As far as I can see the concept of the point particle was originally developed by Roger Boscovich in the 18th century. It was further developed by Immanuel Kant with his concept of a force field where the field would surround a particle without spatial extension. Mathematicians like the concept since it simplies the mathematics of quantum mechanics, like with the usage of the Dirac delta function. This also ties in with a positivist or instrumental philosophy of science which holds that particles only have exact properties, such as location, when measured. I reject these philosophies and instread am a physical realist. However, I do not believe that point particles physically exist. Instead I believe that all physical particles, including ones such as photons and electrons which quantum theory treats as point particles, are spatial. One argumentfor charged particles, such as electrons, being point particles is that if they were spatially extended then they would explode due to electrical repulsion among the charged parts. This assumes though that electrical forces are present within particles per se, and I reject this. Some of this is expanded on in more detail of Chapter Four of my Ebook A Model of the Physical World which can be found at my website quantumrealism.net. Comments are welcome.

Following the view of Robert French (RF) I, Winfried Kaucher, implant some comments (marked with WK) in the text of Robert French (RF). (RF): Mathematicians like the concept since it simplies the mathematics of quantum mechanics, like with the usage of the Dirac delta function. This also ties in with a positivist or instrumental philosophy of science which holds that particles only have exact properties, such as location, when measured. I reject these philosophies and instread am a physical realist. (WK): A "physical realist" is to compare with a "mathematical realist" what I am also doing. I use the “ideal” Pi although I know that this real number can never be represented in any exact manner. It is a pure object of thinking, say model or instrument. We can only write "Pi" as the name of that number, but this is a name for a “procedure” which refers to Geometry of "ideal circles" which also not exist in reality. But we derive formulas with the aid of Pi and we accept that in all cases of terms, where Pi occurs , we use its approximation with some approximation error. This error may be (relatively) very small but in praxis it could happen that the final result deviates (relatively) in a frightening dimension. I can deliver an uncounted number of Examples for this phenomenon (WK is/was an old expert in this subject). What means hereby "realistic"? Should we reject such numbers? (RF): However, I do not believe that point particles physically exist. (WK): Shall I also not blieve that such "real numbers" (such as Pi) do not exist mathematically? or even physically? (RF): Instead I believe that all physical particles, including ones such as photons and electrons which quantum theory treats as point particles, are spatial. One argument for charged particles, such as electrons, being point particles is that if they were spatially extended then they would explode due to electrical repulsion among the charged parts. (WK): This is the decisive point: All numbers in any environment of Pi are not equal Pi and it would lead to false formula if one replaces Pi by such a number! All in mathematics is in principle of this character.- Therefore I will say, we have the similar or comparable situation in Physics. We use Diracs "Delta-function", high discontinous, has no graph, it is no function, it is a distribution. But we use its properties or as an operator. But nevertheless we use this "function" in practice(!) such as a particle is a small ball(sphere), but such a ball is as small as everyone would like.

Note the radius may be R but not zero, what means R > 0 but nobody can give any number for R. It is an open ball. There is no bound. Perhaps R is smaller that each 1/n for n to infinity. Indeed (f(x+R)-f(x))/R = f´(0) for each differentiable function. A Limes is not necessary)! This shows that such a Nature would have strange properties. It is a warning, to be carefully with such a discussion about particles. If you take the ball (interval) B(pi) which contains Pi together with some surrounding numbers of Pi (all approximations of Pi with a defined distance, then in mathematics B(Pi) would be a complete set. If we use B(PI) = B instead of Pi in a formula f(Pi), then we get a Ball (interval) f(B)) = {set of all f(b) with b element of B}. Thus we have in so far a correct representation of the original term f(Pi), since we know at least f(Pi) is contained in F(B). This sounds like a simple triviality but this is the basis of that on what the whole Analysis is built on. In the usual Analysis you find this in use in the in the topological (infinitesimal) proofs, e.g. in the proof of Cauchys convergence of series and the like. However in the higher numerical treatment we avoid every real "point number" p as an exact (unknown) value by replacing p by the set B(p) which contains that number, e.g. P = [2.5 , 3.7] which contains as Interval (although) with rational bounds the transcendent number PI. This corresponds with the ides of “no pint” for a particle. But it shows why the counter-people are not so wrong: (RF) cited:… "the argument for charged particles, such as electrons, being point particles is that if they were spatially extended then they would explode due to electrical repulsion among the charged parts." As is seen, in the range P = [2.5 , 3.7] of Pi there are numbers which have nothing to do with Pi but they may be in (causal) connection with another part of the device and thus such a “particle” “Pi” cannot exist with stability e.g. of cause of “repulsion among the charged parts”. This is only a rough explaining, since a more physical discussion leads to a very difficult theory concerning causality. Also in context e.g. with the proof of existence of zeros of a given function and similar with the proof of existence of solution functions of a differential equations one has to compute a set of functions F = [g(x), h(x) ] such that g(x) < fx) < h(x) for all x in a range X =[a,b] which is a dense set F in the set of all diffenetiable functions. If we are able to transform F by use of the Solution operator T (e.G. GREEN´s Operator) such that T(F) is contained within the set F itself, then it is proved that an exact solution f*(x) exist and is contained in F. This is absolutely necessary since there are other functions in the set F which are no solutions, but it must be proved that at least one function exist which is a solution! (More in my Book “Self-Validating Numerics for Function Space Problems “ Kaucher, Miranker, written in English, Academic Press, 1984, difficult to understand the mathematics). This proves in a side effect that particle cannot be spatially extended, even in no parameter dimension, in any parameter, neither in space nor in charge, nor in mass, nor in time. If a parameter value is constant in a delta time, then the value is constant in that time. Any probability for such a process is nearly zero. Each disturbation of the Fourier expansion of such a constant becomes divergent. Each neighborhood of something would cause a change of existency or causality. A difficult discussion. I intent to publish this later in this blog and shows, how much catastrophic explaining exist in modern physics, in particular the Relativity theory (RT) with its horrible errors.

Continuing the idea of existency and local uniqueness of something (like particle) within a set of “existence”, we proceed to the following consequence also in physics to use sets of processes as an environment of an experience or of observation. Simplified, we see that in a formula like exp(x) we can represent exp(Pi) which is an unknown number by a set Z = exp(P) which contains exp(Pi). If we look at Pi and exp(Pi) as non existing "realistic" numbers which by no means can be represented nor by a drawn circle nor by a floating point number (later we say metamorphic: "measured"). This non-realistic-existence can be over bridged by a mathematic in an Analysis space of Intervals (topological sets). But note, all these objects (Intervals) are concrete represent able objects, denoted by some rational bounds, numbers, and in this sense existing and measurable. In this form we enlarge the term "measurable" in physics to "measure through sure = reliable intervals”, i.e. containing the ideal value of the Nature (Reality). This is essential. At this place I disagree with (RF) in so far, it is rather impossible to built up any physics without the use of essential elements of the Analysis of Mathematics, e.g. “point structures” such as, Delta -function, convergence, ideal circle and the like. The contradictions in most of the no classical physic show that there is need for a "mathematical physic" on the one hand. But it I underline the fact, that most theories are pretty and elegant for itself, but are far away from a natural reality. Their basis are assumptions like constancy of the light velocity, the mass of photons is (exact) zero, the electrical charge of electrons is universal constant or the density of charge e/m is constant, etc. It may be useful to assume such idealistic properties but it could be lead into dead ends like the miss teaches that Nature is not able to decide which true unique value she should offer in some experiments, and the silly consequence, that this the fuzziness of Nature. Nobody will say, the fact that we cannot give a definite represent able number for Pi, (what is a measurement for Pi), that this may imply that Pi is undefined or the Nature of circles is fuzzy and the like. In reality, in that case it is the inexperience or stupidity of such people. Pi as well as other transcendent numbers are nothing else as a way (a procedure) how to be able to approximate them in realty in a chosen coordinate system. In case of Pi one can take the decimal number system and with some algebraic formula one can compute Pi as approximate as you wish, but never exact. But we know from some such numbers some exact properties, e.g. how they interact with other numbers, for example sin(n * Pi) = 0 or 4 * Pi = Pi * 4 and so on. We use in praxis not the number in a floating point system but we use the exact properties. With the Dirac-density function we do the similar. We do not use a graph or a value of that "function", we use its properties (as distribution). Neither Robert French nor I or someone else can dicide, what Nature in reality is doing, but I know, that the idea of Quantum theory is far away from real Nature when they speak about uncertainty. (RF): This assumes though that electrical forces are present within particles per se, and I reject this. (WK): I think (RF) is right in so far as it is uncleared what is inertia and what is mass and do they coincide or not or can it be that mass is changing and the mass of photons is exact zero, ….and so forth. This is a jungle of unclearness. If (RF) feels this is the reason for his “dislikeness” then I agree completely.

As far as I can see the concept of the point particle was originally developed by Roger Boscovich in the 18th century. It was further developed by Immanuel Kant with his concept of a force field where the field would surround a particle without spatial extension. Mathematicians like the concept since it simplies the mathematics of quantum mechanics, like with the usage of the Dirac delta function. This also ties in with a positivist or instrumental philosophy of science which holds that particles only have exact properties, such as location, when measured. I reject these philosophies and instread am a physical realist.

ReplyDeleteHowever, I do not believe that point particles physically exist. Instead I believe that all physical particles, including ones such as photons and electrons which quantum theory treats as point particles, are spatial. One argumentfor charged particles, such as electrons, being point particles is that if they were spatially extended then they would explode due to electrical repulsion among the charged parts. This assumes though that electrical forces are present within particles per se, and I reject this. Some of this is expanded on in more detail of Chapter Four of my Ebook A Model of the Physical World which can be found at my website quantumrealism.net. Comments are welcome.

Following the view of Robert French (RF) I, Winfried Kaucher, implant some comments (marked with WK) in the text of Robert French (RF).

Delete(RF): Mathematicians like the concept since it simplies the mathematics of quantum mechanics, like with the usage of the Dirac delta function. This also ties in with a positivist or instrumental philosophy of science which holds that particles only have exact properties, such as location, when measured. I reject these philosophies and instread am a physical realist.

(WK): A "physical realist" is to compare with a "mathematical realist" what I am also doing. I use the “ideal” Pi although I know that this real number can never be represented in any exact manner. It is a pure object of thinking, say model or instrument. We can only write "Pi" as the name of that number, but this is a name for a “procedure” which refers to Geometry of "ideal circles" which also not exist in reality. But we derive formulas with the aid of Pi and we accept that in all cases of terms, where Pi occurs , we use its approximation with some approximation error. This error may be (relatively) very small but in praxis it could happen that the final result deviates (relatively) in a frightening dimension. I can deliver an uncounted number of Examples for this phenomenon (WK is/was an old expert in this subject). What means hereby "realistic"?

Should we reject such numbers?

(RF): However, I do not believe that point particles physically exist.

(WK): Shall I also not blieve that such "real numbers" (such as Pi) do not exist mathematically? or even physically?

(RF): Instead I believe that all physical particles, including ones such as photons and electrons which quantum theory treats as point particles, are spatial. One argument for charged particles, such as electrons, being point particles is that if they were spatially extended then they would explode due to electrical repulsion among the charged parts.

(WK): This is the decisive point: All numbers in any environment of Pi are not equal Pi and it would lead to false formula if one replaces Pi by such a number! All in mathematics is in principle of this character.- Therefore I will say, we have the similar or comparable situation in Physics. We use Diracs "Delta-function", high discontinous, has no graph, it is no function, it is a distribution. But we use its properties or as an operator. But nevertheless we use this "function" in practice(!) such as a particle is a small ball(sphere), but such a ball is as small as everyone would like.

Note the radius may be R but not zero, what means R > 0 but nobody can give any number for R. It is an open ball. There is no bound. Perhaps R is smaller that each 1/n for n to infinity. Indeed (f(x+R)-f(x))/R = f´(0) for each differentiable function. A Limes is not necessary)! This shows that such a Nature would have strange properties. It is a warning, to be carefully with such a discussion about particles.

ReplyDeleteIf you take the ball (interval) B(pi) which contains Pi together with some surrounding numbers of Pi (all approximations of Pi with a defined distance, then in mathematics B(Pi) would be a complete set. If we use B(PI) = B instead of Pi in a formula f(Pi), then we get a Ball (interval) f(B)) = {set of all f(b) with b element of B}. Thus we have in so far a correct representation of the original term f(Pi), since we know at least f(Pi) is contained in F(B).

This sounds like a simple triviality but this is the basis of that on what the whole Analysis is built on. In the usual Analysis you find this in use in the in the topological (infinitesimal) proofs, e.g. in the proof of Cauchys convergence of series and the like. However in the higher numerical treatment we avoid every real "point number" p as an exact (unknown) value by replacing p by the set B(p) which contains that number, e.g. P = [2.5 , 3.7] which contains as Interval (although) with rational bounds the transcendent number PI.

This corresponds with the ides of “no pint” for a particle. But it shows why the counter-people are not so wrong: (RF) cited:… "the argument for charged particles, such as electrons, being point particles is that if they were spatially extended then they would explode due to electrical repulsion among the charged parts."

As is seen, in the range P = [2.5 , 3.7] of Pi there are numbers which have nothing to do with Pi but they may be in (causal) connection with another part of the device and thus such a “particle” “Pi” cannot exist with stability e.g. of cause of “repulsion among the charged parts”. This is only a rough explaining, since a more physical discussion leads to a very difficult theory concerning causality.

Also in context e.g. with the proof of existence of zeros of a given function and similar with the proof of existence of solution functions of a differential equations one has to compute a set of functions F = [g(x), h(x) ] such that g(x) < fx) < h(x) for all x in a range X =[a,b] which is a dense set F in the set of all diffenetiable functions. If we are able to transform F by use of the Solution operator T (e.G. GREEN´s Operator) such that T(F) is contained within the set F itself, then it is proved that an exact solution f*(x) exist and is contained in F. This is absolutely necessary since there are other functions in the set F which are no solutions, but it must be proved that at least one function exist which is a solution! (More in my Book “Self-Validating Numerics for Function Space Problems “ Kaucher, Miranker, written in English, Academic Press, 1984, difficult to understand the mathematics).

This proves in a side effect that particle cannot be spatially extended, even in no parameter dimension, in any parameter, neither in space nor in charge, nor in mass, nor in time. If a parameter value is constant in a delta time, then the value is constant in that time. Any probability for such a process is nearly zero. Each disturbation of the Fourier expansion of such a constant becomes divergent. Each neighborhood of something would cause a change of existency or causality. A difficult discussion.

I intent to publish this later in this blog and shows, how much catastrophic explaining exist in modern physics, in particular the Relativity theory (RT) with its horrible errors.

Continuing the idea of existency and local uniqueness of something (like particle) within a set of “existence”, we proceed to the following consequence also in physics to use sets of processes as an environment of an experience or of observation.

ReplyDeleteSimplified, we see that in a formula like exp(x) we can represent exp(Pi) which is an unknown number by a set Z = exp(P) which contains exp(Pi). If we look at Pi and exp(Pi) as non existing "realistic" numbers which by no means can be represented nor by a drawn circle nor by a floating point number (later we say metamorphic: "measured"). This non-realistic-existence can be over bridged by a mathematic in an Analysis space of Intervals (topological sets). But note, all these objects (Intervals) are concrete represent able objects, denoted by some rational bounds, numbers, and in this sense existing and measurable. In this form we enlarge the term "measurable" in physics to "measure through sure = reliable intervals”, i.e. containing the ideal value of the Nature (Reality). This is essential.

At this place I disagree with (RF) in so far, it is rather impossible to built up any physics without the use of essential elements of the Analysis of Mathematics, e.g. “point structures” such as, Delta -function, convergence, ideal circle and the like. The contradictions in most of the no classical physic show that there is need for a "mathematical physic" on the one hand. But it I underline the fact, that most theories are pretty and elegant for itself, but are far away from a natural reality. Their basis are assumptions like constancy of the light velocity, the mass of photons is (exact) zero, the electrical charge of electrons is universal constant or the density of charge e/m is constant, etc. It may be useful to assume such idealistic properties but it could be lead into dead ends like the miss teaches that Nature is not able to decide which true unique value she should offer in some experiments, and the silly consequence, that this the fuzziness of Nature.

Nobody will say, the fact that we cannot give a definite represent able number for Pi, (what is a measurement for Pi), that this may imply that Pi is undefined or the Nature of circles is fuzzy and the like. In reality, in that case it is the inexperience or stupidity of such people. Pi as well as other transcendent numbers are nothing else as a way (a procedure) how to be able to approximate them in realty in a chosen coordinate system. In case of Pi one can take the decimal number system and with some algebraic formula one can compute Pi as approximate as you wish, but never exact. But we know from some such numbers some exact properties, e.g. how they interact with other numbers, for example sin(n * Pi) = 0 or 4 * Pi = Pi * 4 and so on. We use in praxis not the number in a floating point system but we use the exact properties. With the Dirac-density function we do the similar. We do not use a graph or a value of that "function", we use its properties (as distribution). Neither Robert French nor I or someone else can dicide, what Nature in reality is doing, but I know, that the idea of Quantum theory is far away from real Nature when they speak about uncertainty.

(RF): This assumes though that electrical forces are present within particles per se, and I reject this.

(WK): I think (RF) is right in so far as it is uncleared what is inertia and what is mass and do they coincide or not or can it be that mass is changing and the mass of photons is exact zero, ….and so forth. This is a jungle of unclearness. If (RF) feels this is the reason for his “dislikeness” then I agree completely.