Can Quantum Jumps Same as Dimension Jumps ???

[Antonio Lao]

By going back to Bohr's stationary orbits of electrons, The idea of quantum jumps was born.

We can hypothesize that matter such as electrons are 4D objects and photons are 3D objects and the continuous space between orbits is a 2D continuum. So what happens when one electron jumps from one orbits to another orbit is that the 4D electron transforms to 3D energy and the 3D energy transforms to a 2D continuum then 2D continuum back to 3D energy and then back to a 4D electron at a lower or higher orbit whichever the case in question. And transformation of 2D continuous space (S) is given by:

S = cE

where c is the speed of light in vacuum and it acts as dimension lowering constant for energy (E).

And by dimensional analysis, the 2D continuous space is proportional to an arbitrary surface area with the proportionality constant as force per unit of time. Futhermore force per unit time is proportional to the time derivative of acceleration with mass as the constant of proportionality.

\frac {Force}{time} = m \frac {\partial a}{\partial t}

[Antonio Lao]

If the surface area changes with time then the gradient of continuous space is given by

\nabla S = m \frac {\partial \vec{a}}{\partial t} \frac{\partial Area}{\partial t}

Futhermore for volume as a function of time, V(t) and given density \rho , the gradient of continuous space is given by

\nabla S = \frac {c^4 \rho_T}{\vec{r}} V(t)

[Antonio Lao]

If we can define the ratio of instantaneous density to total density of the universe as a function of time given by

\frac{1}{\rho_T} \int_{0}^{\infty} \rho_i (t) dt = 1

then the instantaneous density is the sum of kinetic density and potential density given by

\rho_i = \rho_k + \rho_p

If the total mass of the universe is the sum of total potential mass and total kinetic mass

m_T = m_k + m_p

then instantaneous kinetic and potential volumes can be related to the total volume given by

\frac{m_T}{m_p} - \frac{V_k}{V_p} \left ( \frac{V_T - V_p}{V_k - V_T} \right ) = 1

[Antonio Lao]

\frac{m_T}{m_p} \geq 1

and

\frac{V_k}{V_p} \left ( \frac{V_T - V_p}{V_k - V_T} \right ) < 1

if

\frac{m_T}{m_p} = 1

then

\frac{V_k}{V_p} \left ( \frac{V_T - V_p}{V_k - V_T} \right ) = 0

[Antonio Lao]

V_k is the same as the instantaneous volume of radiation. V_p is the instantaneous volume of matter. V_T is the total volume of the universe at each epoch.

[Antonio Lao]

The gradient of continuous space becomes a function of instantaneous density and volume.

\nabla S = \frac{c^4}{\vec{r}} \int_{0}^{\infty} \rho_i (t) V(t) dt

From this, an integral force exist.

F = \frac{c^3}{\vec{r} \times \vec{r}} \int_{0}^{\infty} \rho_i (t) V(t) dt

[Antonio Lao]

This force is infinitely large because the term \vec{r} \times \vec{r} = 0 . But if the r's are orthogonal and comparable to Planck length then the force is just simply large. If orthogonal r's is very large then the force is small.

[Antonio Lao]

Orthogonality seems to be a necessary condition for the force to exist. This implies that orthogonal forces must exist and the extremely high magnitude of this force must be able to quantize spacetime at the local infinitesimal region of a spacetime continuum.

[Antonio Lao]

For low velocity and small mass and together with the invariance, \vec{a} \cdot \vec{r} = c^2 , this force becomes Newton's 2nd law of motion.

[Antonio Lao]

In post #7, the r's can become large only if the individual r's can be added collinearly together.

\vec{r} = \int_{0}^{\infty} \vec{r}_i

[Antonio Lao]

Collinearity implies that \vec{r} = \alpha \vec{r}_i where \alpha is an integer.

[Antonio Lao]

When the linear momentum is zero and when both density and volume are functions of energy and time, the square of mass is given by

m^2 = \int \int \frac{\partial \rho^2}{\partial t} \frac{\partial V^2}{\partial t} dt dt

[Antonio Lao]

Furthermore, the square of mass is a new constant of nature given by

m^2 = \left( \frac{h}{ac} \right)^2

where h is Planck's constant, a is Planck length and c is the speed of light.

[Antonio Lao]

Implication is the existence of real positive and negative root for mass given by

\pm \frac{h}{ac}

without the use of complex number.

[Antonio Lao]

The use of these roots is to constraint the value of

\frac {V_k}{V_p} \left( \frac{V_T - V_p}{V_k - V_T} \right)

or

\frac {V_k}{V_p} \left( \frac{V_p - V_T}{V_T - V_k} \right)

so that the expressions are always less than unity.

[Antonio Lao]

Why is the quantum of mass using \pm \frac{h}{ac} numerically equal to the Planck mass using \sqrt{\frac{hc}{G}}?

If Planck's constant was determined first together with the speed of light and knowing the value of the Planck length, the value of the gravitational constant can be found by

G = \frac{a^2 c^3}{h}

[Antonio Lao]

In a planetary system where one of the earthlike planets is in a perpetual cloud cover, the inhabitants will never be able to have the opportunity of seeing the starry night sky. In this planet, there could be no Galileo nor Kepler and then nor Newton before Maxwell's and Planck's discoveries. Yet the scientists of this planet can derive the laws of electromagnetism and also quantum mechanics first and then discover the law of universal gravitation afterward.

Although Einstein's theories of relativity can clinch the final constancy of the speed of light, these are not necessary for the first order determination of the gravitational constant.

[Epsilon Pi]

Quite clever example of different paradigms and how they will guide our quests in different ways, but I was wondering if you did choose those examples to be discovered first(Maxwell equations without the displacement current concept and Plank's discoveries) because in the last analysis they have to do with more fundamental laws of nature than those of gravitation?

Regards

EP
In a planetary system where one of the earthlike planets is in a perpetual cloud cover, the inhabitants will never be able to have the opportunity of seeing the starry night sky. In this planet, there could be no Galileo nor Kepler and then nor Newton before Maxwell's and Planck's discoveries. Yet the scientists of this planet can derive the laws of electromagnetism and also quantum mechanics first and then discover the law of universal gravitation afterward.

Although Einstein's theories of relativity can clinch the final constancy of the speed of light, these are not necessary for the first order determination of the gravitational constant.

[Antonio Lao]

Epsilon Pi,

Will respond to your reply at the earliest possible time.

[Antonio Lao]

but I was wondering if you did choose those examples to be discovered first

I think all physical constants which are independent on other constants should appear in physical laws that are not depended on the sequential order of their discovery. For example, Planck's constant is depended on Boltzmann's constant therefore Boltzmann's constant must necessarily come first. Maxwell's theory of electromagnetism and quantum theory are in sequential order. One must follow the other since the later one uses some constants found in the earlier theory.

[Antonio Lao]

Unresolve relationships between mass, volume and density for some particles

Planck mass=10e(-5) gram, volume=10e(-99) cm^3, density=10e(+94) g/cm^3
proton mass=10e(-24) gram, volume=10e(-39) cm^3, density=10e(+15) g/cm^3
electron mass=10e(-28) gram, volume=10e(-39) cm^3, density=10e(+11) g/cm^3
photon mass=0 gram (?), volume= ?, density= ?

[Antonio Lao]

different paradigms

I am now reading Thomas S. Kuhn's 'The Structure of Scientific Revolutions' to get some ideas about paradigm shift.

[Epsilon Pi]

By reading and reading T.S.K, I was wondering if the quest for a TOE, is not a futile one after all?
In this respect he wrote, in his classical, when arguing against Popper's falsification argument:
"...no theory ever solves all the puzzles with which it is confronted at a given time; nor are the solutions already achieved often perfect. On the contrary, it is just the incompleteness and imperfection of the existing data-theory fit that, at any given time, define many of all the puzzles that characterize normal science. IF ANY AND EVEN EVERY FAILURE TO FIT WERE GROUND FOR THEORY REJECTION, ALL THEORIES OUGHT TO BE REJECTED AT ALL TIMES".(The Structure of Scientific Revolutions, p.146)

Must not we resolve then first the incommensurability problem we have between the existing paradigms, QM and GTR, before going with any intent? or at least making a great effort in this sense?

Just some inquisitive thoughts in my mind

Regards

EP

I am now reading Thomas S. Kuhn's 'The Structure of Scientific Revolutions' to get some ideas about paradigm shift.

[Antonio Lao]

By reading and reading T.S.K, I was wondering if the quest for a TOE, is not a futile one after all?

If the derivation of a constant of a theory is independent from constants of other theories and furthermore if the theory can describe all other theories with consistency then this theory can be a TOE.

The theory of electromagnetism is the relative TOE to the theory of magnetism and the theory of electricity.

The theory of electroweak interaction is the relative TOE to the theory of electromagnetism and the weak nuclear force.

Although relative TOEs do exist, it is the belief that an absolute TOE can also exist. But as pointed out previously, the gravitational constant can be derived from knowing the speed of light and Planck's constant and maybe guessing by trials and errors the value of the Planck length (needed for determining some quantity of volume) which will require some sort of uncertainty principle (if the mass is quantized) between the concept of density and volume.

My hunch is that the resolution can simply be done in one dimensional space or the quantization of 1D space and 1D time.

The theory of quantized 1D spacetime can resolve the theory of mass by the combined theories of density and volume. This theory implies the existence of two kinds of mass: the potential and the kinetic. It can also clarify the concept of electroweak charges and color charges by invoking a principle of a directional invariance.

So a theory that can describe both mass and charge using the physically revised concepts of density and volume can be called an absolute TOE.

A rough formulation of this uncertainty between density and volume at time=0 is given by

3^{-9} gram \leq \Delta \rho^2 \Delta V^2 \leq 4^{-9} gram

[Antonio Lao]

or

\left| \Delta \rho^2 \Delta V^2 \right| \leq \frac{1}{\pi^9}

[Antonio Lao]

If this is multiplied by square of time rate of change of area, the result is the square of energy.

and the Einstein's field equations can be made equivalent to square of energy by the product of a factor as the speed of light in vacuum.

c \left[ R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R \right] = - 8 \pi \frac{a^2 c^4}{h} T_{\mu \nu}

In empty space at time=0, and considering vacuum fluctuation, the field equations become

cR_{\mu \nu} = \left(\psi_i \cdot \psi_j \right) \left(\phi_i \cdot \phi_j \right) \geq h^2

[Epsilon Pi]

Hi Antonio and thank you very much!

An absolute TOE ? you really mean that? are you serious? in the same mansion of science?
Where is in your... -how would I say without trying to devaluate your great and noble intent- the duality of wave-particle, where is its rationalization?
It seems to me that the uncertainty principle is the one principle that identify in most cases QM, but where is in that description the duality of wave-particle, of space and time, ect?
Are you not seing things just a from classical point of view, or paradigm, just as those classical concepts such as mass and charge?

My best regards

EP

If this is multiplied by square of time rate of change of area, the result is the square of energy.

and the Einstein's field equations can be made equivalent to square of energy by the product of a factor as the speed of light in vacuum.

c \left[ R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R \right] = - 8 \pi \frac{a^2 c^4}{h} T_{\mu \nu}

In empty space at time=0, and considering vacuum fluctuation, the field equations become

cR_{\mu \nu} = \left(\psi_i \cdot \psi_j \right) \left(\phi_i \cdot \phi_j \right) \geq h^2

[Antonio Lao]

Where is in your... -how would I say without trying to devaluate your great and noble intent- the duality of wave-particle, where is its rationalization?


\Delta \psi \Delta \phi \geq h

is the Heisenberg's uncertainty principle of wave-particle duality.

[Antonio Lao]

But the expansion of cR_{\mu \nu} give the following integral equations.
Let \zeta be the product of Planck length and the speed of light c.

\left(m^2 - ma_m dt^2 \right) \left(n^2 - na_n dt^2 \right) \int \int a_m a_n dt dt \geq \zeta^2
\left(m^2 - ma_m dt^2 \right) \left(l^2 - la_l dt^2 \right) \int \int a_m a_l dt dt \geq \zeta^2
\left(n^2 - na_n dt^2 \right) \left(l^2 - la_l dt^2 \right) \int \int a_n a_l dt dt \geq \zeta^2

m, n, l are the quantum number of spacetime metrics.

[Epsilon Pi]

no, that principle does not describe the wave-particle duality; according to what I have learned it was the Schrodinger wave equation -the fundamental and complementary equation- the one that described so well the wave nature of matter-energy, i.e., of physical reality.
As I have understood things the uncertainty principle has to do with that impossibility we have to have an exact, as it were, an absolute description of physical reality.
That reality is after all always paradigm-determined, isn't it?

Best regards

EP
\Delta \psi \Delta \phi \geq h

is the Heisenberg's uncertainty principle of wave-particle duality.

[Antonio Lao]

The a_i where i=m, n, and l are the infinitesimal accelerations due to orthogonal forces.

[Epsilon Pi]

see my point before, please.

Regards
EP
The a_i where i=m, n, and l are the infinitesimal accelerations due to orthogonal forces.

[Antonio Lao]

As I have understood things the uncertainty principle has to do with that impossibility we have to have an exact, as it were, an absolute description of physical reality

If this is the case, then I was wrong all this time in believing that the product of uncertainty in wavelength (a property of wave) and the uncertainty in momentum (a property of particle) is greater than or equal to Planck's constant.

[Antonio Lao]

Anyway, by doubling the uncertainty, multiplied by itself, and at time=0, which is the singularity of the big bang, I get an uncertainty in the square of a time rate of change of an infinitesimal area.

\frac{dA}{dt} = ac

Where A is area, a is Planck length and c is the speed of light.

[Epsilon Pi]

Yes, but this talks about the impossibility we have to have an "ABSOLUTE description" of physical reality.
On the other hand it is Schrodinger wave equation the one that talks us about the wave nature in a precise mathematical description, even though, of course, and here I recognize your point, both:
- the uncertainty principle and
- additionally the Schrodinger wave equation

are fundamentals in describing QM; you cannot describe mathematically, in congruent way, the one without the other.

Regards

EP


If this is the case, then I was wrong all this time in believing that the product of uncertainty in wavelength (a property of wave) and the uncertainty in momentum (a property of particle) is greater than or equal to Planck's constant.

[Antonio Lao]

The uncertainty is between change in area and change in frequency of a wave.

\Delta A \Delta f \geq ac

[Epsilon Pi]

I thought uncertainty was related with that impossibility we have to measure, at the same time, two entities that cannot be reduced one to the other such as, wave and particle, or time and space, or momentum and position? a reason why its description had to be done with a complex differential equation such as the Schrodinger wave equation.

Regards

EP

The uncertainty is between change in area and change in frequency of a wave.

\Delta A \Delta f \geq ac

[Antonio Lao]

Schroedinger's equation is non-relativistic. Dirac's equation is relativistic. The transition is the energy formulation from

E =\frac{p^2}{2m}

to

E^2 = c^2 p^2 + m^2 c^4

I derived the the square of mass by assuming that linear momentum is zero.

[Epsilon Pi]

I really thought Schrodinger's wave equation was an equation that described the behavior of an entity such as the electron that sometimes can have the velocity of light; so just in those cases should we make the corresponding relativistic formulation or correction?

Regards

EP
Schroedinger's equation is non-relativistic. Dirac's equation is relativistic. The transition is the energy formulation from

E =\frac{p^2}{2m}

to

E^2 = c^2 p^2 + m^2 c^4

I derived the the square of mass by assuming that linear momentum is zero.

[Antonio Lao]

thought uncertainty was related with that impossibility we have to measure, at the same time, two entities that cannot be reduced one to the other such as, wave and particle, or time and space

For 1D spacetime quantization, there exists uncertainty in the transformation of 1D space to 2D space (surfaces) to 3D space (volumes) and the product of this uncertainty with inverse uncertainty of time which is uncertainty in frequency of wave is greater or equal to product of Planck length and speed of light. Actually, it is the absolute value of the uncertainty because a negative part also exists as well.

[Antonio Lao]

The area uncertainty comes from its degree of freedom to become 1D or 3D. Like a incremental plane contracting to a line by motion or transforming into a volume. We are not certain how this incremental area would want to do, becomes 1D or 3D?

[Epsilon Pi]

:confused: How can you talk about an ABSOLUTE value of UNCERTAINTY? Is not this a great contradiction?

Regards

EP
Actually, it is the absolute value of the uncertainty because a negative part also exists as well.

[Antonio Lao]

How can you talk about an ABSOLUTE value of UNCERTAINTY? Is not this a great contradiction?

\left| ? \right| \geq n

is the same as

? \geq n

and

? \leq -n

The uncertainty approaches zero from left and right.

[Antonio Lao]

I really thought Schrodinger's wave equation was an equation that described the behavior of an entity such as the electron

Schrodinger's equation was capable of resolving the one-electron hydrogen (neutral atom) spectra. But Dirac's equation is the one for the multiple electrons atoms because of charge and spin in fine structure spectra.

[Antonio Lao]

The positive and negative uncertainties as depicted by the uncertainty in frequency can resolve the concavity and convexity of quantized spacetime structures. One structure, from 2D surface to 1D string to 0D vacuum, resulting in the reality of the graviton with mass=0. The other structure, from 2D to 3D, and quickly to 4D and then bypassing 3D, 2D, and 1D jumps to 0D vacuum, resulting in the reality of the Higgs boson.

[Antonio Lao]

A clearer inequality formulation for the uncertainty in the quantum of mass is

[tex] -\Delta mass \leq -\frac{h}{l_p c} \leq 0 \leq +\frac{h}{l_p c} \leq +\Delta mass [/itex]

where h is Planck's constant, l_p is Planck length, and c is light speed.

[Epsilon Pi]

Oh, thank you, Antonio for another version of the uncertainty inequality, but are you sure there are not others ways, to represent the impossibility we have to measure at the same time those dualities we find at atomic levels, such as wave and particle?
Cannot those dualities be rationalized, as it were, in a complex mathematical description, i.e., a basic unit system which is not anymore an inequality?, but an equation in which the equal sign is not precisely a symbol JUST to reduce the one to the other? a symbol that permits us to include both, yes-or-no, and complementarity?
With such a representation for sure we will have another version of that uncertainty principle and we will have the chance to recover our way to a better description of physical reality not so obscure, which is my main concern in regards to modern physics and which was as I have shown in a paper, the main concern too of Ferdinand de Saussure when was confronted with those dualities of the Linguistic Sign.

Regards

EP


A clearer inequality formulation for the uncertainty in the quantum of mass is

[tex] -\Delta mass \leq -\frac{h}{l_p c} \leq 0 \leq +\frac{h}{l_p c} \leq +\Delta mass [/itex]

where h is Planck's constant, l_p is Planck length, and c is light speed.

[Antonio Lao]

but are you sure there are not others ways, to represent the impossibility we have to measure at the same time those dualities we find at atomic levels

The fault of the quest for the principle of duality lies in the analysis of periodic functions. Given a period T time units, the inverse of T is the frequency. But what is the meaning of time inverse?

Time inverse can appear to be just a velocity magnitude with the distance factor normalized and turned into a dimensionless quantity.

But distance can only be normalized if we assume that there exist a maximum distance to gauge it to.

\frac{1}{d_{max}} \int_{-\infty}^{+\infty} d_i = 1

[Epsilon Pi]

Thank you for your reply! You really have touched a point that concern me the most!
No, the fault of the quest for -what you now call the principle of duality, term, that I have used somewhere to identify: a binary logic, bilateral symmetry as was defined by Hermann Weyl, according to whom, all the theory of relativity rests on it- lies on that need to have a symbolism in which the third is included -not excluded as with the binary logic- or else the use of complex numbers that makes it possible to rationalize duality.
Of course, for our own convenience we have to reduce, in some cases, time to space, by defining a close system and then we will have a mathematical conception of time similar to that of space, that not only has the problem you have described, but additionally, it is a conception of time not in agreement with irreversible processes or the two arrows of time: in this wrong conception of time it can flow in both directions: positive and negative.
Now, thanks to your answer it is even more quite clear for me, why we have such a great difficulty in modern physics to even try to see things in a different way

Regards

EP

The fault of the quest for the principle of duality lies in the analysis of periodic functions. Given a period T time units, the inverse of T is the frequency. But what is the meaning of time inverse?

Time inverse can appear to be just a velocity magnitude with the distance factor normalized and turned into a dimensionless quantity.

But distance can only be normalized if we assume that there exist a maximum distance to gauge it to.

\frac{1}{d_{max}} \int_{-\infty}^{+\infty} d_i = 1

[Antonio Lao]

in this wrong conception of time it can flow in both directions

You might have just rescued me from falling into the trap of further futile analysis in the quantification of the double time integrals

\int_{0}^{-\infty} \int_{+\infty}^{0} E(t) E^{*}(t) dt dt

[Epsilon Pi]

The kind of beautiful integral I liked so much when I was a mathematical student, and it even has a negative time

Thanks!

EP
You might have just rescued me from falling into the trap of further futile analysis in the quantification of the double time integrals

\int_{0}^{-\infty} \int_{+\infty}^{0} E(t) E^{*}(t) dt dt