Natural Philosophy

Mathematics || Particle Physics

[trigonometry-is-my-bitch]

getting exam results tomorrow.

[trigonometry-is-my-bitch]

Getting A level exam results on thursday.

[trigonometry-is-my-bitch]

Getting final A2 results tomorrow.

[trigonometry-is-my-bitch]

Getting my Masters Certificate in Particle Physics in January.

[mysteriousquantumphysics]

Correlations without Correlata?

While Bell's theorems aim at the derivation of a statistical inequality that violates quantum predictions, the physicsts Greenberger, Zeilinger and Horne provided in 1989 a new theorem "without inequalities", therefore yielding a "stronger" result than Bell's, that again heated up the ongoing debate about the philosophical implications of quantum theory. We will focus on a simplified version of the proof given by Mermin [1].

Imagine a gedanken setup (figure above) that consists of a particle source that creates triples of spin 1/2 particles. After the emission, they are sent into areas of spacetime that are far apart from each other - so far that the velocity of light would not suffice to signal information between them while experimenters do nearly simultaneous spin measurements on each of the particles (spacelike separation). Each measurement yields binary results, one can only get +1 or -1, and the measurements are modeled in the quantum mechanical formalism using Pauli spin matrices σ. Their upper case index {1, 2, 3} denotes the number of the particle, whereas the lower case index {i,j,k} denotes the spatial direction in which the spin is measured. One can choose i,j,k=x,y,z independently for each of the particles. In particular the following four "observables" are of interest, meaning that we either measure all combinations of one x-setting and two y-settings or the triple x-setting. Those combinations can be simultaneously mesured because all four operators do indeed commute (though the single particle Pauli matrices do not commute of course).

At this point it is useful to become aware of the fact that the dimension of the whole system's Hilbert space is 2^3=8, since each particle's Hilbert space is 2-dimensional and the dimension of the whole Hilbert space is the product of the constituent's dimensions due to the tensor product. Bell's setup involves only two particles and therefore the Hilbert space dimension is only 2*2=4. Having noted this, the following maximally entangled wavefunction describes the joint state of those three particles:

Where the kets denote the basis of the σ_z Pauli matrix. Some computation yields the following eigenvalues:

Note that the triple x-measurment (last line) does have a negative eigenvalue - this will be the source of the derived contradiction. So far so good, we're done with the part that relies on the purely quantum mechanical formalism. The interesting thing is, that with this particular setup one can, after the measurement of two particles, determine with certainty what the result of the third must be. E.g. we measure the x-direction on particle 1, the y-direction on particle 2 and the y-direction on particle 3 (first line of the equations above). Then the product of all outcomes must be +1. In the case that the outcomes of particle 2 and 3 are -1, we can directly determine that the first particle's outcome must be +1. More generally, if both y-components yield the same result, the x-outcome must be +1. If both y-measurements yield a different result, the x-outcome must be -1. This is in accordance with the quantum mechanical predictions above. The possibility of determining values with certainty makes one suspect that it might be reasonable to assume that there are some predetermined values prior to the measurement. Let's elaborate this train of thought: We denote the predetermined measurement value with the variable m. Its upper case index stands for the particle's number and the lower case index for the spin component. Then, according to our findings, the products of any combination of two y-measurements and one x-measurements yield +1.

If these three equations hold, also the product of these equations must be:

Since all of these are scalars, we should be able to simply rearrange them. Any y-value appears twice, which is why they contribute each a +1 [simply because (-1)^2=(1)^2=1].

But then only the x-values remain and their product must be +1. But this is a clean contradiction to the quantum mechanical prediction! The eigenvalue to the operator corresponding to a triple x-measurement is -1. Since the mathematics is clear, the assumptions leading to the formulation of predetermined values must be flawed somehow - and there's by far no consensus which assumption is responsible for this mess. Either some notion of "locality" must fail, or there are no "elements of reality" - the possibilites are various. Just to mention one conclusion that could be drawn from this: "Correlations have physical reality; that which they correlate does not." [3] So one might give up the idea that we have to deal with "real" correlata - but this is not mandatory. One can save the idea of "reality" with certain restrictions, it's a matter of the own philosophical worldview, so no one can say for sure that the GHZ proof or Bell-type proofs in general are "no go theorems" for hidden variable accounts. Having noted this, I recommend to really understand what's going on in the GHZ proof above, before jumping to conclusions by adopting a certain interpretation.

---

Further reading: [1] Mermin: "What's wrong with these elements of reality?" doi: 10.1063/1.2810588 [2] Greenberger, Zeilinger, Horne: "Bell's theorem without inequalities" doi: 10.1119/1.16243 [3] Mermin: "What is Quantum Mechanics trying to tell us?" arXiv:quant-ph/9801057v2

[trigonometry-is-my-bitch]

This is a proof for why we do not make an assumption on Locality!

If you allow each of your entangled GHZ Kets to be shared by Alice, Bob, and Charlie, lightyears away from each other, each of whom can measure their ket in whichever basis they please, say either bases X or Y, then on the locality assumption, you expect Eigenvalues -1.

The locality assumption states that the choice among the two possible measurement bases X or Y made by Alice cannot affect the measurement result obtained by another observer Bob or Charlie when they are space-like separated, or outside each other’s light cone.

The locality assumption is demanding that it is the choice of basis of one observer which should not influence the measurement outcome of another observer.

In most of the literature i’ve come across, the GHZ state takes the form:

where zero is to mean the up state and 1 the down state.

Then my eigenvalues are yours sign-flipped!

So the quantum mechanics measurement will give you +1, and the locality Assumption is a false one when we accept the quantum measurement to be true.

Since the locality measurement is false, Alice’s choice of basis does affect Bob’s / Charlie’s outcome. This is the so called spooky action at a distance - One observer instantaneously affects the measurement of the others. Adding to that, there is no dynamics involved, its purely a statistical action.

But it does not violate relativistic causality - no information can be sent this way over space-like separated distances between Alice or Bob or Charlie.

Quantum nonlocality is how two or more parties can share entangled states across a quantum channel in Quantum computing! 

[trigonometry-is-my-bitch]

Sign the Petition

Remove the Cecil Rhodes statue at Oxford University

In the light of the racial Injustice we have seen in America, where a Black man named George Floyd was murdered at the hands of the US police, and where Breonna Taylor, a Black EMT, was murdered while sleeping in her home by police without uniform, and many other brutalities take place every day in America unto black people that goes unheard, it lifts my spirits to see people all around the world in support of the Black lives matter movement for racial justice.

I am heartbroken to watch their governments show little to no support for their people protesting against racism, when they have plenty to talk about from their unsettled past actions during the colonial era, as well as institutionalised racism which breathes today in their countries. 

In the years of 1884 - 1914, the Berlin conference saw many of the major nations of europe partake in the colonisation of African countries known as the Scramble for Africa, which set a low bar for the history of mankind. You can and should read all about the atrocities onto African people, such as the Human Zoos, or the genocides which took place under the foot of colonial powers

I ask you to remember the starving to death of 65,000 Herero people, 80% of their number. And the 10,000 Nama people, 50% of their number in German concentration camps like Shark Island. They were the first genocides of the 20th century, we are only in the beginning of the 21st. The effects are still suffered today.

I ask you to remember the Belgium Rubber mines which saw the people of Congo enslaved and tortured. Even today countries like France interfere with African politics, and are forcing 14 african ex-colonies to pay colonial tax since their independence. 85% of these countries’ national reserve is controlled by France. The effects are still suffered today.

The colonial powers did not care to make amends with the ex-colony states after the end of colonialism, which saw many african nations and cultures in total ruin. They carved up borders that were not there, grouping and racially categorising indigenous people, which saw tribes amongst the desolation of a sacked continent at war with each other, and lead to incidents such as those seen in Rwanda.

Now today you hear white supremacist leaders that accomplish nothing for their citizens, calling African nations “shithole countries”. The painting of the african continent is that they cannot develop their cultural identity like Europe can because they are inferior, and not because while they were subjugated, their culture was abolished for being unlike theirs, and surely therefore barbaric. Or they cannot create strong economies like Europe can because they are inferior, while the wealth of those european countries is built on the spoils of a war against Africans, those same Africans did not choose to be engaged in.

And in america, white supremacy will convince you today that black people are lazy, after fighting a civil war to keep them enslaved, to do the hard labour for them because black people were just so good at building roads and the White house.

It is not okay to be silent about racism. It is equally dangerous to humanity to leave mankind’s sins unhealed and conveniently forgotten, because the pain is felt every day by black people and african nations by racism which lives today just as fresh as then in the beginning of Colonialism. The effects of racism don’t go away because a government stops carving up your homeland, or enslaving your people, the problem gets passed down on both sides of the conflict and becomes a generational issue like we are seeing in Britain, Africa, America, and other parts of the world.

I live in the United Kingdom, so I call for the government to come forward and address properly the injustice done to ethnic minorities today in britain, as well as in the past. I call for other European countries to address racism in a similar manner officially, if you are a European citizen, you should call upon your government too. Italy, Spain, Portugal, France,Belgium, Britain are failing to engage the public in conversation about racism.

European Governments are failing the people taking a stand against racism and protesting. Because they are hiding in shame like the biblical adam  who knew he had sinned. If only they put the same level of energy and enthusiasm in healing the wounds of centuries of racism as they did sowing its seeds.

To start, I call for the Statue of British colonialist Cecil Rhodes, proudly displayed at Oriel College, Oxford University, to be taken down and put in a museum. In 2016, a petition to remove the statue which represents British white supremacy, was refused because “the statue was a reminder of the complexity of history and of the legacies of colonialism”. There is nothing complex at all about colonialism, that is only delusional exceptionalism born from white supremacy. It is animalistic barbarism, and failure of compassion that forever left a stain upon humanity, which at least we can learn from - in a museum. If it is indeed history it should be put in a museum.

Sign the petition to remove the Cecil Rhodes statue.

Please, share this, and do all that you can to support the movement for racial justice. Do what you can in your country.

Donate to Black lives matter: https://secure.actblue.com/donate/ms_blm_homepage_2019

[trigonometry-is-my-bitch]

Amazing.. I haven't fallen this deeply into a blog in years. Continue to amaze the world. Salutations :)

thank you! this is a nice message to see after years away from this site. Thought i’d enjoy a little nostalgia :’). Simpler times indeed.

[trigonometry-is-my-bitch]

Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude.

Animation by Martin Holtham:

https://twitter.com/GHSMaths

https://www.desmos.com/calculator/xhwhiztqzd

[trigonometry-is-my-bitch]

FUCKING SWEET JOB PASSING! Also, what are "a level exams"? I think we might be in different parts of the world.

The dreadful exams one must take before they go to higher education. The rites of passage if you will.I did my time.

[trigonometry-is-my-bitch]

Hey did you pass your a levels? :)

Yeah boiiiiiBut my school experienced some "admin issues" and the exams officer didnt publish my maths grade, so now they've had to contact my uni to clarify that i did actually get a maths grade which has taken two days just to get a hold of them (busy i assume) and im still sorting this out :/// good thing im taking a gap year

[trigonometry-is-my-bitch]

Good luck !! What course have you applied for at uni hope you get!!

Thanks!! The course is particle physics, but im planning on deferring for a gap year~~

[trigonometry-is-my-bitch]

getting exam results tomorrow.

[trigonometry-is-my-bitch]

Getting A level exam results on thursday.

[trigonometry-is-my-bitch]

Getting final A2 results tomorrow.

[bigblueboo]

[theemotionmachine]

The growth of knowledge* *Degrees not actually required. Just time, effort, and dedication.

[trigonometry-is-my-bitch]

How have I only just found this blog!?

I keep a low profile 👤👤. glad you like it!

[mathani]

As the disk sector is being completed, the centroid spirals into the center.

[wolframalpha]

Fewer than a dozen lines of Wolfram Language code created this hypnotic view of a Roman Surface: https://wolfr.am/dYwe52BS

[trigonometry-is-my-bitch]

Stereographic of Jupiter as seen from the bottom

[journalofscience]

Beneath the water of Mangroves

“Mangroves are shrubs or small trees that grow in coastal saline or brackish water. The term is also used for tropical coastal vegetation consisting of such species. Mangroves occur worldwide in the tropics and subtropics, mainly between latitudes 25° N and 25° S. In the year 2000, the area of mangroves was 53,190 square miles (137,760 km²), spanning 118 countries and territories.

Mangroves are salt tolerant trees, also called halophytes, and are adapted to life in harsh coastal conditions. They contain a complex salt filtration system and complex root system to cope with salt water immersion and wave action. They are adapted to the low oxygen (anoxic) conditions of waterlogged mud.

The word is used in at least three senses: (1) most broadly to refer to the habitat and entire plant assemblage or mangal, for which the terms mangrove forest biome, mangrove swamp and mangrove forest are also used, (2) to refer to all trees and large shrubs in the mangrove swamp, and (3) narrowly to refer to the mangrove family of plants, the Rhizophoraceae, or even more specifically just to mangrove trees of the genus Rhizophora.

The mangrove biome, or mangal, is a distinct saline woodland or shrubland habitat characterized by depositional coastal environments, where fine sediments (often with high organic content) collect in areas protected from high-energy wave action. The saline conditions tolerated by various mangrove species range from brackish water, through pure seawater (30 to 40 ppt [parts per thousand]), to water concentrated by evaporation to over twice the salinity of ocean seawater (up to 90 ppt).”

 - Wikipedia

[journalofscience]

The code that took America's Apollo 11  to the moon in the 1960's has been published

When programmers at the MIT Instrumentation Laboratory set out to develop the flight software for the Apollo 11 space program in the mid-1960s, the necessary technology did not exist. They had to invent it.

They came up with a new way to store computer programs, called “rope memory,” and created a special version of the assembly programming language. Assembly itself is obscure to many of today’s programmers—it’s very difficult to read, intended to be easily understood by computers, not humans. For the Apollo Guidance Computer (AGC), MIT programmers wrote thousands of lines of that esoteric code.

Here’s a very 1960s data visualization of just how much code they wrote—this is Margaret Hamilton, director of software engineering for the project, standing next to a stack of paper containing the software:

The AGC code has been available to the public for quite a while–it was first uploaded by tech researcher Ron Burkey in 2003, after he’d transcribed it from scanned images of the original hardcopies MIT had put online. That is, he manually typed out each line, one by one.

“It was scanned by an airplane pilot named Gary Neff in Colorado,” Burkey said in an email. “MIT got hold of the scans and put them online in the form of page images, which unfortunately had been mutilated in the process to the point of being unreadable in places.” Burkey reconstructed the unreadable parts, he said, using his engineering skills to fill in the blanks.

  “Quite a bit later, I managed to get some replacement scans from Gary Neff for the unreadable parts and fortunately found out that the parts I filled in were 100% correct!” he said.

As enormous and successful as Burkey’s project has been, however, the code itself remained somewhat obscure to many of today’s software developers. That was until last Thursday (July 7), when former NASA intern Chris Garry uploaded the software in its entirety to GitHub, the code-sharing site where millions of programmers hang out these days.

Within hours, coders began dissecting the software, particularly looking at the code comments the AGC’s original programmers had written. In programming, comments are plain-English descriptions of what task is being performed at a given point. But as the always-sharp joke detectives in Reddit’s r/ProgrammerHumor section found, many of the comments in the AGC code go beyond boring explanations of the software itself. They’re full of light-hearted jokes and messages, and very 1960s references.

One of the source code files, for example, is called BURN_BABY_BURN--MASTER_IGNITION_ROUTINE, and the opening comments explain why:

About 900 lines into that subroutine, a reader can see the playfulness of the original programming team come through, in the first and last comments in this block of code:

In the file called LUNAR_LANDING_GUIDANCE_EQUATIONS.s, it appears that two lines of code were  meant to be temporary ended up being permanent, against the hopes of one programmer:

In the same file, there’s also code that appears to instruct an astronaut to “crank the silly thing around.”

“That code is all about positioning the antenna for the LR (landing radar),” Burkey explained. “I presume that it’s displaying a code to warn the astronaut to reposition it.”

And in the PINBALL_GAME_BUTTONS_AND_LIGHTS.s file, which is described as “the keyboard and display system program … exchanged between the AGC and the computer operator,” there’s a peculiar Shakespeare quote:

This is likely a reference to the AGC programming language itself, as one Reddit user . The language used predetermined “nouns” and “verbs” to execute operations. The verb pointed out 37, for example, means “Run program,” while the noun 33 means “Time to ignition.”

Now that the code is on GitHub, programmers can actually suggest changes and file issues. And, of course, they have

[trigonometry-is-my-bitch]

You can find the official Apollo 11 AGC source code on GitHub here

[source]