#science #physics #ideas The Biggest Ideas in the Universe | 9. Fields

#science #physics #ideas The Biggest Ideas in the Universe | 9. Fields 172,142 viewsMay 19, 2020 2.9K DISLIKE SHARE DOWNLOAD CLIP SAVE Sean Carroll 154K subscribers The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody. This is Idea #9, "Fields." A little bit about classical fields, but mostly concentrating on quantum field theory, and in particular on why a quantized field ends up looking like particles. This one is a bit challenging! My web page: http://www.preposterousuniverse.com/ My YouTube channel: https://www.youtube.com/c/seancarroll Mindscape podcast: http://www.preposterousuniverse.com/p... The Biggest Ideas playlist: https://www.youtube.com/playlist?list... Blog posts for the series: http://www.preposterousuniverse.com/b... #science #physics #ideas #universe #learning #cosmology #philosophy #quantum #fields 366 Comments rongmaw lin Add a comment... Chris Moon Chris Moon 1 year ago (edited) Dr Carrol I'd like to thank you for taking the time to make this interesting series of videos. I'm 70, a retired "blue collar" worker, neither gifted nor bright, nevertheless interested in the world and your lectures/talks have shed light on things which have fascinated me but were beyond my comprehension. Am also enjoying reading "Something Deeply Hidden". You have brought a lot of joy, understanding and pleasure to an old man. Thank you. 196 Amir# Amir# 1 year ago Sean, please, don't stop making this content! 134 Existentialist Dasein Existentialist Dasein 1 year ago This series is a great source of comfort during the current lockdown. Thank you for continuing! 87 Matias Reinoso Matias Reinoso 1 year ago As many already stated, this video series is amazing and so informative & intuitive, please continue it if possible! 17 Director4Sciences Director4Sciences 1 year ago Sean, thank you. What a beautiful balance you've established in this series between faithfulness to the current academic discipline and yet accessibility by the 'unwashed masses'. I find myself in what's probably the sweet spot of your 'ideal viewer profile' - limited but significant general academic background in math & physics + infinite curiosity and a deep & abiding desire to understand the physical world. As such, I greatly appreciate the (clearly sizable) personal investment you're making in synopsizing (and extending) your generous contributions to my cohort with Big Picture, SDH, Particle At The End…, et al. Please know that I am grateful! All the best. -b 8 EverythingQuestion EverythingQuestion 1 year ago "It's beggining to seem....hard" meanwhile I've been pretending to understand for a few videos now. lol. Love these. Great job. 23 Salvatron Prime Salvatron Prime 1 year ago Thank you so much for this series. It's so awesome to get the most current date theory and details in a very concise and easily understood format! 13 Bill G. Bill G. 1 year ago Thank you. This is like a college physics course without having to actually do problems! I'm glad you don't shy away from the math, I want to see it, but don't want to do it. 6 decobocopithec decobocopithec 1 year ago (edited) This is my favorite episode so far. Thank you Sean! 42:49 Question. The string metaphor helped, but I still missed a few steps. Is there a better way to qualitatively understand why are energy levels discrete and equidistantial without having to go through the solution of the wave function of the modes? Looking forward to the part where you explain the interaction among quantum fields. I my naive mind they exist parallel to one another, in a kind of Descartian pluralistic way, which is obviously not realistic :) My guess is that they can be connected through spacetime or maybe mass or equivalent mass, as indicated in the potential energy equation. Waiting for the next episodes to see if I guessed right :) 2 Matthew Krellwitz Matthew Krellwitz 1 year ago 10/10 best content on the interweb. Thanks Sean! 13 Kieran Garland Kieran Garland 1 year ago I absolutely love this series. Will be returning to them again and again. Thank you. 3 Chip Hill Chip Hill 1 year ago Perfect level of mathematics! Thank you - I thought I would never get this far into quantum mechanics! Álvaro Rodríguez Álvaro Rodríguez 1 year ago Thank you! No popular physics explanation has gone so far. Live long and prosper! 3 Ronan Sleep Ronan Sleep 1 year ago Wow. This is hitting exactly the stuff I was missing when I did my physics course in the 60's. Really great introduction. Sandra sandra Sandra sandra 1 year ago Dr Sean, you explain beautifully well, in a pleasant way, and speak very clearly, which is very important for those like me who are learning English 2 Christian Pala Christian Pala 1 year ago Does the amplitude of the wave function have any kind of meaning per se, like displacement for water, or should it just be seen as a mathematical construct to square in order to get the "probability configuration" of observing particles at specific locations? 4 Alex Cherfan Alex Cherfan 1 year ago Thanks for all this wonderful and informative content Sean. You're awesome for doing these. Plus, this is the fist time EVER and I'm not even kidding, that I've seen a youtube video with 0 dislikes. Congrats, even the trolls love you. Haha! 1 Michael Dougherty Michael Dougherty 1 year ago (edited) Thank you Dr. Carroll for making this series. I'm this far in, watching sequentially. This is the level of explanation and intuition that I have been waiting for from among the many of you that do similar work. Cariboo Land Cariboo Land 1 year ago (edited) I just wanna thank you for this wonderful lecture, please keep that kind of content coming. could you make a video about quantum gravity? Quantization? String theory? Emanresu56 Emanresu56 1 year ago Sean, don't drown yourself! There are so many things to live for! 88 Francisco Dias Coelho Francisco Dias Coelho 1 year ago I appreciate these videos Professor Sean, thorough yet comprehensible enough for a layman like me. Thanks Cloud Variable Cloud Variable 1 year ago Thanks so much for this! For the longest time, I had no idea what the hell quantum fields even meant, but this immediately cleared up the conceptual debate for me! And I have to say, the formalism of it all just seems so appealing... 박건욱 박건욱 1 year ago Wow, this was very interesting! Love your content! Vinny V Dali de Monet Vinny V Dali de Monet 9 months ago (edited) @Sean Carroll, your video's have the feel of delivering individualized attention. I watch them with the appreciation as if you're focused on helping me understand these awesome lessons. Thank you so very much. For generously sharing your time, passion, lifes work with all your years of learned experience. Lambda Lambda 1 year ago Thank you!!!! So interesting! How lucky we are to have this. 1 Carolina Ruivo Carolina Ruivo 1 year ago Thank you Sean Carroll for this series and please keep sharing your knowledge! Listening to it while working :) Greetings from Portugal! Samuel Charles Hamilton-Green Samuel Charles Hamilton-Green 1 year ago (edited) Thanks for this I am so glad you have taken the time. I enjoy seeing you get serious and taking us along for the ride. Also happy to take the pause outbreaks needed to keep up with you :) dont slow down 1 TheRebelTreble TheRebelTreble 1 year ago I'd been doing ok with previous videos but I think I've hit my quantum comprehension wall. I'll see if the Q&A clears any of it up 7 Samuel J Samuel J 1 year ago yo this is a really good video, concise yet still a great explanation. i beg of you to bring out the next episode soon, my particle physics exam is only in a couple of days :) Russell Black Russell Black 1 year ago Thank you for making QFT seem approachable. Very well explained. One minor correction at 35:30 - potential energy of a simple harmonic oscillator is proportional to k, not k^2. Gilbert ENGLER Gilbert ENGLER 1 year ago (edited) Just excellent! Thanks a lot👍 Please go on explaining the things so clear. Jorge Figueroa Jorge Figueroa 1 year ago Sean, you are a terrific teacher, you make complicated things so easy to understand, you really have an special way of communicating, I’m enjoying every single one of your lectures. I do like your books, also, I recommend it to everybody out there who loves science. 1 camniv1234 camniv1234 1 year ago Thank you so much for these videos. I am ashamed I made it through a physics degree without this insight into the equations. Yoshi Tatsu Yoshi Tatsu 1 year ago (edited) Thank you for taking time to make these videos. I am a huge armchair physics nerd. I was a philosophy undergraduate and after I Graduated, I became completely obsessed with physics. Wayne Long Wayne Long 1 year ago Thank you for all you have done to help improve our understanding of current perspectives in physics. Regarding the zero vector potential: is not the zero of the potential arbitrary like other potentials? Tom Utube Tom Utube 1 year ago Fantastic series! I am really looking forward to every new video! Dear Sean, I have a few questions which I hope you'd consider addressing during the Q&A: 1. If I understood correctly, since each field mode is an element of a Hilbert space, we could as well described it in a particle number basis, i.e. a superposition of modes that have a definite number of particles. This alternative basis seems much simpler to me: no harmonic oscillator needed, no (absolute) energy for vacuum, and a countable infinite number of basis vectors instead of an uncountable number. Why do we complicate matters with the harmonic oscillator? Is my particle number viewpoint overlooking some aspects perhaps? 2. Since the Schroedinger equation is linear, am I correct in assuming that QFT is a linear theory (with a rather complicated non-linear potential 'landscape' but only involving linear operators)? If so, why don't we try to solve the system of linear operators instead of dealing with path integrals? Path integrals, while pretty, just seem to be complicate matters by turning an operator inversion into an infinite sum/integral which may or may not converge. 3. Since free fields don't interact, does it make sense to talk about particle number for free fields if we can't observe them? I always imagined the definition of 'quantum particle' to be intrinsically linked to a unit of interaction between fields. Am I misguided here? 4. In a previous video you mentioned that some people object to Everett's many worlds view because it is unclear where the (Born?) probabilities would come from. I didn't quite follow that point. Wouldn't the Born rule follow from the inner product in the Hilbert space as a posteriori probabilities? I surely misunderstood something. If I did, are there any (non-philosophical) limitations to Everett's many worlds? Many thanks again for making this series! The unique viewpoints that you bring to these big ideas is very refreshing. Tom 1 Tomáš Šedovič Tomáš Šedovič 1 year ago Thank you so much for these videos! They are the perfect combination of depth and accessibility. What would the full notation of the Schrödinger equation for the "quantum wavefunction of the universe" look like? There seem to be different versions depending on whether it's a single particle or a field. What does it look like for the entire universe? If there are unitary tricks (such as setting c=1 in relativity), will you please write out both the verbose and more compact version? Deep Recce Deep Recce 1 year ago Another great episode..Thanks Prof! Sipplix FPV Sipplix FPV 1 year ago The more physics I absorb and "try" to understand including, perfection in the mathematical description of our reality, speed of light limit, Entanglement, Super Position, measurement problem, wave function, dark matter, Plank scale etc. the more the Simulation hypothesis seems to explain a lot of things. Thanks for this very educational show, Knowledge Onward! Nick Steele Nick Steele 1 year ago Love this series, Sean! I have a general question about quantum fields. When ppl talk about unifying the fields, what exactly do they mean? Are we trying to prove that there is actually only one field that acts like different fields at low energy? Or there are many different fields that used to be one field at the time of the big bang? 10 R C R C 1 year ago The biggest video yet on the Biggest Ideas in the Universe! I am just loving all the bigness including the big smile I get from enjoying these videos. 1 Juan Lizarazo Juan Lizarazo 1 year ago Dr Carrol, thank you very much for this video series. I have a question regarding fields, and hope you could address it. If what we call a particle is a perturbation on a field, then the speed of the particle would be determined by the field, just like the speed of a wave is determined by the media in which it travels. In your lecture, you reconstruct the particle-like behavior by adding an infinite number of planar waves. Wouldn't the speed of those waves, and thus the particle they form, be determined by the media?. In this case the media is the quantum field. Could you give me some pointers of where can I find the answer to this. Best Regards. 1 Csababo Balo Csababo Balo 1 year ago Really appreciating your sharige of knowledge! Ps: fields exists due density? Tenzin Lundrup Tenzin Lundrup 7 months ago The subject of this video is exactly what I have been wondering about, namely, what is a quantum field and how does one get particles from fields. vidar traeland vidar traeland 1 year ago (edited) Hi:) Instead of thinking about fields as waves/values in 3d space, could they be visiulized as dimentions of space, where each field has an axis, and each point has a value in each axis? (I dont know much about this, so its probably a silly question) Thanks for making these videos! Dan Noc Dan Noc 1 year ago Perfect lesson, thanks. Bruno Prates Bruno Prates 1 year ago Hi Sean. Amazing video as always, thank you so much for sharing this knowledge. My questions: (first one is more important) 1 - If there is just one wavefunction for a given field, how come when we measure an electron in a experiment, it doesn't collapse the entire wavefunction of the electron field, therefore collapsing every electron in the world to a particular location? 2 - How many fields are there in QFT? I can't seem to find a consistent answer online... Are there one for each particle in the Standard Model? Do anti-particles have their own fields? What about different handed particles? 3 - Are there different fields for the different generations of fermions? An electron field, a muon field, and a tau field? Or is it just one field? If it is just one field, how does the difference between an electron and a muon appear to us when we observe the field? 7 Matthew Hondrakis Matthew Hondrakis 1 year ago Listening to this reminded me I should review Fourier Series! It has been a while since I saw them in my Diff Equations class hehe! 1 Cooldrums777 Cooldrums777 1 year ago This video takes me back to my undergraduate days studying nuclear engineering. You even threw in a tiny partial diff eq (S wave eq). So much fun to watch this because QFT wasn't even covered in my graduate work. Thanks for this one Prof. BazNard BazNard 1 year ago Amazing videos. Thanks so much for making these. smoozerish smoozerish 1 year ago Keep up the great work Sean 👍 qcxcq qcxcq 1 year ago This is exactly the intermediate cohesive introduction to these topics that I have sought for for years. The last time I've had such insights on new topics was probably in University. Thank you for sharing your knowledge in such an elaborate way. M W M W 1 year ago Very clear introduction into fields, and quantum field theory. Pamela Collins Pamela Collins 1 year ago I have no physics background; I have read a lot of lay person books on particle physics. Thank you so much for making these videos!!!! I’m starting to understand some of this, and it is so fascinating!!!!! And I love learning what these symbols stand for. Thank you thank you thank you 💕💕💕 1 George Rey George Rey 1 year ago @SeanCarroll, I have seen your lecture at "New Scientist Live", (youtu.be/p7XIdFbCQyY), and, this is an excellent speech as far as public speaking techniques go -- but I think it can be greatly enhanced if you add a few words about how fields work:: that there can be fields that are cross-interacting, self-interacting and non-interacting. AND, in this video, you demonstrate that non-self-interacting fields are fairly easy to visualize - very much like the sound waves in a 1D string, or surface waves in a pond. If, in your public speech on Many Worlds you spend a few words on explaining that some events are interacting (and produce a split) and some are non-interacting (waves just travel on their own and move away from each other, maybe even going through each other without distortion) the entire story will be much more believable. You can do it! TetonGemWorks TetonGemWorks 1 year ago I am watching these so you CAN put ideas in my head! Thank you sir. If we watch all the videos and Q&A sessions, can we get a "Licensed Quantum Mechanic" shirt? Andi C Andi C 1 year ago (edited) Are all particles entangled with the single wave function of the universe, even particles that aren't entangled with any other particle? 9 Ricardo Ricardo 1 year ago Hello Dr. Carroll, as with many others, thank you so much for these lessons (actually, thank you is not enough!). I have a question on 14:15 , if we plug in a scalar field function into Schrodinger's equation and then square it to get the probability, should we get the same probability at all points in space regardless of the value of the field (because it's a function of a specific scalar field, or specific field configuration)? Edward Lee Edward Lee 1 year ago To continue .. A layman, me, can't resist to follow it to the very end. Thanks for the efforts and keep up with the good work. From Hker worldwide 1 Iain Mackenzie Iain Mackenzie 1 year ago 35:30 (Sir, I think the k is not squared in this equation: Energy of SHM oscillator = 1/2kx>2) It is not critical for your argumant of course, but just for completeness. (The omega IS squared later at 105:36) I am LOVING your videos. I look forward to them each week ( including the QNA sessions) Thank you so much for your time and energy. It is very much appreciated. ŞABAN DALAMAN ŞABAN DALAMAN 1 year ago (edited) Sean Carroll is a perfect teacher which is the best compliment I can say for who is doing this. And this is not only virtue he has Chris Chris 1 year ago It's important to support the sciences to keep guys like Sean happy, or they might decide to turn the rest of us into dust. Edward Lee Edward Lee 1 year ago Great lecture. Truly heady stuff captivatingly presented. A layman 1 John Fruhwirth John Fruhwirth 1 year ago Does the warping of space due to gravity have any effect on fields? Any idea what happens to fields (any fields) below the event horizon of a black hole? Jean Gatti Jean Gatti 1 year ago Clearly the very best explanation I've ever seen for describing QFT principles, many thanks Christopher Mora Christopher Mora 1 year ago Hey Sean, can you explain why at 17:10, F(R3) is an infinite dimensional vector space when it's just mapping to R which seems 1 dimensional? Bernard Loke Bernard Loke 1 year ago Wow things are beginning to come together now! Thanks Sean!! Matt Watkins Matt Watkins 3 months ago Watching these now a year after they were published and they're exactly what I was looking for: a down-to-the-metal description of how all this stuff works without all the pop-sci fluff that surrounds these topics. Thank you so much for taking the time to make this series. It's a monumental effort. An interesting note: you use the word "quantize" to (I think) refer to taking a classical phenomenon and bringing it into the paradigm of quantum mechanics. This terminology is disorienting to an electrical engineer: "quantize" to me invariably means to take a sampled continuous amplitude signal and place each sample in an amplitude "bin" that can be labelled with a fixed-width binary number. We talk about quantization noise and various ways to mitigate its effects. That seems emphatically to not be what you're talking about here. You're talking about taking something with a single classical configuration and describing it as a superposition of configurations characterized by a probability function, all of which exhibit fully continuous behavior in space, time, and amplitude. DaveDashFTW DaveDashFTW 1 year ago Thanks Sean for making such complex ideas accessible. In regards to the Cosmological Constant problem, isn't this kind of just like the Singularity in general relativity? I.e. the theory works well to describe 99% of conditions, but just at some certain conditions we need new physics to describe that problem. One thing I also get confused over, what's the relationship between the cosmological constant, vacuum energy, dark energy, and virtual particles? Is this something that can be answered yet, or is it still in "folk" theory territory? Stumpy Mason Stumpy Mason 1 year ago (edited) I noticed that in the wave function of a mode.. 49:00 𝜓-0 = 0 particles -- the wave in the above graph did not cross h. 𝜓-1 = 1 particle --- the wave in the above graph crosses the h line 1 time. 𝜓-2 = 2 particles -- the wave in the above graph crosses the h line 2 times, so does that mean.. 𝜓-n - n particles - the wave in the above graph crosses the h line n times? also there is a great vid by 3Blue1Brown about the fourier transform which made this easier to understand. vid link - www.youtube.com/watch?v=spUNpyF58BY 1 Mike Mike 1 year ago when you’re at the beach and Sean pops out of the ocean to hit you with some intellectual pills. 👌 life is good! 17 Per Jespersen Per Jespersen 1 year ago Man this video is good. Many of those dificult questions like particle tracks and the distingtions between the wavefunvtion and the field are answered... and I even passed a QGT course once. Iain Mackenzie Iain Mackenzie 1 year ago For QnA session could you please clarify about the Fourier analysis being applied here? Is it product of sin waves (multiplication) ? or sum (addition of and, therefore, superposition)? Many thanks K J K J 1 year ago So far, this video is the most interesting but most complex of Dr. C's series. Grat Mian Grat Mian 1 year ago Dear Dr. Sean, I wonder what app you use to create this video. I like very much the whiteboard with infinite canvas. Kevin Evans Kevin Evans 1 year ago I'm another one like Chris Moon. I have had it in mind for decades to understand this stuff and although I'm just making a start on listening to you, I can tell that you are the man that's going to do it for me. The Eternal Hermit The Eternal Hermit 1 year ago Would air temperature in a 3D space be an example of a scalar field? Mackenzie Marceau Mackenzie Marceau 1 year ago (edited) I don't know if anyone has ever wondered about this: Even in his "greatest blunder", Einstein was a genius. He formulated a 'cosmological constant' that he later dismissed as an error. But when scientists later discovered that the expansion of the universe was accelerating, they realized Einstein had been right all along and that his cosmological constant was useful after all! 1 llaauuddrruupp llaauuddrruupp 1 year ago These videos are so good! 4 Hattie Lankford Hattie Lankford 1 year ago 18:58 let's acknowledge how huge that is that he had to clarify in the right direction. KAĞAN NASUHBEYOĞLU KAĞAN NASUHBEYOĞLU 1 year ago Prof.Sean Carroll. Thanks a lot for this series. A Z A Z 1 year ago Is it accurate to say antiparticles that pop into existence are simply a down wave in a field, or inversely charged quanta at that location? Jim's Mind Jim's Mind 1 year ago This is getting to the limits of my maths, but you're done a great job explaining it in a way that makes it easier to understand than a lot of explanations I've heard. My question would be at some point could you talk more about the physical experiments scientists are currently doing, especially those on the cutting edge. For example I find it hard to visualise how quantum computers physically create entangled Qbits. joy fergie joy fergie 10 months ago Thank you, Dr. Sean 🙏 Ronan Sleep Ronan Sleep 1 year ago Unresolved for me: Say we have QFT but no fields. What process do we go through to decide to add a new field? How many fields are there? Is this a matter of taste? 1 Chris C Chris C 1 year ago (edited) There is also a reflection of all things electric (sub atomic), both micro and macro; then there's reflections of reflections of quantum reflectors. Eg. We have an eye, a fist and a process of calculated appropriation with an infinite amount of outcomes. The sub atomic being [unlimited] orbiting reflectors. StarMan StarMan 1 year ago Hello Sean and fellow viewers, In keeping with the spirit of this series, here is my newest Big Idea, in the quest for discovering the ultimate reality. I hope you will find this to be of interest: I understand that all of space within our universe, is a 3D field of potential energy, possessing various, specific characteristics, on which the standard model of our physical reality is derived from. As I understand it, a black hole is basically a super compressed portion of the 3D field. What I am proposing is that we need to create the exact opposite of a black hole. Imagine the quantum field is a 3D curtain. We need to 'part' the curtain. We need to literally rip the field itself, so that we can look inside it, or behind it, whichever the case may be. Somehow completely clear a part of it away. I think this would answer a lot of questions. K J K J 1 year ago Again, great subject matter. The theory does great job of explaining everything except for how do we begin with a mental, mathematical construct and end up with something that we can observe in a cloud chamber. I suspect an observation can be made of a path in a cloud chamber with no knowledge of this theory.?? What am I missing ?? Naimul Haq Naimul Haq 1 year ago Are there two different wave function for two different particles, not entangled? Elsewhere Sean claimed there is just one wave function for all the particles in a field. Joel Curtis Joel Curtis 1 year ago Or to put it differently, the height of each mode in the superposition of the quantum field tracks how many particles of the momentum corresponding to that mode's k-value are present? lilit vehuni lilit vehuni 1 year ago Could you please explain scalar fields in more detail? I read that temperature distribution us a scalar field but how can it be scalar if there is a temperature gradient? BakonKing BakonKing 1 year ago Since we live in a simulation, humor me, there can not be an infinite number of these harmonic oscillators. Storage limits + gotta keep an acceptable framerate 😁 Navid Azadi Navid Azadi 1 year ago Dear Prof. Carroll, I have a few questions, I would be most grateful if you reply; 1- Why you made each mode into "particles" and not a single particle? why this interpretation? For example, can't we consider excited energy state of n, corresponding to n paths/trajectories of just a single particle? instead of "n particles" interpretation. Lorio Myoreo Lorio Myoreo 1 year ago This is my fav so far. I came for the voice and stayed for the knowledge. 1 Alex d.alessandro Alex d.alessandro 9 months ago May I say that though I understand little of the talk, I am enchanted by the utter humility and spirit of generosity of this beautiful person. Makes me think of Socrates. Thank you and all the best! A Z A Z 1 year ago Thanks for taking the time to do these videos Iain Mackenzie Iain Mackenzie 1 year ago You mentioned that pi comes up a lot in these equations. For mechanical SHM, this can be attributed to the 'auxiliary circle'. I wonder, therefore... is this 'projected' circular motion simply a 'tool'? A way to support our analysis of SHM? Or might it be pointing us toward something more fundamental about the nature of the field wave function? Isabel AB Isabel AB 1 year ago As I listen to this...it just feels like you are just making this up...and I mean that in the most respectful way! This material is so different than what I do that it is just hard to see how it is real. Thank you for challenging me intellectually...every time! RobertHF RobertHF 1 year ago (edited) Hi, I hope I'm not too late to ask a question about this video. Around 20 mins you start discussing a particular classical field and how to quantize it. What I didn't get from the discussion following is: how do you choose a classical field which, when quantized, will lead to a specific particle type? If e.g. the electromagnetic field leads to photons, which classical fields would lead to electrons, neutrinos etc? Do you e.g. have to "design" a classical field which has to have some set of required properties? Andy Everett Andy Everett 1 year ago Standing ovation, thank you. Navid Azadi Navid Azadi 1 year ago Dear Prof. Carroll, I have a few questions, I would be most grateful if you reply; 1- Why you made each mode into "particles" and not a single particle? why this interpretation? For example, can't we consider excited energy state of n, corresponding to n paths/trajectories of just a single particle? instead of "n particles" interpretation. 2-When you take the potential of a field proportional to the square of field itself, we would have a stable vacuum; While the reality of the universe is that with being the Higgs field, the vacuum should be unstable to decay into a free parameter to produce Higgs boson's mass. So, Wouldn't it be better if the field potential is something other than the square of the height of the modes? 3-How can we have a static universe? Is that possible at all? You said that the cosmological constant entered to having the so-called Einstein static universe. The introduction of the cosmological constant by Einstein, however, was not entirely trouble-free and the solution is unstable, as this together with Hubble’s later discovery led Einstein to abandon the cosmological constant term. 4-In the early universe, for the inflation model in order to have the inflationary acceleration of expansion, the cosmological constant must be very large. However, the value of cosmological constant is very very small in the present universe. How is this contradiction justified in magnitude? (after all, there is only one Universe!) 5- Eventually, the reality of the world is field/wave or particle? The final reality of the world in QFT,in fact, seems to be the field, not the "particles" that are appeared from it. 6-Is it possible to present a kind of theory instead of the QFT in which the wave/field property has emerged from the collective behavior of particles? So that, it might even be possible to solve the cosmological constant problem! Yours Sincerely, M. M-Fard Louis Cohen Louis Cohen 1 year ago What's the most complicated quantum system that can be analyzed numerically with, say, a week or two of current supercomputer time? A collision at the LHC? The bonds in a small molecule? Thanks for a great series. Do you have the wave function that yields the probabilities of a pony tail vs a man bun? Paul Rite Paul Rite 1 year ago Is there reason to believe that the inflaton field is 'real' or is it just a convenience for calculating? 1 bruinflight bruinflight 1 year ago Sean you are so brilliant. No fault that I'm a layman, but I sure wish I could talk about these things with the depth of understanding and intuition you do. It's amazing to listen to you and there are parts I feel I understand more, but the math and subtleties therewith leave me behind. That's unfortunate too because I do love math! Great series, please keep your chats coming! theosib theosib 1 year ago Couldn't you refer to wave height as "wave amplitude" and the value of the wave function at a point the "probability amplitude"? Walker Walker 1 year ago Damn. He is sooooo good. Thank you, Sean. 2 Pavlos Papageorgiou Pavlos Papageorgiou 1 year ago Trying to keep track of what ingredients you've added to bake QFT. I think you assumed 4D spacetime, a kind of locality where the field has a gradient and propagates in space, and a kind of elasticity where it vibrates around a zero point. Was that all? At some point will you give a list of fields? I take it it's not 17 fields one per particle. Navid Azadi Navid Azadi 1 year ago 2-When you take the potential of a field proportional to the square of field itself, we would have a stable vacuum; While the reality of the universe is that with being the Higgs field, the vacuum should be unstable to decay into a free parameter to produce Higgs boson's mass. So, Wouldn't it be better if the field potential is something other than the square of the height of the modes? Navid Azadi Navid Azadi 1 year ago 4-In the early universe, for the inflation model in order to have the inflationary acceleration of expansion, the cosmological constant must be very large. However, the value of cosmological constant is very very small in the present universe. How is this contradiction justified in magnitude? (after all, there is only one Universe!) Tim Seguine Tim Seguine 1 year ago At least in the usage by mathematicians(and as far as I know, physicists use it the same way) a "functional" is specifically a map from an arbitrary space X to the complex plane or the real line. Which is quite a bit different than what you said. These have a special name because they are important in their own right and are in a certain sense a generalization of the space X they came from. There is a robust theory of calculus on functionals for example (which I am pretty sure you must be aware of since this fact is pretty important for both classical and quantum mechanics) Clément Dato Clément Dato 1 year ago In three dim, all functions can be written as a sum of volume wave not plane wave I think. Rinkamime Rinkamime 1 year ago 10:30 is the absolute value done first and then you square or do you square and then take the absolute value? | a + ib |^2 is really confusing me. Shouldn't you foil it and get negative terms in the middle? What is | i |? That's a question I thought I'd never ask xD Or are you not foiling and specifically squaring both terms first and then taking the absolute value? That makes sense, but why isn't there a foil done? If someone knows the math could they explain that? Would be a big help. Navid Azadi Navid Azadi 1 year ago 6-Is it possible to present a kind of theory instead of the QFT in which the wave/field property has emerged from the collective behavior of particles? So that, it might even be possible to solve the cosmological constant problem! Jeff Rogers Jeff Rogers 1 year ago Has anyone seen a discussion of how the idea of fields can be related to the idea of more than four dimensions? If the Kaluza-Klein theory can derive the electromagnetic equations from five dimensional theory, can interactions in eleven dimensional theory account for the observed properties of the fields and their associated particles? I haven't been able to find anything, even speculation, in this area. I wish I could do the math! :-) Brian Tyler Brian Tyler 1 year ago 🙏🙏 2 Vaibhav Vaibhav 1 year ago Sean, the thumbnails are amazing! 5 John Healy John Healy 1 year ago It's all about energy. Is there an alternative to viewing a quantum field as a superposition of an infinite number of harmonic oscillators? And, can we really separate the vacuum from quantum fields? If entangled to some degee? We typically abstract them as superpositions – as like layers; but for an entangled system, the wave function associates amplitudes with the entire configuration of the system, not part-by-part. (And any isolated system is only an approximation.) In some sense, I'd be interested in starting with an entangled interactive energy "fabric" and then how fields emerge and then how those modes get us to talking about particles as stable modes (and observables). If the energy density function is not merely a superposition of separable wave functions – the vacuum-energy wave function and the fermion-energy wave function, ... then there's just one amplitude. Not adding amplitudes of two wave functions. Energy density function = wavefunction(E-vacuum + E-field) Energy density function ≠ wavefunction(E-vacuum) + wavefunction(E-field) No infinite gradients. Then we predict energy density probabilities, which can be decomposed into modes which have modes themselves and which have properties of interest. Do Feynman diagrams, as visual tools, use vacuum energy in an ad hoc way? As an add-on perturbative device which simplifies computing interactions. As such, a successful approximation (at least in QED vs. QCD), but sidestepping entanglement of vacuum and fields? Does entanglement create spacetime? Bernard Whipps Bernard Whipps 1 year ago (edited) Just before the Big Bang was there a Professor like Sean that explained to the universe how it should behave? Olly Wright Olly Wright 1 year ago Epiphany-inducing in places. Thank you Silent Monk Silent Monk 11 months ago I'll not compare you to my teacher nor to someone else, because you are absolute 👌 Navid Azadi Navid Azadi 1 year ago 5- Eventually, the reality of the world is field/wave or particle? The final reality of the world in QFT,in fact, seems to be the field, not the "particles" that are appeared from it. Joel Curtis Joel Curtis 1 year ago (edited) Soooo we can think of each mode as the field corresponding to a particle, and when that mode's height/energy goes up, that amounts to adding a particle which has the momentum associated to the wavelength k of the mode? Alex Alex 1 year ago Hi Sean, love your podcast, you should talk to Pascaline Lapeltier who is not only one of the best wine experts in the world but also a philosopher. You guys will have a blast... Marcel Janssens Marcel Janssens 1 year ago Aren't we all waiting for an episode of Your Biggest Daily Ideas in the Equation Universe? 2 Kami Shinigami Kami Shinigami 1 year ago This man explains successfully in layman's terms some of the most complicated concepts in Physics, yet smiles like a child at Christmas when copy/pasting. "Can we keep him?" But seriously, thank you @Sean Carroll for providing these lectures and keeping our brains busy. Let me see if I get the gist of it at least a little: Quantum field theory asks "Is it possible?" It looks at different energies and wonders about the probability of finding certain energy states at any given moment in time and space or its potential. It does however not tell the actual scale of it? (I'm not a Physicist, can you tell? :)) Tibi Munteanu Tibi Munteanu 1 year ago QUESTION 1: Is "if we choose to measure it" the same as "if the particle gets entangled with something" and the same as "if decoherence happens to it"? OUESTIONs 2: In your books, you say that there is no place for free will in the laws of nature, but I think it depends what you mean by free will. I am not talking about the domain aplicability of concepts. I think you are right that there is no place for free will as a separate entity that derails nature from schrodingers eq. You always talk about the importance of us trying to interpret the laws instead of just using them for gadgets. The many worlds interpretation seems so nice and clean, but there is always that point where they say that when decoherence happens, you end up in one of the worlds, while some other worlds continue to exist.. but never talk about a formula or law that dictates in which of those worlds we end up living in. Can't this be a place for free will? What if consciousness is something that is constantly bombarded with decoherences and is forced to pick which world to be in at each junction? Is there proof that in the rest of the many worlds in which I am not, the configuration of particles that makes up me, is certain to also be conscious? Can't all the quantum mechanics fields and the wave function of the universe just be the canvas on which consciousness paints, or an intricate set of roads that consciousness navigates? If you ever played a song on guitar or piano, what you are doing for those 3 minutes.. appears as choosing the results of each "frame".. you appear to choose to place your finger someplace and not someplace else.. and to pick the strings at precise times, rather than at other random times. But along the way, a large number of decoherences happen, where maybe you chose to advance to one of the many worlds in which you placed the finger at the right fret. So you wouldn't bend the laws of physics with your consciousness.. but rather choose a path through the austere wave function. Константин Новицкий Константин Новицкий 1 year ago (edited) When something happens, the probability function of this event turns into a fact. This is a case. A case can be considered as a quantum of probability, right? That is, the probability function of an event consists of quanta, right? 허유선 허유선 1 year ago I don't understand all. But I think I can understand all in the near future. You are a good lecturer. C R C R 1 year ago can't believe i'd ever be so excited even for the upcoming Q and A! 4 Navid Azadi Navid Azadi 1 year ago 3-How can we have a static universe? Is that possible at all? You said that the cosmological constant entered to having the so-called Einstein static universe. The introduction of the cosmological constant by Einstein, however, was not entirely trouble-free and the solution is unstable, as this together with Hubble’s later discovery led Einstein to abandon the cosmological constant term. Shalkka Shalkka 1 year ago This pushes forward understanding of very important physics that is very valuable contribution. With the term "wavefunction" I have long taken the stance to maybe dance along and maybe I will get the meaning from context. But it seems digging down and still having a mysterious aura of qunatumness probably isn't appropriate. When does a function fail to be a wavefunction? I would imagine it involves things like being continous and not having corners. I have some incling it involves having an imaginary e exponent in it somewhere. So when a physicist invokes the letter psi and uses the term wavefuntion I am left wonering whether there aer some extra "magical semantics" I should read into that. Letters f and g are commonly used for properly unused "unknown/undefined" properties functions. So if I present "Consider the particle having wavefuction abs(x)" have I somehow failed to provide a wavefunction? It is somewhat hard to keep track of what the type signatures of various things are and they seem to changing a little. Getting it exactly concret would risk being stuck in technical details but when the subject area might no tbe intuitive it can be hard to separate out which is a hard unintutive turth that needs digesting and which is an obviously wrong interpretation that needs discarding. The term "Non-interacting" seems to have a meaning-option that is very different that seems to really go on. If have a pool of water what happens in one part of the lake affects what goes in nearby parts of the lakes and this could be called "self-interaction" water acting on water. And one could argue that such watermolecyle to watermolecyle interactions are how waves propagate. Howeveer water also has the property that if you throw 2 stones into it one at a time and record the wave patterns you get the same wavesum than if you threw both stones at the same time. In the context of where we go from wavesnapshots into two waves passing by each other we kind of need mechanics on how the next wavepattern is dependent on what the previous wavepattern was. One could think that "interactionless" version could include a scenario where each individual point did its thing without regard to its neighbours. "Neighbour going in positive direction, I am going to keep going on negative direction because that is what is part of my swinging". When doing a fourier transform into modes the imagination pointers can point to a wavepacket-like image. When a human things of a sound they probably think of a short shout. The modes are associated with frequencies that "constantly stay on", one keeps on shouting the sound. If you shout a short pulse early or late each will have a single unique correct fourier transform but there is no time dependency in those composite modes. A lot of listeners might implictly be adding position or timing information to what the total representation is. I am myself unsure how the same wavelength but different phase is handled. That is 10 hz starting from 0 and 10hz starting from top of peak are probably different modes but it could make sense to handle same hz modes in the same direction in the same "bundle". And this might happen via making "amount of frequency" complex where different phases correspond to how much that phase off-set is included (so 1+0i and 0+1i combine to some multiple of 1+1i which is a scalar rotated by only 45 degrees). Whether this is a connected or unconnected imaginarity source I can't tell. It is unclear how the energy of the modes realtes to the energy of the composition or superposition of the modes. If same formulas (kinetic, gradient and potential) apply to the actual wave and the modes and the mode decomposition preserves the wave then the energy shouldn't change just by decomposing it. Mathematically a function f(x)=0 can be expressed equally well as f(x)=sin(x)-sin(x). Even if wiggliness costs energy and sin(x) is undoutably wiggly it would be inappropriate to add the absolute values of the wigglinesses to get the total wiggliness. Antiwiggliness cancels out! I guess a more formal compaint woud be that psi^2+phi^2 is different from (psi+phi)^2. It is also not clear why that you can do fourier transforms means that you can only observe one mode. I can fourier transform a sound file and when a sound file enters my ear I don't have to pick a frequency to listen to I can use the full range of my ears. And the decoherence ideas could be employed in that the observation is not randomly picked. It is super confusing that the colored lines loan the x axis but do not loan the y axis form the white drawings (and the confusion of assuming x is space instead of h is hard enough as it is). The y axis is amplitude rather than energy. Or was the point ot say there were negative energies? The abstract hard things seems very interesting. Why would the amplitude need to spread out like that? But since it does spread out like that wouldn't that spreading define a number analogous to kinetic energy and gradient energy, say heigth gradient the amount of amplitude lost when moving over a small height difference (in the middle of and sides of yellow the height gradient would be low but where green and yellow intersect it would be high on yellow) There seems to be missing information why f(x)=0 is a dissallowed attempt to be a low energy state for a mode. Kinetic energy is 0, gradient energy is 0 and any number proportional to height is 0. I guess any other mode has a positive change to get a value for any x (minus some crossover points) . But if you don't pluck a string it's a perfectly fine string. Or does it mean somehow that any energetic entity touching truly quiet place will leave it echoing that is not proportional to the touching? Jacques Smeets Jacques Smeets 2 months ago (edited) correction: at 33.00 he states that all energies (kinetic, gradient, potential) per mode are proportional to the square of the height h. This is true for the graient and potential energy. The kinetic energy is proportional to the square of the time derivative of the height h. This is exactly what is required for a simple harmonic oscillator. South Coast South Coast 1 year ago I use this to go to sleep and I love physics 🙃 Ray Guest Ray Guest 1 year ago This cosmological constant problem Sean talked about at the end of the video seems vaguely reminiscent of the black body radiation problem. to me. Max Planck solved that problem by saying that energy could only radiate away in discrete increments given by his constant times the frequency. Here again, it seems that we should have run away infinite energy, but we don't. I don't know how to picture a zero energy state of a quantum field really. But if you picture it as an infinite number of harmonic oscillators that happen to all be in a configuration of their zero potential energy state, then since they are oscillating, they would all be at their maximum kinetic energy state. That picture does not seem to help the problem in any way though. Although, if there are an infinite number of quantum field modes contributing to the total zero point energy, it seems that a lot of them would be randomly trying to push the energy in the negative direction while a lot would be trying to push it in the positive direction and that they would largely cancel out. If I am trying to pluck a guitar string in one direction with my forefinger and trying to pluck it in the opposite direction with my thumb, then I am just holding the string. Smarter people than me think that it is a problem though, so obviously there is much that I am missing. schel sullivan schel sullivan 1 year ago Good job socially isolating. The middle of the ocean. 148 Rust in Peace Rust in Peace 1 year ago (edited) What is a Quantum field made out of, if the particles are a result of it. With the Schrödinger equation saying how the quantum field behaves? Is that question even something science could answer or would we never be able to access that information. 2 Michael G Michael G 1 year ago How are electric and magnetic fields related exactly. At the fundamental level are they the same field ? Maxwells equation just tells us that if there is a moving electric field it generates a magnetic field and vice versa, but why does it behave this way ? And can one exist without the other ? Are there electric fields in a permanent magnet ? Shaun Bartone Shaun Bartone 1 year ago On the Energy of Empty Space (transcription and comment): Empty Space has fields which could have energy Set classical energy of empty space to 0 Quantum field is a product of an infinite number of harmonic oscillators, aka the modes, the plane waves. Put the modes in their lowest energy state (ground state) they still have energy, the quantum bit that is added to the classical state of empty space. Quantum mechanics contributes an infinite amount of energy to empty space in quantum field theory The cosmological constant problem, now known as the energy of empty space. This is ‘bad’? Like it’s a problem. But it’s not a problem if you understand it from the point of view of the Buddhist concept of Emptiness. That empty space could simultaneously contain an infinite amount of energy makes perfect sense in Buddhist cosmology. James Sorenson James Sorenson 1 year ago A naive question: What is energy? In classical physics it is pretty easy to get ones head around. There is Potential energy, Kinetic Energy, Chemical energy and so on. Relativity makes it a little more confusing with E=mc^2. We understand that you can convert some mass and get energy, that is amazing but we can get our head around it. Today you talked about the vacuum energy, but what is it. Presumably this is all the same stuff i.e. energy, I hope you can explain more about what this is. Thank you in advance. Erik Dahlgren Erik Dahlgren 1 year ago Question: so is state of a particle the same as number of particles. (n+1/2)hw. Seems like n is both number of particles and state of the particle? Fixundfertig1 Fixundfertig1 1 year ago It feels like if Sean was sinking in an Ocean field 😂 BakonKing BakonKing 1 year ago So is the idea of particles "popping in and out of existence" the same thing as the state of the field changing with time? @54:30 C B C B 1 year ago Is time just the felt experience of intrinsic momentum as defined by birth locality? Lunatic Cultist Lunatic Cultist 1 year ago On this day Sean Carroll was revealed to be Poseidon, risen from the murky depths to share the wisdom of the abyss with those ready to comprehend it. Cory Pride Cory Pride 1 year ago This 70 year old, unfortunately got left behind here. I love your lectures but I just can't fathom derivatives, vectors and amplitudes. John Długosz John Długosz 1 year ago You showed the wavelength the inverse of the k vector length. But you never said how you get the amplitude. What else is there besides the direction and length of the vector in 3 dimensions? Wenjie Gong Wenjie Gong 1 year ago I'm little confused about "superposition" vs. "product." Don't we integrate over creation & annihilation operators of different "p" in our definition of a quantum field? Is the integration not considered a superposition? Brane 2 Brane 2 1 year ago Great podcast. My favorite. Less great choice for a thumbnail. It looks like you felt off a cruiser in the middle of nowhere. It's perhaps not the best idea to project an image of a nerd that ruins every trip he attends. ;o) milivoj milosevic milivoj milosevic 1 year ago Setting up a subwoofer in my music listening room. Having to deal with room modes. Discrete areas of 3D space where bass frequencies reinforce and and take on the appearance of virtual particles in a sound field Nga Nguyen Nga Nguyen 1 year ago Could ‘Quantum Entanglement’ creates space in empty space? Octavia Octavia 1 year ago my degrees are in the humanities. i could NEVER qualify for physics and yet i am so interested!!! this makes it so accessible and lets us armchair science aficionados get some education from Dr Carroll! (who, let’s be honest, is a fucking rockstar of the science world) Thank you SO much!!! Bill Carey Bill Carey 1 year ago In reference to the Cosmological Constant problem: In your 2008 lectures on Dark Matter/Energy (found in Great Courses Plus), I think you said the problem was approximately in the order of 10 to the 120 power. Now you are saying the difference is infinite because vacuum energy mathematically speaking should equal infinity and not the small value we observe. Or am I badly mixing things up? If I’m not mixing things up, is the new view due to better interpretations of Quantum Field Theory or better understanding of the wave function equations? Or neither? Or corrections? Thank you for great presentations and please keep the Big Ideas coming. Rogério Pereira Rogério Pereira 1 year ago I'm always surprised to see some imbeciles giving thumbs down to these lectures. I wish they would go to the Lunatic Fringe of YouTube, to watch documentaries about aliens, new age energy vortexes, crystals and the like, and leave these amazing lectures for people like me that just want to learn. I admit that I'm disturbed when I see those thumbs down, Always makes me think of that famous Einstein quote: "Only two things are infinite, the Universe and Human stupidity, and I'm not that sure about the Universe". Ditto. Thanks Dr, Carroll, you're doing an amazing work, kudos for you and keep these videos going. Stephen Bryant Stephen Bryant 1 year ago My mood brightened when this came in. Great background image too replica replica 1 year ago (edited) divided by an infinite number of dimensions in acceleration / tension David Seed David Seed 1 year ago 23:10 ,you talk about adding amplitudes to make a fourier analysis but you miss the important point the the amplitude is a complex number which by r and. theta give us the norm of the amplitude and the phase. Czeckie Czeckie 1 year ago is |Ψ|² really the probability or is it rather probability density function? It's just seems that a probability of particle being at any single position could be (in most physical situations) equal to zero, since there are uncountably* many possibilities and we actually can't measure infinitely precisely. It would make sense to me if the probability that the measurement will be in some region is equal to the integral of |Ψ|² over that region. *) in some cases, like measuring spin of particles, the possibilities form a discrete set and the integral would simplify to e.g. just |Ψ(up)|² But I might be wrong. I'm a mathematician with zero physical intuition. I find these lectures extremely helpful. ShinyMike ShinyMike 1 year ago That opening...you did that on porpoise! 2 Константин Новицкий Константин Новицкий 1 year ago (edited) Mathematics reflects our world, right? So If we have a math function, it reflects something real in the World. A lot of people forget about, right? Joao Joao 1 year ago Watching right now! Thanks! =)))) Rajeev Gangal Rajeev Gangal 1 year ago What writing pad does he use? Something made for writing or iPad like general device with appropriate software? Roberto Tomás Roberto Tomás 11 months ago You talk about the Fourier transform as if the collection of sine waves that sum to a function is unique. Is that really true? naively, I think there is a requirement that you seek the smallest such collection: because you can add two of half values of any one wave to replace it. So I feel you might be making things simple for us, which is appreciated. :) if that is so, what does the real world basis for enforcing this uniqueness criteria? Particles cannot be located at the same spot? rage rage 1 year ago Waiting for this topic i have a spare 3 hours 😁 Valdagast Valdagast 1 year ago His hair field is expanding in value. 76 zicada zicada 1 year ago I just have to say i love this thumbnail 23 Volaire oh Volaire oh 1 year ago Just lovely 😊 Jon Olsen Jon Olsen 1 year ago (edited) Are subatomic particles round and If they are, why do they have this geometrical shape? Soul DFS Soul DFS 1 year ago (edited) Can you just explain - infinity ♾ and positive infinity = 0 a little more. (🌊) Kid Mohair Kid Mohair 1 year ago "languorously changing"; it is not often, I suspect, that "languor" has been used in QM btw I was quite taken by the mid-water thumbnail (as others have been) the tie-in is not lost... O, those languorous waves... until the hands disappear above, in a non-linear fashion 1:09:03 1 Daniël Boomsma Daniël Boomsma 1 year ago Nobody said it was easy No one ever said it would be this hard… V T V T 1 year ago Imagine a massive stadium, filled with 100k people, that's the audience for this lecture. James the Other One James the Other One 1 year ago Everyone understands that fields don't exist right? They are just numeric representations of forces or values to enable mathematical modelling. This is something that is often unstated and many people forget or don't realize. Spim Falter Spim Falter 1 year ago LMFAO fields are just a simplification of a probablistic distribution of particles. Charles Fox Charles Fox 1 year ago Wrote this once before but had trouble posting (it didn't take), so will try again ... Always had a mental block with magnetism, stops me fully understanding things like sun spots, CMEs and earths interactions (northern lights). Have never heard a good description, all generally go: it's a field, just like gravity, here's a bar magnet and iron fillings, look at the field lines, and there you go. That tells me nothing. What are the dynamics of the field lines? Is there a 'flow' and if so of what? Why do the iron fillings gather there? and not necessarily pulled to the poles of the magnet? What happens between the field lines? Why are the field lines that specific distance apart? Is it some sort of wavelength distance? Is it an interference pattern? is it the absence of any magnetism? etc. This one mental block is obviously inhibiting me from understanding everything about the universe. Any help would be greatly appreciated. Brian Cannard 1 year ago Thank you! rv706 1 year ago (edited) Sean Carrol is great. But, with all due respect, I tend to find physicists' notations quite dis-functional. Why can't they just write functional dependencies for what they are? Writing Ψ[ϕ(x)] when you really mean Ψ[ϕ] may be confusing to the more mathematically minded. Also, that ϕ_k (h) notation is quite obscure: if it's the Fourier component of ϕ pertaining to k, what does it mean that it "depends on h"? Why is it written as a function of h? -- I'm not suggesting that he should be pedantic in every line (that would be ugly), but at least at the beginning, when he gives the definitions. 2 Maciej Tudek 1 year ago Well, You're as good as Susskind in teaching these topics.. 1 george gonatas 1 year ago Gonna have to watch this one twice Tenzin Lundrup 7 months ago (edited) Question: Here "probability" is really probability density. How is the probability density for a field configuration defined? For probability density rho(x) the probability for x in [x, x+dx] is rho(x)*dx. What is the analog for a field configuration? But I guess with Fourier analysis I don't have to worry because for fixed \vec{k} we are back to rho of the height, h. The classical energy of phi(x, t) is the space integral of what you wrote down (the 3 terms) and for a plane wave that integral is infinite. hm4791 1 year ago composition at the start of the video: 10/10 2 The Biggest Ideas in the Universe! Sean Carroll 18 / 48 1 6:33 NOW PLAYING The Biggest Ideas in the Universe: Introduction Sean Carroll 2 28:17 NOW PLAYING The Biggest Ideas in the Universe | 1. Conservation Sean Carroll 3 40:41 NOW PLAYING The Biggest Ideas in the Universe | Q&A 1 - Conservation Sean Carroll 4 43:17 NOW PLAYING The Biggest Ideas in the Universe | 2. Change Sean Carroll 5 40:58 NOW PLAYING The Biggest Ideas in the Universe | Q&A 2 - Change Sean Carroll 6 56:20 NOW PLAYING The Biggest Ideas in the Universe | 3. Force, Energy, and Action Sean Carroll 7 53:23 NOW PLAYING The Biggest Ideas in the Universe | Q&A 3 - Force, Energy, and Action Sean Carroll 8 1:00:54 NOW PLAYING The Biggest Ideas in the Universe | 4. Space Sean Carroll 9 48:05 NOW PLAYING The Biggest Ideas in the Universe | Q&A 4 - Space Sean Carroll 10 54:31 NOW PLAYING The Biggest Ideas in the Universe | 5. Time Sean Carroll 11 51:11 NOW PLAYING The Biggest Ideas in the Universe | Q&A 5 - Time Sean Carroll 12 1:03:21 NOW PLAYING The Biggest Ideas in the Universe | 6. Spacetime Sean Carroll 13 35:39 NOW PLAYING The Biggest Ideas in the Universe | Q&A 6 - Spacetime Sean Carroll 14 1:05:28 NOW PLAYING The Biggest Ideas in the Universe | 7. Quantum Mechanics Sean Carroll 15 46:06 NOW PLAYING The Biggest Ideas in the Universe | Q&A 7 - Quantum Mechanics Sean Carroll 16 1:20:30 NOW PLAYING The Biggest Ideas in the Universe | 8. Entanglement Sean Carroll 17 59:38 NOW PLAYING The Biggest Ideas in the Universe | Q&A 8 - Entanglement Sean Carroll ▶ 1:16:30 NOW PLAYING The Biggest Ideas in the Universe | 9. Fields Sean Carroll 19 1:05:30 NOW PLAYING The Biggest Ideas in the Universe | Q&A 9 - Fields Sean Carroll 20 1:05:52 NOW PLAYING The Biggest Ideas in the Universe | 10. Interactions Sean Carroll 21 The Biggest Ideas in the Universe | Q&A 10 - Interactions Sean Carroll 22 The Biggest Ideas in the Universe | 11. Renormalization Sean Carroll 23 The Biggest Ideas in the Universe | Q&A 11 - Renormalization Sean Carroll 24 The Biggest Ideas in the Universe | 12. Scale Sean Carroll 25 The Biggest Ideas in the Universe | Q&A 12 - Scale Sean Carroll 26 The Biggest Ideas in the Universe | 13. Geometry and Topology Sean Carroll 27 The Biggest Ideas in the Universe | Q&A 13 - Geometry and Topology Sean Carroll 28 The Biggest Ideas in the Universe | 14. Symmetry Sean Carroll 29 The Biggest Ideas in the Universe | Q&A 14 - Symmetry Sean Carroll 30 The Biggest Ideas in the Universe | 15. Gauge Theory Sean Carroll 31 The Biggest Ideas in the Universe | Q&A 15 - Gauge Theory Sean Carroll 32 The Biggest Ideas in the Universe | 16. Gravity Sean Carroll 33 The Biggest Ideas in the Universe | Q&A 16 - Gravity Sean Carroll 34 The Biggest Ideas in the Universe | 17. Matter Sean Carroll 35 The Biggest Ideas in the Universe | Q&A 17 - Matter Sean Carroll 36 The Biggest Ideas in the Universe | 18. Atoms Sean Carroll 37 The Biggest Ideas in the Universe | Q&A 18 - Atoms Sean Carroll 38 The Biggest Ideas in the Universe | 19. Probability and Randomness Sean Carroll 39 The Biggest Ideas in the Universe | Q&A 19 - Probability and Randomness Sean Carroll 40 The Biggest Ideas in the Universe | 20. Entropy and Information Sean Carroll 41 The Biggest Ideas in the Universe | Q&A 20 - Entropy and Information Sean Carroll 42 The Biggest Ideas in the Universe | 21. Emergence Sean Carroll 43 The Biggest Ideas in the Universe | Q&A 21 - Emergence Sean Carroll 44 The Biggest Ideas in the Universe | 22. Cosmology Sean Carroll 45 The Biggest Ideas in the Universe | Q&A 22 - Cosmology Sean Carroll 46 The Biggest Ideas in the Universe | 23. Criticality and Complexity Sean Carroll 47 The Biggest Ideas in the Universe | Q&A 23 - Criticality and Complexity Sean Carroll 48 The Biggest Ideas in the Universe | 24. Science #science #physics #ideas The Biggest Ideas in the Universe | Q&A 9 - Fields 59,330 viewsMay 24, 2020 Sean Carroll 154K subscribers The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody. This is the Q&A video associated with Idea #9, "Fields." I go once more through why the simple harmonic oscillator has quantized energy levels, explain how the Dirac equation and Klein-Gordon equations are *not* relativistic versions of the Schrödinger equation, and mention entanglement and the Reeh-Schlieder Theorem. My web page: http://www.preposterousuniverse.com/ My YouTube channel: https://www.youtube.com/c/seancarroll Mindscape podcast: http://www.preposterousuniverse.com/p... The Biggest Ideas playlist: https://www.youtube.com/playlist?list... Blog posts for the series: http://www.preposterousuniverse.com/b... #science #physics #ideas #universe #learning #cosmology #philosophy #quantum #fields 146 Comments rongmaw lin Add a comment... Cody Ramseur Cody Ramseur 1 year ago Just want you to know, Sean, this show is everything for me right now. 44 Brad S Brad S 1 year ago (edited) We adore your math and we adore you. Yes we are ignorant but the more times we see it, that ignorance slips away. You do an excellent job of explaining the concepts. 21 BZ BZ 1 year ago Dr. Carroll please keep these coming. I love the perfect amount of detail and math you use for us normies. 2 Chip Hill Chip Hill 1 year ago Thanks for explaining the inclusion of complex numbers in the wave function. What a cool method to keep the probability a positive value. Sen's Sword Sen's Sword 1 year ago Good on you for making content that educates during this time. Being productive during rough times is a good trait. Pablo Ledesma Pablo Ledesma 1 year ago wow great content, thanks for all this work Sean! Jim's Mind Jim's Mind 1 year ago Enjoyed so much! "QFT is hard" You've done a great job of describing it in a way that makes it easier to imagine. Darth Plagueis Darth Plagueis 1 year ago (edited) Thank you for these videos Mr. Carroll. You have really re-ignited my interest for physics. Yeracxam Yeracxam 3 months ago I'm a professional physicist, yet I'm learning so much from these videos! One confusion: how can the spatial modes of the field be entangled if you can write it as a superposition of momentum modes? Shouldn't entanglement be independent of basis? Thanks for all the effort on these! 🙏 Marcos Freijeiro Marcos Freijeiro 1 year ago Thanks sean for doing the math (maths in England ) absolutely love it wish I understood it better but I have to start some where. Love these lectures I have recommended it to lots people. Thumbs up 👍 2 LGB Gábor Lénárt LGB Gábor Lénárt 1 year ago Fantastic videos, I especially love the approach "do not fear of math", and indeed usually there are two kinds of materials to learn more about these for "the average guy": one is the "no math, it's the brain killer" one, and the other, which is more targeted people learning this in universities and such. But not too much between these two cases ... I love the kind of "middle way" approach of these lectures! :D Mehdi Baghbadran Mehdi Baghbadran 1 year ago Dear Sean : thank for your good explanation, as you know I’m experimented physicists, and I explained, things in my own language, not mathematical language, so thanks for your hard work , hope to see you soon.mehdi Alfian Gunawan Alfian Gunawan 1 year ago i like the explanation about the quantum harmonic oscillator in this video, very intuitive Pvte Pyle Pvte Pyle 1 year ago Awesome! I'm studying physics at KIT and just started masters degree. I've nerver heard of the Reeh-Schlieder Theorem and it just blew my mind. Definately have to read into it. Thank you very much sean! :) Xiaochen Li Xiaochen Li 1 year ago I really like the pacing of these videos and the just right amount of information for me. I have been interested in physics and been watching related videos on YouTube. I got to the point where entry level videos by most science communicators are too easy for me, but outright college lecture VODs are too hard. This series has been the first stop gap that I have not been able to find. And his way of explaining really help me wrapping my head around some of the more difficult concepts. 1 Dr10Jeeps Dr10Jeeps 1 year ago If there is a positive to the pandemic it's that I can listen to podcasts from Sean Carroll, Brian Greene, and Lawrence Krauss. These sessions represent what is great about the internet. 1 decobocopithec decobocopithec 1 year ago (edited) note for myself 11:23 28:18 49:27 19:09 replace x with h to get to modes from simple harmonic oscillators. Wave function can be constructed from perpendicular modes (like vectors). Modes are perpendicular if the sum of their overlap is zero. This combined with the restriction that the integral over h of the probability of being in any state has to be equal to 1 determines the shape of the modes, and the discreteness of energy levels. J. Curtis J. Curtis 1 year ago Wonderful work, thank you for hitting this note. Soul DFS Soul DFS 1 year ago Thank you for the big explanation at the end and answering my question. The end of the videos are always far out! Go Mezant Go Mezant 1 year ago There's a theory that there's only one electron. This actually goes hand in hand with field theory. That is, the electron field is THE ONE electron and the appearance of a local electron is just a manifestation of local interactions with the electron field. This is akin to the surface of a lake, think of it as THE electron field, now throw different size rocks at different strengths on the surface in different places and these are the other interactions. But it's still one surface, one field, one electron. Am I crazy? 3 Olle Abrahamsson Olle Abrahamsson 1 year ago Has Sean covered how the Hamiltonian operator is defined and how it works? If not, what would be a good reference to read/watch on my own? Wayne B Wayne B 1 year ago Sean you are SO right about the gap! Please continue with the mathy derivatives! Paul C. Paul C. 1 year ago Thanks for this great series Prof. Carroll. I am a big fan of your lectures, Talks & books. But, I am now in my sixties, and although I am a (BSc Hons) science graduate, I do find some of the material pretty heavy going. This is probably because I never did advanced Maths, nor learnt Calculus, I'm sad to say. Explanations with words are always preferable in my case, but I always watch each episode twice anyway. Take Care & Stay Safe, with best wishes from West Wales. tim hawker tim hawker 1 year ago Thankyou for your time and passion and briiliance.. I love your emotion in explaining your passion Neomadra Neomadra 1 year ago Dear Sean Carrol, thank you very much for this series. I studied physics and doing my PhD now and I'm realizing more and more how sloppy the quantum mechanics and quantum field theory courses at our university are. And I believe to know why many QM/QFT courses are so sloppy: Because they are immensely rushed to get to "applications" as soon as possible. Of course in this rush there is no time to think about foundations of quantum mechanics. In the end the university has produced many Master or sometimes even Bachelor(!) students who are able to do QFT calculations without understanding it on a deeper level. Of course, most students leave academia unsatisfied as a result. CrYou575 CrYou575 1 year ago Great series that goes into the concepts in greater depth than a purely popular presentation. Helps fill the gap, with Lenny Susskind's lecture series being the next rung up the ladder. Leonardo Gacitua Leonardo Gacitua 1 year ago Sean please write quantum textbook, I am a physics undergrad and your GR textbook is the best 22 Pavlos Papageorgiou Pavlos Papageorgiou 1 year ago 50:00 thanks for clarifying that Φ as a sum of modes is not the same thing as a simple harmonic oscillator at every point in space. A lot of what's going on here seems to be about jumping between the frequency domain and the spacetime domain of a Fourier transform. When you talk about emergent spacetime, are you trying to define physics directly in the frequency domain? 1 Roman Travkin Roman Travkin 1 year ago Mathematicians denote complex conjugation by an overline ("bar"); the dagger is for matrices or linear operators (Hermitian transpose). Dean Inada Dean Inada 1 year ago (edited) Could the Reeh–Schlieder theorem resolve the black hole information paradox? Just as charge, mass, and angular momentum don't disappear when objects fall into a black hole, if the information associated with an object is part of the global quantum field, then it wouldn't be lost when the object falls into a black hole. Michael Onstad Michael Onstad 1 year ago I have heard that if quantum tunneling occurs in the Higgs field and the boson migrates to a lower energy, we are done. Jeff K Jeff K 1 year ago Immediately clicked 'like' at the beginning when Sean said he wasn't afraid of using a bit of math to bridge the gap between fluffy discussions of physics and videos down-right dense with math. Thank you. 1 Jeff Bass Jeff Bass 1 year ago I want to make sure I understand this correctly: 1. In order to have "n particles" does every mode need to be in the nth energy state? 2. I'm not sure I understand how it connects back to "ordinary" QM. Do you treat a region of space that is supposed to have one particle as though it's a field with modes in the 1st energy state? Siri Landgren Siri Landgren 1 year ago 3:00 There is indeed a gap, and you're one of the few who fill it! <3 4 Spencer Delallo Spencer Delallo 1 year ago You're like a fuckin rockstar, you definitely play a role in my passion for physics! and had a big influence on what I chose to study. The coolest mad scientist in this day and age Rust in Peace Rust in Peace 1 year ago What makes the Schrödinger equation so difficult to make it Background independent, and do you think this points to it being a wrong direction to go for a TOE? Bvic3 Bvic3 1 year ago 8:10 star is conjugate for scalars, dagger is conjugate of the transposition of a matrix Reiner Wilhelms-Tricarico Reiner Wilhelms-Tricarico 1 year ago OK. Alice might be able to prepare some system so that some spin is up, or prepare a whole bunch of them, and this has consequences for remote systems that are in entanglement with her lab equipment. The question is though in what way this is reciprocal: What if Bob "at the same time" (whatever that means), and not knowing what Alice is doing, "forces" some spin to be the way he likes it. Who wins? Or, how is it decided which actions determine which outcomes? The same question could be asked about "mere" observations: Alice makes a measurement and gets a certain spin state, and Bob also makes a measurement. The next day they compare their results and find out that they are always complementary. There seems no way to establish a direction of causality, if there is no real way to establish "objectively" a temporal order. Apparently a measurement or observation is also at its core an action upon something. The simplistic way out seems to be that everything included Alice and Bob's actions are predetermined (would that be super-determinism?). An aside: Let's not even talk about magic of the kind: some action on some wave function - resulting in thunderclap, smell of sulfur smoke - and voila a Taj Mahal appears on the moon. I am fairly sure that this is complete baloney, and remains nonsense when you put the words "in principle" in front of it. Most things that we may dream up to be plausible and believe "in principle" possible (within some logically coherent theory) are almost certainly - in principle or not - impossible in reality. eefaaf eefaaf 1 year ago I had a bit of a problem interpreting the expression for phi(x) at 13:22 : Sum(phi_k(h),k) (meaning sum over k) Firstly, not sure how it depends on x, Secondly, are the h values the same for every k? Something like: Sum(h_k*k(x),k) where k is phi_k with 'height'=1 would feel more explicit to me. Or would it even be Sum(phi_k(h,x),{k,h}), where you also have to sum (or if h is continuous, integrate) over all values of h? Forrest Orange Forrest Orange 1 year ago Thanks for saying that the last video was hard, makes me feel a lot better about struggling to understand it...... :)) 2 Eliézer José da Silva Rios Eliézer José da Silva Rios 1 year ago I think designing the nature in its basic concept as a field is great idea, but what is a field ? I think field is where energy is carried. As energy increases its density at a particular position we humans see a particle. Makes sense ? Pseudo Nym Pseudo Nym 1 year ago Amen yes amen yes lots of us know some Calculus and are willing to use it! 1 David David 1 year ago You go Sean Carroll! Roman Travkin Roman Travkin 1 year ago 10:42 can't believe a physicist doesn't know by heart the formula for the square of a complex number… there's a minus sign: (a + b i)^2 = a^2 – b^2 + 2ab i David Jordan David Jordan 1 year ago I am a musician who watches this when I take a break from practicing. When I start. "Taking particleness seriously" I am going to tell everyone I know! I am not going to be holding my breath. D. Apple D. Apple 4 months ago Overall great content, but I find myself wishing Sean would be clearer on math notation. He purports to give math equations, but as written they often are extremely ambiguous. Makes the whole video very frustrating even for people with excellent math background. Craig Simpson Craig Simpson 1 year ago Good to see you keeping your head above water there, Sean! ;-) John Bach John Bach 1 year ago Sean, really enjoying this series, and Big Picture and Particle at the End ... Currently at ch 9 of Something Deeply Hidden. I'm wondering if MWI can do better (eventually) about the question of how many worlds exist? (for example, p163 talking about decoherence of a continuous event like the location of a particle hitting a screen, wasn't clear how the choice of how many divisions to make of the continuum should be up to us?). Also, p172 concept of how branches exist simultaneously out of contact with our own - I'm hoping eventually MWI can say more about this. Anyway, enjoying the book and hope to get to From Eternity to Here soon! Jason Jason 1 year ago Perhaps there is another field that causes gravity. This could be where the dark matter is. Maybe there is a second field that causes gravity, where an excitation that doesn't require mass as a mechanism to warp it. Anthony M Anthony M 1 year ago are "fields " intended to be something intrinsic to reality? or are they just a mathematical construct/method/concept in order to perform calculations and make predictions? Bill Garthright Bill Garthright 1 year ago (edited) 17:00 Did Sean Carroll just say that Wikipedia is too technical? I didn't expect that. :) Seriously, I love these videos, but my brain turns off when you use math. In fact, I think I left my brain behind a few videos ago. I hope it can find its own way back home, someday. :) Mr. feynman48 Mr. feynman48 1 year ago Deravatives are simple and easy..even a blind mand can do them...and yea I am blind lol. So, shaun keep using the calc and other maths, I am currently a physic undergrad and would love to hear you teach calc, physics, and anything else you would be profficiant in those fields. John Joseph John Joseph 1 year ago Please continue to sow the show the Maths in detail. Richard Linter Richard Linter 1 year ago It has been forty years since abandoning an MSc for a degree in engineering. Thought I had forgotten it all, but Sean's delivery is as good as Feynman's. Retired now. Downloaded a bunch of maths and physics papers from my local university library and I'm trying to recycle myself back to the point I reached with the BSc. Funny thing is, the stage I bits were hardest to revisit. Sean makes it easy. karmel lewis karmel lewis 1 year ago Thank you. Top 5 teacher list 6 bmoneybby bmoneybby 1 year ago (edited) As an advanced arm chair physicist, this is the point where I remember that I don't know ish. Lol 6 Nigel Brayshaw Nigel Brayshaw 1 year ago I love this series....but I will never understand it in a million billion years x Pi daveffs daveffs 1 year ago I was with you all the way up to "Simply" bondmode bondmode 1 year ago cat entering the room and meowing is adding perfection to perfection 1 Pavlos Papageorgiou Pavlos Papageorgiou 1 year ago 16:50 yeah, I couldn't get what Hilbert Space is from Wikipedia. Decoherence is surprisingly a bit better. 4 Carlos Hernández Carlos Hernández 1 year ago Hey, great series but my question is about the piece of equipment do you use to write like in a black board. What is it? (My wife work as a teacher and it would help her a lot) 👍🏽 Angry Satsuma Angry Satsuma 1 year ago OMG, I never understood before now the contribution of the gradient energy to the ground state. Why wasn't I taught this??? Thanks, Sean Carroll :)) 2 Stumpy Mason Stumpy Mason 1 year ago 25:25 Thank you for answering my question from the last video <3 1 Andrew Andrus Andrew Andrus 1 year ago how to define gradient energy and why do we want "reasonably" sized? I have solved S.E. for Eigenstates w/o regard for gradient energy and get ground energy Eigen state clearly exhibiting distribution he shows for ground state. I just want to understand this idea of gradient energy, I understand gradient but not gradient energy. Sean Carroll Sean Carroll 1 year ago Mistake at 11:10, caught by Patric McHargue: it should be -b^2, not +b^2 in that expression. 45 Jaskooner Singh Jaskooner Singh 1 year ago wow, just great. had to take this vid in bite size chunks though Hawthorne Hill Nature Preserve Hawthorne Hill Nature Preserve 11 months ago Sean Carroll NEVER makes me feel stupid. He doesn’t patronize like so many others but he just elicits intelligence and critical thought from his students. This mans contribution to teaching and higher thought is historic and I believe he is at the forefront of conquering ignorance and lack of education in our species. LOVE THIS GUY! Y Y 1 year ago Thank you sir... 2 Loz Shamler Loz Shamler 1 year ago This is no doubt the wrong time to comment ie I should have commented at quantum particles and not fields. In the delayed quantum eraser experiment, as it uses photons, and photons moving at the speed of light, is it therefore true that as the photons are traveling at the speed of light they experience no time. Therefore anything that happens to a photon would happen to it instantly (especially if it's some kind of distributed structure, something happening at point (a) on that distributed structure could be "communicated" instantly to point (b), because the photon experiences no time. Therefore it looks to us as if the photon has communicated backwards in time because it's instant and we're moving forward in time at unrelativistic speeds. (Like a train pulling out of a station, you're in another carriage in a stationary train, and for a moment you think you're going backwards, but it's the other train going forwards.) (The final) therefore there's no magic. Is any of that right ? Happy to know how this is wrong ! Hope y'all well chris P chris P 1 year ago Sean u talk about Laplace's demon alot! Yet laplace's demon is physically impossible It's like a perpetual motion machine Because in order to measure a physical system u must affect that physical system There is no way to know the position of all of the universe... because in the ACT OF KNOWING u measure and affect, and create a system which is larger and more complicated than what U R trying to measure or KNOW FURTHERMORE there is no way to store the info without having a brain or hard drive which is itself unknowable to some degree Your a super smart guy it frustrates me that what is very obvious to me and seems like an important foundational point of physics is nowhere in your analysis What can I do to get this line of thinking into the convo K K 1 year ago Shame he didn't write a book on QFT like he did for GR. Nice thing he does not reply to anyone Eric Svilpis Eric Svilpis 1 year ago 11 people dislike that you’re not paying enough attention to Ariel. Steve White Steve White 1 year ago wow. that made me want to give up work and go back and do maths. And the physics. And then quantum field theory. And then make Taj Mahals everywhere.. and then... and then ... and then ... :) Bob Garrett Bob Garrett 1 month ago We love you Ariel. Paul Bates Paul Bates 1 year ago So I am watching this late at night. At 34:31 I hear a cat meowing outside my door. I press pause and go outside into the dark to look for it. After five minutes I find nothing so come back in, press play and at 34:34 I find out the meowing is from Sean's Cat, Ariel. doh! 2 karmel lewis karmel lewis 1 year ago Alice can control spin 2 scrpion669 scrpion669 1 year ago Could it be possible that dark matter is just like normal matter but the particles aren't entangled for what ever reason making them "invisible" to us? Christine LaBeach Christine LaBeach 6 months ago Is spacetime a field? David Campos David Campos 1 year ago Suppose I told you I purchased a Luger from a man who's last name was Schrodinger. My question to you is ; Where is Schrodinger's cat box? will2see will2see 1 year ago 10:35 - You have an error, there should be -b^2 BazNard BazNard 1 year ago Superb! Yoda Jimmy Yoda Jimmy 1 year ago (edited) Does Reeh-Schlieder then mean that any tiny little in guy somewhere out in space can cause a glitch so that the whole universe is destroyed? So why we are not doomed yet? Don't tell me anthropic principle. Daniel Karbach Daniel Karbach 1 year ago Isn't (ib)² = -(b²)? Jerry Brown Jerry Brown 3 months ago at 11:00 that should be -b^2 not +b^2 Esa Koivuniemi Esa Koivuniemi 1 year ago Help. Someone rescue Sean. He's drowning. :D Good thumbnail. Brian Cannard Brian Cannard 1 year ago Thaaaaank you! BazNard BazNard 1 year ago Amazing life42theuniverse life42theuniverse 1 year ago (edited) 5:00 and amplitude is height Weird Health News Weird Health News 1 year ago (edited) Similar to a Bob Ross type effect Rick Harold Rick Harold 1 year ago Cool stuff apburner1 apburner1 1 year ago Where have you been? These unannounced disappearances are unacceptable!!! Inter Net Inter Net 1 year ago The thumbnail looks like a fancy word document Snatch n Grab Snatch n Grab 1 year ago (edited) I take it back. Let your hair grow out. It looks good. Sean Carroll UW Summer Camp Ad youth-teen.uw.edu