*Eigenfunctions of the Hamiltonian for Hydrogen with eigenvalues representing the sum of kinetic and potential energies, with Schrödinger’s equation as a smoothed version of the particle dynamics of a harmonic oscillator. *

This is continuation of the previous post How to Make Schrödinger’s Equation Physically Meaningful + Computable. Consider the basic case of the Hydrogen atom with one electron (normalized to unit mass and charge): continue reading…

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The Principle of Least Action (PLA) expressing

*Stationarity of the Action (the**integral in time of the Lagrangian),*

with the Lagrangian the difference between kinetic and potential energies, is cherished by physicists as a deep truth about physics: Tell me the Lagrangian and I will tell you the physics, because a dynamical system will (by reaction to local forces) evolve so as to keep the Action stationary as if led by an invisible hand steering the system towards a final cause of least action.

PLA is similar to the invisible hand of Adam Smith supposedly steering an economy towards a final cause of maximal effectivity or least action (maximal common happiness) by asking each member of the economy to seek to maximize individual profit (individual happiness). This is the essence of the capitalistic system. The idea is that a final cause of maximal effectivity can be reached without telling the members the meaning of the whole thing, just telling each one to seek to maximize his/her own individual profit (happiness).

Today the capitalistic system is shaking and nobody knows how to steer towards a final cause of maximal efficiency. So the PLA of economy seems to be rather empty of content. It may be that similarly the PLA of physics is void of real physics. In particular, the idea of a smallest quantum of action as a basis of quantum mechanics, may well be unphysical.

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*The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure. *(Preface to *Mécanique Analytique*)

The Principle of Least Action formulated by Lagrange in his monumental treatise Mecanique Analytique (1811) collecting 50 years work, is viewed to be the crown jewel of the Calculus of Newton and Leibniz as the mathematical basis of the scientific revolution:

*The equations of motion of a dynamical system are the same equations that express that the action as the integral over time of the difference of kinetic and potential energies, is stationary that is does not change under small variations.*

The basic idea goes back to Leibniz:

*In change of motion, the quantity of action takes on a Maximum or Minimum.*

And to Maupertis (1746):

*Whenever any action occurs in nature, the quantity of action employed in this change is the least possible.*

In mathematical terms, the Principle of Least Action expresses that the trajectory $u(t)$ followed by a dynamical system over a given time interval $I$ with time coordinate $t$, is determined by the condition of stationarity of the action:

- ,

where is kinetic energy and is potential energy of at time , and is an arbitrary perturbation of , combined with an initial condition. In the basic case of a harmonic oscillator;

- with ,
- stationarity is expressed as force balance as Newton’s 2nd law: .

The Principle of Least Action is viewed as a constructive way of deriving the equations of motion expressing force balance according to Newton’s 2nd law, in situations with specific choices of coordinates for which direct establishment of the equations is tricky.

From the success in this respect the Principle of Least Action has been elevated from mathematical trick to physical principle asking Nature to arrange itself so as to keep the action stationary, as if Nature could compare the action integral for different trajectories and choose the trajectory with least action towards a teleological final cause, while in fact Nature can only respond to forces as expressed in equations of motion.

But if Nature does not have the capability of evaluating and comparing action integrals, it can be misleading to think this way. In the worst case it leads to invention of physics without real meaning, which is acknowledged by Lagrange in the Preface to Mecanique Analytique.

The ultimate example is the very foundation of quantum physics as the pillar of modern physics based on a concept of **elementary** (smallest) *quantum of action * denoted by $h$ and named Planck’s constant with dimension $force \times time$. Physicists are trained to view the elementary quantum of action to represent a “quantization” of reality expressed as follows on Wikipedia:

*In physics, a quantum (plural: quanta) is the minimum amount of any physical entity involved in an interaction.**Behind this, one finds the fundamental notion that a physical property may be “quantized,” referred to as “the hypothesis of quantization“.This means that the magnitude can take on only certain discrete values.**A photon is a single quantum of light, and is referred to as a “light quantum”.*

In the quantum world light consists of a stream of discrete light quanta named photons. Although Einstein in his 1905 article on the photoelectric effect found it useful as a heuristic idea to speak about *light quanta, *he later changed mind:

*The quanta really are a hopeless mess.*(to Pauli)*All these fifty years of conscious brooding have brought me no nearer to the answer to the question, ‘What are light quanta?’ Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken.*(1954)

But nobody listened to Einstein and there we are today with an elementary quantum of action which is viewed as the basis of modern physics but has not physical reality. Schrödinger supported by Einstein said:

*There are no particles or quanta. All is waves.*

Connecting to the previous post, note that to compute a solution according the Principle of Least Action typically an iterative method based on relaxation of the equations of motion is used, which has a physical meaning as response to imbalance of forces. This shows the strong connection between computational mathematics as iterative time-stepping and analog physics as motion in time subject to forces, which can be seen as a mindless evolution towards a hidden final cause, as if directed by an invisible hand of a mind understanding the final cause.

*Bohr plotting the Copenhagen Interpretation of quantum mechanics together with Heisenberg and Pauli (left) and Bohr wondering what he did 30 years later (right).*

To view physics as a form of analog computation which can be simulated by digital computation offers resolutions of the following main unsolved problems of modern microscopic and classical macroscopic physics:

- Interaction between subject (experimental apparatus) and object under observation.
- Meaning of smallest quantum of action named Planck’s constant $h$.
- Contradiction between particle and wave qualities. Particle-wave duality.
- Meaning of the 2nd law of thermodynamics and direction of time.
- Meaning of Heisenberg’s Uncertainity Principle.
- Loss of cause-effect relation by resort of microscopic statistics.
- Statistical interpretation of Schrödinger’s multidimensional wave function.
- Meaning of Bohr’s Complementarity Principle.
- Meaning of Least Action Principle.

This view is explored on this blog and on Computational Blackbody Radiation suggesting the following answers to these basic problems:

- Observation by digital simulation is possible without interference with physical object.
- Planck’s constant $h$ can be viewed as a computational mesh size parameter.
- All is wave. There are no particles. No particle-wave duality.
- Dissipation as an effect of finite precision computation gives a 2nd law and direction of time.
- Uncertainty Principle as effect of finite precision computation.
- Statistics replaced by finite precision computation.
- Schrödinger’s wave equation as system in 3d without statistical interpretation.
- No contradiction between complementary properties. No need of Complementarity Principle.
- Least Action Principle as computational mathematical principle without physical reality.

The textbook physics harboring the unsolved problems is well summarized by Bohr:

*There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature…**Everything we call real is made of things that cannot be regarded as real. If quantum mechanics hasn’t profoundly shocked you, you haven’t understood it yet.**We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough.**We must be clear that when it comes to atoms, language can be used only as in poetry. The poet, too, is not nearly so concerned with describing facts as with creating images and establishing mental connections.*

John Bell’s Against Measurement is a direct attack onto the heart of quantum mechanics as expressed in the Copenhagen Interpretation according to Bohr:

*It is wrong to think that the task of physics is to find out how nature***is.**Physics concerns what we**can say**about nature…

Bell poses the following questions:

*What exactly qualifies some physical systems to play the role of “measurer”?**Was the wavefunction of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared?**Or did it have to wait a little longer, for some better qualified system…with a Ph D?*

Physicists of today have no answers, with far-reaching consequences for all of science: If there is no rationality and reality in physics as the most rational and real of all sciences, then there can be no rationality and reality anywhere…If real physics is not about what **is, **then real physics is irrational and irreal…and then…any bubble can inflate to any size…

The story is well described by 1969 Nobel Laureate Murray Gell-Mann:

*Niels Bohr brainwashed a whole generation of theorists into thinking that the job of interpreting quantum theory was done 50 years ago.*

But there is hope today, in digital simulation which offers observation without interference. Solving Schrödinger’s equation by computation gives information about physical states without touching the physics. It opens a road to bring physics back to the rationality of 19th century physics in the quantum nano-world of today…without quantum computing…

Planck introduced Planck’s constant in his proof of Planck’s law using statistics based on an ad hoc assumption that the “smallest packet of energy” or “quantum of action” of a wave of frequency is equal to . Later appeared in Schrödinger’s wave equation as a measure of the diameter of an atom. Planck’s constant thus has two meanings and the connection has remained a mystery, well supported by a statistical interpretation of the wave function as solution to the wave equation and with the photon as a mysterious particle of energy or .

To seek to resolve the mystery let us consider wave mechanics without reference to statistics presented in Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation, which offers a New View of blackbody radiation based on a generic wave model of a blackbody (as a web of atoms) with the following radiation spectrum of the principal form of Planck’s law:

- for ,
- for ,

where is temperature in Kelvin, is frequency and and are two parameters with representing a *high-frequency cut-off.** *Radiative equilibrium between two such blackbodies defined by parameters with the same high frequency cut-off, shows that where is a universal constant. The generic wave model thus effectively depends on one parameter, which we take to be as a representation of Planck’s constant. We here assume that Boltzmann’s constant as a measure of energy per degree Kelvin is normalized to 1.

We observe that the cut-off condition has the form

- .

which gives the parameter in the a wave model in computational form the meaning of a mesh parameter which can be connected to the diameter of the atoms forming the web supporting the wave model. We thus have only one meaning of Planck’s constant as a mesh size parameter.

In the New View based on wave mechanics there is no need to give a different meaning as a mysterious smallest “packet of energy or action” of a wave of frequency and there is no need to speak about a particle named photon. According to Ockham’s razor this should be a step forward.

To view as a measure of the size of atoms is rational and thus not mysterious. To view as a measure of “smallest packet of energy” of a wave of frequency is irrational and thus mysterious.

Observing the radiation spectrum from a peephole of a cavity with graphite walls, as a physical wave model of a backbody, allows and to be determined experimentally.

Leibniz World is a world in such perfect harmony, and as such a Best of Worlds, with everything in resonance with everything else as an expression of Pre-established Perfect Harmony without cause-effect.

Another example is a coldblooded animal with temperature in perfect harmony with that of the environment, again by a resonance phenomenon as analyzed in Computational Black Body Radiation:

Leibniz World is a Classical World in stationary state of resonance with forces in equilibrium, to be compared with a Modern World in rapid change from non-equilibrium of forces.

In a Classical World in a stationary state of resonance, a distinction between cause and effect cannot be made because the directional aspect of time, with a cause necessarily appearing before an effect, is missing. An example is the Newton’s law of gravitation between mass density and gravitational potential , possibly without cause-effect or in any case not with the cause and the effect, as discussed in the previous post and in more detail here.

In a Modern World changing in time the cause-effect relation is of basic importance, in particular for anyone with ambitions to exercise control and there are many. In a Modern World with the physics of control becomes of paramount importance, and then the basic idea of modern physics of force carrying particles comes in handy:

- If you want to control something far away, just send some force carrying particles, like rockets or drones and you will be able to complete your mission.
- A hot body is sending heat energy carrying particles named photons to warm up a cold body.
- Gravitational forces between bodies are established by exchange of force carrying particles named gravitons.
- Electromagnetic forces are transmitted by force carrying particles named photons.
- Strong and weak nuclear forces are transmitted by force carrying particles named gluons and W/Z-bosons.

While there are rockets and drones, the physical existence of particles like photons and gravitons may be more fiction than reality.

Ambitions of control may be huge, while reality of control is another thing. The cause-effect relation may not be want you would like it to be.

Below is a modern physics view of a world filled with force carrying particles:

Modernity is change and cause-effect action expressed as Will to Power by Nietzsche and in the dynamics of a capitalistic system, in contrast to a classical society in equilibrium. But sociology and politics is not physics, and so the modernity of modern physics may be illusion more than reality.

The revolution into modern physics the beginning of the 20th century (relativity and quantum mecahnics) came along with a revolution of technology (car, telephone, electricity, oil) combined with revolutions in politics, art, music, where old systems were overthrown with radical novelty of top priority. But physics is eternal and does not care about novelty, and so a lot of relevant classical physics (Newtonian mechanics and Maxwellian electromagnetics) was dumped in modern physics.

Modernity as motion was expressed in the avant-garde art movement of futurism:

*Unique Forms of Continuity in Space, *by Umberto Boccioni (1913).