A Matter of Semantics

Any scientific description of cause and effect is often subject to a degree of semantic preference of the day, although often predicated on how science has historically defined its fundamental units of measurement, e.g. length [m], time [t], ma ss [m] and charge [C]. Within the confines of its preferred semantics and units, science is then allowed to build mathematical models of space-time in which all physical action might be described.

While we have an intuitive sense of length and time, an earlier discussion of various Doppler effects may lead us to inquire further into the fundamental nature of these units, especially within the scope of a WSE model. So while mass is typically described as a property of a single particle and a fundamental unit of measure, a WSE model might seek to reverse the dependency of energy on mass, e.g.

[1]     

In the first highly speculative form shown in [1], potential energy might ultimately be described as resulting from a displacement [A] of space from its ground state quantified in terms of an elastic constant [k]. We might also question whether the idea of mass [m] can ever be directly equated to energy, as a scalar quantity, if it implicitly suggests a required volume [V], i.e. mass infers an energy-density.

Note: A WSE model might suggest that all energy is a manifestation of potential energy, where the concept of kinetic energy would result from the potential energy confined within a specified volume [V] of space changing its position [x] with time [t], such that it would have a velocity [v=x/t]. However, it is also recognised that the manifestation of potential energy might also be described differently in terms of larger scaled effects, e.g. gravitation or electric charge.

At this point, we should extend the discussion of energy to include two accepted forms of potential energy, while possibly inquiring further into the nature of two constants, e.g. [G,K] required to balance the equations in [2] in terms of basic units of measurement introduced above.

[2]     

Note: While the ratio of K/G suggests a figure in the order of 1020, a more comparative ratio of the electric force [FE] and gravitation [FG] between a proton and electron would be in the order 1040. We might reasonably assume that this huge difference in the electric and gravitational force must be attributable to the potential energy that underpin the manifestation of these forces at a given distance based on the same force inverse square law.

Like mass [m], the concept of charge is also often attributed to a single particle, e.g. electron, where the unit of measure is the Coulomb [C] in the SI system. As shown in [2], both potential energy equations are very similar in form and both require a constant, i.e. [G] and [K]. However, the units used to defined [G,K] do not really seem to impart any physical meaning, such that their function appears primarily to resolve the units of the overall equations back into energy, as required by the SI system.

Note: In [2], the definition of both [G] and [K] are both dependent on mass [kg], which are assumed subject to relativistic effects dependent on the frame of reference in which the mass is measured. Likewise, given that the last discussion has questioned the division of total energy in terms of potential and kinetic energy, on which it is assumed mass is dependent, we might also have to question how such constants fit into a WSE model.

We might also inquire as to whether the idea of gravity or charge is best described as an attribute of a single ‘ particle ’ or rather as a field of potential energy between two concentrations of energy-density in space. Of course, there is also the mechanism of cause and effect by which the potential energy defined in [2] propagates between the distance [r]. For if either of masses [m1,m2] or charges [q1,q2] change in position, i.e. causing the radial distance [r] to change, then there will be a corresponding change in the potential energy. However, this change cannot take place immediately, if the propagation of gravitational or electric potential is also constrained by a finite velocity [c].

What might this tell us about the fundamental nature of cause and effect?

If special relativity does not allow any particle with mass [m] to propagate with velocity [c], irrespective of the idea of it carrying charge, we might reasonably assume that some other mechanism is required to propagate the implied change in scalar potential energy between two points in space now separated by distance [r+∆r]. As stated on many occasions, the only known mechanism that might explain this effect is a wave mechanism, which can propagate energy at velocity [c] without any direct reference to mass [m]. Of course, this does not necessarily imply that the energy is propagated through the media of space, as accepted science appears to prefer the semantics of an ‘undulation’ or ‘excitation’ in a gravitational or electric field or possibly the abstracted semantics of some form of quantum field.

Note: Beyond the generalisation of classical physics, it might be accepted that the propagation of potential energy, of the forms described in [2], cannot be explained using particles with mass. We might also consider the idea that kinetic energy is simply a localisation of potential energy in some given volume [V], i.e. an energy-density, which can move in space as a function of time with a velocity [v] below [c]. If so, the debate may not be about the idea of the wave mechanism itself, but rather what these waves propagate through and how.

In a previous discussion, the idea of a simple harmonic oscillator was used to model a point in space, by which we might explain how energy is propagated. However, this model was then qualified in terms of quantum mechanics, which introduced the issue of the quantization of energy as defined by the Planck constant [h], which is historically linked to the problem of blackbody radiation . However, the oscillator motion was initially based on the mathematical description of a sine wave and the classical interchange of potential and kinetic energy, as described by the following equation.

[3]     

In the subsequent development of the quantum model, the energy of the oscillator was eventually expressed in terms of the Planck energy equation [E=hf], which we might attempt to reconcile in [4], where every expression must be resolved in terms of the fundamental units, i.e. mass [kg], length [m] and time [t].

[4]     

The form of [4] ignores numeric constants, such that we can focus on how different variables might be used to quantify the potential energy [Ep] in different ways. The first form infers a displacement [A] of the oscillator as a function of an elastic constant [k], while the second form has transposed the value of [k] into a conceptual mass [m] with  angular velocity [ ω]. The third form simply converts angular velocity [ω] back into a linear frequency [f] of oscillation, which is finally equated to the Planck energy equation [E=hf]. As such, we might possibly clarify the construct of units that define the Planck constant [h].

[5]     

Without the breakdown of the detail shown in [4] and [5], it might not be immediately obvious as to why the Planck energy equation [E=hf] makes no reference to the oscillation amplitude [A] or why it is only proportional to frequency [f], not [f2]. However, in the specific case of the oscillator model, [4] and [5] show that the amplitude [A2] and one of the frequency [f] terms has been subsumed into the definition of [h]. Again, like the constants [G,K] in [2], the Planck constant [h] might simply be described as a requirement to resolve the equation, both in terms of the fundamental units and the preferred semantics of the accepted scientific model.

OK, but what is the point of this exercise?

Well, in the context of a WSE model, it might be highlighted that the wave nature of energy did not necessarily disappear in the formulation of [E=hf], but has simply been masked within the Planck constant [h]. Of course, this is not to say that there is not an important rationale behind the constant [h] linked to blackbody radiation that needs to be taken into consideration. For it was Planck’s work on the blackbody radiation problem in 1900 that led to the idea that energy is quantized and the subsequent development of quantum mechanics.

Note: Today, the semantic scope of quantization is now described in many forms, although it might be rationalised in terms of the first and second quantizations. However, the definition of the semantics used in most of these descriptions might be better described as mathematical abstraction, where the idea of physical cause and effect is often difficult to ascertain. Therefore, the next section of the discussion attempts to consider how the quantization of energy might be physically explained by a WSE model.

As indicated, the idea of energy quantization is historically linked to the problem of blackbody radiation, which was a ‘hot’ topic at the end of the 19th century. The semantics of a ‘blackbody radiator’ is used to describe a material body that can absorb and emit electromagnetic (EM) energy. While the scope of this discussion will try to avoid too much historic and mathematical details that led Planck to the development of an equation that first introduced the Planck constant [h], it is often considered the foundation stone that underpins quantum mechanics. Therefore, some retelling of the salient details is possibly an important line of inquiry to investigate and starts with the concept that if you heat up a blackbody, it will radiate energy with a total power per unit area proportional to the fourth power of temperature in degrees Kelvin, which had been defined in terms of the Stefan-Boltzmann law established in 1879.

[6]     

However, it was recognised that the EM energy emitted from a blackbody correlated to the wavelength [λ] of the energy. As such, the radiant power [P] was experimentally determined as a function of wavelength, where the total power per unit area might be described as the integrated value over all wavelengths. Based on experimental data, the blackbody spectrum curve was plotted for different temperatures, where each curve has the characteristic shape as shown in the diagram right for different temperatures in degrees Kelvin. However, the diagram also shows that more power [P] is emitted as the temperature [T] of the blackbody increases, while the peak of the spectrum shifts to the left, i.e. shorter wavelengths, higher frequency. However, explaining this diagram has to be initially understood in the timeframe of the late 19th century, when physicists had only acquired a general understanding of the mechanisms involved , i.e. heat was assumed to cause the atoms of a solid to vibrate , although atoms were only vaguely understood as a structure involving electric charge.

Note: At this time, Planck’s scientific worldview was anchored in a definition of the second law of thermodynamics that led him to doubt the existence of atoms. However, a new interpretation of the second law had been forwarded by Boltzmann suggesting that the law was only statistically valid. It should also be recognised that J.J.Thompson only discovered the existence of the electron in 1897, while science would have to wait until 1911 for Rutherford to show that an atom must contain a central nucleus. Some of this history context is necessary in order to assess the limitations of Planck’s understanding, especially when connected to the subsequent development of quantum mechanics.

Based on earlier experiments, Maxwell’s theory of electromagnetic radiation linked to an oscillating charge, e.g. an electron, had been generally established and shown to propagate at the speed of light [c]. As such, it was logical to assume that the observed power spectrum of emitted energy was linked to oscillating charges within the material of the blackbody. Likewise, a similar process would occur in reverse, where EM radiation in the form of heat would be absorbed, which caused the charges to oscillate, which maintained the emission of EM radiation being observed. In 1893, Wilhem Wien had formulated a relationship between the wavelength of the peak of the spectrum and its temperature.

[7]     

By 1896, Wien had also formulated a possible solution to the observed spectrum based on thermodynamic principles with the assumption that the gas molecules obeyed the Maxwell-Boltzmann speed distribution for atoms in a gas, where the energy in a given wavelength interval [dλ[ was given by:

[8]     

Wien then extended [8] by using the Maxwell-Boltzmann distribution law for the speed of atoms in a gas, which he substituted back into [8]

[9]     

At this stage, [A] and [a] were just constants to be determined by experimental evaluation, while the form of [9] might also be transposed in terms of frequency [f] using the non-dispersive relationship [c=fλ], while noting that [dλ] has also to be transposed into [df] as follows:

[10]   

Substituting [10] back into [9] we get:

[11]   

In [11], the revised constants [A’,a’] have subsumed the value of the constant [c]. However, while Wien’s equation was a very close match to the observed blackbody spectrum, it was shown to deviate on the long-wavelength side of the peak of the curve. While we might simply introduce Planck’s equation, as normally presented today, it may be informative to show the original form of Planck’s law published in 1900, which might be more easily compared to Wien’s solution.

[12]   

It is probably fair to say that Planck had essentially worked back from the required solution demanded by observation. For in many respects Planck initially had no explanation as to why his equation worked and possibly more telling there is no appearance of the Planck constant [h] in the initial form of [12]. However, the final revised form of the equations in [12] are now normally presented in terms of the radiance [R] per wavelength interval [dλ] or frequency interval [df], where radiance [R] might be defined as the energy-density associated with wavelength or frequency.

[13]             

So, in the form of [13], we see the introduction of the Planck constant [h], which Planck calculated based on the available experimental data, i.e. the spectrum curves, plus the insertion of Boltzmann’s constant [kB].

OK, so how did Planck eventually explain the revisions in [13]?

Now, at this point, many of the explanations appear to be written with a certain degree of deference to Planck’s being cited as the ‘ father of quantum mechanics ’, although this can hardly have been the case in 1900. For, at this time, Planck had only just started to accept the general notion of an atom without any knowledge of its sub-structure consisting of positively charged protons surrounded by negative charged electron orbitals.

Note: In 1900, Lord Rayleigh was also looking at the problem of blackbody radiation, but assumed the radiation might be a superposition of standing waves confined within an enclosure, where the electric wave component of the EM wave had to be zero at the walls of the cavity. In this model, only wavelengths that were small enough to fit within the enclosure would be allowed. Although this would still allow an almost infinite number of shorter wavelengths, the idea of ‘equipartition’ required any system in thermal equilibrium to assign equal energy to each degree of freedom, i.e. xyz spatial dimensions. Of course, we might question this idea of the ground that standing wave do not propagate energy.

Planck knew, based on classical electromagnetic theory, that an oscillating electron would radiate EM energy, such that when the cavity was in thermal equilibrium, charged electrons in the walls of the cavity would oscillate and become the source of the observed radiation. Planck therefore decided to try to model the electrons as simple harmonic oscillators with different elasticity [k] in order to produce the different frequencies observed in the spectrum. In classical physics, the energy of a harmonic oscillator depends on both the square of the amplitude [A2] and the frequency [f2], where the potential energy of the oscillator might be attributed to the temperature [T]. Once in motion, the oscillator could emit EM radiation into the cavity and absorb it from the cavity. However, at this point, it is stated that Planck could not derive his formula using the assumption of classical physics based on the oscillator having amplitude [A2] and the frequency [f2], but rather a discrete energy defined by the Planck energy equation [E=hf] and in so doing signalled in the era of quantum mechanics without necessarily realising the implications. However, we might want to inquire further into the actual reality of these oscillators.

What did the oscillators in Planck’s model actually represent?

These oscillators were atoms in the blackbody that we now know have a discrete electron orbital structure, which Planck knew nothing about in 1900. So while Planck’s equation are known to be empirically accurate, his oscillator model might not necessarily provide a physical description of energy quantization.

OK, so how else might energy quantization be explained?

This question will be left open for now as this section of discussions has only been following various lines of inquiry, primarily anchored in the semantics of classical physics. However, at the beginning of the 20th century, new ideas were starting to emerge that would question the fundamental nature of all physical structure.