@@genghisgalahad8465 basically a Delta function is something that is zero everywhere, except for one point where it spikes to infinity. He’s saying the learning curve is as steep as it gets
@@icodestuff6241I remember the naive bygone days of “place value” and “calculating the determinant of binary relativistic electroweak flux given an alpha of 2=e=pi+1=g-8”.Bygone are those days; poor is the understanding I think I have. All that remains (save one island of hope that is *Reading Eggs)* is the dust of all great intellectual studies from Eensteen’s plagiarism of Newtonian mechanics to Tesla’s Pigeonhole Paradox to even the great Plateo’s (a humble brand of crunchy mind-numbing cereal; an indivisible atomi of a philosophers balanced diet): Yes, Indeed, all that remains (deem it the greatest triumph of man or the steepest fall from grace) is *NUMBER BONDS.*
Once you've finished the course, could you make a compare and contrast video between the 2nd quantization and Feynman path integral formulations of QFT? I've always felt super weird using the latter
Hey guys, I just want to share my favourite physics book which covers the topics which Andrew mentioned to be difficult to grasp. The book is 'Physics from Symmetry' by Jakob Schwichtenberg. It covers group representation in different dimensions and Lie Algebra of SO(3) and Lorentz group pretty well. My favourite of this book is the way it covers Classic field theory. He derives the dirac equation and gamma matrices for spinor fields, Klien Gordon equation for scalar fields and Proca equation for vector fields and He makes it look very simple. I highly recommend that book to all Physics Aspirants.
I was a number theorist by training, and I took a first course to QFT, 2 years ago. Needless to say, the vacuum of rigor was quite shocking to me at the time. Now I have a better sense of how to approach the subject, partly by seeing it as a collection of calculation techniques held together by heuristics.
@@AndrewDotsonvideos yah, but Uzu Lim is not wrong. If you ever try to practice QFT with rigour you will die of rigourmortus before contributing anything useful to physics.
@@shivanshusiyanwal296 It is not lacking in mathematical rigour really. It is lacking in tight axiomatic development. No serious quantum field theorist will tolerate any mathematical mistakes, so you have 100% rigour in that sense, but when procedures are not well defined and loads of approximations are done, to a mathematician this _looks like_ breakdown of axiomatic development. They are right, but so what? If physicists were always waiting for a mathematician to let them in the door we'd get nowhere.
"Rigorous" in mathematics usually means having all of the objects well-defined and each inference being supported by an airtight mathematical proof. In that sense, I heard that many calculations in QFT are not "mathematically rigorous". Or at least, a first course certainly doesn't come with a full set of well-defined objects; taking integrals of annihilation / creation operators with its Dirac deltas are not dealt with the appropriate axiomatic framework that a mathematics students would normally want to refer to. . I'll mention, however, that quite a large collection of QFT material have been mathematically formalized, see for example "PCT, Statistics, and All That" and modern mathematical treatments of Feynman integral, e.g. works by Matilde Marcolli. . I'm not sure what exactly are the concepts in QFT that haven't been fully formalized mathematically at the moment (2021). I think the Yang-Mills existence and Mass Gap problem is one example showing that the theory isn't fully mathematically supported, but I'm actually not sure. The main point I wanted to get across is that a first course in QFT is unfriendly to a student trained in the typical way of pure mathematics.
The classification of particles via unitary irreps of the Poincaré group is honestly to me one of the most mindblowing things in physics. It's amazing that we can show that quantities such as mass and spin (as well as other charges) are necessary properties of fundamental states that obey the symmetries of the universe. So elegant! That's why I like the "particle-to-field" approach Weinberg takes in his QFT Vol. 1. That book is great for getting a firm grasp on the foundations of QFT.
"but I'll spend hours thinking about why that step makes sense" literally me when it comes to A levels Physics. Glad to know I'm not alone with this habit.
A lot of things will start making even more sense once you get to higher level Physics (even more so if you do an undergrad Physics). But soon enough everything will get messed up again and you'll go back to thinking about why these things make sense, and the cycle repeats!
We did second quantization in our bachelors hahahaha one of the earliest homeworks was to prove the bogoliougov transformations. A true learning delta function
For a more rigorous mathematical view (for example, constructing classical Yang-Mills theory, but heaps of other stuff too) I would strongly recommend Mark Hamilton - Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics Super good book. Without it my thesis would be mush
Hey, i just wanna say thank you, you really are the one reason i got so invested in math and physics in 8th grade, I’m now planning to do a physics degree pretty much as a result of watching your videos
Concerning 04:05, equation (13) in your PDF tells us that the combination \bar\psi\gamma^\mu\psi (one type of Dirac bilinear) transforms as a four-vector. If you take eq. (13) and put a \bar\psi on the left and a \psi on the right (on both sides of the equation), then you get the equivalent of x^\mu = \Lambda^\mu_ u x^ u. This is because \psi transforms with \Lambda_1/2 and \bar\psi with \Lambda_1/2^\dagger under a Lorentz transformation (also note that the \gamma^0 inside the \bar\psi is necessary for this to work since the \Lambda_1/2 are not unitary/the generators are not Hermitian. For more information, see the discussion above eq. (3.33) in Peskin&Schroeder). Looking forward to your next video on QFT ☺️👍
Hey thanks for the comment! If I understand your comment properly, it looks like you're saying how to construct vectors fromspinors and gamma matrices, but I was moreso trying to understand if gamma mu itself is a proper four-vector. More concretely, I figured the Lorentz transform see's all indices and transforms them accordingly. To transform the gamma matrices, one would need to hit the lorentz index with the vector rep of the Lorentz transform, and then hit once with the spin 1/2 rep (one of them inverse) for each spin index. This just solves the isomorphism relation (13) for gamma mu, but the combination of all 3 Lorentz transformations leaves the gamma matrix unchanged which was why I started calling it an "invariant four-vector", but you can't just transform the Lorentz index and think you're done, whereas other people might say that the isomorphism is not a transformation rule, but just an identity relating representations. I'm not convinced my interpretation is wrong just yet because it's obvi their anti-commutator gives something proportional to the metric which is also Lorentz invariant in flat space (times an identity matrix in spin space).
@@AndrewDotsonvideos Oh, I see. I usually think about the gamma matrices that they do not transform under Lorentz transformations, since they're just Pauli matrices (i.e., numbers) in disguise :p However, they bring along an identity which we can use to ensure that things like \bar\psi \gamma^\mu \partial_\mu \psi transform properly. Your approach of viewing the gamma matrices as objects with three indices that each get their own Lorentz transformation matrix (in their resp. representation) is interesting though! Maybe I'll have to think about this a bit more..
Every time I watch one of these explanation videos I'm both fascinated and horrified that there's so much complex physics out there. And I'm in my final year of undergrad
Congrats! Just say QFT 50 times before every test instead of actually studying. I hear it makes you smarter. Didn't even see the cat on the shelf at first. 'Learning delta function' lol
@@shivanshusiyanwal296 Just saying the name of the class literally makes you a smarter person. I didn't even study in undergrad, I said quantum field theory 4000 times and here I am.
the gamma matrices are just matrices, they have no transformation laws under the lorentz group (so are technically scalars). the \mu is just a label. when stuck inside of spinors, however, for example j^\mu = \overline{\psi} \gamma^\mu \psi, they mix up the spinor degrees of freedom in such a way that you get a vector current, that is j^\mu transforms like a vector. they do a similar thing in the dirac equation, by mixing up the spinor degrees of freedom of psi, they make a vector to contract with the covector \partial_\mu. so i think i agree with your interpretation of equation 13, in that they bridge two representations of the Lorentz group. that is an interesting way of thinking about it.
If you haven't seen these, I can highly recommend the lectures given by Carl Bender at PSI in 2012, ruclips.net/p/PLOFVFbzrQ49TNlDOxxCAjC7kbnorAR1MU. These 15 lectures cover a lot of how to solve differential equations found in QFT using divergent series and asymptotics.
Probably the most rigorous way of talking about gamma-matrices is that it is an intertwining operator (which in this case is just fancy word for isomorphism) between two representations of so(3,1) -- (1/2,0)\otimes(0,1/2) and vector one.
That tensor playlist is being so helpful, I stated studying physics like a month ago (all by myself) and it's a nice challenge while I'm practicing more basic calculus since I took those classes eons ago. Thanks for making it!
We had to study two different textbook for the QFT class. One used subscripted gamma another used superscripted gamma. Rules of commutation were different for different kind of gamma. I spent huge amount of time in figuring out similarities and differences between these gammas. Beside this, QFT class went really well.
QFT is a massive subject. The year of graduate courses is really the tip of the iceberg. This area of physics has become deeply mathematical and has blown up over the past few decades in very unexpected ways. This is a subject that mathematicians and physicists are collaborating on now more than ever.
Im with you there Andrew when you say you spend too long thinking about points of a question that aren't the main focus. I actually had a question for an assignment of mine recently to do with 2 body central forces, and long story short the textbook I was reading (which had the process to partially solve the question) merely stated some constants without proving them. Most of my classmates accepted that constant and moved on but I was "who what why how dare you assume".... Spent the next hour or 2 deriving that constant and coding into Latex. Gotta explain everything and not assume stuff man.
Second quantization feels much more natural to me, and as a solid state theorist it seems more applicable. Still pretty shaky on the path integral formalism
@@twistedsector unfortunately my current work is in semiclassical picture, but I will check it out. I'm hoping to extend this current project to include more pure quantum phenomena
Lol man I'm in my 4th year of undergrad (astrophysics specialization) and this very thing you're saying you do plagues me; I'm the same way. Maybe not to the degree you are or maybe I'm not going through exactly what you are but I'm always being ocd about the nit picky details about stuff like that (that's not necessarily related) and I'm over here then wondering why it takes me way longer to get my assignment done than other people.
The gammas confuse me too, because there seems to be multiple ways to interpret them. On one hand, I think you're basically correct when you say they're "bridging the two representations of the Lorentz group"... You might be interested in reading about the "Van der Waerden symbols": en.wikipedia.org/wiki/Infeld%E2%80%93Van_der_Waerden_symbols. In some ways, this is just a fancy name for the sigma matrices... but I think the article emphasizes that the sigma matrices are really a mapping from a vector space V (where vector/tensor-like objects live) to a pair of spinor spaces S⊗S (where spinor pairs live). In other words, "1 vector index" is equivalent to "2 spinor indices", and the symbols give you the right coefficients for converting between tensor and spinor indices. I think this is what people are getting at when they say "a spinor is like the square root of a vector", because a product-pair of spinors is equivalent to a vector, via these symbols. But you can also view the 4 gammas as the 4 spacetime basis vectors in the Clifford Algebra Cl(1,3). When you do a summation of coefficients and gammas over spacetime indices, you're really just writing a linear combination for a spacetime 4-vector... One thing that's still very confusing to me about Clifford Algebras is that scalars, vectors, spinors, and rotation operators all live together inside the same algebra... so you can add scalars and vectors together (that's basically what a quaternion is), or multiply vectors and spinors together, as when you multiply the dirac spinor "psi" by one of the gammas. But being able to churn through additions/multiplications in CA doesn't really illuminate what's actually going on, unfortunately. I wish I understood this stuff better.
Yeah that sounds reasonable but I really don't understand insisting that transformations don't apply to them (not that you were necessarily implying that). The anticommutator {\gamma^\mu,\gamma^ u}=2g^{\mu u}*I seems to imply that they do transform, they just transform trivially just like the metric in flat space. If that assumption is true, which I'm having a really hard time convincing myself that it isn't, then the isomorphism would have to count as defining the full transformation (of both lorentz and spin indices) of the gammas I think. ""1 vector index" is equivalent to "2 spinor indices", and the symbols give you the right coefficients for converting between tensor and spinor indices. I think this is what people are getting at when they say "a spinor is like the square root of a vector"": That comment blew my mind more than it had any right to.
@@AndrewDotsonvideos So I think that anti-commutator formula makes a lot more sense when you use the "gammas = spacetime basis vectors" interpretation. The standard way of defining the metric components is using the dot products of the basis vectors, so g_μν = e_μ·e_ν. Since the dot product is commutative, this could also be written as 2g_μν = (e_μ·e_ν) + (e_ν·e_μ) = {e_μ,e_ν} ... which is basically identical to the anti-commutator formula you wrote. Now, I'm leaving out the detail that the product of gammas isn't EXACTLY the same as the dot product... the product of gammas is actually the "Clifford product", aka the "dot-product-plus-the-wedge-product": (γ^μ)(γ^ν) = (γ^μ·γ^ν) + (γ^μ ∧ γ^ν), but the wedge product part goes to zero in the anti-communtator since the wedge product gets a negative sign when we swap the order in the anti-communtator formula. So I think the spinnor components of the gammas (i.e. the matrix entries) will indeed transform when we change basis, but the change is always trivial if the basis you're changing to is orthonormal. If you change to a non-orthonormal basis, the numbers inside the matrices will change. It only seems like the gammas don't transform because most of the time you only ever deal with Lorentz transformations in physics, which move from one orthonormal basis to another orthonormal basis. But if you, say, double the length of the basis vectors (that's a more pure-math transformation, not one you'd see in physics), then the gamma matrix entries will also multiply by 2 (and the metric components will multiply by 4) as a result. I think the reason γ^0 matrix different in the "Weyl" basis and the "Dirac" basis because it's a literal change of basis-spinors in the target S⊗S space. Actually, thinking of it this way, it makes sense to me that γ^μ is a basis vector and γ^μ_ab is a set of coefficients for a linear map from V to S⊗S when you map the spacetime vector γ^μ to the spinor-pair s_a⊗s_b. The fact that they use the same gamma symbol for both basis vectors and the linear map coefficients is what throws me off. One final comment on the "1 vector index = 2 spinor indices" bit: it's possible to re-write all of relavitiy using spinors instead of tensors... you just double the number of indices for everything (so 4-vectors have 2 spinor indices, the metric has 4 spinor indices, the Riemann tensor has 8 spinor indices, etc.). It's just an alternative way of writing the same objects.
As someone only in my 2nd semester of physics and in calculus 3, this is equal part terrifying and equal part intriguing. There's soooo much I don't know yet
You should definitely read the first chapter of the book by Kristen and Müller titled "Introduction to Supersymmetry". The first chapter deals with the properties of Lorentz and Poincaré Group and their representations in detail. I think you will find a very satisfactory answer to your question about gamma matrices.
I think QFT is a kind of subject that one learns over a long period of time. I have already taken one and two based on peskin but I still have to take time everyday to learn it well
Big Dotson looks like you're enjoying classes now that you're finally over the hump 🔥. Dietrich labs has some great videos on QFT. Wish I took this stuff before leaving grad school. 😩
Yo Andrew, you should check out geometric algebra! It gives a really clean interpretation of the gamma matrices as a matrix representation of the basis vectors of spacetime. Similarly, the Pauli matrices are just the x, y, and z basis vectors of space. The slash derivative is just a 4-gradient (or d'alembertian - I forget)!
The operator in the Dirac equation is an example of a Dirac Operator, something related to the study of Clifford algebras and spinor bundles. If you really want to understand the math behind this operator you will need to get acquainted with concepts like fibre bundles, etc. Nakahara's 'Geometry, Topology and Physics' is probably a good start.
there's an anti commutation relation which kinda relates the gamma matrices to the space-time metric:: eta_{\mu u}. So the spin space is in some way related to space time and the gamma matrices transform under the spin representation of the Lorentz group. Gamma_{mu} itself doesn't transform as a normal 4-vector, but the particular combination (psybar gamma^mu psy) you see in the Dirac lagrangian does transform as a normal spacetime 4 vector, so the 'mu' index you see in this particular combination (psybar gamma^mu psy,) corresponds to a real space time index, but when you see a just gamma^mu, then I wouldn't think of the 'mu' being a spacetime index, but then again i'm no QFT expert. I am simply a level 7.5 QFT boi xp
A string is revealed to be a twisted cord when viewed up close. Both Matter and Energy described as "Quanta" of Spatial Curvature. Is there an alternative interpretation of "Asymptotic Freedom"? What if Quarks are actually made up of twisted tubes which become physically entangled with two other twisted tubes to produce a proton? Instead of the Strong Force being mediated by the constant exchange of gluons, it would be mediated by the physical entanglement of these twisted tubes. When only two twisted tubules are entangled, a meson is produced which is unstable and rapidly unwinds (decays) into something else. A proton would be analogous to three twisted rubber bands becoming entangled and the "Quarks" would be the places where the tubes are tangled together. The behavior would be the same as rubber balls (representing the Quarks) connected with twisted rubber bands being separated from each other or placed closer together producing the exact same phenomenon as "Asymptotic Freedom" in protons and neutrons. The force would become greater as the balls are separated, but the force would become less if the balls were placed closer together. Therefore, the gluon is a synthetic particle (zero mass, zero charge) invented to explain the Strong Force. An artificial Christmas tree can hold the ornaments in place, but it is not a real tree. String Theory was not a waste of time, because Geometry is the key to Math and Physics. However, can we describe Standard Model interactions using only one extra spatial dimension? What did some of the old clockmakers use to store the energy to power the clock? Was it a string or was it a spring? What if we describe subatomic particles as spatial curvature, instead of trying to describe General Relativity as being mediated by particles? Fixing the Standard Model with more particles is like trying to mend a torn fishing net with small rubber balls, instead of a piece of twisted twine. Quantum Entangled Twisted Tubules: “We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct.” Neils Bohr (lecture on a theory of elementary particles given by Wolfgang Pauli in New York, c. 1957-8, in Scientific American vol. 199, no. 3, 1958) The following is meant to be a generalized framework for an extension of Kaluza-Klein Theory. Does it agree with some aspects of the “Twistor Theory” of Roger Penrose, and the work of Eric Weinstein on “Geometric Unity”, and the work of Dr. Lisa Randall on the possibility of one extra spatial dimension? During the early history of mankind, the twisting of fibers was used to produce thread, and this thread was used to produce fabrics. The twist of the thread is locked up within these fabrics. Is matter made up of twisted 3D-4D structures which store spatial curvature that we describe as “particles"? Are the twist cycles the "quanta" of Quantum Mechanics? When we draw a sine wave on a blackboard, we are representing spatial curvature. Does a photon transfer spatial curvature from one location to another? Wrap a piece of wire around a pencil and it can produce a 3D coil of wire, much like a spring. When viewed from the side it can look like a two-dimensional sine wave. You could coil the wire with either a right-hand twist, or with a left-hand twist. Could Planck's Constant be proportional to the twist cycles. A photon with a higher frequency has more energy. ( E=hf, More spatial curvature as the frequency increases = more Energy ). What if Quark/Gluons are actually made up of these twisted tubes which become entangled with other tubes to produce quarks where the tubes are entangled? (In the same way twisted electrical extension cords can become entangled.) Therefore, the gluons are a part of the quarks. Quarks cannot exist without gluons, and vice-versa. Mesons are made up of two entangled tubes (Quarks/Gluons), while protons and neutrons would be made up of three entangled tubes. (Quarks/Gluons) The "Color Charge" would be related to the XYZ coordinates (orientation) of entanglement. "Asymptotic Freedom", and "flux tubes" are logically based on this concept. The Dirac “belt trick” also reveals the concept of twist in the ½ spin of subatomic particles. If each twist cycle is proportional to h, we have identified the source of Quantum Mechanics as a consequence twist cycle geometry. Modern physicists say the Strong Force is mediated by a constant exchange of Gluons. The diagrams produced by some modern physicists actually represent the Strong Force like a spring connecting the two quarks. Asymptotic Freedom acts like real springs. Their drawing is actually more correct than their theory and matches perfectly to what I am saying in this model. You cannot separate the Gluons from the Quarks because they are a part of the same thing. The Quarks are the places where the Gluons are entangled with each other. Neutrinos would be made up of a twisted torus (like a twisted donut) within this model. The twist in the torus can either be Right-Hand or Left-Hand. Some twisted donuts can be larger than others, which can produce three different types of neutrinos. If a twisted tube winds up on one end and unwinds on the other end as it moves through space, this would help explain the “spin” of normal particles, and perhaps also the “Higgs Field”. However, if the end of the twisted tube joins to the other end of the twisted tube forming a twisted torus (neutrino), would this help explain “Parity Symmetry” violation in Beta Decay? Could the conversion of twist cycles to writhe cycles through the process of supercoiling help explain “neutrino oscillations”? Spatial curvature (mass) would be conserved, but the structure could change. ===================== Gravity is a result of a very small curvature imbalance within atoms. (This is why the force of gravity is so small.) Instead of attempting to explain matter as "particles", this concept attempts to explain matter more in the manner of our current understanding of the space-time curvature of gravity. If an electron has qualities of both a particle and a wave, it cannot be either one. It must be something else. Therefore, a "particle" is actually a structure which stores spatial curvature. Can an electron-positron pair (which are made up of opposite directions of twist) annihilate each other by unwinding into each other producing Gamma Ray photons? Does an electron travel through space like a threaded nut traveling down a threaded rod, with each twist cycle proportional to Planck’s Constant? Does it wind up on one end, while unwinding on the other end? Is this related to the Higgs field? Does this help explain the strange ½ spin of many subatomic particles? Does the 720 degree rotation of a 1/2 spin particle require at least one extra dimension? Alpha decay occurs when the two protons and two neutrons (which are bound together by entangled tubes), become un-entangled from the rest of the nucleons . Beta decay occurs when the tube of a down quark/gluon in a neutron becomes overtwisted and breaks producing a twisted torus (neutrino) and an up quark, and the ejected electron. The production of the torus may help explain the “Symmetry Violation” in Beta Decay, because one end of the broken tube section is connected to the other end of the tube produced, like a snake eating its tail. The phenomenon of Supercoiling involving twist and writhe cycles may reveal how overtwisted quarks can produce these new particles. The conversion of twists into writhes, and vice-versa, is an interesting process, which is also found in DNA molecules. Could the production of multiple writhe cycles help explain the three generations of quarks and neutrinos? If the twist cycles increase, the writhe cycles would also have a tendency to increase. Gamma photons are produced when a tube unwinds producing electromagnetic waves. ( Mass=1/Length ) The “Electric Charge” of electrons or positrons would be the result of one twist cycle being displayed at the 3D-4D surface interface of the particle. The physical entanglement of twisted tubes in quarks within protons and neutrons and mesons displays an overall external surface charge of an integer number. Because the neutrinos do not have open tube ends, (They are a twisted torus.) they have no overall electric charge. Within this model a black hole could represent a quantum of gravity, because it is one cycle of spatial gravitational curvature. Therefore, instead of a graviton being a subatomic particle it could be considered to be a black hole. The overall gravitational attraction would be caused by a very tiny curvature imbalance within atoms. In this model Alpha equals the compactification ratio within the twistor cone, which is approximately 1/137. 1= Hypertubule diameter at 4D interface 137= Cone’s larger end diameter at 3D interface where the photons are absorbed or emitted. The 4D twisted Hypertubule gets longer or shorter as twisting or untwisting occurs. (720 degrees per twist cycle.) How many neutrinos are left over from the Big Bang? They have a small mass, but they could be very large in number. Could this help explain Dark Matter? Why did Paul Dirac use the twist in a belt to help explain particle spin? Is Dirac’s belt trick related to this model? Is the “Quantum” unit based on twist cycles? I started out imagining a subatomic Einstein-Rosen Bridge whose internal surface is twisted with either a Right-Hand twist, or a Left-Hand twist producing a twisted 3D/4D membrane. This topological Soliton model grew out of that simple idea. I was also trying to imagine a way to stuff the curvature of a 3 D sine wave into subatomic particles. .--------
I'd be really interested in your take on what the heck renormalization is all about and why we're able to get away with it. We blew through path integrals in like 3 weeks (not recommended) in my "Special topics in QM" class, and it seemed reallllllly sus to me. Like, assigning finite values to infinite products level of sus.
Did you read this book ? Ashok Das, Susumu Okubo - Lie Groups and Lie Algebras for Physicists-World Scientific Publishing Company. You should be learn lie and Lorentz algebra before start to QFT1
I dont get your curricula. In our theoretical physics program we did GR (Carroll) and QM on the 3rd year of bachelor, and QFT 1,2,3, string theory, lattice field theory, AdS/CFT, on the 4th and 5th year during masters.
On the same boat too. Took differentiable manifolds and general relativity and found it to be really elegant and clear mathematically. ON THE OTHER HAND, QFT just doesn't make any sense to me, path integral is somewhat fine, second quantisation was just so random (maybe it's just me, but I can't find any reasoning behind why to do so). I always have an image (I know it's very wrong to say this) of QFT being a large piece of cloth with so many patches and stitches, that it's so hard to see what it's suppose to be. It's like mumbo jumbo being thrown together. Does it works? Yeah. Does it have a good mathematical formulation? I don't think so, but please correct me and inspire me.
Hi, there Andy!!! Try Klauber's QFT vol 1. You will understand everything in QFT. His QFT-2 is appearing soon, maybe by the end of the year. I don't want to boast but we are pals and that's what he told me. The title is A Student Friendly Introduction to Quantum Field Theory.
Reading your note makes me anxious, eventhough I think that have some insight about QFT 2nd quantization. I highly recommend David Tong's lecture note and Schwartz's QFT book
Here I am feeling confident as an undergrad freshman self teaching my self Intro to ED by Griffiths and learning tensor calc and I thought I was smart until I didn’t understand anything on my mans hw except the Dirac equation by itself😒
If you really want to get into the nitty-gritty part of Renormalisation in scalar field theory, you should consider "Renormalization and Effective Field Theory" by Kevin Costello. Should you be more interested in really understanding the mathematical mechanics behind Field Theory in general, especially gauge fixing, the BV-formalism might be a good goal. Hmu if you need an introductory script on the mathematical foundations of scalar and general QFT entailing renormalisation and the BV formalism ;)
From the homework example you discuss around 3:00, it looks like your difficulties are with the metaphysics, or at least ontology, not the physics! Some things tie you up in knots (in ancient Greek, aporia), and sorting the difference between physical entities and their representations is a crucial one. I don't know what the real issues are with your example or the rest of QFT, but it sounds like category-theoretic tools, not just groups but groupoids and algebroids, might be relevant. I'm not a physicist, but I am a fan of the nlab wiki on everything n-categorical.
I wonder how will you explain renormalization/regularization in 30 minutes. That will certainly be interesting, at least if you don't want to just give some handwavy overview of the idea. Maybe you can just talk about the running of the coupling constants... 🤔
In my simple world, the γ matrices are constant matrices that do not change under any transformation. Transformations only ever change the fields themselves. What makes ψbar γ^μ ψ a vector is the combined transformation of ψ and ψbar. Vectors have many properties that γ doesn't share; for example γ^μ γ^ν is not a symmetric tensor. Of course this is just my preferred way of looking at it.
My E&M course is only covering chapters 1-7 of Griffiths (the reasons are complicated). Which sections should I self-study in preparation for grad school and the PGRE?
If you want to know the details lool to Weinberg's bool, volume 1. What you want to know about gamma matrix is the chapter 5.4 . It's really the best book if you want to understand why qft is this way and what are the details.
Andrew, could you make video or give us your skript of your lesson in divergent series renormalization. I am asking since i am a math student doing my master degree and I wrote my bachelor thesis about how to renormalize alternating divergent series, by integrating Taylor series. Best regards.
Clifford algebra helped me to better understand the Dirac matrices. There is a nice book by Hestenes. The other book that helped was Cartan's book on spinors.
@Andrew Dotson I am curious about how you manage your finances as a phd student? Like do you earn money while studying, is it sufficient for a reasonable standard of living?
Hello Andrew I'm waiting for your math and physics video. I thought you start to teach QFT to us (please do it) ... it seems you star! (Please please start to teach QFT) Thank you
Maybe it helps to think of the Dirac gamma as a sort of 'metric' or transformation from the spin representation space to the frame bundle? In this way it acts like a (1,0) Lorentz tensor but also a (1,1) spin tensor, and the spin transformations carry a spin and Lorentz index as well. Maybe this is t a new thought to you but in this context it seems like a useful way to interpret the nature of your spinor parts
Hi Andrew, love your work ethic. I have QFT this term and an exam at the end of the year, which textbooks do you recommend I read to get a better intuition of the concepts? Thanks :D
Seems really exciting, knowing I was too stupid for anything beyond the basic theoretical physics lectures ^^ Settled for teaching and miss uni every day.
Interesting and worthwhile video. Most second or third year grad students in your field have significant familiarity with renormalization techniques. You'll find some good overview papers on said techniques on the arXiv.
Learning delta function is the most accurate description I have ever heard
Hello there!
The references/insider puns are way above my grade!
Hello there!
@@genghisgalahad8465 basically a Delta function is something that is zero everywhere, except for one point where it spikes to infinity. He’s saying the learning curve is as steep as it gets
@@martinjose6273 U a bot or a real person? XD
9th!! Love, Mom 💕💕
Awww
Your son is a great physicist , god bless you Andrew and your family .
@@downsundrome1645 many thanks. Have a great Holiday!
Andrew lives in a non-linear timeline where you have to take E&M, then Classical Mechanics, then GR, then QFT 2, then kinematics, then QFT1.
I can’t wait to take pre calculus next semester!
@@AndrewDotsonvideos Oh boy wait till you see algebra
@@ianism1103 hardcore lie algebra is no joke!
@@AndrewDotsonvideos teach me how to do quantum
@@icodestuff6241I remember the naive bygone days of “place value” and “calculating the determinant of binary relativistic electroweak flux given an alpha of 2=e=pi+1=g-8”.Bygone are those days; poor is the understanding I think I have. All that remains (save one island of hope that is *Reading Eggs)* is the dust of all great intellectual studies from Eensteen’s plagiarism of Newtonian mechanics to Tesla’s Pigeonhole Paradox to even the great Plateo’s (a humble brand of crunchy mind-numbing cereal; an indivisible atomi of a philosophers balanced diet): Yes, Indeed, all that remains (deem it the greatest triumph of man or the steepest fall from grace) is
*NUMBER BONDS.*
3:00 ah yes I got the same thing.
Wait you also study QFT?
@@omniwazowski5075 it’s a joke
You probably know more about this stuff than I do at this point with your videos on group theory 😂
@@AndrewDotsonvideos When is the next meme review?
Hi Andrew, thank you so much for inspiring me to change my degree and swap to physics. Best decision I've ever made :)
Welcome to the dark side:)
Physics is the best man. A complete joy to learn.
@@AndrewDotsonvideos You aren't wrong. Most of stuff in Universe is Dark.
gEt a LoAD oF tHiS gUy
What a nerd
Once you've finished the course, could you make a compare and contrast video between the 2nd quantization and Feynman path integral formulations of QFT? I've always felt super weird using the latter
That’s a great idea
@@AndrewDotsonvideos what is the max level physics
@@AndrewDotsonvideos can you teach max level physics
2nd quantization was so ratchet lmao
@@masternobody1896 if you refer to it as “max level physics,” you aren’t ready to learn “max level physics.”
The video quality is strangely professional
Thanks!
QFT's making him more of a Chad that he was before
Look at the dudes neck man
What about NFTs
Hey guys, I just want to share my favourite physics book which covers the topics which Andrew mentioned to be difficult to grasp. The book is 'Physics from Symmetry' by Jakob Schwichtenberg. It covers group representation in different dimensions and Lie Algebra of SO(3) and Lorentz group pretty well. My favourite of this book is the way it covers Classic field theory. He derives the dirac equation and gamma matrices for spinor fields, Klien Gordon equation for scalar fields and Proca equation for vector fields and He makes it look very simple. I highly recommend that book to all Physics Aspirants.
I was a number theorist by training, and I took a first course to QFT, 2 years ago. Needless to say, the vacuum of rigor was quite shocking to me at the time. Now I have a better sense of how to approach the subject, partly by seeing it as a collection of calculation techniques held together by heuristics.
Woah. As a Physics Student, seeing someone saying that QFT lacks mathematical rigor is mind blowing to me. What are you researching on right now btw?
“Calculation techniques held together by heuristics” That would be a sick burn to give after someone’s theory talk haha.
@@AndrewDotsonvideos yah, but Uzu Lim is not wrong. If you ever try to practice QFT with rigour you will die of rigourmortus before contributing anything useful to physics.
@@shivanshusiyanwal296 It is not lacking in mathematical rigour really. It is lacking in tight axiomatic development. No serious quantum field theorist will tolerate any mathematical mistakes, so you have 100% rigour in that sense, but when procedures are not well defined and loads of approximations are done, to a mathematician this _looks like_ breakdown of axiomatic development. They are right, but so what? If physicists were always waiting for a mathematician to let them in the door we'd get nowhere.
"Rigorous" in mathematics usually means having all of the objects well-defined and each inference being supported by an airtight mathematical proof. In that sense, I heard that many calculations in QFT are not "mathematically rigorous". Or at least, a first course certainly doesn't come with a full set of well-defined objects; taking integrals of annihilation / creation operators with its Dirac deltas are not dealt with the appropriate axiomatic framework that a mathematics students would normally want to refer to.
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I'll mention, however, that quite a large collection of QFT material have been mathematically formalized, see for example "PCT, Statistics, and All That" and modern mathematical treatments of Feynman integral, e.g. works by Matilde Marcolli.
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I'm not sure what exactly are the concepts in QFT that haven't been fully formalized mathematically at the moment (2021). I think the Yang-Mills existence and Mass Gap problem is one example showing that the theory isn't fully mathematically supported, but I'm actually not sure. The main point I wanted to get across is that a first course in QFT is unfriendly to a student trained in the typical way of pure mathematics.
The classification of particles via unitary irreps of the Poincaré group is honestly to me one of the most mindblowing things in physics. It's amazing that we can show that quantities such as mass and spin (as well as other charges) are necessary properties of fundamental states that obey the symmetries of the universe. So elegant! That's why I like the "particle-to-field" approach Weinberg takes in his QFT Vol. 1. That book is great for getting a firm grasp on the foundations of QFT.
"but I'll spend hours thinking about why that step makes sense" literally me when it comes to A levels Physics. Glad to know I'm not alone with this habit.
A lot of things will start making even more sense once you get to higher level Physics (even more so if you do an undergrad Physics). But soon enough everything will get messed up again and you'll go back to thinking about why these things make sense, and the cycle repeats!
We did second quantization in our bachelors hahahaha one of the earliest homeworks was to prove the bogoliougov transformations. A true learning delta function
That delta function joke is hella funny I LOLed at that!
For a more rigorous mathematical view (for example, constructing classical Yang-Mills theory, but heaps of other stuff too) I would strongly recommend Mark Hamilton - Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics
Super good book. Without it my thesis would be mush
Hey, i just wanna say thank you, you really are the one reason i got so invested in math and physics in 8th grade, I’m now planning to do a physics degree pretty much as a result of watching your videos
Concerning 04:05, equation (13) in your PDF tells us that the combination \bar\psi\gamma^\mu\psi (one type of Dirac bilinear) transforms as a four-vector. If you take eq. (13) and put a \bar\psi on the left and a \psi on the right (on both sides of the equation), then you get the equivalent of x^\mu = \Lambda^\mu_
u x^
u. This is because \psi transforms with \Lambda_1/2 and \bar\psi with \Lambda_1/2^\dagger under a Lorentz transformation (also note that the \gamma^0 inside the \bar\psi is necessary for this to work since the \Lambda_1/2 are not unitary/the generators are not Hermitian. For more information, see the discussion above eq. (3.33) in Peskin&Schroeder). Looking forward to your next video on QFT ☺️👍
Hey thanks for the comment! If I understand your comment properly, it looks like you're saying how to construct vectors fromspinors and gamma matrices, but I was moreso trying to understand if gamma mu itself is a proper four-vector. More concretely, I figured the Lorentz transform see's all indices and transforms them accordingly. To transform the gamma matrices, one would need to hit the lorentz index with the vector rep of the Lorentz transform, and then hit once with the spin 1/2 rep (one of them inverse) for each spin index. This just solves the isomorphism relation (13) for gamma mu, but the combination of all 3 Lorentz transformations leaves the gamma matrix unchanged which was why I started calling it an "invariant four-vector", but you can't just transform the Lorentz index and think you're done, whereas other people might say that the isomorphism is not a transformation rule, but just an identity relating representations. I'm not convinced my interpretation is wrong just yet because it's obvi their anti-commutator gives something proportional to the metric which is also Lorentz invariant in flat space (times an identity matrix in spin space).
@@AndrewDotsonvideos Oh, I see. I usually think about the gamma matrices that they do not transform under Lorentz transformations, since they're just Pauli matrices (i.e., numbers) in disguise :p However, they bring along an identity which we can use to ensure that things like \bar\psi \gamma^\mu \partial_\mu \psi transform properly. Your approach of viewing the gamma matrices as objects with three indices that each get their own Lorentz transformation matrix (in their resp. representation) is interesting though! Maybe I'll have to think about this a bit more..
Hey Andrew, I’m taking QFT this term too! We haven’t got up to the Dirac equation yet though
Praise the sun \[T]/
Me too! Which textbooks do you recommend I read to get a better intuition of the concepts? Thanks :D
@@notspc1548 Try Ryder, Quantum Field Theory, I'm taking QFT this semester and my professor said it's a quite good book to study from.
The one guy that disliked has no clue what you are talking about and is jealous that all of us (likes) know what your are talking about.
Every time I watch one of these explanation videos I'm both fascinated and horrified that there's so much complex physics out there. And I'm in my final year of undergrad
Congrats! Just say QFT 50 times before every test instead of actually studying. I hear it makes you smarter.
Didn't even see the cat on the shelf at first. 'Learning delta function' lol
I hope that's sarcastic.
@@shivanshusiyanwal296 Just saying the name of the class literally makes you a smarter person. I didn't even study in undergrad, I said quantum field theory 4000 times and here I am.
Already 4 years in? Time is flying so fast
I’m glad I’m not the only one who spends a lot of time on trying to understand every single step in a proof
the gamma matrices are just matrices, they have no transformation laws under the lorentz group (so are technically scalars). the \mu is just a label. when stuck inside of spinors, however, for example j^\mu = \overline{\psi} \gamma^\mu \psi, they mix up the spinor degrees of freedom in such a way that you get a vector current, that is j^\mu transforms like a vector.
they do a similar thing in the dirac equation, by mixing up the spinor degrees of freedom of psi, they make a vector to contract with the covector \partial_\mu. so i think i agree with your interpretation of equation 13, in that they bridge two representations of the Lorentz group. that is an interesting way of thinking about it.
If you haven't seen these, I can highly recommend the lectures given by Carl Bender at PSI in 2012, ruclips.net/p/PLOFVFbzrQ49TNlDOxxCAjC7kbnorAR1MU. These 15 lectures cover a lot of how to solve differential equations found in QFT using divergent series and asymptotics.
Where can I read ur Research paper sir?
Sir I have also written Research paper on graviton ..How can I send u?
Probably the most rigorous way of talking about gamma-matrices is that it is an intertwining operator (which in this case is just fancy word for isomorphism) between two representations of so(3,1) -- (1/2,0)\otimes(0,1/2) and vector one.
"Intertwining operator" Is new to me, thanks!
That tensor playlist is being so helpful, I stated studying physics like a month ago (all by myself) and it's a nice challenge while I'm practicing more basic calculus since I took those classes eons ago. Thanks for making it!
We had to study two different textbook for the QFT class. One used subscripted gamma another used superscripted gamma. Rules of commutation were different for different kind of gamma. I spent huge amount of time in figuring out similarities and differences between these gammas. Beside this, QFT class went really well.
Will you get a Ph.D or a master's degree when you finish your graduate school?
He’ll be getting his PHD :)
Thank you Ms. Dotson@@maureendotson4634
QFT is a massive subject. The year of graduate courses is really the tip of the iceberg. This area of physics has become deeply mathematical and has blown up over the past few decades in very unexpected ways. This is a subject that mathematicians and physicists are collaborating on now more than ever.
hey Andrew
nice work as usual ,
i wanted to ask what do u do for living ?
Im with you there Andrew when you say you spend too long thinking about points of a question that aren't the main focus.
I actually had a question for an assignment of mine recently to do with 2 body central forces, and long story short the textbook I was reading (which had the process to partially solve the question) merely stated some constants without proving them. Most of my classmates accepted that constant and moved on but I was "who what why how dare you assume"....
Spent the next hour or 2 deriving that constant and coding into Latex. Gotta explain everything and not assume stuff man.
Yep, canonical quantisation has a very steep learning curve
Hey, there is a cat in some collapsed state on the background.
Second quantization feels much more natural to me, and as a solid state theorist it seems more applicable. Still pretty shaky on the path integral formalism
Have you read Altland and Simons? It bridges the gap between the canonical formalism and the path integral in solid state theory.
@@twistedsector unfortunately my current work is in semiclassical picture, but I will check it out. I'm hoping to extend this current project to include more pure quantum phenomena
There is a book: *Relativistic quantum mechanics. Wave equations* - _Greiner_ ...
Lol man I'm in my 4th year of undergrad (astrophysics specialization) and this very thing you're saying you do plagues me; I'm the same way. Maybe not to the degree you are or maybe I'm not going through exactly what you are but I'm always being ocd about the nit picky details about stuff like that (that's not necessarily related) and I'm over here then wondering why it takes me way longer to get my assignment done than other people.
That means you really care.
Second quantization: So basically, creation and annihilation operator
Learning delta function made me chuckle. That's a good one.
The gammas confuse me too, because there seems to be multiple ways to interpret them. On one hand, I think you're basically correct when you say they're "bridging the two representations of the Lorentz group"... You might be interested in reading about the "Van der Waerden symbols": en.wikipedia.org/wiki/Infeld%E2%80%93Van_der_Waerden_symbols. In some ways, this is just a fancy name for the sigma matrices... but I think the article emphasizes that the sigma matrices are really a mapping from a vector space V (where vector/tensor-like objects live) to a pair of spinor spaces S⊗S (where spinor pairs live). In other words, "1 vector index" is equivalent to "2 spinor indices", and the symbols give you the right coefficients for converting between tensor and spinor indices. I think this is what people are getting at when they say "a spinor is like the square root of a vector", because a product-pair of spinors is equivalent to a vector, via these symbols.
But you can also view the 4 gammas as the 4 spacetime basis vectors in the Clifford Algebra Cl(1,3). When you do a summation of coefficients and gammas over spacetime indices, you're really just writing a linear combination for a spacetime 4-vector... One thing that's still very confusing to me about Clifford Algebras is that scalars, vectors, spinors, and rotation operators all live together inside the same algebra... so you can add scalars and vectors together (that's basically what a quaternion is), or multiply vectors and spinors together, as when you multiply the dirac spinor "psi" by one of the gammas. But being able to churn through additions/multiplications in CA doesn't really illuminate what's actually going on, unfortunately. I wish I understood this stuff better.
Yeah that sounds reasonable but I really don't understand insisting that transformations don't apply to them (not that you were necessarily implying that). The anticommutator {\gamma^\mu,\gamma^
u}=2g^{\mu
u}*I seems to imply that they do transform, they just transform trivially just like the metric in flat space. If that assumption is true, which I'm having a really hard time convincing myself that it isn't, then the isomorphism would have to count as defining the full transformation (of both lorentz and spin indices) of the gammas I think.
""1 vector index" is equivalent to "2 spinor indices", and the symbols give you the right coefficients for converting between tensor and spinor indices. I think this is what people are getting at when they say "a spinor is like the square root of a vector"": That comment blew my mind more than it had any right to.
@@AndrewDotsonvideos So I think that anti-commutator formula makes a lot more sense when you use the "gammas = spacetime basis vectors" interpretation. The standard way of defining the metric components is using the dot products of the basis vectors, so g_μν = e_μ·e_ν. Since the dot product is commutative, this could also be written as 2g_μν = (e_μ·e_ν) + (e_ν·e_μ) = {e_μ,e_ν} ... which is basically identical to the anti-commutator formula you wrote. Now, I'm leaving out the detail that the product of gammas isn't EXACTLY the same as the dot product... the product of gammas is actually the "Clifford product", aka the "dot-product-plus-the-wedge-product": (γ^μ)(γ^ν) = (γ^μ·γ^ν) + (γ^μ ∧ γ^ν), but the wedge product part goes to zero in the anti-communtator since the wedge product gets a negative sign when we swap the order in the anti-communtator formula. So I think the spinnor components of the gammas (i.e. the matrix entries) will indeed transform when we change basis, but the change is always trivial if the basis you're changing to is orthonormal. If you change to a non-orthonormal basis, the numbers inside the matrices will change. It only seems like the gammas don't transform because most of the time you only ever deal with Lorentz transformations in physics, which move from one orthonormal basis to another orthonormal basis. But if you, say, double the length of the basis vectors (that's a more pure-math transformation, not one you'd see in physics), then the gamma matrix entries will also multiply by 2 (and the metric components will multiply by 4) as a result. I think the reason γ^0 matrix different in the "Weyl" basis and the "Dirac" basis because it's a literal change of basis-spinors in the target S⊗S space. Actually, thinking of it this way, it makes sense to me that γ^μ is a basis vector and γ^μ_ab is a set of coefficients for a linear map from V to S⊗S when you map the spacetime vector γ^μ to the spinor-pair s_a⊗s_b. The fact that they use the same gamma symbol for both basis vectors and the linear map coefficients is what throws me off.
One final comment on the "1 vector index = 2 spinor indices" bit: it's possible to re-write all of relavitiy using spinors instead of tensors... you just double the number of indices for everything (so 4-vectors have 2 spinor indices, the metric has 4 spinor indices, the Riemann tensor has 8 spinor indices, etc.). It's just an alternative way of writing the same objects.
I'm starting a grad program in particle physics, and am currently teaching myself QFT as well! Great to know I'm in good company.
As someone only in my 2nd semester of physics and in calculus 3, this is equal part terrifying and equal part intriguing. There's soooo much I don't know yet
You should definitely read the first chapter of the book by Kristen and Müller titled "Introduction to Supersymmetry". The first chapter deals with the properties of Lorentz and Poincaré Group and their representations in detail. I think you will find a very satisfactory answer to your question about gamma matrices.
Currently doing the canonical route. I don't know how to feel about it, Im finding it interesting while simultaneously wanting to pull my hair out.
Learning delta function really got me. Both a great joke and a really apt description of how it feels to get into many advances topics.
I started learning qft last week after seeing your other video on deriving the Dirac equation. This is gonna be fun😃
I think QFT is a kind of subject that one learns over a long period of time. I have already taken one and two based on peskin but I still have to take time everyday to learn it well
Congratulations, Dr Dotson, now you are almost an expert biologist :D
Big Dotson looks like you're enjoying classes now that you're finally over the hump 🔥. Dietrich labs has some great videos on QFT. Wish I took this stuff before leaving grad school. 😩
His videos are great for sure
Hey Andrew do you still workout?
4:04 Sorry but you got it all wrong. (I’m 9 yrs old and 6’7” btw)
This is my favorite comment
Yo Andrew, you should check out geometric algebra! It gives a really clean interpretation of the gamma matrices as a matrix representation of the basis vectors of spacetime. Similarly, the Pauli matrices are just the x, y, and z basis vectors of space. The slash derivative is just a 4-gradient (or d'alembertian - I forget)!
The operator in the Dirac equation is an example of a Dirac Operator, something related to the study of Clifford algebras and spinor bundles. If you really want to understand the math behind this operator you will need to get acquainted with concepts like fibre bundles, etc. Nakahara's 'Geometry, Topology and Physics' is probably a good start.
there's an anti commutation relation which kinda relates the gamma matrices to the space-time metric:: eta_{\mu
u}. So the spin space is in some way related to space time and the gamma matrices transform under the spin representation of the Lorentz group. Gamma_{mu} itself doesn't transform as a normal 4-vector, but the particular combination (psybar gamma^mu psy) you see in the Dirac lagrangian does transform as a normal spacetime 4 vector, so the 'mu' index you see in this particular combination (psybar gamma^mu psy,) corresponds to a real space time index, but when you see a just gamma^mu, then I wouldn't think of the 'mu' being a spacetime index, but then again i'm no QFT expert. I am simply a level 7.5 QFT boi xp
A string is revealed to be a twisted cord when viewed up close.
Both Matter and Energy described as "Quanta" of Spatial Curvature.
Is there an alternative interpretation of "Asymptotic Freedom"? What if Quarks are actually made up of twisted tubes which become physically entangled with two other twisted tubes to produce a proton? Instead of the Strong Force being mediated by the constant exchange of gluons, it would be mediated by the physical entanglement of these twisted tubes. When only two twisted tubules are entangled, a meson is produced which is unstable and rapidly unwinds (decays) into something else. A proton would be analogous to three twisted rubber bands becoming entangled and the "Quarks" would be the places where the tubes are tangled together. The behavior would be the same as rubber balls (representing the Quarks) connected with twisted rubber bands being separated from each other or placed closer together producing the exact same phenomenon as "Asymptotic Freedom" in protons and neutrons. The force would become greater as the balls are separated, but the force would become less if the balls were placed closer together. Therefore, the gluon is a synthetic particle (zero mass, zero charge) invented to explain the Strong Force. An artificial Christmas tree can hold the ornaments in place, but it is not a real tree.
String Theory was not a waste of time, because Geometry is the key to Math and Physics. However, can we describe Standard Model interactions using only one extra spatial dimension? What did some of the old clockmakers use to store the energy to power the clock? Was it a string or was it a spring?
What if we describe subatomic particles as spatial curvature, instead of trying to describe General Relativity as being mediated by particles? Fixing the Standard Model with more particles is like trying to mend a torn fishing net with small rubber balls, instead of a piece of twisted twine.
Quantum Entangled Twisted Tubules:
“We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct.” Neils Bohr
(lecture on a theory of elementary particles given by Wolfgang Pauli in New York, c. 1957-8, in Scientific American vol. 199, no. 3, 1958)
The following is meant to be a generalized framework for an extension of Kaluza-Klein Theory. Does it agree with some aspects of the “Twistor Theory” of Roger Penrose, and the work of Eric Weinstein on “Geometric Unity”, and the work of Dr. Lisa Randall on the possibility of one extra spatial dimension? During the early history of mankind, the twisting of fibers was used to produce thread, and this thread was used to produce fabrics. The twist of the thread is locked up within these fabrics. Is matter made up of twisted 3D-4D structures which store spatial curvature that we describe as “particles"? Are the twist cycles the "quanta" of Quantum Mechanics?
When we draw a sine wave on a blackboard, we are representing spatial curvature. Does a photon transfer spatial curvature from one location to another? Wrap a piece of wire around a pencil and it can produce a 3D coil of wire, much like a spring. When viewed from the side it can look like a two-dimensional sine wave. You could coil the wire with either a right-hand twist, or with a left-hand twist. Could Planck's Constant be proportional to the twist cycles. A photon with a higher frequency has more energy. ( E=hf, More spatial curvature as the frequency increases = more Energy ). What if Quark/Gluons are actually made up of these twisted tubes which become entangled with other tubes to produce quarks where the tubes are entangled? (In the same way twisted electrical extension cords can become entangled.) Therefore, the gluons are a part of the quarks. Quarks cannot exist without gluons, and vice-versa. Mesons are made up of two entangled tubes (Quarks/Gluons), while protons and neutrons would be made up of three entangled tubes. (Quarks/Gluons) The "Color Charge" would be related to the XYZ coordinates (orientation) of entanglement. "Asymptotic Freedom", and "flux tubes" are logically based on this concept. The Dirac “belt trick” also reveals the concept of twist in the ½ spin of subatomic particles. If each twist cycle is proportional to h, we have identified the source of Quantum Mechanics as a consequence twist cycle geometry.
Modern physicists say the Strong Force is mediated by a constant exchange of Gluons. The diagrams produced by some modern physicists actually represent the Strong Force like a spring connecting the two quarks. Asymptotic Freedom acts like real springs. Their drawing is actually more correct than their theory and matches perfectly to what I am saying in this model. You cannot separate the Gluons from the Quarks because they are a part of the same thing. The Quarks are the places where the Gluons are entangled with each other.
Neutrinos would be made up of a twisted torus (like a twisted donut) within this model. The twist in the torus can either be Right-Hand or Left-Hand. Some twisted donuts can be larger than others, which can produce three different types of neutrinos. If a twisted tube winds up on one end and unwinds on the other end as it moves through space, this would help explain the “spin” of normal particles, and perhaps also the “Higgs Field”. However, if the end of the twisted tube joins to the other end of the twisted tube forming a twisted torus (neutrino), would this help explain “Parity Symmetry” violation in Beta Decay? Could the conversion of twist cycles to writhe cycles through the process of supercoiling help explain “neutrino oscillations”? Spatial curvature (mass) would be conserved, but the structure could change.
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Gravity is a result of a very small curvature imbalance within atoms. (This is why the force of gravity is so small.) Instead of attempting to explain matter as "particles", this concept attempts to explain matter more in the manner of our current understanding of the space-time curvature of gravity. If an electron has qualities of both a particle and a wave, it cannot be either one. It must be something else. Therefore, a "particle" is actually a structure which stores spatial curvature. Can an electron-positron pair (which are made up of opposite directions of twist) annihilate each other by unwinding into each other producing Gamma Ray photons?
Does an electron travel through space like a threaded nut traveling down a threaded rod, with each twist cycle proportional to Planck’s Constant? Does it wind up on one end, while unwinding on the other end? Is this related to the Higgs field? Does this help explain the strange ½ spin of many subatomic particles? Does the 720 degree rotation of a 1/2 spin particle require at least one extra dimension?
Alpha decay occurs when the two protons and two neutrons (which are bound together by entangled tubes), become un-entangled from the rest of the nucleons
. Beta decay occurs when the tube of a down quark/gluon in a neutron becomes overtwisted and breaks producing a twisted torus (neutrino) and an up quark, and the ejected electron. The production of the torus may help explain the “Symmetry Violation” in Beta Decay, because one end of the broken tube section is connected to the other end of the tube produced, like a snake eating its tail. The phenomenon of Supercoiling involving twist and writhe cycles may reveal how overtwisted quarks can produce these new particles. The conversion of twists into writhes, and vice-versa, is an interesting process, which is also found in DNA molecules. Could the production of multiple writhe cycles help explain the three generations of quarks and neutrinos? If the twist cycles increase, the writhe cycles would also have a tendency to increase.
Gamma photons are produced when a tube unwinds producing electromagnetic waves. ( Mass=1/Length )
The “Electric Charge” of electrons or positrons would be the result of one twist cycle being displayed at the 3D-4D surface interface of the particle. The physical entanglement of twisted tubes in quarks within protons and neutrons and mesons displays an overall external surface charge of an integer number. Because the neutrinos do not have open tube ends, (They are a twisted torus.) they have no overall electric charge.
Within this model a black hole could represent a quantum of gravity, because it is one cycle of spatial gravitational curvature. Therefore, instead of a graviton being a subatomic particle it could be considered to be a black hole. The overall gravitational attraction would be caused by a very tiny curvature imbalance within atoms.
In this model Alpha equals the compactification ratio within the twistor cone, which is approximately 1/137.
1= Hypertubule diameter at 4D interface
137= Cone’s larger end diameter at 3D interface where the photons are absorbed or emitted.
The 4D twisted Hypertubule gets longer or shorter as twisting or untwisting occurs. (720 degrees per twist cycle.)
How many neutrinos are left over from the Big Bang? They have a small mass, but they could be very large in number. Could this help explain Dark Matter?
Why did Paul Dirac use the twist in a belt to help explain particle spin? Is Dirac’s belt trick related to this model? Is the “Quantum” unit based on twist cycles?
I started out imagining a subatomic Einstein-Rosen Bridge whose internal surface is twisted with either a Right-Hand twist, or a Left-Hand twist producing a twisted 3D/4D membrane. This topological Soliton model grew out of that simple idea. I was also trying to imagine a way to stuff the curvature of a 3 D sine wave into subatomic particles.
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Invite Simon Clark for group study, he has learnt QFT himself. He'll help you!
dude simon is an environmental physicist. Do you even know wot the hell u r talking abt 😂
Me, on my last year of physics undergrad, just starting to learn quantum mechanics: (chuckles) I'm in danger
I'd be really interested in your take on what the heck renormalization is all about and why we're able to get away with it. We blew through path integrals in like 3 weeks (not recommended) in my "Special topics in QM" class, and it seemed reallllllly sus to me. Like, assigning finite values to infinite products level of sus.
Randomly stumbled upon this video, watching it remembering my own QFT experien... 1:49 jesus! That's cat!!!
Did you read this book ? Ashok Das, Susumu Okubo - Lie Groups and Lie Algebras for Physicists-World Scientific Publishing Company. You should be learn lie and Lorentz algebra before start to QFT1
Why does gluons inside a proton and neutron carry Quantum chromodynamic binding energy?
I dont get your curricula. In our theoretical physics program we did GR (Carroll) and QM on the 3rd year of bachelor, and QFT 1,2,3, string theory, lattice field theory, AdS/CFT, on the 4th and 5th year during masters.
Any good video lectures suggestion for this(second quantization approach)?
On the same boat too. Took differentiable manifolds and general relativity and found it to be really elegant and clear mathematically. ON THE OTHER HAND, QFT just doesn't make any sense to me, path integral is somewhat fine, second quantisation was just so random (maybe it's just me, but I can't find any reasoning behind why to do so).
I always have an image (I know it's very wrong to say this) of QFT being a large piece of cloth with so many patches and stitches, that it's so hard to see what it's suppose to be. It's like mumbo jumbo being thrown together. Does it works? Yeah. Does it have a good mathematical formulation? I don't think so, but please correct me and inspire me.
Hi, there Andy!!! Try Klauber's QFT vol 1. You will understand everything in QFT. His QFT-2 is appearing soon, maybe by the end of the year. I don't want to boast but we are pals and that's what he told me.
The title is A Student Friendly Introduction to Quantum Field Theory.
Reading your note makes me anxious, eventhough I think that have some insight about QFT 2nd quantization.
I highly recommend David Tong's lecture note and Schwartz's QFT book
Here I am feeling confident as an undergrad freshman self teaching my self Intro to ED by Griffiths and learning tensor calc and I thought I was smart until I didn’t understand anything on my mans hw except the Dirac equation by itself😒
If you really want to get into the nitty-gritty part of Renormalisation in scalar field theory, you should consider "Renormalization and Effective Field Theory" by Kevin Costello.
Should you be more interested in really understanding the mathematical mechanics behind Field Theory in general, especially gauge fixing, the BV-formalism might be a good goal. Hmu if you need an introductory script on the mathematical foundations of scalar and general QFT entailing renormalisation and the BV formalism ;)
From the homework example you discuss around 3:00, it looks like your difficulties are with the metaphysics, or at least ontology, not the physics! Some things tie you up in knots (in ancient Greek, aporia), and sorting the difference between physical entities and their representations is a crucial one. I don't know what the real issues are with your example or the rest of QFT, but it sounds like category-theoretic tools, not just groups but groupoids and algebroids, might be relevant. I'm not a physicist, but I am a fan of the nlab wiki on everything n-categorical.
I wonder how will you explain renormalization/regularization in 30 minutes. That will certainly be interesting, at least if you don't want to just give some handwavy overview of the idea.
Maybe you can just talk about the running of the coupling constants... 🤔
In my simple world, the γ matrices are constant matrices that do not change under any transformation. Transformations only ever change the fields themselves. What makes ψbar γ^μ ψ a vector is the combined transformation of ψ and ψbar. Vectors have many properties that γ doesn't share; for example γ^μ γ^ν is not a symmetric tensor. Of course this is just my preferred way of looking at it.
My E&M course is only covering chapters 1-7 of Griffiths (the reasons are complicated). Which sections should I self-study in preparation for grad school and the PGRE?
Cat on the shelf!
If you want to know the details lool to Weinberg's bool, volume 1. What you want to know about gamma matrix is the chapter 5.4 . It's really the best book if you want to understand why qft is this way and what are the details.
I'm in the middle of taking Part1 of QFT now... I'm in way over my head during my 2nd year lol
Dirac wouldn't like to see you going down the path of Renormalization.
Andrew, could you make video or give us your skript of your lesson in divergent series renormalization. I am asking since i am a math student doing my master degree and I wrote my bachelor thesis about how to renormalize alternating divergent series, by integrating Taylor series.
Best regards.
Clifford algebra helped me to better understand the Dirac matrices. There is a nice book by Hestenes. The other book that helped was Cartan's book on spinors.
@Andrew Dotson I am curious about how you manage your finances as a phd student? Like do you earn money while studying, is it sufficient for a reasonable standard of living?
Glad to see he’s taking good care of Schrödinger's cat.
Young Melissa Thomas Edward Wilson Sandra
Man QFT was a bit rough for me. I passed just fine but it was like drinking from a firehose
QFT be lively but you know what's more lively? The QT in the bookshelf.
Dear Andrew, please make a video about *Weak Energy Condition and Strong Energy Condition*
Thank you
can you please tell me what software you use in your research work......
......
good to see you again bro
Hello Andrew
I'm waiting for your math and physics video.
I thought you start to teach QFT to us (please do it) ... it seems you star! (Please please start to teach QFT)
Thank you
Physics
So true king
I am in 1st year of my undergraduate course and Could understand it but still watching it 😂
4:25 yeah I don’t understand, but I like listening to you talk about physics.
Maybe it helps to think of the Dirac gamma as a sort of 'metric' or transformation from the spin representation space to the frame bundle? In this way it acts like a (1,0) Lorentz tensor but also a (1,1) spin tensor, and the spin transformations carry a spin and Lorentz index as well. Maybe this is t a new thought to you but in this context it seems like a useful way to interpret the nature of your spinor parts
QFT is love QFT is life...
On that isomorphism of the Lorentz group, it may fail in general but true if complexified or something like that.
Hi Andrew, love your work ethic. I have QFT this term and an exam at the end of the year, which textbooks do you recommend I read to get a better intuition of the concepts? Thanks :D
AHHHH doing second quantization rn, 1st yr master student in Amsterdam here
Seems really exciting, knowing I was too stupid for anything beyond the basic theoretical physics lectures ^^ Settled for teaching and miss uni every day.
could you perhaps make a video about how your research is going? I would love to hear about it.
Interesting and worthwhile video. Most second or third year grad students in your field have significant familiarity with renormalization techniques. You'll find some good overview papers on said techniques on the arXiv.
The most universal physics channel, thanks dude.