The complex numbers in physics

THE COMPLEX NUMBERS IN PHYSICS INDEX 1. THE COMPLEX NUMBRES IN PHYSICS The four-dimensional space of Minkowsk Alternating current circuits 2. WAVE PARTICLE DUALITY The double slit experiment and the interference phenomenon Emission of black bodies The spectrum of hydrogen. Bohr´s model Photoelectric effect

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3. PLANCK EQUATION Planck length, time, energy and mass Unification of the forces of the Universe. Standard model The quantum theory extended to the electromagnetic field The fourth state of matter: the plasma The fifth state of matter: the Bose-Einstein condensate The dilemma of two models: Heisenberg versus Schrödinger

11 15 15 17 18 19 21

4. SCHRÖDINGER EQUATION Analysis of the Schrödinger equation Interpretation according to De Broglie wave particle duality Interpretation according to the Schrödinger wave intensity Interpretation according to the probability of Born Interpretation of Copenhagen. Bohr's quantum thinking Deterministic interpretation. The quantum thinking of Einstein Heterodox interpretation from the analysis of complex variables Need for the utilization of complex variables Experimental and instrumental uncertainty

21 23 23 23 24 26 27 30 32 35

5. UNCERTAINTY PRINCIPLE OF HEISENBERG

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6. BIBLIOGRAPHY

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The laws governing the behavior of the world at its tiniest scales are fundamentally governed by the complex-number system1 Penrose 1. THE COMPLEX NUMBRES IN PHYSICS The British mathematician of global reputation Roger Penrose, in his book The Road to Reality, states that complex numbers "find a unity with nature truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-numbers system as we are ourselves, and has entrusted to these numbers the precise operations of her world al its minutest scales”2.

Penrose refers to the microscopic world of quantum physics, and leaves out of the scope of complex numbers • to general relativity, thought for speeds close to light and for energy densities and large masses, such as black holes.

• to the Newtonian laws, very sure in their predictions in a world neither too small nor too big, of large particles and very small velocities.

But the importance of complex numbers transcends the tiniest scale announced by Penrose. The complex numbers, formed by the combination of a real number and an imaginary one, are used in physics for pure practical reasons in principle, to facilitate the representation of the propagation phenomena of electromagnetic waves. It gives us an idea of the extent of its use the presence in subjects as different as the mathematical formalization of Minkowski's space-time, created by the theory of relativity, or in alternating current circuits, or in the representative equations of mechanics quantum, or even in the theory of Hartle-Hawking about the beginnings of the universe, which supposes a cosmos without limits and an imaginary time3. In the formulation of all these diverse contents it is shared the presence of the pulsation ɷ of an electromagnetic wave, or of the frequency υ equivalent, and both always appear accompanied by the imaginary unit i.

Thus, the resource of complex variables obeys interpretative needs of physical phenomena in which we must treat mathematically with ɷ and υ. But in the treatment the imaginary unit i automatically appears. The mere presence of i causes the opening of an angular gap φ between the imaginary component and the complex variable itself (sum of the real and imaginary component). The angular gap is presented systematically and always introduces into the equation of which is part geometric distortions, and sometimes even uncertainties of calculation of unpredictable effect.

Summing up, the use of complex variables in the equations responds to demands in the mathematical treatment of the pulsation ɷ and the frequency υ. This pair of variables always participates in the world of physics next to the imaginary unit i. When the complex variable (a + bi) is formed, the angular gap φ appears, which separates the components a and (a + bi). For the participation in the world of physics initiated from the formation of 1

Penrose, R. (2006) The road to reality, eight printing, Publisher Alfred A. Knopf, page 73.

Ibíd., page 73.

3 Hawking and Hartle demonstrated the absence of borders when calculating the initial state of the Universe by means of an approximation of quantum theory of gravity. If the proposal of absence of limits were correct, there would be no singularity, and the laws of Science would always be valid, even at the beginning of the Universe. The Hartle-Hawking model discovered a singularity of the Big-Bang when it was applied the classic general relativity, that disappeared in the quantum version. Hawking, S. W. (2005) The theory of everything. The origin and fate of the universe, Phenix Books.

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ɷi and υi works efficiently through equations formulated with complex variables, it is necessary to find the geometry that integrates the angular gap φ. In my opinion, this whole process represents a genuine sample of nature's confidence in complex numbers, which Penrose has probably considered in his quote at the beginning of this section.

The four-dimensional space of Minkowski The four-dimensional space-time of Minkowski was constructed by the theory of general relativity. The equation [M9] of the file Mathematics in Physics defines a space-time interval between two events. ds2 = dx2 + dy2 + dz2 – c2dt2 [M9]4 The equation includes a real spatial component (dx2 + dy2 + dz2) and another imaginary temporal component (cidt)2. Both make a complex variable, which implies: • The imaginary component is a function of the pulsation ɷ.

• An angular gap φ is opened between the imaginary component cidt and the complex variable ds.

The light is an electromagnetic wave defined by its pulsation ɷ according to the equation c = λɷ /2π, where λ is the wavelength. The imaginary component ci in [M9] is equivalent to the component ɷi, since c = f (ɷ). What Penrose calls the magical effect of complex numbers is verified here; the mere presence of the imaginary component ci causes the angular gap φ.

It can be concluded therefore that the two variables, φ and ɷ, are linked to the imaginary component cidt and to Minkowski's own space-time structure.

In Figure M4 of the file Mathematics in Physics, the cones of light that limit the field of observable phenomena5 in Minkowski's four-dimensional space have been geometrically represented. The angle formed by the generatrix of the cone and the time axis itself coincides with the angular gap φ.

Figure M4 In this case, the open angular gap φ limits the observation horizon. As detailed in the Mathematics in Physics file, the conical surface of Figure M4 defined by equation [M10] limits the horizon of an observer from its vertex. ds = (dx2 + dy2 + dz2)1/2 + (cidt)2 = 0 [M10] [M10] is fulfilled when (dx2 + dy2 + dz2) = cdt, that is for φ = π/4.

4

To facilitate the reading of the text, equations and Figures from other files will be inserted.

To expand the information of the observable fields, see the section The new geometry of special relativity, from the file Mathematics in Physics.

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Alternating current circuits The use of complex variables in alternating current circuits is typical. In synthesis, these circuits work in the following way: the current that circulates I causes a potential drop V with two components. The first RI component is real and is caused by the passage of I through the resistance R; the second component XI, produced by traversing I the reactance X, is imaginary. The balance of potentials is expressed by the following equation: V = ZI = (R + Xi) I = (R + i (Lω - 1/ωC))I [C1]6 The impedance Z of the circuit is formed by two components: the resistance R, which is a real variable, and the reactance X, which is imaginary. At the same time, the reactance X is divided into an inductive reactance, XL = Lω, and another capacitive, XC = 1/ωC, both imaginary. To complete the computation of variables that shape the equation [C1], L and C are the induction and capacity of the circuit, and ω its frequency.

The elements included in the circuit are the resistance R, the inductive reactance XL and the capacitive reactance XC. The resistance opposes the passage of the intensity through the circuit. It acts as a passive element, independent of the frequency of the current that passes through it, R ≠ f (ɷ), and dissipates an energy in the form of heat equal to RI2. The reactance X is the opposition offered by the coils and capacitors of the circuit to the circulation of the intensity depending on the frequency, X = f (ɷ).

The reactance is composed of an inductive reactance XL and a capacitive reactance XC. The inductive reactance is created by a passive element (coil), which stores the energy 1/2LI2 in the form of a magnetic field by induction. Its function is to protect the circuit from abrupt changes in intensity and to ensure that the oscillations of the intensity stabilize. That's right, the inductive reactance generates a voltage V(t) = LdI/dt which is opposite to that caused by the current change.

The capacitive reactance originates when a capacitor is introduced into a circuit. It is a passive element, capable of storing the electrical energy 1/2CV2 that it receives during the period of intensity loading I (t) = CdV/dt; the same energy is ceded afterwards during the discharge period. Its function is to regulate the voltage of the circuit; when it increases, the capacitor accumulates electric charge; when it decreases, the capacitor returns the accumulated charge to the circuit.

In the alternating current circuit, electric waves (stored in the capacitor) and magnetic waves (stored in the coil) are combined. Their joint action interacts between both waves orthogonal and π/2 out-of-phase with each other and produces a resultant vector, whose direction suffers a gap φ with respect to the electric wave. So, the orthogonality between the electric and magnetic fields, which requires the use of complex variables, corresponds to the appearance of a systematic geometric phase gap φ.

The triangle of potentials whose balance is reflected in the equation [C1] is represented in Figure C1. The complex character of the variable ZI implies: • an angular gap φ between ZI and its real component RI, or the complementary gap (π/2 – φ) between the imaginary component XI and the complex variable ZI. • that the imaginary component is a function of the pulsation ɷ, that is, XI = f (ɷ).

Therefore, when complex variables are used to describe physical phenomena, the double condition indicated in the previous section is met: the existence of an angular gap φ and the linking of the frequency ω to the imaginary component XI. In this case, the angular gap has geometric representation in the plane, whose effect is a simple rotation from position RI to position ZI. 6

From here the correlative numbering of the series of equations [C] begins.

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ZI XI φ RI Figure C1 It has special interest the analysis of electrical resonance, a phenomenon that occurs in a circuit in which there are only coils and capacitors crossed by an alternating current of a frequency such that the X reactance is annulled. The electrical resonance originates when the inductive reactance XL cancels the capacitive reactance XC, that is when Lɡ = 1 / ɡC. So: 1 X = XL - XC = Lɡ - 1/ɡC → đ?œ”đ?‘… = √đ??żđ??ś

1

The resonance pulse ωR corresponds to the resonance frequency đ?‘“đ?‘… = 2đ?œ‹âˆšđ??żđ??ś, to which the coil liberates the energy 1/2LI2, exactly equal to that absorbed by the capacitor: 1/2CV2.

That is, during the first half of a complete cycle, the coil absorbs all the energy discharged by the capacitor; during the second half of the cycle, the capacitor recaptures the energy liberated by the coil. This oscillatory state is known as resonance.

If you think you understand quantum theory... you don’t understand quantum theory 7 Richard Feynman 2. WAVE PARTICLE DUALITY In section 2 of the Theory of special relativity, from the file Mathematics in Physics, the unification established by Maxwell of the electric and magnetic field in the electromagnetic field supported the concept of wave light. Figure M3 of the same section has schematized the combined evolution of an electric field perpendicular to another magnetic, composing an electromagnetic wave8 propagated in a direction orthogonal to both. In [A2.6] and [A2.8] of Annex 2 Maxwell equations, the mathematical relationship of this combined evolution has been expressed. Both equations represented for Einstein the support of his theory of special relativity. The treatment of light as a wave seemed to solve the problem that the diffraction of light posed to Newton's corpuscular theory.

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Quote from Richard Feynman, well-known American researcher in quantum electrodynamics and super fluids, in Dawkins R. (2006) God delusion, Transworld Publishers, page 365.

8 The electromagnetic wave has two vector components perpendicular to the direction of propagation, although the vectors can oscillate in different directions and vary their amplitude with time. The sum of both transversal components is tracing a geometric figure that defines the type of polarization of the wave; linear polarization corresponds to the straight line; to the circle, the circular polarization; and to the ellipse, the elliptical polarization.

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⃑

⃑ Ă— đ??¸âƒ‘ = − đ?œ•đ??ľ ∇ đ?œ•đ?‘Ą

[A2.6] 9 ⃑

⃑ Ă—đ??ľ ⃑ = đ?œ‡0 đ??˝âƒ‘ + đ?œ‡0 đ?œ€0 đ?œ•đ??¸ ∇ đ?œ•đ?‘Ą

[A2.8]

The double slit experiment and the interference phenomenon The decisive empirical test of the undulatory nature of light was provided by the English physicist Thomas Young, who devised the amazing experiment of the double slit. The trial showed for the first time that matter was influenced by observation.

The experiment of the double slit consisted in placing a photosensitive screen at a certain distance from two metal plates: the first, with a slit in the center, was the closest to the photosensitive plate; the second, with two slits, was further away. When Young projected a beam of light on the metal plates, when crossing the first plate nothing extraordinary happened, but when crossing the double slit of the second plate the light waves intercepted each other; some were reinforced, when the maximums of the waves coincided; others were annulled, as the maximums concurred with the minimums. This phenomenon of interference was registered by the photosensitive plate located behind the double slit. The succession of waves reinforced and annulled at the exit of the double slit corresponded with the unexpected image of a pattern of bright and dark interference band10 became visible on the photosensitive screen.

Up to here Young’s interference experiment clearly explained the wave nature of light.

But when replacing the beam of light with a flow of electrons something strange happened. When crossing the double slit, the phenomenon of interference was repeated.

That was not expected, because the electron was associated with the nature of a particle, not a wave. It was thought that the experiment was distorted by the occurrence of unforeseen shocks between the electrons. So, the test was repeated, this time projecting electrons over the slit one by one. The interference phenomenon appeared again. But when the image of the electrons was projected on the photosensitive plate before passing the double slit, the vertical lines disappeared, alternating light and dark points. That is, the electron behaved as a wave after passing through the double slit, and as a particle before passing.

In this way the dual wave particle nature of the electrons was demonstrated. Nevertheless, the extraordinary thing of the experimental result was the change of the nature of the matter by the simple fact of being observed. It seemed as if the electron decided its behaviour depending on the place and the moment it was observed, or that its different natures (wave and particle) were interconnected in some way.

If the interference phenomenon was definitive to prove the undulatory nature of light, three counter-tests refuted that same wave theory: the emission of black bodies at different wavelengths, the absorption spectrum of hydrogen and the photoelectric effect. The three anomalies contradicted the hypothesis that matter and energy were continuous.

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Here the correlative numbering of equations of the series [C] is interrupted by the insertion of equations of the Annexes.

10 Considering the movement of light waves composed of indivisible photons, how was the phenomenon of diffraction by a slit possible? If you tried to close the passage of photons by narrowing the slit, the manoeuvre was useless, since the photons passed through the narrowest spaces. However, the behavior of the photon discriminated between a narrow slit, in which it was diffracted, of a wide one, through which it passed without deviating. In the other hand, the slit produced diffraction only if its width exceeded a few times the wavelength of the light, which indicated that some property of the photon was related to the wavelength. Planck would solve this mystery by relating the energy of the photon with its wavelength.

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Emission of black bodies Until the late nineteenth century it was believed that a hot body emitted continuous energy with a constant intensity for all frequencies. The Rayleigh and Jeans equation11, applied by classical theory, determined that the energy intensity was directly proportional to the temperature and the square of the frequency. According to this relationship, the high frequency emission tended to infinity in the ultraviolet spectrum. But the unlimited increase in energy intensity predicted by Rayleigh and Jeans, known as the ultraviolet catastrophe, contradicted the postulate of conservation of energy.

Max Planck postulated in 1900 that electromagnetic waves could only be emitted in discrete quantities and that each quantum of energy depended on its frequency, that is, high frequencies implied high energy quanta. Since the frequency could not be infinite, the radiation emitted by anybody linked to it had to be finite. In conclusion, a black body had to lose energy from a critical frequency.

Planck's revolutionary idea gave an explanation on the radiation emitted by a black body for all frequencies and radically replaced the concept of continuous emission of energy by intermittent emission in packets called quanta.

The spectrum of hydrogen. Bohr´s model The emission spectrum of an atomic element is the electromagnetic trace marked after being subjected to a thermal agitation sufficient for the atoms to break their bonds and jump to a state of higher energy. When the thermal agitation ceases, each element returns to a lower energetic state that characterizes it.

The traces of discontinued lines of the hydrogen spectrum were not predicted by the classical wave model and came into contradiction with Maxwell's theory. The explanation that the electromagnetic absorption of hydrogen was spaced in discrete lines was solved with Planck's new paradigm, which prognosticated a discontinuous spectrum caused by quantum packets12.

Bohr's atomic model was based on information from the spectral series of the hydrogen atom wavelengths that Maxwell's theory of classical electromagnetism could not explain.

It was proposed in 1913, from the simple model for the hydrogen atom, and its content could be summarized in a couple of main postulates: • When an electron jumps from one orbital state to another always absorbs or emits electromagnetic energy E in discontinuous and complete quanta, according to the equation E = hυ13; h is the Planck constant and υ the frequency of the wave.

• Between the potential orbits of the electron, only angular momentum nh/2π are stable, where n is an integer. The electron inside these orbits does not emit energy.

From the first postulate, Bohr deduced that the orbits of the atom were circular, since the centripetal acceleration, function of the orbital radius, was incompatible with the idea of the continuous emission of the electromagnetic radiation. The second postulate was based

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Baron Rayleigh received the Nobel Prize in 1904 for the discovery of argon gas. James Jeans was an astronomer and mathematician.

12 Bohr assumed Planck's thesis that atoms could have only discrete energy values and that transactions between them were linked to the emission or absorption in energy quanta. The new theory clarified the fact that gases, like hydrogen, absorbed light sharply only at certain defined frequencies.

13 See equation [C10] of Planck below.

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on the fact that the light emitted in the spectrum of the atom did not contain all the frequencies, but only those defined by angular momentum nh/2Ď€14.

From these postulates, Bohr calculated the radius of the orbit followed by the electron and its total energy. He started from the centripetal force Fc of an electron of mass me that moved at a velocity v in a circular orbit of radius r: Fc = mev2/r [C2]15 According to Coulomb's law, the electrostatic attraction force Fe between the charge electron e and the nucleus, of atomic number (number of protons) Z, is: Fe = 1/4πξ0 Ze2/r2 [C3] ξ0 is the electric permittivity of the vacuum and Ze the electric charge of the nucleus of the atom, of opposite sign to that of the electron.

Equalizing the equations [C2] and [C3], we can clear the velocity v of the electron in its stationary orbit of radius r. đ?‘?đ?‘’ 2

1

đ?‘Ł = √4đ?œ‹đ?œ€

[C4]

0 đ?‘šđ?‘’ đ?‘&#x;

Classical electromagnetism predicted that an electron moving circularly emitted energy; therefore, it had to collapse on the nucleus after a certain time. The Bohr model overcame this approach by discovering that electrons could only occupy specific orbits, defined by their energy level. Each orbit was identified by an integer n, which symbolized the main quantum number and took values from 1 onwards.

How the electrons could not radiate energy while they rotated within their orbits, the energy balance was obtained by equating the angular momentum of rotation with the angular momentum defined in the second principle, that is to say with nh/2Ď€, after replacing the velocity v obtained in [C4]: đ?‘šđ?‘’ đ?‘Łđ?‘&#x; = đ?‘šđ?‘’ đ?‘&#x;√

1 đ?‘?đ?‘’ 2 đ?‘›â„Ž = 4đ?œ‹đ?œ€0 đ?‘šđ?‘’ đ?‘&#x; 2đ?œ‹

From this equation the radius r of the only possible atomic orbits can be cleared: â„Ž 2 đ?œ€0 đ?‘› 2

đ?‘&#x; = đ?œ‹đ?‘š

đ?‘’

đ?‘’2

đ?‘?

= đ?‘Ž0

đ?‘›2

[C5]

đ?‘?

a0 = 0,529 AngstrĂśm is the radius of Bohr, which corresponds to the distance between the orbit of the electron and the nucleus of the hydrogen atom. That is, it is the value of the orbital radius for n = Z = 1, values of the quantum number and protons of hydrogen.

The total energy of the electron E in its orbit is the kinetic energy plus the potential: 1 đ?‘?đ?‘’ 2 đ??¸ = 1â „2 đ?‘šđ?‘’ đ?‘Ł 2 + 4đ?œ‹đ?œ€ đ?‘&#x; [C6] 0 Substituting the value of v from [C4] into [C6], it results: 1 đ?‘?đ?‘’ 2 1 đ?‘?đ?‘’ 2 1 đ?‘?đ?‘’ 2 đ??¸ = 1â „2 đ?‘šđ?‘’ 4đ?œ‹đ?œ€ đ?‘š đ?‘&#x; + 4đ?œ‹đ?œ€ đ?‘&#x; = 4đ?œ‹đ?œ€ 2đ?‘&#x; [C7] 0 đ?‘’ 0 0 Substituting the value of r from [C5] to [C7], we finally get: đ?‘š đ?‘’4 đ?‘?2

đ??¸ = 8â„Žđ?‘’2 đ?œ€2 đ?‘›2

[C8]

0

If a hydrogen electron jumps from an orbital level n1 to another n2, it will absorb or emit the energy of the equation [C8] for Z = 1. The energy is equal to hĎ…, according to the first Bohr postulate. đ?‘šđ?‘’ đ?‘’ 4 8â„Ž2 đ?œ€02

1

đ?‘š đ?‘’4

1

1

1

{đ?‘›2 Âą đ?‘›2 } = â„Žđ?œ? → đ?œ? = 8â„Žđ?‘’3 đ?œ€2 {đ?‘›2 Âą đ?‘›2 } 1

2

0

14

1

[C9]

2

Only if the electron jumps from the momentum orbit h/2Ď€ to the momentum h/Ď€, or from the momentum h/Ď€ to 3h/2Ď€, and so on, the electron emits energy, but never when turning inside of its orbit.

15 From here, the correlative numbering of the series [C] is recovered.

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Equation [C9] calculates the frequency of the electron's orbit in function of the two levels of the jump: n1 and n2. The coefficient of the second member of [C9] coincides with the experimental measurements of the frequency.

The kinematics of the Bohr hydrogen atom was perfectly defined by the velocity of the electron deduced in [C4], by the radius of the potential orbitals of [C5], by the total energy of the electron of [C8] and by the frequency of the energy emitted or absorbed by the electron [C9] when jumping from orbit.

Confirmed by the experimental results, the atomic model of Bohr turned out to be at the time the most reliable base of quantum mechanics. Not only gave explanation of the emission spectrum of hydrogen, but he adapted to quantum postulates while maintaining the model itself in a permanent dynamic state.

In the 1920s, the Bohr-Sommerfeld model gave an account of this dynamic state, replacing Bohr's primitive model with the objective of extending the quantum numbers that defined the atomic orbitals. Its introduction allowed to improve the structure of the Bohr atom and to perfect its adjustment with the empirical results. To explain the new orbital structures, alternatives to the circular ones of Bohr, the new atomic model resorted to five quantum numbers: the main n, already adopted by Bohr and that was defined by the size and the orbital energy of the band occupied by the electron counted from the nucleus; Sommerfeld16 added the azimuthal l, which he linked with the eccentricity of the elliptical orbits.

The three remaining quantum numbers were17: • The magnetic orbital ml, defined by the orbital orientation of the sub band.

• The spin number18 s, defined by the spin of the electron.

• The spin magnetic ms, defined by the orientation of the spin.

Photoelectric effect Another phenomenon that had confused nineteenth-century physicists was the photoelectric effect, which consisted of the emission of electrons from metals exposed to a light of fixed wavelength. Almost twenty years passed from the first tests of Rudolf Hertz19 on the critical distance between two electrodes to make an electric arc jump depending on the frequency until the theoretical explanation of Einstein's photoelectric process. The founder of the relativist paradigm published20 the formulation of photoelectricity that would mark the beginning of Planck's quantum ideas. In this paper, Einstein confirmed that light gave all its energy in the form of quanta to a single electron, and added that quanta behaved like particles called photons. The concept of the photon led to a paradox: light behaved as a continuous wave and as a discrete particle. So,

16

His contributions in atomic physics and quantum physics culminated in the characterization of the electromagnetic interaction by a dimensionless value. Its ability to find dimensionless parameters was decisive in the construction of the atomic model that bears his name.

17 Adding the last two quantum numbers, s and ms, to the previous list of three was necessary to explain the appearance of doublets in the emission spectrum of several chemical elements, including hydrogen. The doublets are horizontal lines of the spectrum produced by the splitting of the orbital energy caused by the magnetic field of the electronic spin.

18 The spin quantum number is sometimes omitted because for its constant value.

19 Discoverer of the photoelectric effect, this German physicist hapless by an early death had the honour that the unit of frequency, the hertz, took his own name as a posthumous homage.

20 Einstein, A. (1905) On a Heuristic Viewpoint Concerning the Production and Transformation of Light, Annalen der Physik 17: 132-148.

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Einstein was partially responsible for the concept of discrete particle (the photon), though he rejected it completely21.

Antecedents of the notion of light particles were already found in the 19th century, during which the phenomenon of the aberration of light was interpreted with difficulty from the wave theory. Bradley22 discovered it the previous century, when he tried to measure the stellar parallax. The famous English Royal Astronomer expected to observe an elliptical movement, parallel and opposite to the terrestrial orbital movement, of an amplitude dependent on the distance of the star observed to the Earth. However, although Bradley noticed that the star moved along an elliptical trajectory, it did so in a direction perpendicular to the Earth's trajectory. The aberration was measured by the small angular deviation formed between the fixed position of the star and the perceived in the direction of the observer and depended on the speed of Bradley himself, that is to say the speed of the Earth in its orbit around the Sun The result obtained in the historical experiment was explained according to the corpuscular notion of light, and so the astronomer himself explained it.

With the discovery of the aberration of light began a traumatic period for the history of physics, during which it was tried to reconcile this phenomenon with the light wave paradigm. The prestige of the wave theory was developed at the beginning of the 19th century, with the study of the phenomenon of Young's interference mentioned above.

Despite the push given by Bradley to the corpuscular theory, it was not very well received by the scientific community. The effect of the experiment of light aberration was muffled by the forcefulness of Young's experiment and by the credit granted to the so often demonstrated wave nature of light.

Until in 1905 arose Einstein's article about the aforementioned photoelectric effect, based on the quantum notion and where light showed its corpuscular behavior. Despite the mathematical apparatus developed to correctly decipher the phenomenon, many physicists opposed considering light as a beam of individual particles. Those who held this position appreciated Einstein's powerful theoretical deployment, but did not change their minds about the indisputable functioning of light as a wave.

Later, Millikan23 confirmed that the quantum energy of the photons projected onto a plate was proportional to the frequency of the light and determined the photoemission process.

For example, if a photon gave energy to an electron above its emission threshold24, the latter escaped from the surface of the metal; below the emission threshold, the electron could not escape. However, the intensity of the light did not modify the energy of the photons and therefore did not affect the photoelectric phenomenon; only the number of photons emitted changed. Solely by varying the frequency of the light, that is to say its colour, the photon energy was altered: if the energy gave by the photon exceeded the emission threshold of the electron, the difference between the ceded energy and the threshold contributed to increase the kinetic energy of the free electron separated from its orbit.

21

Einstein thought that, if general relativity represented physical reality by a continuous field, the concept of particles could not play a fundamental role in that representation. Admitted the appearance of particles in the special case of regions of space where field strength or energy density were particularly high.

22 In addition to the aberration of light, James Bradley detected the nutation of the axis of rotation of the Earth, a light movement caused by the attraction of our satellite.

23 Einstein and Millikan were awarded the Nobel Prizes in 1921 and 1923 respectively for their works on the photoelectric effect.

24 The emission threshold is the work necessary for an electron to jump out of its orbit released from the atomic bond, that is to escape the electrostatic attraction force of the atom's nucleus. Each metal has its threshold frequency: heavy metals require more emission energy than light metals.

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Millikan's oil drop experiment to determine the charge of the electron showed that as the light dimmed, there were fewer photons escaping from the metal, but all of them kept the energy above the emission threshold. The light behaved like particles in coincidence with the Newtonian model. But when sufficiently sensitive experiments were developed to detect a single photon, the wave theory predicted that the electrical signals from the electronic detection sensors would get softer and softer, and they just would occur less and less often. To explain this confusing behavior was resorted to the wave particle duality of light.

Compton's25 empirical test completed the demonstration of the validity of the quantum theory by testing the inelastic scattering of a photon by an electron in the photoelectric experiment. The specific thing about the Compton test was to use high energy photons, that is, very high frequency. When the electron captured the photon, the electron jumped from its orbit according to the photoelectric effect, but it lost part of the energy absorbed by emitting less energetic photons. This phenomenon was recognized as the Compton effect, where the loss of energy of the photon in the transmission was manifested in the experiment by the growth of the wavelength of the electron.

Briefly explained the Compton effect works as follows: the energy of the photon is directly related with the frequency υ and the mass me of the electron. So that E = hυ = mec2, if the angle formed between the incident and scattered photons is π/2. But υ = c/∆λ, where ∆λ is the variation in wavelength of scattered photons. Therefore, hc/∆λ = mec2 and ∆λ = h/mec. As h = 6,63∙10-34 J and the mass of the electron is me = 9,1∙10-31 kg; ∆λ = 2,43∙10-12 m.

Compton's experiment ended up demonstrating the corpuscular nature of light. The scattering of the photon could be explained as the inelastic collision between two particles: the photon, propelled by its kinetic energy, and the electron, spinning in the orbit of its atom.

All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?26 Albert Einstein 3. PLANCK EQUATION Figure C227 of the spectral distribution of the radiation emitted by a blackbody as a function of frequency is a continuous line, with a maximum that varies with the blackbody temperature. According to Planck, the emission works pursuant to the following scheme: 1. Blackbody radiation is in equilibrium with atoms within an enclosure, which behave as harmonic oscillators of frequency υ.

2. Each oscillator absorbs or emits radiation in an amount proportional to υ, that is, its energy can only have the values 0, hυ, 2hυ ,3hυ .... nhυ, where h is the Planck constant28.

According to the second phase of the scheme, the energy of the oscillators is shown in discrete quanta, by the famous Planck equation: E = hυ == ħɷ, [C10] 25

Arthur Compton received the Nobel Prize in 1927 for the discovery of the effect that bears his name, for his research on cosmic rays and on polarization and spectra of X-rays.

26 Einstein A. (1995) Ideas and opinions, Three Rivers Press.

27 Figure taken from https://es.images.search.yahoo.com/search/images.

28 All the reasoning followed so far in this paragraph has been supported by the work of La radiación del cuerpo negro UPV/EHU. http://www.sc.ehu.es/sbweb/fisica/cuantica/negro/radiacion/radiacion.htm.

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ɷ = 2πυ is the pulsation, and the normalized Planck constant ħ = h/2π is the conversion factor between the old unit systems, kJ, and the natural quantum mechanics energy unit, which is the cycle per second.

Figure C2 According to the second phase of the scheme, the energy of the oscillators is shown in discrete quanta, by the famous Planck equation: E = hυ == ħɷ, [C10] ɷ = 2πυ is the pulsation, and the normalized Planck constant ħ = h/2π is the conversion factor between the old unit systems, kJ, and the natural quantum mechanics energy unit, which is the cycle per second.

Planck showed the need to establish a law of heat radiation, that is of infrared waves, in compliance with the experience. It was necessary to use a method of calculation whose incompatibility with the principles of classical physics became clearer. Then he exposed the revolutionary explanation that energy was not radiated as a continuum of the electromagnetic spectrum at any possible frequency, but in discontinuous discrete packets. The number of packets was determined by the frequency of the energy. For example, low-frequency light emission required few low-energy packets, while a large number of high-energy packets were required to emit light at the ultraviolet end of the spectrum.

From the mathematical plane, Planck also made an extraordinary discovery. He found that the law of radiation as a function of temperature could not be derived solely from Maxwell's electrodynamic laws. To arrive at results consistent with the relevant experiments, he had to treat the radiation as if it proceeded from individual entities (photons), carriers of individual energy of value hυ.

In Einstein's view, Planck's law revealed the existence of a kind of atomistic structure of energy, in correspondence with the atomistic structure of matter, which was governed by the universal constant h. According to him, without this discovery that shattered the complete framework of classical mechanics and electrodynamics, it would not have been possible to establish a feasible theory of molecules and atoms and of the energetic processes that govern their transformations. In sum, the exegesis of the functioning of the emission of a black body was a very fundamental conceptual finding, although Einstein never considered it fully satisfactory.

The [C10] equation of Planck was received as a brilliant confirmation of its quantum hypothesis and was revalidated when Einstein explained the photoelectric effect using that same equation. It was ironic that Einstein received the Nobel Prize for demonstrating that the energy of the electrons ejected from his orbit depended on the frequency υ, when the award-winning did not believe in the particles nor in the discrete transfer of energy, which contradicted the theory of the continuous field of the matter that he always defended.

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The Planck energy radiation approach was taken further by Einstein when he affirmed that in the photoelectric effect the discrete energetic packets were administered by nature as particles of light29.

The radical concept of proposing energy as discrete packets30, multiples of hĎ…, culminated in the second quantum postulate, although it did not harmonize with the continuous structure of the relativistic wave and that seemed more affined to the discontinuous Newtonian particle scheme.

The balance of empirical tests and contra tests presented in the four preceding sections to explain the nature of matter and energy can be summarized, after Planck's formulation, in Table C1.

Table C1 Experiment

Nature Light Electrons Energy Double slit Undulatory Wave-particle Electromagnetic wave Emission of black bodies Quantum Spectrum of hydrogen Quantum Photoelectric effect Quantum Prince de Broglie proposed an alternative compromise option in 1924 when he first devised the dual wave particle structure. He started from the particle of a photon and associated it with the equation [C10] of the Planck energy, and the momentum p = h/Îť [C11] Îť is the photon wavelength (see attached Box).

The momentum is deduced from its definition p = mc [C12] m m is the mass of the photon and c is its velocity. For an electromagnetic wave, c = Νν → Ν = c/ν [C13] Ν is the wavelength of the photon. Substituting the variable ν from [C10] into [C13], we obtain Ν = hc / E. But how E = mc2 → Ν = h/mc Substituting mc for its value in [C12], momentum of photon is obtained: p = h/Ν → Ν = h/p [C14] .

In this way, the wave particle was associated with the electromagnetic wave and the photon: the first one determined by the energy [C10] and the photon, by its momentum [C11]. In postulate 7 of Annex 9 SchrĂśdinger equation synthesizes the action wave ⃑ is the wave vector31 of module 2Ď€/Îť. particle in the pair [A9.14], where đ?‘˜ ⃑} {E = ħɡ; đ?‘? = ħđ?‘˜ [A9.14] 32

29

In 1900, Planck showed that the energy was emitted and absorbed in quantum discrete to explain the radiation of a black body; in 1905, Einstein revealed that the energy of light was a function of its frequency; in 1920, de Broglie and SchrĂśdinger introduced the concept of standing waves to explain that these existed only at discrete frequencies and thus in energetic states of light and matter.

30 The discrete nature of variable E of [C10] posed practical problems of precision of measurement.

31 ⃑ provides direction and sense to the momentum, whose component in The introduction of the vector đ?‘˜ any particular direction can be defined as the phase change of the wavefunction when one cm is moved in that direction and sense the point from which the position of the particle is measured.

32 Here the correlative numbering of equations [C] is interrupted again.

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The wave particle structure responds to a dual behavior of the nature of matter. As a wave (E = ħɡ), there are no problems in following its trajectory, when it oscillates and what happens at every point and at every moment; the equation of the wave allows us to ⃑ ), the distinguish its crests, its valleys and its inflection points. But as a particle (đ?‘? = ħđ?‘˜ oscillation ceases suddenly, and only maintains the reference of its position as the center of the oscillation According to Einstein, the photon is the particle produced by the agitation in a very short time (of the order of t ≈ 10-9 seconds) of the electrical charges of a molecule. During that time, the charges generate an electromagnetic wave that propagates at the speed of light, together with the photon. The extension of the wave is limited; it occupies only the distance that the light travels during the agitation of the charges. The calculation of the approximate distance is Îť = ct ≈ 3.1010.10-9 cm ≈ 30 cm But de Broglie's daring idea was to extrapolate the association of the quantum equation of momentum and energy [A9.14] to any particle. This brilliant proposal considered the wave particle duality as a universal state, which was equivalent to recognizing that particles in general behaved like waves.

The genius arose from de Broglie's dissatisfaction with the treatment of the quantum theory of light, which defined the energy of a particle (the photon) as a function of frequency, according to [C10]. But a particle contained nothing that would allow defining a frequency. This contradiction forced De Broglie to introduce the idea of particle next to frequency. On the other hand, the movements of the electrons in the atom responded to the normal mode33 of vibration. This wave behavior suggested to de Broglie the idea that electrons could not be considered as simple particles, but that they had to be assigned a frequency.

Davisson and Germer34 experimentally confirmed De Broglie's hypothesis. The test consisted of bombarding an electron beam towards a nickel crystal plate. Previously the electrons were excited thermally, and then projected to the glass through a vacuum chamber with a determined kinetic energy. The purpose of the experiment was to measure the number of electrons that dispersed elastically. In full observation an incident occurred: air entered in the vacuum chamber that oxidized the electron-receiving plate. To remove the oxide, the nickel crystal was heated at high temperature and the crystalline structure was altered. Then, Davisson and Germer noticed maximum intensity that had no explanation from the scheme of elastic collisions between particles. It was expected that the nickel atoms scattered the electrons in all directions. However, the experiment revealed that a diffraction by electron reflection had been generated with unexpected intensity peaks. The wavelengths of these peaks coincided with the theoretical Planck values of equation [C14].

So it was confirmed De Broglie's theory that particles shared the properties of waves. But the experiment by Davisson and Germer also played an important role in the acceptance of quantum mechanics and the SchrĂśdinger equation among the scientific community.

33

The normal mode of a system is a movement pattern in which all the parts oscillate with the same frequency and with a fixed phase relation, that is with fixed frequencies, which are known as resonance frequencies.

34 In 1927, Davisson and Germer tested the accuracy of the wavelength predicted by de Broglie using the electron diffraction experiment. Davisson received the Nobel Prize in 1937.

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Planck length, time, energy and mass The inclination of Einstein for intuitive wit led him to think that one had to choose between sensory experiences when defining a concept or developing a physical law. His concern with the experimental framework and scale was less. But quantum theory forced him to thoroughly check this type of ideas. Its counterintuitive nature and the small size of its scale escaped the logic of the senses and the postulates of classical mechanics.

Einstein's desire to witness the enormous range of empirical phenomena and make them correspond with the theory was too pretentious. This attitude corresponded to Newtonian Physics, which handled determined magnitudes, of a very different range than Quantum Physics. Both operated within very different temporary kinetic frames and geometries.

When the scale of wavelength and local lengths referred to electronic orbitals, for example, one entered fully into the domain of probability and Heisenberg's uncertainty, from which only macro magnitudes escaped.

Therefore, the quantum theory had to adapt to the metric of magnitudes of subatomic order, like the length, the time, the energy and the mass of Planck.

The Planck length35 lp is the distance below which space-time is expected to stop functioning as a classical geometry due to the appearance of quantum gravity. It is the lowest physical point of reference for subatomic particles considered as punctual. The longitudinal limit lp eliminates the problem of symmetry that results from considering physical points of infinite density and smaller than it, combined factors that fit the microscopic black holes.

The length calculated in Annex 8 Planck Length, time, energy and mass is: lp = 1,6.10-35 m [A8.5] The Planck time tp is the smallest period that can be measured, that is, the time it takes a photon to travel the Planck length at speed c. tp = 5,4∙10-44 s [A8.6] The energy of Planck Ep is the maximum energy that can be contained in a sphere of diameter lp.

Ep = 2∙109 J [A8.7] The mass of Planck mp is that contained in a sphere of radius lp and that generates a density of 1093 g/cm3 that the universe would have at age tp. .mp = 2,2∙10-8kg [A8.8] Unification of the forces of the Universe. Standard model Repeated empirical evidence confirmed that the postulates of general relativity and quantum mechanics were not contradictory. Experiments such as the formation of an interphase with a particle beam were the conclusive proof of the wave character (quantum) of the movement (relativistic). As has been said above, de Broglie conceived the electrons turning around the atomic nucleus as connected with a hypothetical wave train, which conciliated with the discrete character of the orbits confirmed by Bohr by the stationary character of the waves.

Despite this, forming a model compatible with both theories has proved extremely difficult. In Einstein’s opinion the two theory systems seem little adapted to fusion into one unified theory, in whose search he invested a large part of his life without any result.

In particular applications, such as particle physics, it is not urgent to achieve the unification between general relativity and quantum mechanics. But there is no unified 35

The Planck length belongs to the scale of the granular space described by the physicist Carlo Rovelli, one of the founders of the theory of loop quantum gravity.

15

theory of gravity in cosmology, a field of Physics very needy of it. Solving the inconsistencies between both theories has been an essential objective in the XX and XXI centuries.

The combination of photon particles and quantum energy packets, announced by Einstein in his discovery of the photoelectric phenomenon, has been key to the conception of the four fundamental fields of force that are manifested in the Universe: gravity, electromagnetism, weak nuclear and strong nuclear.

The strong nuclear force holds together protons and neutrons by the interaction of the subnuclear particles quarks and gluons, with an effect similar to the binding forces that link atoms to form molecules. Coexists in the atomic nucleus in equilibrium with the force of electromagnetic repulsion between protons. Its action is perceived at very short distances, such as the atomic radius. The weak nuclear force is formed by the exchange of bosons, manifested in radioactive disintegration. Its field strength is 1013 times smaller than the strong nuclear interaction.

Since the end of the last century, attempts to merge the four fundamental forces of Nature have not ceased so that the same law could respond to its behavior anywhere and anytime.

Recalling the first section of the file Mathematics in Physics, the mathematical laws that preserve physical magnitudes without changes are known as symmetry transformations.

So, to find common laws for the different forces is equivalent to maintain the symmetries of the forces that are trying to unify.

Until now, the attempts at fusion between the electromagnetic and weak nuclear forces have been rewarded with success36. In return, the symmetry that unifies the strong gravitational and nuclear forces with the electroweak force is not finished to find, or is broken barely found. It is the same difficulty that appears when it comes to building a quantum model about gravity. In the formulation of a common theory of quantum gravity, there are still arduous incompatibilities between general relativity and some of the main assumptions of quantum theory.

Many leading physicists, such as Stephen Hawking, have attempted to discover an explanatory theory that combines the different models of subatomic physics and unifies the four integrating forces of the standard model of particle physics: strong nuclear force, electromagnetism, weak nuclear force and gravity. This is the Grand Unified Theory, which is looking for the fusion of the four fundamental forces through quantum mechanics. Quantum electrodynamics, which is the most currently contrasted physical theory in competition with general relativity, has successfully reunited the weak nuclear force and electromagnetism in the electroweak force, and there are nowadays works that attempt to conciliate the latter and the strong nuclear force.

In 1995, Edward Witten formulated supersymmetry based on superstring theory, although the lack of verification experiments caused a growing degree of skepticism. That same year, Carlo Rovelli described the properties of loop quantum gravity37, which treated quantum space and time as networks of loops that he called spin networks. The smallest size of this loop structure was the Planck length, approximately 1.616Ă—10−35 m. Rovelli's theory of quantum gravity attributed to matter and space common atomic structure.

Currently, the predictions of the Grand Unified Theory establish that around a power of 1014 GeV the strong nuclear force, electromagnetism and the weak nuclear force can merge in the same field. Beyond this great unification, it is speculated with the possibility of merging gravity with the other three forces around a power of 1019 GeV. While special 36

The electroweak model was developed by Sheldon Lee Glashow, Abdus Salam and Steven Weinberg, for which they won the Nobel Prize in Physics in 1979.

37 In the space of quantum gravity networks of loops are formed that Rovelli names as spin networks and that evolve to the so-called spin foam.

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relativity is expected to be fully incorporated into quantum electrodynamics, general relativity is the best theory describing gravitational force.

In the current balance of unification of the forces present in the Universe, brief triumphs and failures alternate. Until now, the standard model has worked very well; its precision in the experimental results is very high. Its field of action are all the elementary particles and three of the interactions that occur between them: the weak nuclear, the strong nuclear and electromagnetism38. Although not without problems, this model integrates the electroweak interaction through a theory that unifies the electromagnetic force with the weak nuclear one, and leaves the description of the strong nuclear force for the theory of Quantum Chromodynamics39. In principle, with these two theories can be explained all that is observable, with the already mentioned exception of the force of gravity.

The quantum theory extended to the electromagnetic field In 1905 Einstein defended the hypothesis that the emission of quantum packets was a property extendable to the electromagnetic field. This risky position was difficult to accept by the scientific community, since it supposed a return to the corpuscular theory of the light of Newton. However, all the experimental results (the double-slit experiment, the emission of the black bodies, the discontinuous spectrum of the hydrogen and the photoelectric effect) were confirming Einstein's hypothesis.

Einstein gradually developed the idea of light as a particle, between 1905 and 1917, leaning on Planck's works. The culminating point of the progressive evolution in favor of the particles came with the approach of the photon as a quantum unit. The update of the concept facilitated the explanation of the experimental results, which presented anomalies with the classical wave model of light. In the confirmation of the photon hypothesis, three empirical certainties had a special impact: • The photoelectric effect, demonstrated in the experiment that proved that the photon left a metal after overcoming the barrier of an energy threshold, exclusive function of the frequency of light.

• The photoionization of gases by light, produced at sufficiently high frequencies, different for each gas.

• The disconcerting effect of the decrease in specific heat of solids at low temperatures.

The quantum field theory, established on Einstein's idea that light quanta were photons, prospered at a good pace between 1920 and 1965 thanks to the impulse of a batch of prestigious physicists laureates with the Nobel Prize: in 1933 Dirac, for his contributions in quantum theory of radiation or statistical mechanics; in 1945 Pauli, for the discovery of the exclusion principle that bears his name; in 1965 Feynman40, for his work in quantum electrodynamics and its effects on elementary particles.

Known by quantum electrodynamics, the new theory posed the creation of electromagnetic forces as a result of the exchange of photons, as opposed to the classical stationary approach, where each electron maintained a fixed position and contributed to the electromagnetic field as a simple element of the total flow of electrons. Rectifying 38

Gravity is not considered because it is very weak between individual particles.

Quantum Chromodynamics (QCD) is a quantum theory proposed in the 70s by David Politzer, Frank Wilczek and David Gross for which they received the Nobel in 2004.

40 Dirac was considered the founder of quantum electrodynamics; Feynman has been recognized as the maximum exponent of the conceptual pragmatism of language, by using the terms particles or waves as mere options of representation of matter according to a logical behavior, that is to say, depending on how they satisfy mathematical equations even though they do not always represent a real situation.

39

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this approach, in the Einstein's electromagnetic field was produced a continuous interaction of electrons that were exchanged with constantly emitted and absorbed photons, where photons were simple mediating particles of the interaction.

The discipline that Einstein had founded applied the principles of quantum mechanics to classical systems of continuous fields, such as electromagnetic. The new electrodynamic model tried to solve the problem created by the unsatisfactory experimental results with the sole application of relativistic laws.

An extravagant physical phenomenon, that quantum theory could not explain, was the cooperation imposed by nature on two particles. It was the phenomenon called by SchrĂśdinger of entanglement. In the paradox of the three Jewish physicists, Einstein, Podolsky and Rosen, known as the EPR paradox, it was suggested that the idea of quantum entanglement violated a characteristic principle of mechanics: the principle of locality or local theory41.Proposed in 1935, the EPR paradox exposed the following mental experiment: if two particles were linked by the quantum entanglement, two observers separated from each other would receive information from each of them; when an observer measured the momentum of a particle, he would know the momentum of the other, and if he measured the position of the first particle, he could know the position of the second instantaneously.

The EPR paradox contradicted two firm laws of physics: the theory of special relativity and Heisenberg’s uncertainty principle. In the proposed experiment, information was transmitted instantaneously between two particles, breaking the barrier of the speed of light in the transmission. The controversy arose that two events could not influence each other if the distance between them was greater than the distance light could travel in the time available. In other words, only local events inside the light travel could influence each other.

On the other hand, if lapsed a time since the formation of a system of two connected particles across any distance, simultaneous measurements of the linear momentum of one of them and the position of the other were made, the complete information about each particle of the system would be obtained, which would violate Heisenberg’s uncertainty principle.

In any case, despite the difficulties of fitting relativistic and quantum principles, nowadays it is fully trusted on quantum theory, supported by numerous experiments.

Although many of his predictions challenge everyday experience, this theory has demonstrated a very high degree of precision in the description of the microscopic world.

It is also known that the action at a distance, which the EPR paradox tried to put in evidence, is possible. Recently, in the field of quantum computing, it has been achieved to transmit the quantum state between entangled particles instantaneously.

The fourth state of matter: the plasma Used as a usual tool, the electrodynamic interaction has propitiated advances in the two new discovered states of matter: the plasma and the Bose-Einstein condensate. Plasma is the fourth state of matter, characterized by the large volume it occupies. It is an ionized gas, where a good part of the atoms that compose it have lost their electrons partially or totally. It is composed of the atoms that have not lost electrons and cations, transformed by the ionization of the atoms that have lost them.

The plasma state is reached by raising the temperature of a gas enough for the electrons to escape from their orbits around the nucleus of the atom. At high temperatures, such as 41

This principle is a key axiom of Einstein's relativistic quantum field theory and was always questioned

by Bohr.

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those that exist in the Sun, its molecules and atoms move rapidly and suffer very violent collisions, which facilitates the release of electrons from their orbits. That's right, in the solar atmosphere a large part of its atoms is permanently ionized, that is, in the plasma state. There is also interplanetary plasma created by the solar corona, which is almost two million degrees Celsius. At this temperature, almost all the perimeter atoms of the Sun are ionized and the formed plasma escapes from the force of gravity of our star flowing in all directions like a solar wind at the speed of a tornado. The energy of this wind sometimes reaches the edge of our atmosphere, and penetrates it causing unusual phenomena, such as aurora borealis and magnetic storms.

The ionosphere is also in the plasma state. In this case, the energy of agitation comes from the short wavelength of the sunlight itself, in the interval between the X and ultraviolet rays. Before passing the atmospheric filter, that is to say over 70 km of the earth's surface, the atmosphere becomes rarefied by day, which reduces the probability of collisions between electrons and makes their aggregation difficult. After sunset, the atmosphere recovers its usual density and collisions between ions and electrons increase. Then the plasma state disappears, until sunlight returns to restore it. Above 200 km, the collisions are so infrequent that the plasma state carries on day and night at that altitude.

The fifth state of matter: the Bose-Einstein condensate42 In 1920, the Bengali physicist Bose developed a statistic about the probability of particle configurations able to coexist in the same quantum state in thermal equilibrium. The objective of the work was to distinguish the identical photons and the different photons.

Einstein was impressed by the study and supported its publication43. But in addition, he applied Bose's work for atoms. The result was that not all atoms fit the statistic, and that, at very low temperatures, three bosonic atoms condensed at the same energy level. In 1924, both physicists predicted the existence of the fifth state of matter, known as the Bose-Einstein condensate or the superfluid state.

The Bose-Einstein condensate is the state of aggregation of matter that is given in certain materials at temperatures close to 0ºK44. The quantum property that characterizes it is that a macroscopic amount of the total bosons45 passes to the level of minimum energy. The accumulation of atoms in the same place, not one over another but occupying the same physical space, is an alteration of the Bose-Einstein condensate difficult to understand intuitively. The fact is that when mutually adhering atoms and molecules do not collapse under electric forces and achieve a stable state in the form of condensed matter. In this state, the atoms lose their individual action, they are grouped forming a great atom within which their components are indifferentiable and move globally manifesting the same physical properties and equal reactions to external actions.

In addition to the very low temperature, the state of condensed aggregation of matter is also present at very low density, conditions that minimize its kinetic energy. On the other hand, below a critical temperature, the chemical potential of a condensate tends to its minimum energy. Between this temperature and the absolute zero, the bosons descend 42

The specialized literature recognizes a sixth state of matter. It is the fermionic condensate, which is formed by fermionic particles at temperatures lower than the Bose-Einstein condensate.

43 Bose S.N. (1924) Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik, Vol. 26, Nº 1: 178– 181.

44 The state of the Bose-Einstein condensate is reached at very low temperatures, close to absolute zero.

Among all the chemical elements, only helium resists without freezing around that temperature.

45 The bosons are formed by photons, mesons, gluons and other particles. They are characterized by their integer spin number, which corresponds to them according to the Bose-Einstein statistic, and by a symmetric wavefunction that encourages the state of atomic aggregation.

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towards a same quantum state or fundamental state, which defines the minimum energetic level allowed for a particle and that coincides with the state of aggregation of the condensate.

If the energy of a system is reduced to the limit, the constituents of this system reduce the molecular Brownian motion until reaching a quasi-static state. In this phase of high repose, the small molecular movements tolerated depend on the movement of all the constituents of the system, that is, the behavior of the system correlates with that of the components. When the degree of correlation increases, the macro behavior is dominated by the micro proceeding, that is, it remains subject to the quantum laws.

For example, when lowering the temperature near absolute zero, many systems behave as if they lost a dimension, that is, they go from being three-dimensional to twodimensional. To understand the cause of this dimensional reduction, it is necessary to analyze the quantum property of atoms sensitive to magnetic fields: the spin The spin is the magnitude of the angular momentum caused by the rotation of an electron around its own axis. It therefore corresponds to a magnetic momentum, which is represented by an arrow whose sense points out from the south magnetic pole to the north magnetic pole. As well, from a sufficiently low temperature, all the arrows of the spins are confined in a plane, which determines the collective behavior of the system and its passage to a two-dimensional extension This surprising fact of going from a three-dimensional state to a two-dimensional state is known as a topological phase transition, a phenomenon studied by three physicists laureated with the Nobel Prize in Physics 2016: Berenziskii, Kosterlitz and Thouless, known by their initials BKT. To explain how occur phase transitions, BKT mentions the appearance of defects in the spin network caused by the breaking of its symmetry. They even recover the Cartesian idea of the vortex, a concept that has concentrated the worries of physicists of condensed matter. On the quantum vortices, the most stable states are configured at a very low temperature, which coincide with the paired spins of opposite united turns. These vortices linked by pairs of spins behave as a single entity, but as the temperature rises, the vortices separate and give place to a new topological phase. But if the temperature is maintained near absolute zero, the condensed matter behaves like a superconductor, presents untold properties of quantum computing for ultra-precise measurements and functions as a superfluid without viscosity.

The reason for the good conductivity of condensed matter is that at the absolute zero limit an atom remains immobile. Therefore, the position where it is located is very uncertain and occupies a very large volume. According to the quantum theory, in an aleatory position scenario the probability of oscillation of the particles grows, which increases the thermal conductivity of superfluid up to limits that exceed the standard values of the best conductors. Likewise, the oscillating enlargement of the particles facilitates that the condensates of matter can traverse any solid or non-porous surface.

Another differential characteristic of the superfluid is their almost absence of viscosity, which allows them to flow continuously without any friction. The state of condensate is presented at very specific temperatures for each chemical element. For example, in the isotope of helium 4 the temperature fluctuates around 2ยบ K. Below it, the fluid manifests in a dual state: a liquid fraction and another superfluid fraction, in a variable proportion in function of temperature.

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The dilemma of two models: Heisenberg versus Schrödinger The fundamental role played by complex numbers in Physics 46 reached its maximum relevance with the Schrödinger equation, where all the magic potential of complex numbers47 was concentrated. The depth of the conceptual richness of its content inspired multiple interpretations in the scientific community, often contradictory, even provoked in Schrödinger some repentance for having published it. Schrödinger's article48 was well received by the scientific community. Heisenberg, however, did not accept it in principle.

Different criteria were contrasted in the sporadic discussions held between Heisenberg, Bohr and Schrödinger himself. On the side of the first two was Max Born, a profound connoisseur of the quantum model who had already participated in the mechanics of Heisenberg and who would contribute decisively to solve the problems posed by the Schrödinger wave equation49. On the other hand, Einstein, and De Broglie were inclined in favor of Schrödinger. The position of the first regarding the uncertainty principle was clearly revealed with his famous phrase "God does not play dice with the Universe"50.

Schrödinger edited other works in which he solved his equation for simple cases, such as the hydrogen atom. In one of them he demonstrated mathematically that his theory and that of Heisenberg were equivalent51; in another he applied his wave equation to calculate the energy of the electron in the hydrogen atom and obtained exactly the same result as that calculated by the model of the Bohr atom. Since the publication of these articles, most physicists openly bet on Schrödinger's formulation to treat physical systems.

I don't like it, and I'm sorry I ever had anything to do with it 52 Erwin Schrödinger 4. SCHRÖDINGER EQUATION The Schrödinger equation tries to explain how a quantum system evolves over time. Its objective is to define the wavefunction ψ (x, t) associated with a quantum state determined by its position x at time t.

The wave function ψ is characterized by the frequency ν and the wavelength λ. The discovery of Schrödinger was that from ν and λ it was possible to express the energy E (related to ν through the equation [C10]) and the momentum p (related to λ through the equation [C14]) as wave functions. In this way a confined particle, like the electron inside the atom, with the known energy and momentum, could be described with a definite wave function ψ.

From this initial starting point of Schrödinger, it was found later that only certain frequencies were compatible with ψ and that their existence depended on the confining structure of the atom to which the electron was subjected. These were the discrete 46

Penrose, R. (2006) op. cit., page 66.

Chapter 4 of the aforementioned work Magical complex numbers, Ibíd. Page 71.

48 Schrödinger, E. (1926) Quantisierung als Eigenwertproblem, Annalen der Physik, Vol. 385, 13: 437490.

49 In 1926, Born worked in the interpretation of the location of wavefunction at a certain point considering it as a particle. In 1954, he received the Nobel Prize for this work.

50 Harrison, E. (2000) Cosmology: The Science of the Universe. Cambridge University Press. page 239.

51 Although Schrödinger and Heisenberg departed from very different mathematical approaches, at the end the results were identical. According to Schrödinger's mathematical demonstration, the calculation of the speed or energy of an electron was exactly the same through the model of Heisenberg matrix that of the equation of complex variables of Schrödinger.

52 A Quantum Sampler, The New York Times, 26 December 2005.

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frequencies of the light emitted and absorbed by the electrons when breaking the confinement of the atom, in correspondence with the frequencies of standing waves of certain energy states. These states were defined by the SchrĂśdinger equation, which definitively established the importance of quantum theory.

For the particular case of a flat wave, it is started for its development of the energy balance of a particle53: E = Ec + Ep= ½mv2 + V = hν [C15]54 E is the total energy of the particle, Ec, its kinetic energy, Ep, its potential energy, m, the mass of the particle, v, its velocity and V the potential of the particle related to the position it occupies, particularly for each case55.

The definition of momentum is: p = mv [C16] If ω is the angular velocity of the particle, it is accomplished: ω = 2πν [C17] Substituting ν of [C17] in the third member of the equation [C15], and v of [C16] and p of [C14] in the second member of [C15], we get: ωh/2Ď€ = 1/2mh2/Îť2 + V [C18] Substituting ħ = h/2Ď€ and k = 2Ď€/ Îť in [C18], the equation [C19] is finally reached ħω = 1/2mħ2k2 + V [C19] The first member of [C19] is the total energy of the particle, the first addend of the second member, its kinetic energy and the second addend of the second member, its potential energy In correspondence with the SchrĂśdinger wavefunction Ďˆ (x, t), the equation [C19] can be written, for the particular case of a plane wave, in the following way: ħ2 đ?œ•2 đ?›š

đ?œ•đ?›š

đ?›źÄ§ đ?œ•đ?‘Ą = đ?›˝ 2đ?‘š đ?œ•đ?‘Ľ 2 + đ?‘‰đ?›š

[C20]

ι y β are constants.

The typical solution of [C20] for a wave is of the form: Ďˆ = cos (kx – ωt) + isen (kx – ωt) [C21] k is a constant.

Deriving [C21] once with respect to t and twice with respect to x, we have: âˆ‚Ďˆ/∂t = ωsen (kx – ωt) – iωcos (kx - ωt) [C22] ∂2Ďˆ/∂x2 = k2cos (kx - ωt) - k2isen (kx – ωt) [C23] Substituting [C21], [C22] and [C23] in [C20], for Îą = i and β = 1, we obtain the general equation of SchrĂśdinger: đ?œ•đ?›š

ħ2 đ?œ•2 đ?›š

đ?‘–ħ đ?œ•đ?‘Ą = 2đ?‘š đ?œ•đ?‘Ľ 2 + đ?‘‰đ?›š

[C24]

equivalent to [C19] provided that: đ?œ•2 đ?œ“

đ?œ•đ?œ“

đ?œ” ≥ đ?‘– đ?œ•đ?‘Ą , đ?‘˜ 2 ≥ đ?œ•đ?‘Ľ 2 đ?‘Ś đ?‘‰ ≥ đ?‘‰đ?œ“ [C25] From [C24], one can define the energy of a system in quantum mechanics as the phase change of the wavefunction Ďˆ at a time t.

53

For its greater simplicity, the particular case of a flat wave has been analyzed. In the Annex 9 SchrĂśdinger equation, we have proceeded to the detailed development of the general equation.

54 From here, the correlative numbering of the series [C] is recovered.

55 In the case of an electron for example, Ep is the electric potential energy due to the attraction of the nucleus and to the repulsion between the electrons.

22

Analysis of the Schrödinger equation In principle, solving the Schrödinger equation [C24] should not present major problems: from classical mechanics, it would be enough to assign a succession of discrete values to energy E (corresponding to discrete frequencies, through the Planck equation [C10]), depending on the position x and the momentum p; from Schrödinger's own approach, the differential equation [C24] presents a scalar wave function E that evolves with the independent variables of position x and time t.

But the substantial problem with the Schrödinger equation is that although all the variables of the second member of [C24] are real and, in principle, can be measured, the first member is an imaginary variable. The wavefunction ψ is not real, therefore unobservable, and cannot be measured56. If the equation [C24] could be solved, the solution would be a complex number, and we would not know what that means. In summary, the wavefunction is simple to manage mathematically but difficult to interpret physically.

As explained in detail in Annex 9 Schrödinger equation, to avoid the handling of complex variables we must resort to the product of the wavefunction by its conjugate ψψ*, and thus obtain a real value. This mathematical subterfuge57 obliges us to give up the solution of the wavefunction ψ (x, t), which places it precisely in a defined position and time.

Through the product of ψ by ψ* the wavefunction is normalized, which means sacrificing the absolute value of ψ (x, t) and resigning to the option of a probable value within the complete domain where the wavefunction is integrated.

Interpretation according to De Broglie wave particle duality In the particular case of an electron, it is expected that its wavefunction responds to the trajectory described by ψ, in a sense similar to the wave traced by a photon according to Maxwell's equations. But according to De Broglie's wave particle identity principle, the mass and charge of the electron are not concentrated at one point, but diffuse through space. The distribution of mass and charge is defined by ψ, although we can only know the value of the sum of squares ǀψǀ2 of its two components (real and imaginary), which is the probability that the electron is around the nucleus of the atom whose shape is defined by ψ. Therefore, ψψ* = |ψ|2 is interpreted as the density function of the probability that a particle occupies the position x at a time t and with the probability amplitude ψ. In conclusion, through the scalar ǀψǀ2, that is to say a number, we have recovered information from an imaginary variable ψ without solution, in exchange for lowering the qualitative content of the information. In sum, the sense of the physical magnitude of ψ is lost, which is reduced to the statistical parameter ǀψǀ2.

Interpretation according to the Schrödinger wave intensity The interpretation of ǀψǀ2 as a density function did not satisfy Schrödinger, who was an annoying burden the complex nature of the wavefunction ψ. So, the Viennese Nobel scientist in 1933 tried to pose the development of his equation with real functions to solve the problem mathematically, but he did not succeed. He replaced the wave amplitude with The function ψ defines an oscillation of a wave caused by a particle positioned in x at time t, but the function does not have a real value, that is, as a physical quantity it is not representable.

57 The mathematical manipulation of the wavefunction ψ is to obtain information about the position of the particle in question. In Annex 9 Schrödinger equation we must resort to another mathematical maneuver: the change of the energy scalar E by the energy operator Ê to facilitate its development.

56

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the wave intensity, in order to find a real a real number associated with a measurable physical magnitude equivalent to ǀψǀ2. Really, he proposed to equalize the density of charge or mass of the electron in each point with the wave intensity, based on the reasoning that in the areas of greater intensity the density of the electron would grow, and lower in those of less intensity.

In his explanation, Schrödinger had radically changed the essential concept of the problem with respect to De Broglie: from the dual interpretation of the wave particle to the purely wave sense of the function ψ. And again, the limitations of this function came back with the appearance of high speeds compared with light58, where ψ lost its accuracy.

According to the undulatory interpretation of Schrödinger, in any collision of an electron against the nucleus of another atom, the wave effect should be manifested before the shock. As the electron approached the nucleus, much of its charge and mass would be concentrated in the proximity of the atom's nucleus, and the density would decrease as it got away from it. But after the shock, continuing with the same interpretation, the wavefunction would scatter irregularly in all directions. Therefore, the density of charge and mass of the electron would be reduced quickly after the collision with the nucleus and would disperse in all directions. However, the experimental measurements contradicted this interpretation. After the collision, it was empirically verified that the electron was perceived as a particle, not a wave, and that it maintained a definite position, without any dispersion appearing.

Despite all these experimental tests in favor of dual wave particle interpretation, Schrödinger maintained his position that particles were only appearances, that there were no barriers between a perceived and an existing world. He believed that the waves were real and opposed until his death the probabilistic interpretation that Born proposed in 1925, and that was accepted by almost everyone.

Interpretation according to the probability of Born Max Born, who had helped Heisenberg in the formulation of the uncertainty principle by assisting him with the calculation of matrices, applied the notion of probability to the wavefunction to solve the problem of its interpretation. The solution was proposed only one year after the publication of Schrödinger's articles, and pleased Bohr and Heisenberg, but not Einstein or Schrödinger.

Born's ideas about physics were totally opposed to Einstein's. The illustrious mathematician of physics59 recognized that his differences in the conception of Nature with the creator of relativity were fundamental. The conceptual confrontation between these two geniuses had a starting point: Born accepted without reservation the principle of wave particle complementarity of Bohr; Einstein did not want to accept it. The discrepancies on this point were in the background: Born defended the usefulness of the mathematical concepts of particles and fields; Einstein warned about the problems that these useful concepts caused in the interpretation of physical phenomena. Planck's discovery of the discrete energetic states of the electrons within the atoms and of the discrete light quanta emitted and absorbed by them when jumping from orbit posed the solution of the standing waves as a complementary solution to the particles.

58

Schrödinger's formulation, like that of Heisenberg in his day, had been proposed for behind of Einstein's relativity.

59 Born's mathematical training was very solid. He had Hilbert as his mentor and was an assistant to Minkowski in Göttingen. Together with Bohr (esteemed father of quantum theory) and Heisenberg, he has been considered one of the guides of quantum mechanics: the first two looked for physical arguments that unravel quantum complexity; Born looked for mathematical arguments.

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Born thought that the complementarity principle would have an essential importance in the development of modern physics. He also considered that deepening in knowledge of the Universe would be at the cost of uncertainty in the results. He left evidence of his thinking against classical determinism in an article, published in 195560, where he rejected it roundly based on the fact that a small change in the initial conditions of a particle of a gas widely altered its trajectory.

The probabilistic interpretation of Born assumed the indeterminate behavior of an individual measurement of the Schrodinger wavefunction Ďˆ. As mentioned above, he resorted to the product ĎˆĎˆ* to solve the problem of complex variables, normalizing the variable ĎˆĎˆ* in passing, that is, finding a scale such that: ‖đ?œ“‖ = âˆŤ đ?œ“đ?‘Ą đ?œ“đ?‘Ąâˆ— đ?‘‘đ?‘Ł = 1 [C26] The normalization of the variable ĎˆĎˆ* through the integral [C26] was a routine mathematical procedure to solve the problem of lack of knowledge of the function Ďˆ.

Although the states of the environment of the domain Ď… were unknown, the sum of these states in the complete domain was 1 only considering ĎˆĎˆ* a density matrix, in this case of probability density of where the particle would be found. Thus, the following paradox was reached: ĎˆĎˆ* was a real probability function that corresponded to a non-real wave function Ďˆ to maintain the hypothesis of the real particle. In the background, everything was based on the wave particle duality, where the quantum scalar standing waves had become probability waves for the real particle.

In Postulate 3 of Annex 9 SchrĂśdinger's equation is enunciated that this equation allows any linear relationship: Ďˆ (x) = ÎąĎˆ1 (x) + Î˛Ďˆ2 (x) Therefore, any solution of the SchrĂśdinger equation multiplied by an appropriate number n can be adjusted to the normalized scale [C26] without ceasing to be a solution, for being nĎˆ a linear variation of Ďˆ. With this mathematical artifice, we try to avoid the problem presented by equation [C24], posed with operators applied to a complex variable Ďˆ, and replace it with the scalar61 |Ďˆ|2 in order to find a real solution. But in the maneuver, the variable Ďˆ, which confines all information, has been transformed into ĎˆĎˆ* = |Ďˆ|2, a probability function that overrides the physical entity of Ďˆ.

According to Born, |Ďˆ|2 must be interpreted as the probability that a measurement of the position of Ďˆ62 be in a region of the given space: where the wavefunction is maximum, the probability will be maximum; where the function is null, the particle will not be found.

In the analysis of an extreme case, if in the spatial enclosure dĎ… |Ďˆ|2 = 0, it means that the particle is never found inside it; the other extreme case |Ďˆ|2 → 1 denotes the location of the particle within the enclosure dĎ… if the experiment is repeated many times. So |Ďˆ|2 gives information about the frequency with which the particle occupies different positions within the dĎ… enclosure. And in Born's interpretation it is unimaginable to think about this position before performing the experiment.

The integral [C26] is extended to all the space Ď&#x2026; and adds 1, which indicates that the particle is somewhere in Ď&#x2026;. Therefore, it will be true that Ď&#x2C6;Ď&#x2C6;* = |Ď&#x2C6;|2 < 1 in a dĎ&#x2026; environment. But the scalar |Ď&#x2C6;|2 does not measure the charge and the mass of the particle that is in such an environment; this information would belong to Ď&#x2C6;, in the case that the 60

Born M. (1969) Is Classical Mechanics in Fact Deterministic? In: Physics in My Generation. Heidelberg Science Library.

61 Hilbert created a vector space with the structure of a scalar product that functioned as a generalized Euclidean space for any number of dimensions. The space was designed to act on complex numbers, but could only measure scalars: modules and angles formed by two vectors.

62 The same reasoning can be applied to the rest of variables related to the state of the wavefunction. It is possible to calculate the probability that the velocity, energy or any of the magnitudes that define that state are within a defined range of values.

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wavefunction was a real variable. |ψ|2 < 1 is the probability that, at the time of measurement, the particle is within the domain dυ.

Schrödinger did not share Born's interpretation. The first one interpreted the result of |ψ|2 = 0,75 in the following way: 75% of the charge and mass of the particle extended within the space dυ and 25% remained outside it. But in all the experiments carried out, it had not been possible to confirm stable results on the position of the electron inside or outside a specific spatial enclosure. It was hard to Schrödinger to leave classical determinism and the empirical evidence denied the meaning he gave to |ψ|2.

If Born interpreted the same previous example, he would say that if an experiment were repeated many times, 75% of the measurements would confirm that the position of the electron would be inside the enclosure dυ, and 25% would be outside. But in each experiment the whole electron would be detected inside or outside the enclosure; it would not show a percentage of its charge and mass as Schrödinger interpreted. Born had no doubt that when a particle was detected in an experiment it was located in a single point, that if the particle appeared in a certain place it could not appear in another place, insofar as it conceived the wavefunction as a whole indivisible.

In conclusion, from the instant in which the position of ψ was measured, the wavefunction collapsed, without the result of the next measurement depending on it. This meant that the measurement did not condition the distribution function ǀψǀ2 at all, but was independent of it, just as laws of probability demand. Therefore, in each measurement we had to forget all the previous experimental results and measure again. But also, the interpretation of Born supposed a categorical reduction of the expectations in the knowledge of the physical phenomena. According to his position, all that could be aspired in Physics63 was to define the probability with which something would be measured in certain conditions.

Interpretation of Copenhagen. Bohr's quantum thinking The interpretation of Copenhagen has been considered like the orthodox, by the acceptance of the majority of the scientific community since its declaration in 1927. It received its name in homage to the city of residence of Bohr, initial propeller of the quantum concept. Its content could be summarized in three big conclusions: • There is no reality expressed in the Schrödinger wavefunction, but a mere mathematical formalism.

• The probabilistic nature of quantum mechanics is definitive, not temporal, and its replacement by a deterministic theory cannot be expected.

• It is necessary to combine the evidence of different experimental measurements with the well-defined application of the formalism of the wavefunction.

Bohr thought that the Schrödinger wavefunction did not represent a quantum reality, but was a mere description of the knowledge that an observer obtained from a quantum system. According to this interpretation, when a measurement was made, a greater degree of knowledge of the system was acquired, so that when the measurement was concluded, the degree of its perception jumped. This idea of replacing the discontinuous random jump of a wavefunction with the jump of knowledge identified the ontological position of the Copenhagen group like no other.

The discussion of the Copenhagen group focused on the fact that the knowledge of quantum systems gained in depth with the process of experimentation. In this field, the 63

Born's aspiration was to consolidate probabilistic predictions by repeating the same experiment many times. These predictions were confirmed by doing experiments, what did not stop being a tautological explanation.

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group maintained a critical position against the misuse that classical mechanics made of cinematic descriptions. According to Bohr, quantum phenomena had to be experienced, and the process of experimentation had to be fulfilled with the complete description of all its stages: preparations, intermediate phases and final measurement. The description had to be rigorous and meticulous, because she herself was part of the knowledge of the quantum system for the physicist of Copenhagen.

Bohr reprobated the inaccuracies at the time of facing a quantum experiment and tried to correct the macroscopic allusions and the colloquial language of classical mechanics when referring to the new quantum conception. For example, quantum mechanics did not admit that the position and momentum of the initial state of the wavefunction could predict its final state in a deterministic way. On the contrary, in his last writings Bohr conceived the quantum phenomenon as a complete process from the initial to the final condition, and not defined by two instantaneous states, initial and final, linked by a causal relation. From the quantum approach, the initial condition of the wavefunction, by very precise it was, only allowed probabilistic predictions. In summary, Bohr treated the Schrödinger equation not as a method to determine the evolution of the wavefunction, but as a system of measurements of probability.

For Bohr, in the experimental cases several distinct trajectories arose through which a particle could pass from the initial to the final condition. There was no fixed option between the different paths followed by the particle. Only the initial and final conditions were definitive, but the position and momentum of the particle obliged to meet the opposite requirements in order to be observed. For example, for a particle to be experimentally in a favorable position, it had to remain motionless; to experimentally find a stable momentum, the particle had to move freely, and these two conditions were incompatible.

Deterministic interpretation. The quantum thinking of Einstein Einstein did not accept interpretations of quantum theory contrary to determinism and causality. He rejected the idea that the state of a physical system depended on the experimental management of its measurement. He maintained that every phenomenon occurred by itself, regardless of whether it was observed or how it should be observed.

From his relativistic approach, he conceived the quantum state as an invariant under any configuration of the chosen space, that is, under any form of observation. Despite recognizing that the quantum theory had obtained a good representation of an immense variety of phenomenon that otherwise appeared completely incomprehensible, he discovered a point where the quantum failed. It was the impossibility of associating the wave function ψ with the movements of the particles, which, after all, had been the original purpose of Schrödinger. The difficulty seemed insuperable until it was overcome in a way as simple as unexpected. ψ was not interpreted as a mathematical description of an event located in time and space, but as a mathematical description of what could be known about the system. Therefore, the Schrödinger wave function only served, in Einstein's opinion, to establish statistical statements and predictions of the results of all measurements that could be carried out upon the system. This statistical interpretation of the wave function did not describe a state of a single system, but related to an ensemble of systems.

The American physicist Everett propellant of the parallel universes was also totally opposed to the empiricist exegesis of Copenhagen. It conceived a quantum state that evolved presenting coexisting quantum alternatives, not as probabilities, but as

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overlapping options. He thought therefore that in each measurement of the wavefunction alternative results coexisted in a quantum linear superposition.

Formulated in 1956, Everett's theory proposed a universal wavefunction, which corresponded to the interpretation of many alternative universes. He argued that all the possibilities described by the quantum theory occurred simultaneously, as a composition of alternative and independent universes. Everett proposed that all potential and consistent states of the measured system were present in real physics, not just formally as interpreted by the Copenhagen school. From his perspective, the explanation of the collapse of the wave packet at the end of each measurement was unnecessary.

The superposition of combinations of the quantum state has been recognized as a state of entanglement, a physical phenomenon that happens when pairs or groups of particles interact in such a way that the quantum state of each particle cannot be described independently of the others. The phenomenon occurs even when the particles are separated by a large distance and recently results have been obtained at the macroscopic level.

Against entanglement, Einstein argued that the quantum theory should formulate a principle of locality against action at a distance, and attacked along his life the interpretation of Copenhagen, which could not solve the famous paradox of Einstein, Podolsky and Rosen (see above). The experimental concept EPR was used in 1964 by John Bell to formulate a theorem that showed that the statistical predictions of the Schrรถdinger equation and the principle of locality or local theory could not be true both; one of them had to be incorrect, although it was clear that Einstein expected confirmation of the local theory.

Bell's theorem was based on the experimental results obtained by Clauser & Freedman in 1972, Aspect, Dalibard and Roger in 1982, and others. Figure C4 64 shows the scheme of the two-channel optical experiment of Alain Aspect, which consists of a source of two paired photons S obtained from the simultaneous disintegration of two excited atomic states and sent in opposite directions. Each photon finds a detector of two channels D, whose orientation can be fixed by the experimenter through filters a and b. The emergent signals of each channel are detected, counting the coincidences in the CM monitor.

Figure C4 The polarization filters a and b can be set parallel to each other, or at any other angle freely chosen. The experimenters knew that the polarizations of paired photons were always parallel to each other, but random with respect to their surroundings. So, if the detector filters were set parallel, both photons would be detected simultaneously65; if the 64

Figure taken from https://en.wikipedia.org/wiki/Bell_test_experiments.

The simultaneous detections were registered by D+ or D-, establishing four categories: ++, + -, - + and - for the accumulated accounts of photons.

65

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filters were at right angle, the two photons would never be detected together. It remained to establish the detection pattern at intermediate angle adjustments, on which Bell focused his theorem.

The creators of the EPR paradox assumed, like Bell, that the photons arriving at each detector could not know the angular adjustment of the other detector, insofar as the information would have to travel faster than the speed of light. This assumption reflected the principle of locality or local theory, which recognized only the events local to each detector as causes that could affect its behavior, but never the actions at a distance.

Bell supposed, at first, that the relationship between the angular difference in detector adjustments and the coincidences of detected paired photon was linear, like the line shown in Figure C4. He attributed symmetrical and independent functioning to the two detectors D, so that any difference of adjustment at one detector had the same effect in coincidences at the other detector; he supposed also that the relation between angular differences and coincidences was proportional, that is, linear.

Coincidences (%)

Results of the experiment to test Bell's theorem 100 Linear 50 Experiment points 0 0

45

90

Angular difference (⁰)

Figure C4 But the recent experiments of Aspect, Dalibard and Roger did not show a straight line between the angular difference and the coincidences of paired photons, but a curved line as marked by “Experiment points” of Figure C4. The measurements were made for different angular adjustments and the agreement with the calculations obtained by quantum mechanics was excellent66. These results destroyed Bell's previous assumptions and the foundaments of the EPR paradox, whose assumption of the linear relationship was empirically refuted by a difference of coincidences of paired photons of five times the standard deviation.

Finally, Bell's theorem showed that the particles behaved as predicted by quantum mechanics: the coincidences registered of paired photons increased with respect to the classical prediction if the detectors were adjusted to angular differences greater than 45⁰.

The line "Experiment points" in Figure C4 implied that certain photons were intimately connected, connections supported by quantum theory. It was an action at a distance, where a source produced pairs of entangled photons.

The disconcerting anomaly of entanglement is a quantum phenomenon manifested by non-local connections between particles that maintain three important characteristics: • They link events in separate locations, without there being fields that justify these connections.

• Effects do not decrease with distance.

66

Quantum theory was verified within experimental error of about 2%.

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• Act at a speed greater than light.

Recently, a team of researchers from the University of Cambridge has got unlink the state of matter with observation. In particular, it has achieved to specify the trajectory of quantum particles when they are not observed. The paper Evaluation of counterfactuality in counterfactual communication protocols67 has posed the questions advanced in previous experiments: far from our observation, could quantum particles evolve unpredictably when nobody sees them? Could they occupy two positions at the same time? If the answers are affirmative, it would have to be admitted that the information between two sources is transmitted without any relative movement between them. This phenomenon, known as counterfactual68 quantum communication, contradicts the accepted fact (facto) that there is no information but through the environment.

The methodological innovation put forward by Cambridge scientists is to consider the fact that any particle that travels through space cannot stop interacting with its environment. In the paper referred to, they outlined a way to map these “tagging” interactions without looking at them. This technique is useful when scientists only can make measurements at the end of an experiment but want to follow the movements of particles during the full experiment. In fact, the tagging of the interactions allows coding the information in the particles and decoding it at the end of the experiment, when the measurements are made.

The main author of the article, Arvidsson-Shukur, affirms that the results point to the Schrödinger wave function is closely related to the state of the particles at a given time.

He further argues that the work has explored the forbidden domain of quantum mechanics, which tries to know the path of quantum particles when nobody is watching them.

Heterodox interpretation from the analysis of complex variables Mathematics has been used as the scientific language par excellence that has worked undoubtedly, as if the laws of the Universe did not want to express themselves in another alternative way. Without questioning that the unappealable court that approves or denies the consent to the scientific paradigm is its empirical verification69, several questions can be asked to calibrate the role played by Mathematics. Do they really go beyond being an explanatory means of the laws of Nature? Is its function reduced to read the framework of a physical phenomenon, just as a partiture reads the development of a piece of music?

As far as does the degree of its fecundity go? Does not it sometimes seem that Mathematics governs Physics and not the other way around? Or, on the contrary, are all the laws of physics, taken to the limit of empirical evidence, merely statistical and not mathematically perfect and precise? Is Physics, finally, a law without law?

These doubts about the participation limit of the mathematical world and the misgivings and controversies that the Schrödinger equation has raised among the great colossus of physics, encourage us to propose a heterodox interpretation from the analysis of complex variables.

The most interesting of the doubts and permanent discussions brought forward by the interpretation of the Schrödinger equation are the positive collateral effects that have been 67

Arvidsson-Shukur, D. R. M., Gotfiries. A. N. O., and Barnes, C. H. (2017) Evaluation of counterfactuality in counterfactual communication protocols W. Phys. Rev. A, 96, 062316.

68 “Counterfactuals, that is, things that might have happened, although they did not in fact happen”, Penrose, R. (1994) Shadows of the Mind: A Search for the Missing Science of Consciousness, 1st ed., Oxford University Press, page 240.

69 See the section REMEMBERIG of the file Mathematics in Physics.

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caused in practically all fields of Physics. We are referring to the recoils and scientific achievements, to the innovations and conceptual rectifications, continually provoked by the Schrödinger approach. In this field of questioning that seems to never end, the composition of real and imaginary variables of the function ψ of the equation [C24] has had a determining weight.

The central question of the problem is how far the complex nature of the variables integrated in the Schrödinger wavefunction has made difficult their understanding. We spoke above the mathematical artifice developed to circumvent the calculation problem that poses to solve an equation with the complex variable ψ. In a first phase, the astute maneuver of multiplying ψ by its conjugate ψ* is applied to work with real variables. In a second phase, ψ is normalized, since only a normalized wavefunction can have physical meaning. But by substituting the complex variable ψ for the scalar ǀψǀ2 the original physical sense has been lost and the function ψ has been forced to function in the random world of the probability function. In this way, at the end of the two phases that culminate in the substitution of the complex variable ψ for the scalar ǀψǀ2 we find a double paradox: • The physical sense of the wavefunction ψ has been lost.

• The probability function ǀψǀ2 replaces the complex variable ψ, which cannot be interpreted as probability because it is complex.

Supporters of not losing the physical sense of ψ were Schrödinger himself and Einstein; On the contrary, Born defended the probabilistic interpretation. The first two held their position using the following arguments: • The wavefunction describes what the particle is.

• Reality is knowable and the process of observation is interrelated with the mathematics of physics.

• There are no barriers between the existing world and the observed world. Both are the same thing.

The main arguments of Born in favor of the probability function were the following: • The wavefunction contains the probability information of what I will measure when observing a particle.

• The wavefunction is a mathematical description to make predictions of the results of all the measurements that we can perform on the system.

That is to say, Born started from the notion of oscillating particle and that the oscillation was defined by the mathematical function ψ, although this had no representation in the real world. But counting on the severe restriction of the complex nature of ψ, real results could be obtained through the scalar ǀψǀ2, which indicated the probability of experimentally finding the particle in a defined place.

On the other hand, when through an experiment the probability of finding the position of a particle increased, the probability was reduced for other magnitudes related to the position, such as the momentum. This inverse relationship between probabilities of position and momentum, revealed by experimental practice, had its correspondence in the instrumental plane.

Indeed, position and momentum are totally unobservable jointly in practice. The more the measurement equipment is refined and perfected to observe the position of a particle clearly, the more blurred and unstable the measurement of the momentum becomes. This behavior is reciprocal: if an experiment is designed with guarantees in the stable measurement of the momentum, the measurement of the position inevitably becomes unstable.

Although Schrödinger and Einstein maintained a critical common position with the probabilistic sense of ψ, there were differences of nuance between them. The first considered each experimental measurement as a mere result, incommunicable with any 31

other measurement. The German physicist was more moderate. Einstein recognized that Born's interpretation coincided with the empirical results obtained. But the recognition did not suppose the renunciation to find an alternative function to Ç&#x20AC;Ď&#x2C6;Ç&#x20AC;2 as base of the behavior of the physical phenomena, and to this task devoted great part of its life.

Need for the utilization of complex variables Complex variables are necessary in Physics whenever it is necessary to include in an equation the pulsation ɡ or the frequency Ď&#x2026; of a wave. For example, in the exposition of the section above entitled Minkowski's four-dimensional space, the conical surface of Figure M4 that limits the horizon of an observer located at its vertex is defined by: ds2 = (dx2 + dy2 + dz2)1/2 + (cidt)2 = 0 [M10]70 This is the typical case of an equation that has to resort necessarily to complex variables.

The second member of [M10] is formed by a real spatial component, (dx2 + dy2 + dz2)1/2 and another imaginary temporal (cidt).

The use of real and imaginary variables in equation [M10] is adjusted to the explanation needs of general relativity. Its real component is the expression of the curved space and the imaginary component shows up the photons in motion. In this double manifestation the concept of general relativity is simplified, which establishes the interactions of fields and particles and that can be summarized in two essential principles: the curvature of space governs movement and movement defines the curvature of space.

It has also been said in the same section Minkowski's four-dimensional space that the equations of complex variables such as [M10] imply a double condition: â&#x20AC;˘ The imaginary component is a function of the frequency Ď&#x2026;. In effect, c = Νɡ/2Ď&#x20AC; = ÎťĎ&#x2026;, where Îť is the wavelength of the photon. The variable c = f (Ď&#x2026;) must be accompanied therefore of the imaginary unit i in the equation. â&#x20AC;˘ An angular gap Ď&#x2020; is opened between the imaginary component cidt and the complex variable ds. The open angle Ď&#x2020; between the cone generatrix and the time axis of Figure M4 defines the boundary of the observable phenomena.

Another example of the use of complex variables has been presented above in the section Alternating current circuits. In the equation that relates the potential difference of a circuit with its impedance, the imaginary unit i appears again accompanied by the pulsation ɡ.

V = ZI = (R + i (LĎ&#x2030; - 1/Ď&#x2030;C))I [C1] The second condition associated with the equations of complex variables is the opening of the angular gap Ď&#x2020; between the imaginary component and the resultant of the real component plus imaginary component. In a plane geometry such as that determined by equation [C1], the gap Ď&#x2020; between (LĎ&#x2030; - 1 / Ď&#x2030;C) I and ZI is easy to represent, as shown in Figure C1.

Taking into account the two previous antecedents, if we analyze now the first of the three identities linked to the SchrĂśdinger equation [C24], that is to say đ?&#x153;&#x2022;đ?&#x153;&#x201C; đ?&#x153;&#x201D; â&#x2030;Ą đ?&#x2018;&#x2013; đ?&#x153;&#x2022;đ?&#x2018;Ą [C25] it is immediately verified that the prediction that the imaginary variable is a function of ɡ is met again.

On the other hand, the first member of the SchrĂśdinger equation, [C24] which is equivalent to the total energy of the wave, is the imaginary component, and equal Ď&#x2030;ħ by the identity [C25]. The second member of [C24] is the real component, formed this time

70

To follow the reasoning better, the correlative numbering is interrupted here by the insertion of equations translated from previous sections of this file.

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by the first addend, the kinetic energy of the wave, and by the second addend, the potential energy dependent on the conditions of the environment. đ?&#x153;&#x2022;đ?&#x203A;š

ħ2 đ?&#x153;&#x2022;2 đ?&#x203A;š

đ?&#x2018;&#x2013;ħ đ?&#x153;&#x2022;đ?&#x2018;Ą = 2đ?&#x2018;&#x161; đ?&#x153;&#x2022;đ?&#x2018;Ľ 2 + đ?&#x2018;&#x2030;đ?&#x203A;š [C24] From the mathematical formalization of the SchrĂśdinger equation two conclusions are deduced a priori, by the mere fact of incorporating complex variables in [C24]: â&#x20AC;˘ The imaginary term of the equation (total energy of the wave Ď&#x2C6;) is a function of the frequency ɡ. â&#x20AC;˘ There is an angular gap Ď&#x2020; between the imaginary component of the equation (total energy of the wave Ď&#x2C6;) and the integral complex variable (sum of the real component and the imaginary component), which constitutes the energy balance itself.

The enunciate of the second conclusion expresses that the energy balance of the SchrĂśdinger equation is a balance affected by the strange angular gap Ď&#x2020;, which we do not know what it means. The only thing we can assure is that there is an angular opening Ď&#x2020;, of which we cannot have any geometric representation, and a collapse of the SchrĂśdinger energy balance associated with it. The meaning of this angular gap, accompanied by the energy collapse, must be related in some way to the quantum uncertainty. We can refer to the angular gap or aperture, the quantum leap and the energy collapse as different manifestations of an insurmountable mathematical discontinuity, that is, as an irregularity announced a priori by the complex nature of the variables that make up the equation [C24].

As mentioned above, the substitution of the complex variable Ď&#x2C6; by the probability function Ç&#x20AC;Ď&#x2C6;Ç&#x20AC;2 eludes the calculation problem that poses solve an equation with complex variables. But with the substitution, the certainty of the wavefunction Ď&#x2C6; is lost because of the uncertain probability function. Thus, the simple change of Ď&#x2C6; to Ç&#x20AC;Ď&#x2C6;Ç&#x20AC;2 immerses us fully into quantum uncertainty, which is nothing but the incompatibility of jointly specifying the position of the particle x and its momentum p in the SchrĂśdinger wavefunction. The loss of global definition of x and p reaches the theoretical and experimental planes: from the first, the uncertainty principle of Heisenberg recognizes that x and p are subject to quantum uncertainties; at the experimental level, uncertainties are also discovered in the measurements.

Penrose perceives a connection between both planes of uncertainty and talks about the collapse that happens to each measurement. It refers to the mysterious quantum leaps, and accepts that mathematics describes the evolution of a quantum state in the form of wavefunction extended by space, which can then be concentrated in a more localized region: "but then, when a measurement is performed, the state collapses down to something localized and specific. This instant localization happens no matter how spread out the wavefunction may have been before the measurement where after the state again evolves as a SchrĂśdinger-guided wave, starting from this specific localized configuration, usually spreading out again until the next measurement is performed. From the above experimental situations, the impression could be gained that the particle-like aspects of a wave/particle ones that show up between measurementsâ&#x20AC;?71.

With this cycle of successive states, measurement, localized collapse, evolution as wave, describes Penrose the quantum leap. The equivalent description derived from the mathematical formalization with complex variables of the SchrĂśdinger equation would be: opening of the angular gap Ď&#x2020;, collapse of the energy balance, evolution as a wave.

71

Penrose, R. (2006) op. cit., page 516-517.

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These two descriptions point in the same direction as the explanation that emerges from the equation [A9.10] of Annex 9 SchrĂśdinger equation. [xĚ&#x201A;, pĚ&#x201A;] = [xĚ&#x201A;pĚ&#x201A; â&#x2C6;&#x2019; pĚ&#x201A;xĚ&#x201A;] = iħ[đ?&#x;&#x2122;] [A9.10] 72 73 [xĚ&#x201A;, pĚ&#x201A;] is a mathematical commutator operator applied to the position and momentum of the wave function. It is equivalent, according to [A9.10], to the quantum unit of Planck iħ, of an obviously imaginary nature. The notation [đ?&#x;&#x2122;] indicates that it is the identity operator, that is to say that [xĚ&#x201A;, pĚ&#x201A;]Ď&#x2C6; = Ď&#x2C6;.

From the equation [A9.10] it follows: â&#x20AC;˘ Since the commutator of x with p gives an imaginary value, x and p are encapsulated, that is, the position x and the momentum p of a particle cannot be measured alternately. â&#x20AC;˘ The combined action of the operators xĚ&#x201A;, pĚ&#x201A; is the identity operator.

The first conclusion is equivalent to the Heisenberg uncertainty principle. From the second conclusion it follows that the position x and the momentum p provide all the complete information of the evolution of the wavefunction Ď&#x2C6; in time. But there are no matrices that can be computed as the operators [xĚ&#x201A;, pĚ&#x201A;], except matrices of infinite dimension. This impossibility of materializing the calculation of the equation [A9.10] is the mathematical manifestation of quantum mechanics Thus, the explanation of the equation [A9.10] confirms that the quantum leaps alternate the particle state, defined by the momentum p, and the wave state, defined by the position x. But the commutator's explanation goes a little further, by demonstrating [A9.10] mathematically that position and momentum completely define SchrĂśdinger's function.

This means that the position x, in the form of a wave, and the momentum p, in the form of a particle, integrally compose the wavefunction Ď&#x2C6;.

Before closing the topic of quantum leaps that concern the SchrĂśdinger equation, it is convenient to re-examine the second conclusion determined by the own condition of the complex variable of the wavefunction. We said above that there is an open angular gap Ď&#x2020; đ?&#x153;&#x2022;đ?&#x203A;š between the total energy of the wave, imaginary term đ?&#x2018;&#x2013;ħ đ?&#x153;&#x2022;đ?&#x2018;Ą of the equation [C24], and the balance established by SchrĂśdinger, sum of the real component and imaginary composition of [ C24]. It has also been said that the angular gap Ď&#x2020; represents a quantum uncertainty that is transmitted to the energy balance of the SchrĂśdinger equation in the form of collapse. This transmission is the way in which geometry participates, through the structure of complex variables, in the world of Physics. The participation scenario has its own dynamics, as vital as SchrĂśdinger's wave-particle dynamics.

To interpret the meaning of the angular gap and the energy collapse we would have to have a geometry that would allow to contrast, according to Newton's idea74, the inexactitude of mechanics (concerned with how the lines that form the basis of the geometry are traced) with the precision of the geometry (which postulates that layout).

Whatever the metric of this imagined geometry capable of explaining the mystery of the angular gap and of the energy collapse, its scale should be quantum, of the order of the Planck length calculated in [A8.5], that is, around at 1,6.10-35 m. With regard to magnitudes of a quantum field, Penrose says: 72

Here equations of the Annexes are reinserted and the correlative numbering of equations [C] is interrupted.

73 Born paid a final tribute to the commutative operator ordering to inscribe on his tombstone the equation pq - qp = h/2Ď&#x20AC;i, identical to the [A9.10] previous opportune changes of notation and making the normalized Planck constant ħ = h/2Ď&#x20AC;.

74 â&#x20AC;&#x153;It comes to pass that Mechanics is so distinguished from Geometry what is perfectly accurate is called Geometrical, what is less so is called Mechanicalâ&#x20AC;?, appointment from Newton, I. (1968) The Mathematical Principles of Natural philosophy, Dawson of Pall Mall, Authorâ&#x20AC;&#x2122;s Preface, Vol 1.

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"if one tries to measure the value of a quantum field in some very small region to great the accuracy, this will lead to a very large uncertainty in other related field quantities… Thus, the very act of ascertaining the precise value of some field quantity will result in that quantity fluctuating wildly. This quantity could be some component of the spacetime metric, so we see that any attempt at measuring the metric precisely will result in enormous changes in that metric. It was considerations such as these that led John Wheler in the 1950s, to argue that the nature of spacetime at the Planck scale of 10-33 cm would be a wildly fluctuating 'foam’ "75.

The quantum foam is a concept created by John Wheeler76 in 1955 to describe the turbulences of space-time that occur at tiniest scales such as the Planck length. The problem posed by quantum indeterminacy is that different spatial-temporal geometrical magnitudes do not compute with each other. Instead, according to the image of Wheeler, it is possible to conceive a vast superposition of different geometries, many of which would deviate greatly from flat geometry and therefore should have the foamy character that he imagines.

Within the quantum proportions, Wheeler poses a space-time where the particles and the energy exist briefly subjected to the uncertainty principle, to annihilate later without violating the laws of conservation of mass and energy. With this premise of double conservation, Wheeler tries to maintain the continuity of a space-time that evolves from quantum scales to macro scales, to even the great curvatures of space-time.

Wheeler's approach does not stop being a speculative try, inspired by the fact that energy fluctuations can be raised enough from small scales to significant energy scales. This mechanism of energy fluctuations would give the spatiotemporal frame a foamy character, which within the gravitational field could result a bubble of foam with multiple changes of topology.

Experimental and instrumental uncertainty Returning to the expression [A9.10] of Annex 9 Schrödinger equation, the commutator operator [x̂, p̂] does not clarify the mystery of how two physical magnitudes, position and momentum, which can be measured in a laboratory, give the product x̂p̂ by result, different to the product p̂x̂, and much less clarifies how the imaginary unit is present in [A9.10].

The only verisimilar explanation is that quantum uncertainty corresponds to an experimental and instrumental uncertainty. Equation [A9.10] already warns that the imaginary value of the second member means that x and p are encapsulated, that is, it announces that the position and momentum of a particle cannot be measured alternately.

Therefore, the commutating operator warns a priori, like Heisenberg's indeterminacy principle, of the uncertainty in the measurement of the position and momentum of a wavefunction.

On the other hand, the collapse of the energy balance of the Schrödinger equation, caused by the angular gap, is also related to quantum uncertainty, and also announces the experimental uncertainty. The energy balance of equation [C24] presents in the first member the evolution of the momentum p; in the second one, the variation of the position x. So, the angular gap φ, which transmits the collapse of Schrödinger's energy balance,

75

Penrose, R. (2006) op cit., page 861.

Recently dead, Wheeler is known for being part of the Manhattan Project and for coining the term black hole, among other things.

76

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coincides with the localized collapse that transmits the measurement, and finally announces an experimental uncertainty77.

Penrose refers to the paradox of the measure. The rarity that involves the experimental procedure of the Schrödinger wavefunction is linked to the quantum conflict, an open conflict between the deterministic process of evolution of ψ and the reduction of the quantum state caused at the moment of making a measurement. The quantum reduction pushes the wavefunction to give a discontinuous random jump and the definition of ψ is lost in the jump. In this critical state of the process the only possible option is to resort to the probability of the results of the measurements, considered by Penrose as simple parts of a quantum entangled state.

"The idea is that any measurement process, the quantum system under considerations cannot be taken in isolation from its surroundings. Thus, when a measurement is performed each different outcome does not constitute a quantum state on its own, but must be considered as part an entangled state, where each alternative outcome is entangled with a different state of the environment"78.

De Broglie solves the mystery of the duality of behavior of matter, as a wave or as a particle, depending on the specific experiment to which it submits. That is, the absurd of the dual wave-particle nature is resolved in the experimental facts. It is not the quantum uncertainty that anticipates the uncertainty of the measurement, but the waves are particles and the particles are waves until the experimenter finds the momentum p or the position x of ψ. In the first case, the measurement of the momentum dictates that ψ is a particle; in the second case, the measurement of the position dictates that ψ is a wave. In this way, when the experimenter measures the position, ψ (x, t) collapses in ψ (x), and with it collapses the balance of the equation [C24] by action of the angular gap φ; with the measurement of the momentum, ψ (x, t) collapses in ψ (t), and equation [C24] breaks again because of the same angular gap.

Finally, experimental uncertainty culminates in instrumental uncertainty. As every observation requires an energy exchange of photon to create the observed data, part of the energy of the observed wave has to be altered. Thus, the mere observation has a discrete effect on what is measured and we change the experiment by observing it.

For example, to measure the position of a particle, it is necessary to use short wave light, that is, high frequency, so that the diffraction does not blur the images of small wavelength. Therefore, the more precision we require from the measurement of the position of the particle, we need a shorter wavelength of the photon, which corresponds to a greater momentum. So, using high-frequency light means equipment with highmomentum photons, that when colliding with the observed particle, necessarily make it go back. The recoil implies a distortion of the measurement of the position. In this way, the adaptation of the instrument to obtain clear images of the position of a particle distorts the measurement itself.

A similar case occurs when the measure of the momentum is adjusted. During the same previous experiment, as a consequence of the photon impact a fraction of the momentum is transferred to the particle. Therefore, when the photon momentum increases, its transference grows, as well as the distortion of its measurement.79.

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To gain the precision lost in each measurement by the experimental uncertainty it is necessary to repeat measurements many times. There is no other option than the statistical calculation for any treatment of the observation data of electromagnetic waves. It is hard to believe that to establish the trajectory of an electromagnetic wave with a defined wavefunction ψ (x, t), you have to settle for the statistical solution.

78 Penrose, R. (2006) op. cit., page 785.

79 Weinberg S. (1993) Dreams of a final theory, Vintage, page 58.

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Knowledge of one aspect a system precludes knowledge of certain other aspects80 Heisenberg 5. UNCERTAINTY PRINCIPLE OF HEISENBERG When talking about probability in the SchrĂśdinger equation, the impossibility of specifying the position of the particle x and its momentum p in an experiment is recognized. We said above that from equation [A9.10] of Annex 9 SchrĂśdinger equation it follows that the position x and the momentum p are encapsulated, which means that the position and momentum of a particle cannot be measured alternately [xĚ&#x201A;, pĚ&#x201A;] = [xĚ&#x201A;pĚ&#x201A; â&#x2C6;&#x2019; pĚ&#x201A;xĚ&#x201A;] = iħ[đ?&#x;&#x2122;] [A9.10] The equation [A9.10] was the key explanatory of quantum oscillations, and penetrated fully into the uncertainty principle. Heisenberg was interested in the quantum magnitude measurements in a small environment and noticed the uncertainty they caused in other magnitudes related to it. He guessed that the mere fact of determining the precise value of a quantum magnitude caused an uncontrolled fluctuation of its measurement.

Heisenberg was referring to the precision in the measurements of non-commutative variables, such as those linked by the commutator operator [xĚ&#x201A;, pĚ&#x201A;] in [A9.10]. He confirmed the impossibility of knowing together the exact measurement of position x and momentum p because they are not commutative variables. Its postulate said that it was impossible to simultaneously specify the position and the momentum with arbitrary accuracy. The uncertainty principle or principle of indeterminacy was limited by the following inequality: Î&#x201D;xÎ&#x201D;p â&#x2030;Ľ ħ/2 [C27]81 Î&#x201D;x is the dispersion of the position of the particle and Î&#x201D;p, the momentum dispersion.

[C27] indicates that there is a limit on the precision of the measurements of the pair of position-momentum variables. But what it does not indicate is that the observation of a quantum system introduces an uncontrollable disturbance that distorts the values of these measurements. Not much less can be deduced from it that the position and the momentum change erratically, without any connection, making it impossible to establish a law that relates the dispersions of both magnitudes Inversely, the inequality [C27] establishes an accurate delimitation of the dispersion of a wave packet. In the most favorable case that the value of the dispersion Î&#x201D;x could be determined with total exactitude, the indeterminacy of the momentum Î&#x201D;p would increase by an amount â&#x2030;Ľ ħ/2. And reciprocally, if the dispersion Î&#x201D;p were an exact value, the imprecision of the position Î&#x201D;x would suffer an increase â&#x2030;Ľ ħ/2. In the extreme case of a pure momentum state Î&#x201D;p = 0 implies Î&#x201D;x â&#x2020;&#x2019; â&#x2C6;&#x17E;. And on the contrary, a state of pure position Î&#x201D;x = 0 implies Î&#x201D;p â&#x2020;&#x2019; â&#x2C6;&#x17E;.

Heisenberg included in his principle of uncertainty the equation [C28], where the Planck constant ħ, with units of energy-time, marked as in [C27] the limit of the indetermination. Î&#x201D;EÎ&#x201D;t â&#x2030;Ľ ħ/2 [C28] If the equation [C27] is measured in units of quantity of movement-space, the [C28] is measured in units of energy-time, both equivalent and comparable simultaneously to the units of action. As the record of the action is in charge in physics of the measurement, the principle of uncertainty is directed mainly to the empirical field. The enunciate in equation [C27] says that it is not possible to simultaneously and without error determine the exact value of a particle's momentum and position; equation [C28] indicates that it is 80 81

IbĂ­d, page 58.

The correlative numbering of the series of equations [C] is restored.

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impossible to accurately measure the energy of a process and the moment when the process takes place.

The relationship [C28] has a different treatment from the more familiar positionmomentum uncertainty relation [C27]. The reason for this difference is that time is considered an external parameter to the dynamic variable of energy. The interpretation of [C28] is that if the energy of a quantum system is determined by a measurement made in a range Δt, there is an uncertainty ΔE that this inequality must satisfy. This is especially relevant for unstable atomic particles or nucleus. For example, the half-life of the uranium nucleus U238 is Δt = 109 years. Since ħ = 1,054571628 × 10 -34 Js, the uncertainty of the energy according to [C28] is: ΔE ≤ 1,054571628/2.10-34 Js./109 years/365/24/3600 s/year ≈ 10-51 J Starting from Einstein's universal energy equation [A3.4] of Annex 3 Einstein's general equation of energy, we can obtain the equivalent uncertainty of the mass Δm: E = mc2 [A3.4] 82 2 ΔE = Δmc [C29] 83 Δm ≤ 10-51/9/1010 ≈ 10-62 kg 6. BIBLIOGRAPHY Arvidsson-Shukur, D. R. M., Gotfiries. A. N. O., and Barnes, C. H. (2017) Evaluation of counterfactuality in counterfactual communication protocols, W. Phys. Rev. A 96, 062316.

Born M. (1969) Is Classical Mechanics in Fact Deterministic? In: Physics in My Generation. Heidelberg, Science Library Springer.

Bose S.N. (1924) Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik, Vol. 26, Nº1: 178–181.

Dawkins R. (2006) God delusion, Transworld Publishers.

Einstein, A. (1905) On a Heuristic Viewpoint Concerning the Production and Transformation of Light, Annalen der Physik, 17: 132-148.

Einstein, A. (1995) Ideas and opinions, Three Rivers Press.

Harrison, E. (2000) Cosmology: The Science of the Universe. Cambridge University Press.

Hawking, S. W. (2005) The theory of everything. The origin and fate of the Universe, Phoenix Book. https://en.wikipedia.org/wiki/Bell_test_experiments https://es.images.search.yahoo.com/search/images.

La radiación del cuerpo negro UPV/EHU. http://www.sc.ehu.es/sbweb/fisica/cuantica/negro/radiacion/radiacion.htm.

Newton, I. (1968) The Mathematical Principles of Natural Philosophy, Dawson of Pall Mall, Vol 1.

Penrose, R. (1994) Shadows of the Mind: A Search for the Missing Science of Consciousness, 1st ed., Oxford University Press.

Penrose, R. (2006) The road to reality, eight printing, Publisher Alfred A. Knopf.

Schrödinger, E. (1926) Quantisierung als Eigenwertproblem, Annalen der Physik, Vol. 385, 13: 437-490.

The New York Times, 26 December 2005, A Quantum Sampler.

Weinberg, S. (1993) Dreams of a final theory, Vintage.

82 83

Here the correlative numbering of equations [C] is interrupted again.

The correlative numbering of the series of equations [C] is restored.

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