Form Meets Function Polar Graphing Project: Fill & Download for Free

I think there will always be a lot of programming jobs that don’t involve a great deal of math. Those areas are growing, too. Things like user-interface, data management, communications, and more, will always have a core whose focus is on making life easier, more robust, etc.That said … do you want the interesting jobs? The ones that do something new, something that lasts?45 years ago, I was introduced to Macsyma (now Maxima -- GPL CAS based on DOE-MACSYMA), a computer algebra system. You can download and run it today — a 50 year old software package. I have it on my laptop. I have it on my phone. Maxima on Android - Android Apps on Google PlayMeanwhile, at the University of Utah, the ground was being broken on computer graphics.(That’s the real, original teapot, on display at the Computer History Museum that was used as the model for early rendering).Today, it’s not a major industry. It’s an entire cluster of industries. Movies. Games. 3D design and printing and CNC manufacturing. Data visualization. In her long career in animation and games, my wife has done everything from physical simulations for the Coca-Cola polar bears’ Christmas Tree, to the dynamic behavior of the cables in the pod racers in Star Wars EP1, to the Navier-Stokes equations for fluid simulation, for the film Pearl Harbor.At MIT and elsewhere, we were forging the beginnings of computer networks — and graph theory was critical there and a myriad other places — such as Google employment quizzes. Here’s the net when I started:It now looks like this:[CC-BY-NA 4.0 © 2014 by LyonLabs, LLC and Barrett Lyon. The Opte Project]So yeah, probably should learn some graph theory.Do you want a well-paying job? Hell, do you want to get rich? Among my cohort, some of the AI people with the best math backgrounds ended up being recruited by Wall Street for financial modelling. Probably retired on their private island by now… The Quants Are Taking Over Wall StreetOr are you more interested in making the world a better place? You could help stave off the next plague by helping find an answer to antibiotic resistance:Big Data harnessed to tackle antibiotic resistanceBig data is not just about lots of data. It’s about techniques to analyze it mathematically to discover insights through math and statistics that we can get no other way.Or maybe you want a field that has the potential to save lives every day, addressing a risk most of us face very day, even infants. You can teach cars to drive themselves. Computer vision. Sensor integration. Physics meets perception meets action. Math every step of the way.Do you like music? In 1973, looking at all the drawbacks of the synthesizers of the day, the ARPs and the Moogs — unstable analog circuits wired with patch cords, and conceived of a workstation that could take stored wave forms, multiply them together under computer control, with an actual user interface.So why didn’t I turn it into a product? Lots of reasons, but one that stands out for our purposes is, I didn’t yet know enough math to really understand the spectral implications of FM synthesis. The work had already been done (by computer, before it could be done in real time) at Stanford by John Chowning, who spent years developing and refining the capabilities — and in 1974 licensed it to Yamaha. Who, in 1983, released the DX-7 — the first commercially successful digital synthesizer, and one of the best selling of all time.I eagerly got on the waiting list to buy one. I still have it — but today I use an improved emulation of it.(Rock on!)You’ve heard of “Big O notation” and complexity theory? It’s not complicated math, but it is the one big area I find people who learn programming but don’t study computer science, end up missing.[math]\mathit{\mathbf{O}}(n^{2})[/math]Perhaps the most esoteric branch of mathematics out there is Category Theory. In part, it’s an extension of topology. In math, it describes how different transformations preserve mathematical structure.Perhaps not surprisingly by now — it has a myriad applications to software. Everything from analyzing and extending the semantic structure of data and its transformations, to doing the same to the software itself, to some of the basic precepts of Functional Programming, from functional composition to monads, and beyond.This combines nicely with graph theory, BTW.By all means — learn what math you can! It opens up a much broader array of interesting jobs. Don’t be intimidated by it. You can pick up what you need as you go, building on what you’ve learned before!But the better your start, the more opportunities lie ahead. Just don’t let “my math isn’t good enough” hold you back. I’ve worked with math my entire career — but I’m no mathematician. I struggle with it at times, relearning math I’ve forgotten. Yet it has become an integral part of how I view the world, and how I view programming.So it’s not all that important to programming and Computer Science — until it is. Graph theory, category theory,But what should you study? I offer two guides: Build on what you know, and focus on what you find interesting.You’ll never find enough time to learn enough math, so focus on the things that relate to the problems you want to work on. I hope this gives you a sense of the breadth of possibilities, but ultimately, math is a tool, and it is your career, your choice.Don’t learn math just because “you need math for CS”.Pick a direction — then seek out the math that supports it!—-[Edit: I just noticed the part of your question that got stuffed into a comment. OK, you’re starting late. My advice doesn’t change, but I can reassure you, I’ve seen a lot of people come into software from other fields. Your math mostly won’t be a barrier to learning CS. For university courses, you’ll want to be at least ready for Calculus; ideally you’ll have studied it, but it wasn’t offered at my high school. Regard it as a parallel track that opens up new territory!]

I've spent two months sitting on this question, turning it over in my head, thinking of all the factors and all the possibilities because the idea of such a world intrigues me so much. Also, because this is pertaining to my age group(I turn seventeen in only a few days), I am genuinely curious how we would end up.The visions that adults seem to have of our competence are disgustingly condescending. We know more and are capable of more than you give us credit for(as I am about to demonstrate a level of deep thinking many wouldn't have thought possible.)A lot of these answers are so very dismissive. It's like people think we're are determined to annihilate ourselves because it's "in our human nature" (I.e. Lord of the Flies.) Really??? Does that sound like the species that has managed to pull itself together for millions of years? People don't really seem to understand that humans are vastly adaptable creatures capable of cooperation to pursue a positive end. Whenever an adverse situation is encountered, people have a tendency to casually suggest that we're going to doom ourselves with self destructive stupidity. History shows they are dead wrong. So despite what cynics(always the adults) would say about how we are absolute toast, I have taken an attitude towards this concept that is defiantly optimistic in the end in comparison to their own grim and cursory "analysis".In total, my idea of what would ensue would be just l like others said:utter chaos.But only initially.Let's think about what these kids have to work with:pipelines (especially ones connected to water)HousesHardware storesMetalA whole bunch of conveniently available tools and no one to shout at them if they take it from their property.RoadsCrop seeds taken from places across the globe that they can growLots of newly available clothingAnd probably the most valuable resources to their success: libraries full of information regarding science and history and how to's, and an abundance of information connected to the thousands of years of research and development of knowledge from the adults before us.Brains that tuned to high risk, high reward experimentation as part of their stage of development. (And younger kids in stages where they are beginning to explore consequences of actions, and even younger kids in stages of short term fantasizing. All are great for trying to carve a new path, but I assume we are focusing more on the teen's importance to this scenario.)Things they do not have:ExperienceSkillKnowledgeOh look, those poor kids are toast aren't they? They wouldn't even know which way is up if it wasn't pointed out to them! They don't have the knowledge and experience to accomplish anything!Wait, they have up to grade 12 education?And everything left behind that had been created before them?And the knowledge imparted to them from their parents and older peers?And a broad understanding of how the world they lived in was created and maintained by the people that used to inhabit the world with them?And not everyone lives in big cities, tended to by the constant import of food from afar, and were already capable of sustaining themselves without adults?Oh, okay, for a second I thought you implied that it was going to be an absolute global meltdown! Whew! You had me worried there for a second!Really, they have everything they could need to rebuild a civilization at about the same level as it was before, given a bit of time. But that's a long ways away.But perhaps more realistically, that's not likely to happen. It would take a massive collective, happy, socialist, "we're all in this together" sort of attitude from everyone on earth to participate and a stunningly well organized global project to achieve it in a narrow frame of time it would take to meet anyone's best hopes; a feat that is laughably impossible to muster, especially considering the technical limitations. So, more realistically, you're likely to find small pockets of prospering organizations that grow to take over the regions that failed to bloom.But let's ignore what happens later on for right now. Let's look at what happens right at the start. What will that mean for them?The major hurdles they will have to face:The power going out: indoor climate control, water, and food refrigeration all go out the window.No one left with the technical skills to grow crops(poorer places where kids participate in growing crops will most certainly be better off in this way. Developed countries are most vulnerable.)The initial time frame after the adults dissapear will be a pessimist's dream world. Everything will go to hell. And it will take decades to get anything back to the way it was before. But I'm sorry, you pessimists that answered before me, you seem to misunderstand just how stubbornly adaptable humans are, especially teenagers and kids. With the nearly adult-like cognitive abilities that can adapt to the change better than a full grown adult can, they will certainly find a way to fight the elements and stay alive.Hour 1 after The DisappearanceWhat happens at this point may differ depending on the time of day where the kids are. If it's midday, the reaction will be immediate. Most kids will still be in close proximity to each other in schools or in activities after school. So they will be able to unite and organize more quickly. The slackers at school will certainly take this as an excuse to leave immediately(though it is most definitely not in their best interests.)If it is in the middle of the night, most aren't going to notice until morning because they are likely sleeping. But being isolated and fragmented, each in their own homes, there will probably be more chaos and confusion initially than there may be for those who are in daylight.Cars will lose their steering and control and will crash and many will burn. The babies that may have been in the backseat will die. Every road will be a huge pile up accident. In places where it was rush hour traffic, roads will be completely impassible. This will by far be the biggest road-block (pun very much intended) to any adventurous kids being able to travel and do anything.Kids at home will likely wander outside into the desolation and panic when they can't find any adults. Maybe they'll find a few other kids. Fights may break out, or they might immediately try to cooperate because they recognize the adversity of their current situation. An infinite variety of conflict problem solving methods may take place. Some older kids will watch over the younger ones if they manage to find them.Twitter will be ablaze with news from all over the world with kids panicking, exchanging rumors and maybe some true information about what happened, if any solid facts can be deduced amidst the chaos. Twitter servers will crash because of the load of Internet traffic, and the other platforms of communication may crash as well when they turn to other platforms to communicate. Many people will realize that it will some be time to say goodbye to the friends they knew only through the Internet, possibly forever.Nightfall after day 1 of the disappearanceThe emotions of the day are starting to settle in. Whatever networks of organization that some kids may have created has little sense of direction, but ideas, questions, and plans for how to proceed begin to circulate. As kids settle in to the idea of this new world, they may start whispering to each other wild rumors and mythologies about the cause of this occurrence. The wild fantasies the younger kids generate and circulate may become the foundation of what may develop into doomsday cults.After week 1 of the disappearance.Babies left behind by mothers alone without supervision will starve and die. Those without older siblings are even more likely to die. Those that do have older siblings are likely to be rescued by them as soon as possible, possibly including all the other abandoned babies they may find at the daycare.The older kids may have the sense to start looking for each other and organize as fast as possible. There will be tens of thousands—no—millions of micro states, more like tribes, scattered through every town and city in the world. In slums, or other troubled places, they will likely fight for territorial dominance in some areas where the environment they were in before the disappearance had some conflict. In more developed places, they may not conflict so much simply because they have more access to goods. But that may change sooner or later as food runs dry.Whichever places where there are a few kids who haven't formed any coalition—in daycares not found by older kids, sparsely populated areas, etc— will remain in fear and darkness. Eventually, if these kids don't find a way to live on their own or other kids to tag along with, they will starve and perish.Pets neglected and trapped in their humans' homes will die off slowly; their increasingly desperate barks, howls, meows, chirps, and inaudible squeaks will fade away, and many neighborhoods will be faintly permeated by the smell of death.Depending on the source of power of some places, this will be the time that the power goes out, and so does the Internet for those places. If they are powered by wind, the power would have gone out almost the same day that the maintenance workers disappeared(wind generators require a surprising amount of human work on them, despite how simple the machinery seems in concept.) Coal fired power plants may be next, as the trains that supplied coal stop running, and the people that did all the work to keep it going are gone. I'm not certain of how long it may take hydro-electric power to go out, but it may take the next month to go out. Nuclear power plants will run for a while without human maintenance, but not forever. Micheal grant's books imagine that the nuclear power plants will stay active potentially forever until they were forcibly shut down, but I think this isnt possible, because some monitoring is always needed on hand. M not sure how long it will last without human surveillance. Perhaps someone who works at a nuclear power plant could comment, if you could find them to request their opinion. But without their input, I would say that it would last around three weeks before something happened to destabilize the reaction and cause it to shut itself down. Internet cable junctions will cease to function when the local power source will go out, so even if one place still has power they may not been able to access internet in the unpowered place. Slowly, the world will find itself increasingly powerless (pun intended) and isolated from the globe.Livestock, abandoned by their ranchers, starve and die. A lucky few may escape their enclosures, but they live in environment that is nothing close to their native habitat from where they were originally domesticated, and some are so completely domesticated, like cows, that they are completely incapable of living without human assistance. They will certainly die. Horses will live, but chickens, pigs, and other livestock aren't so certain.Month 1The food in refrigerators has run dry. There may have been a few cunning leaders wise enough to anticipate this, and have the knowledge and resources to be able to get people to go out and grow their own food. However, in places where they didn't have those brave leaders, this will be the start of a major decline.But besides the growing problem of food shortage, another problem will certainly be a big strain on young people's lives: medical supply shortage. The kids that were kept alive by insulin shots and other refrigerated medication are going to find themselves without it. Lots of kids are going to die at this point if they hadn't before due to lack of ability to adapt or strategize. Later, the pills that keep some alive, like hormones, psychotic drugs, and biological stimulants and suppressants will run out, and there will be another, slower wave of tragic death as some are worn out by the elements of nature and simply cannot compete, or will be driven mad without the drugs that kept them sane and will commit suicide.Up to year 1This is going to be a deciding time for the success of many people. In less than a year after their disappearance, a cold season or a hot season will have swept over the land. Places in the polar latitudes or in the tropics are all likely to have kids die of heat stroke or hypothermia. They didn't have the fur lines coats and other ways of life of the people in centuries before. They've lived in heated and air conditioned houses their whole life, and the way of life that their parents lived has adjusted to that comfortable way of life, so they do not have any knowledge of how to live without hearing and cooling systems. They will die exposed to the elements if they weren't resourceful enough to evade it. There will be a clever few who anticipated and planned ahead for this catastrophe, but they'll find themselves living with very few people left around them, especially in the polar regions, or any place that has cold weather.By now, people are beginning to experience the last few things before they've seen about half of what they need to experience to be able to successfully continue their existence. They've figured out the general way of life that they need to live, and will most certainly hone their techniques over the years, but overall this is about where the population stop declining so sharply. I estimate about 1/2 of the population left on my most generous estimates, and around 1/8th of the original population in my most pessimistic estimates.General forward outlookFrom here forward, they will begin the steady climb higher from the low place they were. It's a long journey, but I expect that they'll figure out how to live.After knowing and experiencing the environmental harm of their old ways of sustaining themselves, they will certainly give up on fossil fuels for power, and will likely use other methods instead, like wind energy, and abandon the cars that they had used before, becoming a greener world than before. Also, they will give up on fertilizers. Partly because they no longer have access to the sources of these commodities, and when they do finally get connections to the sources they will have found more local ways of fueling their economy.Using the information learned in the libraries, they will begin pursuing jobs that will improve their lives. They'll start handling factory machinery, make businesses that operate in full capacity using the equipment left behind, and build new things using the convenient standardized metric system that was invented before them.Lots of land will be reclaimed by the wild. Fences will deteriorate and cease to be functional, so wild animals will skyrocket in population, farmland is turned back into forest and grassland, burrowing mammals and grazers will take over what used to be people's yards, and the once carefully maintained lawns that you put so much care into destroying "weeds" will succumb to the inevitable as native plants and animals take residence once again. Many houses will lie vacant, inviting animals to make it their own house to raise their tiny fluffy children without interference.The small tribes of kid humans that had been able to live out the first year will grow in size, annexing other tribes, and giving birth to more people and become small nations, and develop their own identities.The social demons of the past—patriarchy, racism, and economic tyranny— will suddenly take a sudden surge in prominence. But such things are just signs of a society in massive distress and change, which tend to minimize as societies begin to stabilize and stop changing.Over time, the new generation will stabilize and regenerate the world they once had. The population will stop declining within the first decade and will begin growing again as they start having children of their own. They will relearn the information left for them in equipment guides, formulas, blueprints, research papers, graphs, and what ever.The Internet will be back up and running, and the global connectedness and the era of readily available and exchangeable facts that the world never had before now will prevent us from going back in the Middle Ages.We'll be free from the environmental terrorism we had before. After learning from our past mistakes that would continue to hang over our heads in the coming the centuries, we as a collective society won't go back to such colossal destruction.However, one thing that is certain to hinder their ability to adapt in the early stages before they properly reestablish all the old securities they had before is something they might never see coming: epidemics.They have always lived in a time when hand sanitizer is always on the shelf, and food is pasteurized, vacuum -sealed, and dehydrated, and when immunization shots and doctors were always available to them. The security they had from sickness was so surrounding that it wasn't ever a second thought to them. They'll not have any idea that they are dangerously exposed to loads of pathogens. To say that they are at risk to new disease is a bit of an understatement. After the initial threat of starvation, unpredictable continental scale outbreaks of disease are the next biggest threat to their existence and may wipe out another huge portion of the population. And this isn't anything they can readily prevent.In my head, I like to imagine what may happen if the people that disappeared ever magically come back. It can be a good indication of how much things begin to change over time, as they might not be so obvious as things progress. They are definitely not going to have an easy time adjusting to the new balance of the world. The longer they are away, the more things will change.At first, maybe within the first two years, the change on the world won't be too big to handle. The kids wouldn't have settled into their new way of life quite yet, and they would gladly go back to having the new people to take care of them.But as time goes on into the years and decades following, they will find it harder and harder to reassert themselves into the new world. The places that they used to work would have been raided, and be in various states of disrepair; the food that used to be growing out in fields and stocked in their refrigerators is all gone, so the new comers would starve for quite a while; the way of life that the kids found would be too rigid to allow several billion more people to join in without lots of turmoil, so you could expect lots of fighting for dominance between the old world leaders and the new world's leaders. Heck, there might be global civil war as people rip themselves apart trying to hold either of their ways of life intact. Ultimately, having more knowledge of the world they live in than their predecessors had, the children will win out in one way or another, while the older generation is forced to live out the rest of their lives begrudgingly under their control.Mainly, this answer was written in regards to where I live, in the Great Plains of North America, with highly developed infrastructure, and high standard of living, with the assumption that the disappearance happened sometime during mid-springtime, giving kids enough time to organize and strategize before winter. All factors will differ depending on when and where you are: time of day, latitude, climate, economic standing of your region, etc. I neglected to think of what may happen for places and times like France if it happened at 19:00(7:00PM) in late autumn, or Pakistan at 4:00(AM) midsummer, or Russia 11:00(AM) midwinter, or any islands like Iceland or Hawaii. Given the climate and availability of some things and the daily patterns of behavior, all factors are subject to variation. And given how impossible it is to predict the future of the world, I found it quite hard to chronologically outline the upward progress of the globe.If anyone has any revision, feel free to comment, or edit it into the answer yourself if you can(not sure how that function works yet). This isn't a complete analysis, but maybe you could think of something I hadn't thought of, I would love to add it to this.

( Update : a little explanation for chirality added)Edit upgraded to technical level within layman limits[ Higgs mechanism for fermions & bosons covered ]Higgs boson was such an elusive particle to discover it got the nickname God Particle. A book bearing the name God Particle triggered popular belief that Higgs boson gives mass to matter thus making matter ‘real' to . But experts are exhorting it's but a popular misconception and now in the scientific circles it's also referred to as Godamn Particle.Because not all elementary particles seem to get their mass from Higgs field and more importantly all mass of matter in general does not come from mass of elementary particles alone but also from the energy they possess (E = mc^2).Mass of protonFor instance consider the mass of a composite particle like proton. A proton consists of three elementary particles called quarks which are held tightly by constant exchange of strong force carrying particles called gluons. Now according to recent, precise calculation only 1% of mass of proton comes from rest mass of constituent elementary particles and rest comes mostly from binding energy. Because E = mc^2. It seems it's a very complex calculation. Because the picture inside a proton is a very dynamic one with quarks and gluons constantly changing their colour and flavour. Not to speak of the constant bubbling of virtual particles. Computer simulation of the interior of a proton resembles boiling water.Colour is the charge associated with strong force-field just like the familiar electric charge is associated with electromagnetic field. And flavour refers to types of quarks.Image crdit WikipediaAll the massive particles in the Standard Model of particle physics, with a couple of exceptions, derive their mass from their interaction with Higgs field. In case Higgs field didn't exist, these particles would be massless and hence move with speed of light. This would result in electrons, quarks etc flying off like radiation and matter as we know would cease to exist. Universe would have only radiation and neutrinos, neutrinos being particles that do not seem to obtain their mass from Higgs field.Standard ModelIn the Standard Model of particle physics there are fermions and bosons. Fermions are matter particles, bosons are particles that carry force between matter particles.Fermions consist of six quarks and six leptons. Bosons consist of gluons (strong force), photons (electromagnetic force), W+,W- & Z° (weak force) and Higgs boson.Most important quarks are up and down quarks that make up protons and neutrons. Others are heavier unstable cousins.Important leptons are electrons and electron neutrinos. Others are heavier, unstable cousins.God particle's 'God' like featuresFrom this perspective perhaps there's a justification for calling Higgs boson ‘God Particle’ as Higgs field is responsible for materialising the universe from sheer radiation and neutrinos. Another Godlike feature of Higgs field is its Omnipresence. In every point of space-time, even in sheer empty vacuum space there exist an average positive value of Higgs field. It makes sheer empty space pregnant with its presence. In vacuum space average values of all other quantum fields like electromagnetic, strong, weak hover around zero, although quantum energy fluctuations remain resulting in particle-antiparticle pairs continuously popping in and out of existence. In fact the whole universe is filled with virtual particles bubbling in and out of existence.Also Higgs field has no source where as all other quantum fields require a source. Remove that electron, all the surrounding field vanishes. Higgs field is truly universal, self-existing field like no other.Higgs boson is the only force-particle with no spin (spin = 0). So it has no spin-vector pointing in any direction. That's why Higgs is called a scalar boson and Higgs field, a scalar field.All other bosons have a well defined integer spin of 1 and hence possess a spin with specific direction. That's why they're called vector bosons and the respective fields vector fields.However particle physicists seem to loath the name ‘God Particle’ notwithstanding unique peculiarities of Higgs field and boson. For them Higgs boson is just another elementary particle, the last piece of the jigsaw puzzle that was discovered after a long effort.Funny thing is Higgs boson itself doesn't get its mass from Higgs field.Quantum Fields and HiggsA word about quantum fields and particles. Particles are ‘excitations’ in a field. Like a guitar string being struck or stone thrown into a pond. Any disturbance in the field creates a small ripple of wave which appears as the virtual particle. The waves are the excitations of the field, the quantum fluctuations appearing as virtual particles.When particles that interact with Higgs field move, they create excitations in the Higgs field creating virtual particles called Higgs bosons. Higgs field cannot be detected directly but the discovery of Higgs boson confirms its existence.Higgs boson has spin zero. Higgs field is a scalar field as opposed to vector field. That means it has no preferred direction. It's rotationally invariant. (In whichever direction an interacting particle moves in High field the interaction will be the same). Higgs boson has no electromagnetic charge, or colour charge of strong force but has weak hypercharge. Weak hypercharge is charge associated with weak force-field. Actually Higgs field is an infinite ocean of weak hypercharge called weak hypercharge condensate whose sum total of hypercharge remains unchanged with gain or lose of hypercharge while interacting with a particle. It's like “earthing” in electric wiring.Weak hypercharge is present only in the left-handed chiral matter particles (fermions) similar to left-handed spin.HelicityLet me explain. Angular momentum of a particle is called spin. When a particle is spinning clockwise around the direction of motion of the particle as its axis, its spin vector points in the direction of the motion and when counter-clockwise, spin-vector points in opposite direction. The particle is said to possess left-handed and right-handed helicity respectively. Please note, of the two, spin and motion, changing the direction of one without changing the other will change helicity.Right-handed helicityLeft-handed helicityImage credit - Quantum DiariesRight-hand thumb ruleBut helicity is not an inherent property of a massive particle. Because it changes from one frame of reference to another.Let me explain. Suppose in one frame of reference a massive particle is moving to the right and at the same time spinning clockwise to motion then it possesses left-handed helicity. Now in another frame of reference that's moving faster than the particle, the same particle appears to be moving to left without any change in spin resulting in change of helicity to right-handed helicity. Thus helicity is a property that's relative and not an inherent property of the particle.Right-handed helicity when observed from a frame moving slower than the particleHelicity changes to left-handed helicity when seen from a frame moving faster than the particleImage credit : Quantum DiariesHowever massless particles move at the speed of light. Since no frame can move faster than light helicity cannot change. Helicity remains same in all frames of reference for massless particles.ChiralityBut there's a property called Chirality of a particle related to helicity that's frame-independent and hence an inherent property of the particle. Chirality is not a physical property but an abstract, mathematical concept involving rotation in a complex-plane in maths. So there are left-chiral and right-chiral particles.Change of helicity of massive particles does not change Chirality. Thus for instance, a left-handed chiral particle can have either left-handed or right-handed helicity. Same with right-handed chiral particle.In the case of massless particles since helicity never changes helicity and chirality are the same.I'll just add a couple of words on chirality.A particle's quantum state is given by its wave-function. When a fermion spins by 360° the phase of the wave-function rotates only by half ie 180°. So one full rotation won't bring the fermion back to its former quantum state. It only brings it by half. Hence its spin is 1/2. It requires two rotations to back to the same state.Image credit : Quantum DiariesNow the phase can change by rotating clockwise or counter-clockwise. If the phase changes by clockwise motion then the fermion possesses right-handed chirality and if it's counter-clockwise then it possesses left-handed chiralityImage credit : Quantum DiariesHiggs Mechanism for fermionsFirst let's consider only the fermions or matter particles. All fermions possess mass. Let's see how they gain mass from Higgs field.Now only the left chiral particles posess weak hypercharge. If a left chiral particle loses the hypercharge it becomes a right-chiral particle. It so happens a matter particle (fermion) that interacts with Higgs field constantly oscillates between left and right chirality by losing and gaining weak hypercharge to and from Higgs field. Due to this continuous oscillation, the particle gains relativistic mass as given by Dirac equation. This relativistic mass gained is the particle’s rest/invariant mass,which prevents the particle from traveling at the speed of light. Greater the frequency of oscillation larger the mass of the particle. Frequency of oscillation is proportional to what's called the coupling strength of Higgs boson to the particle. This is the Higgs Mechanism for fermions. Relativistic mass gain due to motion of the particle is an add-on on the top of rest mass.The same thing could be interpreted in another way. The oscillations of chirality from left to right and back could be thought of as the outcome of the particle bumping into a Higgs boson, present in the Higgs field as quantization of the field and by losing a weak hypercharge to it, and gaining it back by bumping into another Higgs boson. This bumpy ride ricocheting from one Higgs boson to another slows the particle from moving at the speed of light in vacuum. There's no generation of mass out of nothing as that's a violation. There's no hard uncuttable substance called mass. There's only the inability of particle to move at the speed of light and that's called rest mass. This could be thought of as relativistic mass gain.The familiar Higgs bosons, the fermions bump into are quantizations of the ground state of Higgs field called vacuum expectation value (vev). Let's call this Higgs boson 'h' as we'll refer to it again.Greater the Higgs boson couples to the fermion greater will be the rest mass of the fermion.Particle with greater rest mass bumps into Higgs bosons moreParticle with less rest mass bumps into Higgs bosons lessImage credit : Quantum DiariesThis is the mechanism how, with the exception of neutrinos, all fermions ie six flavours of quarks and three flavours of leptons ie electron, muon and tau particles gain mass.Neutrinos also oscillate and hence possess a little mass. But the mass gain doesn't seem to be from Higgs field. There are many models regarding neutrino mass origin but not yet a proven one.Higgs Mechanism for bosons - 1Now let's consider bosons, the force carrying particles.Photons and gluons are massless. W+, W- and Z° bosons, the force carriers of weak force are massive and they gain mass from Higgs field but the mechanism is different from that of the fermions described above.The familiar Higgs boson h that interacts with the fermions happens to be only one of the four Higgs bosons! Yes there are four Higgs bosons H+, H-, H° and h! As you might guess the other three bosons H+, H- and H° are related to W+, W- and Z° bosons.The four HiggsesImage credit : Quantum DiariesBut where are they?Answer is inside the 'belly' of weak bosons! Yes they're absorbed (“eaten” or paired up) by the weak bosons. To be specific W+, W- and Z° bosons get their charge and mass by absorbing H+, H- and H° Higgs bosons respectively.Weak bosons “eating” three Higgses called Goldstone bosonsBut you'll be surprised if I say the three Higgses are themselves massless! And yet W+, W- and Z° bosons (weak bosons) get their mass from them! Then what's the mechanism?Electroweak symmetry breakingWell there's more than that meets the eyes.We know electricity and magnetism are one and the same. When we see a natural magnet, magnetism seems like different from electric force as there are no wires attached. But when we see a current going in circles producing a north and south poles just like a magnet we realise electricity and magnetism are one and the same. Within a natural magnet, alignment of electrons spin produce a net magnetic field. Magnetic field around a live wire is actually electric field born of relativistic effects. Anyway the bottom line is electricity and magnetism are one and the same force called electromagnetism.Next level of unification comes between electromagnetic and weak force. They're in fact derivatives of a single force that could be called electroweak force. Electromagnetic and weak force appeared as separate forces after what's called breaking of electroweak symmetry.At higher energies (temperatures) in the early Big Bang, there were actually four unified massless W1, WE, W3 and B bosons representing electroweak force. However W and B bosons don't really look very unified and for a true, complete unification we must we must look up GUT, Grand Unified Theories, where B and W bosons as well as eight gluons of strong force find mathematical unification.Unified weak bosons before symmetry breakingImage credit : Quantum DiariesThen there were the four massless Higgses H+, H-, H° and h. 'h' is the Higgs boson we are familiar with.Now the massless W1and W2 combine individually with massless H+ and H- to form massive W+ and W- bosons respectively.W1 + H+ → W+(massless) + (massless) → (massive) (???)W2 + H- → W-(massless) + (massless) → (massive) (???)Image credit : Quantum DiariesNext the massless W3 and B bosons together combine with massless H° to form massive Z° and the massless photon.W3 + B + H° → Z° + photon(m-less) + (m-less) + (m-less)→(massive) + (m-less)(???)Image credit : Quantum DiariesThe mystery is in the italics, how combinations of massless bosons end up with massive bosons. That's the Higgs Mechanism for bosons.Higgs Mechanism for bosons - 2Transverse and longitudinal polarizationsFirst we must understand the difference in one feature between a massless boson and a massive boson. That feature is polarization.We have already seen the concepts of spin, helicity and chirality for fermions. Now let's understand a similar related concept called polarization for bosons.Polarization means different ways a quantum particle can spin.Both massive and massless bosons can spin clockwise or counter-clockwise around the direction of motion of the particle. This is our familiar left and right-handed spins or helicity. Also sometimes called spin-up and spin-down. These are the two “degrees of freedom” for polarization of both massive and massless bosons. Also called transverse polarizations.There's another degree of freedom called longitudinal polarization, only polarization of massive particles can enjoy. Massless bosons can't have this.Consider a massless boson spinning forwards in the same direction as its motion like a ball rolling. If the particle is moving from left to right, its left-handed spin would be perpendicular to this page, projecting towards the reader, or away from him if it were a right-handed spin. If it was spinning backwards in the opposite direction to its motion the spin would be right-handed, perpendicular to page projecting towards the reader or away from him if it were a left-handed spin. This degree of freedom of polarization is called longitudinal polarization.Because as the massless boson moves at the speed of light we encounter a problem. The top part of the particle would be moving in same direction as the motion but at a greater speed than light, and the lower part moving opposite to the motion at a speed less than light.This is an anomaly and against relativity. The conclusion is massless bosons do not have longitudinal polarization.But it poses no problem with massive bosons as their speed would always be less than that of light. They can indeed possess longitudinal polarization.Extra longitudinal polarization is the main difference between a massless boson and a massive boson.So how or from where can massless bosons gain the extra degree of freedom?The answer is by absorbing another particle.So when a massless boson X ‘eats’ or obsorbs another massless boson Y, if it results in addition of longitudinal polarization to X that would signify mass gain.Let's see what qualities of Y would bestow the extra degree of freedom to X.Any massless boson Y with a spin would have two degrees of freedom (transverse polarizations) and hence can't add one extra freedom (longitudinal polarization) to X.To add an extra degree of freedom to X, the massless boson Y must be spinless (ie spin = 0). Spinless particles without spin vectors are called scalar particles.Coming back to our unified weak Wi B bosons, Y boson must be charged inorder to interact with them.Further the Y boson must obtain vacuum expectation value, which is the minimum quantum field value expected in vacuum. Which happens to be, positive (non-zero).Goldstone bosonsTo sum up, the unified massless weak Wi and B bosons can become massive by absorbing massless bosons, provided the absorbed massless bosons areSpinless scalar particles (spin = 0)Possess chargeObtain vacuum expectation value (vev)The three Higgses H+, H- and H° happens to be exactly such particles. They are referred to as Goldstone bosons after Jeffrey Goldstone who gave a theorem related to this.In general, the particles that combine with massless force particles to form massive force particles are called Goldstone bosons.In a nutshell we can sayFermions ‘gained’ mass by slowing down their speed from that of light. They achieved this by continuous bouncing off 'h' Higgs bosons and changing chirality.Bosons 'gained' mass by adding an extra degree of freedom (longitudinal polarization) to the two degrees of freedom (transverse polarizations) they already possessed. They achieved this by absorbing (eating or pairing up) Goldstone bosons H+,H- and H°.Now one last question remains. From where did these four Higgses H+,H-,H° and h suddenly appear?Spontaneous symmetry breakingHiggs field is said to be a doublet meaning it has two components - one is neutral and another charged. Each component is represented by a complex number. A complex number has two parts, one real (Re), another imaginary (Im = i = √-1). Higgs field can vary along two axes - one real, another imaginary of a component. Since there are two components, Higgs field can vary along four axes, also called degrees of freedom.At every point in space-time one can think of a quantum field and its potential energy, simply called potential. One can think of its graph.First let's simply imagine a quantum field ϕ with four components and see how its graph of potential energy against field looks like.Since such a graph requires five dimensions (5D hyperspace) let's reduce the components from four to two - one real (Re), another imaginary (Im) and plot it against the potential (V) to see how it looks.Image credit : Quantum DiariesTendency of a particle in a given field ϕ is to roll down to a state of minimum potential V. In the graph above when the potential happens to be zero the field also happens to be zero ie V = 0, Re = 0, Im = 0. Which means the field cannot obtain vacuum expectation value (vev). Vev is the minimum non-zero (positive) field value a field is expected to attain in vacuum.The particle is now sitting at the origin where everything is zero. The only degree of freedom the particle has is to rotate. But rotating doesn't change the picture (field configuration) in any way. So we can say symmetry is unbroken. It remains so even with the original, abstract, 5D hyperspace graph.What we considered was an imaginary field with four components. Real quantum fields like electromagnetism may have different components but their average minimum potential and field values are zero in vacuum. Before symmetry breaking Higgs field was similar with four Higgses enjoying same degree of freedom as to remain identical.Now let's consider graph of Higgs field ϕ, its potential V, plotted against field values. Here Higgs field means field of combined four Higgses. Like before only two components of field ϕ, one real (Re) and another imaginary (Im), against potential V is plotted. The original, abstract, hyperspace graph shall be considered later.Image credit : Quantum DiariesIn the above graph, often referred to as the Mexican hat, at the origin where field values are zero (Re = 0, Im = 0) the potential V remains non-zero (+ve). We can see particle sitting there on the top of the potential hill.Here on the top of the potential hill, the particle has only one degree of freedom, that of rotation. Rotation doesn't change the picture (field configuration). Symmetry remains unbroken. All the four Higgses sitting on the hill will have the same degree of freedom to rotate, hence they all remain identical.Image credit : Quantum DiariesAs it's a very unstable position it spends its energy and rolls down to stable zero potential at the bottom of the potential hill as shown in the graph above. But here as one can see from graph, at the position where the particle lies at the bottom of the (hill) Mexican hat, the field values are non-zero. That is the particle has obtained vacuum expectation value, vev. As we know, there are four Higgs bosons that obtain vacuum expectation values this way.We also notice if we rotate x-y plane (Re,Im) around z axis (V) the particle moves in circle drawn in green colour. If you turn it by 180° the particle on the right will come to left (blue arrow). Thus we see symmetry is broken. And the particle has earned a degree of freedom to move along the circle where the potential remains zero. Earlier it could only rotate sitting at the origin. The circle actually represents field excitations pushed along values of zero potential. The particle with this freedom is massless.In the original abstract hyperspace graph there are actually three such circles with zero potential corresponding to the three massless Higgses H+, H- and H° stemming from three of the four field components.Image credit : Quantum DiariesThere's another degree of freedom the particle can move along. It can climb the potential hill, radially from the origin, as shown as the blue line in the graph above. The particle with this freedom happens to be the familiar, massive Higgs boson h. As you can guess the positive potential hill it climbs somehow has something to do with the its mass. May be it's the potential energy stored in 'h' as its mass. Where as the other three Higgses are massless because the potential energy is zero. But the exact cause of rest mass of h is still a mystery!As it's very clear now, there are two distinct types of degrees of freedom. One going round in circles of which there are three. Another going up the hill of which there is only one. Hence the symmetry is broken. Unified one type of Higgs bosons before symmetry breaking have become four distinct Higgs bosons after symmetry breaking.This symmetry breaking occurred spontaneously in the early Big Bang universe when Higgs obtained vacuum expectation value, vev, at every point in space-time when the temperature (energy) dropped below a certain threshold energy. Like phase transition occurs when gas cools down to liquid, spontaneous breaking of symmetry below threshold energy resulted in Higgs field forming a weak hypercharge condensate.Before symmetry breaking the unified weak Wi and B gauge bosons were whizzing about at the speed of light. Spontaneous symmetry breaking enabled unified massless weak Wi and B gauge bosons to combine with the three Higgses as explained before and become massive weak W+, W- and Z° gauge bosons. Mass gain occurred when the three Higgses lost their degrees of freedom to add a new degree of freedom called longitudinal polarization.Similarly massless fermions whizzing about at the speed of light suddenly had the familiar fourth Higgs boson h in their tracks to tackle. As explained before, the fermions had to start bouncing about the h Higgses by constantly changing their chirality. This reduced their speed from that of light and added rest mass.W and Z bosons are extremely massive particles hence their range and life are also extremely short. It's understood that prior to absorption the weak bosons were massless. The discovery of weak bosons meant indirect discovery of the three Higgs bosons provided there was proof of Higgs field's existence. Otherwise one can imagine weak bosons getting mass and charge by some other means. Existence of Higgs field could be confirmed only when the familiar fourth Higgs boson 'h' was detected. Which was duly done at CERN in 2011 after decades of fruitless searching.Please check the following links for further details.YouTube :Watch "How Small Is It - 05 - The Higgs Boson (1080p)" on Youtubehttps://Youtu.be/xG.YtASz7gYQuantum Diaries (helicity & chirality)Quantum Diaries (WZ bosons - Higgs mechanism)Quantum Diaries (electroweak symetry breaking-1)Quantum Diaries (electroweak symetry breaking- 2)Why Eat Goldstone Boson?Origin of rest mass of Higgs boson h itself ?Origin of rest mass of Higgs boson h itself doesn't come from Higgs field. From where does it come is still a mystery.For excellent article on this topic check the links below.Professor Matt Strassler's excellent article :Does the Higgs Field Give the Higgs Particle Its Mass, or Not?The Hierarchy Problem