What is Spinor: Definition and 113 Discussions

In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. However, when a sequence of such small rotations is composed (integrated) to form an overall final rotation, the resulting spinor transformation depends on which sequence of small rotations was used. Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms).
It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group). There are two topologically distinguishable classes (homotopy classes) of paths through rotations that result in the same overall rotation, as illustrated by the belt trick puzzle. These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class. It doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class. In mathematical terms, spinors are described by a double-valued projective representation of the rotation group SO(3).
Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with. A Clifford space operates on a spinor space, and the elements of a spinor space are spinors. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.

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  1. baba26

    A Proof of covariant derivative of spinor

    I have read that we can define covariant derivative for spinors using the spin connection. But I can't see its proof in any textbook. Can anyone point to a reference where it is proved that such a definition indeed transforms covariantly ?
  2. Baela

    A Covariant derivative of Weyl spinor

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

    A Lagrangian density for the spinor fields

    hi, i have seen lagrangian density for spin 0 , spin 1/2, spin 1 , but i am not getting from where these langrangian densities comes in at a first place. kindly give me the hint. thanks
  4. wnvl2

    I Experimenting with Spinor Rotations & Sign Changes

    When a spinor is rotated through 360◦, it is returned to its original direction, but it also picks up an overall sign change. This sign has no consequence when spinors are examined one at a time, but it can be relevant when one spinor is compared with another. Is there an experiment to make an...
  5. justpeeking

    B How does a spinor affect a wave function?

    How do spinors affect wave function solutions? Like how is the output different
  6. LCSphysicist

    From spinor to ket space: Equivalents eigen equations

    "##\sigma . n X = 1*X##" to "##S. n| S. n; +\rangle = \frac{h}{4\pi}| S .n; +\rangle ##" X is a spinor n is any unitary vector sigma are the pauli matrices ##(\sigma 0, \sigma x,\sigma y,\sigma z)## S is the spin vector. It was claimed that both equations are equivalent, but i couldn't see why.
  7. V

    A Adjoint representation and spinor field valued in the Lie algebra

    I'm following the lecture notes by https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf. On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM. The fundamental representation makes sense to me. For example, for ##SU(3)##, we define the...
  8. S

    A Covering Group of SO(g) & Understanding Spinors on Curved Spacetime

    I'd like to better understand spinors on curved spacetime, but started wandering along the following tangent. I've looked at but not particularly understood the sections on spinors in the texts by Penrose and (Misner, Thorne and Wheeler). Let ##g_{ij}## be a spacetime metric (a symmetric...
  9. M

    I Condtion on transformation to solve the Dirac equation

    The problem is given in the summary. My attempt: Assume that ##\psi^\prime (x^\prime)## is a solution of the Dirac equation in the primed frame, given the transformation ##x\mapsto x^\prime :=\Lambda^{-1}x## and ##\psi^\prime (x^\prime)=S\psi(x)##, we have $$ \begin{align*} 0&=(\gamma^\mu...
  10. M

    I How to determine Spinor in Feynman diagram

    Consider Moller scattering, that is $$e^-(\vec p_1, \alpha)+e^-(\vec p_2, \beta) \quad\longrightarrow\quad e^-(\vec q_1, \gamma)+e^-(\vec q_2, \delta),$$ where the ##\vec{p}_i,\vec q_i## label the momenta of the in and outgoing electrons and the greek letter the spin state. The two relevant...
  11. Garlic

    I Lorentz transformation of the "bilinear spinor matrixelement"

    Dear reader, there is a physics problem where I couldn't understand what the solutions. It is about the lorentz transformation of a bilinear spinor matrix element thing. So the blue colored equation signs are the parts which I couldn't figure out how. There must be some steps in between which...
  12. park

    I The Dirac Equation: Understanding Spinors and Approximations

    I'm studying about dirac equation and it's solution. When we starts with the equation (2.75), I can understand that it is possible to set 2 kinds of spinor. But my question is... 1. After the assumption (2.100), how can we set the equation like (2.101) 2. I can't get (2.113) from (2.111)...
  13. filip97

    A Why Spinors Are Irreducible if Gamma-Traceless: Explained

    I read this question https://physics.stackexchange.com/questions/95970/under-what-conditions-is-a-vector-spinor-gamma-trace-free . Also I read Sexl and Urbantke book about groups. But I don't understand why spinors is irreducible if these are gamma-tracelees. Also I read many papers about...
  14. 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}...
  15. M

    Proving Even Parity for this Expression

    My idea was straight forward calculation: $$\begin{align*}\bar { \psi }' ( x' ) \gamma ^ { \mu } \partial _ { \mu }' \psi ( x' ) &= \psi^\dagger\gamma^{0\dagger}\gamma^0\gamma^\mu \partial_\mu'\gamma^0\psi = \bar\psi\underbrace{\gamma^0\gamma^\mu\gamma^0}_{=\gamma^{\mu\dagger}=-\gamma^\mu}...
  16. M

    Calculating field transformation

    Homework Statement Let ##\psi(x)=u(p)e^{-ipx}##, where $$ u((m,0)) = \sqrt{m}\begin{pmatrix} \xi\\\xi \end{pmatrix}\quad\text{where}\quad \xi = \sum_{s\in \{+,-\}}c_s\xi^s\quad \text{and}\quad \xi^+\equiv\begin{pmatrix} 1\\ 0 \end{pmatrix}\quad \xi^-\equiv\begin{pmatrix} 0\\ 1 \end{pmatrix}, $$...
  17. Dewgale

    Anti-commutation of Dirac Spinor and Gamma-5

    Homework Statement Given an interaction Lagrangian $$ \mathcal{L}_{int} = \lambda \phi \bar{\psi} \gamma^5 \psi,$$ where ##\psi## are Dirac spinors, and ##\phi## is a bosonic pseudoscalar, I've been asked to find the second order scattering amplitude for ##\psi\psi \to \psi\psi## scattering...
  18. M

    Find the spinor-state for a given expectation value

    Homework Statement Let ##\vec{e}\in\mathbb{R}^3## be any unit vector. A spin ##1/2## particle is in state ##|\chi \rangle## for which $$\langle\vec{\sigma}\rangle =\vec{e},$$ where ##\vec{\sigma}## are the Pauli-Matrices. Find the state ##|\chi\rangle## Homework Equations :[/B] are all given...
  19. A

    I Real function instead of spinor field in Yang-Mills field

    An old thread (https://www.physicsforums.com/threads/state-observable-duality-john-baez-series.451101/) triggered a lively debate on whether complex functions are necessary for quantum theory or real functions (but not pairs of real functions) can be sufficient for it. I argued that one real...
  20. D

    A Lorentz invariance from Dirac spinor

    I have a really naive question that I didn't manage to explain to myself. If I consider SUSY theory without R-parity conservation there exist an operator that mediates proton decay. This operator is $$u^c d^c \tilde d^c $$ where ##\tilde d## is the scalar superpartner of down quark. Now...
  21. A

    A Measuring the spin of a moving Dirac spinor particle

    Hello, I would like to ask about the process of measuring the Spin of a Dirac 4-spinor Ψ that is not in the rest frame. Note that even though there is plenty of information about what a Dirac spinor is, what reasoning lead to its discovery and how it can be expressed in terms of particle and...
  22. G

    Problem Proving a Spinor Identity

    Homework Statement Given the spinors: \Psi_{1}=\frac{1}{\sqrt{2}}\left(\psi-\psi^{c}\right) \Psi_{2}=\frac{1}{\sqrt{2}}\left(\psi+\psi^{c}\right) Where c denotes charge conjugation, show that for a vector boson #A_{\mu}#; A_{\mu}\overline{\Psi_{1}}\gamma^{\mu}\Psi_{2} +...
  23. M

    A Spinor product in Peskin-Schroeder problem 5.3

    Hello, I am currently stuck on problem 5.3 (c) about spinor products in PS, where one needs to prove the Fierz identity: $$ \bar{u}_{L}(p_{1}) \gamma^{\mu} {u}_{L}(p_{2}) [\gamma_{\mu}]_{ab} = 2 [u_{L}(p_{2})\bar{u}_{L}(p_1) +u_{R}(p_{1})\bar{u}_{R}(p_2) ]_{ab} $$ They say that a Dirac matric M...
  24. T

    A What's the idea behind propagators

    I'm studying QFT by David Tong's lecture notes. When he discusses causility with real scalar fields, he defines the propagator as (p.38) $$D(x-y)=\left\langle0\right| \phi(x)\phi(y)\left|0\right\rangle=\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_{\vec{p}}}e^{-ip\cdot(x-y)},$$ then he shows that the...
  25. J

    A Understanding Dirac Equation Spinor Boosts

    Hi, I'm recently learning the Dirac equation and we're following the more historical approaching working in the Dirac basis. At first it seems OK that the upper two components are interpreted as positive energy and the lower two negative. However, when I learned that after a boost the spinor...
  26. J

    I Normalisation constant expansion of spinor field

    Hi, I'm reading about the wave packet solution to the dirac equation but in the book I'm reading it states that \int \frac {d^3p} {(2\pi)^3 2E} [a u e^{-ipx} + b^\dagger \bar{v} e^{ipx} The normalisation constant confuses me. I guess the 2pi^3 is reasonalbe. However, the 1/2E seems a bit...
  27. K

    Eigenstates of Rashba Spin-Orbit Hamiltonian

    Homework Statement I am given the Rashba Hamiltonian which describes a 2D electron gas interacting with a perpendicular electric field, of the form $$H = \frac{p^2}{2m^2} + \frac{\alpha}{\hbar}\left(p_x \sigma_y - p_y \sigma_x\right)$$ I am asked to find the energy eigenvalues and...
  28. U

    Functional Derivative with respect to Dirac Spinors

    Homework Statement I am currently working on an exercise list where I need to calculate the second functional derivative with respect to Grassmann valued fields. $$ \dfrac{\overrightarrow{\delta}}{\delta \psi_{\alpha} (-p)} \left( \int_{x} \widetilde{\bar{\psi}}_{\mu} (x) i \partial_{s}^{\mu...
  29. B

    Dual spinor and gamma matrices

    Here it is a simple problem which is giving me an headache,Recall from class that in order to build an invariant out of spinors we had to introduce a somewhat unexpected form for the dual spinor, i.e. ߰ψ = ψ†⋅γ0 Then showing that ߰ is invariant depends on the result that (ei/4⋅σμν⋅ωμν)† ⋅γ0 =...
  30. P

    Massive spin-s representations of the Poincare group

    Context The following is from the book "Ideas and methods in supersymmetry and supergravity" by I.L. Buchbinder and S.M Kuzenko, pg 56-60. It is about realizing the irreducible massive representations of the Poincare group as spin tensor fields which transform under certain representations of...
  31. F

    I Spinor Representation of Lorentz Transformations: Solving the Puzzle

    I've been working my way through Peskin and Schroeder and am currently on the sub-section about how spinors transform under Lorentz transformation. As I understand it, under a Lorentz transformation, a spinor ##\psi## transforms as $$\psi\rightarrow S(\Lambda)\psi$$ where...
  32. DeathbyGreen

    A Nambu Spinor Notation in Kitaev spinless p-wave model

    Hey all! Thanks for reading. I'm currently following along in some reading and had some trouble with re-writing a Hamiltonian in Bogluibov-de Gennes form using Nambu notation (Nambu spinors). Here is the low down: Say we have a Hamiltonian: \frac{1}{2} \sum_{i=1}^{N} c_{i}^{\dagger} D c_{i} +...
  33. S

    A Spinor indices on Yukawa coupling terms in electroweak sector

    In the electroweak sector, we define the left-handed Weyl fields ##l## and ##\bar{e}## in the representations ##(2,-1/2)## and ##(1,+1)## of ##SU(2) \times U(1)##. Here, ##l## is an ##SU(2)## doublet: ##l = \begin{pmatrix} \nu\\ e \end{pmatrix}.## The Yukawa coupling in the electroweak sector...
  34. B

    A Spinor Lorentz Transform via Vectors - Cross Product Issue

    The Lorentz transformation operator acting on an undotted, i.e. right-handed, spinor can be expressed as $$e^{-\frac{1}{2} \sigma \cdot \mathbf{\phi} + i\frac{1}{2} \sigma \cdot \mathbf{\theta}}.$$ There is a very cool, almost childlike, derivation of this expression in Landau Vol. 4 S. 18 I've...
  35. 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...
  36. A

    I Understanding Spinors - Geometry & Usage

    Okay, I have read on spinors here and there but I really don't understand geometrically or intuitively what it is. Can someone please explain it to me and how/when it is used? Thanks!
  37. D

    I What kind of space is the space of spinors?

    Hi, i don't find much about spinor spaces. I can think in that spaces like a vector space above the field of complex numbers (a complex vector space)? sorry if what i saying is a non-sense, but i really want to understand better the math behind the concept of a spinor. thanks
  38. R

    I The Naked Spinor and related stuff

    Hello, studying deeply the Dirac equation in its different aspects, I came across the series of book "The Naked Spinor", and all their related sibliings, made by Dennis Morris. It's a quite new series of book, which states how current physics in ALL its aspects could be easily made born from a...
  39. N

    A Understanding Dirac Adjoint Derivation & Spinor Transformations

    I am trying to understand the derivation of the Dirac adjoint. I understand the derivation of the following identities involving Spinors, the Gamma matrices and Lorentz transformations: (Sμν)† = γ0Sμνγ0 s[Λ] = exp(ΩμνSμν/2) s[Λ]† = exp(Ωμν(Sμν/2)†) The part I'm having trouble with is...
  40. EnigmaticField

    A Why must a polar vector parameterized by spinor be timelike?

    I heard (somebody told me and I also read from some paper) that a polar vector whose components are parameterized by the Dirac spinor \bar\psi\gamma^\mu\psi must be a timelike vector. Why is so? I think a general polar vector can either be timelike or spacelike, isn't it? Is that because a...
  41. G

    I Understanding Spinor Rotations

    Hi, I am confused on a very basic fact. I can write \xi = (\xi_{1}, \xi_{2}) and a spin rotation matrix as U = \left( \begin{array}{ccc} e^{-\frac{i}{2}\phi} & 0 \\ 0 & e^{\frac{i}{2}\phi} \end{array} \right) A spinor rotates under a 2\pi rotation as \xi ' = \left( \begin{array}{ccc}...
  42. H

    Clarification of spinor solutions in Srednicki

    On page 235 of srednicki (print) it says to plug a solution of the form $$ \textbf{$\Psi$} (x) = u(\textbf{p})e^{ipx} + v(\textbf{p})e^{-ipx}$$ into the dirac equation $$ (-i\gamma^{\mu} \partial_{\mu}+m)\textbf{$\Psi$}=0 $$ To get $$(p_{\mu}\gamma^{\mu} + m)u(\textbf{p})e^{ipx} +...
  43. G

    Can the usual inner product be defined on spinor space?

    I would like to gain a more formal mathematical understanding of a construct relating to spinors. When I write down Dirac spinors in the Weyl basis, I see why if I multiply the adjoint (conjugate transpose) of a spinor with the original spinor I don't get a SL(2,C) scalar. It just doesn't work...
  44. D

    System of ODE - comparison with paper

    I have the following system of differential equations, for the functions ##A(r)## and ##B(r)##: ##A'-\frac{m}{r}A=(\epsilon+1)B## and ##-B' -\frac{m+1}{r}B=(\epsilon-1)A## ##m## and ##\epsilon## are constants, with ##\epsilon<1##. The functions ##A## and ##B## are the two components of a...
  45. S

    Why chiral fermions don't exist in odd dimensions?

    In four dimensions, left and right chiral fermion can be written as \psi_L= \begin{pmatrix} \psi_+\\ 0 \end{pmatrix},\qquad \psi_R= \begin{pmatrix} 0\\ \psi_- \end{pmatrix}, respectively, where \psi_+ and \psi_- are some two components spinors(Weyl spinors?). In this representation, the...
  46. Spinnor

    Spinor in 1+3 spacetime-flag+pole+entanglement rel.,in 1+1?

    In 1+3 dimensional space time a spinor can be thought of as a flagpole+flag+entanglement relationship. Is there some similar construction in 1+1 dimensional space time? Thanks for any help!
  47. stevendaryl

    How Do Spinors Fit in With Differential Geometry

    When I studied General Relativity using Misner, Thorne and Wheeler's "Gravitation", it was eye-opening to me to learn the geometric meanings of vectors, tensors, etc. The way such objects were taught in introductory physics classes were heavily dependent on coordinates: "A vector is a collection...
  48. Spinnor

    Dirac spinor in 1+1d, do the 2 components represent spin?

    The Dirac equation in 3+1 space-time yields spin, is this still true in 1+1d space-time? If not what do the 2 components of the spinor represent? Do we still have intrinsic spin in 1+1d space-time? Thanks for any help!
  49. Spinnor

    1+1D Dirac eq., low energy 2 components of spinor roughly =?

    I found a paper that derives the Dirac equation in 1 + 1 dimensional space-time. It is equation 8, here, http://academic.reed.edu/physics/faculty/wheeler/documents/Classical%20Field%20Theory/Miscellaneous%20Essays/A.%202D%20Dirac%20Equation.pdf and here...