The Quest for Quantum Gravity: The Stubborn Offspring of Quantum Field Theory and General Relativity

Meet the Parents of Quantum Gravity: Quantum Field Theory and General Relativity

The Quest for Quantum Gravity: The Stubborn Offspring of Quantum Field Theory and General Relativity: imageQuantum Field Theory (QFT) and General Relativity (GR) form the theoretical and mathematical foundations for modern physics and cosmology.  QFT is an extension of Quantum Mechanics (QM), accounting for creation and annihilation of particles.  The primary entities in QFT are fields rather than particles, and it can be expressed in a Lorentz-invariant form, consistent with Einstein’s Special Theory of Relativity.  GR is, of course, Einstein’s brain child that explains gravity as the curvature of space and time, induced by matter and energy.  GR enabled Einstein to correctly calculate the magnitude of the precession of Mercury’s perihelion and the deflection of light by the Sun, and almost enabled him to predict the expansion of the cosmos.

These two paradigms, QFT and GR, have enjoyed unprecedented success in their range of validity, precision of experimental verification, and the amazing technologies that they have made possible.  However, many questions remain unanswered.  Puzzles include: what was the physics of the early universe and the pre-universe, what is dark matter, what is dark energy, what is the origin and nature of spacetime, what goes on at the horizon of a black hole and at a black hole singularity, how can gravity be united with the other three forces in a unified theory, what is the role of gravity in quantum decoherence?  Answering these questions may require finding a more general theory that merges QFT and GR into a unified framework encompassing both paradigms, a theory known as Quantum Gravity (QG).

QFT is essentially the theory of the very small, where quantum effects dominate and gravity can be ignored because it is so weak. GR is essentially the theory of the very large or heavy, where gravity dominates and quantum effects disappear.  A theory of QG must be able to predict and explain situations where both quantum effects and strong-field gravity are important.  Quantum Gravity in under five minutes:

The Apparently Incompatible Natures of Quantum Gravity’s Parents

QFT and GR are founded on seemingly different premises for how the universe works.  For example, in QFT, particle fields are embedded in the flat (Minkowski) spacetime of Special Relativity.  In GR, time flows at different rates depending on the spacetime geometry.  And gravity is due to the curvature of spacetime, which changes as gravitational masses move.  The most straight-forward ways of combining the two theories by quantizing gravity are non-renormalizable.  This means that calculations run away to infinity and cannot be tamed through a redefinition of certain parameters, as is done in QFT.

This problem is related to the fact that all particles attract each other gravitationally, and energy as well as mass create spacetime curvature.  When quantizing gravity, there are infinitely many independent parameters needed to define the theory.  At low energies, this form of quantum gravity reduces to the usual GR.  But, at high energies (small distance scales), all of the infinitely many unknown parameters are important and predictions become impossible.

A workable theory of quantum gravity must make use of some deep principle that reduces the infinitely many unknown parameters to a finite and measureable number.  Attempts at a workable theory of quantum gravity include string theory, loop quantum gravity, non-commutative geometry, causal dynamical triangulation, and a holographic universe.  Of course, which hypothesis you prefer is not a decision to be taken lightly:

The challenge of uniting QFT and GR is further compounded by the lack of experimental results that could point to a breakdown of either QFT or GR; or results from experiments that are sensitive to both theories.   Scientists are turning to a variety of astrophysical as well as table top experiments to address this issue.

Searching for Common Ground Between Quantum Field Theory and General Relativity

Testing the predictions of quantum theory on macroscopic scales is one of the outstanding challenges for modern physics.  Some experiments are not tests of a specific theory of quantum gravity, per se.  Rather, they look for a deviation from some fundamental tenet of either QFT or GR, with the hope that this will guide theorists in how to supplant either QFT or GR.  Other experiments attempt to create or observe conditions that are sensitive to both theories, to see how they play together.

Common to many philosophical or phenomenological approaches to QG is the possibility that fundamental symmetries, essential in our current understanding of the universe, may not hold at extremely small distance scales or high energy scales, due to a discrete structure of spacetime. Or, perhaps these symmetries do not hold in a highly curved spacetime with boundaries, such as in the vicinity of a microscopic black hole or the cosmological horizon of an inflationary universe.

These symmetries include Lorentz Invariance (LI) and CPT symmetry (charge conjugation – parity transformation – time reversal).  Lorentz invariance means that a property or process remains invariant under a Lorentz transformation. That is to say, it is independent of the coordinate system and independent of the location or motion of the observer, and the location or motion of the system.  CPT symmetry requires that all physical phenomenon are invariant under the combined operations of charge conjugation (swapping matter and antimatter), parity transformation (reflection in a mirror), and time reversal (viewing the process in reverse).

The IceCube South Pole Neutrino Observatory has weighed in on this issue, setting extremely tight limits on a possible violation of Lorentz Invariance.  Neutrinos, lacking strong or electromagnetic interactions and moving at essentially the speed of light (due to their teeny, tiny, and as-yet un-measureable, mass), are sensitive probes of these effects.  IceCube uses data from interactions of high energy atmospheric and astrophysical neutrinos in the South Pole ice.  See Search for a Lorentz-violating sidereal signal with atmospheric neutrinos in IceCube”, Stringent constraint on neutrino Lorentz-invariance violation from the two IceCube PeV neutrinos, and Probing Planck scale physics with IceCube.

The Fermi Gamma-ray Space Telescope is also a member of this club, using photons rather than neutrinos: Constraints on Lorentz Invariance Violation with Fermi-LAT Observations of Gamma-Ray Bursts and Constraints on Lorentz Invariance Violation from Fermi -Large Area Telescope Observations of Gamma-Ray Bursts.

Another sweet spot is the equivalence principle (EP), which provides the foundational basis for GR. The EP is the idea that the effects of acceleration are indistinguishable from the effects of a uniform gravitational field. The EP requires that gravitational and inertial mass are equivalent; that a particle’s coupling to a gravitational field is equal to its inertial mass.  See, for example, Expanded solar-system limits on violations of the equivalence principleor A millisecond pulsar in a stellar triple system.

Foundational Principles of Quantum Mechanics and the Cosmic Microwave Background

I have previously discussed the resurgence of de Broglie-Bohm mechanics, despite its historical neglect, in Hydrodynamic Quantum Analogs”.

In Beyond the Quantum, Antony Valentini follows the logical consequences of Louis de Broglie’s pilot wave theory to predict evidence of quantum non-equilibrium in the Cosmic Microwave Background (CMB).   Pilot-wave theory makes use of hidden variables.  The canonical interpretation of quantum mechanics says that there are no well-defined trajectories.  But in pilot-wave theory, these hidden variables describe the trajectories for whatever particles or fields a system may contain.  They can also explain the apparently random outcomes of quantum measurements.

Pilot-wave theory gives the same observable results as conventional quantum theory if the hidden variables have a particular distribution, a quantum equilibrium distribution, analogous to an ensemble of particles being in a thermal equilibrium.  But, as Valentini points out, there is nothing in de Broglie’s dynamics that requires this assumption to be made.  When the hidden variables have an equilibrium distribution, superluminal signaling is not possible; any attempted non-local signals would average out to zero.  However, if the hidden variables are not in an equilibrium distribution, superluminal signals may become controllable and observable! Relativity theory would be violated; time would be absolute rather than relative to each observer!

To help understand this, Valentini provides an analogy with classical physics:

“…For a box of gas, there is no reason to think that the molecules must be distributed uniformly within the box with a thermal spread in their speeds. That would amount to restricting classical physics to thermal equilibrium, when in fact classical physics is a much wider theory. Similarly, in pilot-wave theory, the `quantum equilibrium’ distribution – with particle positions distributed according to the Born rule – is only a special case. In principle, the theory allows other `quantum non-equilibrium’ distributions, for which the statistical predictions of quantum theory are violated – just as, for a classical box of gas out of thermal equilibrium, predictions for pressure fluctuations will differ from the thermal case. Quantum equilibrium has the same status in pilot-wave dynamics as thermal equilibrium has in classical dynamics. Equilibrium is a mere contingency, not a law.

…It seems natural to assume that the universe began in a non-equilibrium state, with relaxation to quantum equilibrium taking place during the violence of the Big Bang.

…The crucial question is whether the early non-equilibrium state could have left traces or remnants that are observable today.”

Quantum non-equilibrium at the onset of inflation would modify the spectrum of anisotropies (differences from place-to-place) in the CMB sky.  Hence, measurements of the CMB can test for the presence of quantum non-equilibrium during the inflationary phase.

See also: Samuel Colin and Antony Valentini, Mechanism for the suppression of quantum noise at large scales on expanding space, where the authors present numerical simulations showing how the expansion of space can slow down the relaxation to quantum equilibrium in the super-Hubble regime:

“Given these results it is natural to expect a suppression of quantum noise at super-Hubble wavelengths. Such suppression could have taken place in a pre-inflationary era, resulting in a large-scale power deficit in the cosmic microwave background”.

A variety of tests of fundamental physics, conceivable with artificial satellites in Earth orbit and elsewhere in the solar system, are discussed in David Rideout, et al., Fundamental quantum optics experiments conceivable with satellites — reaching relativistic distances and velocities:

“We propose to push direct tests of quantum theory to larger and larger length scales, approaching that of the radius of curvature of spacetime, where we begin to probe the interaction between gravity and quantum phenomena. …the potential to determine the applicability of quantum theory at larger length scales, eliminate various alternative physical theories, and place bounds on phenomenological models motivated by ideas about spacetime microstructure from quantum gravity.”

Table-Top Tests of Quantum Mechanics and General Relativity

The question of simultaneously observing the effects of quantum physics and GR in a table-top experiment can be framed as simply as this: The idea that particles can be in superpositions of multiple states (states with different trajectories, different spins, different energies, etc.) is an essential feature of quantum mechanics.  If a particle is in a superposition of states with different paths through a gravitational field, for example, the different superpositions should be effected differently by the different trajectories through spacetime.  If a particle is in a superposition of different energy states, these different superpositions should create different gravitational fields.  If a macroscopic object could be placed in a superposition of oscillating and non-oscillating, for example, its gravitational field should also split into a superposition.  What does a superposition of gravitational fields look like and how does it behave?

Unfortunately, quantum superpositions are very delicate. As soon as a particle in a superposition interacts with the environment, it appears to collapse into a definite state (see Decoherence and the Quantum to Classical Transition; or Why We Don’t See Cats that are Both Dead and Alive).  Only tiny particle-sized entities can be maintained in quantum superpositions for any significant period of time.  However, only macroscopic objects have detectable gravitational fields.  So this presents immense technical challenges for experimentalists.  People are working very hard to improve upon these limitations.  See, for example, Brian Pepper, et al, Optomechanical superpositions via nested interferometry and Macroscopic superpositions via nested interferometry: finite temperature and decoherence considerations.

Magdalena Zych and his colleagues are searching for evidence of gravitationally-induced time dilation and its effects on the phase of a quantum state: Quantum interferometric visibility as a witness of general relativistic proper time, (also available here).  They propose using a Mach-Zehnder interferometer (MZI) in a gravitational field.  According to GR, proper time flows at different rates in different regions of spacetime.  Their proposed experiment requires a particle with evolving internal degrees of freedom, such as spin or internal vibrations, that can act as a clock.  And the two different legs of the MZI are at different gravitational potentials.

Table-Top Tests of Quantum Mechanics and General Relativity: image

Difference between probabilities to find the particle in different outputs of the Mach–Zehnder interferometer as a function of the time ΔT for which the particle travels in a superposition of two trajectories (corresponds to changing the length of the interferometric arms). Without the ‘clock’ degrees of freedom, the dashed, black line would be the result. With the ‘clock’ and the predictions of GR, the predicted result is the blue line. From “Quantum interferometric visibility as a witness of general relativistic proper time”.

If there is a difference in proper time elapsed along the two legs of the interferometer, the particle’s internal clock will evolve into two different quantum states.  This is a consequence of the prediction that the clock ticks at different rates when placed in different gravitational potentials.  As a result of the quantum complementarity between interference and which-path information (in the form of the different internal clock values), the general relativistic time dilation will cause a decrease in the interferometric visibility (see the adjacent figure).

“Such a reduction in the visibility is a direct consequence of the general relativistic time dilation, which follows from the Einstein equivalence principle. Seeing the Einstein equivalence principle as a corner stone of general relativity, observation of the predicted loss of the interference contrast would be the first confirmation of a genuine general relativistic effect in quantum mechanics.”


This has been just a sampling of the work underway to pry nature’s secrets from her grasp.  For theorists and experimentalists, working on the interplay between QFT and GR with the ultimate goal of creating a theory of QG, is one of the most challenging and stimulating areas of research in fundamental physics.  If this brief discussion has piqued your interest, let me know.  I can point you towards more resources concerning the theoretical and experimental work taking place on the road to quantum gravity.

Hydrodynamic Quantum Analogs

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; hydrodynamic quantum analogs image

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.

Welcome to “The Fun is Real!” (Fun with Physics, that is)

Welcome to “The Fun Is Real“, a new blog that will explore wonders and mysteries of physics.  In particular, I am interested in the questions that are not yet understood.  These questions may be due to new experimental evidence that out-paces the theorists, like dark matter, dark energy, neutrino anomalies, etc.  Or it may be areas where the theory works, but we don’t have a conceptual understanding of how/why the universe does what it does.  One example of this is quantum physics and  quantum non-locality.

The Fun Is Real: wave-particle duality imageThe predictions of quantum mechanics have been confirmed, time and time again, by experimentalists, with greater precision than any other theory in the history of physics.  In the history of science, for that matter!  Additionally, the engineering breakthroughs that have created our information society, and the current trajectory of our technology, are dependent upon quantum mechanics.  Yet, we do not understand how the universe pulls off some of the tricks inherent in quantum physics.  We don’t understand why certain things are quantized.  And entangled particles seem to be able to affect each other over arbitrary distances, without regard to time.  I will expand more on these issues in future blogs.  I also invite your inputs and ideas on the discussions.

In addition to quantum non-locality, examples of other areas that you will see discussed here in the coming months include: (1) Given that a charged particle undergoing acceleration gives off electromagnetic radiation (i.e. emits photons), and a gravitational field is equivalent to acceleration, then why don’t charged particles emit photons simply due to being in a gravitational field? Or do they? (2) Would time exist if there were no matter? (3) Why does the universe insist upon the use of “imaginary”, or complex, numbers to communicate it’s behavior? (4) Where does inertia come from and why does gravitational mass appear to be the same as inertial mass?

I don’t accept anthropomorphic explanations.  That is, I don’t accept as adequate an argument that states “we would not be here if it were not so”.  That does not contribute to our understanding of the how/why of the universe.  I also don’t accept “the theory has to be that way to be consistent with the evidence”.  I want to understand.  I want to know why.  I want to know how.  I want to know how a particle can impact measurements done on it’s entangled partner, in apparent violation of locality and the speed of light; not just how to do the calculations.

This is a new website.  I am trying to make it interesting and accessible.  Let me know if you see problems or if you have ideas to make it better.  Remember, the physics may be theoretical, but “The Fun Is Real“.