**1. Introduction**

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Quantum Physics has its historical beginnings with Planck's derivation of his formula for blackbody radiation, more than one hundred years ago. In his derivation, Planck used what latter became known as *energy quanta*. In spite of the best efforts at the time and for decades later, a more *continuous approach* to derive this formula had not been found. Along with Einstein's *Photon Hypothesis*, the *Quantization of Energy Hypothesis* thus became the foundations for much of the Physics that followed. This *physical view* has shaped our understanding of the Universe and has resulted in mathematical certainties that are counterintuitive and contrary to our experience.

Physics provides *mathematical models* that seek to describe *what is* the Universe. We believe mathematical models of *what is* -- as with past metaphysical attempts -- are a never ending search getting us deeper and deeper into the 'rabbit's hole' [Frank 2010]. We show in this Chapter that a *quantum-view* of the Universe is not necessary. We argue that *a world without quanta* is not only possible, but desirable. We do not argue, however, with the mathematical formalism of Physics -- just the *physical view* attached to this.

We will present in this Chapter a mathematical derivation of *Planck's Law* that uses simple continuous processes, without needing *energy quanta* and *discrete statistics*. This *Law* is not true by Nature, but by Math. In our view, *Planck's Law* becomes a *Rosetta Stone* that enables us to translate known physics into simple and sensible formulations. To this end the quantity *eta* we introduce is fundamental. This is the time integral of energy that is used in our mathematical derivation of *Planck's Law*. In terms of this *prime physis* quantity *eta* (acronym for *e*nergy-*t*ime-*a*ction), we are able to define such physical quantities as energy, force, momentum, temperature and entropy. Planck's constant *h* (in units of energy-time) is such a quantity *eta*. Whereas currently *h* is thought as *action,* in our derivation of *Planck's Law* it is more naturally viewed as *accumulation of energy*. And while *h* is a constant, the quantity *eta* that appears in our formulation is a variable. Starting with *eta*, Basic Law can be mathematically derived and not be physically posited.

Is the Universe *continuous* or *discrete*? In my humble opinion this is a false dichotomy. It presents us with an impossible choice between two absolute views. And as it is always the case, making one side *absolute* leads to endless fabrications denying the opposite side. The Universe is neither *continuous* nor *discrete* because the Universe is both *continuous* and *discrete*. Our *view* of the Universe is *not* the Universe. The Universe simply *is*. In The Interaction of

<sup>1</sup> cragaza@lawrenceville.org

The Thermodynamics *in* Planck's Law 697

*e* 

> *av <sup>E</sup> r t*

> > *Pr*

*E* 

are constants, often a rate of growth or frequency

*Pr E s <sup>e</sup>*

*Pr E s <sup>e</sup>*

*<sup>E</sup> E s*

*<sup>e</sup> is invariant with t* 

*Pr E s <sup>e</sup>*

<sup>T</sup> (1)

(2)

is a scalar constant

 

*Dx* indicates differentiation with respect to *x*

T T where

<sup>1</sup>

*r* ,

Note that *Eav*

section as,

used in this Chapter.

T

where *E*0 is the intensity of radiation,

*Characterization 2:*<sup>0</sup> ( ) *rt Et Ee if and only if* ( ) ( ) <sup>1</sup> *rt s*

*Characterization 2a:*<sup>0</sup> ( ) *rt Et Ee if and only if* ( ) <sup>1</sup> *Pr Eav*

*Characterization 3:*<sup>0</sup> ( ) *rt Et Ee if and only if* ( ) <sup>1</sup> *E Eav*

*Theorem 1a:* <sup>0</sup> ( ) *rt Et Ee if and only if* <sup>1</sup> *Pr Eav*

<sup>0</sup> ( ) *<sup>t</sup> Et Ee*

*Planck's Law* for blackbody radiation states that, 0 <sup>1</sup> *h kT*

*Result I: A 'Planck-like' characterization of simple exponential functions* 

<sup>0</sup> ( ) *<sup>t</sup> Et Ee*

Using *Theorem 2* above we can drop the condition that 0 ( ) *<sup>t</sup> Et Ee*

*Result II: A 'Planck-like' limit of any integrable function* 

*Theorem 2: For any integrable function E(t),* lim ( ) <sup>1</sup> *r t t s*

. We can re-write *Characterization 2a* above as,

For any *integrable* function *E t*( ), 0 <sup>0</sup>

We list below for reference some helpful variations of these mathematical results that will be

*if and only if* <sup>0</sup> <sup>1</sup>

temperature of the blackbody, while *h* is Planck's constant and *k* is Boltzmann's constant. [Planck 1901, *Eqn 11*]. Clearly (1) and (2) have the exact same mathematical form, including the type of quantities that appear in each of these equations. We state the main results of this

 *if and only if* <sup>0</sup> <sup>1</sup>

*E*

*E*

*e* 

lim

 <sup>T</sup>

<sup>T</sup>

> and get,

*<sup>t</sup>* 1

*e* 

*E*

*e* 

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

is the frequency of radiation and *T* is the (Kelvin)

*Characterization 1:* 0 ( ) *rt Et Ee if and only if E Pr*

*Characterization 4:* <sup>0</sup> ( ) *rt Et Ee if and only if* 

**2.1 'Planck-like' characterizations** [Ragazas 2010a]

Measurement [Ragazas, 2010h] we argue with mathematical certainty that we cannot know through direct measurements what a physical quantity *E(t)* is as a function of time.

Since we are limited by our measurements of *'what is'*, we should consider these as the beginning and end of our knowledge of *'what is'*. Everything else is just *'theory'*. There is nothing real about theory! As the ancient Greeks knew and as the very word 'theory' implies. In Planck's Law is an Exact Mathematical Identity [Ragazas 2010f] we show *Planck's Law* is a mathematical truism that describes the *interaction of measurement*. We show that *Planck's Formula* can be *continuously* derived. But also we are able to explain *discrete* 'energy quanta'. In our view, *energy propagates continuously but interacts discretely*. Before there is *discrete manifestation* we argue there is *continuous accumulation* of energy. And this is based on the *interaction of measurement*.

Mathematics is a tool. It is a language of objective reasoning. But mathematical 'truths' are always 'conditional'. They depend on our presuppositions and our premises. They also depend, in my opinion, on the mental images we use to think. We phrase our explanations the same as we frame our experiments. In the single electron emission double-slit experiment, for example, it is assumed that the electron emitted at the source is the same electron detected at the screen. Our explanation of this experiment considers that these two electrons may be separate events. Not directly connected by some trajectory from *source* to *sensor*. [Ragazas 2010j]

We can have beautiful mathematics based on *any* view of the Universe we have. Consider the Ptolemy with their epicycles! But if the view leads to physical explanations which are counter-intuitive and defy common sense, or become too abstract and too removed from life and so not support life, than we must not confuse mathematical deductions with *physical realism*. Rather, we should change our view! And just as we can write bad literature using good English, we can also write bad physics using good math. In either case we do not fault the language for the story. We can't fault Math for the failings of Physics.

The failure of Modern Physics, in my humble opinion, is in not providing us with *physical explanations* that make sense; a *physical view* that is consistent with our experiences. A *view* that will not put us at odds with ourselves, with our understanding of our world and our lives. Math may not be adequate. Sense may be a better guide.
