Spinor Definition and 105 Threads

Hello everyone. Help me solve the problem. I don't understand how to handle this type of task. Find the energy levels of a spin 1/2 particle in a potential well: V(r)+W(r)*(l,s), where V(r<a)=-U, V(r>a)=0, W(r) = q*δ(r-a)

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 ?

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

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

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...

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

REMOVED pending revision

"##\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.

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...

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...

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...

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...

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...

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)...

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...

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}...

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}...

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

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...

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...

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...

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...

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} +...

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...

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...

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...

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...

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...

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 =...

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...

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...

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} +...

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...

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...

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...

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!

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

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...

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...

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...

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}...

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} +...

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...

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...

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...

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...

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!

I was reading in this book: Supergravity for Daniel Freedman and was checking the part that has to do with Extremal Reissner Nordstrom Black Hole. He was using killing spinors (that I am very new to). I was understanding the theory until he stated with the calculations: He said that the...

I would like to take the trace over spinorial indices of the following expression: (\gamma_{\mu}\gamma^{0})_{\alpha}^{\beta}=(\gamma_{\mu})_{\alpha}^{\gamma}(\gamma^{0})_{\gamma}^{\beta}. How do I go about doing this? I reckon I could expand the trace out (let's say I want to do this in 4D)...

Can you give me an intuitive understanding of the following: "The spin states of massive and massless Majorana spinors transform in representations of SO(D-1) and SO(D-2), respectively". I see the similarity with vectors bosons, where massive vectors have d-1 degrees of freedom and massless...