would be seen only with weak interactions

Jean Claude;

”Lévy-Leblond – Balibar” is not an alternative to QM or even offering a new interpretation. They introduce nonconventional terminology and proceeds pragmatically by focusing on examples rather than presenting a coherent theory. It has flaws and I do not defend it.

Nontheless their illustrations point at multifarious ways to realize “spin” from examples of the Stern-Gerlach experiment to spin statistics and the exclusion principle. We all know that Pauli did not support actual spin motions (Der Kronig hätt’ den Spin entdeckt, hätt’ Pauli ihn nicht abgeschreckt) in connection with the two-valuedness of electrons.

Claude Pierre has already pointed out that quantum spin exist already on the non-relativistic level. In addition to group theoretic representations one can also mention spin-orbit coupling, the aufbau principle etc. The experimental results during the last 100 years could not be understood without employing the concept of spin. Although the Dirac equation provides a deeper basis for its origin, we are still within the quantum framework.

Note that symmetries and constants of motion characterize the properties of the actual physical states of fundamental microscopic theories and this is also commensurate with the introduction of relativistic geometries. Hence spin by definition must be a quantum effect.

Niels G. Gresnigt

To Jean Claude Dutailly,

Thank you for your comments. In response to some of the things you mentioned:

1. Are particles located at a definite position at a given time? I think QM teaches us that this is impossible to determine, or at least measure. Even if a particle was at a specific position at a definite time, measuring it would disturb it. For example, firing a photon at some test particle is going to give that test particle some momentum. Distinguishing between a particle and a field may not be a big issue if we consider the particle to be some local topological feature of a global field such as, for example, a knot. A particle would then in a certain sense look like the source of the field (for example via Gauss’s law).

2. Related to point number 3. Forgive my ignorance here. Could you give me some examples of the obvious discontinuities that we would see at a macroscopic scale?

3. Regarding your comment about science being about finding satisfying and efficient models of reality, I agree. The concept of a point particle in many instances is efficient, although I do not find it satisfying to attach all the properties of a particle (mass, spin, electric fields, dipole moments) to a single point in space. Would it not be more satisfying if it were possible to explain how these properties of particles arise from some underlying (possibly dynamical) structure?

To Demetris

Do we need the concept of a test particle if we define a force from a field via Hamilton’s principle? I suppose to observe the force we would need to supply something that will experience it. If we consider the force as the gradient of the field (via Hamilton’s principle) then we can see why the concept of force-field is irreplaceable. I think the idea I discussed in my previous answer also sheds some light on this. If particles are replaced by their fields, then the (nonlinear) energy density of the superposition of fields gives rise to (what I call) interaction terms. In a sense therefore the fields themselves interact with each other throughout all space, and it is this total global interaction that through Hamilton’s principle gives rise to a force. The charged particles are thus essentially delocalized and replaced by their fields. This is similar to how particles are replaced by their wave functions in quantum mechanics.

As for the quote from Levy-Leblond that Erkki provided about spin being purely quantum mechanical: if we consider the Poincare algebra, it admits two Casimir invariants corresponding to mass and spin. But there is nothing quantum mechanics about the Poincare algebra. The Poincare algebra simply encodes the set of transformations that preserve the spacetime interval. Yet it gives rise to the idea of spin. I do not understand how this fits in with the quote that spin is purely quantum mechanical…

Eric Lord

A ‘point’ is a mathematical abstraction. There are no points in Nature. In Newtonian mechanics we can calculate the trajectory of a cannonball by treating it as a ‘point particle’. That is an idealization that allows the physical situation to be dealt with mathematically – an approximation to the reality. Our habitual way of thinking mathematically in terms of ‘points’ misleads us into imagining that we can do the same for elementary particles such as electrons. That is to project our mathematical habits of thought onto Nature. The uncertainty principle of QM tells us that the location of the position of a particle ‘at a point’ is a meaningless concept. (Even the ‘points’ of space and time that we label when using coordinate systems in physics are imaginary – space and time are assumed to be a continuum. At the very smallest scales (Planck length…) that is very likely incorrect.)

As noted in the question, a ‘point charge’ leads to infinite energies. In reality a charged particle is a distribution of charge density over a region.

To Niels G. Gresnit,

I would like to express my viewpoint on your interesting considerations.

1. You point out well the limits of QM that states, via the Uncertainty Principle, an absolute indeterminacy in nature. I have demonstrated (see "Deterministic Quantum Physics") this conclusion is the result of an unsatisfactory mathematical model used in QM that doesn't consider the most interesting error theory. The Planck length, like the speed of light, is a limit that the new contemporary physics allows to surmount.

2. The particle (on microscopic scale) and the body (on macroscopic scale) are the sources of the field. In the case of the particle, if it is placed in the origin of a reference frame supposed at rest, its position (barycentre) is well defined and its representation through a function wave is useful only to complicate things. I think instead of the dualism wave-corpuscle, it would be more useful to consider an equivalence wave-corpuscle.

3. It is true, the concept of point particle is only a convenience of representation that often is used also for macroscopic bodies, always making use of the concept of barycentre.

4. Regarding the definition of field via Hamilton's Principle I think it is more correct to represent the field as the gradient of the scalar potential than the force as the gradient of the field. The relationship between force and field is linear but non-differential.

5. I think then it isn't appropriate to replace particles with their fields, because anyway fields are defined in the space outside the particle and in this space a principle of superposition can be valid, like you observe. It is just this the main difference between the concept of field and the concept of wave function that instead, in concordance with the dualism wave-corpuscle, replaces completely the particle.

6. My research convinces me to conclude the spin of charged elementary particles has electrodynamic nature, i.e. whether mechanical or electromagnetic. This way in the Non-Standard Model I have demonstrated, via the Theorem of Charge and Spin, the spin is related mathematically and physically to the electric charge.

Dear Eric, you have hit the target: We are always starting from our math convenience and, provided that we have success, then we start to believe that the used math convention is a 'living entity' (recall one of my previous questions about

https://www.researchgate.net/post/Is_Probability_a_self-existent_concept_quantity_or_whatever )

So, even if the concept of a field with a scalar potential giving force as gradient is a math convenience for us, this does not necessarily means that the field exists in reality. This is the puzzle that I am trying to investigate in the current question.

To the point nihilists, I recall that there have been experiments for testing the point nature of the electron, which gave very small upper bounds (~10^-18 m.) There is a big confusion here, between the measurement precision of the position, and the successful mathematical theory describing a quantized point particle. Until this confusion is dissipated, no valuable discussion can take place.

Niels G. Gresnigt

To Daniele Sasso,

Thank you for your insightful comments and giving me more to think about. In response to some of your comments:

1. My understanding is that there exists a fundamental connection between the homogeneity and continuity of physical space and the Heisenberg algebra. In the general quantum mechanical framework Isham (Lecture on Quantum Theory, 1995) shows that if space is homogeneous and continuous then it necessarily follows that [x,p_x]=ihbar. The uncertainty principle follows.

2. How exactly is the Planck length a new limit? I am not sure what you mean by this statement. Do you mean that it is a new invariant scale just like c? If so then this is presumably similar to theories of Doubly Special Relativity where one in addition to c incorporates an additional invariant scale. I think this is a good direction to take although I would argue a mathematically sounder framework is given by Lie-algebraic stability considerations. Here invariant scales are the result of deforming unstable Lie algebras into stable Lie algebras. For example, deforming the unstable Galilean algebra into the stable (in the tangent space) Poincare algebra introduces c as the deformation parameter.

3. Regarding your comments on Hamilton’s principle; you are absolutely right (I think) and I used the wrong words. What I meant to say was applying Hamilton’s principle to the electromagnetic energy density integrated over all space gives rise to the electromagnetic force between two particles. The electric fields combine via a dot product to give the energy density which is a scalar field.

4. I do not understand the point you are trying to make in your point 5.

5. As for your last comment, I think you may well be right about the nature of spin! Although I would have to say that for me this is just a hunch at this stage. Certainly looking into further!

To Claude Pierre Massé,

"To the point nihilists, I recall that there have been experiments for testing the point nature of the electron, which gave very small upper bounds (~10^-18 m.) There is a big confusion here, between the measurement precision of the position, and the successful mathematical theory describing a quantized point particle. Until this confusion is dissipated, no valuable discussion can take place."

If you relate ("big confusion") to my considerations I will strive to be clearer. If the "successful mathematical theory describing a quantized point particle" is Quantum Mechanics I confirm it is based on an unsatisfactory mathematical model that represents photon like a wave function defined in an infinite space. As per De Broglie's dualism then this same mathematical model is considered also for massive particles like electron. I have demonstrated this model is unsatisfactory and it isn't nihilism but scientific criticism. If the scientific criticism is nihilism then we must go back to the geocentric model.

In actuality your assertion " there have been experiments for testing the point nature of the electron, which gave very small upper bounds (~10^-18 m)", that I agree, confirms the electron doesn't have point nature and also it doesn' extend on all the infinite space, but it has smallest dimensions (just about 10^-18m).

After clarifying that from my viewpoint QM is now an obsolete theory I wrote the uncertainty is really present because of our measurement methods that present always an error (described by the error theory). Consequently the indetermination/uncertainty isn't a law of nature but only a consequence of our methods.

If your words aren't related to my considerations, this my comment is valid in general.

Best regards.

To Daniele

The idea that the meaning, or validity, of a physical concept is linked to the way it can be measured comes from Dirac.It is clear that, if a quantity cannot be measured, it cannot enter into a scientific theory. But this should be true for the wave function and many others. To have a location is a defining characteristic of material body, and if onegives up this characteristic, one has to tell how to define matter (or any equivalent concept). With location comes speed (whatever its form). We cannot measure simultaneously the speed and the location of a particle ? and so what ? for centuries artillery men computed both for their shells, even if they were not able to measure it. The importance given to the simultaneity of measures has no relevance, and is justified only by formal arguments coming from operator algebras. Anyway speed, whatever its form, is linked

So I feel much more comfortable to keep the usual definition of matter, location, speed, rooted in our experiments at a macroscopic scale, than to invent strange properties for particles, based only on a formal system, postulated but not understood. But the case of the spin is somewhat different.

To Errki

"Spin" : it is clear that the concept of skin comes from our notion of rotation. We try to transpose to particles what we have at a macroscopic scale for solids. The problem is that the notion of rotation is not so obvious : it is well defined for solid, and there is no solid in relativity, the usual image of roatating frames attached to a body is no more than an image, and rotation, even at a macroscopic scale, is more a kinematic concept than a geometric concept (we can measure the rotation of a ball, we cannot see it). When we go to the relativist geometry, the usual formula in a change of frames are based on a rotation but it is obvious that they mix spatial rotation and translation. The usual use of the Poincarré group is not pertinent (it requires 10 parameters to define a motion, but obviously 6 are enough). So we need to rethink the definition of motion, and this leads to the spinor. What you have in QTF are spinors, which are geometric quantities, that can be defined properly in Clifford algebra. And this has nothing to do with QM. Spinors can be used at a macroscopic level (Clifford algebras are a classic way to describe the deformation of solids in automation or graphic imaging).

In astrophysics a big issue is the motion of stars in galaxies, which has lead to the introduction of dark matter. But in a GR context it is difficult to deal with the rotation of bodies. Using spinors this can be done. And indeed the rotational kinetic energy of stars and planetary systems, which could probably not neglected, could account for a good deal ofmissing mass.

Daniele, no, 10^-18 m is an *upper bound,* not a measure of anything, save for the experimental imprecision, and the value 0 is still compatible with the data. It is anyway much smaller than the classical radius, for which the electromagnetic energy of an electron is equal to its rest mass. As yet we have no successful theory with elementary particles of finite size. Whatever the intuition we have about that, it may be the wrong way to pose the problem, because of the observations themselves.

Finally, another big confusion: the uncertainty principle of Heisenberg isn't an issue of imperfect measurement procedure. In addition, it doesn't prevent a particle can have a well defined position (uncertainty 0) OR a well defined momentum. The uncertainty principle is observed every day in experimental quantum optics, and in the very existence of stable atoms.

Dear all,

Although the discussion provides highly different opinions we seem to agree (except Claude Pierre) that the mathematical point is unphysical – Horst and Eric did express this much better than I did.

Nevertheless we find points and trajectories to be excellent ways to communicate classical physics and in the same manner we find the mathematical convenience of the concept of fields. The crux is always in the interpretations given to specific cases. In this ballpark many physicists may prefer the Feynman (and Lévy-Leblond) pragmatic tradition, which to many theoreticians including Jean Claude appear b..

In contrast we do not agree on how to view “spin”. Here it is very important to distinguish (as always!!) what is mathematics and what is physics. When I say the “spin” is a quantum effect I mean first of all how it is measured (see my comments above) and then how to interpret the results in the framework of contemporary physical theories.

A physical effect that constitutes a law of nature seems a less profound notion today than in the past. Note that the reason Einstein did not receive the Nobel for his relativity theories was strictly based on the fact that theoretical physicists at the time were not on par with experimentalists unless they could explain or predict a physical law. For the Nobel Committee therefore he had to wait an extra year before he was honoured “….. especially for his discovery of the law of the photoelectric effect"!

Today, with the accuracy of the GPS etc. we can actually talk about the Einstein laws of relativity as the effects of relativity are physically measurable. Would anybody disagree and say that e.g. SR is the consequence of the Poincaré algebra and therefore is not purely a result of relativity physics. This is again is mixing math and physics.

In this vein (Daniele), the saying that QM is an obsolete theory, are to the best of my valuation based on speculations that heavily conflates mathematics and physics and the same judgment befalls on the assertion that “indetermination/uncertainty isn't a law of nature but only a consequence of our methods”.

Claude, it seems me now that we say the same things by different words.

I agree that the "upper bound" of our ability to measure a length will tend to decrease in time and therefore we will near always more the Planck length (about 10^-35m), that then will be certainly exceeded. If you are search for a new theory for atomic systems and particles I could propose the following article

https://www.researchgate.net/publication/211874241_Basic_Principles_of_Deterministic_Quantum_Physics

I would want to propose the same paper also to other kind commenters so that I can answer to prospective scientific criticisms rather than answering in simplistic way to all interesting questions.

Article Basic Principles of Deterministic Quantum Physics

Erkki, there is nothing physical or not, there are theories, and the relationship of their predictions with experiment. The standard model based on pointlike particles is the theory that works best. The string theory has no relationship at all with experiment. Unless a better theory is found, we can say nothing at all about the status of the "mathematical point." Anyway, there is no experimental evidence that something elementary would have a spatial extention. Intuition and reality are distinct, and thinking about extended little bodies would mean making a big step backward to old classical mechanics that has been falsifyed by so many experiments.

Eric Lord

Claude ~

“The standard model based on pointlike particles is the theory that works best.”

> The standard model is not based on pointlike particles. Baryons are known to be structured entities. The nature of leptons it is less well-understood, but there’s no evidence at all that they are ‘pointlike’.

“Anyway, there is no experimental evidence that something elementary would have a spatial extention.”

> There is no experimental evidence that something elementary would be pointlike.

“…thinking about extended little bodies would mean making a big step backward to old classical mechanics that has been falsified by so many experiments.”

> Which experiments falsify the idea that elementary particles are extended objects? I’ve never heard of any such experiments.

It seems to me that the concept of 'point particles' is something that modern physics has inherited from 'old classical mechanics'. We no longer think of the electrons in an atom as 'point particles', for example, but as extended probability distributions.

What is meant by 'elementary' in this context anyway? It can be argued that the 'elementary' constituents of physical reality are fields, not 'particles'.

Following Eric's line of reasoning: the electrons building up the states of the elements of the periodic table cannot be modelled as "small" planetary systems with the "planets" in well-defined trajectories around the "sun"(nucleus). This was a wellknown fact almost 100 years ago – who is backwards?

It is manifest that is convenient to represent extended physical systems (particles, bodies, etc..) through conventional geometric representations that make use of the point concept of barycentre. Zero volume has no physical meaning.

Eric, nevertheless I disagree also with your deduction "It can be argued that the 'elementary' constituents of physical reality are fields, not 'particles'". In fact the concept of field is based upon the concept of source and consequently any type of field, from macroscopic scale to microscopic scale, involves the presence of a source.

I understand you would want to reconcile the concept of field with the extended probability distribution but I see it is very hard because the concept of field involves only a mathematical potential and also the potential energy involves the presence of another physical system that is omogeneous with the source. It would be then still harder to explain physical realities like mass, charge, etc.. that I think you don't want to deny.

Steve Faulkner

I like the concept of fields because fields satisfy the Field Axioms. And the Field Axioms are the rules of arithmetic for scalars -- that is the rules of elementary algebra. Within this algebra is an inherent logical independence (mathematical undecidability) that can help us understand indeterminacy we see in experiments involving wave packets, prior to measurement.

Eric Lord

Daniele ~

Every elementary particle has an associated field equation or set of field equations (Maxwell equations, Dirac equation, etc), coupled through interaction terms. This can all even be thought about and formulated classically, before quantization. All is then expressed in terms differential equations of fields. No “particles” in evidence at this level of thinking. When thought of like that fields do seem to be primary, particles secondary manifestations. That’s all I meant. For instance an elecromagnetic field has charges and currents as sources, but the protons (or their constituent quarks and gluons) and the electrons that constitute the sources are themselves manifestations of field equations. As for “explaining” things like mass and charge – they occur as parameters in the field equations, like coupling constants. What they actually are is a mystery!