Quantum Interpretations (2019 Edition, 2020 QTT Update)

Contents

The Scientific Approach

The scientific methodology (often called "Effective Theory") adopts essentially a reductionist's approach, which break down the investigation into many levels mainly defined by size and from small to large (see a few examples in Figure 01). Usually, the higher level does not involve the detail of the lower one, which just provides a few parameters as linkage between the two. These parameters serve as macroscopic description of the more

Figure 01 Reductionist's Systems [view large image]

complicated microscopic domain, and often can be measured by experiment or observation.

Particle and Wave

Galaxy - The galaxies in the universe become points in the pressure-less fluid. Cosmology is a macroscopic domain in which those points are lumped into the density = Nm/V, where m is the averaged mass of the galaxies, N the total number, and V the volume of the universe. The size to system scale ratio R ~ 10-6. The dynamics is governed by the "Friedmann Equation". Planet - Its behavior is summarized in the Kepler's Law derived from Newtonian mechanics, and characterized by its mass m with the ratio R ~ 10-7. Electron - The electrons are considered as points in the hydrogen atom and other atomic nuclei. It is parameterized by its mass, charge, and spin. It is governed by the Schrodinger Equation, and can be solved analytically in hydrogen atom. The ratio R ~ 10-7 similar to the planetary motion since the interaction is long range for both cases with 1/r2 dependence. Quark - The point-like quarks are mediated by gluons to form nucleon. It is supposed to be

Figure 02 Particle, Definition [view large image]

characterized by its electric charge, mass, spin, and color charge. The ratio R ~ 10-3, which is much larger because the strong interaction in this case is a short range force. Actually, not much is known about this system.

		random directions as shown in Figure 03. The thermal energy becomes work if they can be coaxed to move in an unison direction such as in the "Heat Engine". It is called a wave when the organized motion is propagating through the medium such as the sound wave in Figure 04. Thus, particle and wave seem to be very different. Each particle is a single entity with certain characteristics, while the wave involves coherent motion of a collection of particles at macroscopic level as it is understood until the advent of quantum theory which treats these two entities as the flip side of each others and creates severe conceptual problem.
Figure 03 Thermal Motion [view large image]	Figure 04 Sound Wave [view large image]

		The wave nature of light (as Electro-Magnetic, EM wave) has been demonstrated conclusively by its diffraction pattern since 1801 in Young's experiment (Figure 05). The particle aspect of light was first revealed by Max Planck's "quanta of light" to account for the discrepancy in blackbody radiation; it was finally confirmed by the discovery of photo-electric effect (Figure 06) in 1905. It was found subsequently that electrons can produce similar diffraction as light. The diffraction pattern is attributed to matter wave (or probability wave, see Figure 07) similar in form to the EM wave but with different origin.
Figure 05 Diffraction of Light (EM Wave)	Figure 06 Photo-electric Effect [view large image]	The EM wave is actually at macroscopic level - one step away from quantum.

In 1924, de Broglie derived a relation between the wave length of the matter wave and momentum p of a particle. Starting from the relativistic version of the kinetic energy Ek = pc, which also can be expressed in term of quanta in quantum theory as Ek = hf, where the frequency f ~ c/ if the wave travels at relativistic speed, and h = 6.625x10-27 erg-sec is the Planck constant. By putting the two formulas together, we obtain the de Broglie relation :                                                             p = h/.

Figure 07 Matter Wave (Probability Wave) [view large image]

In Figure 07, P1 and P2 are the probabilities of finding the particle (from slit 1 or 2) along the screen , and P12 is the result of interference - hence the nomenclature of "probability wave".

Quantum Mechanics and Quantum Fields

Figure 08a Traveling and Standing Waves [view large image]	It shows that starting from a non-linear equation for r, i.e., (m/2)d2r/dt2 + V(r) = E, quantization has it linearized into the Schrodinger equation for the wave function which admits superposition of its solutions (as another form of solution). Thus, it is the theoretical design and experimental evidences that bring about the weird world of superpostion.

Figure 08b Standing Waves [view large image]	Figure 08c Superposition	See "Harmonic Oscillator" and "Infinite Square Well".

		There is a deeper level of quantum theory than the particle and wave as mentioned above. In the Standard Model (SM) of elementary particles, it is the quantum fields that are ubiquitous in the formulation. In contrary to the classical fields such as the electro-magnetic fields in Maxwell's equations and the metric tensors in General Relativity, in which there is always a source such as charge and current or mass-energy tensor to generate the fields, the quantum fields in SM has no source - it is assumed to be everywhere and at all time in this Universe. That's why there is always a
Figure 09 Quantum Fields [view large image]	Figure 10 Virtual Particles [view large image]	suspicion about an even deeper level to describe the detail of their origin - according to the the reductionist's view point. See "Quantum Fields and Vacuum Energy Density" for more on vacuum state.

Quantum Field Characteristics :


• As shown in Figure 09, the various quantum fields permeate throughout the Universe at ground state with virtual particles popping out incessantly according to the Uncertainty Principle (Figure 10). The particle becomes real when enough excitation energy is injected into the field. It is suggested that at the end of the inflation era, the inflaton field (not yet identified) decayed from false vacuum to true vacuum, all its energy was converted to the particles in this world. Since then, there is the Higgs field, which also existed in false vacuum state and then settled down to the true vacuum level of 125 Gev above the zero point energy. It permeats throughout the universe uniformly and isotropically, and endows mass to other particles. The interactions between the particles and the Higgs maintain a meta-stable state which could end anytime according to one calculation (see "Stability of the Universe").



There are 12 fermion and 12 boson fields in the Standard Model. The Largranian L for leptons is shown in Figure 11. It contains all the lepton fields (waves) and the interactions between these fermions and bosons. The Largranian L1 contains 1 EM , 2 weak boson fields; L2 has 6 leptonic fermion fields; L3 is for the Higgs boson. At this level, we are dealing with waves excited from the underlaying fields, i.e., one step up from the more or less featureless elements (see Figure 09). The interaction is expressed in the covariant derivative D. There are altogether 3 coupling constants - g and g' are associated with the interaction between the gauge bosons W and B and the Higgs respectively, Ge is the Yukawa coupling constant, which defines the interaction between the Higgs and the lepton. The first three terms in Eq.(41c) are responsible for generating the mass of the gauge bosons, while the last term takes care of the fermion mass.

Figure 11 Lepton Largranian for WS Model of SM [view large image]

The quarks have a slightly different form of the Lagrangians by the title of quantum chromodynamics (QCD).

• The fields in SM actually are traveling waves. The governing equations for them can be derived from the Largranians. The explicit forms are: the Higgs Boson Equation for scalar wave, and the Dirac Equation for the 1/2 spin fermion waves (with replaced by D for interaction with the gauge boson wave). Similar to the interpretation of "wave" in quantum mechanics, the "wave" in quantum field theory can be the traveling kind propagating in free space or the standing variety being spatially confined.


• The waves are usually expressed in 4-momentum space via Fourier transform in which the coefficients become q-number, i.e., they are operators obeying the commutative relations for boson and anti-commutative relations for fermion which carries extra-label such as spin. The extra-label for spin 1 wave is the polarization. This process is called quantization or 2nd quantization to transform waves into particles. The particles are thus endowed with energy-momentum plus any extra-label. The system is characterized by the state vector to indicate the number of particles of certain value of 4-momentum, spin, etc.



QCD has solved the highly complicated mathematics in SM numerically by using super-computer. By the method of Lattice Theory, the theoretical concept of quark confinement is visualized pictorially in Figure 12, in which the quark wave has been transformed to particle (the small sphere) while the massless gluons are represented by flux tubes (see some QCD animations in "Visualizations of Quantum Chromodynamics").

Figure 12 Quarks and Gluons [view large image]

• Another way to extract some useful information from SM is through Perturbation Theory. It is particularly suitable for Quantum Electrodynamics (QED) with smaller coupling constant than the strong interaction in QCD. The "quantum field" in this method is considered to be freely traveling wave, which is quantized to turn into particle. Such particle can have very high energy useful for probing the interior of very small size object in term of cross section, life time etc. The interaction in this scheme is not in the covariant derivative anymore. It is treated as a small correction in the calculation of transition probability.


• There are well defined Feynman rules to evaluate the transition probability or S-matrix and ultimately the cross section for various processes. The S-matrix element S_if defines the probability of a process from an initial state "i" going to final state "f". Usually, one initial state can produce one or more final states as shown in Figure 13, where three different initial states are taken as examples - namely, the electron positron scattering, the Compton scattering and the deep inelastic scattering. Each of this process produces many final states, but only a few have been shown just for illustration purpose. The S-matrix is a complex number in general. It has to satisfy the unitary condition, i.e., _f S*_ifS_if = 1, which guarantees that probability is conserved in the process. Such relationship

  indicates that the matrix element Sif has an inverse, which in turn implies that it is possible to return to the initial state from the final state at least in principle although the probability is almost zero in practice so that the second law of thermodynamics is "almost" never violated. This property is also related to the so-called "conservation of information", which caused so much trouble for Stephen Hawking in the "Black Hole Information Paradox".

  Figure 13 S-Matrix [view large image]

  Note: For example in the electron positron scattering process, there are three possible finally states as illustrated in Figure 13. The sum of the probabilities for each one of these has to be: S*11S11 + S*21S21 + S*31S31 = 1 to insure that the final state is in one of the three possibilities.

Superposition and Entanglement

		Similar to the superposition of waves in quantum mechanics, the unitary condition f S*ifSif = 1 is another kind of superposition. That is, given an initial condition there are many possible outcomes until an observation or measurement is made. It is not clear why people would have problem with such process and relates it to "animated suspension". They seem to avert at seeing such process in black and white or in formula. But the realization of probabilities happens all the time as shown in Figure 14 and the chance of lottery win (Figure 15).
Figure 14 Possibility in Life [view large image]	Figure 15 Probability in Lottery [view large image]	The unitary condition for Lottery can be expressed as : f Pf = 1, where the subscript f stands for the 6 digits out of 49, and Pf = 1/13983816 is the chance of winning for each f in fair play, 13983816 is the number of combination.

	For example, in case the game is rigged such that only 6 balls (digits) would fit the exiting hole (representing the hidden cause to be discovered or invented), then the unitary condition is still : f Pf = 1, but now all Pf = 0, except Pw = 1 for the winning combination w, i.e., this outcome is certain.
Figure 16 Bell's Theorem [view large image]	This is an extreme example of Bell's Theorem, which was proposed to test if there is any (not yet discovered) hidden elements in quantum theory. Since then, many tests have been performed to show there is no hidden variables in quantum theory (no cheating).

		What irked Einstein and friends even more is the phenomena of entanglement which is the superposition of states between two or more particles (Figure 17). It bothered them to witness the simultaneous collapse of corresponding states even when the detectors are separated beyond the distance allowed by the speed of light (called non-locality). They blamed the in-completeness of quantum theory and proposed the "Hidden Variables" interpretation as mentioned above.
Figure 17 Entangled States	Figure 18 Entanglement, Non-locality [view large image]	Traditional definition of "probability" is : "likelihood", "chance", ... always associated with incomplete knownledge. In quantum theory however, "probability" can be calculated from well-defined equation and becomes deterministic, which is the oft-missed difference.

After all is said and done up to this point, there is still plenty of doubt about how to make sense of the quantum theory. For example, Sean Carroll in his 2019 book "Something Deeply Hidden" states that "the Copenhagen approach is difficult to swallow for several reasons. Among them is the fact that the wave function is unobservable, the predictions are probabilistic and what makes the function collapse is mysterious." It seems that the decoherent interpretation is still ignored by many people, physicist included. They still believe that the "Many-Worlds interpretation" is more creditable (Figure 18a). In another 2019 article "Proof of Parallel Universes ?", it is argued that instead of proofing non-locality is correct; the Bell test could mean the existence of parallel universes (while local causality is preserved).

Figure 18a Parallel Universes [view large image]

Then in his 2019 book titled "Einstein's Unfinished Revolution" Lee Smolin mentions that decoherence cannot be derived from the Schrodinger equation because the latter is reversible while former is irreversible (as measurement cannot be undone). The problem can be resolved by invoking the "Quantum Poincare Recurrrence Theroem", which permits the reversibility of measurement under certain rarely occurred circumstance. The other way is to invent another equation such as the Pilot Wave Theory" by de Broglie to describe the measurement process. Quantum mechanics is not complete without such supplement to address this so-called "measurement problem" - his said.

The same Lee Smolin has a even more radical idea to formulate a new theory which would merge General Relativity with Quantum Theory, i.e., another "Theory of Everything". He repeats the same kind of criticism of quantum theory in an August 24, 2019 New Scientist article "Beyond Weird". In his view, the faults of quantum theory are in its mysterious nature (superposition, entanglement), the phenomena it doesn't explain (dark matter, dark energy), the answers it doesn't give (why gravity

Figure 18b New Reality [view large image]

is left out ???), and the measurement problem as stated earlier. The foundation of his new theory is based on a "Manifesto" including 6 hypotheses as shown in Figure 18b. He claims certain successful attempts so far, but there is still a long way to go.

phenomena is different from "Quantum Theory" for small scale particle. Problem occurs only at the beginning of the universe, which becomes very small and requires quantum treatments. Actually, such combination has been successful implemented in "Origin of Time (2019 Version)" by "Quantization of the Friedmann Equation (Matter-only)". Lee Smolin's obsession to unify the large and small domains would be cooling down if he accepts the view that mathematics has its own limitation; it is not designed to represent the reality of the world. It is to the credit of theoretical physicists who managed to work out some equations at different levels that can be stitched together to portray a picture of the whole (see "The Eight Most Important Equations in Physics").

Figure 18c Theory Of Everything (TOE) [view large image]

Progress in theoretical physics can be served better by building on existing foundation, rather than starting anew from scrap turning everything upside down.

Decoherence and Quantum-Classical Boundary

Way back in 2004, half-a-dozen experiments have been designed to determine the boundary between the classical and quantum world. One experiment shown in Figure 19 fires C70 fullerene balls (70 carbon atoms in the soccerball-like crystal of about 1 nm across) at 190 m/sec toward two diffraction gratings. The first grating creates the matter wave from the fullerenes. The wave is then diffracted by the second grating and the interference pattern is formed on the detecting screen demonstrating the wave property of the fullerenes. However, the interference pattern fades away if the fullerenes are heated by a laser heater (to about 2700oC) or collide with gas (leaking into the vacuum chamber of the experiment). No one has a definitive answer for how the striking photons or molecules switch the quantum to classical domain. One explanation is that the interaction causes an uncertainty in the position of the fullerenes, blurring the interference pattern. Another argument asserts that the disappearance of the quantum property is caused by entanglement between the photons/molecules, the fullerenes, and the rest of the world (the wall of the chamber).

Figure 19 Matter-Wave Experiment and quantum-classical boundary

See explanation by "Decoherence".

Interpretations

In an article commemorates the 50th anniversary of the "New Scientist" magazine, Roger Penrose suggests that there are three kinds of reality: the physical, the mental and the mathematical, with something (as yet unknown) profoundly mysterious in the relations between them. According to this view, the various "Quantum Interpretations" are attempts to link the mathematical reality to the physical or mental reality. Figure 20 shows the mathematical reality as the patterns of interference computed from a mathematical formula, while the physical reality is in the form of photographic plate with the darker strip corresponding to the higher value of the curve. The mental reality is the image of dark and white strips formed in the retina and perceived by our consciousness.

Figure 20 The Reality [view large image]

Table 01 Status of Various Quantum Interpretations

Quantum Trajectory Theory (QTT) - Resolution of the Quantum Measurement Problem? (2020)

		A modern interpretation around early 21st century suggests that the measurement is a continuous process similar to the diffusion of an air particle going from one spot to another making collisions with other particles many times on its way. In the quantum world, the particle is identified to the quantum state while the interacting particles in the intervening space are replaced by the Hilbert states of the environment (such as the the detector for the measurement), which is a collection of huge number of particles (see a much simplified illustration in Figure 22). This theory about quantum paths is called Quantum Trajectory Theory (QTT), while the process is what used to be called "decoherence".
Figure 21 Two States Superposition	Figure 22 Quantum Trajectory Theory (QTT) [view large image]	Meanwhile in 21st century, the technology in time scale measurement has been refined to the level of nono-second. It enables experiment to trace the time development of the collapsing process as shown in Figure 23,a.

The 2019 experiment uses an artificial atom made of superconducting qubits having a ground state G and an excited state D. Microwave is used to excite the "atom" from ground to excited state. It is found that back-action (actually decohence) keeps sending the superposition (in the form of 2 linear probability functions for G and D) back to the Ground state. Then transitions to the excited state occur after a quiet interval (see Figure 23,a). The sequence of events happen in time scale of about few hundred micro-second - definitely not instant as thought 100 years ago.

Figure 23 Modern Quantum Jump Experiment (ill.) [view large image]

Here's the details (see the original article "To Catch and Reverse a Quantum Jump Mid-flight") :

• The artificial atom actually has three energy levels (Figure 23,b), i.e., the ground level G (black dots in Figure 26), excited level D (for Dark, red dots) and an auxiliary level B (for Bright, blue dots). The G-B transition running in the background is to monitor whether a transition has occurred from G to D. It keeps on re-emitting the (G-B)microwaves (the black lines in Figure 26 with time scale ~ 1 s) in absence of D level occupancy, otherwise there is a switch to turn it off. Such processing is necessary to avoid disturbing (decohering) the G to D evolution while monitoring.



Rabi Drive is used to prepare two-lavel G-D superpositions in cycles of about 350 s (Figure 24). For illustration purpose, the superposition of the G and D states is assumed to be the sum of 2 linear probability functions subjected to the conservation of probability PG + PD = 1 as shown in Figure 23,a and similarly in Figure 24. The quantum trajectory is any one of those evolution tracks (in Figure 23,a).

Figure 24 Two-Level Rabi Drive

• It is found that the G-D transitions in the early phase are interrupted by the back-action (decoherence) with the re-emission of
  (G-D)microwaves (the black lines in Figure 26) . The interval between the lines is typically about few s. This phase of the evolution is shown in the upper left quadrant of Figure 23,a (the time interval there between the lines is much exaggerated).


• This kind of action goes on for about 200 s until the mixture of G/D probability is close to 50-50%, then the system becomes quiet for a duration (designated as t_catch, see Figure 26) more than those for the successive blue or black levels.


• After this silent period, the process resumes to complete the excitation via decoherence. It lasts for about 100 s (designated by t_D, see Figure 26) as shown in upper right quadrant of Figure 23,a and the red dots (from the excitation detector) in Figure 26.


• Figure 23,c shows the jump rate from B to G and from G to D (hence the "not-B"). The BG rate is faster and numerous, while the G to D counts decline to 1000 times fewer toward the end of the cycle (counts normalized from 3.2 seconds of data gathering).


• Figure 23,a does not show the process of "collapse" (de-excitation) from the D level back to the ground state (with a time scale ~
  30 s). The sequence for collapse is similar to excitation with back-action in the beginning and eventually return to G level with the emission of (G-D) microwave (the black lines amid the red dots in Figure 26).



A time-record of the experiment is shown in Figure 26. It displays the RF signals demodulated to pair of In-phase and Quadrature components (Irec, Qrec) which record the state of the atom every 0.26 s by 10 photons or least. This kinds of signaling can retrieve more information because there is two ways to encode the data instead of only one to do it (see Figure 25 for a very simple illustration of its operation).

Figure 25 RF Signal and I, Q Basebands [view large image]

	Beside proving the case for QTT and the process of decoherence, the researchers are delighted by discovery of the tcatch interval, which can be used to increase the coherence time (for entanglement with another qubit) in quantum computing by reversing the process upon the detection of tcatch.
Figure 26 Neo-Quantum-Jump Experiment, Time-Record [view large image]	See "Has Quantum Theory's Greatest Mystery Been Solved ?" for an Introduction to QTT and the Experiment.

Such effect has been observed experimentally by observing the "entanglement measure" for the electrons in various diatomic molecules. It is drastically shown in Figure 03k where the H2 molecule having 2 electrons attains much higher entanglement than the Cl2 molecule having 34 electrons. Thus, the collapse to an unique state by decoherence can be attributed to the weakening of the entanglement measure to zero by all the Hilbert states in the environment.

Figure 27 Entanglement [view large image]

See a detailed treatment of " Quantum Decoherence".