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This Insight takes a look at how it is possible to calculate the spin of Sagittarius A*, the supermassive black hole at the center of the Milky Way using some data and a few equations derived from Kerr metric. The following paper was used as a source- Spin and mass of the nearest supermassive black…
Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be thrown away because they are negligible. That way, when defining the derivative, for example, you do not run into 0/0, but when…
Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be thrown away because they are negligible. That way, when defining the derivative, for example, you do not run into 0/0, but when…
Abstract My school teacher used to say “Everybody can differentiate, but it takes an artist to integrate.” The mathematical reason behind this phrase is, that differentiation is the calculation of a limit $$ f'(x)=\lim_{v\to 0} g(v) $$ for which we have many rules and theorems at hand. And if nothing else helps, we still can…
Abstract I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments that can be found in many excellent standard textbooks. Analysis is about differentiation. Hence, complex differentiation will be my starting point. It is simultaneously my finish line because its…
Introduction A commonly encountered school-level Physics practical is the determination of the internal resistance of a battery – typically an AA or D cell. Typically this is based around a simple model of such a cell as a source emf in series with a small resistor. The cell is connected to a resistive load and…
Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham Cohen, and Emmy Noether how Lie groups became a central topic in physics. Physics, in contrast to mathematics, didn’t experience the Bourbakian transition so the language of for example differential geometry didn’t…
Part 1: Overview Part 2: Mathematical Details Part 3: Implications In the last article in this series, we finished up with a metric for the Oppenheimer-Snyder collapse: $$ ds^2 = – d\tau^2 + A^2 \left( \eta \right) \left( \frac{dR^2}{1 – 2M \frac{R_-^2}{R_b^2} \frac{1}{R_+}} + R^2 d\Omega^2 \right) $$ Now we will look at some of…
Part 1: Overview Part 2: Mathematical Details Part 3: Implications In a previous article, I described in general terms the model of gravitational collapse of a spherically symmetric massive object, first published by Oppenheimer and Snyder in their classic 1939 paper. In this follow-up article, I will give further mathematical details about the model, using…
An entire book could easily be written about the history of numbers from ancient Babylon and India, over Abu Dscha’far Muhammad ibn Musa al-Chwarizmi (##\sim ## 780 – 845), Gerbert of Aurillac aka pope Silvester II. (##\sim ## 950 – 1003), Leonardo da Pisa Fibonacci (##\sim## 1170 – 1240), Johann Carl Friedrich Gauß (1777 –…
* Oct. 25th, 1811 † May 31st, 1832 … or why squaring the circle is doomed. Galois died in a duel at the age of twenty. Yet, he gave us what we now call Galois theory. It decides all three ancient classical problems, squaring the circle, doubling the cube, and partitioning angles into three…
I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist, a classical geometer, an algebraist, a differential geometer, and a physicist “What is a metric?” then we get five different answers. I mean it is all about distances, isn’t it? “Yes”…
P or NP This article deals with the complexity of calculations and in particular the meaning of ##P\stackrel{?}{\neq}NP## Before we explain what P and NP actually are, we have to solve a far bigger problem: What is a calculation? And how do we measure its complexity? Many people might answer, that a calculation is an…
Introduction to Quantum Computing This introduction to quantum computing is intended for everyone and especially those who have no knowledge of this relatively new technology. This discussion will be as simple as possible. A quantum computer can process a particular type of information much faster than can a ‘conventional’ computer. Large companies including Google, Microsoft,…
This article is the second of a two-part series that presents two distinctly different approaches to parallel programming. In the two articles, I use different approaches to solve the same problem: finding the best-fitting line (or regression line) for a set of points. The two different approaches to parallel programming presented in this and the…
This article is the first of a two-part series that presents two distinctly different approaches to parallel programming. In the two articles, I use different approaches to solve the same problem: finding the best-fitting line (or regression line) for a set of points. The two different approaches to parallel programming presented in this and the…
Introduction In this insight article, we will build all the machinery necessary to evaluate unit hypercube integrals by a novel technique. We will first state a theorem on Dirichlet integrals, second develop a sequence of nested sets that point-wise converges to a unit hypercube, and thirdly make these two pieces compatible by means of a…
Riemann Hypothesis and Ramanujan’s Sum Explanation RH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All zeros of L-functions to complex Dirichlet characters of finite cyclic groups within the critical strip lie on the critical line. Related Article: The History and Importance of the Riemann Hypothesis The goal of this…
We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that leads us to Integration reverses differentiation. Hence let us start with differentiation. Weierstraß Definition of Derivatives ##f## is differentiable at ##x## if there is a linear map ##D_{x}f##, such that \begin{equation*} \underbrace{D_{x}(f)}_{\text{Derivative}}\cdot…
(based on https://www.physicsforums.com/threads/teaching-sr-without-simultaneity.1011051/post-6588952 and https://physics.stackexchange.com/a/689291/148184 ) In my earlier Insight Spacetime Diagrams of Light Clocks, I stated without proof that the areas of all light-clock diamonds are equal. In my article, “Relativity on Rotated Graph Paper”, Am. J. Phys. 84, 344 (2016) I provided algebraic proof. In the penultimate draft, I had a non-algebraic motivating…
Two of my favorite areas of study are linear algebra and computer programming. In this article I combine these areas by using Python to confirm that a given matrix satisfies the Cayley-Hamilton theorem. The theorem due to Arthur Cayley and William Hamilton states that if ##f(\lambda) = \lambda^n + c_{n-1}\lambda^{n-1} + \dots + c_1\lambda +…
Introduction The theory of geodesic congruences is extensively covered in many textbooks (see References); what follows in the introduction is a brief summary. Consider a 1-parameter family of timelike geodesics ##\gamma_s(\lambda)##, where ##s## labels each geodesic in the family whilst ##\lambda## is an affine parameter along each ##\gamma_s##. Then the vector field ##\xi \equiv \partial / \partial…
Introduction In a previous Physics Forums article entitled “How to Master Projectile Motion Without Quadratics”, PF user @kuruman brought to our attention the vector equation ##\frac{|V_0 \times V_f|}{g} = R## and lamented the fact that: “Equally unused, untaught and apparently not even assigned as a “show that” exercise is Equation (4) that identifies the range as the…
The title is admittedly clickbait. Or a joke. Or a provocation. It depends on with whom you speak, or who reads it with which expectation. Well, I cannot influence any of that. I can only tell how I mean it, namely as an entertaining collection of simple truths which later on turn out to be…
Euler’s amazing identity The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous equation is ##e^{i \pi} = -1##, which is one of the most important equations in modern mathematics, but unfortunately, it wasn’t invented by Euler.Something that is original with Euler is this amazing identity: Equation 1: ##1…
This Insight was inspired by the discussion in “electric field seen by an observer in motion“, which tries to understand the relation between two expressions: the definition of the electric field as seen by an observer (expressed as an observer-dependent 4-vector, as decomposed from the Maxwell field tensor ##E_{a}=F_{ab}v^b##, as found in Wald’s General Relativity…
Confessions of a moderate Bayesian, part 4 Bayesian statistics by and for non-statisticians Read part 1: How to Get Started with Bayesian Statistics Read part 2: Frequentist Probability vs Bayesian Probability Read part 3: How Bayesian Inference Works in the Context of Science Predictive distributions A predictive distribution is a distribution that we expect for…
What is the ambiguous case? In high school geometry, the idea of proofs is often first introduced to American students. A common task is to use a basic set of rules to prove two triangles are congruent, which is a fancy way of saying they have the same side lengths and the same angles. Phrases…
Confessions of a moderate Bayesian, part 1 Bayesian statistics by and for non-statisticians https://www.cafepress.com/physicsforums.13265286 Background I am a statistics enthusiast, although I am not a statistician. My training is in biomedical engineering, and I have been heavily involved in the research and development of novel medical imaging technologies for the bulk of my career. Due…
Main Point: The electric vector potential offers a means of determining the non-conservative component of mixed stationary or quasi-stationary electric field While many are familiar with the static magnetic vector potential ## \bf A = (\mu/4\pi) \int \bf j~ dv/s ## with ## \bf j ## = current (area) density within differential volume ## dv…
I’m sure anyone who has hung out long enough here on Physics Forums has encountered threads that go something like this (I’ll use an example based on threads I’ve seen and participated in in the relativity forum, but I’m sure similar things occur in other forums as well): Original Poster: I don’t understand how black…
ref: asi.edu.au/programs/past-exams/physics-olympiad-past-exams/ Classification of issues: [1*] Quibble regarding the solution. [2*] Clearer problem description suggested. [3*] Alternative solution should be accepted [4*] Significant error, likely leading to incorrect marking. Tally matrix: Year 1* 2* 3* 4* 2019 3 0 0 0 2017 2 2 0 2 2016 1 1 2 0 2015 2 2 1…
The “twin paradox” is often discussed in the introductory treatment of special relativity. Under “twin paradox” we understand the fact that if two twins start from the same place with synchronized clocks, traveling in an arbitrary way and then meet again at the same spacetime point, where they compare their clocks, in general, they find…
In this article, I will show you how to create a ringtone picker using the RingtoneManager class in Android. You will be able to get the list of tones, display them as RadioButton and let the user pick one tone from the list. I expect you to have a basic knowledge of programming in Android….
We asked our PF Advisors “How do you see the rise in A.I. affecting STEM in the lab, classroom, industry and or in everyday society?”. We got so many great responses we need to split them into parts, here are the first several. Enjoy! Ranger Mike If things run true as before, and I have…
Introduction In a previous article entitled “Alternate Approach to 2D Collisions” we analyzed collisions between a moving and stationary object by defining the co-ordinate axes as being respectively parallel and perpendicular to the post-collision direction of motion of the stationary object. In this article, we will be adopting the same approach to analyze the well…
Introduction In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler’s Series Transformation (zeta) and a generalization thereof, the E-process (Lerch). Dirichlet Series is mentioned as a steppingstone. The continuations are given but not shown to be convergent by any means, though if you…
In Part 1 of this 3-part series, I explained kinematics in Einstien-deSitter (EdS) cosmology and in Part 2, I explained kinematics in ##\Lambda##CDM cosmology (essentially EdS plus a cosmological constant ##\Lambda##). Now I will bring those models to bear on the Supernova Cosmology Project (SCP) Union2.1 type Ia supernova data. This will show clearly how…
This is Part 2 of a 3-part series explaining evidence for so-called “dark energy” leading to a current positive cosmological acceleration. The evidence comes from fitting the SCP Union2.1 type Ia supernova data which indicates the existence of a cosmological constant ##\Lambda## (read “Lambda”, thus ##\Lambda##CDM is sometimes written LCDM) in Einstein’s equations (EEs) of…
Introduction At the beginning of the 20th century, it was thought that all matter consisted of only three particles: the electron, the neutron, and the proton. The major outstanding question was how the neutrons held the positively charged protons together within the atomic nucleus. In the search to answer that question, however, it was found…
Key Points Black Holes (BH’s) have a gravitational pull so strong that nothing, not even light, can escape it BH’s can be divided into two main classes: Stellar–mass and Super Massive Black Holes (SMBH’s) BH’s are observed indirectly through the flow they accrete and emit, also by the supersonic collimated jets they produce and by…
Introduction This bit is what new thing you can learn reading this:) As for original content, I only have hope that the method of using the sets $$C_N^n: = \left\{ { \vec x \in {\mathbb{R}^n}|{x_i} \ge 0\forall i,\sum\limits_{k = 1}^n {x_k^{2N}} < n – 1 } \right\}$$ and Dirichlet integrals to evaluate certain integrals of…
Introduction Collisions are very much a stock item in any school physics curriculum and students are generally taught about the use of the principles of conservation of momentum and energy for solving simple collision problems in one dimension. In this article we will be examining a very common type of collision problem: the inelastic collision….
Contents 1. Getting a C++ Compiler and Compiling Your First Program 2. Simple Datatypes and Declarations 3. Operators and Expressions 4. Input and Output (I/O) 5. Selection Statements 6. Iterative Statements 7. Arrays 8. Functions 1. Getting a C++ Compiler and Compiling Your First Program Getting a C++ Compiler For Windows users, especially beginners, I…
Introduction In previous articles relating to various transition energies in Hydrogen, Helium and Deuterium we have employed the following formula for electron energy given a particular primary quantum number n: $$ E_{n}=\mu c^2\sqrt{1-\frac{Z^2\alpha^2}{n^2}} $$ where ## \alpha ## is the fine structure constant and ## \mu ## the reduced electron mass for a single electron…
Preface My first experience with derivatives was seeing how they are obtained from the usual definition $$f'(x)=\underset{\text{$\Delta $x}\to 0}{\text{Lim}}\frac{f (\text{$x+\Delta $x})-f (x)}{\text{$\Delta $x}}.$$ I accepted the binomial theorem derivation in the case of polynomials and the small-angle explanation in the case of sines and cosines until my math instructor asserted, without justification, that the derivative…
In a recent thread, I outlined how to compute the correlation function for the Bell basis states \begin{equation}\begin{split}|\psi_-\rangle &= \frac{|ud\rangle \,- |du\rangle}{\sqrt{2}}\\ |\psi_+\rangle &= \frac{|ud\rangle + |du\rangle}{\sqrt{2}}\\ |\phi_-\rangle &= \frac{|uu\rangle \,- |dd\rangle}{\sqrt{2}}\\ |\phi_+\rangle &= \frac{|uu\rangle + |dd\rangle}{\sqrt{2}} \end{split}\label{BellStates}\end{equation} when they represent spin states. The first state ##|\psi_-\rangle## is called the “spin singlet state” and it…
Recursion is actually quite simple. It’s a subroutine calling itself. It’s surprising but some problems that look quite hard can be trivial using recursion – but be wary – as I hope to explain there are traps you need to consider especially if working in a team environment. When I studied Computer Science the first…
Definition/Summary The four laws of black hole thermodynamics are as follows… The Zeroth Law Surface gravity [itex](\kappa)[/itex] is constant over a black holes event horizon. The First Law ‘This law deals with the mass (energy) change, dM when a black hole switches from one stationary state to another.’ The following (in natural units) applies- [tex]dM=\frac{\kappa}{8\pi}\,dA\,+\,\Omega\,dJ\,+\,\Phi\,dQ[/tex]…
We get a lot of “what if” questions here on Physics Forums. This article will explore three different types and then some suggestions for students who feel their question may fall into one of those types. “what if” questions that contradict physics as we know it “what if” questions that are themselves self-contradictory “what if”…
Definition/Summary Abstract from Poisson and Israel’s 1990 paper, ‘Internal structure of black holes’- ‘The gravitational effects associated with the radiative tail produced by a gravitational collapse with rotation are investigated. It is shown that the infinite blueshift of the tail’s energy density occurring at the Cauchy horizon of the resulting black hole causes classically unbounded…
Preface While browsing through unanswered posts in the Classical Physics Workshop, I came across a gem at the link shown below. For the reader’s convenience, I have included (in italics) the OP’s statement of the question. https://www.physicsforums.com/threads/spacecraft-path-with-polar-coordinates.683210/ There is a circular gate rotating at a constant angular speed of ##\omega##. The circular gate has a…
Introduction Most readers acquainted with the hydrogen spectrum will be familiar with the set of lines in the visible spectrum representing transitions of electrons from energy levels 3,4,5 and 6 (H alpha, beta, gamma, and delta respectively) of atomic hydrogen to energy level 2 – the Balmer series lines. The picture below shows 3 of…
This will be the first of several articles which will provide, for reference, useful equations for static, spherically symmetric spacetimes. This is a common special case that is studied in General Relativity, and it has the advantage of having a general solution for the Einstein Field equation that can be expressed in closed form equations….
Proofs in mathematics are what mathematics is all about. They are subject to entire books, created entire theories like Fermat’s last theorem, are hard to understand like currently Mochizuki’s proof of the ABC conjecture, or need computer assistance like the 4-color-theorem. They are sometimes even missing, although everybody believes in the statement like the Riemann…
The Classical Limit of Commutator (without fancy mathematics) Quantum mechanics occupies a very unusual place among physical theories: It contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation. Many textbooks on elementary QM show you how the Hamilton-Jacobi [tex]\frac{\partial S}{\partial t} + H…
The Absolute Derivative In relativity we typically deal with two types of quantities: fields, which are defined everywhere, and particle properties, which are defined only along a curve or world line. The familiar covariant derivative is appropriate when we need to differentiate a field. A field is a function of all four coordinates, and the…
This Insight develops equations of state that are useful in calculations about cosmology and about the insides of stars. The first calculation is for a photon gas and the second is for a ‘relativistic’ gas of particles with mass. Photon Gas Exercise 22 on p108 of Bernard Schutz’s ‘The first course in General Relativity (Second…
Definition/Summary The double-slit is a simple configuration used to demonstrate interference effects in waves. Equations At distances that are large compared to the spacing between the slits (i.e. the far-field), the interference maxima (bright fringes) occur at angles such that [tex]d \ \sin\theta \ = \ m \ \lambda[/tex] where [tex]\begin{align*}d & = \text{the slit…
For reference purposes and to help focus discussions on Physics Forums in interpretation questions on the real issues, there is a need for fixing the common ground. There is no consensus about the interpretation of quantum mechanics, and – not surprisingly – there is disagreement even among the mentors and science advisors here on Physics…
Commonly there is a lot of imprecision in talking about ”indistinguishable” (or ”identical”) particles, even in serious work. This Insight article clarifies the issues involved in a conceptually precise way. Classical mechanics. Historically, indistinguishable particles were introduced in order to explain the failure of the thermodynamics of a Newtonian ##N##-particle system to describe the absence…
Definition/Summary A Lie algebra (“Lee”) is a set of generators of a Lie group. It is a basis of the tangent space around a Lie group’s identity element, the space of differences between elements close to the identity element and the identity element itself. Lie algebras include a binary, bi-linear, anti-symmetric operation: commutation. The commutator…
Here is another version of proof of Maupertuis’s principle. This version is pure Hamiltonian and independent of the Lagrangian approach. The proof is based upon the Hamiltonian version of the Vector Field Straightening Theorem. It seems that such a style of exposition simplifies the understanding of this non-trivial construction. First, recall and briefly discuss…
This Insight article is part of my paper Foundations of quantum physics III. Measurement, featuring the thermal interpretation of quantum physics. See this discussion. $$\def\D{\displaystyle} % display style \def\SS {{\bf S}} \def\Bsymbol#1{{\bm #1}} \def\Bsigma{{\sigma}} \def\half{\frac{1}{2}} \def\<{\langle} % expectation \def\>{\rangle} % expectation \def\fct#1{\mathop{\rm #1}} % e.g., \fct{tr} \def\fns#1{{\mbox{\rm \scriptsize#1}}} % subscript text \def\Tr{{\fct{Tr}}}$$ Stokes’ quantum counter. One…
In the first post in this series, we introduced the concepts of frame field, Fermi-Walker transport, and the “Fermi derivative” of a frame field, and developed some basic machinery for dealing with them. In this post, we will use that machinery to look at two congruences in Schwarzschild spacetime, to see how things differ from…
This is the first of several posts that will develop some mathematical machinery for studying Fermi-Walker transport. In this first post, we focus on Minkowski spacetime in order to introduce the basic concepts without having to deal with the complications introduced by spacetime curvature. Before looking at Fermi-Walker transport, we first need to introduce the…
I argue that all interpretations of quantum mechanics (QM) are incomplete, each for its own reason. I also point out that for some interpretations (those marked with (*)) this incompleteness is in fact a good thing because in principle this incompleteness may be resolved experimentally. Shut up and calculate logical positivism: It’s OK to talk…
In the first part of this article (AVX-512 Assembly Programing – Opmask Registers for Conditional Arithmetic), we looked at how opmask registers can be used to perform conditional arithmetic. That article ended with two 512-bit ZMM registers that each contained 16 integer values. One of the registers contained 16 partial sums of the positive numbers…
Learning Objectives * Execute a specific experimental procedure to test a specific hypothesis. * Use the Tracker video analysis software for a simple experiment. * Analyze the acquired data with a spreadsheet to test the hypothesis. * Explain in one’s own words whether the experimental data supported the hypothesis, and (if so), how well. *Use…
Part III: Representations 10. Sums and Products. Frobenius began in ##1896## to generalize Weber’s group characters and soon investigated homomorphisms from finite groups into general linear groups ##GL(V)##, supported by earlier considerations from Dedekind. Representation theory was born, and it developed fast in the following decades. The basic object of interest, however, has…
Learning Objectives Gain confidence and experimental care in making accurate measurements. Understand the relationship between force and spring stretch. Use a neat and orderly lab notebook in which data are recorded. Execute a specific experimental procedure to test a specific hypothesis. Analyze the acquired data with a spreadsheet and graphing program to test the hypothesis….
Part II: Structures 5. Decompositions. Lie algebra theory is to a large extend the classification of the semisimple Lie algebras which are direct sums of the simple algebras listed in the previous paragraph, i.e. to show that those are all simple Lie algebras there are. Their counterpart are solvable Lie algebras, e.g. the Heisenberg algebra ##\mathfrak{H}=\langle…
Part I: Basics 1. Introduction. This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems, and a brief overview of the subject. Physicists usually call the elements of Lie algebras generators, as for them they are merely differentials of trajectories, tangent vector fields generated by some operators. Thus…
Introduction In the first part of this series, I talked about some fundamental notions in the world of algorithms. Beyond the definition of an algorithm, we saw the criteria, ways to represent an algorithm, the importance of correctness, and elements of the classification and analysis of algorithms. It has also become evident that it is…
Introduction Bernoulli’s equation is one of the most useful equations in the field of fluid mechanics, owing to its simplicity and broad applicability. That applicability does have limits, however, which has led to it being one of the most misunderstood and also misused equations in fluid mechanics. The goal of this post is to present…
In this Insight, I’ll go over implementing basic orbital mechanics simulations in the Unity game engine as well as an approach to scaling the simulation for Augmented Reality applications. Unity is a great tool for prototyping games but also for animating physics models. The physics gets more interesting when you can watch all the interactions…
Wheeler’s opening statement in his 1986 paper, “How Come the Quantum?” holds as true today as it did then [1] The necessity of the quantum in the construction of existence: out of what deeper requirement does it arise? Behind it all is surely an idea so simple, so beautiful, so compelling that when — in…
Much has lately been said regarding this paradox which first appeared in one of W. Lewin’s MIT lecture series on ##{YouTube}^{(1)}##. This lecture was recently critiqued by C. Mabilde in a second YouTube video and submitted as a post in a PF ##{thread}^{(2)}##. The latter cited the third source, that of K. T. McDonald of…
Many threads here at PF include some questions about how to learn to program. This is asked by Physics students who want to learn programming in order to help their study, work on some project or create some simulation among other things but also by students in other fields of science and by other members,…
A few months ago, pervect pointed me to a post by Chris Hillman which is an introduction to the usage of Maxima for General Relativity. Maxima is a free (both as in beer and as in speech) symbolic algebra package, and it includes a library called censor that handles tensor components and looks to have…
Operators. The Maze Of Definitions. We will use the conventions of part I (Basics), which are ##\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}##, ##z \mapsto \overline{z}## for the complex conjugate, ##\tau## for transposing matrices or vectors, which we interpret as written in a column if given a basis, and ##\dagger## for the combination of conjugation and transposition, the adjoint…
Basics Language first: There is no such thing as the Hilbert space. Hilbert spaces can look rather different, and which one is used in certain cases is by no means self-evident. To refer to Hilbert spaces by a definite article is like saying the moon when talking about Jupiter, or the car on an automotive…
This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 14. Free quantum fields. The next chapter is 16. Renormalization. 15. Interacting quantum fields In this chapter we discuss the following topics: Free field vacua Perturbative S-matrices Conceptual remarks Interacting field observables Time-ordered products (“Re”-)Normalization Feynman perturbation series Effective…
In this Insight we will discuss the general Robertson–Walker (RW) universe, in which a set of co-moving observers with proper time ##t## observe the universe to be homogeneous and isotropic. In such a universe, the line element takes the form $$ds^2 = dt^2 – a(t)^2 [dr^2 + f(r)^2 d\Omega^2] \equiv dt^2 – a(t)^2 dS^2,$$ where…
Intransitive dice are sets of dice that don’t follow the usual rules for “is better/larger than”. If A<B and B<C, then A<C. If Bob runs faster than Alice and Charlie runs faster than Bob, then Charlie runs faster than Alice. If die B wins against die A (larger number wins) and die C wins against…
How do we use units? You may see one of these speed limit signs, nearly every day. Even though neither of them displays units, drivers know they are implied. In the United States, a Speed Limit 25 sign implies that 25 miles per hour are the maximum legal speed. The other type of speed limit…
The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 6. Symmetries. The next chapter is 8. Phase space. 7. Observables In this chapter we discuss these topics: General observables Polynomial off-shell observables and Distributions Polynomial on-shell observables and Distributional solutions to PDEs Local observables and Transgression Infinitesimal…
Key Points Published paper studied how knots in polymers respond to an external flow field and how the knot affects the response of the polymer Experiments and simulations of knotted polymer chains being stretched by an extensional flow field The coil–stretch transition occurs when the flow is weak enough for the molecule to relax to…
This is the first chapter in a series on Mathematical Quantum Field Theory. The next chapter is 2. Spacetime. 1. Geometry The geometry of physics is differential geometry. This is the flavor of geometry which is modeled on Cartesian spaces ##\mathbb{R}^n## with smooth functions between them. Here we briefly review the basics of differential geometry…
Part 1 Representations Image source: [24] 6. Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author to author: the numbering of the Pauli matrices ##(\text{I 4}), (\ref{Pauli-I})##, the linear combinations of them in the definition to basis vectors ##\mathfrak{B}## of ##\mathfrak{su}(2,\mathbb{C}) \; (\text{I 5}), (\ref{Pauli-II}), (\ref{Pauli-III})##, the embedding of the orthogonal groups ##(\text{I…
Part 2 Differentiation, Spheres, and Fiber Bundles Image source: [24] The special unitary groups play a significant role in the standard model in physics. Why? An elaborate answer would likely involve a lot of technical terms as Lie groups, Riemannian manifolds or Hilbert spaces, wave functions, generators, Casimir elements, or irreps. This already reveals…
The CMB establishes a record of ancient acoustic oscillations in the baryon-photon plasma. We’ve been studying how these primordial sound waves evolve, and how to analyze the last scattering surface to learn about them. Now it’s time to confront their origin: what process composed the cosmic symphony? A few different proposals have been advanced…
All these concepts belong to the toolbox of physicists. I read them quite often on our forum and their usage is sometimes a bit confusing. Physicists learn how to apply them, but occasionally I get the impression, that the concepts behind are forgotten. So what are they? Especially when it comes to the adjoint…
Introduction: In this Insight article, Parseval’s theorem will be applied to a sinusoidal signal that lasts a finite period of time. It will be shown that it necessarily follows that ## (\frac{\sin(2 x_o)}{x_o})( \frac{\pi}{2})=\int\limits_{-\infty}^{+\infty} \frac{\sin(x-x_o) \sin(x+x_o)}{(x-x_o)(x+x_o)} \, dx ##. Note: This integral result was computed by this author in 2009. I have not done…
Part 1: Quantum Mechanics [This article expects the reader to have some experience with quantum field theory and path integrals] [I should also make a disclaimer here, some parts of this article use some results from conformal field theory which is beyond my knowledge. So I have to just state the results in those parts….
Let me start with a counter-question. What is a number? Before you laugh, there is more to this question than one might think. A number can be something we use to count or more advanced an element of a field like the real numbers. Students might answer that a number is a scalar. This is…
In the early days of the personal computer revolution, computers were small, simple, and easy to operate. It was always great fun to write BASIC games on my Commodore 64 using its amazing sound generator, cool sprite graphics, and joystick features. With only a few lines of code, well okay maybe more than a few…
Quantum Mechanics(QM) is one of the greatest intellectual achievements in human history. Not only because it describes the world at the microscopic level and in turn, provides us with the technological advancement that we enjoy today, but also because it shows us how little we know about the world we’ve been living in for so…
Although Special Relativity was formulated by Einstein (1905), and given a spacetime interpretation by Minkowski (1908) [which helped make special relativity more accessible and acceptable], it could be argued that these approaches are still too abstract and too mathematical for most students. In the early 1960s, Hermann Bondi advocated a presentation of special relativity (called…
Traditional presentations of special relativity place emphasis on “velocity”, which of course has an important physical interpretation… carried over from Galilean physics. However, in many situations, working only with velocities can be tedious in calculations and may obscure the underlying geometry. First, a story (inspired by my post in the recent Relativistic Relative Velocity thread)….