I do not know if useful is the correct word, but your videos are very helpful in understanding that I have no clue regarding this subject! Thanks a lot!
Hi Edwin, off topic: Do you know anything about Lubos Motl? His blog is not accessable for some time now and there does not appear any online activity by him, e.g.on facebook.
Thank you for your valuable work. Finaly got the time to watch and indeed enjoyed a lot. In one dimension, there are left and right moving components. I guess in three dimensions this translates into forward and backward along each of three linearly independent directions, right? Also, if these moving modes ultimately represent particles, than these particles never colide. Is that correct?
Thanks, mi haj! The left- and right-movers don't quite generalize like that. While there are waves moving in opposite directions in higher dimensions, the difference is that in D > 2, there are spatial rotations, i.e. *continuous* Lorentz symmetry transformations that can turn a wave moving towards, say, +x_1 into a wave moving into the opposite direction (say -x_1). This is different from D=2, where only a *discrete* symmetry transformation (reflection of space) turns a left-mover into a right-mover. Because of this, you can easily have D=2 theories where the left-movers behave quite differently from the right-movers (see for example heterotic string theories) but this does not work in (rotationally symmetric) D>2 theories. There is something a bit similar in higher dimension, though: Considering spin, massless particles can have right or left helicity and there is no continuous transformation that turns one into the other. BTW, in D>2, you need to consider *all* directions, not just a basis of orthogonal ones, because a particle moving e.g. at 45° to the x,y-axes is *not* a superposition of a particle moving along x and a particle moving along y. (The x- and the y-momentum are operators that commute with each other, so they can have independently well-defined values.) Regarding collisions: Yes, that's right. In a free theory, there cannot be any particle collisions. All particle behave completely independently from each other, as if the other particles weren't there.
Content wise it is not clear to me why you want to immediately relate to string theory, when your focus is bosonic fields. That includes focus on zero modes, light cone coordinates and the whole alignment with Zwiebach's book mentioning the string constant etc. Just imagine the field in a cavity, take the field to zero at the boundary and get rid of the zero mode or ignore it altogether like other QFT books do. A constant field linearly increasing everywhere seems just like a strange choice of boundary conditions. So to make this more useful I think you should decide what physical model motivates the discussion, a field like the electric field or the description of a string.
The connections to string theory are optional and maybe they take up a bit too much space here, but people can skip them using the chapters. In my mind, string theory is the most interesting reason to study the free boson, so I did not want to leave the connections out. I hope that some viewers will be interested in them. Still, the immediate motivation for this series is to get to the vertex algebra structure from the physical point of view and starting from string theory would require many more episodes than I planned for this series. If I would get rid of the zero mode by boundary conditions, I'd lose the compatibility of what I present with string theory, making the material in my estimation less useful rather than more.
Please, keep it up! The way you make complicated material so digestible motivates to study!
Great content! Please continue this series!!!!
I do not know if useful is the correct word, but your videos are very helpful in understanding that I have no clue regarding this subject! Thanks a lot!
Excellent!
I've been watching the series and I feel like it helped me deepen my knowledge. Thank you
Hi Edwin, off topic: Do you know anything about Lubos Motl? His blog is not accessable for some time now and there does not appear any online activity by him, e.g.on facebook.
All I know is that Luboš had to shut down his blog due to censorship problems with Google.
Thank you for your valuable work. Finaly got the time to watch and indeed enjoyed a lot. In one dimension, there are left and right moving components. I guess in three dimensions this translates into forward and backward along each of three linearly independent directions, right? Also, if these moving modes ultimately represent particles, than these particles never colide. Is that correct?
Thanks, mi haj! The left- and right-movers don't quite generalize like that. While there are waves moving in opposite directions in higher dimensions, the difference is that in D > 2, there are spatial rotations, i.e. *continuous* Lorentz symmetry transformations that can turn a wave moving towards, say, +x_1 into a wave moving into the opposite direction (say -x_1). This is different from D=2, where only a *discrete* symmetry transformation (reflection of space) turns a left-mover into a right-mover. Because of this, you can easily have D=2 theories where the left-movers behave quite differently from the right-movers (see for example heterotic string theories) but this does not work in (rotationally symmetric) D>2 theories. There is something a bit similar in higher dimension, though: Considering spin, massless particles can have right or left helicity and there is no continuous transformation that turns one into the other.
BTW, in D>2, you need to consider *all* directions, not just a basis of orthogonal ones, because a particle moving e.g. at 45° to the x,y-axes is *not* a superposition of a particle moving along x and a particle moving along y. (The x- and the y-momentum are operators that commute with each other, so they can have independently well-defined values.)
Regarding collisions: Yes, that's right. In a free theory, there cannot be any particle collisions. All particle behave completely independently from each other, as if the other particles weren't there.
Content wise it is not clear to me why you want to immediately relate to string theory, when your focus is bosonic fields. That includes focus on zero modes, light cone coordinates and the whole alignment with Zwiebach's book mentioning the string constant etc. Just imagine the field in a cavity, take the field to zero at the boundary and get rid of the zero mode or ignore it altogether like other QFT books do. A constant field linearly increasing everywhere seems just like a strange choice of boundary conditions. So to make this more useful I think you should decide what physical model motivates the discussion, a field like the electric field or the description of a string.
The connections to string theory are optional and maybe they take up a bit too much space here, but people can skip them using the chapters. In my mind, string theory is the most interesting reason to study the free boson, so I did not want to leave the connections out. I hope that some viewers will be interested in them. Still, the immediate motivation for this series is to get to the vertex algebra structure from the physical point of view and starting from string theory would require many more episodes than I planned for this series.
If I would get rid of the zero mode by boundary conditions, I'd lose the compatibility of what I present with string theory, making the material in my estimation less useful rather than more.