# A Historical Method Approach to Teaching Kepler’s 2nd law

Kepler’s 2nd law, the law of the areas, is usually taught in passing, between the 1st and the 3rd laws, to be explained “later on” as a consequence of angular momentum conservation. The 1st and 3rd laws receive the bulk of attention; the 1st law because of the paradigm shift significance in overhauling the previous circular models with epicycles of both Ptolemy and Copernicus, the 3rd because of its convenience to the standard curriculum in having a simple mathematical statement that allows for quantitative homework assignments and exams. In this work I advance a method for teaching the 2nd law that combines the paradigm-shift significance of the 1st and the mathematical proclivity of the 3rd. The approach is rooted in the historical method, indeed, placed in its historical context, Kepler’s 2nd is as revolutionary as the 1st: as the 1st law does away with the epicycle, the 2nd law does away with the equant. This way of teaching the 2nd law also formulates the “time=area” statement quantitatively, in the way of Kepler’s equation, M = E – e sin E (relating mean anomaly M, eccentric anomaly E, and eccentricity e), where the left-hand side is time and the right-hand side is area. In doing so, it naturally paves the way to finishing the module with an active learning computational exercise, for instance, to calculate the timing and location of Mars’ next opposition. This method is partially based on Kepler’s original thought, and should thus best be applied to research-oriented students, such as junior and senior physics/astronomy undergraduates, or graduate students.

# Sidney Coleman’s Dirac Lecture “Quantum Mechanics in Your Face”

This is a write-up of Sidney Coleman’s classic lecture first given as a Dirac Lecture at Cambridge University and later recorded when repeated at the New England sectional meeting of the American Physical Society (April 9, 1994). My sources have been this recording and a copy of the slides Sidney send to me after he gave the lecture as a Physics Colloquium at Stanford University some time between 1995 and 1998. To preserve both the scientific content and most of the charm, I have kept the editing to a minimum, but did add a bibliography containing the references Sidney mentioned.–Martin Greiter

# Quantum clocks observe classical and quantum time dilation

Alexander R. H. Smith & Mehdi Ahmadi
At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime. The probability that one clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of-mass of these clocks move in localized momentum wave packets they observe classical time dilation. We then illustrate a quantum correction to the time dilation observed by a clock moving in a superposition of localized momentum wave packets that has the potential to be observed in experiment. The Helstrom-Holevo lower bound is used to derive a proper time-energy/mass uncertainty relation.

# Black Holes and Quantum Gravity

Aurélien Barrau
ALTHOUGH BLACK HOLES were first imagined in the late eighteenth century, it was not until Karl Schwarzchild devised a solution to Einstein’s field equations in 1915 that they were accurately described. Despite Schwarzchild’s pioneering work, black holes were still widely thought to be purely theoretical, and so devoid of physical meaning. This view persisted until recent decades, an accumulation of observational evidence removing any lingering doubts about their existence. Beyond their obvious interest as astrophysical phenomena, black holes may, in time, come to be considered a laboratory for new physics. It is conceivable that black holes could be used to study quantum gravity; and a complete and consistent theory of quantum gravity remains the most elusive goal in theoretical physics…

# Reflections on a Revolution

John Iliopoulos, reply by Sheldon Lee Glashow
In response to “The Yang–Mills Model” (Vol. 5, No. 2).

To the editors:

It is an honor and a pleasure to comment on Sheldon Lee Glashow’s magisterial essay on the Yang–Mills model. His essay is all the more special for describing one of the greatest chapters in the history of physics, the story of a beautiful mathematical concept transformed into a theory for all seasons, the Standard Model of particle physics. It was of course Glashow who played a central role in the electroweak part of the story by coming up with his masterpiece, the SU(2) × U(1) gauge theory. And with the added masterstroke from Steven Weinberg, his high-school buddy, as he reminds us, we now have not just a theory of all relevant particle interactions at today’s energies, but also a theory of the origin of the masses of elementary particles. Simply, mass turned into a dynamical variable, the knowledge of which enables us to unambiguously predict the associated Higgs boson decay into the relevant particles. And we know with certainty that the W and Z bosons, predicted by Glashow, and the third-generation fermions receive their masses from the Higgs mechanism….