Anybody interested in Dirac's equation?

[Bombadil]

Massless is “without mass”:i.e., the mass term is zero. When I was a student, the photon was the only known massless boson and the neutrino was a massless fermion. Now it seems that they consider the neutrino to have a very small mass (personally I wonder if that could be a subtle consequence of Pauli exclusion). Of course the supposed graviton was assumed to also be a massless boson but I am not aware of a usable graviton theory outside my work.

 

Might it not be the case though that the definition that you have put forth for a photon might not be derived directly from the fundamental equation for a massless boson? It really doesn’t seem to far fetched from how I’m understanding it. The point being isn’t it possible that a photon might be the only massless boson.

 

If so wouldn’t this have implications for any possible explanation when considering the transfer of interactions from one element to another by means of a massless element?

 

The alpha and beta operators are defined as attached to specific numerical reference labels (those variables referred to as x). The alpha operators are directly attached to a specific reference whereas that beta operators are attached to a specific pair of references. If those variables are taken to be independent (i.e., seen as different directions in a geometry) then the fact that n reference variables are seen as a vector (in an n dimensional orthogonal space; four dimensional in my presentation) then the attached alpha and beta operators are also collected in terms of that vector representation.

 

Ok so [math]\vec{\alpha}[/math] is a vector in which every element in it is from the same space as [math]\vec{\Psi}[/math] and the dot product [math]\vec{\alpha}_i \cdot \vec{\nabla}_i\vec{\Psi}[/math] in the fundamental equation is a scalar in the space that [math]\vec{\Psi}[/math] is defined in. This can also be looked at as being a matrix in an abstract space. I use the word matrix here only to emphasize the point that if [math]\vec{\Psi}[/math] were defined to be a vector space then [math]\alpha[/math] would be a matrix and [math]\vec{\alpha}_i \cdot \vec{\nabla}_i\vec{\Psi}[/math] would be a vector. Looking at it in this way I think that I can understand how the dot product is being applied. Likewise the [math]\beta_{ij}[/math] operator can be seen as a matrix in the same abstract space.

 

Now also will the fact that we have not yet removed the [math]\vec{\alpha}[/math] and [math]\beta[/math] operators entirely from the equation in deriving the Diarc equation mean that any possible solution to the Diarc equation will have the [math]\vec{\alpha}[/math] and [math]\beta[/math] operators as part of any possible solution?

 

In my picture, an electron is not a fundamental entity. It is rather a description of a phenomena involving interactions between fermions brought about by bosons. Without the existence of photons, electrons could not exist.

 

I’ll take this statement as true for the time being as I suspect that it would take us slightly off topic to conform that every approximation that is made has no effect on determining that you have not defined one of the elements to be a electron and not a arbitrary particle. But, is all that we would have to do is conform this for the derivation of the Diarc equation or would we also have to do this for the second equation as well? I think that we would actually only have to do this for the second equation as the first equation says nothing about what the fermion is, it is the second equation that tells how it must be explained.

 

With that in mind then, are the constraints that are placed on the equation in your derivation actually constraints placed on the rest of the universe that is not under consideration or how do we correspond to the idea that there are elements not considered to be electrons. Is this just a consequence of the idea of a photon, and other types of bosons would imply other types of fermions? Maybe this question is to far off topic for the time being.

 

Laying that issue aside for the moment, there is still a subtle difficulty with the result I have achieved. I have shown that my fundamental equation yields a result quite analogous to Dirac's equation: i.e., Dirac's equation presumes the electromagnetic potentials are given and, in my deduction of my fundamental equation, I proved there always exists a potential function which will yield the observed behavior. Actually all this proves is that my expression above “could be” a valid expression of the electromagnetic potentials. That really isn't sufficient to identify my result with Dirac's equation as the electromagnetic potentials are specifically defined.

 

The problem here is that, from the perspective of my fundamental equation (and the work above), the electromagnetic potential is a many body problem and, as such, is a problem we cannot solve. That being the case, let me take the same attack to discover the form of [imath]\vec{\Psi}_2[/imath] which I used to discover the form of [imath]\vec{\Psi}_1[/imath]: i.e., presume a solution for [imath]\vec{\Psi}_1[/imath] and examine (via the interaction term) the kind of equation which the expectation value of [imath]\vec{\gamma}[/imath] must obey.

 

I’m not sure that I fully understand the problem here. Is it that we really have not defined how the second element must behave and in order to consider the first derivation to truly derive Dirac’s equation the second element must obey a particular equation otherwise the solution to the first equation may behave vary differently from what is expected from Dirac’s equation? That is, it is only Dirac’s equation if [math]\gamma_\tau \vec{\Psi}_2^\dagger(\vec{x},t)\cdot \vec{\Psi}_2(\vec{x},t)[/math] has a particular form.

 

If this is the case I really don’t understand why you have called the second element a electron. Won’t all elements behave this way when interacting with a photon, unless the rest of the universe really is of importance and we have inadvertently considered something about it? But still the only thing that it seems calling it a electron might do is maintain consistence with a more conventional view of physics as it is not really a electron but an arbitrary fermion.

[Doctordick]

Bombadil, on reading your post I get a feeling you are missing the central issue here. There are two very different things being discussed. The central issue is examining exactly what can be deduced from my fundamental equation. Since the fundamental equation is a many body differential equation, it should be clear to you that it is impossible to produce a general solution. Nevertheless, since I have proved the equation must be valid it is interesting to look at approximations which yield an equation which can be solved. That is to say that the only course open to us is to come up with approximations which might apply to situations we can comprehend.

 

This thread is titled, “Anybody interested in Dirac's equation? Thus what I am saying is, what results do we obtain when we make the approximations which define Dirac’s equation. That comment has to do with the approximations essential to use of Dirac’s equation in modern physics. My equation is entirely general and applies to any collection of information (and its application actually requires no approximations); however, in the absence of approximations we can’t solve it so it is pretty worthless as a general equation.

 

Thus it is that our first interest is to lay out exactly what the standard approximation modern physics makes when it goes to defend the correctness of Dirac’s equation. It should be clear to you that Dirac’s equation applies to “electrons” and how their states are influenced by electromagnetic fields. Electrons and electromagnetic fields are things defined in modern physics. What must be understood is that these modern physics definitions are actually descriptions of circumstances which qualify as being cases where Dirac’s equation is applicable. That is to say, they specifically express the approximations embedded in validity of Dirac’s equation (when and where it can be used).

 

The first approximation is that one entity be “an electron”. Exactly what is an electron? First, it is a fermion which means its wave function (the function which yields the probability of finding one) is antisymmetric with respect to exchange of identical things. This means that, when one goes to look at solutions to my equation, one is only interested in those solutions which are antisymmetric with respect to exchange of "electrons".

 

But, in classical modern physics, there are many fermions which are not electrons. Thus it is that we need an approximation which separates electrons from all those other fermions. Another classical modern physics approximation used to identify an electron is that it is electrically charged. This means that all “electrons” can be seen as a source of an electro-magnetic field which is the classical consequence of interactions with many photons. (Classically, photons are quantized electro-magnetic fields which certainly are not fundamental entities per my definition).

 

But there are still a great number of charged fermions which are not electrons so we need another approximation which will eliminate those fermions. That approximation (and one really has to understand some important aspects of classical modern physics to comprehend the fact that this approximation does indeed provide the needed result) can be put in several different ways that essentially make provide the same result (that result being, making Dirac’s equation applicable). The classical electron must reduce to a point particle, the entire mass of the electron can be seen as produced by the associated electromagnetic field or its wave function, [imath]\vec{\Psi}[/imath] must be limited to four abstract dimensions (the alpha and beta operators thus can be seen as four by four matrix operators or two by two matrices of Pauli spin matrices).

 

So, what do I do? I make those approximations (which are exactly the approximation made by modern physics) on my fundamental equations and, low and behold, what do I get? I get exactly Dirac’s equation. What this proves is that Dirac’s equation is an approximation to my equation (given that those approximations are the ones to be used). Since I have proved my equation is valid for any collection of information, this proves that Dirac’s equation is valid for any collection of information so long as the approximations modern physics uses to define its use.

 

Hopefully this will make things a bit clearer to you. To really get it clear, you would have to study modern physics. Counter to what everyone seems to believe, I am not overthrowing modern physics; what I am doing is displaying the underlying logic (the meta-physics: i.e., what is beyond physics) which defends the accuracy of modern physics.

 

Have fun -- Dick

[AnssiH]

That's a nice and clear post from DD there. I was about to comment to Bombadil also, just to keep the eye on the ball, that the set of specific definitions we call "modern physics" is just one possibility for "explaining" (structuring/comprehending/understanding) reality.

 

The fundamental equation is the expression of the entirely general constraints on one's worldview, springing from the need to stay away from logical contradictions, but not having anything to do with the actual nature of the "data to be explained".

 

When you have the fundamental equation, the question is, what gets us to such and such definitions/relationships expressed by modern physics. In the post above, DD talks about the approximations that were made in this thread. Note that they are approximations/definitions that can always be made, because they are related only to how you end up categorizing the data, not related to what the data actually is.

 

The problem people seem to have is that they don't believe purely logical arguments can yield modern physics. They think the validity of modern physics is a function of how reality actually is, and that information from different sorts of realities could not be structured in these ways. That turns out to be a bit hasty conclusion.

 

This isn't a matter of taking into account some tiny "what if" possibilities existing somewhere in the margins of our ontological knowledge about reality. This is a matter of taking into account, how do we arrive at sensical (self-coherent) object definitions in epistemological sense.

 

-Anssi

[Bombadil]

The first approximation is that one entity be “an electron”. Exactly what is an electron? First, it is a fermion which means its wave function (the function which yields the probability of finding one) is antisymmetric with respect to exchange of identical things. This means that, when one goes to look at solutions to my equation, one is only interested in those solutions which are antisymmetric with respect to exchange of "electrons".

 

On your stating these requirements it seems clear that they are exactly what you have done to derive the function that [math]\vec{\Psi}_2[/math] must obey (although I’m not entirely sure where you defined the element to be electrically charged) on the other hand it looks like you have not made these approximations in deriving the first equation which is Dirac’s equation instead you have used the definitions for a photon. This does not seem to me to be an entirely trivial thing as it seems to mean that whenever an element interacts with an element that satisfies the definition that has been put forth for a photon then Dirac’s equation can be applied. If this is the case is this the normal view of physics? If it is not the common view then it seems that it may not be an entirely trivial statement on its own.

 

Also haven’t you made the assumption that “(short term violations of conservation of energy) can not take place.” which sounds like it is also a completely expected approximation for the equations that you are deriving.

 

What is of some interest is that when we also define the second element as you have done to be considered an electron then we obtain the Maxwell equations as explaining its behavior (this is only of interest though because these are well known equations that approximate the fundamental equation) it also seems interesting to me to note that you have already derived a more general form of this equation in your derivation of the Schrödinger equation actually you derived all of these equations in more general forms but just ignored the form of the interaction term. And so we can gather that these equations are still relativistically correct. This seems though only to be of real interest in that it means that it should still be scale invariant. Which may not be an entirely trivial statement and may be of some passing interest.

 

What seems to be of most importance outside of the implications to physics which we are not interested in except in that it means that physics can be deduced by choosing the proper constraints for an arbitrary explanation is that you have demonstrated how in the absence of possible solutions to the fundamental equation we can still find subsets of the universe for which we can find equations that are far easer to work with and in fact some solutions can be found for simple cases. You also have shown how you can define elements to simplify the fundamental equation.

 

But, in classical modern physics, there are many fermions which are not electrons. Thus it is that we need an approximation which separates electrons from all those other fermions. Another classical modern physics approximation used to identify an electron is that it is electrically charged. This means that all “electrons” can be seen as a source of an electro-magnetic field which is the classical consequence of interactions with many photons. (Classically, photons are quantized electro-magnetic fields which certainly are not fundamental entities per my definition).

 

That is a photon is an invalid element and only supplies a means by which elements can interact. The point being that a photon only supplies an interaction between elements. They tell us that an element is charged, it says nothing about what the particle is that it interacts with other then that it is charged? So that Dirac’s equation tells us how a photon interacts with other elements or how a charge behaves.

 

It is the second equation that then supplies the remainder of the needed definitions for the second element to be an electron. Together they show that you have derived the equations governing the behavior of electromagnetic fields as a special case of the fundamental equation.

 

I am wondering at this point can all of the study of electromagnetic fields be derived from these equations or is this just a trivial case that tells us that all of the physics relating to electromagnetic fields could be derived from the fundamental equation if we wanted to

 

All this tells us is that the equations governing electromagnetism approximate the fundamental equation.

 

That being the case, under the assumption that interaction between photons is negligible and that any number of bosons may occupy the same state (the same function [imath]\psi(\vec{x},t)[/imath]) we may clearly include as many photons (the name I have used to refer to as element number two in the above deduction) as we wish. Thus it is that “electromagnetic field potentials” controlling the behavior of the electron could be defined by the expressions

[math]\Phi(\vec{x},t)=-i\frac{\hbar c}{e}\sum_{i=2}^\infty \left[\gamma_\tau \vec{\Psi}_i^\dagger(\vec{x},t) \cdot \vec{\Psi}_i(\vec{x},t)\right][/math]

 

and

 

[math]\vec{A}(\vec{x},t)=i\frac{\hbar c}{e}\sum_{i=2}^\infty \left[\vec{\gamma}\vec{\Psi}_i^\dagger(\vec{x},t) \cdot \vec{\Psi}_i(\vec{x},t)\right][/math].

 

Is this once again a standard definition from physics or is this just a first step in defining a field from the fundamental equation? And will ignoring interactions between photons only remove a single term in the equation that arrives from integration over the delta function of the additional terms or will there be other terms that are added to the equation. I don’t think that any additional terms will be added to the equation. So you are only ignoring a single sum and I have to wonder if such a term will even appear in the Dirac equation.

 

Either way would we expect much change to the behavior of possible solutions to take place from this approximation or would we expect this to be just the first approximation needed if we wished to derive modern field theory from the fundamental equation? Which is something that is really not of interest to us.

[Doctordick]

Bombadil, you don’t seem to comprehend the fact that all I am doing is using the standard definition of an electron to constrain the generality of my fundamental equation. The order with which I make the constraints implied by that definition is really of no consequence; all constraints implied by the standard definition must be implemented or else I am not looking at the solution for an electron. Once all those constraints are applied (including the constraint that we are looking at a single electron), the fundamental equation becomes identical to Dirac’s equation. Your analysis does not take into account all of these constraints prior to coming to conclusions concerning the result. That attack is invalid because, until you complete all the constraints implied by the definition of an electron, you are not talking about an electron.

Also haven’t you made the assumption that “(short term violations of conservation of energy) can not take place.” which sounds like it is also a completely expected approximation for the equations that you are deriving.

That has to do with the definition of the electric field of an electron. The classical electric field of an electron (i.e., the definition of the classical electron) does not include the possibility of short term violations of conservation of energy). In the classical picture, the electric field goes to infinity at the location of the electron; something which could not occur if one allowed short term violations of conservation of energy.

What is of some interest is that when we also define the second element as you have done to be considered an electron then we obtain the Maxwell equations as explaining its behavior (this is only of interest though because these are well known equations that approximate the fundamental equation) it also seems interesting to me to note that you have already derived a more general form of this equation in your derivation of the Schrödinger equation actually you derived all of these equations in more general forms but just ignored the form of the interaction term. And so we can gather that these equations are still relativistically correct.

Then you didn’t follow the deduction of Schrödinger’s equation because there is one step in that deduction which explicitly presumes relativistic velocities are not included.

You also have shown how you can define elements to simplify the fundamental equation.

I would instead say that what I have done is shown that modern physics has defined their elements such that the relationships simplify the fundamental equation. They didn’t know they were doing such a thing but they were doing it none the less.

That is a photon is an invalid element and only supplies a means by which elements can interact.

Watch out about using that term “invalid element”. When I use the term, I mean that it is not representing a bona fide nomena (the undefined real “things” our explanations are created to explain) but is rather an entity invented by the explanation itself and that explanations may exist which do not require it.. When a scientist uses the term “invalid element” he will mean that he can prove “his” explanation does not require it. These are very different concepts. He means the entity does not obey his rules. When I define an “invalid element”, I require it to obey the rules embedded in the fundamental equation exactly. As a consequence, my equation is a constraint on explanations and not a constraint on reality. The possibility always exists that the entire collection of elements in an explanation are “invalid”: i.e., figments of our imagination. Thus my equation still stands as perfectly valid even for an explanation based upon solipsism.

It is the second equation that then supplies the remainder of the needed definitions for the second element to be an electron. Together they show that you have derived the equations governing the behavior of electromagnetic fields as a special case of the fundamental equation.

And are required by the definition of “an electron”.

I am wondering at this point can all of the study of electromagnetic fields be derived from these equations or is this just a trivial case that tells us that all of the physics relating to electromagnetic fields could be derived from the fundamental equation if we wanted to

No, there are aspects of QED which are not directly embedded in Maxwell’s equations (which I have derived). Now I have proved that all internally consistent explanations must obey my fundamental equation (or rather there exists an interpretation of those explanations which does) but I have not proved that every solution of my fundamental equation represents something which can be found in reality; though I have a very strong suspicion that it is true.

Is this once again a standard definition from physics or is this just a first step in defining a field from the fundamental equation? And will ignoring interactions between photons only remove a single term in the equation that arrives from integration over the delta function of the additional terms or will there be other terms that are added to the equation. I don’t think that any additional terms will be added to the equation. So you are only ignoring a single sum and I have to wonder if such a term will even appear in the Dirac equation.

That depends upon how and what the Dirac equation is written to represent. Modern physics uses past notation to represent new things through minor changes in definition all the time.

Either way would we expect much change to the behavior of possible solutions to take place from this approximation or would we expect this to be just the first approximation needed if we wished to derive modern field theory from the fundamental equation? Which is something that is really not of interest to us.

Before one can expect to derive “modern field theory” one must first thoroughly understand modern field theory and the definitions upon which it is based.

 

What I have said many times is that there must exist an interpretation of any explanation which conforms to my fundamental equation. That means the problem is a translation problem. Of course, one could find approximations to that equation and then search reality to see if a circumstance consistent with that approximation exists; however, I wouldn’t expect that attack to be particularly fruitful: i.e., I expect the standard approach (being led by unexpected experimental results) to be more productive.

 

Have fun -- Dick

[Bombadil]

Bombadil, you don’t seem to comprehend the fact that all I am doing is using the standard definition of an electron to constrain the generality of my fundamental equation. The order with which I make the constraints implied by that definition is really of no consequence; all constraints implied by the standard definition must be implemented or else I am not looking at the solution for an electron. Once all those constraints are applied (including the constraint that we are looking at a single electron), the fundamental equation becomes identical to Dirac’s equation. Your analysis does not take into account all of these constraints prior to coming to conclusions concerning the result. That attack is invalid because, until you complete all the constraints implied by the definition of an electron, you are not talking about an electron.

 

But wouldn’t any possible solution to the fundamental equation after all of the definitions have been made still have to satisfy any of the equations at a prior step in which less definitions have been applied, the converse then being false. Of course we don’t have solutions to more general equations but if we had one wouldn’t a solution to one of them have a form that would satisfy the equation after the remainder of the definitions have been placed directly upon the solution?

 

My point being how do we know that all of the requirements that are used to define a photon or electron are needed and that we just haven’t seen a case in which effects arising from not having these definitions takes place.

 

Then you didn’t follow the deduction of Schrödinger’s equation because there is one step in that deduction which explicitly presumes relativistic velocities are not included.

 

Perhaps I didn’t make myself clear in what I meant. When you derived the Schrödinger equation you made several approximations but the one that I think you must be referring to is the approximation

 

[math]K\sqrt{2}\frac{\partial}{\partial t}\vec{\Phi} \approx -iq\vec{\Phi}[/math]

 

Which can also be written as

 

[math]i\hbar\frac{\partial}{\partial t}\vec{\Phi}\approx q\hbar c \vec{\Phi}= \left( \frac{q\hbar}{c}\right) c^2\vec{\Phi} = mc^2\vec{\Phi}.[/math]

 

Clearly as you have pointed out this makes relativistic relationships of the solutions of any further equations impossible. Or is there some other approximation that you are referring to as the question still may be open as to whether or not any previous equations are relativistic but I have thought that they most likely were for some time. But when I saw this page on the Klein-Gordon equation I couldn’t help but notice the similarity to the equation

 

[math]

\left\{\frac{\partial^2}{\partial x^2} - q^2 + G(x)\right\}\vec{\Phi}(x,t)= 2K^2\frac{\partial^2}{\partial t^2}\vec{\Phi}(x,t).[/math]

 

Which you derived in deriving the Schrödinger equation, at this point I couldn’t help myself but to make the same substitutions as you made in deriving the Schrödinger equation. The result is of course the only difference between them being that yours has a G(X) term while the Klein-Gordon equation does not. I have to wonder why they haven’t got an alternative version that does, but that page almost makes the Klein-Gordon equation sound obsolete. But I really don’t know anything more then what that page says which is not much.

 

Anyhow, what I was referring to was that the equation

 

[math]

\left\{\frac{\partial^2}{\partial x^2} - q^2 + G(x)\right\}\vec{\Phi}(x,t)= 2K^2\frac{\partial^2}{\partial t^2}\vec{\Phi}(x,t).[/math]

 

From you derivation of Schrödinger equation can clearly be compared to the equation

 

[math]

\nabla^2\Phi -\frac{m_2^2c^2}{\hbar^2}\Phi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\Phi= -4\pi\rho

[/math]

 

From this thread if you consider a particular form for the G(X) term. Similar results seem to hold with other equations that you arrived at in your derivation of Schrödinger’s equation but I think that the introduction of Dirac’s notation hides this because they can no longer be directly related. So the Dirac equation is clearly not directly comparable, but it is just a few steps back when you are still using your notation.

 

Watch out about using that term “invalid element”. When I use the term, I mean that it is not representing a bona fide nomena (the undefined real “things” our explanations are created to explain) but is rather an entity invented by the explanation itself and that explanations may exist which do not require it.. When a scientist uses the term “invalid element” he will mean that he can prove “his” explanation does not require it. These are very different concepts. He means the entity does not obey his rules. When I define an “invalid element”, I require it to obey the rules embedded in the fundamental equation exactly. As a consequence, my equation is a constraint on explanations and not a constraint on reality. The possibility always exists that the entire collection of elements in an explanation are “invalid”: i.e., figments of our imagination. Thus my equation still stands as perfectly valid even for an explanation based upon solipsism.

 

But will a photon as you have defined it even satisfy all of the constraints originally placed on the fundamental equation in particular if the constraint

 

[math]\sum_{i\neq j}\delta (\vec{x}_i-\vec{x}_j)\vec\Psi =0[/math]

 

was satisfied wouldn’t the interaction term in the Dirac equation vanish? If this is the case wouldn’t the element have to be invalid. Actually if this is the case I’m somewhat puzzled by how we can continue to use this explanation. Shoulder’s as you are suggesting all elements satisfy this relation if we are going to include them in our explanation?

 

No, there are aspects of QED which are not directly embedded in Maxwell’s equations (which I have derived). Now I have proved that all internally consistent explanations must obey my fundamental equation (or rather there exists an interpretation of those explanations which does) but I have not proved that every solution of my fundamental equation represents something which can be found in reality; though I have a very strong suspicion that it is true.

 

You also have not proved then that all of physics is consistent or that all of the consequences of the definitions that they use have been included in the derivation of modern physics. So there is still the possibility of subtle errors that have been over looked in physics or perhaps even definitions that have been inadvertently made.

 

What I have said many times is that there must exist an interpretation of any explanation which conforms to my fundamental equation. That means the problem is a translation problem. Of course, one could find approximations to that equation and then search reality to see if a circumstance consistent with that approximation exists; however, I wouldn’t expect that attack to be particularly fruitful: i.e., I expect the standard approach (being led by unexpected experimental results) to be more productive.

 

But might it not be possible to find a way to form ones definitions directly off of the fundamental equation if one knows how they want their defined element to behave in different conditions? So that instead of applying definitions for an unknown element to the fundamental equation in the hopes of finding something useful and instead of trying to find the equations explaining the behavior of an element by applying different definitions used for the element. We might find the definitions of an element by knowing how it behaves under certain conditions and then deriving the necessary definitions for the element.

 

Might this not even be of some interest in how it could be done as we would be considering the constraints that the fundamental equation places on possible definitions as well as what constraints different definitions place on the fundamental equation.

[Doctordick]

But wouldn’t any possible solution to the fundamental equation after all of the definitions have been made still have to satisfy any of the equations at a prior step in which less definitions have been applied, the converse then being false.

No. There may exist no exact solution which satisfies all the definitions of interest. What I have shown is that Dirac’s equation yields approximate solutions to my fundamental equation; therefore, Dirac’s equation is indeed an approximation to my equation which is valid under specific defined circumstances.

Of course we don’t have solutions to more general equations but if we had one wouldn’t a solution to one of them have a form that would satisfy the equation after the remainder of the definitions have been placed directly upon the solution?

I don’t know; I would have to have one of those “exact” solutions before I could make any such judgement.

My point being how do we know that all of the requirements that are used to define a photon or electron are needed and that we just haven’t seen a case in which effects arising from not having these definitions takes place.

I have no idea as to how to settle that issue.

Or is there some other approximation that you are referring to as the question still may be open as to whether or not any previous equations are relativistic but I have thought that they most likely were for some time.

No, that is exactly the approximation I was referring to.

The result is of course the only difference between them being that yours has a G(X) term while the Klein-Gordon equation does not. I have to wonder why they haven’t got an alternative version that does, but that page almost makes the Klein-Gordon equation sound obsolete. But I really don’t know anything more then what that page says which is not much.

Actually, you shouldn’t be surprised by the resemblance. The Klein-Gordon equation was hypothesized because of the success of the

Schrödinger equation (via the “guess and by golly” method used by modern science that I complain about all the time). My fundamental equation is a holistic expression which takes into account the entire universe of possibilities. As apposed to that, modern physics never really concerns itself with many body relationships (why should they, they couldn’t solve them if they had them).

 

That G(x) arises purely out of the rest of the universe’s impact on the solution for that single entity described by “x”. In essence, I specify exactly where that “G(x)” comes from: the rest of that many body equation. Since they are trying to describe a single entity, they won’t put in such a function unless they can specify the interaction which yields that function. In this respect, Dirac’s equation includes the electromagnetic effects. They do this because they had the very successful Maxswell’s equation (a complex “guess and by golly” result) to work with which already connected electrons to electromagnetic effects. When it comes to the Klein-Gordon equation, they had no fundamental universal interaction to include. It was a “guess and by golly” attack on the relativistic problems in Schrödinger equation and, as such, produced difficulties even before they began to guess about how to handle interactions.

Similar results seem to hold with other equations that you arrived at in your derivation of Schrödinger’s equation but I think that the introduction of Dirac’s notation hides this because they can no longer be directly related. So the Dirac equation is clearly not directly comparable, but it is just a few steps back when you are still using your notation.

When I derived Dirac’s equation, I never made the approximation

[math]K\sqrt{2}\frac{\partial}{\partial t}\vec{\Phi} \approx -iq\vec{\Phi}.[/math]

 

Thus the equation was still relativistic. The central issue in that approximation is that it requires mc² to be such a large portion of the energy (associated with the particle of interest) such that the relativistic contribution due to mass is constant. This amounts to a simple constant shift in energy due to that mass (so long as one is not in a relativistic situation) which can be separated by the simple adjustment to [imath]\vec{\Psi}[/imath].

 

When I go to look at the solution for a photon, that approximation is not necessary at all since the mass of a photon is zero (essentially that amounts to q=0 in the above approximation). In a sense, Maxwell’s equations do not require relativity; what they do require is a fixed velocity which brings on the problem of relativity but that is not exactly the same thing. Constant velocity itself is a violation of Newtonian relativity. What is important is to note that relativity is not necessary if a preferred coordinate system exists. In fact, once one takes into account the specific problems of setting up that preferred coordinate system and defining clocks, the standard relativistic transformations are enevitable.

 

Before Einstein came up with his theory of relativity, Lorentz and Fitzgerald had already come up with a solution to the problem (that solution is called “Lorentz-Fitzgerald contraction” which I am sure you have heard of). They clearly showed that all objects, whose structure was derived from electromagnetic phenomena, would display exactly such contraction when moved. The problem was that electromagnetic interactions are not sufficient to explain everything and there was no reason to apply such a thing to other interactions. If you look at that problem carefully you will discover it arises entirely from mass effects. By the way, problems with general relativity (calculating transformations to accelerating frames of reference), as apposed to special relativity (transformations in the absence of acceleration) also arise entirely from those very same mass effects. As a consequence, GR is much simpler in my paradigm than it is in Einstein’s paradigm as mine is entirely built from massless entities (mass being a quantum mechanical effect).

 

At any rate, since the photon is massless, momentum in the tau direction is actually zero and that component does not come into the expression: i.e., I am essentially using the correct approximation for a specific one body solution to my equation in order to develop the correct value of G(x) to use in Dirac’s equation.

But will a photon as you have defined it even satisfy all of the constraints originally placed on the fundamental equation in particular if the constraint

 

[math]\sum_{i\neq j}\delta (\vec{x}_i-\vec{x}_j)\vec\Psi =0[/math]

 

was satisfied wouldn’t the interaction term in the Dirac equation vanish?

That sum, in the fundamental equation, goes over all the ontological elements in the entire universe. Instead of using G(x) we are using the standard solutions obtained from Maxwell’s equation. Those solutions presume most of the entities in the universe have no impact at all (a standard modern physics presumption). In fact, the only interactions the Maxwell’s equation presumes are those with electrons or collections of electrons: i.e., electromagnetic fields. That is exactly why I showed that my fundamental equation (in the specific situation considered in Dirac’s equation) did in fact reduce to Maxwell’s equation when the associated G(x) was limited to collections of electrons.

 

Later I extended that representation of G(x) to include an infinite number of photons (the collection yielding the appropriate electromagnetic fields). In this respect, interactions between photons are ignored; another common approximation used in physics of electromagnetic phenomena.

If this is the case wouldn’t the element have to be invalid. Actually if this is the case I’m somewhat puzzled by how we can continue to use this explanation. Shoulder’s as you are suggesting all elements satisfy this relation if we are going to include them in our explanation?

I have no idea what you mean by “Shoulder’s”. With regard to the adjective “invalid”, you are apparently making exactly the error I pointed out in my post. In my paradigm, invalid does not mean it cannot be used in the explanation. What it means is that it is not a necessary part to explaining the nomena behind the information being explained. In essence, you cannot prove that you have ever encountered a photon without making use of electromagnetic theory. Essentially, even in your world view, all you can be sure of is that some physical change occurred. That the change was due to a photon is a presumption of your specific explanation. In my paradigm, you are free to use such a thing because it is totally consistent with all the rules and existing elements in your explanation.

 

As I said, I do not at all mean what the common scientific community would mean by their use of the word. That is exactly why I say my fundamental equation is valid for any and all philosophic positions on what exists; even Solipsism (which is that all the elements of your world view are fictitious creations of your conscious and/or unconscious mind). In fact most scientists simply presume philosophic arguments concerning what may or may not exist are irrelevant to their work. Through the validity of my fundamental equation, I have proved that presumption is indeed valid.

You also have not proved then that all of physics is consistent or that all of the consequences of the definitions that they use have been included in the derivation of modern physics.

No I haven’t, but I have certainly shown that a great amount is.

So there is still the possibility of subtle errors that have been over looked in physics or perhaps even definitions that have been inadvertently made.

That is quite true and it comes up quite strongly when I elaborate on general relativity. That is one reason I currently avoid the subject of general relativity. My results are slightly different from Einstein’s and I am fully confident of the fact that the scientific community will jump upon that fact as a good reason to classify me as a crackpot. I would rather that some rational people at least begin to take my paradigm and its arguments seriously before I bring up something which differs with the standard modern catechism. They would prefer to ignore me even if my paradigm yielded exactly what they say is true. When I say mass is a direct consequence of quantum mechanics, I face exactly the same situation Galileo faced when he said the planets went around the sun instead of the sun going around the earth. Mass obviously cannot be seen as momentum in a direction orthogonal to our commonly conceived three dimensional space and anyone who would suggest such a thing is obviously a crackpot.

Might this not even be of some interest in how it could be done as we would be considering the constraints that the fundamental equation places on possible definitions as well as what constraints different definitions place on the fundamental equation.

I have no idea as to what “could be done”. That is an open question. In a sense, that could be a characterization of my work and my experience is that such a processes is not obvious.

 

I realized that my equation had to be true back in 1968 but was unable to prove it for a number of years (had many problems with little details). I was finally able to prove that it had to be true a few years later but I was totally at a loss to find any way of solving it. Without a way of solving it, the whole idea seemed rather worthless assertion. It was not until over ten years later that first was able to prove that Schrödinger’s equation was in fact an approximation to my fundamental equation. Once I had done that, proving that other physics equations were also approximations to my equation was rather straight forward.

 

Have fun -- Dick

[Bombadil]

I have no idea as to how to settle that issue.

 

I have a suspicion that there may be two separate issues here that I am confusing. In particular what I would call the consistency of physics and the validity of physics. The first meaning that we want to know if physics is consistent with the definitions that they make and if they are consistent with approximate solutions to the fundamental equation. The second being if the definitions that they use form a useful explanation of what it is that they are trying to explain. The first one we are interested in because it gives us information about approximate solutions to the fundamental equation, the second one while the fundamental equation might give us a unique view of and we could approach it in a way that has not likely been looked at, I don’t think that it is really of interest to us because the fundamental equation tells us nothing about what is being explained which is what physics tries to do.

 

When I go to look at the solution for a photon, that approximation is not necessary at all since the mass of a photon is zero (essentially that amounts to q=0 in the above approximation). In a sense, Maxwell’s equations do not require relativity; what they do require is a fixed velocity which brings on the problem of relativity but that is not exactly the same thing. Constant velocity itself is a violation of Newtonian relativity. What is important is to note that relativity is not necessary if a preferred coordinate system exists. In fact, once one takes into account the specific problems of setting up that preferred coordinate system and defining clocks, the standard relativistic transformations are enevitable.

 

I seem to remember that you mentioned something about this earlier but I can’t seem to find where, anyhow what it means is that as long as we use the correct [math]\vec{\Psi}[/math]. Which gives us the correct result in our preferred reference frame we don’t have to consider the issue of relativity in our explanation or make any transformations to our explanation as everything is part of the solution already. On the other hand the defining of a reference frame will require the defining of length and a clock which results in relativistic transformations taking place a subtly different issue.

 

Now in the condition that we are looking at, the solution to the Dirac equation will supply the correct [math]\vec{\Psi}[/math] and will require the existence of a photon to be used so we can construct a clock in the same way as how you have already laid out in detail else-where so I won’t go into it right know as it is off topic.

 

It is actually all unnecessary though to consider relativistic changes for what we are doing, as we can by a simple change in our reference frame that we are using make the solution to Dirac’s equation be in a reference frame in which relativistic effects don‘t even need to be considered as part of the solution.

 

I have no idea what you mean by “Shoulder’s”. With regard to the adjective “invalid”, you are apparently making exactly the error I pointed out in my post. In my paradigm, invalid does not mean it cannot be used in the explanation. What it means is that it is not a necessary part to explaining the nomena behind the information being explained. In essence, you cannot prove that you have ever encountered a photon without making use of electromagnetic theory. Essentially, even in your world view, all you can be sure of is that some physical change occurred. That the change was due to a photon is a presumption of your specific explanation. In my paradigm, you are free to use such a thing because it is totally consistent with all the rules and existing elements in your explanation.

 

I have no idea how the word shoulder’s got in there it should have been shouldn’t but I think that you figured that out.

 

Then in order to use Dirac’s equation to explain something we have to assume the existence of a photon. And in order for us to say that a particular element is a photon we have to say that an element that you have defined as an electron interacted with some other element in a particular way. So in a sense the photon only supplies a means by which electrons can interact with each other. Whether or not a photon is a valid or invalid element is of no concern to us as what we are doing is supplying a means by which elements in our explanation can interact.

 

But, won’t this mean that it really doesn’t matter to us if we can tell if an element is invalid as all bosons must be invalid. But, that there is no way to tell if an element is valid because a fermion can be either valid or invalid.

[Doctordick]

I have a suspicion that there may be two separate issues here that I am confusing. In particular what I would call the consistency of physics and the validity of physics.

Well, I am of the opinion that, if one actually discovers an inconsistency in some physics theory, that fact marks the theory as invalid. From the other perspective you bring up, I cannot comprehend anyone having an interest in an invalid theory. Essentially my interest is only in consistency. I would like to point out that, although my equation is solely based on absolute internal consistency, finding a physical theory which can not be interpreted in a way such that satisfies my equation (which would imply an inconsistency) does not specify exactly where that inconsistency occurs.

 

Secondly, the fact that all internally consistent explanations must satisfy my equation does not require the reverse. One might find an approximate solution to my equation which might not describe any observed physics phenomena. Such a result would be interesting as it would imply the existence of real physical constraints not imposed merely by consistency itself.

The first one we are interested in because it gives us information about approximate solutions to the fundamental equation, the second one while the fundamental equation might give us a unique view of and we could approach it in a way that has not likely been looked at, I don’t think that it is really of interest to us because the fundamental equation tells us nothing about what is being explained which is what physics tries to do.

Now that is what physics claims to be doing but it is clearly failing in that purpose. My equation expresses only constraints required by internal consistency. The fact that these constraints (together with the definitions used by physics) imply the validity of Newtonian mechanics, Schrödinger's equation, Dirac's equation and Maxwell's equation is pretty strong evidence that these areas of physics tell us nothing about what is being explained. They are clearly little more than internally consistent tautologies. They are indeed useful tautologies as they tell us what to expect given the surrounding circumstances (as defined). Essentially, they are elaborate memory devices allowing us to remember similar circumstances: i.e., they bring the power of "induction" into our world view.

I seem to remember that you mentioned something about this earlier but I can’t seem to find where, anyhow what it means is that as long as we use the correct [math]\vec{\Psi}[/math]. Which gives us the correct result in our preferred reference frame we don’t have to consider the issue of relativity in our explanation or make any transformations to our explanation as everything is part of the solution already. On the other hand the defining of a reference frame will require the defining of length and a clock which results in relativistic transformations taking place a subtly different issue.

So long as everyone uses the same coordinate system (call it the preferred reference frame) relativistic transformations serve no purpose whatsoever. It is only when two different frames are defined that transformations are required. The required consistency is discussed in my post,”An “analytical-metaphysical” take on Special Relativity!”

The solution to this problem lies with the scaling of the geometry between the two systems: there must exist a consistent way of converting a solution in one system to a solution in the other independent of any influence between the two.

Your comment, “On the other hand ...” has no place here. In any defined frame one is required to define lengths and a clock. Setting clocks to agree is a serious issue concerning the definition of “agree”.

It is actually all unnecessary though to consider relativistic changes for what we are doing, as we can by a simple change in our reference frame that we are using make the solution to Dirac’s equation be in a reference frame in which relativistic effects don‘t even need to be considered as part of the solution.

What you seem to be missing here is that my fundamental equation is a many body wave equation with fixed velocity of that probability wave. As such, relativity is inevitably embedded in the equation in exactly the same manner as relativity is embedded in Maxwell's equation: i.e., exactly the same issues arise when it comes to defining lengths and clocks.

But, won’t this mean that it really doesn’t matter to us if we can tell if an element is invalid as all bosons must be invalid. But, that there is no way to tell if an element is valid because a fermion can be either valid or invalid.

Again, you seem to lose sight of the fact that my definition of valid is quite different from the common standard definition. You are apparently using the interpretation that “invalid” means “not true”. My definition essentially says there may exist explanations which do not require the element: i.e., I allow for the possibility that there may actually exist aspects of reality which are “real” and not pure figments of one's imagination. That is why I keep bringing up Solipsism. Solipsism, by its nature as a philosophy, can neither be proved nor disproved. My concern is exactly what can I prove is true!

 

Have fun --Dick

[Bombadil]

Now that is what physics claims to be doing but it is clearly failing in that purpose. My equation expresses only constraints required by internal consistency. The fact that these constraints (together with the definitions used by physics) imply the validity of Newtonian mechanics, Schrödinger's equation, Dirac's equation and Maxwell's equation is pretty strong evidence that these areas of physics tell us nothing about what is being explained. They are clearly little more than internally consistent tautologies. They are indeed useful tautologies as they tell us what to expect given the surrounding circumstances (as defined). Essentially, they are elaborate memory devices allowing us to remember similar circumstances: i.e., they bring the power of "induction" into our world view.

 

This is what I am referring to, only from the perspective of, are the definitions that are used needed to explain the behavior of a particular set of elements. Or can less definitions be made and still describe the behavior of the same elements? Is there really that much difference in saying that the definitions that you use say something about reality and saying that the definitions that you use are consequences of simplifying the conditions that are being explained or observed. Or in other words, just what is being remembered about the circumstances. At this point I don’t see any way to tell the difference between the two possibilities, as even if you could derive an equation from the fundamental equation which has not been observed we have no way to know if this is because of constraints on reality or if it is because we have never seen the conditions needed for the equation to be true.

 

Again, you seem to lose sight of the fact that my definition of valid is quite different from the common standard definition. You are apparently using the interpretation that “invalid” means “not true”. My definition essentially says there may exist explanations which do not require the element: i.e., I allow for the possibility that there may actually exist aspects of reality which are “real” and not pure figments of one's imagination. That is why I keep bringing up Solipsism. Solipsism, by its nature as a philosophy, can neither be proved nor disproved. My concern is exactly what can I prove is true!

 

So the issue here is that there exists an infinite number of explanations that satisfy our observations, some of these will of course require invalid elements. But seeing as we can’t know what elements are invalid there is no problem including invalid elements in our explanation. But if we can tell that some elements in our explanation are invalid then we know that there are possible explanations that won’t require these elements.

 

This is why you derived both the Dirac equation and the Maxwell equations, in doing so we know that we are looking at a photon and a electron if we are looking at a magnetic field. If we are not looking at a magnetic field there is no way to know if we are looking at a photon. Likewise by definition a magnetic field implies the existence of a photon.

 

This does not imply that there are not other explanations for the same observations, quite to the contrary, there are an infinite number of different explanations for the same observations. However, how I understand this it is a flaw free explanation the you have derived because it can be derived directly from the fundamental equation and so is error free.

 

This equation implies that, with regard to the function [imath]\vec{\Psi}_2[/imath], the expression

[math]\left\{-i\hbar c\vec{\alpha}\cdot \vec{\nabla}_2 -2i\hbar c \beta \psi^\dagger(\vec{x}_2,t)\psi(\vec{x}_2,t) \right\}=\frac{i\hbar}{\sqrt{2}}\frac{\partial}{\partial t}[/math].

 

can be seen as an operator identity.

 

Is there any reason that we could not continue to use this as an operator identity and continue to apply it resulting in further differential equations? And the possibility of mixed partial derivatives in the resulting equations. This seems to imply that there would be an infinite number of equations that approximate the Maxwell equations and your fundamental equation, some of which will be of increasing complexity. As all equations found as a result of using this as an operator identity will have to approximate the previous equations.

 

There are a few issues which deserve a little attention. First, the issue of the substitution [imath]\nabla^2\left(\vec{\Psi}^\dagger \cdot \vec{\Psi}\right)=4\vec{\Psi}^\dagger\cdot\nabla^2\vec{\Psi}[/imath] essentially says that the resultant equations (and that would be Maxwell's equations) are only valid so long as the energy of the virtual photon is small enough such that quantum fluctuations can be ignored. That explains the old classical electron radius problem from a slightly different perspective. Maxwell's equations are an approximation to the actual situation. The correct solution requires one include additional elements (those quantum fluctuations which arise when the energy exceeds a certain threshold). It could very well be that the energy must be below the energy necessary to create a fluctuation equal to the energy of the electron: the field solutions must include photon-photon interactions before the "classical electron radius" is reached. It is also possible that inclusion of the fluctuations could lead to massive boson creation and another solution. The problem with that fact is that I have not discovered a way to approximate a solution to the required many body problem.

 

What do you mean by virtual photon? Are you referring to only being able to know if it is a photon if we are looking at a magnetic field or is there some other property that defines a particle as a virtual particle?

 

Also are you referring to interactions between photons caused from the creation of a new particle? If so how can this be included in the fundamental equation isn’t the number of elements fixed for any given explanation. Or is there a way to include element creation in the fundamental equation, or is it already part of it we just have to approximate the equation and the necessary conditions? If it is already part of it how do we know when it is taking place? Also is it possible for photons to interact in other ways such as from the sum of the differentials in the fundamental equation having to sum to zero, if so won’t both fermions and bosons interact in this way?

[Doctordick]

This is what I am referring to, only from the perspective of, are the definitions that are used needed to explain the behavior of a particular set of elements. Or can less definitions be made and still describe the behavior of the same elements?

I do not know! Do you have an explanation which requires fewer definitions? If you think about it a little, you should be able to comprehend that, under my paridigm, the only issue of significance is “how much does the supposed future resemble the known past”. The higher that probability, the more probable a supposed future should be expected to be. The only purpose of my definitions is to express that very similarity.

At this point I don’t see any way to tell the difference between the two possibilities, as even if you could derive an equation from the fundamental equation which has not been observed we have no way to know if this is because of constraints on reality or if it is because we have never seen the conditions needed for the equation to be true.

That is exactly why I brought in the standard definitions of physics. We have had millions of years to lead us to the fact that those concepts are very powerful. If you are going to come up with some no one has yet thought of; you have posed yourself a rather difficult problem.

 

The fact that my simple (and it is conceptually quite simple) paradigm reproduces all of modern science is rather sufficient to satisfy me.

So the issue here is that there exists an infinite number of explanations that satisfy our observations, some of these will of course require invalid elements. But seeing as we can’t know what elements are invalid there is no problem including invalid elements in our explanation. But if we can tell that some elements in our explanation are invalid then we know that there are possible explanations that won’t require these elements.

First, I have not proved “that there exists an infinite number of explanation that satisfy our observations”. It may very well be that every one of those fully flaw free possibilities simply map directly into the paradigm I have presented as they would certainly have to satisfy my fundamental equation. In that case, can you really say there are more than one valid explanation? Notice that my fundamental equation was derived by the mechanism of introducing those “invalid” elements. If you think you can make a simpler expression embodying the “similarity” I talk about above, I would love to see it: i.e., the introduction of those specific “invalid” elements is a very powerful step in simplifying the mathematical expression of that similarity.

 

By the way, I have changed my mind about calling the two different proposed nomena “valid” and “invalid”. I instead think the common concepts inferred by “real” and “presumed” come closer to fitting the circumstance I am talking about here.

This is why you derived both the Dirac equation and the Maxwell equations, in doing so we know that we are looking at a photon and a electron if we are looking at a magnetic field. If we are not looking at a magnetic field there is no way to know if we are looking at a photon. Likewise by definition a magnetic field implies the existence of a photon.

Let us say rather that electrons, electric fields, magnetic fields and photons are a rather inherently set combinations of definitions clearly interdependent upon one another.

This does not imply that there are not other explanations for the same observations, quite to the contrary, there are an infinite number of different explanations for the same observations.

Again with this “infinite number of explanations”. What I have shown is that every possible flaw-free explanation can be interpreted in a way which satisfies my equation. That means that every one of those explanations (so long as they are flaw-free) can be interpreted as explanations under my paradigm. Thus are they the same explanations or are they different explanation? Certainly they can be expressed in a different language but does that really make them different explanations?

Is there any reason that we could not continue to use this as an operator identity and continue to apply it resulting in further differential equations?

And exactly why would you want to do that? Linear differential equations are difficult enough to solve already.

But the issue here is, “keep it simple!” Erasmus00, you were apparently very disturbed by constraining my examination to a first order linear differential equation.

What do you mean by virtual photon? Are you referring to only being able to know if it is a photon if we are looking at a magnetic field or is there some other property that defines a particle as a virtual particle?

The term “virtual” refers to physical elements which are not “real”: i.e., do not obey the rules of classical Newtonian objects . Take a look at the various entries you obtain if you google “virtual particles physics”

Also are you referring to interactions between photons caused from the creation of a new particle? If so how can this be included in the fundamental equation isn’t the number of elements fixed for any given explanation. Or is there a way to include element creation in the fundamental equation, or is it already part of it we just have to approximate the equation and the necessary conditions? If it is already part of it how do we know when it is taking place? Also is it possible for photons to interact in other ways such as from the sum of the differentials in the fundamental equation having to sum to zero, if so won’t both fermions and bosons interact in this way?

Let us just say that “presumed elements” may come and go. Remember, the number of elements were fixed for the simple reason that normally defined mathematical functions don't change the number of arguments. We can get around that “problem” by simply stating that, before creation, these created elements had no influence on the remaining elements and afterwards, they do (after all, the number of arguments under my paradigm is something like “infinite” so a few coming and going don't seem to have any serious consequences. There are some long and subtle discussions which can be had on this subject. For the time being I am not interested in the issue.

 

Have fun -- Dick

[Bombadil]

I do not know! Do you have an explanation which requires fewer definitions? If you think about it a little, you should be able to comprehend that, under my paridigm, the only issue of significance is “how much does the supposed future resemble the known past”. The higher that probability, the more probable a supposed future should be expected to be. The only purpose of my definitions is to express that very similarity.

 

If all that we are interested in is explaining our observations then your definitions seem very useful and the question of “how much does the supposed future resemble the known past” seems very important, but it seems to bring into question what we mean when we say that the past resembles the future. Does this mean that the same collections of elements making up an object in the past will continue in the future, or that the same elements that react to other elements in the past will continue in the future, or will it mean that there is an explanation that will explain the future if the past is its initial conditions?

 

Now the definitions that you have made also seem quite useful from the prospective that they seem to emphasize a particular characteristics of possible solutions to the fundamental equation. That is, a single element will have to still obey the Dirac equation even when interacting in ways not explained by the Dirac equation, but other interactions will have to be considered as well. But there still seems to be no requirement that all explanations will have any of these characteristics or how these characteristics will show up in an explanation. Isn’t part of what we are interested in, is the consequences of our explanation that we might not be able to observe but will be required by our explanation all the same.

 

I think that this is going to continue to bug me for some time as the issues don’t seem to have a straight forward answer. The major ones being, are these definitions that you are making required by the fundamental equation to have a flaw free explanation? (clearly the explanation itself must be flaw free); In what way will the characteristics that defining an element in some way, appear in other derivations? Are there other explanations or is there a unique explanation which can be derived form any of the equations that you have derived so far? At this point I think there are other explanations that can be derived from the fundamental equation only because you have had to make these definitions (if there where no other explanations then I would think that they would not be needed to derive the Dirac equation).

 

In short you have so far shown some powerful existence formulas and I am wondering if there are some uniqueness formulas to go along with them.

 

So I think that for the time being we would be best off overlooking these questions as I think that they are currently outside of my abilities to try and answer and outside of your interest to try and go further into right now. Or at the very least they are sufficiently off topic to not be worth going into right now.

 

First, I have not proved “that there exists an infinite number of explanation that satisfy our observations”. It may very well be that every one of those fully flaw free possibilities simply map directly into the paradigm I have presented as they would certainly have to satisfy my fundamental equation. In that case, can you really say there are more than one valid explanation? Notice that my fundamental equation was derived by the mechanism of introducing those “invalid” elements. If you think you can make a simpler expression embodying the “similarity” I talk about above, I would love to see it: i.e., the introduction of those specific “invalid” elements is a very powerful step in simplifying the mathematical expression of that similarity.

 

By the way, I have changed my mind about calling the two different proposed nomena “valid” and “invalid”. I instead think the common concepts inferred by “real” and “presumed” come closer to fitting the circumstance I am talking about here.

 

This actually seems to be an interesting question and the first problem that I come to is that I don’t know when to consider two explanations to be the same. I don’t think that I would consider two explanations to be the same just because we are using different initial conditions. But you did consider only two elements to be of interest in the beginning, one valid and one invalid or as you are now calling them one real and one presumed. (I will now use these terms taking it that a valid element is now a real element and an invalid element is now a presumed element.) and then latter you expanded this to an infinite number of presumed elements. Not what I would consider an altogether equivalent representation of my observations as you have required all presumed elements to be defined in the same way.

 

On the other hand, would we consider it a different explanation if all we did was add more real or presumed elements defined in the same way, or could we even call it a valid explanation if we defined them in a different way? We would certainly then have difficulty making our explanation be of any use for explaining the same collection of elements.

 

No matter how we consider the above I would consider the Dirac equation to be a less fundamental equation then the Schrödinger equation if we can no longer derive something from it that we can derive from the Schrödinger equation. Likewise I would consider the Schrödinger equation to be less fundamental if we can derive something from the Dirac equation that we can’t from the Schrödinger equation. If both are the case then I would consider them to be different explanations.

 

And exactly why would you want to do that? Linear differential equations are difficult enough to solve already.

 

Yes and I’m only now beginning to understand just how difficult they are to solve, but wouldn’t it give us new ways of understanding the equation and new ways to approach the problem if we have more then one differential equation that the function of interest must satisfy?

 

A second issue concerns the existence of magnetic monopoles. In this development of Maxwell's equations, the symmetry between the electric and magnetic fields does not exist and likewise, “magnetic monopoles” do not exist.

 

I don’t know exactly what you are referring to here. Unless you are referring to the purely mathematical symmetry of the equations. Also I’m not sure as to why “magnetic monopoles” aren’t possible in this derivation of the equations. Unless it is that both poles of a magnetic field are defined by the Dirac equation which requires only two elements both of which are required to define a magnetic field while this is not the case in defining a charge.

 

Actually, the analysis just done implies considerable more than Classical Electrodynamics: Quantum Electro Dynamics is a direct consequence of adjusting for additional terms in the fundamental equation (quantum fluctuation). It is also of significance that, in the fundamental equation, the gradient operator was explicitly defined to be a four dimensional entity. So long as the second element (the boson interaction) is a massless element, the tau component of the gradient will vanish and resulting deduced equations will correspond exactly to Maxwell's equations; however, if the interacting boson generating the field is not massless we end up with an additional term in these supposed Maxwell's equations.

 

That is, you only derived Maxwell’s equations because you made one of the elements massless you could have defined it to have some mass and arrived at a deferent equation. If you had done this would we have to conclude that the real element was no longer an electron or would this still be required by the previous definitions? Either way in such a case there would be no magnetic field as the Maxwell equations are needed to define one as well as the Dirac equation so we would have arrived at a different force. I’m not sure that I like the word force here perhaps interaction would be a better term to use although perhaps force or even field is more of the standard view?

[Doctordick]

Bombadil, the only serious comment about your post which I can make is that you simply do not comprehend the essence of my presentation.

If all that we are interested in is explaining our observations then your definitions seem very useful and the question of “how much does the supposed future resemble the known past” seems very important, but it seems to bring into question what we mean when we say that the past resembles the future.

It shouldn't! If you have an explanation of the past (think of, “a functioning world view”) then the statement is no more than an assertion that your expectations are that the immediate future will not destroy that world view (the future is expected to resemble the past, what you have come to see as your world view).

Now the definitions that you have made...

These definitions were not made by me; they are the definitions concocted by the physics community. I use those definitions for the simple reason that they lead to the validity of Dirac's equation: i.e., the implication is that modern physics is a tautology and actually tells us no more about reality than did the pronouncements of the theologians of the middle ages.

 

Your whole approach seems to be, “what can Dick's fundamental equation tell us about reality?” The answer to that question is simple: the answer is “NOTHING”. That is why this is being posted to “Philosophy of Science” and not to “Physics and Mathematics”.

I think that this is going to continue to bug me for some time as the issues don’t seem to have a straight forward answer. The major ones being, are these definitions that you are making required by the fundamental equation to have a flaw free explanation?

Again, “I” am not making these definitions (here I mean the critical ones necessary to the deductions of Schrödinger's, Dirac's or Maxwell's equations), these are definitions the scientific community has come up with. The same goes with regard to the definition of momentum except for the fact that the partial (which is, in many respects, the essence of their definition) being a fundamental expression in my fundamental equation.

 

I do define time in a manner which ends up being a defined thing no matter what explanation is being modeled but that is actually not original with me. The concept time is taken as understood by the science community in spite of the fact that they actually never take the trouble to define it. The same thing goes for space. It is another concept of science has actually left undefined. I take it as no more than a totally undefined mechanism for plotting those arbitrary numerical references to the concepts underlying the explanation being modeled; which really isn't a very constraining definition at all. Totally unconstrained as far as I can tell but it is always possible that I have made an error.

I think there are other explanations that can be derived from the fundamental equation only because you have had to make these definitions (if there where no other explanations then I would think that they would not be needed to derive the Dirac equation).

Sure there are other flaw free explanations. What happens happens because that is what the gods want to happen is an excellent flaw free explanation of everything; however, it isn't very valuable for predicting what to expect in the immediate future. It kind of leaves everything up in the air (unless you want to believe the priests who tell you what the gods want)!

 

Indeed, I, on the other hand, have presented an explanation which seems to work fine. “Reality is a totally unknowable thing which obeys no rules whatsoever!” All the supposed rules we have come up with are no more than data compression algorithms which makes the known past deductible from a reasonably small collection of concepts. What I have proved is that the mechanisms used to perform that explanation (or that deduction) are mere tautologies: i.e., they are no more than consequences of those definitions the scientific community has put forth.

On the other hand, would we consider it a different explanation if all we did was add more real or presumed elements defined in the same way, or could we even call it a valid explanation if we defined them in a different way? We would certainly then have difficulty making our explanation be of any use for explaining the same collection of elements.

As I have said above, all explanations are actually little more than data compression algorithms and, as such, there are no such things as “valid explanations”. Every explanation contains within it the possibility that future information can invalidate it. In fact it is easy to show that there are an infinite number of algorithms which will give exactly the past it was designed to explain and yet give a different answer for the next possibility. That is why I use the term “flaw-free” which I have defined to mean that there exists no data from the past which can invalidate it.

I don’t know exactly what you are referring to here. Unless you are referring to the purely mathematical symmetry of the equations.

Magnetic monopoles are conceived of as a possibility because of the fact that the nature of Maxwell's equations are unchanged if [imath]\vec{E}[/imath] and [imath]\vec{B}[/imath] are simply interchanged. Since it is experimental fact that electric monopoles (i.e., electrons) are observed it is thus presumed that there exists no reason for the absence of magnetic monopoles other than the fact that we have not seen them.

 

In my analysis, Maxwell's equations are mere approximations to my fundamental equations and thus not actual solutions. Since the process of creating that approximation involves different definitions of electric and magnetic fields, the observation of electrons cannot be taken to imply magnetic monopoles are possible. Not as elemental entities anyway. There certainly may be the possibility of specific complex situations being interpreted as magnetic monopole interactions but that would not qualify as a universal fundamental entity equivalent to a “magnetic” version of an electron.

 

I am presuming that continuation of this discussion of Dirac's equation is a rather worthless effort therefor I am proceeding in my development of a representation of a general relativistic transformation in my (x,y,z,tau) representation of a reality of no rules. I do this for Anssi. The only reason I will post it on the forum is the rather fond hope that someone out there beyond Anssi might begin to fathom what I have been saying.

 

Have fun -- Dick

[Bombadil]

These definitions were not made by me; they are the definitions concocted by the physics community. I use those definitions for the simple reason that they lead to the validity of Dirac's equation: i.e., the implication is that modern physics is a tautology and actually tells us no more about reality than did the pronouncements of the theologians of the middle ages.

 

Your whole approach seems to be, “what can Dick's fundamental equation tell us about reality?” The answer to that question is simple: the answer is “NOTHING”. That is why this is being posted to “Philosophy of Science” and not to “Physics and Mathematics”.

 

It’s not what your equation says about reality that I wonder about. Clearly it can say nothing about reality as it was derived from symmetry constraints required by an explanation and then used to derive the fundamental equation. All that it can do is allow us to study the implications of those symmetries. But what I can’t understand is what the definitions are doing, clearly they say that physics can be seen as a tautology that is, the definitions that are made are completely equivalent to the equations that are used in theoretical physics.

 

The problem is that I can’t understand why it is that physicists arrived at the definitions that lead to the equations that you derive from the fundamental equation. Would any tautology that could be arrived at from the fundamental equation have worked for what they were doing and they just happened to arrive at this one, or do the definitions that they use tell use something about the symmetries that they where studying, “ I want to emphasize that I am not suggesting that reality has these symmetries but rather that they assumed that some symmetry existed in what they where trying to explain and as a result they unknowingly stumbled on a tautology” or is there some other reason that I am over looking. Or do you consider what the definitions are doing to be of no interest?

 

I think that the intent of this thread was to show that all of modern physics that are based on the Maxwell and Dirac equations combined with the definitions of a electron and photon form a tautology .

In which I think that it was successful. That is the construct is true by means of the vary definitions that are used in modern physics. However I think that I have strayed from this intent and perhaps it would be best to continue discussions in other threads. So unless you have any objections or you want to continue here or somewhere else on a different topic you think I should examine I will wait for you to post your thread on the general relativistic transformation’s and try to stay on topic and help examine the derivations.

 

I’m am still trying to understand what you are presenting even if I keep straying from the topic under discussion this is not intentional. I just don’t see the topic of what the definitions are as so different.

[AnssiH]

Maybe I could try to explain what's going on in this thread...

 

It’s not what your equation says about reality that I wonder about. Clearly it can say nothing about reality as it was derived from symmetry constraints required by an explanation and then used to derive the fundamental equation. All that it can do is allow us to study the implications of those symmetries. But what I can’t understand is what the definitions are doing...

 

...are the definitions that are used needed to explain the behavior of a particular set of elements. Or can less definitions be made and still describe the behavior of the same elements? Is there really that much difference in saying that the definitions that you use say something about reality and saying that the definitions that you use are consequences of simplifying the conditions that are being explained or observed.

 

The problem is that I can’t understand why it is that physicists arrived at the definitions that lead to the equations that you derive from the fundamental equation.

 

You can view those definitions as definitions that can always be made. It needs to be made clear, that they are NOT dependent on the content of the "data to be explained" at all, i.e. against the common perception, they are not something that were found to be "true" because of reality around us having been built to function like that. They are rather "valid" because they are features of a framework (or a mechanism) capable of interpreting unknown data patterns in terms of predictably behaving persistent entities.

 

I can see that some parts of this analysis can be very unintuitive to most people, so it is certainly appreciated that you are looking at this closely. I'll just walk through briefly what were the arguments that got us here, starting from the fundamental equation.

 

The fundamental equation itself was a statement of constraints that are universal to all valid explanations (to all self-coherent explanations for unknown data patterns).

 

As far as being able to make any predictions about the future of some data patterns, those constraints are fairly useless (the view is still very complicated). But they do tell you already that it is always possible to interpret any valid explanation in terms of point particles moving in an [imath]x,y,z,\tau[/imath]-space.

 

There is nothing in the "data to be explained" itself that makes that sort of interpretation possible. Meaning, the data itself could be absolutely anything at all. What makes this sort of interpretation "always possible" is the fact that the prediction function of a valid explanation must obey the symmetry requirements. That means, that the elements defined by that explanation must behave that way, not the data itself. (Or rather that the elements of an explanation must be such that they can always be validly interpreted that way)

 

How exactly those defined elements are "related" or "attached" to the patterns or features of the "data to be explained", is a different matter entirely. You can think of all this as a matter of becoming able to pick up specific sorts of circumstances or patterns in some data, and being able to interpret it as an evidence of an existence of some defined element. Or you can think of this as a mechanism for picking up recurring features for prediction purposes. It is as if you already had the definitions before the data itself, and once you get the data, you simply interpret it in terms of those pre-existing definitions.

 

In this same vein, in the OP about the Schrödinger's equation, we are seeing some rather simple definitions that can always be made; energy, momentum and mass. For instance, getting a definition of "mass" was simply a matter of starting to call the velocity along the [imath]\tau[/imath]-axis as "mass".

 

Being able to make that definition obviously has got nothing to do with what the underlying "data to be explained" was (or whether such a property as "mass" exists in ontological reality). Still that move, as it turns out, gives us a defined quantity that behaves exactly as "mass" does in conventional physics; i.e. it is related to other definitions (like momentum and energy) in exactly the same way as these things are related to each others in conventional physics.

 

You should think about that a bit. We are talking about properties that are normally viewed as some pretty "tangible" features of the reality around us, but we arrived at these definitions as features of a data ordering mechanism itself, i.e. as decisions for how we could choose to view some data. Their validity is in no way a function of the actual data we are "ordering"; we have simply chose to order the data in terms of those definitions.

 

Once again, you can view this as a matter of already having those definitions, and then becoming able to see certain circumstances in terms of those definitions. No one forced us to use these definitions, but they are rather the definitions that make up a useful predictive world view. It is largely the ways our self-made definitions are related to each others that makes them so useful. They sort of make up a self-coherent circle of beliefs, so when you know some part of the circle, you can make useful expectations of the other parts, so to speak :) (The point of this analysis being, that it turns out that very many things are defined exclusively in terms of each others, instead of being grounded to ontological reality)

 

That gets us to the OP of this thread; we are interested of what stands between the fundamental equation, and "Dirac's equation for an electron coupled to an electromagnetic field". Is there something in the "data to be explained", i.e. in reality itself, that makes Dirac's equation valid, or can we arrive at it via definitions that can always be made, which would make the whole expression simply a feature of a particular data ordering scheme; something we can simply choose to use for usefulness sake.

 

Dirac's equation in the OP is a statement about the expectations for an electron coupled to an electromagnetic field, but the fundamental equation does not contain a definition for an electron or for an electromagnetic view. What DD is doing is he is trying to recognize the sorts of circumstances that in conventional physics would be interpreted as "a photon" or "an electron" or "an electromagnetic field". What allows him to say anything about what those circumstances are like in terms of his [imath]x,y,z,\tau[/imath]-framework, are those definitions that were in the earlier steps established as being in line with the conventional physics definitions.

 

...Dirac's [imath]\vec{p}=-i\hbar\left\{\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z}\right\} [/imath], is exactly the same momentum operator defined in my deduction of Schrödinger's equation...

 

...In Dirac's equation, [imath]\vec{A}[/imath] and [imath]\Phi[/imath] are electromagnetic potentials. In my equation, the potentials, V(x,t), are obtained by integrating over the expectations of some specific set of known solutions. Since, in common physics, the electromagnetic potentials arise through the existence of photons, it seems quite reasonable that the electromagnetic potentials (from my fundamental equation) would arise from integrating over the expectations of known massless solutions...

 

I.e. since we have a definition for mass, and since we know that in common physics definitions "electromagnetic potential" is a defined thing that is related to "photons" (that are by their definition massless entities) in very specific way, that gives us some leverage in our attempt of plugging in the common definitions for "electrons" and "electromagnetic potentials" into the equation.

 

The important bit of each step in the OP is that none of the definitions that are used, are such that they would require something in the "data to be explained" to be of particular kind, but rather they are definitions that could be applied onto any unknown data patterns.

 

So, to answer to your question, you could say that it has been partially an accident that physicists made those definitions in conventional physics. I am saying "partially" because there are reasons why those definitions would be very useful features of a world view, but on the other hand it is always possible to come up with another set of definitions that is equally useful, or more useful for particular situations.

 

That is in fact what is going on with physics all the time; there often are many different ways of understanding the same circumstances. That is the essence of semantics, and it is something that is made possible by the fact that very many things in our world views are defined exclusively in terms of each others, i.e. not actually grounded to any explicit knowledge about ontological reality, but made for usefulness sake.

 

So the only reason DD is using those physics definitions in this thread is because we are interested of the epistemological roots of Dirac's equation, and Dirac's equation is using those definitions.

 

I hope that clears things up a bit.

 

...I will wait for you to post your thread on the general relativistic transformation’s and try to stay on topic and help examine the derivations.

 

Yes, any help in examining the derivations is greatly appreciated :)

 

You know the whole interesting thing about all this is that once you can see how exactly these physics definitions are grounded to epistemological issues (i.e. data ordering mechanisms we are using for usefulness sake) rather than ontological reality, it explains some very elusive features of those associated theories. For instance, quantum theory produces experimentally valid expectation, but at the same time its validity requires that some assumptions in our world views cannot be real ("Bell inequalities are violated" as they put it). Well, if you can prove that our object definitions are indeed arising from the mechanisms that order unknown data patterns into predictable views, and those mechanisms also require QM expectations to be valid, you can safely say that it is the object definitions that are not real (i.e. whatever you conceive as a persistent object, is rather a data pattern you have a name for). We are merely talking about "labels" that we have placed upon certain circumstances, and that is exactly why Bell inequalities are and must be violated, when the circumstance is such as is described by the QM Bell experiments.

 

That to me is by far the most "realistic" explanation to quantum mechanical phenomena I know of; it is the only explanation I know of that doesn't have either idealistic features to it, or static or advanced (from future to past) time features.

 

Or much in the same vein, when Einstein put forward his paper about special relativity, a lot of people were kicking and screaming against it because they believed their view of space and time was grounded on ontological reality, and simply could not be questioned. Why Einstein was so sure of himself even prior to any experimental proof was that he had arrived at those relationships as a required consequence of other relationships that were known to be valid (definitions behind electromagnetism). Of course the same relationships could be expressed in many different ways, that each imply different reality, but the interpretation of SR that Minknowski put forward, i.e. spacetime & Lorentz transformation operating on it, is still viewed as something that is somehow related to ontological reality, even though we are actually talking about the impact that our ability to freely choose an inertial frame has got to our expectations. When you understand the epistemological roots/requirements for the relativistic transformations upon our valid definitions (definitions for "ordering reality"), suddenly all the elusive implications of relativity are little more than features of the definitions of our choice. That fact is just little bit hidden by the fact that you have to understand analytically where the definitions underlying relativity came from, and where those definitions came from, etc, etc...

 

Oops, sorry this turned out a bit long :P

 

-Anssi

[Bombadil]

You should think about that a bit. We are talking about properties that are normally viewed as some pretty "tangible" features of the reality around us, but we arrived at these definitions as features of a data ordering mechanism itself, i.e. as decisions for how we could choose to view some data. Their validity is in no way a function of the actual data we are "ordering"; we have simply chose to order the data in terms of those definitions.

 

So we are looking at the ways in which elements can be defined and the resulting ways that the ontological elements must behave if such a definition is made and the resulting order that results in the data being explained. In a sense we are actually talking about the requirements of the mapping for the ontological elements to the coordinate system that was constructed in the derivation of the fundamental equation that [imath]x,y,z,\tau,t[/imath] coordinate system.

 

If this is the case then the issue of how and why some definition might be made is really a senseless question from the beginning because there is nothing to explain if we don’t make a definition. And so from the start we can only assure that our definitions are self consistent which is what was being done when the fundamental equation was being derived.

 

Although this says nothing about if any definition can be used to explain any set of elements. The question of if we can use any definition kind of has two answers. One, from the perspective of if we have a mapping to our coordinate system we have already made some definitions and we must continue to perform further definitions in a way that is consistent with the definitions that have been made and with the mapping that we are using. Or two, we can look at it from the perspective that if we haven’t made any definitions than we are free to define an element in any way that we want and the definitions that we make will have there own required mapping of elements that results from using those definitions.

 

It seems though that the first perspective is taken by modern physics. That is, they assume that they have a valid mapping of the elements of interest and then they try and come up with a set of definitions that explain the behavior of the elements. While our interest is in the second possibility.

 

The important bit of each step in the OP is that none of the definitions that are used, are such that they would require something in the "data to be explained" to be of particular kind, but rather they are definitions that could be applied onto any unknown data patterns.

 

I don’t see how we know that this is true, if the very definitions that we are using result in the mappings used to map the elements to our coordinate system, how can we say that the definitions that we make don’t require something in the data being explained. It’s just that those patterns always exist if we are using those definitions likewise without those definitions we would never see that pattern in our data and so have no need for that definition.

[AnssiH]

So we are looking at the ways in which elements can be defined and the resulting ways that the ontological elements must behave if such a definition is made and the resulting order that results in the data being explained. In a sense we are actually talking about the requirements of the mapping for the ontological elements to the coordinate system that was constructed in the derivation of the fundamental equation that [imath]x,y,z,\tau,t[/imath] coordinate system.

 

Yes, little bit cumbersome wording but I think you have an idea of what this is about. As long as a particular set of definitions is self-coherent, it obeys the fundamental equation (=the symmetry constraints). The derivations of standard physics definitions are demonstrations of how those definitions are fundamentally related to those symmetry constraints, instead of to the actual nature of the information that was explained.

 

It all boils down to mechanisms that can be used to yield an interpretation about some information whose characteristics are entirely unknown.

 

Like you are trying to say;

 

If this is the case then the issue of how and why some definition might be made is really a senseless question from the beginning because there is nothing to explain if we don’t make a definition. And so from the start we can only assure that our definitions are self consistent which is what was being done when the fundamental equation was being derived.

 

Again a bit cumbersome wording, but I think you are basically thinking of the correct issue. "...there is nothing to explain if we don't make a definition", I guess with that you are trying to get to the fact that whatever it is that is explained, is something whose characteristics are not known at all, prior to definitions that were made. And hence, what we know about an explanation is simply that it is self-coherent, i.e. it obeys the symmetry arguments, i.e. it obeys the fundamental equation.

 

I just posted a message here about this:

http://hypography.com/forums/philosophy-of-science/18457-explanation-what-i-am-talking-about-9.html#post295318

(Post #89)

 

Let me know if that sounds like we are talking about the same issue. Also you might be interested to browse that thread backwards a bit as I feel there are parallels there with my previous post to this thread.

 

Although this says nothing about if any definition can be used to explain any set of elements.

 

Instead of saying "to explain a set of elements", it's better to say "to explain some information". Simply because "a set of elements" implies there are elements that have been defined, i.e. "a set of elements" is a characteristic of an explanation.

 

The question of if we can use any definition kind of has two answers. One, from the perspective of if we have a mapping to our coordinate system we have already made some definitions and we must continue to perform further definitions in a way that is consistent with the definitions that have been made and with the mapping that we are using. Or two, we can look at it from the perspective that if we haven’t made any definitions than we are free to define an element in any way that we want and the definitions that we make will have there own required mapping of elements that results from using those definitions.

 

It seems though that the first perspective is taken by modern physics. That is, they assume that they have a valid mapping of the elements of interest and then they try and come up with a set of definitions that explain the behavior of the elements. While our interest is in the second possibility.

 

Well not really, I mean sometimes standard physics takes steps backwards in its definitions also, if things start to get overly complicated. But the point to be picked up from this treatment is in how the definitions are related to the symmetry constraints and how some strange features of modern physics are features of the explanation, not features of reality per se.

 

Regarding this, the last bit is the most important one;

 

The important bit of each step in the OP is that none of the definitions that are used, are such that they would require something in the "data to be explained" to be of particular kind, but rather they are definitions that could be applied onto any unknown data patterns.

I don’t see how we know that this is true, if the very definitions that we are using result in the mappings used to map the elements to our coordinate system, how can we say that the definitions that we make don’t require something in the data being explained.

 

It's because it is those symmetry constraints (arising from self-coherence requirement) that yield the kinds of behaviours that these definitions can be placed on.

 

I'll try to explain it in my own words.

 

There's some information to be explained, and since nothing about the information is known, any explanation must operate via definitions that are based on familiarity to recurring patterns of some sort, with no actual knowledge about why those patterns are recurring. But any type of (semi)recurring activity will give the possibility to create valid definitions for (semi)predictably behaving persistent entities.

 

It is always possible to map those defined entities onto an [imath]x,y,z,\tau[/imath]-space with [imath]t[/imath] for time evolution.

 

The [imath]x,y,z,\tau[/imath] coordinate system itself is "immaterial"; the "information to be explained" did not contain its definitions; we just made it up for convenience. That means, your expectations about the future of a particular element cannot be based on its position in the [imath]x,y,z,\tau[/imath]-space. Your expectations must be based on its position in relation to the rest of the defined universe.

 

Which is to say, the location of the origin of the [imath]x,y,z,\tau[/imath]-space can be freely chosen, which is to say a valid [imath]\Psi[/imath] obeys shift symmetry.

 

If you have a valid [imath]\Psi[/imath] for a sub-set of the defined universe, it means you have a solution where you can ignore the rest of the universe. That solution must also be shift-symmetrical for the same reasons as the entire universe. And of course a useful world view should be expected to contain definitions that are useful without having to take into account the entire known universe, which is getting us to a situation where the expectations for some small sub-sets of the defined universe can be seen to "persist" through the evolution of "t"; i.e. you'd expect that an element conserves its momentum. Like a definition for a tennis ball.

 

Let me re-iterate; we got to the validity of the conservation of momentum via being forced to define elements based on recurring patterns, and having an ability to map those definitions in terms of an [imath]x,y,z,\tau[/imath]-space. In other words, the conservation of momentum is a feature of an explanation, which merely ordered some information that way.

 

Which is to say, the conservation of momentum is tautologous to the shift symmetry arguments, and did not require anything in the "information to be explained" to be of particular kind. It is an interpretation, that can always be made.

 

Now from that point, it's probably easy for you to see how the definitions for mass and energy etc, are also simply statements about a particular circumstance that can always be detected in the [imath]x,y,z,\tau[/imath]-space.

 

It’s just that those patterns always exist if we are using those definitions likewise without those definitions we would never see that pattern in our data and so have no need for that definition.

 

Yes, nothing is forcing us to use those exact definitions. The point is exactly that nothing in the "information to be explained" requires us to use those definitions, but they can always be useful as data ordering mechanisms. The focus here should be in the connections between those definitions, which end up being pretty much exactly the connections found by modern physics. Except that for modern physics, some of those connections were not immediately obvious at all, and some continue to seem quite strange. I mean, focus onto the fact that the physics assumptions about what constitutes "a persistent object" are, after all, based on some sort of recurring familiarity to some information that we are interpreting according to our world view.

 

So in the end, when you just take all those definitions together, they form one big tautologous structure that is fundamentally based on the symmetry requirements of an explanation, as oppose to an explicit information about reality. It is not so much about how valid those definitions might be for some information by chance, it is rather about what sorts of features of some information can be useful for prediction purposes at all, and how they fall inside these definitions.

 

Personally, I think that fact is visible also from the way that any physics definition must be understood in terms of other definitions, and vice versa. I.e. from how they define each others.

 

-Anssi