## Is Quantum Mechanics Just a Special Case of Classical Mechanics?

The bouncing is antiphase in (a) and (c) and in-phase in (b) and (d). From “Why bouncing droplets are a pretty good model of quantum mechanics.”

If you enjoyed my post from about three months ago on Hydrodynamic Quantum Analogs, or perhaps even if you didn’t, you will likely enjoy this new paper by Robert Brady and Ross Anderson at the University of Cambridge: “Why bouncing droplets are a pretty good model of quantum mechanics“.

They discuss the recent experimental work with silicon oil droplets bouncing on vibrating trays and the behaviors exhibited by these systems; behaviors normally associated with quantum mechanical systems.  They quantify the effective forces between the droplets and their environment.  Then, they go on to show how the classical physical and mathematical description is equivalent to the Schrodinger equation, with the substitution of a surrogate parameter for Planck’s constant.  See “Droplets moving on a fluid surface: interference pattern from two slits” for a similar discussion.

## Quantum Mechanics and Spin Statistics

What is perhaps most intriguing about this new paper is the demonstration of how spin-half behavior can arise in these classical systems.  One of the central characteristics of quantum physics, and indeed an essential feature of our Universe, is the difference between fermions (particles with half-integral spin; electrons, neutrinos, protons, etc.) and bosons (particles with integer spin; photons, W and Z bosons, etc.).  This feature results in these two classes of particles having completely different statistical properties; Fermi-Dirac statistics for fermions, and Bose-Einstein statistics for bosons.  It is what leads to the Pauli exclusion principle and the stability of atoms.  The overall wavefunction for a boson is an even function and the overall wavefunction for a fermion is an odd function.

A peculiar feature of fermions that is reproduced in the hydrodynamic wave field is the fact that, if the direction of a fermion’s angular momentum is rotated through 360 degrees, its wavefunction changes sign.  This has typically been assumed to be an exclusive behavior of the quantum realm.  But here it is, in a table-top, classical experiment.

## Is Quantum Mechanics Just a Special Case of Classical Mechanics?

These experiments continue to provide tantalizing and provocative insights into the quantum world, challenging our notions and assumptions.  Some of the questions that come to mind as I ponder the implications for interpretations of quantum mechanics in general, and de Broglie-Bohm pilot wave theory in particular, include:

Is Quantum mechanics just a special case of classical mechanics?  Is quantum physics simply a subclass of events where we recognize certain behaviors over other noise and interference?

What are the quantum parallels for the effective external forces in these hydrodynamic quantum analogs, i.e. gravity and the vibrations of the table?  Not all particles carry electric charge, or weak or color charge.  But they are all effected by gravity.  Is their a connection here to gravity? Quantum gravity?

In addition to helping us understand quantum mechanics, can these or similar experiments help us understand general relativity as an effective force?

What are the technical challenges to doing these experiments in a microgravity environment, like the International Space Station?  What about somehow curving or warping the oil surface?

Why does the quantum world seem to have no energy (or frequency or wavelength) dependence for the limiting speed c (contrary to hydrodynamic quantum analogs)?

Until next time, have fun pondering!

# What Would Happen if a Quantum Cheshire Cat Were to Visit the Leisure Hive?

Happy Holidays, Everyone!  Today’s article, just in time for your New Year’s Eve party, is on something extremely cool.  It has to do with a paradox that is completely unintuitive and that is only revealed by weak measurements.  A particle and its properties can be in different locations!

In the classic Doctor Who episode Leisure Hive, a so-called “science of tachyonics” serves as the basis for entertaining guests at a resort.  A person enters a booth and their head and limbs are seemingly separated from their body, yet remain animated and are then harmlessly reattached.

That is, of course, full-fledged science fiction.  However, a quantum particle such as a photon, an electron, or an atom, apparently can have its properties located in a position separate from the particle itself.

Recent theoretical and experimental work has invigorated the search for “quantum Cheshire cats”.  Before I continue, however, I want to stress that the reference to cats is strictly metaphorical.  Just as with the case of Schrödinger’s cat (Decoherence and the Quantum to Classical Transition; or Why We Don’t See Cats that are Both Dead and Alive), decoherence prevents macroscopic objects from displaying these quantum mechanical properties.

# In Search of a Quantum Cheshire Cat

The authors of Quantum Cheshire Cats (also available here) define a “quantum Cheshire cat” as a photon that is in one location while its circular polarization is in another.  The metaphor comes from the Cheshire cat in the story of Alice in Wonderland, whose smile persists independent of the cat:

The “cat” is the photon and its “smile” is the photon’s circular polarization state.  The photon is in one of two possible locations, the left or right side of a modified Mach-Zehnder interferometer.  Using weak measurements, including cleverly chosen pre-selected and post-selected states, leads to a sample of events where the photon went through the left arm with certainty.  However, a polarization detector in the right arm can still see a signal!

“We seem to see what Alice saw—a grin without a cat! We know with certainty that the photon went through the left arm, yet we find angular momentum in the right arm.”

The paradox is removed if conventional, strong measurements of position and polarization are performed.  The inevitable and apparent wave function collapse occurs and the photon’s position and angular momentum are found to be co-located.  This is analogous to directly measuring which slit the particle goes through in a double slit experiment, which prevents an interference pattern from forming.  Strong measurements are analogous to turning the light on and letting the cockroaches quickly scurry into hiding.  Everything looks normal.  But, weak measurements are like peering at what is going on in the dark, without scaring the roaches away.

Using weak measurements (The Strength of Weak Measurements in Quantum Physics), the disturbance on the state of the system can be reduced by accepting less precision.  Then, the measurement is repeated many, many times to achieve the desired accuracy.  This reveals that the circular polarization was in fact in the right leg of the interferometer while the photon was in the left, for certain pre- and post-selected events.

# What Do We Do With a Quantum Cheshire Cat Once We Catch One?

Conventional wisdom is that when you look at, or measure, a quantum system, the wave function collapses into something that makes sense from a classical level.  That is to say, strange or apparently contradictory paradoxes disappear.  However, that assumes strong measurements.  Until weak measurements were explored theoretically and experimentally in recent years, the distinction between strong and weak measurements was not appreciated.

Contemplating the implications of quantum Cheshire cats opens up several mind-boggling possibilities and opportunities.  Separating physical properties, such as mass, energy, charge, magnetic moment, etc., from what we conventionally understand to be a particle could lead to new and more precise measurements, new technologies, new materials…  Additionally, it has profound implications for our conceptual understanding of quantum physics and what a quantum system is up to between measurements or between interactions.  Scientists will be exploring this amazing field for many years to come.

In Quantum Cheshire Cats, the authors discuss a couple modifications (beyond the reach of existing technology, but should be possible eventually), where the signature of a quantum Cheshire cat should be unambiguous; ensembles of electrons, for example.

Proposed modifications to the setup discussed above, i.e. using entangled pre- and post-selected states to allow the linear as well as the circular polarization states to be separated from the photon, are discussed in The Complete Quantum Cheshire Cat.

Possible hints of the metaphorical quantum Cheshire cat have been seen: Observation of a quantum Cheshire Cat in a matter wave interferometer experiment  “…using a neutron interferometer… The experimental results suggest that the system behaves as if the neutrons went through one beam path, while their spin travelled along the other.”

The quantum Cheshire cat is an example of an interaction free measurement.  Another example is the Elitzur–Vaidman bomb tester, also known as a quantum mechanical bomb tester.  Also see Using quantum mechanics to detect bombs.

Masters student Catherine Holloway lectures on the science behind a quantum bomb detector at the Quantum Cryptography School for Young Students, held at the Institute for Quantum Computing, University of Waterloo:

## The Strength of Weak Measurements in Quantum Physics

1940 Charles Addams cartoon for the New Yorker.
Which way did the skier take around the tree?

You may recall being told by your parents, as you were growing up, outmoded ideas or outright misconceptions about quantum mechanics.  Examples may have included: the uncertainty principle is due to momentum imparted by photons as you measure a particle’s position; in any given experiment you can observe wave or particle properties but not both; the wave function is a mathematical tool and not part of objective reality; you cannot ask what a particle is doing between measurements; you cannot simultaneously determine position and momentum; the phase of a wave function is not observable; you cannot discuss reality separately from what you choose to measure; you cannot ask what is really there when no one is looking; “just shut up and calculate”.

Maybe it was while you were in college rather than while you were growing up.  And perhaps it was your quantum mechanics (QM) instructor rather than your parents.  Nonetheless, much of what has been written and taught about QM since its inception has misled students, teachers, researchers, and the general public about the implications it has for reality and observation.

The above notions, and many other bits of nonsensical interpretational issues are being clarified and sometimes overturned by talented theorists and experimentalists.  These explorers continue to peel back the curtains to see what is really going on behind the cloak of quantum weirdness.   The techniques of weak measurements have become vital tools in this quest.  The article I wrote a couple weeks ago, Discontinuous Trajectories in Quantum Mechanics, was an example of weak measurements.  Today, I discuss weak measurements used to reconstruct particle trajectories in a double-slit experiment.

# Imagining Weak Measurements

Weak measurements were initially proposed by Yakir Aharonov, David Albert, and Lev Vaidman about twenty five years ago.  The idea is basically this: you prepare particles in some particular initial state (pre-selection) and later detect some of them in a particular final state (post-selection).  It may be that only a small subset of your particles end up in the particular final state that you make the selection on, but that is ok, you are going to repeat the experiment many, many times.

You want to know what your particles are doing between these initial and final states; how they get from point A to point B, for example.  So you need to do some measurements.  However, that would lead to apparent collapse of the state vector into a particular eigenstate, essentially re-setting the experiment.  If you measure the particle’s position somewhere along its trajectory, the momentum becomes uncertain and uncorrelated with any initial momentum it may have had.

The key insight that these gentlemen had was to make the disturbance from this intermediate measurement as small as necessary, so as not to disturb the wavefunction too much.  When measuring the position of a particle in a weak measurement, the velocity does not become random.  However, the uncertainty in the position measurement is large.  So, the second trick is to average over a very large number of trials.  This leads to precise information about the wavefunction itself.

# Weak Measurements in Action

In a Physics World article, In Praise of Weakness, Aephraim Steinberg and his colleagues discussed their use of weak measurements to map particle trajectories in a double-slit experiment.  Their article is also available here.  The green 3D plot below shows where a quantum particle is most likely to be found as it passes through the double-slit apparatus while behaving as a wave. The black lines on top of the green 3D surface are the average paths that the particles take through the experiment, as reconstructed from weak measurements.

Obtained through weak measurements, this 3D plot shows where a quantum particle is most likely to be found as it passes through a Young’s double-slit apparatus and exhibits wave-like behaviour. The lines overlaid on top of the 3D surface are the experimentally reconstructed average paths that the particles take through the experiment. (Courtesy: Krister Shalm and Boris Braverman).
Figure and caption from “In Praise of Weakness”: http://physicsworld.com/cws/article/print/2013/mar/07/in-praise-of-weakness.

From Steinberg, et al:

“…it is striking that the average result of such a measurement will yield exactly what common sense would have suggested. What we are arguing – and this admittedly is a controversial point – is that weak measurements provide the clearest operational definition for quantities such as “the average velocity of the electrons that are going to arrive at x = 1”. It is very tempting to say that this value, this hypothetical measurement result, is describing something that’s “really out there”, whether or not a measurement is performed. We should stress: this is for now only a temptation, albeit a tantalizing one. The question of what the “reality” behind a quantum state is – if such a question is even fair game for physics – remains a huge open problem.”

Aephraim Steinberg spoke about quantum mechanics and weak measurements in the Perimeter Institute Public Lecture Series.  The video of his talk: In Praise of Weakness Public Lecture.

# Exploring Weak Measurements Further

Dressel, et al, provide a review of the mathematics and applications of weak measurements in their recent paper: Understanding Quantum Weak Values: Basics and ApplicationsThey discuss three different types of experimental applications that are revolutionizing our ability to study and manipulate quantum systems using weak measurements: (1) amplifying a signal, enabling the sensitive estimation of unknown evolution parameters, such as beam detection, phase shifts, frequency shifts, time shifts, temperature shifts, etc.; (2) measuring the real and imaginary parts of a complex-valued parameter, enabling new methods for reconstructing the quantum state, including relative phase of the complex value; (3) finding conditioned averages of generalized observable eigenvalues, providing a window into non-classical features of a quantum mechanical system.

For additional discussion of the theory and mathematical background for directly measuring the wavefunction of a quantum system, it is worthwhile to read Direct Measurement of the Quantum Wavefunction and Direct measurement of general quantum states using weak measurement.

So where will all this lead?  This is still, very much, an evolving field of study.  In an area as unintuitive as quantum physics, you cannot just take one or two experimental results and assume you understand what is going on.  Perhaps the wavefunction is not just a mathematical tool, but rather something that is real and can be directly measured.  Perhaps these experiments will clarify the relationship between quantum and classical behaviors.  Perhaps these experiments will help reduce the confusion and misunderstanding concerning the meaning of measurement and observation in quantum mechanics.  The insights gained from weak measurements will certainly lead to a deeper conceptual understanding of the quantum realm.

# “It’s 10:00 pm, do you know where your photons are?”

As parents, we try to know where our kids are at all times.  We teach them that, when they want to play outside or at a friend’s house, they need to let us know where they will be.  If we were to ever find out that they were not where we expected them to be, we would go ballistic!  Well, could you imagine being the parent of a photon?

A group of scientists at Tel-Aviv University performed an experiment that shows you may not get a sensible answer when you ask a photon where it has been.  Soon to appear in Physical Review Letters, Asking photons where they have been,” demonstrates that the past of a photon cannot be represented by a continuous trajectory, or even by the superposition of continuous trajectories.  To quote from their paper:

“The photons tell us that they have been in the parts of the interferometer through which they could not pass!”

# Interrogating Discontinuous Trajectories in Quantum Mechanics

I have previously mentioned two puzzling aspects of the quantum universe: non-locality and the intrinsic, probabilistic nature of outcomes.  I typically caveat these two properties with “apparent”, i.e. apparent non-locality and apparent probabilistic nature.  That is because it is still possible (although not certain) that there could be some underlying causal and realist explanation.  In fact, explanations may already be available, we just do not know how to validate them.

Another mystery that our classical brains struggle with is the apparent discontinuous nature of the trajectories of quantum particles.  What I love about this new result from the Tel-Aviv group (Danan, Farfurnik, Bar-Ad, and Vaidman) is that, not only does it demonstrate a unique and important property of quantum physics, it does so with a straight-forward and conceptually easy to understand experiment.

The scientists used a nested Mach-Zehnder interferometer (MZI).  You may recall that I discussed a MZI in “The Transactional Interpretation of Quantum Mechanics.”  However, in this case, they nested two MZI together – one leg of a MZI includes another MZI nested within one of it’s legs (see the figure below).

Schematic of the nested Mach-Zehnder interferometer used to interrogate photons as to their whereabouts. From “Asking photons where they have been,” http://arxiv.org/pdf/1304.7469.pdf.

Photons enter the apparatus from the source in the upper left corner of the figure.  The unlabeled squares represent beam splitters.  After passing through either the lower leg or the nested interferometer in the upper leg, photons are detected by a quad-cell photo-detector (D).  The unique and essential feature of this experiment is that the mirrors (A, B, C, E, and F) vibrate around their horizontal axes at different frequencies fA, fB, fC, fE, and fF, with very small amplitudes.  This induces oscillations of the vertical positions of photons after they encounter each particular mirror along their path.  Hence, each photon carries a record that describes which mirrors they encountered, and thus which path they took through the apparatus.

Each photon’s signature is extracted by measuring (at D) its position coming out of the interferometer. This data is then Fourier-analyzed to produce a power spectrum of the different frequencies present in the output signal.  When the vibration frequency of a certain mirror appears in the power spectrum, the scientists logically conclude that at least some of the photons have been near that particular mirror.

# Trajectories That Appear to be Continuous

The first run of the experiment that we will consider is the one depicted above.  One third of the beam power was sent into the lower arm and two thirds of the beam power was sent to the nested MZI in the upper arm (the beam splitter used to split the two legs of the outer interferometer was specifically designed for this one-third, two-thirds spilt; other beam splitters produced a 1:1 split).

The interferometers were aligned to ensure that all the photons ended up at the detector. The power spectrum showed peaks at all frequencies, as intuitively expected.  The peaks at fE and fF were higher due to the larger fraction of photons in contact with them.  The power spectrum at the output of the experiment shows, unsurprisingly, frequencies from all five mirrors.

# So Much for Common Sense: Discontinuous Trajectories in Quantum Mechanics

The surprising result was obtained when the interferometer was modified to be a “which-way” experiment.  By slightly shifting mirror B, the nested MZI was aligned so that there would be complete destructive interference between the light reaching mirror F from A and the light reaching mirror F from B (see the figure below but ignore the red and green lines for now).

So, in effect, there were no photons at F.  Hence, there were no photons that could possibly reach the detector D from the upper leg, right?  By that reasonable bit of logic, any photons detected at D should have come from the lower arm of the interferometer.  We would therefore expect that any photons reaching the detector would have interacted only with mirror C.  The punch-line is that the scientists observed three peaks in the power spectrum: the expected one at frequency fC, and two more peaks at frequencies fA and fB.

Nested Mach-Zender interferometer, tuned so that photons arriving at mirror F interfere destructively. Red and green (dashed) lines are explained in the text. From “Asking photons where they have been,” http://arxiv.org/pdf/1304.7469.pdf.

Common sense tells us that any photons passing through the inner interferometer (so that they could encounter mirror A or B and pickup oscillations at frequency fA and fB) must by necessity have also encountered mirrors E and F.  However, frequencies fE, and fF were not seen in the output power spectrum.  How did photons pick up oscillations at frequencies fA or fB, associated with mirrors A or B, and make it to the detector without also encountering mirror E or F?

# Interpreting Discontinuous Trajectories in Quantum Mechanics

Although the conventional interpretation of quantum mechanics can predict the correct outcome for this experiment, it offers little insight into what is going on.  The authors offer an alternative that provides an improved conceptual understanding.  The interpretation they prefer is the two-state vector formalism.  This is a time-symmetric interpretation of quantum mechanics; both forwards and backwards evolving quantum states are required to describe a quantum system. This includes a state vector that evolves from the initial conditions towards the future, and a second state vector that evolves backwards in time from the final conditions of the experiment.  That is to say, the state vector describing the pre-selected state as well as the state vector for the post-selected state are both required to fully describe the system.  This highlights another intrinsic aspect of QM that makes it distinct from classical physics: the past of a quantum particle does not uniquely determine its future.  Past and future measurements, taken together, provide complete information about the system.

In the present experiment (see the above figure), a standard forward evolving quantum state is depicted by the red line and a backward evolving quantum state is depicted by the green dashed line. There is no continuous path for the forward evolving state to proceed through the inner MZI and reach the detector.  However, there is a non-zero probability for the photon to have existed anywhere that both forward and backward quantum wave functions are present.  Hence, this includes the nested MZI in the upper leg, inside mirrors E and F, but in the region of mirrors A and B.

The transactional interpretation (TI) also provides a conceptual explanation of this experiment.  Additionally, at least in my opinion, the TI provides a more straight-forward way of calculating the probability for the photon to be in the inner interferometer, and hence simplifies the prediction of the power spectrum at the output.

# Will We Ever Understand Particle Trajectories, Much Less Quantum Physics?

Making progress towards the goal of fully understanding what nature is up to in the quantum world requires that you have a full grasp of the variety of experimental evidence and theoretical results.  If you have been reading my posts up to now, hopefully I have been filling in some gaps.  Don’t get too comfortable, yet.  There is a lot more to the story about trajectories, a story that is being told through “weak measurements”.

Considerations of pre- and post-selected systems lead to the theory and practice of weak measurements.  In an upcoming article, I plan to discuss what I mean by weak measurements, and how they are being used to survey and reconstruct the properties of quantum particles between pre-selection and post-selection measurements.  The results of these measurements are amazing just due to the fact that they are possible, as well as due to the enlightening results that they provide.  These experiments give me confidence that, as a result of the amazing work of skilled quantum physicists, we are making steady progress along the road to a proper conceptual understanding of our quantum universe.

# Hollywood and Black Hole Analogs

The Big Bang Theory’s end-of-season cliff-hanger referred to a similarity between the equations of hydrodynamics and the equations of black holes, and the usefulness of hydrodynamic simulations to understand black holes.  Leonard joined a team put together by Stephen Hawking to search for the equivalent of Unruh radiation in water (at sea).  Here he is telling Penny that he will be on an ocean research vessel for four months:

## Hawking and Unruh: Radiation from the Vacuum

One of Hawking’s many contributions to our understanding of black holes is his prediction of Hawking radiation.  By combining concepts and math from General Relativity and Quantum Mechanics, Hawking showed that black holes have a surface temperature and radiate particles.  The possibility of actually detecting the equivalent of Unruh radiation in the ocean may have been an exaggeration.  But, as is typical for The Big Bang series, the physics that appears is based on actual physics and is inspired by current events in science.  See, for example, Black Hole Analogue Discovered in South Atlantic Ocean”  and Coherent Lagrangian Vortices: Our Oceans Have Their Own Kind Of Black Holes.”

According to the equivalence principle, physics in a uniform gravitational field should be the same as that in a uniformly accelerating reference frame.  So, a particle or object undergoing uniform acceleration should also emit (thermal) radiation, analogous to the surface of a black hole.  This is essentially what Unruh radiation is.  Unruh radiation has the same mathematical relationship as Hawking radiation, except it is proportional to the uniform acceleration rather than gravity.  To reach detectable levels, the acceleration needs to be pretty drastic.  Experimentalists hope to use intense lasers to accelerate electrons sufficiently to detect Unruh radiation.  Unruh radiation is different from the usual radiation emitted by accelerated charged particles.  It is independent of the particle’s mass and charge and also has a different frequency distribution and angular distribution, features that will be used to identify its presence.

# Hydrodynamic Quantum Analogs

Now, on to the subject of hydrodynamic quantum analogs.  In a previous post, I mentioned experiments with silicon droplets that were mimicking quantum physics:  The Folly of Physics: Interpretations of Quantum Physics, Part 1:  De Broglie-Bohm mechanics at work?”  If you missed it, take a look at this amazing clip from the Science Channel’s Through the Wormhole:

This Science Channel video shows how the results of the canonical double slit experiment can be reproduced by a silicon droplet (the “particle”) riding on an actual, physical wave (a “pilot-wave”, reminiscent of de Broglie-Bohm pilot-wave theory).  Physicists and mathematicians continue to explore this rich environment to further advance our understanding of nature.

## Getting into the Experimental Details of Hydrodynamic Quantum Analogs

You can take a look at the MIT web page of John Bush for highlights of some of his group’s work on hydrodynamic quantum analogs.   Their work is further discussed in “Wavelike statistics from pilot-wave dynamics in a circular corral”, which is also available here“Exotic states of bouncing and walking droplets” (also available at this location), explains the experimental setup in more detail and digs deeper into the theory and math.

A silicon-filled tray is placed on a vibrating table.  The depth and geometry of the tray are chosen to enable studying the desired behavior or phenomenon.  The intensity of the vibration is adjusted to just below the threshold at which waves would be generated on the surface of the fluid by the vibrations.  When a droplet of silicon is then placed on the surface of the vibrating fluid, a cushion of air between the drop and the fluid bath prevents the drop from coalescing. The droplet bounces and “walks” on the vibrating surface.  This bouncing causes a wave field to be generated on the surface of the bath, similar to skipping a rock on a pond.  The wave field becomes more and more complex as waves from subsequent bounces interfere with each other and reflect off of the boundaries of the surface (or off of other obstacles placed in the fluid bath).

The motion of the particle depends on its current location as well as its history, due to the complex wave field generated by previous bounces.  The motion also depends on the environment; the geometry and depth of the tray, depth changes, boundaries and obstacles, etc.  In addition to a vertical component, there is a horizontal component to the force on the droplet.  This is due to the droplet landing on a sloping part of a wave.  Under the right conditions, the droplet achieves resonance with its self-generated wave field and is propelled horizontally along the surface.  This two-dimensional motion displays properties of a microscopic quantum system.  The trajectories that are observed, and the probability distributions mapped out by the areas in which the droplet spends the majority of its time, are equivalent to the results of quantum physics experiments with microscopic particles.

## Visualizing Hydrodynamic Quantum Analogs

Take a look at this YouTube video, provided by MIT, to help visualize what is going on.  It is important to note that in the images where you see the droplet walking across the surface, the camera is being strobed in synch with the bouncing – so you just see the horizontal motion, not the vertical bouncing.

If the silicon bath is rotating, in addition to vibrating vertically, the droplet will lock into an orbit determined by the troughs of its self-generated wave pattern.  This is precisely a demonstration of “quantization” of the allowed orbitals for a subatomic particle confined in a potential.

Similar experiments have demonstrated other behaviors that are typically assumed to be exclusive to the quantum realm.  These phenomenon include diffraction, tunneling, quantized orbits, orbital level splitting, and more (see “Wavelike statistics from pilot-wave dynamics in a circular corral” and references therein).  To mimic tunneling, for example, a walking droplet approaching a barrier that it will on most occasions simply bounce off of, will once in a while receive enough energy from the wave enabling it to jump over the barrier.

## Interpreting Hydrodynamic Quantum Analogs

The authors of “Wavelike statistics from pilot-wave dynamics in a circular corral” state that:

“Our study indicates that this hydrodynamic system is closely related to the physical picture of quantum dynamics envisaged by de Broglie, in which rapid oscillations originating in the particle give rise to a guiding wave field.”

Louis de Broglie is perhaps best recognized for postulating in his PhD thesis that all matter (not just photons) has wave properties.  He received the Nobel Prize in Physics in 1929, “for his discovery of the wave nature of electrons”.  Clinton Davisson and George Paget were jointly awarded the Nobel Prize in Physics in 1937, “for their experimental discovery of the diffraction of electrons by crystals”.

Louis de Broglie generalized Einstein’s theory of the photon to propose that all matter has wave-like behaviors. The story of his pilot-wave theory is one that is still being written. (image from Wikipedia)

de Broglie presented his theory of pilot waves at the famous Solvay conference in 1927.  However, his idea lost out to the personalities of Bohr, Heisenberg, and others, in favor of the Copenhagen Interpretation (CI) of Quantum Mechanics.  There is a substantial debate in the literature over whether the adoption of the CI was the result of personalities, politics, and personal ambitions, rather than a deliberative and unbiased review of the available alternatives.  See, for example, “Quantum Theory at the Crossroads.  My personal opinion is that the CI was adopted prematurely and went unquestioned by the bulk of the physics community for far too long. As a result, experimental and theoretical progress towards a fundamental conceptual understanding of the universe has been delayed.  I plan to address this issue in more detail, from a historical and current events perspective, eventually; either in this blog or in a book.  Nonetheless, it is intriguing to consider what conclusions would have come out of the Solvay Conference if de Broglie could have shown the above video.

Pilot-wave mechanics was abandoned until David Bohm independently re-discovered something very similar to it in the 1950’s.  The theory has subsequently become known as Bohmian Mechanics, or de Broglie-Bohm Pilot-Wave Theory.  According to this model, particles are objective point-like objects with deterministic trajectories.  These trajectories are guided by pilot waves, which also objectively exist.  The pilot waves are described by the wave function.  Wave function collapse never happens (contrary to the assumption of the CI).  Hence, pilot-wave theory removes the measurement paradox.  It also provides a mechanism for explaining and visualizing wave-particle duality.  It is easy to see how the movement of a particle can appear to be determined by the interference of waves, because it is directly!

The mathematics used to describe damped harmonic oscillators and RLC circuits are equivalent.  Variables from one regime (such as displacement, mass, spring constant, and damping coefficient) can be mapped to the other regime (charge, inductance, capacitance, and resistance).  However, this does not mean that an RLC circuit is a mass on a spring oscillating in some viscous damping medium.  It just means that the two systems share similar dynamical properties.  It also means that you can use one system to study or visualize the other.  However, beyond the similarities, there remain significant differences between electromagnetism and classical mechanics.

# Implications of Hydrodynamic Quantum Analogs

Nonetheless, the equivalence between the motions mapped out by these classical droplet-wave systems and quantum mechanics is jaw-dropping.  And there is certainly a lot that we can learn from them.  These recent findings should help revive the question of whether there is a more fundamental dynamics in quantum physics.  Whether the correct conclusion is that the illusion of quantum mechanics is just that, and the quantum realm is nothing new (compared to classical systems) is yet to be seen.  There are certain phenomenon in quantum experiments, such as (apparently) discontinuous particle trajectories, for which the classical analog is not yet clear.  Additionally, in de Broglie-Bohm mechanics, there is no dynamic mechanism for the particle to influence the wave field as in the case of hydrodynamic quantum analogs.  But perhaps an extension of de Broglie-Bohm mechanics should account for this feedback?

Randomness is an intrinsic feature of the quantum world.  After reading these papers and watching the videos, it can be tempting to attribute this (apparent?) randomness to chaos theory.  Chaos theory applies to dynamical systems that are extremely sensitive to initial conditions.  Tiny differences in initial conditions lead to huge differences in future outcomes.  The idea to apply it to quantum theory would essentially involve assuming that there is some hidden information about a particle’s initial state that we cannot know well enough to enable a precise prediction for the future.  Hence, the best we can do is predict probabilities.  What bothers me about this idea, however, is that the intrinsic and unavoidable randomness in quantum mechanics is closely tied to non-locality.  Without the intrinsic and unavoidable randomness, problems with causality and relativity quickly show up. On the other hand, if it were true, that there is an underlying explanation for the intrinsic randomness in QM using chaos theory and hidden variables, the technological and conceptual breakthroughs would be astounding, I’m sure!

There are a lot of details that go into these experiments, including how the apparatus is set up and how it is filmed.  So they are definitely not a proof or refutation of any particular interpretation of QM at this point.  However, they are intriguing, and they offer an irresistible visualization that begs further investigation.  Quantum physics is typically presented as a mystical and bizarre subject, involving multiple universes, superimposed cats, and conscious minds deciding reality.  These experiments should push us to recognize that a belief in the mystical aspects of quantum mechanics is a choice and not a necessity.

# Encountering the Many Worlds Interpretation

Several years ago, I looked into the Ma­ny Worlds Interpretation (MWI) of quantum mechanics and concluded that it was not on the right track.  It seemed to be creating more conceptual and technical problems than it solved.  However, I frequently come across mention of it in the physics literature and in documentaries.  Several leading scientists refer to it as a ‘viable’ alternative to the canonical Copenhagen Interpretation (CI); some even calling it the ‘preferred’ interpretation.  So, I recently decided to take another look at the MWI.  Perhaps there was something I missed, or something important that I did not understand on the first go-around.

My initial instincts have been validated.  Reading about the MWI, including papers by its proponents as well as by its detractors, reminded me of the Hans Christian Andersen story called The Emperor’s New Clothes The Emperor and his ministers believe the hype about a fabric that is allegedly invisible to anyone who is unfit for their position.  They pretend that they can see the fabric so as not to feel left out.  While the Emperor is parading naked through the town, believing that he is wearing the best suit of clothes, a naïve young boy blurts out that the Emperor is naked!  Perhaps I can be that naïve young boy when it comes to untestable ideas like the MWI.  I may not be young, but bear with me.

## So what is the Many Worlds Interpretation?

As advertised, the main advantage of the MWI is that it solves the measurement problem. I discussed the measurement problem in two previous posts: Quantum Weirdness: The unbridled ability of quantum physics to shock us and Contrary to Popular Belief, Einstein Was Not Mistaken About Quantum Mechanics.  The measurement problem results from the apparent need for two distinct processes for the evolution of the state vector: (1) continuous and deterministic evolution according to the Schrödinger equation when no one is looking, followed by (2) spontaneous non-unitary evolution, or collapse, of the state vector upon measurement of an observable.  What constitutes a measurement and the dynamics of wave function collapse are not defined in the CI.  Additionally, special status is assigned to an intelligent observer who is treated as being outside the quantum system.

As an added bonus, proponents of MWI claim that it enables independent derivation of quantum probability distributions without assuming the Born rule.  The Born rule for computing the probability of potential outcomes of a quantum event is an additional postulate of canonical quantum mechanics.  According to this rule (which has enjoyed phenomenal experimental verification time and time again throughout the past roughly ninety years), the probability for each potential outcome to become the realized outcome is given by the amplitude squared from the applicable terms in the state vector.

Hugh Everett developed the relative state formulation in his dissertation and his subsequent publication of  “Relative State” Formulation of Quantum Mechanics (also available at this link).  It was later given new life by Bryce DeWitt in 1970, with his work applying rational decision theory and game theory to quantum mechanics; see Quantum Mechanics and reality.  Since then, dozens of papers have been written attempting to patch holes in the theory, or to take it apart.

See the recent article by Sabine Hossenfelder, “The Multiverse is not a paradigm and it’s not shifting anything” for another perspective on multiverses in general.

The MWI hypothesis avoids the measurement problem by assuming that wave function collapse never happens.  A single result never emerges from an interaction or quantum measurement.  Instead, all possibilities are realized. Each possibility is manifested in a new branching universe.  With each observation, measurement, or interaction, the observer state branches into a number of different states, each on a separate branch of a multiverse.  All branches exist simultaneously and each branch is ‘equally real’.  All potential outcomes are realized, regardless of how small their probabilities.

# What is wrong with the Many Worlds Interpretation?

If you have read my earlier post Three Roads to What Lies Beyond Quantum Mechanics, you have already glimpsed my discontent with MWI.  You will find statements in the literature that claim MWI solves the paradoxes of the CI, and that it derives quantum probabilities without the use of an ad hoc assumption (as in the case of the Born rule in the CI).  Hugh Everett’s main goals when he gave birth to the ‘relative state formulation’, which subsequently became known as the MWI, were to get rid of non-unitary wave function collapse and to relegate the observer to just another part of the quantum system.  Unfortunately, MWI and its many variants does not live up to the product’s claims.

The MWI hypothesis requires an unimaginably large, perhaps infinite, number of universes, each spawned essentially instantaneously in a fully evolved state from it’s parent.  Your present universe is constantly branching, sprouting multiple universes at a fantastic rate.  Each new universe is identical to its parent IN EVERY WAY, except for the record of a single quantum event.  I don’t just mean in one you are the Queen or King of your senior prom, and in another you decide not to run for prom royalty.  Every quantum interaction, every quantum measurement, a countless infinity of which happen every day in what we conventionally call the universe, leads to multiple new universes.

According to Bryce DeWitt in Quantum Mechanics and reality,

“…every quantum transition taking place on every star, in every galaxy, in every remote corner of the universe is splitting our local world on earth into myriads of copies of itself.”

Cloning and quantum teleportation Star Trek-style should be a breeze if quantum mechanics allows cloning the entire universe a countless number of times each second!  This may make for interesting and fun science fiction, but without testable predictions it is not physics.

This multiverse evolves in a continuous and deterministic way.  The apparent randomness that an observer in a particular universe (branch) perceives is in his/her mind; a consequence of the particular branch he/she finds him/herself in.  The emergence of macroscopic uniqueness, a consequence of state vector collapse in the CI, is just an illusion in the MWI.  That sounds like progress, right?  But wait.

## There’s More

The different branches are incoherent; they do not interfere with each other and observers in one branch cannot detect the existence of any of the other branches (this is the “no-communication” hypothesis).  The wave function collapse hypothesis has been replaced by the no-communication hypothesis.  Quantum decoherence has been used to justify and explain the no-communication hypothesis, with varying success.  But, it has also been used to justify and explain the wave function collapse hypothesis.  So there is nothing gained here by postulating a countless number of universes branching out from all of the interactions occurring throughout our universe.

As John Bell stated (while writing about the MWI, see p. 133 of Speakable and Unspeakable in Quantum Mechanics):

“Now it seems to me that this multiplication of universes is extravagant, and serves no real purpose in the theory, and can simply be dropped without repercussions.”

# Probabilities in the Many Worlds Interpretation

Everett sets out to show that the Born probability rule can be derived from within his model, as opposed to having to assume it.  He does this by assuming that the square of the amplitudes (from the state vector, same values that the Born rule uses) represent the ‘measure’ that should be assigned to each of the branches.  When an observer repeats the same experiment a large number of times, multiple branches appear corresponding to each of the possible outcomes for each performance of the experiment.  A particular observer will traverse a particular series of branches out of all the possible combinations of outcomes from all the trials.  By applying his weighting scheme, Everett shows that, in most cases, the observer is part of a branch where the relative frequency of the observed results agrees with the Born rule.

What exactly does it mean for different branches to have different weights, if each and every branch is ‘equally real’?  Are we to assume that the number of realizations of branches associated with a particular outcome of a particular measurement or interaction is proportional to the branch weight?  You may naively think that the probabilities of various outcomes should be related to the number of branches with that outcome (a simple counting measure).  What would then happen if the probability was an irrational number?  Combinatorial methods fail.  Even if you could use simple combinatorial methods, many observers would see outcome distributions that conflict with the Born rule.  The Born probability rule has been validated in countless experiments over the past 87 years.  Why have we never witnessed a deviation from it in any of the uncountable combinations of branches we have traversed to get where we are today?

In Everett’s theorem, the observer is considered as a purely physical system.  This is a central part of his relative state formulation.  The observer is just one subsystem in the overall system under consideration.  Once one state is chosen for one part of the overall system, then the rest of the system is in a relative state; state X given that the one subsystem is in state Y.  This was, initially, an advantage of the MWI compared to the CI. However, attempts to patch some of the holes in the theory have relied heavily on rational decision theory and game theory, thrusting a conscious observer back into the spotlight.

## Throwing in Rational Decision Theory and Game Theory

Unfortunately, Everett’s approach to deriving the Born rule has been taken apart due to its use of circular reasoning.  David Deutsch used decision theory and game theory to derive the Born rule; see Quantum Theory of Probability and Decisions.  He demonstrated that if the amplitude squared measure is applied to each branch, then this value is also the probability measure for those branches.  He did this by arguing that it represents the preferences of a rational agent.  He considered the behavior of a rational decision maker who is making decisions about future quantum measurements.  By rational, he meant that the decision maker’s preferences must be transitive: if he/she prefers A to B, and B to C, then he/she must also prefer A to C.  (On a side note – many psychology studies have shown that personal preferences of so-called rational agents in the macro world are often not transitive).

According to Deutsch, if a rational decision maker believes all of quantum theory with the exception of assuming a probability postulate, he/she necessarily will make decisions (behave) as if the canonical probability rule is true.  I am not an expert on decision theory, but it seems to me that the strategy chosen by Deutsch’s rational observer is not unique; it just happens to be the one that correlates with the desired end point – the Born probability rule when the amplitude squared values are used as branch weights.  Additionally, if you accept Deutsch’s reasoning, methodology, and assumptions, I should think his results could equally well be used to demonstrate why the Born probability rule works in the CI, as well as in the MWI.

## Attempts to Make it Consistent

Many attempts to formulate a consistent and defensible version of Everett’s initial ideas have been discussed in the literature since Deutsch’s work.  Adrian Kent addresses many of them in One world versus many: the inadequacy of Everettian accounts of evolution, probability, and scientific confirmation.  Kent points out some of the inconsistencies and contradictions that these attempts fall victim to, either when compared to each other or within themselves.  Given that every potential outcome is actually realized in a branch, regardless of likelihood, a rather tortured path has to be taken to explain the meaning of probability and uncertainty when applying decision theory.  Additionally, Kent is concerned by the lack of uniqueness in the assumptions and conclusions that can be made about the so-called rational decision-maker.  To apply decision theory or game theory reasoning to quantum mechanical events seems rather surreal to me.  But regardless of whether you take the approach seriously, there is little gained from it, unless you want to get extremely metaphysical about the role of consciousness. Which I do not.

# So Where Does This Leave Us With Respect to the Many Worlds Interpretation?

“…no matter how high you pile considerations upon nothing, and extend the boundaries of nothing, to nothing it must come at last”
Writings and speeches of Alvan Stewart, on slavery. Ed. by Luther Rawson Marsh. Stewart, Alvan, 1790-1849., Marsh, Luther Rawson, ed. 1813-1902.

The MWI does not deliver on its promises.  In particular, it does not solve the measurement problem unless you ignore the extra baggage that comes with the theory, such as the no communication hypothesis, the song and dance concerning rational decision theory, and the surreal role of the observer.  Nonetheless, the idea of countless multiple universes has mesmerized popular culture and theoretical physics.  The image of an infinite number of copies of ourselves, with slight variations in each universe, is quite tempting.  Some people claim that multiverses must be real because we are getting hints of one from multiple theories, including superstring theory, inflationary cosmology, and anthropic reasoning.  But each of these predictions are perched upon a mountain of assumptions.  And each posits a different cause for the multiverse.  It is not at all clear to me that satisfying the multiverse hypothesis of one model would necessarily satisfy that of the others.

The idea that the MWI is the only viable alternative to the CI is a myth.  Other viable alternatives already exist; and it is premature to assume no one will ever discover another.  These alternatives, such as de Broglie-Bohm mechanics and the Transactional Interpretation, need more work.  But at the very least, they serve as proof of concept that we should not be so eager to believe any wild idea offered to us, without evidence.  So, if you come across someone endorsing the Many Worlds Interpretation of quantum mechanics, remember the story of the Emperor’s New Clothes.  Let them know that you are aware the emperor is naked.  MWI does not provide a unique and independent derivation of probability, it does not remove the special treatment of the observer, and it replaces the collapse hypothesis with run-away multiverse branching and the no-communication hypothesis.

My upcoming posts will include:

• Discussion of hydrodynamic quantum analogues.  These experiments demonstrate how phenomenon and probability distributions normally associated only with the quantum world can be produced by macroscopic systems and classical dynamics.
• So-called weak measurements that are allowing physicists to directly measure the quantum wave function itself, and monitor its evolution.
• Introduction to de Broglie-Bohm mechanics.  Incidentally, wave function collapse does not occur in de Broglie-Bohm mechanics, and it does not require an infinite number of universes (just empty waves…).

# Alternative Paths to an Interpretation of Quantum Mechanics

This is the real reason why we have not been able to discover a theory more fundamental than quantum mechanics. We no longer wear three piece suits to physics conferences.

If you have read my earlier posts, you have likely concluded that I am a realist with respect to the foundations of quantum mechanics (QM).  I believe there is some deeper reality behind the equations of QM.  This deeper reality, or more fundamental theory, will account for the apparent nonlocality that we see in quantum events.  It will also explain the intrinsic randomness in quantum theory, and how nature decides which option to choose.  In my opinion, the wave function (or state vector) represents more than just our knowledge of a system.  However, it may be just an approximation for the correct representation or description of reality.  I also believe that our preconceived notions of reality will need to be altered, just as they were when Einstein developed the Special (SR) and General (GR) Theories of Relativity.

My views contrast with those who believe that QM is the ultimate mathematical formulation.  Among those that believe QM is complete, there are two classes – those that believe it is not even appropriate to ask “why”, “how”, or “what”; and those that seek a set of principles that lead directly to QM.  Members of the first group hold that there is no such thing as objective reality.  Their position is that QM merely encodes what we know (or can not know) about the quantum world.  All we can talk about are the results of measurements that we make.  They argue that it is not even appropriate to ask nature to reveal a deeper understanding, because there is not one.  The second group looks for a deeper understanding, but they look for this conceptual or philosophical foundation within the framework of the current quantum theory.  I think both of these approaches are flawed.  In my opinion, the first approach amounts to forfeiture and the second is a dead end.  To move beyond QM and find a deeper conceptual understanding of the universe, we need to re-zero our preconceived notions and come at the problem from a new direction.

A search of the internet, or YouTube in particular, reveals multitudes of blogs and videos about QM.  There are many high quality documentaries and video clips, with excellent production qualities.  But, the vast majority parrot the ideas that “spooky action at a distance”, entanglement, wave function collapse, the mysterious role of an intelligent observer, etc., are intrinsic magical mysteries of the universe.  They give the impression that our knowledge of the quantum world is secure, and that we might as well accept this bizarre status quo.  Worse, I have seen expert speakers on these documentaries claim that it is not even correct to question these aspects of QM, these are not the “correct questions to ask”.  Metaphysical BS such as “each of us constructs our own version of reality” really gets me going.  No, we are not constructing alternate realities in our heads.  Each of us is constructing a different, flawed and approximated, representation or model of reality in our heads (some more flawed and approximated than others).  But this is not the same as creating a separate reality.  I believe there is some higher level theory, of which QM is an approximation, that will explain these issues.

# Quantum Theory as a Labor of Love (Vice Money)

There are several theoretical and experimental physicists doing work on interpretations of QM.  It is hard work.  Generations of physicists have gone before you and have not been able to understand quantum theory.  Additionally, you are working up hill against a field that is only slowly realizing the importance and applicability of your work.  Funding is hard to come by, as you compete with paradigms that have sapped man power and resources for decades.  Frustratingly, you are forced to compete in isolation with these fields despite their lack of actual physics results.  We need more people, and more funding, looking into the foundations and extensions of quantum mechanics.  Back in the 1980’s, many physicists believed that an end run had been found.  They believed that string theory was the theory of everything, from particle physics to cosmology, from the quantum world to black holes.  All the hype has not manifested itself in reality.  It is time to re-tool theoretical physics.  This may mean taking smaller steps, ensuring that we understand where our footing is at each hurdle.

The potential payoffs are huge.  This includes understanding and resolving open issues in Quantum Field Theory, which serves as the basis for the Standard Model of Particle Physics.  It may lead to a theory of quantum gravity, and explanations of dark matter and dark energy.  This would lead to a replacement for the Standard Model of Cosmology (the “Lamda Cold Dark Matter” model – beyond the scope of this post, perhaps more on that later).  These models that aspire to explain all of particle physics and all of cosmology have many free parameters.  The ability to tune these parameters to match the observations weakens our confidence in the uniqueness and correctness of these theories.  Hopefully, a new extension of QM will help fix some of these free parameters.

# Getting More Specific About What Lies Beyond Quantum Mechanics

I am not a fan of models that simply add additional mathematical structures to the Schrödinger equation in order to reproduce observed behavior.  Modified Schrodinger dynamics is one example of such attempts.  I am also not a fan of interpretations that ask us to accept too much without conceptual reimbursement, such as the Many Worlds Interpretation (MWI).  In the MWI, each quantum event creates a new universe.  With countless quantum events occurring throughout the universe every moment, the instantaneous proliferation of fully developed universes is staggering, to say the least.

Sean Carroll is a talented and prolific physicist and a wonderful writer. See, for example, his book From Eternity to Here: The Quest for the Ultimate Theory of Time, also linked from my Recommended Reading page.  Dr. Carroll recently gave an interview on interpretations of QM:

I agree with his assessment that the failure of physics to develop a conceptual understanding of quantum mechanics is an embarrassment.  However, I differ with him over his endorsement of the MWI.  There are other interpretations that can resolve the paradoxes of the canonical (Copenhagen) interpretation of QM without such wild assertions and assumptions.  Examples of these alternatives include de Broglie-Bohm mechanics, the Transactional Interpretation, or the Two-State Vector Formalism.  This does not mean that one of these other interpretations are necessarily correct.  It does mean that we should not be so eager and willing to swallow MWI.  The MWI will have to provide a better return on my investment before I start to take it seriously.

I have read several papers that attempt to derive quantum mechanics from invariance laws associated with information, probability, causality, contextuality, composability, etc., (name your principle).  While interesting, I just don’t have the sense that these approaches are on track to capture the essence of reality.  Perhaps the problem is that they are starting with the end in mind.  Although “begin with the end in mind” is one of Stephen Covey’s key tenets (The 7 Habits of Highly Effective People), in this case it may be a handicap.

Compare this situation with how Einstein discovered SR and GR.  Einstein was not trying to reproduce Newton’s absolute space and time or Newtonian gravitational theory.  He saw evidence in the world around him and in his thought experiments that led him to postulate a new invariance law.  For SR, it was the constancy of the speed of light.  For GR, it was the equivalence of gravitational and inertial mass.  He then followed these postulates to where ever they took him.  With his new theories, Einstein was able to explain experimental results that contradicted existing theories; the experiment by Michelson and Morley that failed to detect the ether and the anomalous precession of the orbit of Mercury, for examples.  More importantly, he was able to make additional testable predictions.

# Stating the Goal: Discovering the Path to What Lies Beyond Quantum Mechanics

So, another approach is to discover some fundamental physical principle(s) that leads to some new theory.  String theory is an example of this.  State a hypothesis: things are made up of strings rather than points.  Then, see where that leads.  However, you have to be willing and able to recognize when your hypothesis is not productive.  Hopefully, the new theory that explains QM will make some testable predictions that contradict other interpretations of conventional QM, so that we can tell which is right and which is wrong.  As if that would not be awesome enough, an even more amazing discovery would be a theory that reduces to QM in one limit, and GR in another.

I want to stress that I think this more fundamental theory will likely be founded on principles that seem, at first blush, to not have much to do with classical QM or GR.  And the math may look very different until approximations are made.  There may be additional hidden variables that are not accounted for in the current set of equations.  Something totally new.  Extra dimensions perhaps.  Not necessarily extra space or time dimensions.  But, some extra freedom for the “force carriers”, for entanglement, or for whatever advanced and retarded wave functions are representing.

Maybe some applicable insights can be found in the Aharanov-Bohm effect, Significance of Electromagnetic Potentials in the Quantum Theory.  Aharanov and Bohm showed that, contrary to classical mechanics and classical electromagnetism, the electromagnetic four-potential can have an observable effect on charged particles, even in regions where the electric and magnetic fields cancel (where there are no forces on the particle).

Fields such as a gravitational field, an electric field, or a magnetic field can be described in terms of a potential.  In the case of electromagnetism, you need a scalar and a vector potential.  Gravitational fields require a tensor potential.  The fields can be derived from the potentials.  However, the potentials can not be uniquely determined from the fields.  That is getting more in depth than we should for the present.  Suffice it to say, however, that prior to the prediction (and subsequent experimental verification) of the Aharanov-Bohm effect, physicists believed that all physically relevant dynamics could be expressed in terms of the fields.

In the Aharanov-Bohm effect, an electrically charged particle is affected by an electromagnetic field despite being confined to a region in which both the magnetic field and electric field are zero. This is an apparent nonlocality of the field interactions.  However, it can also be interpreted as the coupling of the electromagnetic four-potential with the phase of a charged particle’s wave function.  The effect is observed in interference experiments, where the phase difference of two interfering wave functions (state vectors) modifies the interference pattern.  The Aharonov–Bohm effect shows that the electric and magnetic fields do not contain full information about the physics.  The electromagnetic four-potential offers a more complete description of electromagnetism than the electric and magnetic fields alone can.

# Seeing Only the Shadows of the Quantum Universe

Like the prisoners in Plato’s Allegory of the Cave, what we see taking place at the quantum level may be just shadows of the true reality.

The title of this post, “Three Roads to What Lies Beyond Quantum Mechanics”, is a play on the title of Lee Smolin’s quantum gravity book: Three Roads To Quantum Gravity (Science Masters).