What is Weyl: Definition and 90 Discussions

Hermann Klaus Hugo Weyl, (German: [vaɪl]; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.
His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him.

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  1. E

    General relativity - Using Ricc and Weyl tensor to find the area

    I have the following question to solve:Use the metric:$$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$ Test bodies are arranged in a circle on the metric at rest at ##t=0##. The circle define as $$x^2 +y^2 \leq R^2$$ The bodies start to move on geodesic when we have $$a(0)=0$$ a. we have to...
  2. Baela

    A Covariant derivative of Weyl spinor

    What is the expression for the covariant derivative of a Weyl spinor?
  3. S

    I Propagator of massless Weyl field

    I have this Lagrangian for a free massless left Weyl spinor, so it’s just the kinetic term, that can be written embedding the field into a larger Dirac spinor and then taking the left projector in this way: $$i \bar{\psi} \cancel{\partial} P_L \psi$$ Srednicki says that the momentum space...
  4. G

    A Principal Invariants of the Weyl Tensor

    It's possible that this may be a better fit for the Differential Geometry forum (in which case, please do let me know). However, I'm curious to know whether anyone is aware of any standard naming convention for the two principal invariants of the Weyl tensor. For the Riemann tensor, the names of...
  5. P

    A Weyl Fermion in an infinite well

    Hello everyone, I have a problem with bounds states of the 1D Weyl equation. I want to solve the Dirac equation ##−i\hbar \partial _x\Psi+m(x)\sigma _z \Psi=E\Psi## with the mass ##m(x)=0,0<x<a##, ##m(x)=\infty,x<0,x>a##. ##\Psi=(\Psi_1,\Psi_2)^T## is a two component spinor. Outside the well...
  6. G

    A Britgrav Weyl Tensor Research: Find Paper Authors, Title & Accessibility

    In the earlier years of Britgrav there were sometimes longer presentations of research done at the host university. In one such, about 2005 or so, a presentation showed that an isolated region of space could be rotated through 180 degrees by the action of extreme waves in the Weyl tensor...
  7. JD_PM

    Showing that the Weyl tensor is invariant under conformal symmetries

    The Weyl tensor is given by (Carroll's EQ 3.147) \begin{align*} C_{\rho \sigma \mu \nu} &= R_{\rho \sigma \mu \nu} - \frac{2}{n-2}\left(g_{\rho [\mu}R_{\nu]\sigma} - g_{\sigma [\mu}R_{\nu]\rho}\right) \\ &+ \frac{2}{(n-1)(n-2)}g_{\rho [\mu}g_{\nu]\sigma}R \end{align*} Where ##n## are...
  8. Pouramat

    Weyl Spinors Transformation, QFT1, Peskin, Chapter 3

    \begin{align} \psi_L \rightarrow (1-i \vec{\theta} . \frac{{\vec\sigma}}{2} - \vec\beta . \frac{\vec\sigma}{2}) \psi_L \\ \psi_R \rightarrow (1-i \vec{\theta} . \frac{{\vec\sigma}}{2} + \vec\beta . \frac{\vec\sigma}{2}) \psi_R \end{align} I really cannot evaluate these from boost and rotation...
  9. U

    I Calculating Surface Area of Schwarzschild Black Hole w/Weyl Coordinates

    Recently, I was tasked to find the surface area of the Schwarzschild Black Hole. I have managed to do so using spherical and prolate spheroidal coordinates. However, my lecturer insists on only using Weyl canonical coordinates to directly calculate the surface area. The apparent problem arises...
  10. Q

    A Why does the Kähler Potential only contain left handed Weyl spinors?

    Why aren't the right handed Weyl spinors included?
  11. Rabindranath

    A Weyl transformation of connection and curvature tensors

    Given a Weyl transformation of the metric ##g_{\mu\nu} \rightarrow g'_{\mu\nu} = e^{\Omega(x)} g_{\mu\nu}##, I'm trying to find the corresponding connection ##\Gamma'^{\lambda}_{\mu\nu}##, and from that ##-## via the Riemann tensor ##R'^{\lambda}_{\mu\nu\kappa}## ##-## the Ricci tensor...
  12. TeethWhitener

    A QFT Srednicki Chap 6: Weyl Ordering

    In Srednicki’s QFT Chapter 6 (intro to path integrals), he introduces Weyl ordering of the quantum Hamiltonian: $$H(P,Q)=\int{\frac{dx}{2\pi}\frac{dk}{2\pi} e^{ixP+ikQ}}\int{dp \text{ }dq\text{ }e^{-ixp-ikq}H(p,q)}$$ where ##P,Q## are momentum and position operators and ##H(p,q)## is the...
  13. A

    I Is the Chirality Projection Operator Misused in This Scenario?

    Hello everybody! I have a doubt in using the chiral projection operators. In principle, it should be ##P_L \psi = \psi_L##. $$ P_L = \frac{1-\gamma^5}{2} = \frac{1}{2} \begin{pmatrix} \mathbb{I} & -\mathbb{I} \\ -\mathbb{I} & \mathbb{I} \end{pmatrix} $$ If I consider ##\psi = \begin{pmatrix}...
  14. Safinaz

    I Weyl Transformation of Metric: Cosmological Inflation

    Hello, I’m new to the cosmological inflation so in this paper: https://arxiv.org/abs/1809.09975 Has some one an idea how to make the Weyl transformation of the metric ## g_{\mu\nu}## Equation (3) , and how to get the potential (4) from the action (3) by this transformation as explained after...
  15. K

    I Weyl Spinors & Helicity: Explained

    Hello! So Weyl spinors are 2 dimensional spinors which describe massless particles and have definite helicities. So if we have a right handed Weyl spinor going along the positive x-axis, it's spin will always point along the positive x-axis too. I am a bit confused how can an object have 2 spin...
  16. K

    I Making GR Weyl Invariant: Attempts and Possibilities

    General relativity is not invariant under Weyl transformations. What would it take to make it Weyl invariant? And what are all the previous attempts to make it such (can you enumerate them)?
  17. Milsomonk

    The Dirac equation in Weyl representation

    Homework Statement Compute the antiparticle spinor solutions of the free Dirac equation whilst working in the Weyl representation.Homework Equations Dirac equation $$(\gamma^\mu P_\mu +m)v_{(p)}=0$$ Dirac matrices in the Weyl representation $$ \gamma^\mu= \begin{bmatrix} 0 & \sigma^i \\...
  18. Milsomonk

    I Dirac equation solved in Weyl representation

    Hi guys :) I'm just wondering if anyone knows of a book that has the Dirac equation solved in the Weyl basis in it? I'd like to check my method to make sure I'm on the correct lines. Thanks
  19. J

    Solid State Discover the Best Books on Weyl Semimetals: A Comprehensive Guide

    Hello! Does anybody know of any pedagogical material(like a book or chapter(s) of a book) on Weyl semimetals? Thanks you!
  20. J

    Solid State Books: Weyl semimetals, Topological Insulators

    Hello! What are some good sources(preferably textbooks) to learn about Weyl semimetals? I also want some sources to learn about topological insulators and anything containing the Integer Quantum Hall effect would be great. As an aside, if you have any good book on theoretical condensed matter...
  21. G

    A Can Weyl Tensor Look Like Negative Mass?

    The smoothed Weyl tensor can look like space that contains a non-zero Einstein tensor. To verify this, consider that gravitational waves carry mass away from (say) a rotating binary, so the apparent mass at infinity of a large sphere containing a radiating binary will be greater than the mass...
  22. D

    I Why and how Dirac cones are "tilted"?

    Given a Weyl Hamiltonian, at rest, \begin{align} H = \vec \sigma \cdot \vec{p} \end{align} A Lorentz boost in the x-direction returns \begin{align} H = \vec\sigma\cdot\vec{p} - \gamma\sigma_0 p_x \end{align} The second term gives rise to a tilt in the "light" cone of graphene. My doubts...
  23. Akineton

    I Transformation matrix from Dirac to Weyl

    Hello friends, I'm trying to construct transformation matrix S such that it transforms Dirac representations of gamma matrices into Chiral ones. I know that this S should be hermitian and unitary and from this I arrived an equation with 2 matrices on the LHS (a known matrix multiplied by S from...
  24. Math Amateur

    MHB First Weyl Algebras .... A_1 ....

    I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some remarks of Bresar on first Weyl Algebras ... Bresar's remarks on Weyl Algebras are as...
  25. Math Amateur

    I First Weyl Algebras .... A_1 ....

    I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some remarks of Bresar on first Weyl Algebras ... Bresar's remarks on Weyl Algebras are as follows: In the above...
  26. S

    A Transformation of the spinor indices of the Weyl operator under the Lorentz group

    The left-handed Weyl operator is defined by the ##2\times 2## matrix $$p_{\mu}\bar{\sigma}_{\dot{\beta}\alpha}^{\mu} = \begin{pmatrix} p^0 +p^3 & p^1 - i p^2\\ p^1 + ip^2 & p^0 - p^3 \end{pmatrix},$$ where ##\bar{\sigma}^{\mu}=(1,-\vec{\sigma})## are sigma matrices.One can use the sigma...
  27. N

    A Is Graphene a Two-Dimensional Weyl Semimetal with Spin Degenerate Dirac Bands?

    I'm recently interested in the topological/Weyl semimetals, but I'm not an expert on the theory. Most papers just define Weyl semimetal as a material that have pairs of Weyl points with opposite Berry curvature. Here in graphene, the Berry curvature of the Dirac cones at K and K' point is also...
  28. redtree

    A Weyl Vacua Solutions to GR: Derivation from Riemann Tensor

    Where can I find a derivation of the vacuum solution for GR directly from the Riemann tensor of zero trace, i.e., Weyl tensor, instead of the more traditional Schwarzschild derivation?
  29. S

    A Calculate the Weyl and Ricci scalars for a given metric

    hello dear, I need to calculate the Weyl and Ricci scalars for a given metric. let's assume for a kerr metric. by using the grtensor-II package in maple i am not able to get the results. would anyone help me out. for ordinary metric like schwarzschild metric, However its not working for Kerr...
  30. N

    A Ricci Flow and Weyl Transformations

    I am trying to my head around these two things in the context of string theory. The Polyakov action becomes simpler to solve in the conformal gauge which, as I understand it, makes the manifold locally Ricci flat in 2D. In Professor Susskind's lectures on String Theory he introduces the concept...
  31. S

    A Weyl Tensor Gravity propagation

    I read Weyl tensor helps on propagating gravitational effects. Ricci is local depending on mass energy at that point and would vanish at other points. Weyl propogates the gravity effects (for example gravity at any point between Earth Moon is due to Weyl Tensor). I didn't quite get it...
  32. A

    Weyl Tensor invariant under conformal transformations

    Homework Statement As the title says, I need to show this. A conformal transformation is made by changing the metric: ##g_{\mu\nu}\mapsto\omega(x)^{2}g_{\mu\nu}=\tilde{g}_{\mu\nu}## Homework Equations The Weyl tensor is given in four dimensions as: ##...
  33. B

    De Donder Weyl: someone give an example question?

    Homework Statement Could someone give an example question in De Donder-Weyl theory (the multisymplectic theory where the hamiltonian is not parametrized by time, it's on wikipedia). Homework Equations The relevant equations should not be too complicated, just complicated enough to use the...
  34. Rumo

    Weyls representation of a propagating (z-v*t) spherical wave

    Hello! The following wave solves the 3D wave equation: $$ \frac{\sin\left(k\sqrt{x^2+y^2+\frac{(z-vt)^2}{1-\frac{v^2}{c^2}}}\right)}{\sqrt{x^2+y^2+\frac{(z-vt)^2}{1-\frac{v^2}{c^2}}}}\cos\left(w\frac{t-\frac{vz}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}\right) $$ This is a propagating standing spherical...
  35. C

    Projection operators and Weyl spinors

    I am working through some course notes where the aim is to derive the equations of motion satisfied by the left handed and right handed components of the Dirac spinor ##\psi##. From the Dirac lagrangian, we have $$\mathcal L = \bar \psi (i \not \partial P_L - m P_L)\psi_L + \bar \psi (i \not...
  36. P

    Lorentz Transformation on Left & Right Chiral Spinors

    I will start with a summary of my confusion: I came across seemingly contradictory transformation rules for left and right chiral spinor in 2 books, and am unable to understand what part is Physics and what part is convention. Or is it that one of the two books incorrectly writes the...
  37. stevendaryl

    Coarse-Grained Einstein Tensor from Weyl Tensor

    Here's a question that has bugged me for a while. The full Riemann curvature tensor R^\mu_{\nu \lambda \sigma} can be split into the Einstein tensor, G_{\mu \nu}, which vanishes in vacuum, and the Weyl tensor C^\mu_{\nu \lambda \sigma}, which does not. (I'm a little unclear on whether R^\mu_{\nu...
  38. S

    Weyl tensor for the Godel metric interpretation

    I have recently derived both the purely covariant Riemann tensor as well as the purely covariant Weyl tensor for the Gödel solution to Einstein's field equations. Here is a wiki for the Gödel metric if you need it: http://en.wikipedia.org/wiki/Gödel_metric There you can see the line element I...
  39. S

    Can someone verify this definition for Weyl tensor?

    I just want to make sure I have this right because when I go to different sites, it seems to look different every time. This is the Weyl tensor: Cabcd = Rabcd + (1/2) [- Racgbd + Radgbc + Rbcgad - Rbdgac + (1/3) (gacgbd - gadgbc)R] Is this correct?
  40. TrickyDicky

    Equivalence between Weyl relations and CCR

    Due to the fact that the operators in the canonical commutation relations(CCR) cannot be both bounded, in order to prove the Stone-von Neuman theorem one must resort to the Weyl relations. Now the Weyl relations imply the CCR, but the opposite is not true, the CCR don't imply the Weyl relations...
  41. Spinnor

    Graphically rep. 1+1D Weyl fermions, does this work?

    Does the following construction allow one to represent both the spinor and spacetime parts of the wavefunctions of the four massless Weyl fermions of a given magnitude of momentum p? In 3 dimensional space let there be some x,y,z coordinate system. Let the x and y-axis represent the complex...
  42. twistor

    Is the Weyl Curvature Hypothesis a Viable Alternative to Inflation?

    Hello guys. I was thinking about alternatives to inflation, especially old ones (such as the hawking-hartle state and imaginary time) and I remebered a theory put foward by Penrose, in which his relatively new CCC is based. Called the Weyl Curvature Hypothesis. No idea of what it is. Could you...
  43. Markus Hanke

    Is Weyl Curvature Present in Interior Spacetimes?

    I am just wondering - is space-time curvature in the presence of energy-momentum ( i.e. in interior solutions to the EFEs ) always pure Ricci in nature ? I had a discussion recently with someone who claimed that, but personally I would suspect that not to be the case in general, since I see no...
  44. B

    Geometrical meaning of Weyl tensor

    Can anyone give me a geometrical interpretation of the weyl curvature tensor?
  45. N

    Poisson bracket and Electric and Magnetic Weyl tensor in GR

    In order to understand how related are the theories of General Relativity and Electromagnetism, I am looking at the electric and magnetic parts of the Weyl tensor (in the ADM formalism) and compare them with the ones from Maxwell's theory. I have tried to look at the Poisson bracket, but the...
  46. Z

    Weyl tensor in 2 dimensions- confused

    hello, The Weyl tensor is: http://ars.els-cdn.com/content/image/1-s2.0-S0550321305002828-si53.gif In 2 dimensions , the Riemann tensor is (see MTW ex 14.2): Rabcd = K( gacgbd - gadgbc ) [R] Now the Weyl tensor must vanish in 2 dimensions. However, working with the g g = [-1 0 0...
  47. V

    Inverse Weyl quantization of the projection operator.

    I am trying to solve the following problem on an old Quantum Mechanics exam as an exercise. Homework Statement Homework Equations I know that the trace of an operator is the integral of its kernel. \begin{equation} Tr[K(x,y)] = \int K(x,x) dx \end{equation} The Attempt at a...
  48. A

    Evaluate Wigner Weyl Transforms for xp+px/2

    For Wigner transforming the function of operators x and p : (xp+px)/2 we need to evaluate something like: g(x,p) = ∫dy <x - y/2 | (xp+px)/2 | x+y/2> e(ipy/h) where h is h/2π. Now I am not sure how to evaluate <x - y/2 | (xp+px)/2 | x+y/2> . I mean what I did was think of |x+y/2> as a...
  49. A

    What can be derived from a vanishing Weyl tensor

    I was just wondering what physical conclusions could be made about a spacetime which possesses a vanishing Weyl curvature tensor, aside from the spacetime being conformally flat. By this question I am simply interested if any inferences can be made about the metric describing the spacetime, or...
  50. A

    Geometry of the Riemann, Ricci, and Weyl Tensors

    Hi, I was wondering if someone wouldn't mind breaking down the geometrical differences between the Riemann, Ricci, and Weyl tensor. My current understanding is that the Ricci tensor describes the change in volume of a n-dimensional object in curved space from flat Euclidean space and that if we...