A Heuristic Derivation of a Planck-Heisenberg Energy Formula

We derive an equation of the energy of a particle equal to Planck h times the frequency plus/minus the Heisenberg uncertainty energy as based on our previously constructed “combined spacetime four-manifold.” At the same time, we provide more details to our spacetime geometry, in particular, that of the “wave universe.”


Introduction
This paper seeks to derive a single equation that explains the energy of a particle equal to that as accounted for by the Planck frequency formula plus that as accounted for by the Heisenberg uncertainty principle.The significance of the Planck formula lies in its therein contained Planck constant 6.62606957 29 10 • (CODATA 2010) (for a recent theoretical derivation of h, cf.(Brodsky & Hover, 2011]).Historically the formula, E=hv, arose from Planck's derivation of the energy density u(v)of the blackbody radiation in 1900 (cf.[Longair, 1984], pp. 201, 205-206 for an elaboration on the analysis in [Planck, 1901]), when Planck related E (Equation (6), ibid.)tothev in Wien's equation (cited by Planck as Equation (10), ibid.) through a joint expression of u (v) Here we can see that there is a discrepancy of the extra term of (-1) in the denominator in equating E with hv.[Einstein, 1905] gave another derivation of E=hv, where he made use of the quantity (in his Section 4), (as cast in [Longair, 1984], p.231); (1.1) here again, had Einstein used   Planck u  , he would not have been able to obtain the above expression of 1  T needed for the conclusion of E=hv.It must be stressed however that Einstein did note the readers about the limitations of E=hv.("Wirlegendiese Formelunseren Rechnungenzugrunde, behaltenaberim Sinne, dassunsere Resultatenurinnerhalbgewisser Grenzengelten."Ibid., Section 4.) Yet from our literature research the validity of this equation has never been examined (cf.e.g., [Zeidler, 2006], p. 141, for how firmly and universally this formula has been established), and additional derivations of the formula do not appear to exist (except for a recently proposed generalization by [Jou & Mongiovi, 2011]).
Coincidentally, we encountered a similar situation in our previous article (Light, 2011, APR), where we found an extra term j E  added onto   1 j h to account for j E ˆ, and this j E  originated from the term of 1 in our previousequation, We had conjectured that this j E  was the Heisenberg's uncertainty energy (see [Samuel, 1927] for an earlier inquiry), and indeed here we will present a proof to confirm our conjecture.
In the ensuing Section 2.1, we will derive our "Planck + Heisenberg" energy formula.We note that the background of this section will be the spacetime before the Big Bang, composed purely of electromagnetic waves in a universe referred to as   2 M in our previous article.Then in Section 2.2, we will essentially cast the recognized post-Big-Bang universein our model and derive Then in Section 2.3, we will prove a lemma that is actually involved in our energy formula.Finally in Section 3 we will conclude with some summary remarks.

A single electromagnetic wave  by Maxwell Equations in a universe
be given, and assume that relative to S there exists an electromagnetic field made up of varying with coordinate systems {byartificial spatial-temporal translations but otherwise Lorentz (geometrically) invariant with all inertial framesand with the resultantPoynting vector Here we note that the superscript of an asterisk as attached to the permittivity constant denotes its pre-Big Bang value and that such superscript representations will consistently be adopted in the sequel, with their relationships to the present laboratory values clarified in Section 2.2.
We now solve the equation to form a quotient space as referenced by a frame traveling with [0, λ], we have and thus a loss of information of Note that it is here in Equation (2.7) that we pass the frequency as referenced by frame S in Equation (2.0) to frame .
with the associated potential energy (2.9) implying in particular that the average potential energy . 2 Denoting the Lorentz factor by L  , we conduct a Lorentz transformation of the above equation: (2.14) Thus, form-invariance is observed, and  carries a fixed amount of energy of  .

/ 2 * s h
(Here we note in passing that the relativistic mass L m in the above Lorentz transformation (2.14) is valid only for slow moving inertial frames; for all inertial frames the relativistic mass should be m divided by the smaller of the two eigenvalues of the Lorentz transformation, i.e., a replacement of L  with the inverse of the smaller eigenvalue of the Lorentz transformation everywhere in (2.14) to result in the generally correct Lorentz transformation of the frequency ; cf.[Light, 2011, InTech].)Now the average kinetic energy = the heat energy in  + a heat converted mechanical energy that is lost for upholding the same constant oscillation frequency v, where This energy E in Equation (2.16), in accordance with the moving frame , is identical to the energy E in accordance with the frame S of the originating Maxwell Equations by virtue of sharing the same frequency as specially noted after Equation (2.7).That is, measuring frequency linearly along the x-axis by the multiple of the wave length λ is identical to wrapping λ around a circle on the (y,z)-plane and counting the number of rotations.Applying the (time x time) component of the metric tensor g in Einstein Field Equations, we proceed to examine the gravitational effect of E on the boundary of  in the universe   2 M , which has a large gravitational constant , we have simultaneously established: i.e., an agreement between electromagnetism and gravity.(Cf.Lemma below in Subsection 2.3; here we also note that chronologically the above Equation (2.17) should be placed in the next Subsection 2.2 of the post-Big-Bang era, when the ratio of the proper time in   2 M to the proper time in   1 M becomes imaginary.)

The transformation of E of  into
and is attracted to the center of B by gravity.We hypothesize that at the center, the infinite gravity transforms  into a particle-wave of energy (see [Feynman, 1963], [Light, 2011, APR]) distributed over (photon  , electromagnetic wave  ).In the laboratories of Here we draw the reader's special attention to the above formulas of conversion that convert any laboratory-measured quantities (as denoted by an overhead carat) to their true quantities in , we have a revised (cf.Equations (2.17), (2.18)) here (see [Light, 2011, APR]) the gravitational constant recognized in the laboratory is For the Poynting vector, we have .The wavefunction is then where Here, we note that as an inertial frame moves away from a light source with speed approaching c, the wave length of the light approaches infinity; therefore, a photon is to appear within any wave length  with probability 1.To be complete, the probability current density is then (2.26) We hypothesize that any representative point mass m in   in accordance with the probability densities   2 0 , 0 , , x t  of Equations (2.24) and (2.25).Furthermore, Equation (2.23) has an average of . This implies that if the photon γ carries an average energy density of (5/16)ε 0 , then it must travel (c second) in one second; however, if γ carries an average energy density of ε 0 , then it only travels [(5/16)c second] in one second but this must mean (5/16) second.That is, the time duration of γ in   , 1 (λ/c).As a result, our later derived Equation (2.32) of the Heisenberg uncertainty energy ΔE becomes an inequality: ΔE 1 2 ⁄ (ħ/Δt).
(2.27)This is a black hole in   (2.28) Thus by Einstein Relativity for all inertial frames   2 E must be calculated , a statement that will beformally established in the following Lemma.
)= the uncertainty energy E  asa remainder from the division algorithm and moreover (2.30) By the symmetry in simple harmonic motions, (2.32) (cf.e.g., [Zeidler, 2009, p. 477] for the statement that harmonic oscillators achieve the Heisenberg energy lower bound, and also, [Toscano, 2006] and [Jizba, 2003] for uncertainties on small scales and examinations of the lower bounds).

A lemma
We prove the following Lemma: (2.33) be given, where the subscript j indicates the reference to a particular frame.Define  Rouhani, 2000] and [Rovelli, 1991] for spacetimeparametrizations).Then we have (2.35) Define the average z(r) to be Thus, the nonlinear-terms related quantity NLT in the derivative is: , which is now derived as follows: Let an electromagnetic wave  relative to all reference frames be given.Assume that  has a radius mmeter r 1  (where the misspelling is intentional, to mean actually "a unit of space distance") and energy jjoulee E 1  (again, with the intentional misspelling to mean "a unit of energy").Denote the average energy density of  over As such, we have provided a treatment of the pervasive problem of (1/r) (cf. a recent study by [Gralla, 2011]).

Summary Remark
In this paper we have contributed an integrated formula of the energy of a particle as based on our previous model of a combined spacetime four-manifold   We thus present the above equation as a test of our model, e.g., substituting it into the Planck formula in Equation (1.0).Otherwise, we have provided additional details of the involved geometry of our model.In particular, we surmise that the ratio   e 2 /  in the above Lemma may have to do with the fine-structure constant (cf.e.g., [Bouchendira, 2011] and [Tomilin, 1999]).Overall we have augmented the quantum probability setup withour combined four-dimensional spacetime geometry, and by providing a clearer temporal-spatial structure of our world we hope to facilitate a future development of quantum mechanical engineering.

E
must reside in W Y  .Since the Lorentz factor L  can assume any value from 1 to infinity, there exists a reference frame relative to which the single point in a quotient space.As such, the radius of  for calculating  