If you really want to, I suppose in classical electrodynamics, you can choose to measure charge using units of mass if you wish. In this case, the Coulomb force will have a coupling constant with the same dimensions as Newton's constant for gravity. You would have to adjust the dimensions of other quantities of course, so that indeed, the electrostatic potential would have the dimensions of velocity squared. But there's nothing wrong with that.

Not sure what you are trying to accomplish with this approach, though. Even if you choose to measure charge using units of mass, charge is NOT mass. For instance, a positron and a proton have the same charge yet wildly different masses. The proton and the neutron have the same mass but different charges. And so on.

So in the end, when you describe a charged particle for instance, you would still need separate terms in its Lagrangian for its kinetic energy (dependent on its mass) and electromagnetic potential energy (dependent on its charge). So you'd still end up with a relativistic Lagrangian like L = muμuμ + quμAμ, where the mass m and the charge q are distinct quantities regardless of the units chosen to measure them. And none of this has much to do with the non-renormalizability of quantum gravity, which at the root of the unification problem.

Hi Max, appart frm the topics mentioned by Toth (gravity is fatally atractive and electrostatic action can be atractive or repulsive) electric force is 10^36 times stronger than gravitational force. Since the mass of a Proton is 10^(-27)kg, you must have larger than 10^9 kg inside the nucleus of a single H atom for gravitational interaction to override electric interaction between p+ and e-.

The fundamental unit of electric charge is sqrt(hbar*c*epsilon_0), in terms of which the electron charge is -0.30282... --- regardless of which (consistent) set of units you choose to express hbar, c and epsilon_0 in.

There are also fundamental units of resistance and magnetic flux.

Mass is a measure of the amount of matter in a system. Charge on the other hand is more closely related to density. They are not the same thing. In Maxwell's early papers, charge does not appear. His equations involve 'per unit volume'. I maintain that charge is a measure of the tension or the pressure in the aether that surrounds particles. Negative charge is the tension that arises due to inflow into a sink. Positive charge is the pressure that arises due to outflow from a source. Both are vorticity dependent. Vorticity tightens sinks and loosens sources, hence explaining the fundamental origins of centrifugal force. See "The Double Helix Theory of the Magnetic Field" at this link,

http://gsjournal.net/Science-Journals/Research%20Papers-Mechanics%20/%20Electrodynamics/Download/252

The short answer is No. The reason is that mass can be defined using Newton's constant, Planck's constant and the velocity of light, while electric charge doesn't have anything to do with gravity. In addition, the finite structure constant, α=e^2/(2ε_0hc), where e is the electric charge,  h is Planck's constant and ε_0 is the vacuum dielectric constant, can be shown to be dimensionless. Therefore the dimensions of e/sqrt(ε_0) are equal to sqrt(hc)=sqrt(M L T^(-1) L L T^(-1)) = M^(1/2) L^(3/2) T^(-1), which can be checked from Coulomb's law, F=M L T^(-2)= (e^2/ε_0) L^(-2). 

Indeed, the quantitative description of quantum electrodynamics relies on the fact that all electromagnetic effects can be expressed in terms of the dimensionless coupling constant, α. Similarly, the consistent quantum description of the weak interactions and their unification with the electromagnetic interactions involves realizing that Fermi's constant, G_F, has dimensions-therefore can't describe quantum effects consistently, but it is possible to describe the weak interactions in terms of two dimensionless coupling constants and Fermi's constant is expressed in terms of the vacuum expectation value of the Higgs field. Similarly, the strong interactions are described, using QCD, by a dimensionless coupling constant. The problem with the gravitational interactions is, indeed, that it isn't known how to write it using a dimensionless coupling constant and how Newton's constant can be defined in a way similar to that of Fermi's constant.

Gravity may well have everything to do with electric charge. Gravity could be a case of two large scale aether sinks attracting. You would then ask why do two sinks repel in electrostatics, and you would further assume that there could be no solution to this apparent contradiction, but the solution begins by acknowledging that motion causes convective pressures in the aether. Fill all of space with tiny aether whirlpools that are pressing against each other with centrifugal force while striving to dilate. The inflow into large sinks would set the tiny whirlpools precessing with their precession axes aligned along the radial flow lines. This would generate centrifugal pressure to act perpendicularly to the radial flow lines. Since gravity is inverse square law and since centrifugal force is inverse cube law, an increase in the attractive force along the radial field lines would cause an increase in the centrifugal repulsive force acting at right angles to the field lines, and this induced increase would be at a greater rate. If the charge is strong enough, a reversal threshold will be reached and two sinks will repel each other. In other words, gravity may simply be weak negative electric charge.

In gauge theories the state of particles is represented by a tensor between a spinor, which embodies the kinematic characteristics of the particle, and a vector which corresponds to the charge (how the particle behaves with force fields). There is no specific unit for the charge : actually we do not have units for the color, the comparison is made between some specific elementary particle (say an electron) and another. The coupling constant are not actually charges. When we look only at the first generation of force fields, the EM field is represented by U(1), and so it is possible to represent the EM charge directly with the spinor (by a phase factor, a complex number of module one) which is the genuine motivation for the introduction of rays in QM. This way elementary particles are represented by spinors vectors, which incorporate both the kinematic and EM charge, and antiparticles have charge opposite to particles.

Beyond the formalism,  I believe that there is a deep relation between the EM field and the gravitational field. For instance they are the only fields which have an infinite range.

Akira> There is no obvious way to unify these two contradicting laws.

You should abandon this nonsense.

Thierry> So, charge is mass.

Og mor-lille er en sten.

As it seems, you answered yourself! Regarding the theory of "everything", at the last there is nothing else but "ENERGY". In principle this real statement is simply understandable, great difficulties appearing only when we try to put this in quantitatively equations.

why this?

I have h=c=m=1, and the only "dimension" is length!

Max,

Units are quite conventional but sometimes the convention is only for convenience supporting certain types of measurement.

It is quite natural to have a system where electric charge and mass have the same units of measure while being different things. I have done it some time ago as an exercise in order to eliminate kilogram from the fundamental units. Maxwell was speculating on this long time ago.

The mass unit would be in m3/s2  and the electric charge as well.

In such system Gravitational constant simply disappears. Electric field and acceleration have the same units and induction the same  unit as angular velocity.

It is however difficult to precisely determine what is the current kilogram worth in those units because it can only be done by measuring gravitational force and that is not so precise for small objects.

Change of units does not change physics however it may help in clear thinking to see relationships between different physical quantities.

Based on my understanding, mass is how much material in a system. Charge is a property of material under certain structure. For example, an electron and a positron has the same mass, but their charges are opposite. This is because they have different structure.

There is repulsion in gravity. But it is not included in Newton's law. For example, an apple falls because the gravity. But when the apple land on the ground, both the apple and earth are neutral. What balances the gravity? We defined other forces to balance it. Then the problem is: when and where do the other forces start to affect the system? Based on my understanding, gravity is an alternative electric force. 

For relative theory, the tricky part of relative theory is most results are right under certain condition, but the theory is not correct. People need to be very clear in which condition you can use it. A very simple contradict part is the theory is based on movement is relative, then the light speed is constant to all the observer. To cover this, Einstein introduce time and space into his theory. These totally misled people in a wrong direction.

People always defense the theory the believed by the approved test result. First, some test result is correct under some certain level accuracy. But incorrect under more accurate measurement. Second, many theory works well in the test condition only, when you change the condition, it does not work any more. 

We need to truly understand how the material acts first before we use any math equation to derive it. Material follows its own property to act, not math equation.

Yutao> But when the apple land on the ground, both the apple and earth are neutral. What balances the gravity? We defined other forces to balance it. Then the problem is: when and where do the other forces start to affect the system?

I don't understand why you refer to it as a problem! It isn't, if you bother to read some introductory physics. Or, even simpler, why not the famous lectures, with back-of-an-envelope calculations, by Vicky Weisskopf. Uploaded here.

@Kare. It is a problem. Let get back to the concept of force. What is the force? Did any body define any other force between apple and earth except gravity? Then, why does the apple stop. You can calculate the attraction with Newton's law. What about the repulsion when the apple lands on the ground? If you agree with the repulsion, then what cause the repulsion and when and where the repulsion start? 

Newton's law never explain what is force or what causes force. Most people think there is force between particles. Then calculate the force with some equation. If the particles are totally separated, why do they interact? If you analyze the particle more detail, you have to realize that there is field around particle. Field moves with particle. It means field is part of particle or extension of particle. Field is not a media to transfer force. Force is caused by field connection between particles. Field alignment between two particles causes force. When particles are far away, the attract dominates. When they get closer, they repel each other because the field structure changing with distance. The repulsion gradually increases with distances and finally balanced with attraction. Newton's law only describes the net interaction when particle are far away. The repulsion is ignored.

The fundamental concept of force in current physics is not clear or wrong. 

I do not know why people treat field and particle separately. They coexist together all the time.  Pair production experiment also proves that particle and field can exchange. This clearly tells us that particle and field are made up of the same thing. Particle is formed by field under some certain structure. This logic is so simple.

Most people are not clear what happens when we say something changes. The basic component of the world never be changed. You can not generate it from others and it will not be changed to others. It just exists. It may change structure or form to creates a new thing. For example, we can make H2O from H2 and O2. But H and O is still there, the only change is the electron position changes. This opinion is also the base of mass conservation law.

Akira:

Your question about am=f/m and aM=f/M is because you changed the reference point from m to M. The reference points, m and M, move in different speed. If you choose a right reference point, you will find m moves faster than M. For example, when you use a rope to pull a basket ball on the ground, if you choose a point on the ground as reference point, you will find the basket ball is moving faster than you.

@Kanda: "This same mutual relative speed is the most basic assumption which came from our empirical experiences. When you violate this you are not an empirical scientists anymore."

This is wrong. Whether the mutual speed of two inertial systems is the same or not, is entirely dependent on the synchronziation gauge chosen by the observers. If they choose Einstein synchronization, the mutual velocity is the same (up to the sign), if they choose another synchronization, for example the one suggested by Tangherlini and later by Selleri, they will not find the same mutual speeds. This is most trivial mathematics.

That different choices for the synchronization gauge can be made, is a consequence of special relativity. Of course, everybody uses Einstein synchronization when dealing with inertial systems. With rotating systems, it is however often advisable to choose a different way to synchronize clocks.

"Newton knew that moving reference frame will cause contradiction and avoided the relativity theory. He was against Galileo's relativity theory through out."

I think this is a misrepresentation. Newton was well aware that his space was not fully absolute (i.e. that rotation was absolute but translation at constant velocity was not).

Relative velocity (``mutual relative speed'' is the norm of the relative velocity) can be defined in flat spacetime, where inertial frames are globally defined. It can't be defined in curved space-time, because the parallel transport of vectors in curved spacetime depends on the path chosen. So it isn't an invariant quantity. There's no point arguing over quantities that can be proven to be non-gauge-invariant; similarly over coordinate system choices. On the other hand, it's hard to see what this discussion has to do with electric charge or mass. Electric charge is a Lorentz invariant quantity whose conservation is the result of an internal symmetry, mass is another Lorentz invariant quantity,  one of the Casimirs of the Poincaré group, which acts as  a spacetime symmetry, and in curved spacetime can be defined only at infinity. In this latter case it is true that electric charge can, also, be only defined at infinity; but that both are defined through a corresponding Gauss' law doesn't mean they are related.  

@Kanda: "I am discussion the classical physics. You should learn how Einstein's theory was developed. Then these issues I pointed out will nail his theory of relativity. They what you are saying looses all its ground. The problem with physics community is that nobody think logically."

Well, to me it seems that you cannot think logically -- at times at least.  And you have too little knowledge about the theory to discuss it meaningfully with you. To define a velocity, you need to know what it means to have the same time at two different places. Newton  states this as an axiom that unfortunately does not agree with experience. Once you cannot use that axiom anymore, you cannot meaningfully talk about system B having velocity -v wrt to system A, if A has velocity v wrt to B. That depends on how A and B have clocks syncchronized in their respective inertial systems. (It is true for Einstein synchronization, it is wrong for "Newtonian synchronization" in the real world.) Most people do not realize that, even those that have a decent knowledge of special relativity. I can hardly expect that you know it.

Special relativity expresses  the invariance of the equations of motion under Lorentz transformations, which are those transformations that map solutions of the wave equation to other solutions of the same wave equation and do not depend on the spacetime point-so the velocity parameter is the same. In fact special relativity is the special case of this statement for the case of electromagnetic waves in vacuum, where the velocity is the velocity of light in vacuum. These statements are consistent mathematically, since the Lorentz transformations so defined, can be proved to form a group and they can lead to experiments, that have been realized for more than a century, to increasing degrees of precision. However it doesn't make sense to confuse the two issues, of mathematical consistency and of experimental relevance. 

Similarly, general relativity is based on the invariance of the equations of motion under general coordinate transformations, that can, also, be shown to form a group.  It can be shown that this means that the equations of motion are invariant under Lorentz transformations, whose parameters may depend on the spacetime point-they're local, not global, transformations. While the mathematical issues are considerably more elaborate than for special relativity, nonetheless a lot has been learned and  the existence of  a mathematically consistent description is well established-in particular of the appearence of singularities, cf. for instance the monograph of Christodoulou, http://arxiv.org/abs/0805.3880, for a technical presentation of the formation of black holes and has, also, been subject to experimental tests, cf. http://relativity.livingreviews.org/Articles/lrr-2006-3/download/lrr-2006-3Color.pdf . There is a lot of work being carried out on better understanding how such singulariities form, but the basic theorems have been proven since many decades. 

Their physical relevance is wholly independent from their consistent mathematical treatment. 

Once more, mathematical consistency is totally distinct from experimental issues, such as how to realize clock synchronization, for example-what matters is that such an operation is meaningful. The former is required, in order for the latter to even make sense. While the understanding of these subjects is a historical process, it doesn't make sense recalling all the *wrong* ways these subjects were studied, except in the context of history of science; it's much better, from the perspective of physics and mathematics, to learn how the correct understanding can be formulated. And while both subjects do owe a lot to the work of Einstein, they're not the property of any one person and can be discussed in a totally impersonal way. 

Akira,

I am glad that you brought up the geometry.  Although it is convenient to use the idea to describe what it looks like from the point of the moving body and how to calculate the trajectories of things falling through gravity it has no real meaning when looking at the situation from a distant point. 

We have given up on the idea that we may be able to calculate something in flat geometry.  This just means that we are so enthralled with the idea of giving up on being able to measure things as they move that we are willing to make stuff up to make it work.     There exists a point in space that will always show if there is true simultaneity of two or more events regardless of weather one person or the other seems to think they happened at the same time or different times.  This however does not change time but only our ability to tell if the events happened at the same time.

From the relativistic standpoint there is always a point in space that will prove that no two events happened at the same time regardless of if they did or did not happen at the same time.

This is an argument over the inability to measure things in a strong field.  Relativity is just the ability to measure things given the strong field it is not reality in a flat geometry.  With our minds we can understand both. 

The problem is that we have spend the last one hundred years or more thinking that flat space is curved.  

Akira,

I think you are correct.  How do we proceed when the logical truth is not considered in the science??????

Akira> with some substantial publications in the top mathematical logic journals

That must be some top journals which are not indexed by ISI... 

One writes the equation that describes the parallel transport of a vector along a parametric curve from one point A to another point B. And one then deduces that the vector at point B depends on the specific parametric curve used to ``parallel-transport'' it from point A to point B, if the space, labeled by the points and on which the vector is defined, is curved. These are all mathematically abstract and well-defined notions. If one now applies these results to spacetime, since they are valid, also, for pseudo-Riemannian, as well as Riemannian, manifolds, one deduces that velocity 4-vectors cannot be consistently compared, if they are not defined in the same frame. The relative velocity of two frames, that refer to different spacetime points,  is, precisely, given by such a difference. In flat spacetime there isn't any problem, because parallel transport does not depend on the path; in curved spacetime it does, therefore relative velocity can't be invariantly defined. 

 In any spacetime, whether Galilean or Lorentzian, motion of points is described by parametric curves, one-dimensional flows that map the spacetime to itself. The maps are defined, in classical physics, by the evolution operators, deduced from the equations of motion.  Nothing mysterious about it. 

Of course a selection has been made-it's called the differential geometry of manifolds, for the topic that's being discussed in the last few messages, so there are assumptions about smoothness and so on. The subject is well known and taught in  courses and available in textbooks. The question of whether the mathematical description can be used to describe physical phenomena, that are known to a certain degree of precision, is a totally independent one-and, has, of course, been addressed in the links provided in previous messages. The mathematical description and the results of the experimental studies  aren't opinions, precisely because it's possible to check their validity independently of who is providing them. It is possible to prove that one can define ``invariant quantities'' and calculate and measure those.  

The equation x^2 + y^2 + z^2 = (ct)^2 defines the light cone, in four variables,  since it is one constraint for four variables. It is a sphere for fixed t, but t does transform under Lorentz transformations and it is easily shown that (x')^2 + (y')^2 + (z')^2 = (ct')^2, where the primes are the transforms: the light cone is preserved under Lorentz transformations. 

There's nothing ``wrong'' (or ``right'')  with an equation not being invariant under Galilean, or Lorentz, transformations. The consequences matter. The whole point is that Maxwell's equations are not invariant under Galilean transformations, while Newton's equations are. Maxwell's equations are invariant under Lorentz transformations, Newton's equations are not. So the question arises, whether Lorentz trasformations are, in fact, Galilean transformations in disguise, or not. Now, for sound waves, it's possible to prove that the invariance of the wave equation, under Lorentz transformations that preserve the velocity of sound, is a ``coordinate artifact'', since it is possible-though complicated-to identify the constituents, whose complicated interactions give rise to the ``simpler'' motion described by the modes of propagation of the wave. The wave equation for sound waves is a consequence of Newton's second law for the constituents. What's remarkable is that if one tries to find the corresponding constituents for electromagnetic waves, one fails: there don't exist material constituents, whose interactions can give rise to the propagation of electromagnetic waves. This is an experimental fact, that can't be deduced from anything else-it's a new input, that comes about by trying to devise experiments that can test the consequences-which is what Michelson and Morley did. There's no a priori reason, why such constituents couldn't exist. The invariance under Lorentz transformations is, therefore, not a ``coordinate artifact'' for electromagnetic waves.  Once experiment showed what the consequences were, it's necessary to try to understand this result-and, of course, show, by further experiments, that this result is not an artifact of the experiment itself.

The reason  can be understood as the fact that sound waves are coherent superpositions of the motion of matter particles, that behave as classical objects at  the resolution of sound waves, they obey Boltzmann statistics, whereas electromagnetic waves are coherent superpositions of the motion of particles that obey Bose-Einstein statistics. 

Such paradoxes arise only if non-invariant quantities are discussed. It can be proved that there exist quantities, that are independent of the coordinate frame used and, using these quantities, there are no paradoxes or ambiguities. Lengths and time intervals are not invariant under Lorentz transformations; spacetime intervals can be proven to be invariant under them. Therefore it's meaningless to use non-invariant quantities for anything else but for the calculation of the invariant combinations.  One  reason Galilean transformations are useful is that Newton's equation of motion are invariant under them; one reason Lorentz transformations are useful is that the wave equation in general and Maxwell's equations in particular are invariant under them. In each case while coordinate choices must be made, for specific calculations, the only quantities,that can be given a physical meaning, are those that are invariant under the corresponding transformations.  

I think most people are not very clear about the relationship between time, space and material. Time and space are math concepts. Material can interact with material only. Light curves when it passes massive particle. This is because the field around the particle interacts with photon. If light follows the space curvature, why do we need massive particle? 

The universe is made up by material. Material has absolute meaning. We can use "yes" or "no" to describe there is material or not. Movement is because the material loses balance. When it try to reach a new balance, it has to adjust its position. We call it movement. Movement has its absolute meaning. It reflects how unbalance a body is. Because all the material in the universe is entangled by field, all the materials are moving. We cannot find an absolute static reference point. We have to use another object as reference. 

We normally use a coordinate system to describe movement. We have to be very clear, once you choose an object as an reference point, the X, Y, Z in your coordinate system are not space any more, they are the distance between the event to the reference point. Einstein wrongfully use X, Y, Z as space. This totally mess up the relationship among space, time and material. But when we deal with real problem, we have to choose an object as reference point. That is why some of the result from relative theory can be proved. 

About particle concept. Let us see a simple example, an electron. We always think electron is a solid ball which has diameter or it is made up of some other small particles which has diameter. Actually, electron has electric field around it. The electric field moves with electron. The field is part of the electron. It means electron does not have a clear edge.  Its field extends far away. Then, the question is why the field attached to the electron. Pair production experiment tells us that electron and positron can change to electromagnetic field. This means electron and positron are made up of electromagnetic field. All other particle have field around it too. Then the picture is so clear now. Particle is a form of field. The higher density part is called particle, the lower density part is still called field. That is why field always starts from particle. 

The main goal of physics is to find how the material acts. Since all the materials in the world are connected and they are relatively moving, it is very hard to use a math equation to describe everything. But in a certain condition, math is a very useful tool for calculation. But we must be very clear how the material acts first, then choose a math equation to describe it. Most people try to derive a math equation to predict how the material behave. This is not going to work. Material is the target, math is a tool.

The existence of invariant quantities under Galilean or Lorentz transformations is a mathematical statement, can be proved using standard mathematical tools and, normally, is background knowledge for the discussion here-as is a working knowledge of classical electromagnetism and general relativity. No sensible discussion can be made, without such background knowledge. The resolution of the topic under discussion, in fact, has been carried out in quantum electrodynamics, where the scaling dimension of electric charge is defined and calculated, within perturbation theory, at least. Using numerical simulations it's possible to obtain information on the scaling dimension beyond perturbation theory; however the lattice artifacts are quite non-trivial. 

This discussion is very interesting but it has drifted away from the main point:

Could the fundamental dimension of Electric Charge be Mass only?

The answer is YES. It it is quite possible to have mass and charge be expressed in the same units. For example mass can be expressed in two fundamental units only - say meters and seconds. And mass can be measeured in m3 /s2  after eliminating the kilogram from fundamental units. After this the charge becomes automatically in m3 /s2

And guess what is the non dimensional ratio of electron's charge to its mass ....

For some reason Stam Nicolis does not think it is possible to have common units for mass and charge.

When it comes to relativity discussions here, they have nothing to do with the original question. 

I am running a thread on the fake nature of relative simultaneity where it would be more appropriate:

https://www.researchgate.net/post/Is_relative_simultaneity_a_misinterpretation_of_Special_Relativity

Units reflect conventions-it's dimensions that express physics. Electric charge enters physical processes, at the quantum level,  in the dimensionless combination known as the fine structure constant, in four spacetime dimensions.  So it doesn't have the dimensions of mass in any consistent description of such physical phenomena. In *three* spacetime dimensions, the electromagnetic coupling constant does have dimensions of inverse length-which are  the dimensions of mass, up to a conversion factor. 

In classical physics electric charge, as an additional attribute of matter,  has distinct dimension from mass-in Coulomb's law it's the dielectric constant that expresses the the fact that charges can give rise to forces, that are defined by Newton's law to have dimensions of mass x length / time^2. 

So, how many m3/s2 is a kilogram?

Kåre,

The approximate conversion factor is 1.498291182x1010 kg/(m3/s2)

Its much bigger than imperial 14.593903 kg/slug  :) and more rational as well

then the neutral body will have zero mass.

Akira

"If mass and charge have the same dimension, why is it that mass and charge have different laws regarding force. For mass we have \newton's second law and for charge we have Lorentz force. They contradict each other"

For this question only Nature can give an answer, but I can only say that this does make sense.

Max - the owner of this thread - posted this question based on his analysis, and I have accidentally discovered this possibility many years ago.

I did not aim to equate the charge and mass. It did not even cross my mind. My question was "what if we could eliminate the ugly and physically irrational kilogram from the units of measure".

One of many possible systems is when mass is measured in (m3/s2)

That makes the gravitational constant an exact non-dimensional mathematical constant. The rest is just simple algebra which at the end removes the Couloumb, Amper etc leaving only meters and seconds or L T in general..

The dimension of charge in result, emerges as (m3/s2)

Both, charge and mass are properties of particles and both are the cause of particle acceleration albeit in a different way.

There is the same form of the law of force between two masses and two charges, hence their dimensions can be the same.

The same goes for forces themselves. Gravity and electrostatic force are of a different origin but they are still forces and have the same units of measure, and no one is surprised about that.

The calculation of the ``engineering dimensions'' of mass and charge in any spacetime dimension is a completely straightforward exercise and is based on the fact that phases are dimensionless. The action in quantum mechanics and quantum field theory is defined in this way, since it enters any physical quantity as exp(iS), so Planck's constant is, already, taken into account. 

The kinetic terms in quantum electrodynamics define the engineering dimensions of the fields:

d^D x ψbar d.γ ψ should be dimensionless, therefore D-1 + 2[ψ] = 0 [ψ] = -(D-1)/2 in terms of  length. From the Fourier transform one deduces that k.x should be dimensionless, therefore [k]=-[x] and [ψ]= (D-1)/2 in terms of energy, momentum or mass dimensions. 

Similarly, the kinetic term for the gauge potential, d^D x F^2, implies that D-2 + 2[A]=0 [A] = -(D-2)/2 in terms of length or [A] = (D-2)/2 in terms of energy, momentum or mass. 

The mass has dimension 1 in all dimensions: 

d^D x M ψbar ψ dimensionless implies that -D + [M] + D-1 = 0 [M] = 1, in terms of energy, momentum or mass dimension, in any dimension.

The engineering dimension of electric charge is deduced from the interaction term:

d^D x e ψbar A.γ ψ implies that -D + [e] + (D-2)/2 + D-1 = 0 [e] + (D/2)-2 = 0

[e] = 2-(D/2)

therefore

[e^2] = 4-D

which satisfies  [e] = 0=[e^2] in 4 spcetime dimensions and [e]=1/2 [e^2]=1 in 3 spacetime dimensions. 

We remark that [e]=1=[M] in D=2, only, but there aren't any electromagnetic waves in 1+1 dimensions. Electric charges are confined in dipoles, due to the fact that the potential is V(x)=const.|x| in 1+1 dimensions, thus two charges feel a constant force, independent of the separation.

Of course one can reinstate the dimensions of Planck's constant by a trivial transformation.

A phase is dimensionless, since it's an angle: k.x-ωt is dimensionless. The dimension of the wavefunction is, in fact, arbitrary, since it cancels out, when performing the calculation for the Schrödinger equation and,  since it defines average values of any observable, O,  through 

= /

 the dimension, [ψ], of the wavefunction, isn't defined by any physical process, which is, of course, consistent with the fact that the wavefunction itself isn't a physical quantity-it is useful because it allows the definition of other quantities, the average values, that are physical. 

An angle is the ratio of two lengths, the length of arc of circle and the length of the radius of the circle and is, therefore, dimensionless. The phase of a wave is |k|.|x| -ωt, with |k| = 2π/λ and ω=2π/T. [x] = [λ]=length, [t]=[T]= time. Therefore the phase is dimensionless. It doesn't matter what anyone ``thinks'', ``believes'' or ``states''-what matters is what can be proved in an impersonal way. And it doesn't matter what anyone thinks about the wave function, the statement that its dimension is indefinite is a mathematical statement that can be proved as shown in a previous message. That many people haven't learned classical mechanics, electromagnetism and quantum mechanics isn't surprising-that they spend time and effort in making trivially wrong and/or metaphysical statements about these subjects is an interesting sociological phenomenon, since it's not obvious what such meaningless statements can be helpful for. Mathematical statements aren't ``good'' or ``bad''-they're decidable or undecidable.  The mathematical statements at issue here are all, trivially, decidable. The engineering dimension of mass is, always, equal to 1 and the engineering dimension of electrical charge is, always, equal to 2-(D/2), where D=dimension of spacetime. The former  does not depend on the spacetime dimension, the latter does. What's interesting is that these numbers describe useful properties in classical physics. When quantum effects are taken into account it turns out that scaling properties of physical quantities are described by numbers that can be identified with dimensions, called scaling dimensions, and the scaling dimension of the electric charge is no longer equal to its engineering dimension-whereas in classical physics the two are equal.  

it isn't true that the equality of mass  and charge, for BPS states,  implies that the scaling dimension of the coupling constant is that  of a mass. Here the question is about the engineering/scaling dimension of the coupling constant.  Indeed for the case, where the duality was introduced, in four spacetime dimensions, the scaling dimension of the coupling constant  is equal to the engineering dimension and is equal to zero, since the coupling constant is dimensionless. 

Akira> What is the dimension of the wave functions of Schrodinger

In non-relativistic quantum mechanics, according to the Born probability interpretation of psi**2, the wave function has dimension length**(-3/2) in three dimensions. Which is what you will find when checking properly normalized example solutions in introductory books on quantum mechanics.

This is true if one imposes the condition that the integral of the probability density be dimensionless-and is a matter of convention, not physics. Since the only physical quantities are the expectation values of observables, the dimension of the integral of the probability density isn't observable itself and the dimension of the wavefunction cancels out in the expression /; the dimension of this expression is the dimension of O. 

In other words-the dimension of the partition function (which the integral of the probability density is), would be the argument of an ``absolute phase'', that's unobservable-and another way to see this is to  realize that a physical dimension (including the value zero!)  of the partition function  would be an ``anomaly'', an indication that the theory isn't unitary and must be completed by other degrees of freedom, whose contribution would make the phase unobservable. 

I think substantial contributions to mathematics were made by physicists. So it is not a question of one group against the other, if we start discussing at the level of ideas about who is lunatic and who is not.

Logic is not something that only mathematicians can use and solipsism is certainly not a position that would allow one to draw conclusions about the state of mind of others. That is clearly a logically self-defeating position. The problem is not about where everybody went wrong, the question is about where you yourself went wrong.

Akira,

There is no flaw in classical physics the way you describe. In a fixed reference frame two gravitating spherical objects linearly heading into one another, away from other forces are ruled by the two ordinary coupled differential equations:

M*d2(X)/dt2 = GMm/(X-x)2

m*d2(x)/dt2 = -GMm/(X-x)2

From this we have:

M*d2(X)/dt2 =- m*d2(x)/dt2

Each body has a different acceleration d2(X)/dt2  or - d2(x)/dt2 and obviously there is only one absolute value of relative velocity. When you accelerate towards Earth at 9.81 m/s2 while in a free fall, the Earth acceleration due to person's gravity field is roughly 1.644x10-22  m/s2