# Networks .. and those who made them possible

#### In an Euler Cycle, the Path ends where the Path began. In a Hamilton Cycle, the Path, likewise, ends where it began, such that the initial and final vertices are identical (the only allowable repeated vertices in a Hamilton Cycle or Path)

##### Notice that in Euler Paths we are looking at crossing (following) edges. In Hamilton Paths it’s vertices we are looking at.  ##### Light and Radio Waves, Photons, Black Bodies, Planck’s Experiment, Einstein’s ‘thought experiments’, Electrons, Quantum Mechanics, Phonons, Mass, Energy, Bosons and Gravitation
###### E & H are examples of  dependent “Vector Fields”. At every point in a vector field there exists a force on a body with magnitude and direction ie a vector. Another example of a vector field is Earth’s familiar Gravitational Force Field (approximately pointing to the centre of our planet). Fields of temperature, mass or energy values are examples, on the other hand, of “Scalar” fields. Multi-dimensional “Tensor fields” exist in physics (eg Mechanical Stress in solids and viscous fluids, and space-time curvature Tensors in General Relativity).

.. Just for clarity, Special Relativity (1905) is concerned with reconciling physics to the space-time transformation ‘metric’ revealed in Hertz’s Experiment’s verification of the existence of radio waves, but predictable from Maxwell’s equations before him. A ‘metric’ here refers to finding a formulaic way we may consistently model relative velocities, and other physical properties, between 2 observers traveling separately in space-time, in “light” of the fact that there is really nowhere to be taken as a zero velocity point and that we may only consistently measure velocities relative to the local velocity of light (radio waves equally). Einstein discovered the metric inherent in Maxwell’s equations, relating to electromagnetism, (building on earlier work completed by Hendrik Lorentz) and argued that there can be only one metric in this universe, and that therefore Newton, a man who is rumoured to have walked out of his first opera, was wrong (in 1687) to assume a Euclidean ‘orthogonal’ space for physical reality, which naturally seemed to separate time from space. The metric of Maxwell’s equations revealed skewed axes in 4 dimensions where the upper limit to universal velocities is the speed of light. By progressing from this point, Einstein was able to unite our concepts of Electricity and Magnetism as well as uniting Energy and Mass.  Our concepts of Energy and Mass were unified by Einstein applying the Conservation of Energy Principle followed by the Conservation of Momentum Principle (in special relativistic formats) to a collision (in a thought-experiment) where a so-called Black Body absorbs a colliding photon fully. By equating the total Energy and Momentum prior to the collision with the total Energy and Momentum after, and utilising the Planck Relation for Energy of a photon E = hf, Albert was able to show with simple algebra and calculus that the total Energy of that system post collision E = mc2, where m is the mass of the black body (the photon has zero mass itself) and c is the velocity of light.

Planck’s experiment some few years after Hertz’s experiment, revealed to Einstein an interpretation of results stating light (and thus radio waves) is actually composed of light “particles” or quanta called photons. Albert deduced this publicly in a paper in 1904, giving birth to a field of research still continuing today, called quantum mechanics. As noted above, Planck’s 1904 experiment helped Einstein deduce the relationship between Energy and Mass of a body by giving him the concept of a “photon”, which he invented himself to account for the results of Max Planck’s experiment. Due to the incompatibility of the Classical Wave based theory of light with a particle based theory at the time, the effort to find the linking equations between a new ‘quantum mechanics’ and the classical theory (adequate until Planck’s Experiment) led scientists to a probabilistic theory which Einstein always disowned. Incidentally Erwin Schrödinger, one of the inventors of quantum mechanics, also believed that there exists a deterministic underlying continuous theory possible in physics. The possibility that an event could happen simultaneously in many spaces is required for the theory to work. “Strings”?

The answer in any case seems to lie in the success Schrӧdinger had in 1926, (and Werner Heisenberg at the same time), with an approach that replaced the (classical) value for total field Energy (E) with (quantum) hf in a single frequency (laser) light field’s (classical) theoretical “Work Function”. The basic experimental relation E = hf (Energy of a photon = its frequency f multiplied by Plank’s constant h) is reliable and verifiable by anyone who wants to repeat Planck’s Experiment. The result of Schrӧdinger’s substitution in the classical field equation gave only the quantum field equations for the particular case of a “geometrical optics” light field. Schrӧdinger had to uncover a Partial Differential Equation (which would be the General Quantum Mechanical Wave Equation) whose solution space allowed these “Quantum Mechanical-Geometrical Optics” Field Equations as solutions, even though they were at that stage particular to a generalised (massless and unbound) photon field (not to other particles, such as electrons). The solution had to approach the behaviour of the Classical Wave Equation, as the value of Planck’s Constant is made to approach zero. (This requirement is a way of mimicking the idea that Energy in a classical light field is taken as independent of frequency). The equations required also had to provide solutions which closely match observed experimental results for other particles when applied as theoretical models of those particles in experiments.

In 1926, Schrӧdinger, and at the same time, independently, Heisenberg, succeeded in finding slightly different versions of the same Equations. The results showed the required tendency towards classical behaviour (say, at more human-sized scales) as Planck’s constant was forced towards zero (in the theory).

The theoretical Schrӧdinger and Heisenberg treatments also matched actual results from practical energy absorption spectral experiments with Hydrogen, in a model of the Hydrogen Atom, with its single electron as a wave-particle, obeying the new Wave Equation, absorbing energy in stable quantum stages as predicted. This meant the Quantum Wave Equation applied to electrons as well as photons (theoretically).

Then, the wavelike nature of electron beams themselves was experimentally established when Electron diffraction, in fact, was observed (1927) by C.J. Davisson and L.H. Germer in New York and by G.P. Thomson in Aberdeen, Scotland, thus supporting an underlying principle of quantum mechanics, “Wave Particle Duality”.

The Quantum Mechanical Wave Equation also applies to Sound/Pressure/Stress waves in the limit, and thus there exist “Phonons” or stress/pressure particles, since the Principles of Energy Quantisation apply equally to sound/pressure/stress energy, including both types of seismic wave: the direct wave (or primary shock) and the shear waveform (or aftershock); similarly to the sonic boom and aftershock occurring with supersonic objects (where the shear wave also arrives last). These waves (all sound/pressure and stress) are carried in the final result by phonons in rays spreading out from the source(s) at the speed of sound in the medium. (Incidentally, heat energy is also, at the submicroscopic level, a quantised phenomenon, being stored and transferred in the form of phonons no different to sound/pressure/stress waves. It’s all about vibrating matter with phonons.) The full classical treatment of mechanical waves involves the three dimensional Stress Tensor T in a space and time continuum (Euclidean). A Tensor Field in space has 3 perpendicular “normal” or “principal” stress values at a point and 3 perpendicular “shear” stress values at the same point. There are thus 6 stress values per point in space in the Tensor Field. Shear corresponds to rotation or torsion (the aftershock) and normal refers to tensile or compressive forces (the primary wave). The elements of the stress tensor at each point in the spatial field vary (“vibrate”) in time and space. A “solution” to a particular Tensor Wave Field Differential Equation (the particular Classical Wave Equation in the medium or continuum) is required as a “function” specifying the 6 values of stress at every point in the spatial & temporal field. The treatment needed to involve and connect the molecular, atomic and sub-atomic levels (Quantum vibrations and phonons) to the higher level continuum mechanics treatment (classical stress waves – think dynamic structural or fluid loading and forcing) is more complicated.

In the early days of Quantum Mechanics, everything was done in pseudo-Euclidean Spaces (although involving “imaginary” numbers and “complex planes”), however Paul Dirac was influential in pushing back boundaries towards reconciling General Relativity with Quantum Mechanics. Albert’s Theory of General Relativity (1915) had gone further than his Special Theory, as Special Relativity still rested upon a ‘flat’ or ‘inertial’ (non-accelerating) space-time cosmology, whereas the General Theory concerned itself with further revelations, now about Gravitation: specifically, that it is linearly related to the local magnitude-and-direction-of-curvature (a vector, perpendicular in space-time, to the tangent hyperplane on the curved surface of our universe at the local point to be measured) of ‘our’ space-time inside a “hyper-volume” (possibly a multiverse) of a larger number of dimensions (larger than 4, but otherwise unspecified) which is outside our universe. Much of the reasoning around the nature of Gravity and Acceleration came down to the question as to why should a body’s inertial mass be identical to its gravitational mass? In Newton’s terms: why should the ‘m’ in F = ma (where ‘a’ represents a real temporal rate of change of velocity) be the same as the ‘m’ in F = mg (where g represents a potential, giving a body’s “weight” in a field of Gravity)? There is no doubt that the identical “m”s are reliable facts, so backtracking from truth to cause was what was called for. Another little human feat accomplished.

As a result of Einstein’s deliberations and reasoning, he was able to develop equations enabling him to accurately predict the amount by which the planet Mercury would appear earlier than astronomers expected (in a Euclidean Space with no Gravitational influence on the path of light rays) from behind the Sun due to the curvature of Space-Time – and the light rays in it – caused by the Sun. See Geodesics in General relativity.

This actual curvature of space-time is caused by the presence of matter (such as the earth or the sun or a pencil or a galaxy), and Einstein gave equations which accurately predict the behaviour of our solar system as well as real galaxies, contrary to Newton’s inconsistent predictions.  (Although, Einstein was never good with pencils .. they never weigh enough and they move too slowly – moreover it was Newton’s Mechanics that enabled the development of the fundamental Impulse/Momentum Equation of Rocketry (refer to our page Computers as Machines regarding “the girls” who programmed the first computer – not using software as it did not exist – and the solution they produced) and the now ubiquitous Finite Element Method of Stress and Strain analysis, for the Apollo rocketships, that took men to the moon successfully. In addition, any navigator since well before Newton would have been happy to plot the course to the moon given the available technology in 1969 – as it was done in Euclidean space, using regular timekeeping devices and astronomical maps based on observations little different to those of prior centuries). Incidentally there was a similar “zeitgeist” moment between Isaac Newton in England and Gottfried Leibniz in France (as for example between Schrӧdinger and Heisenberg), where both men appear to have invented the same ‘Calculus’ ideas at similar times, but developed them slightly differently. Actually Leibniz’ formulation lends itself more readily than Newton’s to Finite Element Analysis, and to many other areas of physics and engineering, as it employs generalised co-ordinates from the outset.

However, much has happened in physics since publication of the General Theory of Relativity in 1915 ..  starting with quantum mechanics, Schrӧdinger, Heisenberg, Dirac and Stephen Hawking’s life devoted to reaching past Einstein (predicting the existence of Black Holes, now confirmed, and even their “evaporation” with the return of matter and “information” to this universe!) .. go search .. and remember that although God may not play dice, people may be required to, in physics, because the human intellect needs a way to comprehend wave-particle duality, and many other probabilistic phenomena, such as the question of how is the particle, the Higgs boson, the particle thought to be responsible for conferring “mass” on some sub-atomic objects (such as electrons and quarks), related to the universal curvature-of-space-time tensor in a generally-relativistic quantum mechanics, or, how would a Higgs boson be related to Gravitons (the quantum particle, existence confirmed, associated with Gravitational disturbances or gravity waves), as emitted by the action of pulsars?