**4. Prime physis eta and the derivation of Basic Law** [Ragazas 2010d]

In our derivation of *Planck's Formula* the quantity played a prominent role. In this derivation is the *time-integral of energy*. We consider this quantity as *prime physis,* and define in terms of it other physical quantities. And thus mathematically derive Basic Law. Planck's constant *h* is such a quantity , measured in units of *energy-time*. But whereas *h* is a constant, is a variable in our formulation.

*Definitions: For fixed* **x0** ,*t*<sup>0</sup> *and along the x-axis for simplicity,* 

*Prime physis: = eta (energy-time-action)* 

$$\text{Energy:}\quad \mathbf{E} = \frac{\partial \eta}{\partial t} \tag{7}$$

$$\text{Momentum:} \quad p\_x = \frac{\partial \eta}{\partial \mathbf{x}}\tag{8}$$

$$\text{Force:} \quad F\_x = \frac{\hat{\partial}^2 \eta}{\hat{\partial} \hat{x} \hat{\partial} t} \tag{9}$$

Note that the quantity *eta* is undefined. But it can be thought as '*e*nergy-*t*ime-*a*ction' in units of *energy-time*. *Eta* is both *action* as well as *accumulation of energy*. We make only the following assumption about .

*Identity of Eta Principle: For the same physical process, the quantity is one and the same.*  Note: This *Principle* is somewhat analogous to a physical system being described by the *wave function*. Hayrani Öz has also used originally and consequentially similar ideas in [Öz 2002, 2005, 2008, 2010].

#### **4.1 Mathematical derivation of Basic Law**

Using the above definitions, and known mathematical theorems, we are able to derive the following Basic Law of Physics:


, , *<sup>x</sup> p E x t* . Since all gradient vector fields are *conservative,* we have the

*Conservation of Energy and Momentum*.

 *Newton's Second law of Motion.* The second Law of motion states that *F ma* . From definition (9) above we have,

$$F = \frac{\partial^2 \eta}{\partial \mathbf{x} \partial t} = \frac{\partial^2 \eta}{\partial t \partial \mathbf{x}} = \frac{\partial p\_x}{\partial t} = \frac{\partial}{\partial t} (m\upsilon) = ma\ \text{ since } p\_x = mv\ \text{ .} $$

define in terms of it other physical quantities. And thus mathematically derive Basic Law.

*= eta (energy-time-action)* 

*t* 

2 *Fx x t* 

Note that the quantity *eta* is undefined. But it can be thought as '*e*nergy-*t*ime-*a*ction' in units of *energy-time*. *Eta* is both *action* as well as *accumulation of energy*. We make only the

Note: This *Principle* is somewhat analogous to a physical system being described by the *wave function*. Hayrani Öz has also used originally and consequentially similar ideas in [Öz 2002,

Using the above definitions, and known mathematical theorems, we are able to derive the

, is a mathematical truism **(Section 3.0)**

*Newton's Second law of Motion.* The second Law of motion states that *F ma* . From

. Since all gradient vector fields are *conservative,* we have the

, since *<sup>x</sup> <sup>p</sup> mv* .

*Conservation of Energy and Momentum.* The gradient of

*<sup>x</sup> <sup>p</sup> <sup>F</sup> mv ma xt tx t t* 

*x* 

played a prominent role. In this

as *prime physis,* and

(7)

(8)

(9)

 *is one and the same.* 

**x**,*t* is

**(Section 3.0)**

, measured in units of *energy-time*. But whereas *h* is

**4. Prime physis eta and the derivation of Basic Law** [Ragazas 2010d]

is the *time-integral of energy*. We consider this quantity

In our derivation of *Planck's Formula* the quantity

 *Energy: E*

 *Momentum: <sup>x</sup> p*

 . *Identity of Eta Principle: For the same physical process, the quantity* 

 *Force:* 

**4.1 Mathematical derivation of Basic Law** 

*<sup>h</sup> <sup>E</sup> e* 

, , *<sup>x</sup> p E*

definition (9) above we have,

*Conservation of Energy and Momentum*.

2 2

 

The Quantization of Energy Hypothesis, *E nh*

following assumption about

following Basic Law of Physics:

Planck's Law, <sup>0</sup> <sup>1</sup> *h kT*

*x t* 

 

2005, 2008, 2010].

 is a variable in our formulation. *Definitions: For fixed* **x0** ,*t*<sup>0</sup> *and along the x-axis for simplicity,* 

*Prime physis:* 

derivation

a constant,

Planck's constant *h* is such a quantity

 *Energy-momentum Equivalence.* From the definition of energy *E t* and of momentum

$$p\_x = \frac{\partial \eta}{\partial x} \text{ we have that, } \eta = \int\_0^t E(u) du \text{ and } \quad \eta = \int\_{x\_0}^x p\_x(u) du.$$
 
$$\text{Using the } I \text{-} Identit \text{ of } \text{Eta } \text{Principle. The } \text{quantity } \eta \text{ in } \text{th}$$

Using the *Identity of Eta Principle*, the quantity in these is one and the same. Therefore, 0 0 () () *t x x t x E u du <sup>p</sup> u du* . Differentiating with respect to *t*, we obtain,

$$E(t) = p\_x(\mathbf{x}) \cdot \frac{d\mathbf{x}}{dt} \quad \text{or more simply, } E = p\_x \upsilon \quad \text{(energy-momentum equivalence)}$$


<sup>1</sup> *Et h* . Or equivalently, for 1 *av E E* , we again have *Et h* , since

*h* is the *minimal eta that can be manifested.* Note that since *av <sup>E</sup> <sup>t</sup> E* (*Characteristic 5)*, we

have 1 *av E E if and only if* <sup>1</sup> *<sup>t</sup>* . Since *E kT av* and entropy is defined as *<sup>E</sup> <sup>S</sup> T* , we have that


Starting with our *Planck's Law* formulation, 0 <sup>1</sup> *E Eav <sup>E</sup> <sup>E</sup> e* in (3) above and re-writing this equivalently we have, 0 0 <sup>1</sup> *E Eav E E e E E* and so, 0 ln *av E E E E* . Using the definition of

The Thermodynamics *in* Planck's Law 703

Thus if *temperature* is twice as high, the accumulation of energy will be twice as fast, and visa-versa. This *characterization of temperature* agrees well with our physical sense of temperature. It is also in agreement with *temperature* as being the average kinetic energy of

. *Conversely*, for a given <sup>T</sup> as characterized above, we will have <sup>1</sup>

 *, we have* <sup>1</sup> 

Planck's constant *h* is a fundamental universal constant of Physics. And although we can experimentally determine its value to great precision, the reason for its existence and what it really means is still a mystery. Quantum Mechanics has adapted it in its mathematical formalism. But QM does not explain the meaning of *h* or prove why it must exist. Why does the Universe need *h* and *energy quanta?* Why does the mathematical formalism of QM so accurately reflect physical phenomena and predict these with great precision? Ask any physicists and uniformly the answer is "that's how the Universe works". The units of *h* are in *energy-time* and the conventional interpretation of *h* is as a *quantum of action*. We interpret *h* as *the minimal accumulation of energy* that can be manifested. Certainly the units of *h* agree with such interpretation. Based on our results above we provide an explanation for the existence of Planck's constant -- what it means and how it comes about. We show that the existence of *Planck's constant* is not necessary for the Universe to exist but rather *h* exists by

Using *eta* we defined in **Section 5.0** above the *temperature of radiation* as being proportional to the ratio of *eta/time*. To obtain a *temperature scale*, however, we need to fix *eta* as a standard for measurement. We show below that the fixed *eta* that determines the Kelvin *temperature* 

In The Interaction of Measurement [Ragazas 2010h] we argue that direct measurement of a physical quantity *E t*( ) involves a physical interaction between the *source* and the *sensor*. For measurement to occur an interval of time *t* must have lapsed and an incremental amount *E* of the quantity will be absorbed by the *sensor*. This happens when there is an *equilibrium* between the *source* and the *sensor*. At *equilibrium*, the 'average quantity *Eav* from the source' will equal to the 'average quantity *Eav* at the sensor'. *Nothing in our observable World can exist without time, when the entity 'is' in equilibrium with its environment and its 'presence' can be observed and measured.* Furthermore as we showed above in **Section 3.0** the *interaction of* 

**6. The meaning and existence of Planck's constant h** [Ragazas 2010c]

Mathematical necessity and inner consistency of our system of measurements.

 

we get

<sup>T</sup> , which will be unique up to an arbitrary scalar

<sup>T</sup> *, where* 

 

T T *, for some fixed* 

   *is some arbitrary scalar* 

 *and arbitrary scalar* 

1

 

T T . We have the

*and arbitrary scalar constant* 

T , where

the motion of molecules.

, we can define <sup>1</sup>

is a proportionality constant. By setting

*constant. Conversely, given* <sup>T</sup> *we have* <sup>1</sup>

following *temperature-eta* correspondence:

*Temperature-eta Correspondence: Given*

 

*. Any temperature scale. therefore, will have some fixed* 

 *associated with it.* 

*scale* is Planck's constant *h*.

*measurement* is described by *Planck's Formula.* 

For fixed

constant

*constant* 

thermodynamic entropy we get *<sup>E</sup> <sup>S</sup> T* <sup>=</sup> *av <sup>E</sup> <sup>k</sup> E* = 0 ln *<sup>E</sup> <sup>k</sup> E* . If ( )*t* represents the number of microstates of the system at time t, then *Et A t* () () , for some constant *A* . Thus, we get *Boltzmann's Entropy Equation*, *S k* ln .

*Conversely*, starting with *Boltzmann's Entropy Equation*, 0 *S k* ln <sup>0</sup> ln *<sup>E</sup> <sup>k</sup> E* .

Since *<sup>E</sup> <sup>S</sup> T* we can rewrite this equivalently as 0 ln *av E E E E* and so <sup>1</sup> *E Eav E E e* . From this we have, *Planck's Law,* <sup>0</sup> <sup>1</sup> *E Eav <sup>E</sup> <sup>E</sup>* in (3) above.


*U W <sup>S</sup> T T* . All the terms in this equation are various entropy quantities. The

fundamental thermodynamic relation can be interpreted thus as saying, *"the total change of entropy of a system equals the sum of the change in the internal (unmanifested) plus the change in the external (manifested) entropy of the system"*. Considering the *entropy-time relationship* above, this can be rephrased more intuitively as saying *"the total lapsed time for a physical process equals the time for the 'accumulation of energy' plus the time for the 'manifestation of energy' for the process"*. This relationship along with *The Second Law of Thermodynamics* establish a *duration of time* over which there is *accumulation of energy before manifestation of energy* – one of our main results in this Chapter and a premise to our explanation of the double-slit experiment. [Ragazas 2010j]
