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My darndest attempt at explaining renormalization to a general audience. Please let me know what you think!

Quantum field theory is the theoretical framework of particle physics. Without it, we never could have worked out what an atom is made of, understood the forces that govern its content, or predicted the Higgs boson.

But when it was first being established in the first half of the 20th century, it came across an apparently fatal flaw. It was plagued with infinities. And infinities don’t belong in physics. Following the rules of quantum field theory, you could end up predicting an electron having an infinite electric charge. Gasp. It’s resolution lead to a revolutionised way of thinking that now underpins all of particle physics.

I may go into a little bit of maths, but don’t worry it’s all easy. Promise.

Infinitely Probable Events

Science is about making predictions, given initial conditions.

If our system is in state A at time 1, what is the probability of it being in state B at time 2?

In particle physics, we read “system” to mean the universe at its most bare-bones fundamental level. The question becomes the following:

At time 1, there exists a given set of particles, each with a particular momentum. What is the probability of a new set of particles, each with a new particular momentum, at time 2?

Quantum field theory is intended to be the machinery one can use to answer such a question. A nice simple challenge we can give it is this: given two electrons, hurtling towards each other at momenta p1 and p2, what is the likelihood of them ricocheting off each other and coming out of the collision with the new momenta q1 and q2?

FullSizeRender (2).jpgFig 1: Feynmann Diagram of two electrons exchanging a photon

The easiest (most likely) way for this to happen is shown in fig. 1, this thing is called a Feynman diagram. Electron 1 emits a photon (the particle responsible for the electromagnetic force). This flies over to electron 2 with momentum k, and gets absorbed. We can use the principle of conservation of momentum to uniquely determine k. The principle states that total momentum must be the same at the beginning and end of all events. Applying this to electron 1 emitting the photon, initial momentum = final momentum implies p1 = q1 + k. Then, rearranging gets us to k = p1q1. Since we’re given p1 and q1, we can use this equation to work out exactly what k will be.

Quantum field theory can be used to work out the probability of each individual part of the Feynman diagram. Electron 1 emitting the photon, the photon travelling from electron 1 to 2 with momentum k, and electron 2 absorbing it. This produces the so-called Feynman rules, a translation between parts of the diagram and probabilities of each part taking place. The probability of the entire event can be found by just multiplying probabilities of each component event. The probability of the photon emission, multiplied by the probability of it’s travel to electron 2, multiplied by the probability of it’s absorption, gets you the overall probability. Nobel prizes all ’round.

But wait. This is not the only way you can put in two electrons of momenta p1 and p2 and get two electrons out with momenta q1 and q2. There are a number of different ways the two electrons could interact, in order to produce the same outcome. For example, this:

FullSizeRender (3).jpgFig.2: Feynmann Diagram of two electrons exchanging a photon which splits into an electron positron pair on the way.

The photon splits into two new particles, which recombine to return the photon. Similarly to before we know exactly what the photon momentum k is, using k = p1 – q1, and the values for p1 and q which we are given in the problem. But now, there is no guiding principle to decide what the momenta of the electron and the positron in the middle will have. We know that k1 + k2= k from conservation of momentum, but this is one equation containing two unknowns. Compare it to how we worked out k in the first diagram, in which case there was only one unknown, so we could use all the other known values (p1 and q1) to get the unknown one (k). If we fix k2 by saying k2k – k1, we have one unfixed degree of freedom left, k1, which could take on any value. k1 could even have negative values, these represent the electron moving in the opposite direction to all the other particles.

k1 is not uniquely determined by the given initial and final momenta of the electrons. This becomes significant when working out the overall probability of fig.2 occurring.

To work out the overall probability, one needs to use the Feynmann rules to translate each part of the diagram into a probability, then combine them. The probability of electron 1 emitting photon, multiplied by the probability of photon moving to where it splits up, multiplied by the probability of photon splitting into the electron & positron, etc.  But this time, since the middle electron could have any momentum, one needs to add up the probability of that part for all values of k1. There is an infinite spectrum of possible k1 values so there are an infinite number of ways fig.2 could occur.

Let’s step back for a moment. In general, if there are lots of different events (call them E1, E2, ….) that could cause the same overall outcome O to occur, then the probability of Oprob(O), is

prob(O) = prob(E1) + prob(E2) + …

If there are an infinite number of ways O could occur, then it becomes and infinite sum of probabilities, and as long as each of the probabilities are not zero, then prob(O) becomes infinite.

This is what happens with our particles. Since there is an infinite number of momentum values the middle electron could have, there is an infinite number of probabilities that must be added up to get the probability of fig.2 occurring, so the probability of fig.2 is infinite.

What could that even mean? A probability should be a number between 0 (definitely won’t happen) and 1 (definitely will happen). Such predictions of infinite probabilities renders a theory useless, quantum field theory is doomed. The Higgs boson is a conspiracy invented by the Chinese.

Renormalization or How to ignore all your problems

This wasn’t the end of quantum field theory- since there is a way of resolving this problem. Kind of. The solution, or rather the family of solutions, are referred to as renormalization. It comes in many different manifestations, but it all boils down to something along the lines of the following. We pretend that k1, our unconstrained electron momentum, can only have a value below some maximum allowed size we’ll call Λ. Then, we don’t need to add up probabilities from situations where k1 goes arbitrarily high. We’re left with a finite number of possibilites, therefore a finite probability for the whole event. More generally, we can solve all problems like this by making Λ a universal maximum momentum for all particles involved in an interaction. Λ is called a momentum cutoff.

This solves the issue, we end up with sensible predictions for all processes. And as long as we make Λ suitably larger than the momentum of the initial and final electrons, the answer matches results of experiments to high precision. But I’ll understand if you feel a little unsatisfied by this. How come we can just ignore the possibility of electrons having momentum higher than Λ? To win you over, I’ll tell you a bit about what Λ physically means.

In quantum mechanics, an electron is both a particle and a wave. One of the first realisations in quantum mechanics was that the wavelength of an electron wave is inversely proportional to it’s momentum; wavelength = 1/momentum. A high momentum corresponds to a small wavelength, and vice versa. Ignoring particles with momentum higher than Λ, is the same as ignoring waves with wavelength smaller than 1/Λ. Since all particles can also be seen as waves, the universe is made completely  out of waves. If you ignore all waves of wavelength smaller than 1/Λ, you’re effectively ignoring “all physics” at lengths smaller than 1/Λ.

Renormalization is a “coarse graining” or “pixelation” of our description of space, the calculation has swept details smaller than 1/Λ under the rug.

Making exceptions like this have in fact been a feature of all models of nature throughout history. When you’re in physics class doing experiments with pendulums, you know that the gravitational pull of Jupiter isn’t going to effect the outcome of your experiment, so broadly speaking, long-range interactions aren’t relevant. You also know that the exact nature of the bonds between atoms in the weight of your pendulum isn’t worth thinking about, so short-range interactions also aren’t relevant. The swing of the pendulum can be modelled accurately by considering only physics at the same scale as it, stuff happening on the much larger and much smaller scale can be ignored. In essence you are also using renormalization.

Renormalization is just a mathematically explicit formulation of this principle.

The Gradual Probing of Scales

Renormalization teaches us how to think about the discovery of new laws of physics.

The fact that experiments on the pendulum aren’t effected by small scales means we cannot use the pendulum to test small scale theories like quantum mechanics. In order to find out what’s happening at small scales, you need to study small things.

Since particles became a thing, physicists have been building more and more powerful particle accelerators, which accelerate particles to high momenta and watch them interact. As momenta increase, the wavelength of the particles get smaller, and the results of the experiments are probing smaller and smaller length scales. Each time a bigger accelerator is required in order to accelerate particles to higher speeds, and each jump is a huge engineering challenge. This race to the small scales has culminated in the gargantuan 27km ring buried under Geneva called the Large Hadron Collider (LHC). This has achieved particle momenta high enough to probe distances of around 10 zeptometers (0.000000000000000000001 meters), the current world record.

Galileo didn’t know anything about quantum mechanics when he did his pioneering pendulum experiments. But it didn’t stop him from understanding those pendulums damn well. In the present day, we still don’t know how physics works at distances under 10 zeptometers, but we can still make calculations about electrons interacting.

From this point of view, it seems like we absolutely should impose a maximum momentum/minimum distance when working out the probabilities of Feynmann diagrams. We don’t know what’s going on at distances smaller than 1/Λ. We need to remain humble and always have in mind that any theory of nature we build is only right within its regime of validity. If we didn’t involve this momentum cutoff, we would be claiming that our theory still works at those smaller scales, which we don’t know to be true. Making such a mistake causes infinite probabilities, which suggests that there is indeed something lurking in those small scales that is beyond what we know now…

The road to the Planck scale

There are currently a bunch of theories about what is going on at the small untested length scales. We can make educated guesses about what scales these prospective new features of nature should become detectable at.

FullSizeRender (1).jpg

Fig. 3: Length scales

There has been a fashionable theory floating around for a while called supersymmetrywhich says, broadly, that matter and the forces between bits of matter are in a sense interchangeable. It’s some well sick theoretical physics that I won’t go into here. The effects of this theory is believed to become visible at scales only slightly smaller than the ones we’ve already tested. It may even be detected at the LHC!

There’s a family of theories pertaining to even smaller sizes, called grand unified theories. These claim that if we can see processes at some way smaller scale, many of the fundemental forces will be revealed to be manifestations of a single unified force. The expected scale where this happens is about a billion billion times smaller than what we’ve currently tested, so will take a billion billion times more energy to probe. Don’t hold your breath on that one.

Finally, there’s reason to believe that there exists a smallest possible scale. This is known as the Planck length. If any object is smaller than the planck length, it would collapse into a quantum black hole, then immediately evaporate, removing any evidence of its existence. This is the scale where the quantum nature of gravity becomes important, and if you want to test that, you’ll need a particle collider 100 billion billion times more powerful than the LHC.

If we want to learn about these mysterious smaller scales, we’re going to need some mighty big colliders. Perhaps impossibly big. Maybe we need some new innovation which makes the probing of scales easier. Maybe the challenge for the next generation of particle physicists will be a rethink of how we test particle physics all together.

More on drawing Feynmann Diagrams

More on renormalisation

Supersymmetry

Grand unified theories

Planck length

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My darndest attempt at explaining renormalization to a general audience. Please let me know what you think! submitted by /u/emc031 to /r/Physics
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Space Dashboard – A website that has multiple widgets that let you view the ISS live streams, various live data feeds about earth, the current locations of the planets, and more

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[How-To] Building the FreeBSD OS from scratch

User Siseneg shows us how to get the FreeBSD operating system set up from the bottom up and details each step along the way. See the link below for the full instructions. We’ll be building FreeBSD from scratch, which starts you off with the base system and a terminal. No flashy graphics, no desktop, no […]

The post [How-To] Building the FreeBSD OS from scratch appeared first on FreeBSDNews.com.

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Is That ‘Bump’ a New Particle?

What happens to quantum information inside black holes? Why is the universe so big and uniform? How did life begin on Earth? How does symmetry shape the laws of nature? Quanta Magazine’s In Theory video series (also available on YouTube), featuring David Kaplan, a theoretical particle physicist at Johns Hopkins University and the producer of the award-winning documentary Particle Fever, offers a simple, visual introduction to some of the biggest mysteries in physics, biology and mathematics.

Learn more about particle physics.

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Δείτε 10 απίστευτα sites που ίσως δεν ξέρατε καν ότι υπάρχουν!!

Δείτε 10 απίστευτα sites που ίσως δεν ξέρατε καν ότι υπάρχουν!! 10-amazing-web-sites10-amazing-web-sites sites

Όλοι μας surfάρουμε καθημερινά και αδιαλείπτως αλλά η πλειοψηφία επισκέπτεται  websites λίγο πολύ ευρέως γνωστά και δημοφιλή σε όλους μας. Μερικές φορές τυχαίνει να πέφτουμε πάνω σ’ ένα website που μας δίνει καλές πληροφορίες. Συχνά, οι περισσότεροι από εμάς κάνουμε bookmark αυτά τα websites και τα μοιραζόμαστε με τους φίλους και γνωστούς στα social media.

Σήμερα θα σας παρουσιάσουμε μερικά άκρως ενδιαφέροντα Websites που ίσως να μην ξέρατε καν την υπάρξή τους.

Το πρώτο EthiΗak live Contest είναι ΓΕΓΟΝΟΣ!!!!! Infocom 2016 – EthiHak Contest – Δήλωσε συμμετοχή ΣΗΜΕΡΑ!

Πάμε να τα δούμε;

 

1. Hackertyper

HACKERTYPE sites

Το Hackertyper είναι ένα φανταστικό website!  Μπορείτε να σπάσετε πλάκα με τους φίλους σας δείχνοντάς τους πόσο γρήγορα προγραμματίζετε! � Απλά ανοίξτε το website και πατήστε οποιοδήποτε κουμπί! Τα υπόλοιπα αφήστε τα πάνω του….

 

2. 10Minutemail.com

10minutemail sites

Το 10Minutemail είναι μια υπηρεσία που σας επιτρέπει να δημιουργήσετε μια διεύθυνση ηλεκτρονικού ταχυδρομείου μόνο για 10 λεπτά. Μπορείτε να χρησιμοποιήσετε αυτή την ιστοσελίδα για να αποκτήσετε ένα email id για εγγραφή ή είσοδο με fake id σε οποιαδήποτε ιστοσελίδα.

Η ομορφιά αυτού του website  είναι ότι όλα τα e-mail σας θα αυτοκαταστραφούν σε 10 λεπτά. Ένα επιπλέον πλεονέκτημα? Δεν χρειάζεται να εγγραφείτε, αφού απλά ανοίγοντας την ιστοσελίδα έχετε ήδη ένα προσωρινό email address και μόλις 10 λεπτά να ολοκληρώσετε αυτό που θέλετε…. τικ τακ τικ τακ…. ο χρόνος περνά! �

 

3. Fake Name Generator

fakenamegenerator sites

Εάν είστε απλά ντροπαλός, ή λάτρης της ιδιωτικότητας, και δεν θέλετε με τίποτα να μοιραστείτε το πραγματικό σας όνομα, τότε αυτό το website είναι η σωτηρία σας! Θα σας βοηθήσει να δημιουργήσετε ένα fake full ID με κάθε λεπτομέρεια!

 

4. Down for Everyone or Just Me

downforeveryoneorjustme sites

Επισκεφθείτε αυτό το website αν αναρωτιέστε αν κάποια ιστοσελίδα είναι down ή απλά δεν ανοίγει στο computer σας.

Δείτε ακόμα: Οι ξένες γλώσσες του μέλλοντος θα είναι… οι γλώσσες προγραμματισμού!

 

5. Date to Date Calculator

timeanddate sites

Υπολογίστε την διαφορά μεταξύ δύο ημερομηνιών… κάτι που προσπαθήσαμε όλοι μας τουλάχιστον μια φορά να το κάνουμε χειροκίνητα… όχι;

6. Take Screenshot of Any Webpage

web-capture sites

Μπορείτε να πάρετε ένα full HD screenshot οποιουδήποτε webpage και να μετατρέψετε το screenshot σε JPG/ JPEG, PNG και PDF format.

7. Use Google without Country Restriction

Όταν πληκτρολογείτε Google.com , η Google σας ανακατευθύνει στο country domain όπως πχ google.gr ή google.co.uk. Eάν θέλετε να χρησιμοποιήσετε το Google χωρίς country restrictions, απλά πληκτρολογήστε google.com/ncr . Στην συνέχεια αναζητήστε ότι θέλετε χωρίς περιορισμούς ανά χώρα.

Διαβάστε: Mazar BOT|Android Malware Rootάρει τη συσκευή σας& διαγράφει τα πάντα!

 

8. Virustotal

virustotal sites

Εάν ο φίλος σας σας στείλει ένα ύποπτο file ή κατεβάσατε μόλις ένα αρχείο από το internet, μπορείτε να τσεκάρετε εάν περιλαμβάνει κάποιον ιό σε αυτό εδώ το site. Το Virustotal  είναι ένα free online virus scanner.

 

9. Live Hacking Attack Map

map.norsecorp sites

Αυτό το website σας επιτρέπει να δείτε live τα DDoS attacks που λαμβάνουν χώρα ανά τον κόσμο. Μπορείτε να δείτε τα IPs, τα  attackers address, τα  attack types, και πολλές άλλες πληροφορίες! Enjoy!

 

0. You can Destroy My Website

netdisaster sites

Αυτό το πανέμορφο μικρούλι Javascript κόλπο σας επιτρέπει να καταστρέψετε ιστοσελίδες. Όταν ανοίξετε τον σύνδεσμο, ένα βέλος θα ανοίξει στην οθόνη. Τώρα Πατήστε space και απολαύστε το shooting. Ελέγξτε το βέλος με τα arrows keys. �

Διαβάστε περισσότερα: Ποιες είναι οι διασημότερες Programming Languages @Hackathons ?

Βρήκατε το άρθρο ενδιαφέρον? Υπάρχει κάποιο site που θα θέλατε να προσθέσουμε στην λίστα μας? Περιμένουμε τα σχόλιά σας παρακάτω! :) sites

 

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HacKat

Daemon, they call me. I perform action without user interaction. Monitoring, logging, notifications. Primal urges, repressed memories, unconscious habits. I'm always there, always active.

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13 κλασσικά video games που μπορείς να παίξεις τώρα στον browser σου!


Μήπως έχεις πεθυμήσει κάποια video games που έπαιζες πριν πολλά χρόνια; Το Archive.org είναι μια απίστευτη πηγή που έχει αποθηκεύσει κομμάτια των ΜΜΕ που έχουν χαθεί με τον χρόνο!

video games video games

Ουσιαστικά είναι μια βιβλιοθήκη του παλιού Διαδικτύου! Η ιστοσελίδα έχει κάνει μεγάλες συλλογές από έργα τέχνης, ταινίες και μουσική διαθέσιμα για οποιονδήποτε έχει πρόσβαση σε έναν υπολογιστή και στο Διαδίκτυο.

Πέρυσι πάνω από 2.400 old-school video games προστέθηκαν στη συλλογή και όλα παίζονται μέσω του web browser σου! Παρακάτω θα δείτε μερικά από τα πιο αγαπημένα παιχνίδια από μια εποχή πριν το Διαδίκτυο! Πατήστε πάνω στον κάθε τίτλο για να παίξετε κάθε παιχνίδι.

«The Oregon Trail»

the-oregon-trail video games

Ένα από τα πρώτα video games που παιζόταν ευρέως, το The Oregon Trail εισήγαγε σε πολλά παιδιά του 20ου αιώνα το ταξίδι στην έρημο και τη λέξη δυσεντερία!   655286-250 video games

«Prince of Persia»

video games video games

Καμιά σχέση με το Prince of Persia που κυκλοφορεί τώρα, το αρχικό παιχνίδι αντιπροσωπεύει τα παιχνίδια πλατφόρμας σε όλο τους το μεγαλείο!

«Sim City»

sim-city video games

Ένα από τα πρώτα παιχνίδια που ουσιαστικά σου επέτρεπε να παίξεις το Θεό, στο Sim City είχες τα ηνία μιας πόλης. Την έχτιζες, την κατέστρεφε, η επιλογή ήταν δική σου!

«Super Street Fighter II»

video games video games

Η τέταρτη έκδοση της σειράς Street Fighter II (ναι υπήρχαν αρκετές εκδόσεις για να δημιουργήσουν μια σειρά), το Super Street Fighter II ήταν γεμάτο με επιπλέον περιεχόμενο όπως πχ περισσότερους μαχητές από το πρωτότυπο.

Τα βίαια video games σκοτώνουν (;)

«Ms. Pac-Man»

ms-pac-man video games

Πιο τεχνολογικά προηγμένο από το αρχικό παιχνίδι, το Ms Pac-man έχει ghost paths που απαιτούν πολύ καλά αντανακλαστικά!

«Tetris»

tetris video games

Το Tetris ήταν ότι καλύτερο υπήρχε για το αρχικό Game Boy αλλά οι παίχτες είχαν περάσει αρκετά χρόνια σπάζοντας τουβλάκια και από το σπίτι τους!

«Maniac Mansion»

maniac-mansion video games

Ένα από τα κλασικά παιχνίδια περιπέτειας της Lucasfilm, το αριστούργημα του Ron Gilbert μπορεί να απέχει από τα σημερινά standards αλλά πραγματικά αξίζεις να τα παίξεις!

Δημιουργήστε τα δικά σας «Mario Maker» wallpapers μέσω Nintendo app!

«Donkey Kong»

donkey-kong video games

Σίγουρα ένα από τα πιο δημοφιλή video games! Ήταν και το πρώτο παιχνίδι που εμφανίστηκε ο Mario κι ένα κλασσικό παιχνίδι Arcade γι’ αυτό δεν πρέπει να απορεί κανείς γιατί τελικά μεταφέρθηκε σε DOS το 1983.

Ορισμένα παιχνίδια Android κρύβουν κακόβουλο κώδικα σε εικόνες

«Teenage Mutant Ninja Turtles: Manhattan Missions»

teenage-mutant-ninja-turtles-manhattan-missions video games

Τα αγαπημένα μας Χελωνονιντζάκια είχαν κάνει άλμα από βιβλίο κόμικ σε κινούμενα σχέδια και από εκεί σε video games. To «Teenage Mutant Ninja Turtles: Manhattan Missions» ήταν το πρώτο των video games τους αλλά είχε τη κακή φήμη ότι το άλμα ήταν αδύνατο σε αυτή την έκδοση!

«Bubble Bobble»

bubble-bobble video games

Πυροβολήστε εχθρούς, συλλέξτε φρούτα και σπάστε φυσαλίδες σε αυτό το κλασσικό παιχνίδι puzzle.

«Gauntlet»

gauntlet video games

Πιάστε ένα τσεκούρι, ένα σπαθί ή κάποιο άλλο όπλο και διαλύστε τους εχθρούς σας στο Gauntlet, ένα από τα πρώτα RPG παιχνίδια που κυκλοφόρησαν.

«Mega Man»

mega-man video games

Όπως και το Donkey Kong, πολλοί συνδέουν το Mega Man με τα αρχικά συστήματα της Nintendo αλλά σχεδόν όλα έχουν βγει αρχικά για Windows ή σε αυτή την περίπτωση για DOS.

«Ultima VI: The False Prophet»

ultima-vi-the-false-prophet video games

Αν είσαι λάτρης των RPGs αυτό το παιχνίδι θα σου χαρίσει ώρες διασκέδασης με την προϋπόθεση βέβαια να αφήσετε ανοιχτό τον browser σας για να το ολοκληρώσετε!

60 παιχνίδια με malware κυκλοφορούν στο PlayStore. Δείτε τα! [Εικόνες]

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Crystal

If at first you dont succeed...Call it version 1.0!

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Spun out of proportion: The Proton Spin Crisis

In the late 1980s, as particle colliders probed deeper into the building blocks of nature, there were hints of a strange and paradoxical behaviour in the heart of atoms. Fundamental particles have a curious quantum mechanical property known as “spin”, which the electron carries in magnitude ½. While the description of electron’s spin is fairly simple, protons are made up of many particles whose “spins” can add together in complicated ways and yet remarkably, its total spin turns out to be the same as the electron: ½. This led to one of the great mysteries of modern physics: how do all the particles inside the proton conspire together to give it a ½ spin? And what might this mean for our understanding of hadrons, the particles that make up most of the visible universe?

[This article is largely intended for a lay-audience and contains an introduction to foundational ideas such as spin. If you’ve had a basic introduction to Quantum Mechanics before, you may wish to skip to section marked —— ]

We’ve known about the proton’s existence for nearly a hundred years, so you’d be forgiven for thinking that we knew all there was to know about it. For many of us, our last exposure to the word “proton” was in high school chemistry, where they were described as a little sphere of positive charge that clumps with neutrons to make atomic nuclei, around which negatively charged electrons orbit to create all the atoms, which make up Life, the Universe and Everything1.

2000px-Proton.svg

The simple, three-quark model of a proton (each coloured circle is a type of “quark”). 

Like many ideas in science, this is a simplified model that serves as a good introduction to a topic, but skips over the gory details and the bizarre, underlying reality of nature. In this article, we’ll focus on one particular aspect, the quantum mechanical “spin” of the proton. The quest to measure its origin has sparked discovery, controversy and speculation that has lasted 30 years, the answer to which is currently being sought at a unique particle collider in New York.

The first thing to note is that protons, unlike electrons2, are composite particles, made up from lots of other particles. The usual description is that the proton is made up of three smaller “quarks” which, as far as we know, can’t be broken down any further. This picture works remarkably well at low energies but it turns out at very high energies, like those being reached at the at the LHC, this description turns out to be inadequate. At that point, we have to get into the nitty-gritty and consider things like quark-antiquark pairs that live inside the proton interacting dynamically with other quarks without changing its the overall charge. Furthermore, there are particles called gluons that are exchanged between quarks, making them “stick” together in the proton and playing a crucial role in providing an accurate description for particle physics experiments.

So on closer inspection, our little sphere of positive charge turns out to be a buzzing hive of activity, with quarks and gluons all shuffling about, conspiring to create what we call the proton. It is by inferring the nature of these particles within the proton that a successful model of the strong nuclear force, known as Quantum Chromodynamics (QCD), was developed. The gluons were predicted and verfied to be the carriers of this force between quarks. More on them later.

Proton structure

A more detailed model of the proton. The golden chains between the quarks (the coloured spheres) are representations of gluons, transferred between them. Quark anti-quark pairs are also visible with arrows representing spins.

That’s the proton, but what exactly is spin? It’s often compared to angular momentum, like the objects in our everyday experience might have. Everyone who’s ever messed around on an office chair knows that once you get spun around in one, it often takes you a bit of effort to stop because the angular momentum you’ve built up keeps you going. If you did this a lot, you might have noticed that if you started spinning with your legs/arms outstretched and brought them inwards while you were spinning, you’d begin to spin faster! This is because angular momentum (L) is proportional to the radial (r) distribution of matter (i.e. how far out things are from the axis of rotation) multiplied by the speed of rotation3 (v). To put it mathematically L = m × v × r where m is just your constant mass. Since L is constant, as you decrease r (by bringing your arms/legs inwards), v (the speed at which you’re spinning) increases to compensate. All fairly simple stuff.

So clearly, for something to have angular momentum it needs to be distributed radially. Surely r has to be greater than 0 for L to be greater than 0. This is true, but it turns out that’s not all there is to the story. A full description of angular momentum at the quantum (atomic) level is given by something we denote as “J”. I’ll skip the details, but it turns out J = L + S, where L is orbital angular momentum, in a fashion similar to what we’ve discussed, and S? S is a slightly different beast.

Both L and S can only take on discrete values at the microscopic level, that is, they have quantised values. But whereas a point-like particle cannot have L > 0 in its rest frame (since if it isn’t moving around and v = 0, then L = 0), S will have a non-zero value even when the particle isn’t moving. S is what we call Spin. For the electron and quarks, it takes on the value of ½ in natural units.

Spin has a lot of very strange properties. You can think of it like a little arrow pointing in a direction in space but it’s not something we can truly visualise. One is tempted to think of the electron like the Earth, a sphere spinning about some kind of axis, but the electron is not a sphere, it’s a point-like particle with no “structure” in space. While an electron can have many different values of L depending on its energy (and atomic structure depends on these values), it only has one intrinsic magnitude of spin: ½. However, since spin can be thought of as an arrow, we have some flexibility. Loosely speaking, spin can point in many different directions but we’ll consider it as pointing “up” (+½) or “down” (- ½). If we try to measure it along a particular axis, we’re bound to find it in one of these states relative to our direction of measurement.

Spin250

Focus on one of the red faces. When the cube rotates every 360 degrees, the red ribbon appears to go above and below the cube alternatively! Because the cube is coupled to its environment, it takes 720 degrees to return it to it’s original orientation.


One of the peculiar things about spin-½ is that it causes the wave-function of the electron to exhibit some mind bending properties. For example, you’d think rotating any object by 360 degrees would put it back into exactly the same state as it was, but it turns out that doesn’t hold true for electrons. For electrons, rotating them by 360 degrees introduces a negative sign into their wave-function! You have to spin it another 360 degrees to get it back into the same state! There are ways to visualise systems with similar behaviour (see right) but that’s just a sort of “metaphor” for what really happens to the electron. This links into the famous conclusion of Pauli’s that no two identical particles with spin-½ (or any other half-integer spin) can share the same quantum mechanical state.

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Spin is an important property of matter that only really manifests on the quantum scale, and while we can’t visualise it, it ends up being important for the structure of atoms and how all solid objects obtain the properties they do. The other important property it has is that the spin of a free particle likes to align with magnetic fields4 (and the bigger the spin, the greater the magnetic coupling to the field). By using this property, it was discovered that the proton also had angular momentum J = ½. Since the proton is a stable particle, it was modelled to be in a low energy state with L = 0 and hence J = S = ½ (that is to say, the orbital angular momentum is assumed to be zero and hence we may simply call J, the “spin”). The fact the proton has spin and that spin aligns with magnetic fields, is a crucial element to what makes MRI machines work.

Once we got a firm handle on quarks in the late 1960s, the spin structure of the proton was thought to be fairly simple. The proton has spin-½. Quarks, from scattering experiments and symmetry considerations, were also inferred to have spin-½. Therefore, if the three quarks that make up the proton were in an “up-down-up” configuration, the spin of the proton naturally comes out as ½ – ½ + ½ = ½. Not only does this add up to the measured spin, but it also gives a pleasant symmetry to the quantum description of the proton, consistent with the Pauli exclusion principle (it doesn’t matter which of the three quarks is the “down” quark). But hang on, didn’t I say that the three-quarks story was incomplete? At high energies, there should be a lot more quark-antiquark pairs (sea quarks) involved, messing everything up! Even so, theorists predicted that these quark-antiquark pairs would tend not to be polarised, that is, have a preferred direction, and hence would not contribute to the total spin of the proton.

If you can get the entirety of the proton spinning in a particular direction (i.e. polarising it), it turns out the scattering of an electron against its constituent quarks should be sensitive to their spin! Thus, by scattering electrons at high energy, one could check the predictions of theorists about how the quarks’ spin contributes to the proton.

In a series of perfectly conducted experiments, the theory was found to be absolutely spot on with no discrepancy whatsoever. Several Nobel prizes were handed out and the entire incident was considered resolved, now just a footnote in history. OK, not really.

In truth, the total opposite happened. Although the experiments had a reasonable amount of uncertainty due to the inherent difficulty of polarising protons, a landmark paper by the European Muon Collaboration found results consistent with the quarks contributing absolutely no overall spin to the proton whatsoever! The measurements could be interpreted with the overall spin from the quarks being zero5. This was a complete shock to most physicists who were expecting verification from what was supposed to be a fairly straightforward measurement. Credit where it is due, there were theorists who had predicted that the assumption about orbital angular momentum (L = 0) had been rather ad-hoc and that L > 0 could account for some of the missing spin. Scarcely anyone would have expected, however, that the quarks would carry so little of the spin. Although the nuclear strong force, which governs how quarks and gluons combine to form the proton, has been tested to remarkable accuracy, the nature of its self-interaction makes it incredibly difficult to draw predictions from.

The feynman diagram for Deep Inelastic Scattering (electron line at the top, proton on the bottom). This type of scattering is sensitive to quark spin.

The Feynman diagram for Deep Inelastic Scattering (electron line at the top, proton on the bottom, with a photon exchanged between them). This type of scattering is sensitive to quark spin.

Future experiments (led by father and son rivals, Vernon and Emlyn Hughes6 of CERN and SLAC respectively) managed to bring this to a marginally less shocking proposal. The greater accuracy of the measurements from these collaborations had found that the total spin contributions from the quarks was actually closer to ~30%. An important discovery was that the sea quarks, thought not to be important, were actually found to have measurable polarisation. Although it cleared up some of the discrepancy, it still left 60-70% of spin unaccounted for. Today, following much more experimental activity in Deep Inelastic Scattering and precision low-energy elastic scattering, the situation has not changed in terms of the raw numbers. The best estimates still peg the quarks’ spin as constituting only about 30% of the total.

Remarkably, there are theoretical proposals to resolve the problem that were hinted at long before experiments were even conducted. As mentioned previously, although currently impossible to test experimentally, the quarks may carry orbital angular momentum (L) that could compensate for some of the missing spin. Furthermore, we have failed to mention the contribution of gluons to the proton spin. Gluons are spin-1 particles, and were thought to arrange themselves such that their total contribution to the proton spin was nearly non-existent.

BNL AERIALS

The Brookhaven National Laboratory where RHIC is based (seen as the circle, top right).


The Relativistic Heavy Ion Collider (RHIC) in New York is currently the only spin-polarised proton collider in the world. This gives it a unique sensitivity to the spin structure of the proton. In 2014, an analysis of the data collected at RHIC indicated that the gluons (whose spin contribution can be inferred from polarised proton-proton collisions) could potentially account for up to 30 of the missing 70% of proton spin! About the same as the quarks. This would bring the “missing” amount down to about 40%, which could be accounted for by the unmeasurable orbital angular momentum of both quarks and gluons.

As 2016 kicks into gear, RHIC will be collecting data at a much faster rate than ever after a recent technical upgrade that should double it’s luminosity (loosely speaking, the rate at which proton collisions occur). With the increased statistics, we should be able to get an even greater handle on the exact origin of proton spin. 


The astute reader, provided they have not already wandered off, dizzy from all this talk of spinning protons, may be tempted to ask “Why on earth does it matter where the total spin comes from? Isn’t this just abstract accountancy?” This is a fair question and I think the answer is a good one. Protons, like all other hadrons (similar, composite particles made of quarks and gluons) are not very well understood at all. A peculiar feature of QCD called confinement binds individual quarks together so that they are never observed in isolation, only bound up in particles such as the proton. Understanding the spin structure of the proton can inform our theoretical models for understanding this phenomenon.

This has important implications, one being that 98% of the mass of all visible matter does not come from the Higgs Boson. It comes from the binding energy of protons! And the exact nature of confinement and precise properties of QCD have implications for the cosmology of the early universe. Finally, scattering experiments with protons have already revealed so much to fundamental physics, such as the comprehension of one of the fundamental forces of nature. As one of our most reliable probes of nature, currently in use at the LHC, understanding them better will almost certainly aid our attempts to unearth future discoveries.

Kind regards to Sebastian Bending (UCL) for several suggestions (all mistakes are unreservedly my own).

 

[1] …excluding dark matter and dark energy which constitute the dark ~95% of the universe.

[2] To the best of our knowledge.

[3] Strictly speaking the component of velocity perpendicular to the radial direction.

[4] Sometimes, spins in a medium like water like to align against magnetic fields, causing an opposite magnetic moment (known as diamagnetism). Since frogs are mostly water, this effect can and has been used to levitate frogs.

[5] A lot of the information here has been summarised from this excellent article by Robert Jaffe, whose collaboration with John Ellis on the Ellis-Jaffe rule led to many of the predictions discussed here.

[6] Emlyn was actually the spokesperson for SLAC, though he is listed as one of the primary authors on the SLAC papers regarding the spin structure of the proton.

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