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Quantum Interpretations (2019 Edition, 2020 QTT Update)


The Scientific Approach
Particle and Wave
Quantum Mechanics and Quantum Fields
Superposition and Entanglement
Decoherence and Quantum-Classical Boundary
Quantum Trajectory Theory (QTT) - Resolution of the Quantum Measurement Problem? (2020)

NB - Sometimes, different interpretations are different perspectives on the same thing. It is just different point of views. However, there could be instance when different interpretations are actually not looking at the same things or some views are distorted.

The Scientific Approach

Many scientific theories are mis-interpreted due to mis-understanding of "how science work". It often leads to absurd conclusion and is proclaimed as evidence to invalidate the theory.

Reductionist's Systems 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.

The human knowledge of nature were limited by our five senses over thousands of years. Science has gradually expanded the view upward in the field of astronomy and downward to unseen molecules and at levels much much smaller. It is at such level of tiny size that our intuitive point of view break down. It turns out that the quantum theory can provide a platform to match theory with experimental data faultlessly. However the mathematic formulas do not have a correspondence of a world confined by our five senses. Since then people had tried to make sense of this newly discovered world according to our limited experience often leading to paradox. Thought experiments had been connoted and publicized by mixing up the working levels because it seems funny, but it is not science.


Particle and Wave

Since both classical and quantum theories are developed under the assumption of dealing with particles, it is necessary to clarify "what does a particle mean".

A particle is a point mass with no spatial extent and thus no structure. There is a more practical approach on the definition of particle in physics such that a galaxy, a planet, electron, quark can be considered as a point as long as the scale of the system under deliberation is much larger than the size of the object. Figure 02 shows some examples with the ratio of object-size/system-extent = r. Even though the structure of the "point" is not important, it is often specified by some parameters such as mass, charge, spin, ... It is actually a reductionist's approach by ignoring finer details in the lower levels (of description) in order to learn something about the higher levels (see "Effective Theories"). The examples in Figure 02 are described briefly below :
    Particle, Definition
  1. 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".
  2. 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.
  3. 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.
  4. Quark - The point-like quarks are mediated by gluons to form nucleon. It is supposed to be
  5. 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.
An ensemble of particles is called a medium which is considered to be macroscopic. The motion is thermal if the particles move in
Thermal Motion Sound Wave 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]

Diffraction of Wave Photo-electric Effect 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.

Matter Wave 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".
See a "footnote" on the mathematics of the Double-slit Experiment.


Quantum Mechanics and Quantum Fields

The starting point for quantum mechanics is the non-relativistic energy equation E = p2/2m + V, where the linear momentum p = mv and V represents the potential energy. As such, it it suitable only for low energy (non-relativistic) physics. It is written down mostly for only one particle, the generalization to many particles is possible in form but useless in practice because it is too complicated to solve. Quantization turns the formulation into describing either traveling wave or standing wave (see Figure 08a, which depicts an incoming traveling wave turning into standing wave after it becomes trapped in a potential well). Such wave is interpreted as the probability of finding the particle in certain spatial point and other variables like spin, ... Quantum mechanics is most useful in describing the structure of atoms and molecules as the motion is usually non-relativistic (v << c). It would yield inconsistent results for high energy particle collision as the formulation is not Lorentz invariant, the motion would appear to be different from different inertial frame of reference.

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.

Standing Waves Superposition

Figure 08b Standing Waves [view large image]

Figure 08c Superposition

See "Harmonic Oscillator" and "Infinite Square Well".
The particle usually occupies a stationary state (i.e., the eigen state) at ground level. Superposition is created by interaction with some kind of force. The wave functions in the superposition move in unison (coherence). It only breaks up (decoherence) in the process of measurement. The detector would pick out one of the states according to the probability. This is called the collapse of the wave function in the jargon of "Copenhagen Interpretation". Such concept seems to be alien for some people, and in particular to the pure theorists and philosophers, who are very un-comfortable with such arbitrary rule, i.e., getting the result of a measurement or perception by chance.

Quantum Fields Virtual Particles 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.
The excited quantum field (quantum wave) is expressed as superpostion of traveling waves (in 4-D spacetime, see "Second Quantization and Feynman Diagram") :

The coefficients ck and (ck)* are quantized to become operators operating on the (Fock or Hilbert) space of "particle number" to determine the creation, annihilation, and number of real particles (see "Second Quantization and Feynman Diagram" again). Thus, either in Quantum Mechanics or Quantum Field, the wave aspect always involves superpostion - the same as all the other kinds of wave. The problem with quantum wave seems to be related to the label of "probability", because the senses of human are deterministic, i.e., they always consider the received information to be un-ambiguous.


Superposition and Entanglement

Possibility in Life Probability in Lottery 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.
Proponent of the "Hidden Variable" interpretation claims that if we can discover the underlying mechanism of the quantum theory, that theory would describe the dynamics in a deterministic way (similar to the good old classical theory), not just some probability.
Bell's Theorem 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).

Entangled States Entanglement, Non-locality 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.

Parallel Universes 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]

Unfinished Revolution 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.
Actually, the Schrodinger equation is applicable only to single particle or at most a few more using approximation. Decoherence involves so many particles turning the system into thermodynamic-like. Thus, the process is similar to the Second Law of Thermodynamic, i.e., it is nearly irreversible - as invoking the "Quantum Poincare Recurrrence Theroem" for certain condition mentioned above.

New Reality 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.

By definition, "Theory of Everything" (TOE, Figure 18c) is incompatible with the concept of "Effective Theory", which adopts different description for different system (at different size level for example). There is no problem if "General Relativity" for large size scale
Theory Of Everyone 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

The explanation on realization of the quantum probability by the previous examples may have committed the same fallacy of mixing the reductionist's levels to cases in Figures 14, 15. The idea of "Decoherence" is more convincing using mathematics and microscopic object.

Rather to repeat the whole section here, see the original article in "Decoherence".

Matter Wave 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".

Report in 2007 indicates that quantum effects have been detected in objects as large as buckyballs, but not viruses. The boundary seems to be located at about 10 nm (see Figure 19). It is found that beside decoherence, the coarseness of detecting instruments also plays a role to induce transition from quantum to classical reality. Nevertheless, there are many more attempts to apply quantum theory to macroscopic world such the "Many World Interpretation" and the "Quantum Universe" with dubious validity.



The older version of various quantum interpretations is not going to be iterated here except from some links in Table 01. Here's some comments which seem to be rather enduring with the passage of time.

Reality 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]

Another scheme is to consider the physical reality as primary, while the mental and mathematical realities as secondary. In this view, some processing steps are required to arrive at the secondary reality such as neuro-activity and computation. But the secondary reality does not always produce a corresponding primary reality. For examples, dreams and other altered mental states are not real; and mathematical formulas can generate result, which has no match in reality (unless the concept of multiverse is invoked).

As the late Stephen Hawking points out in an article from "Extreme Physics" (published by Scientific American, 2013) theory at different level may have its own version of reality and none of them can be said to be more real than any other. It is similar to the concept of "Effective Theory", which employs different method to describe different system.

In its January 22-28, 2011 issue, "NewScientist" has an article about various quantum interpretations with current status (as listed in Table 01 below). Only the most popular ones are included. The others with status such as uncertain, endangered, rare, highly contentious, and popularity limited, are left out.

Interpretation Feature Status
Copenhagen Measurement plays a key role in decoherence of quantum states Healthy
Hidden Variables Hidden variables carry missing information about quantum states Challenged
Many Worlds All quantum possibilities play out in a multiplicity of universes Dubious
Penrose Outcome of experiments is a result of gravitational interactions Under investigation

Table 01 Status of Various Quantum Interpretations

Anyway, all the interpretations seem to be redundant now as the process of decoherence is one level up to explain the so-called "measurement problem" (the discontinuous jump from superposition of many states to just one outcome by measurement). In the sense that it can be expressed in mathematical formula and can be related to observation (such as dependence on environment for the coherence time of the entangled qubits).


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

Quantum mechanics is a very successful theory passing every check and verification. One of its features is superposition of wave functions. Figure 21 shows the superposition of two wave functions for the case of "Infinite Square Well Potential". The superposition varies over certain cycle (as indicated by color codes for the time), it would repeat the cycle over and over without end according to theory. The superposition only "collapses" to a definite eigen-state (unique state) by a measurement. The Copenhagen interpretation asserts that the collapse happens instantly without delay (dubbed quantum jump) - an idea rejected by both Einstein and Schrödinger.

QTT QTT 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.

QTT 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 process of decoherence can be further elaborated in term of entanglement. Tow qubits has a maximal degree of entanglement. It becomes weaker when there is a third one joins in (see Figure 27), i.e., the more partners the weaker. The general formula is :
E(A|B|C1|C2|C3| ... Cn) = E(A|BC1C2C3...Cn) - E(A|B) - E(A|C1) - E(A|C2) - E(A|C3) - ... E(A|Cn) 0,
where E stands for "Entanglement Measure". In terms of the two-level artificial atom, "A" can be identified to the ground state "G", "B" to the excited state "D", and the "C's" represent the Hibert states in the environment.
Entanglement 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".