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. A

    Proving the weyl tensor is zero problem

    Homework Statement Show that all Robertson - Walker models are conformally flat. Homework Equations Robertson Walker Metric: ds^{2}=a^{2}(t)\left(\frac{dr^{2}}{1-Kr^{2}}+r^{2}(d\theta^{2}+(sin\theta)^{2}d\phi^{2} )\right)-dt^{2} Ricci Tensor: R_{\alpha\beta}=2Kg_{\alpha\beta} Ricci...
  2. O

    Weyl spinor notation co/contravariant and un/dotted

    Hello, sorry for my english.. I have a problem with weyl's spinors notation. I'm confused, becouse i read more books (like Landau, Srednicki and Peskin) and it's seems to me that all of them use different and incompatible notations.. If i define...
  3. T

    The meaning of Weyl curvature caused by gravitational waves

    In his article The Ricci and Weyl Tensors John Baez states that the tidal stretching and squashing caused by gravitational waves would not change the volume as there is 'only' Weyl- but no Ricci-curvature. No additional meaning is mentioned. But, beeing not an expert I still have no good...
  4. naima

    A question about simple Weyl reflections

    I am readin Belinte's book about Lie algebras (I have also the Cahn) . And I try to understand this. He writes "Each basic weight is invariant under all but one of the simple Weyl reflections since w_i l_j = l_j for i<>j while w_i l_i = l_i - alpha_i (alpha_i is simple by definition of...
  5. C

    Building a Lagrangian out of Weyl spinors

    I've been watching Sidney Coleman's QFT lectures (http://www.physics.harvard.edu/about/Phys253.html, with notes at http://arxiv.org/pdf/1110.5013.pdf), and I'm now on to the spin 1/2 part of the course. We've gone through all the mechanics of constructing irreducible representations D^{(s1,s2)}...
  6. 3

    Grassmann variables and Weyl spinors

    I just started studying supersymmetry, but I am a little bit confused with the superspace and superfield formalism. When expanding the vector superfield in components, one obtains therms of the form \theta^{\alpha}\chi_{\alpha}, where \theta is a Grassmann number and \chi is a Weyl vector. I...
  7. T

    Tensor products of representation - Weyl spinors and 4vectors

    Hi guys! I'm having some problems in understanding the direct products of representation in group theory. For example, take two right weyl spinors. We can then write\tau_{0\frac{1}{2}}\otimes\tau_{0\frac{1}{2}}=\tau_{00}\oplus\tau_{01} Now they make me see that...
  8. J

    Weyl ordering of the hamiltonian

    Hi , I can't understand the general formula for weyl ordering of the hamiltonian . It is written in Srednicki field theory book in page 68 . Can someone explain how to derive this formula ?
  9. Y

    Weyl version of the Rarita-schwinger equation

    Hello, Can someone please give me the form of the "Weyl" version of the Rarita-Schwinger equation. Thanks
  10. N

    Calculations with Weyl Spinor Indices in QFT

    Homework Statement The task is to show the invariance of a given Lagrangian (http://www.fysast.uu.se/~leupold/qft-2011/tasks.pdf" ), but my problem is just in one step (which i got from Peskin & Schröder, page 70) which i can not reproduce due to my lack of knowledge regarding spinors. The...
  11. jfy4

    Weyl Tensor Notation in Dimension 4

    Hi, I'm getting used to the anti-symmetric bracket notation used with indices and I can't seem to find the Weyl Tensor written fully out. So I want to make sure I get it. Here is my attempt in dimension 4...
  12. Y

    How is Weyl Tensor associated with tidal force ?

    How is Weyl Tensor associated with tidal force? I checked my book, the acceleration in tidal effect can be expressed as: ac=-RabdcZawbZd Note: Za is the tangent of geodesics, wb is the separation vector I cannot see from this equation how Weyl Tensor affects tidal force. It is...
  13. A

    Gravitational mass defect, weyl metric

    hello everyone, following the book of Landau&Lifsitz I managed to understand the Schwarzschild solution. At the end, it finds this formula for the mass of the spherical body generating the gravitational field: M=\frac{4\pi}{c^2} \int^a_0 \epsilon(r) r^2 dr in which \epsilon(r) is...
  14. MathematicalPhysicist

    Understanding Weyl Rule: A Comprehensive Guide with References and Equations

    I hope someone can help me cite the right reference that explains how to arrive at Weyl rule in the next paper, in the second page (eq. (1)). http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry340.pdf Thanks in advance.
  15. G

    Left and right-handed Weyl spinors

    Hi, I'm new on this forum. I have a doubt regarding helicity and Weyl spinors: I can't understand when I have to use left or right-handed Weyl spinors in order to describe particles or antiparticles. What i have understood is that a charged current is described by left-handed Weyl fields...
  16. E

    Weyl Spinors, SO(1,3) algebra and calculations

    Hey guys, something that puzzles me everytime I stumble across spinors is the following: I know that i can express Dirac spinors in terms of2-component Weyl spinors (dotted/undotted spinors). Now, if i do that, i can reexpress for instance the Lorentz or conformal algebra in terms of Weyl...
  17. Z

    Difference Between Weyl & Majorana Spinnors

    Can anyone explain to me what is the difference between a Weyl spinnor and a Majorana spinnor? Thanks
  18. S

    Decomposition of SL(2,C) Weyl Spinors

    Homework Statement Using (\sigma^{\mu \nu})^{\beta}_{\alpha} (\sigma_{\mu \nu})^{\delta}_{\gamma} = \epsilon_{\alpha \gamma} \epsilon^{\beta \delta} + \delta^{\delta}_{\alpha} \delta^{\beta}_{\gamma} show that \Psi_{\alpha} X_{\beta} = \frac{1}{2} \epsilon_{\alpha \beta} (\Psi X) +...
  19. arivero

    So what's the deal with Majorana, Weyl, and Dirac particles in N dimensions?

    Amusingly, a search on these three words here in PF does not show a lot of postings, so I am creating this thread so you can ask all your doubts about N-dimensional Majorana, Weyl and Dirac particles, their representations, their Lagragians, masses, and whatever you have always wanted to know...
  20. L

    Decomposing the Dirac Lagrangian into Weyl Spinors

    If we take the the Dirac Lagrangian and decompose into Weyl spinors we find \mathcal{L} = \bar{\psi} ( i \gamma^\mu \partial_\mu - m ) \psi = i U^\dagger_- \sigma^\mu \partial_\mu u_- + i u^\dagger_+ \bar{\sigma}^\mu \partial_\mu u_+ - m(u^\dagger_+ u_- + u^\dagger_- u_+ ) =0 So far I have...
  21. P

    What is the difference between left and right Weyl spinors in particle physics?

    What is the difference between left and right Weyl spinors? (probably they transform differently under boosts or rotations). Thanks for answer.
  22. TrickyDicky

    Weyl curvature and tidal forces

    I'm a bit confused about this and would like for someone to help me get this straight. I read in wikipedia that a manifold with more than three dimensions, like spacetime, is conformally flat if its Weyl tensor vanishes. I think all FRW metrics are conformally flat, so I guess our universe is...
  23. E

    Exploring Weyl Spinors: A Question from earth2mars

    Hey guys, I have a question about said spinors. In supersymmetry introductions one finds (e.g. for two left-handed spinors \eta , \nu ) that \eta\nu=\nu\eta due to their Grassmannian character and the antisymmetry of the spinor product. If I look, however, at modern field theoretical...
  24. S

    Can Conformal Weyl Gravity be a Viable Cosmology?

    Can Conformal Weyl Gravity be considered a viable cosmological theory?
  25. P

    Weyl Tensor Components in n-Dim Manifold: N=3?

    I found the formula for the number of independent components of Weyl tensor in n-dimensional manifold: (N+1)N/2 - \binom{n}{4} - n(n+1)/2~~~~~N=(n-1)n/2 This expression implies that in 3 dimension Weyl tensor has 0 independent components, so it's 0. Does it implies that any three-dimensional...
  26. snoopies622

    About the Rindler metric and the Weyl tensor

    This question is a follow-up to the one I asked last week in the thread called, "about tidal forces". In that thread the question came up: what would happen to a sphere of free-falling particles (a "ball of coffee grounds") in a gravitational field described by the Rindler metric? After some...
  27. C

    BRS: The Weyl Vacuums. I. Definition, Geometrical Properties, Symmetries

    In discussions of questions related to gtr, it is often useful to know that one can in fact "create solutions to order" in gtr, when one wishes to model specific physical scenarios. Sort of, not really--- and herein lies a tale which illustrates some of the many thorny technical and conceptual...
  28. P

    Peskin Schroeder Problem 5.6 (b) Weyl spinors

    Hey! I have a problem with problem 5.6 (b) from Peskin + Schroeder. Maybe I just don't see how it works, but I hope somebody can help me! Homework Statement We are asked to calculate the amplitude for the annihilation of a positron electron pair into two photons in the high-energy limit. The...
  29. R

    A Dirac field can be written as two Weyl fields

    A Dirac field can be written as two Weyl fields stacked on top of each other: \Psi= \left( \begin{array}{cc} \psi \\ \zeta^{\dagger} \end{array}\right) , where the particle field is \psi and the antiparticle field is \zeta. So a term like P_L\Psi=.5(1-\gamma^5)\Psi=\left( \begin{array}{cc}...
  30. N

    Time in a black hole and Weyl curvature

    Kip Thorne says (Lecture in 1993 Warping Spacetime, at Stephan Hawking's 60th birthday celebration, Cambridge, England,) Comments, interpretations, appreciated. I thought classical time was always symmetric ...apparently not. Is this same description applicable to a "big crunch" as...
  31. N

    Do Weyl Spinors Have Well-Defined Helicities Across Different Masses?

    Weyl spinors are not eigenstates of the helicity operator when the mass is not zero. But they have well-defined chiralities no matter what the mass is. Yet, it seems to me that references keep talking of Weyl spinors as if they have well-defined helicities, regardless of the mass...
  32. S

    Weyl invariant scalar field theory

    I'm not sure if this is the right place for this question, so feel free to move it. Anyway, my question is, is there any good reason why the following field theory should be Weyl invariant in an arbitrary dimension d>1: S = \int d^d x \sqrt{g} \left( g^{\mu \nu} \partial_\mu \phi \partial_\nu...
  33. N

    Simple questions about weyl rescaling/Conformal transfo

    This is a very very simple question and I am sure it will look dumb because I won't be using the correct terminology but here I go. Consider the points in a manifold. Now we assign coordinates to those points. ne thing that I find confusing about any type of transformation is whether a)...
  34. S

    How Is the Weyl Tensor Derived?

    Could someone show me a derivation of the Weyl tensor or link me to a good site?
  35. S

    Weyl tensor on 3-dimensional manifold

    Hello, I wish to show that on 3-dimensional manifolds, the weyl tensor vanishes. In other words, I want to show that the curvature tensor, the ricci tensor and curvature scalar hold the relation Please, if anyone knows how I can prove this relation or refer to a place which proves the...
  36. S

    Proving the Relation Between Weyl Tensor, Ricci Tensor & Scalar

    Hello, I wish to show that on 3-dimensional manifolds, the weyl tensor vanishes. In other words, I want to show that the curvature tensor, the ricci tensor and curvature scalar hold the relation Please, if anyone knows how I can prove this relation or refer to a place which proves the...
  37. S

    A Weyl theory of dark matter

    A "Weyl" theory of dark matter http://web.mit.edu/people/cabi/index.html by Hung Cheng of MIT, showing that if physics is locally conformal (independent of scale choice) then there is a vector particle he calls S which couples to a scalar particle like the hypothetical Higgs, or to a tensor...
  38. R

    Weyl Curvature Hypothesis and entropy

    What's the actual status of the Weyl curvature hypothesis? Is there any best explanation to the early low entropy?
  39. R

    Weyl Curvature, Mach's Principle, and Heisenberg Uncertainty?

    I have been reading that the quantity called "Weyl curvature" can exist independently of any matter, or energy, in the universe? :confused: This seems to contradict Heisenberg uncertainty which says there can be no 100% vacuum, because uncertainty in position and uncertainty in momentum...
  40. A

    Weyl Transformation and Scalar Product

    I was reading about Weyl Transformations in Polchinski's book and I have a little doubt: Is it correct to say that under a Weyl transformation the scalars are invariant, i.e., that a weyl transformation preserves the scalar product?
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