tag:blogger.com,1999:blog-33255702017-02-21T10:27:58.227-05:00strings of ideasjournaljoseph palazzonoreply@blogger.comBlogger50125tag:blogger.com,1999:blog-3325570.post-48632580729195791142016-07-20T17:48:00.000-04:002016-09-02T16:47:52.180-04:00ARE THE LAWS OF PHYSICS WEIRD?See info on my recent book: ARE THE LAWS OF PHYSICS WEIRD?
It's available here. joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-41870373827110880972015-09-04T09:11:00.000-04:002015-09-05T06:24:52.667-04:00Superposition and Quantum States
A lot of confusion in Quantum Mechanics is the result from not being able to differentiate between the real world and the Hilbert Space. Vectors in real space – like velocities, accelerations, forces, etc. – are objects one can actually measure in the real world. On the other hand, quantum states are represented by vectors (more precisely by rays) in a Hilbert space, but these are NOT joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-37016622856312842332015-08-26T20:08:00.000-04:002015-08-30T04:09:38.730-04:00Entanglement and the Uncertainty Principle
Here's your typical experiment. Suppose these particles are electrons/positrons and they are sent one pair at a time. At O, the two particles are released from rest and sent in opposite directions, and of course because of the conservation of angular momentum, they will have opposite spins as they move toward A and B, where observers are stationed.
A..............................←●O●→......joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-40976770023379249622015-05-20T12:33:00.000-04:002015-07-19T16:32:22.728-04:00Killing Vectors and Hawking Radiation
Preliminary
We start out with the interval (see equations (6) to (23) in Relativistic Doppler Effect ),
(1) ds2 = -dt2 + dx2 + dy2 + dz2
We define the metric as the coefficient of each of the terms in the above:
(2) η00 = -1, η11 = 1,η22 = 1,η33 = 1,and ηij = 0 for i≠j
We can rewrite equation (1) in the general form,
(3) ds2 = ηαβdxα dxβ
The proper time τ is,
(4) dτ2 =joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-79472567952629025122014-12-20T15:38:00.002-05:002015-08-28T05:58:35.850-04:00The Equivalence Principle: BBT or SUTEinstein made the following reasoning:
he took the Doppler Effect and using the Equivalence Principle, turned it into a Gravitational Shift. And then from there, he derived his eponymous Field Equations.
See The Essential General Relativity.
By reversing this reasoning I've shown the following:
from the Gravitational Shift and using the Equivalence Principle, I've turned that into a joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-23758896856906690882014-12-05T09:16:00.001-05:002014-12-05T09:55:56.637-05:00Big Bang Theory Versus Static Universe TheoryIn this blog, I will compare the Static Universe Theory (SUT) versus the Big Bang Theory (BBT) in regard to their respective assumptions.
As it has been mentioned in Another argument against the BBT, it is believed that only a Hot Big Bang scenario can explain the CMB, and any other explanation would have to be joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-1147082517707938482014-12-01T09:34:00.000-05:002014-12-03T10:00:22.345-05:00Olbers' Paradox
Olbers' paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe.
Let n be the average number density of galaxies in the universe. Let L be the average stellar luminosity. The flux f(r) received on earth from a galaxy at a distant r is,
(1) f(r) = L/(4πr2)
Consider now a thin spherical shell of galaxies of joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-89391465721959559172014-11-19T11:21:00.000-05:002014-11-19T11:32:15.268-05:00Another argument against the BBT
In The Equivalence Principle and the Big Bang Theory, we explored the idea that the galaxies are exhibiting a redshift that can be interpreted either as a Doppler effect, that is, they are moving away, or as a gravitational shift, in which case we can safely say they are at rest with each other. An argument for the BBT that every observer in the universe will see every other galaxy moving joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-58286656039743438882014-11-01T11:37:00.000-04:002014-12-03T15:28:27.867-05:00The Equivalence Principle and the Big Bang Theory
In this blog I explore the idea that Hubble failed to notice an important aspect of Einstein’s Equivalence Principle.
Consider this thought experiment. Different emitters are placed at different heights from the ground. The earth plays the role of the source of gravity.
They emit light, which from an earth observer, would be blueshifted. According to Einstein’s Equivalence Principle:joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-7657782151816967392014-10-28T10:10:00.000-04:002014-11-22T10:39:23.092-05:00Riemannian Geometry and the Big Bang Theory
Actually it was Gauss who proposed how to describe the inhabitants on a sphere as if they were unaware of the third dimension. Of course, they would need only two coordinates, why it’s called a 2-sphere. Here’s an example with spherical coordinates φ and θ:
But it was Riemann, Gauss’ student, who extended this idea to higher dimensions, hence why it’s generally known as Riemannian joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-61302077451458581032014-07-14T05:26:00.000-04:002014-07-14T05:29:06.037-04:00Quantum Vacuum Fluctuations - The BasicsQuantum vacuum fluctuations are facts in the real world - they were a fundamental prediction of Quantum Mechanics revealed in several processes. The spectrum of quantum fluctuations is a neat mathematical formulation that embodies this concept. It serves as a pivotal point between the very large (cosmological scale) and the very small (subatomic scale). No cosmological model that aspires to joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-26540049427424234512014-06-17T07:53:00.000-04:002014-11-04T08:32:30.671-05:00Quantum Fields in Curved Space-Time
(1) ds2 = dt2 - a2(t)δijdxidxj
(See equation 10 in The Essential General Relativity )
(2) Define the conformal time dη(t) ≡ dt/a(t)
Substitute into 1, we get,
(3) ds2 = a2(η)[dη2 - δijdxidxj]
= a2(η)ημνdxμdxν
(4) where
Borrowing equations 12,13 in The Essential Quantum Field Theory
(5) ℒ = ½ ημν∂μϕ∂νϕ − ½ m2ϕ2
(6) ∂μ∂μϕ + m2ϕ = 0
The corresponding action is,
(7joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-70220376032543807562014-06-03T07:55:00.000-04:002014-06-15T07:35:18.178-04:00Effective Field Theory Made Simple
Math Background
A functional is a function of a function: F[x(t)] is a function of x, which is a function of t. The use of square brackets is standard practice.
Also δ F[x(t)]/δx(t) will denote the derivative of F[x(t)] with respect to x(t).
A Wick rotation is given by t → −iτ . If we substitute this into non-Euclidean geometry, more specifically, a Minkowski geometry with signature (−+joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-38945043267447512342014-05-12T07:21:00.000-04:002014-05-13T06:48:01.878-04:00Path Integral Simplified
Transition Amplitude
Consider a normalized wavefunction ψ,
(1) ψ = A1ψ1 + A2ψ2 + A3ψ3
Where A1 is the amplitude of ψ1, A2 is the amplitude of ψ2, etc.
Given the rules of QM, the probability of measuring ψ1 is
(2) P1 = A1 A1* = │ A1│2
See equations 23 and 28 in The Essential Quantum Mechanics
The notion is that we started with ψ initially, then after measuring ψ1, the joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-61695793812410739732014-04-28T19:24:00.001-04:002015-05-19T16:53:03.608-04:00The Essential Quantum Field Theory
Classically, the electron is fundamentally different from a photon. In QM, we see them not that much differently as particles/waves. How is this to be reconciled?
There are two possible ways to interpret this: one, you could envision that the particles are fundamental. The photons are particles, and if you pack enough of them, that gives rise to a classical field; two, you could imagine thatjoseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-81739118098727646532014-03-27T10:30:00.000-04:002014-06-16T09:56:24.165-04:00Harmonic Oscillators, Vacuum Energy, Pauli Exclusion Principle
Preliminary
There are two reasons why the harmonic oscillator plays such a pivotal role in Quantum Mechanics (QM).
i) It is one of the few problems for which there exists an exact solution.
ii) The whole mathematical apparatus is transferred directly to Quantum Field Theory (QFT).
Sidney Coleman famously said: “The career of a young theoretical physicist consists of treating the harmonicjoseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-45101939746319411592014-02-21T07:33:00.000-05:002014-07-01T19:20:07.639-04:00The Essential Quantum Mechanics
Mathematical formulation of QM
We will focus our attention on the wavefunction, the operative part is "function", as this object is mathematical in nature, and not to be taken as a real wave. If you keep that in mind, a lot of confusion about QM will dissipate.
(1) We will use the Dirac notation, that is, the wavefunction is a vector V denoted by a "ket", | V >.
(2) A ket can be joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-22716161562742822552014-01-22T16:07:00.000-05:002014-04-12T09:15:19.165-04:00The Essential General Relativity
Preliminary
What are the equations needed to go from Newtonian Physics to General Relativity? In Is Newtonian Gravitational Fields Valid Within General Relativity? we showed in the case of a weak gravitational field and low velocity how the Einstein Field Equations (EFE)
(A) Gμν = 8πGc-4Tμν
yield Newton's law of gravity,
(B) ∇2φ = 4πGρ
However, historically, Einstein guessed equation 16joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-44470166918069191902013-10-23T09:37:00.000-04:002014-03-12T15:33:18.388-04:00Description of Reality - The EPR Paper Revisited
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In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-48484946982729436462013-07-07T11:18:00.000-04:002014-05-29T11:34:56.156-04:00The Unruh Effect<!DOCTYPE html>
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In this blog, ℏ=c=kb = 1 where ℏ is the reduced Planck constant, c is the speed of light and kb is Boltzmann’s constant.
Preliminary:
In the early 1970’s there were several applications of QFT to GR, the most famous one was done by Hawking in which he derived that a Black Hole has entropy proportional to its joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-1834295700956532012013-06-10T06:44:00.000-04:002014-07-12T10:13:25.860-04:00Relativistic Doppler Effect
One of the assumption of Special Relativity is that the velocity of light is constant in every inertial frame of reference. Consider two frames: one at rest (t,x,y,z); and a second one (t',x',y',z') is in motion with velocity V with respect to the first one. We've omitted the z- and z'-axis as representation of 4 dimensions on a 2-D surface is not possible.
Now suppose a light was joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-29155131764718340782013-06-01T04:29:00.002-04:002015-05-05T10:47:50.627-04:00Einstein's Derivation of the Famous Equation, E=mc2
Here’s a quick rundown on one of the most famous equation in physics, E = mc2. Einstein knew from experiments previously done that a particle could decay and release gamma rays. He reasoned that when this happened, the particle would lose kinetic energy, and this could only be accounted by a loss of mass. So how did he come to that conclusion? He analysed the situation both in a rest frame and joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-72934762942650300202013-05-29T04:31:00.000-04:002013-05-29T16:21:52.311-04:00Is Newtonian Gravitational Fields Valid Within General Relativity?
We will demonstrate that the Einstein Field Equations reduce to Newton's Law of Gravity in the case of a weak field and slow-motion of a particle (v less than the speed of light, c).
As we have already seen, Newton's Gravitation Law can be written as
(1) ∇2φ = 4πGρ, Equation (16) in Newton's Law of Gravity.
In free-fall, a particle satisfies,
(2) d2x/dt2 = F/m= –∇φ, Equations (4joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-89510456750572705352013-05-28T13:51:00.000-04:002014-11-01T11:34:44.352-04:00Newton's Law of Gravity
Newton’s law of gravity
Fig 1
(1) F = –(GmM)/R2
In vector form, this reads as:
(2) F = – [(GmM)/R2]er
Consider a system of particles. We label the particles 1, 2, 3... i...j...
Fig 2
(3)F = – ∑i≠j[(GmiMj)/Rij2]er
Define Gravitational joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0tag:blogger.com,1999:blog-3325570.post-75356938911878251992013-05-20T08:09:00.000-04:002015-08-30T10:10:08.042-04:00Mach–Zehnder interferometer Particle or Wave?
This is a continuation of what we have covered in the two-slit experiment. If you have forgotten the main concepts, kindly review them.
We will try to shed some light on what makes up light: particles or waves.
In papers written about this subject, such as of the Mach-Zehnder interferometer, you will often read: "The explanation for this result is that it appears a single photon travels joseph palazzohttps://plus.google.com/113648989123557845774noreply@blogger.com0