Tag Archive for: Graduate

Entangled Photon Polarization Qubits

Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality per Quantum Information Theory

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In her YouTube video Bell's Theorem Experiments on Entangled Photons, Dr. Fugate shows how polarization-entangled photons violate Bell's inequality. In…
einstein field equations

Einstein Field Equations – Definition and Fast Facts

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Definition/Summary The Einstein Field Equations (EFE) are a set of ten interrelated differential equations that form the core of Einstein's general theory…
Lambert W Function in Finance

The Lambert W Function in Finance

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Preamble The classical mathematician practically by instinct views the continuous process as the "real" process, and the discrete process as an approximation…
world of algebras

Introduction to the World of Algebras

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Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois…
quantum information

How Quantum Information Theory Solves “the only mystery” of Quantum Mechanics

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In Chapter 37 of "The Feynman Lectures on Physics Volume 1," Richard Feynman famously wrote that the mystery of wave-particle duality in the double-slit…
Infinitesimals

What Are Infinitesimals – Advanced Version

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Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…
lie group physics

When Lie Groups Became Physics

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Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham…
gravity collapse

Oppenheimer-Snyder Model of Gravitational Collapse: Implications

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Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn the last article in this series, we finished up with a metric for the Oppenheimer-Snyder…
gravity collapse

Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details

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Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn a previous article, I described in general terms the model of gravitational…
gravity collapse

The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview

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Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsMost people who have spent any time at all studying GR are familiar with the…
history of numbers

Counting to p-adic Calculus: All Number Systems That We Have

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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…
definition differences

Yardsticks to Metric Tensor Fields

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I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist,…
Riemann Hypothesis History

The History and Importance of the Riemann Hypothesis

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Riemann Hypothesis History The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field…
hypercube integral

A Novel Technique of Calculating Unit Hypercube Integrals

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Introduction In this insight article, we will build all the machinery necessary to evaluate unit hypercube integrals by a novel technique. We will first…
Riemann Hypothesis

The Extended Riemann Hypothesis and Ramanujan’s Sum

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Riemann Hypothesis and Ramanujan's Sum ExplanationRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All…
mermin device quantum entanglement

Superdeterminism and the Mermin Device

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Superdeterminism as a way to resolve the mystery of quantum entanglement is generally not taken seriously in the foundations community, as explained in…
quantum mechanics new approach

Quantum Physics via Quantum Tomography: A New Approach to Quantum Mechanics

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176 Comments
This Insight article presents the main features of a conceptual foundation of quantum physics with the same characteristic features as classical physics…
Kerr Spacetime

Geodesic Congruences in FRW, Schwarzschild and Kerr Spacetimes

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Introduction The theory of geodesic congruences is extensively covered in many textbooks (see References); what follows in the introduction is a brief…
quantum information

How Quantum Information Theorists Revealed the Relativity Principle at the Foundation of Quantum Mechanics

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This Insight is a condensed version of No Preferred Reference Frame at the Foundation of Quantum Mechanics. Reference numbers here correspond to that paper.…
euler sums

Investigating Some Euler Sums

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So, why only odd powers? Mostly because the even powers were solved by Leonard Euler in the 18th century. Since the “mathematical toolbox” at that…
tolman law beginner

Tolman Law in a Nutshell

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The Tolman law describes how the temperature in a fixed gravitational field depends on the position (see https://arxiv.org/abs/1803.04106 for a pedagogic…
electromagnetic computations duality

Building a Numerical FEM Solver for Duality in EM Fields

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In the previous insights article (How to Use Duality in Computational Electromagnetic Problems), I covered some uniqueness theorems for the Riemann-Silberstein…
Electric Field Seen by an Observer

The Electric Field Seen by an Observer: A Relativistic Calculation with Tensors

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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:…
electromagnetic computations

How to Use Duality in Computational Electromagnetic Problems

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Some weeks ago I happened across a post that caught my eye. Dale asked a question about the number of photons in an electromagnetic field. His question…
electric potential

Electric Vector Potential: Detect Non-Conservative Fields

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Main Point The electric vector potential offers a means of determining the non-conservative component of a mixed stationary or quasi-stationary electric…
kerr spacetime centrifugal force

Intro to Centrifugal Force Reversal in Kerr Spacetime

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In this article, we will analyze "centrifugal force reversal" in Kerr spacetime, similar to what was done for Schwarzschild spacetime in a previous Insights…
electromagentic waves traverse

Are Electromagnetic Waves Always Transverse? Full Explanation

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In this insight, we will explore classical electrodynamics and examine whether electromagnetic (EM) waves are always transverse. We use Jefimenko's equations…
pressure is a source of gravity

Is Pressure A Source Of Gravity?

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In a previous series of articles, I posed the question "Does Gravity Gravitate?" and explained how, depending on how you interpreted the terms "gravity"…
lerch and zeta functions

The Analytic Continuation of the Lerch and the Zeta Functions

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Introduction In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler's…
dark energy supernova

Dark Energy Part 3: Fitting the SCP Union 2.1 Supernova Data

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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…
dark energy cosmology lcdm

Dark Energy Part 2: LCDM Cosmology

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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…
symmetric spacetime

Slowly Lowering an Object in a Static, Spherically Symmetric Spacetime

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In the first two articles in this series, we looked at the Einstein Field Equation and Maxwell's Equations in a static, spherically symmetric spacetime.…
Dark Energy Cosmology Part 1

Dark Energy Part 1: Einstein-deSitter Cosmology

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In this 3-part series, I want to motivate the (re)introduction of the cosmological constant ##\Lambda## into Einstein's equations of general relativity…
Cosmic Web Connections

Cosmic Web Connectivity: How It Shapes Galaxy Evolution

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Introduction The universe was not perfectly uniform at the beginning; some regions had higher density than others. Over time these higher-density regions…
Gaia search for black holes

Gaia Astrometry: Detecting Gravitational Waves Guide

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Overview Gravitational waves (GWs) are disturbances in spacetime produced by massive objects moving asymmetrically. Only the most massive and most relativistic…
black holes are not empty

VHE Gamma Rays from Sgr A*: Black Holes and CTA Observations

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Black Hole Key PointsBlack holes (BHs) have a gravitational pull so strong that nothing, not even light, can escape. BHs can be divided into two…
Integral Representations of Some Special Functions

The Orin Fractional Calculus

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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:…
Odd Sums

Explore the Fascinating Sums of Odd Powers of 1/n

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3 Comments
The goal is to get a little bit closer to the values of the zeta function (ζ(s)) and the eta function (η(s)) for some odd values of s. This insight is…
quantum entanglement

Answering Mermin’s Challenge with the Relativity Principle

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89 Comments
Note: This Insight was previously titled, "Answering Mermin's Challenge with Wilczek's Challenge." While that version of this Insight did not involve any…
Conservation of Spin Angular Momentum

Exploring Bell States and Conservation of Spin Angular Momentum

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In a recent thread, I outlined how to compute the correlation function for the Bell basis states\begin{equation}\begin{split}|\psi_-\rangle &=…
Kuhnian Revolution

Modern Physics Understood as an Unrecognized Kuhnian Revolution

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People often claim on Physics Forums and in the foundations community proper that quantum mechanics is "incomplete." Indeed, Lee Smolin recently wrote…
DC straight wire

Relativistic Treatment of the DC Conducting Straight Wire

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Introduction The direct-current-conducting infinitely long wire is often discussed in the context of relativistic electrodynamics. It is of course a completely…
Deuterium Lyman Alpha Line

Revisiting The Deuterium Lyman Alpha Line Experiment

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Introduction In this article, we will be revisiting a somewhat understudied (and seemingly unrepeated) experiment to measure the Deuterium Lyman Alpha…
what are the black hole thermodynamics

Black Hole Thermodynamics: Four Laws Explained — Guide

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Overview Black hole thermodynamics is summarized by four laws that closely parallel the laws of ordinary thermodynamics. Below I present each law, the…
What is Mass Inflation

Mass Inflation Inside Black Holes — Explanation & Equations

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Definition and summary Abstract from Poisson and Israel (1990), "Internal structure of black holes": "The gravitational effects associated with the radiative…
spacetime maxwell

Maxwell’s Equations in a Static, Spherically Symmetric Spacetime

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In the first article in this series, we looked at the Einstein Field Equations in a static, spherically symmetric spacetime. In this article, we are going…
Einstein Field Equation

The Einstein Field Equation in a Static, Spherically Symmetric Spacetime

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This will be the first of several articles which will provide, for reference, useful equations for static, spherically symmetric spacetimes. This is a…
Quantum Mechanical Commutator

The Classical Limit of Quantum Mechanical Commutator

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The Classical Limit of Commutator (without fancy mathematics) Quantum mechanics occupies a very unusual place among physical theories: It contains classical…
Minkowski Spacetime

Understanding Precession in Special and General Relativity

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The Absolute Derivative In relativity we typically deal with two types of quantities: fields, which are defined everywhere, and particle properties, which…
What is a fibre bundle

Fibre Bundle: Definition, Examples & Intuitive Guide

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Definition / Summary Fibre bundle — intuitively, a fibre bundle is a space E that locally looks like a product B × F but may have a different global…
Star Equations

Equations of State for Photon Gas and Relativistic Electron Gas

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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…
Quantum Particles

Clarifying Common Issues with Indistinguishable Particles

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Introduction Commonly there is a lot of imprecision in talking about ''indistinguishable'' (or ''identical'') particles, even in serious work. This Insight…
isotropy definition

A Formal Definition of Large-Scale Isotropy

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This Insight is part of my attempt to develop a formal definition of 'large-scale isotropy', a concept that is fundamental to most cosmology, but that…
Lie Algebra Intro

Lie Algebra Basics: Definitions, Equations & Examples

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Definition / Summary A Lie algebra (pronounced "Lee") is the tangent-space algebra of a Lie group at its identity element. Concretely, it is a vector…
Maupertuis Principle

A Pure Hamiltonian Proof of the Maupertuis Principle

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Here is another version of proof of Maupertuis's principle. This version is pure Hamiltonian and independent of the Lagrangian approach.The proof…
Kerr Spacetime

Exploring Fermi-Walker Transport in Kerr Spacetime

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In the last two posts in this series, we developed some tools for looking at Fermi-Walker transport in Minkowski spacetime and then applied them in Schwarzschild…
quantum qubit

A Classical View of the Qubit

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This Insight article is part of my paper Foundations of quantum physics III. Measurement, featuring the thermal interpretation of quantum physics. See…
Minkowski_Spacetime_2

Learning Fermi-Walker Transport in Schwarzschild Spacetime

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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…
Minkowski Spacetime

Fermi-Walker Transport in Minkowski Spacetime

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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…
What are virtual particles

Virtual Particles Explained — Quantum Field Theory

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Definition / Summary Virtual particles are a mathematical device used in perturbation expansions of the S-operator (transition matrix) for interactions…
quantum mechanics is incomplete

9 Reasons Quantum Mechanics is Incomplete

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Incomplete interpretations of quantum mechanics I argue that all interpretations of quantum mechanics (QM) are incomplete, each for its own reason. I…
against quantum interpretations

Against “interpretation”

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195 Comments
Against interpretations in QM I am against "interpretations" of Quantum Mechanics (QM) in a sense in which John Bell [1] was against measurement in QM…
lie algebra representations

Learn Lie Algebras: A Walkthrough – The Representations

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  Part III: Representations  Sums and Products. Frobenius began in ##1896## to generalize Weber's group characters and soon investigated…
Lie Algebra Structure

Learn Lie Algebras: A Walkthrough – The Structures

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  Part II: StructuresDecompositions.Lie algebra theory is to a large extend the classification of the semisimple Lie algebras which…
lie algebra basics

Learn Lie Algebras: A Walkthrough – The Basics

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  Part I: Basics Introduction. This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems,…
Quantum Eraser

The Quantum Mystery of Wigner’s Friend

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In this Insight I will introduce the quantum mystery called ``Wigner's friend'' using Healey's version [1] of Frauchiger and Renner's (FR's) version [2]…
Super Quantum

The Unreasonable Effectiveness of the Popescu-Rohrlich Correlations

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In this Insight, I will show how the Popescu-Rohrlich (superquantum) correlations provide an unreasonable advantage in a particular "quantum guessing game''…
rotational mechanics

An Example of Servo-Constraints in Mechanics

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Servo-constraint was invented by Henri Beghin in his Ph.D. thesis in 1922. For details see the celebrated monograph in rational mechanics by Paul Appell.To…
killing Fields and Black Holes

Killing Vector Field & Surface Gravity in Kerr BH Explained

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Killing Vector Field The Killing vector field is a vector field on a differentiable manifold that preserves the spacetime metric. In other words, it generates…
why quantum

Why the Quantum | A Response to Wheeler’s 1986 Paper

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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…
Sagittarius A

Calculate Sgr A* Spin from QPOs — Kerr Metric Derivation

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Introduction This Insight explains how it is possible to calculate the spin of Sagittarius A* — the supermassive black hole at the center of the Milky…
maxima

How to Solve Einstein’s Field Equations in Maxima

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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…
hilbertspaces2

Learn the Basics of Hilbert Spaces and Their Relatives: Operators

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  Operators. The Maze Of Definitions. We will use the conventions of part I (Basics), which are ##\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}##,…
qft_renormalization

Learn Renormalization in Mathematical Quantum Field Theory

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This is one chapter in a series on Mathematical Quantum Field TheoryThe previous chapter is: 15. Interacting quantum fields. 16. Renormalization In…

Learn Interacting Quantum Fields in Mathematical Quantum Field Theory

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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.…
qft_freequantum

Learn Free Quantum Fields in Mathematical Quantum Field Theory

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This is one chapter in a series on Mathematical Quantum Field TheoryThe previous chapter is 13. Quantization.The next chapter is 15. Interacting…
qtf_quantization

Learn Quantization in Mathematical Quantum Field Theory

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14 Comments
The following is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 12. Gauge fixing.The next chapter is 14.…
expandinguniverse

Explore Coordinate Dependent Statements in an Expanding Universe

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4 Comments
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…
gps3

The Schwarzschild Metric: A Newtonian Comparison

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A Short Proof of Birkoff's Theorem derived the Schwarzschild metric in units of ##G = c = 1##:\begin{equation} ds^2 = -\left(1 - \frac{2M}{r}\right)dt^2…
gps2

The Schwarzschild Metric: The Photon Sphere

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 A Short Proof of Birkoff's Theorem derived the Schwarzschild metric in units of ##G = c = 1##:\begin{equation} ds^2 = -\left(1 - \frac{2M}{r}\right)dt^2…
qft_gauge

Learn Gauge Fixing in Mathematical Quantum Field Theory

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The following is one chapter in a series of Mathematical Quantum Field Theory.The previous chapter is 11. Reduced phase space.The next chapter…
qtf_phased_space

Learn Reduced Phase Space in Mathematical Quantum Field Theory

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4 Comments
The following is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 10. Gauge symmetries.The next chapter is…
qtf_guagesym

Learn Gauge Symmetries in Mathematical Quantum Field Theory

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18 Comments
This is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 9. Propagators.The next chapter is 11. Reduced phase…
qft_prapagators

Learn Propagators in Mathematical Quantum Field Theory

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This is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 8. Phase space.The next chapter is 10. Gauge symmetries. 9.…
qtf_phasespace

Learn Phase Space in Mathematical Quantum Field Theory

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7 Comments
The following is one chapter of a series on Mathematical Quantum Field Theory.The previous chapter is 7. Observables.The next chapter is 9. Propagators. 8.…
qtf_observables

Learn Observables in Mathematical Quantum Field Theory

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8 Comments
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…
dna_stretch

Knotted DNA in Extensional Flow: Coil–Stretch and Untying

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4 Comments
Macromolecules Paper overview A joint computational-experimental paper that I worked on was just published in MacroLetters, a journal that covers all…
qtf_symmerities

Learn Symmetries in Mathematical Quantum Field Theory

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4 Comments
The following is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 5. Lagrangians.The next chapter is 7. Observables.  6.…
qft_lang

Learn Lagrangians in Mathematical Quantum Field Theory

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42 Comments
This is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 4. Field Variations.The next chapter is 6. Symmetries.  5.…
qtf_field_variations2

Learn Field Variations in Mathematical Quantum Field Theory

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6 Comments
This is chapter 4 of a series on Mathematical Quantum Field Theory.The previous chapter is 3. Fields.The next chapter is 5. Lagrangians.  4.…
qtf_fields

Learn the Fields of Mathematical Quantum Field Theory

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82 Comments
The following is one chapter in a series on Mathematical Quantum Field TheoryThe previous chapter is 2. Spacetime.The next chapter is 4. Field…
qft_spacetime

Learn Spacetime in Mathematical Quantum Field Theory

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9 Comments
The following is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 1. Geometry.The next chapter is 3. Fields  2.…
qtf_geometry

Learn the Geometry of Mathematical Quantum Field Theory

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33 Comments
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…
quantumfieldtheory

A First Idea of Quantum Field Theory – 20 Part Series

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These notes mean to give an expository but rigorous introduction to the basic concepts of relativistic perturbative quantum field theories, specifically…
manifold2

A Journey to The Manifold SU(2): Representations

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3 Comments
Part 1 Image source: [23]  Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author to author: the…

A Journey to The Manifold SU(2): Differentiation, Spheres, and Fiber Bundles

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Part 2  Differentiation, Spheres, and Fiber Bundles Image source: [24]The special unitary groups play a significant role in the standard…
perturbation

Introduction to Perturbative Quantum Field Theory

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110 Comments
This is the beginning of a series that gives an introduction to perturbative quantum field theory (pQFT) on Lorentzian spacetime backgrounds in its…
quantumpeace

How I Stopped Worrying and Learned to Love Orthodox Quantum Mechanics

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289 Comments
Why I'm a Bohmian and Questioning Orthodox QM Many people here know that I am a "Bohmian", i.e. an adherent of a very non-orthodox interpretation of quantum…
inflation5

A Poor Man’s CMB Primer: Quantum Seeds

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 The CMB establishes a record of ancient acoustic oscillations in the baryon-photon plasma. We've been studying how these primordial sound…
mathoperators

How to Tell Operations, Operators, Functionals, and Representations Apart

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 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.…