Questions tagged [spinors]

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Bhabha scattering in the spinor-helicity formalism

I am trying to calculate the square amplitude for Bhabha scattering $e^-(p_1)e^+(p_2)\rightarrow e^-(p_3)e^+(p_4)$ using the spinor-helicity formalism but one of the interference terms just will not ...
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Rotation operator from angular momentum or spin operator

My instructor on quantum physics just stated that the total angular momentum operator, $\hat{J}$, can be expressed as $\hat{J}=\hat{L}+\hat{S}$, where $\hat{L}$ is angular momentum operator ...
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What are the Feynman rules for the spinor-helicity formalism?

If we do not work with helicity amplitudes, there are Feynman rules for the external legs of a Feynman diagram, i.e. $u_s(k),\overline{v}_s(k),\epsilon_r(k)$ for an incoming fermion, antifermion and ...
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1answer
215 views

Does Wightman's unitary $U(\Lambda)$ really exist for Lorentz boost?

This question is related to another question here. But I am asking a more fundamental question about the existence of Wightman's unitary $U(\Lambda)$ for Lorentz transformation. Let $\psi^\alpha$ be a ...
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40 views

What do these matrices represent physically? Are they related to Majorana spinors?

I am studying $\mathfrak{so}(1,3)$ representations and I found this claim that $(m,n)\oplus(n,m)$ representations have a real structure (for which I asked a separate question on math.SE). I tried to ...
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49 views

Commutator and Anti-commutator

In spinor field representation I found that for any scalar $\#$, following happens: $[\Sigma^{ij},\#\gamma^0]=0$ and $\{\Sigma^{i0},\#\gamma^0\}=0$ Is there any analogue for vector representation too? ...
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2answers
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Spatial inversion and Time reversal

On the spinor field $\psi^{\mu}(x)$, I found the action of $\psi^{\mu}(x)$ on spatial inversion $P$ by postulating $\psi^{\mu}_{P}(x)=P^{\mu}_{\nu}\psi^{\nu}(P^{-1}x)=P^{\mu}_{\nu}\psi^{\nu}(t,-x)$, ...
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Does this argument prove that all fermionic states have zero norm?

The following argument seems to show that all states created by a fermionic field have zero norm. This would surely cause problems in QFT, so I believe there must be an error somewhere, but I can't ...
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59 views

An essential exercise involving spin composition

Suppose we have two particles with spin $1/2$. They have $S^{tot}=1$ and $S^{tot}_y=0$. How can we write the state of the system in terms of the eigenstates of $S_{1z},S_{2z}$? My attempt: I would ...
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21 views

Fierz Identity calculation

While reading an article, it's said that to simplify the following Dirac structure $$\left(P_Lv_j^d\bar{v}^s_kP_R\right)_{\alpha\beta}\tag{1}\label{1}$$ where $j,k$ are color indices and $d,s$ ...
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1answer
72 views

Spinors and Tensors: what is the form of spin transformation matrix?

The (covariant) vector transformation law is given by: $$V^{'}_{\mu} = t^{\nu}\hspace{0.1mm}_{\mu'}V_{\nu} =\frac{\partial x^{\nu}}{\partial x'^{\mu}}V_{\nu} \tag{1}$$ where the transformation is ...
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3answers
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Problem regarding Eigenfunctions of Identical Particles in Quantum Physics

Suppose I have 2 electrons under an harmonic potential: $$H=\frac{1}{2m}(p_1^2+p_2^2)+\frac{1}{2}m\omega ^2 (x_1^2+x_2^2)$$ Now let's think about the eigenvalue and the eigenfunctions of energy of the ...
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57 views

Resources for general relativity (especially spinors) without abstract index notation

I'm taking a course on general relativity and try to find a good source on spinors. I'm using "General relativity" by Wald as a guideline, but I'm struggling with his notation. I already ...
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1answer
71 views

How these two approaches to spinors in curved spacetimes relate?

Regarding spinors in curved spacetimes I have seem basically two approaches. In a set of lecture notes by a Physicist at my department he works with spinors in a curved spacetime $(M,g)$ by picking a ...
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1answer
82 views

Lorentz generators $J_i,K_i$ and Casimir invariants in bispinor representation

Lorentz generators satisfy the Lie algebra $$[J_i,J_j]=i\epsilon_{ij}^kJ_k, ~~~~[J_i,K_j]=i\epsilon_{ij}^kK_k, ~~~~[K_i,K_j]=-i\epsilon_{ij}^kJ_k.$$ Now, define $$A_i=\frac{J_i+iK_i}{2},~~~~B_i=\frac{...
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75 views

Little-group and spinor helicity

I'm struggling a bit to understand the little group in the context of massless momenta and the spinor-helicity formalism. I'll clarify notation and my understanding through a brief recap, and put the ...
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1answer
81 views

How to prove these relations for Pauli matrices?

I am reading Schwartz's QFT book and I am trying to verify (10.141) and (10.142). σ means Pauli matrix and $ϵ:=−iσ_2$. How to prove these relations? $$\sigma^{\mu}_{\alpha\dot{\alpha}}\sigma^{\nu}_{\...
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Spin combinations forming representation of $\text{SU}(2)$ [closed]

I'm currently reading Griffiths' book about particle physics. On page 183, after showing a basis for a system of three spin-1/2 particles (tripe product space of up and down states), he says In the ...
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1answer
44 views

Obtaining the Pauli spinor from the Dirac spinor

While I'm fairly familiar with non-relativistic quantum mechanics, I have just recently started delving into relativistic QM, my main reference being Sakurai's "Advanced Quantum Mechanics" (...
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1answer
85 views

Lorentz indices to label rotation irreducible representations

Consider the $A_\mu\in\left(\frac{1}{2},\frac{1}{2}\right)$, the vector representation of the restricted Lorentz group. One can decompose this vector under spatial rotations as $A_\mu\in 0\oplus 1$ ...
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1answer
67 views

Why are the 4-vector and bispinor representation of the Lorentz algebra in particular so related?

When learning about the Dirac equation, there are several indications that the fundamental (4-vector) representation and the bispinor representations are connected in some way. To give an example, the ...
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46 views

How is the Weyl lagrangian zero for Weyl spinors?

The action from which Weyl equation can be derived is $$S=i\int{\psi_{L/R}^\dagger\bar{\sigma}^\mu\partial_\mu\psi_{L/R}}$$ where $\bar{\sigma}^\mu=(1,\vec{\sigma})$. Imposing $\delta S=0$ we arrive ...
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1answer
68 views

Why to introduce spinor fields we need this map in the definition of a spin structure?

Let me start with what I currently understand. Let ${\rm SO}(1,3)$ be the proper ortochronous Lorentz group. Its universal cover is ${\rm SL}(2,\mathbb{C})$. The representations of its universal cover ...
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2answers
76 views

How would the eigenstates of a particle with spin 3/2 look like?

I learnt in an introductory course about quantum mecanics how to work with spin 1/2 particles. I saw how the algebra is almost the same as for angular momentum, but no one ever told me about particles ...
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36 views

Closed superstrings IIA and IIB (7.22) of Clifford Johnson's book

In (7.22) the massless sectors is shown as \begin{equation} \text{IIA}:\; (\bf{8}_\text{v}\oplus\bf{8}_\text{s})\otimes (\bf{8}_\text{v}\oplus\bf{8}_\text{c});\;\; \text{IIB}:\; (\bf{8}_\text{v}...
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Summing of neutrino spins in Muon decay

This might be a silly question, but I am just a beginner so please bear with me. When we average the matrix element over spins, do we also have to sum over neutrino spins since neutrinos are only Left-...
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1answer
31 views

Finite spinor transformation

I am currently studying the finite spinor transformations in QFT. There is equation which i do not fully understand. Rather i don't understand the notation and what it represents: In the script, we ...
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1answer
71 views

Resource recommendation for Vector/Tensor/Operator and Spinor geometric understanding

I am a Bachelor's Physics student,currently on my 5th Semester (3rd Year 1st semester). We are at a point in Physics where the mathematics is getting seriously hard. Our Bachelor program in itself is ...
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53 views

Derivation of the Dirac spinor expressed as a Fourier transform

Introduction For the Klein Gordon Field, the equations of motion are described by the equation $$(\partial_{\mu}\partial^{\mu} + m^2)\phi(\vec{x},t)=0$$ Which when the field is expressed as a Fourier ...
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Help with a step in a derivation in Peskin & Schroeder

In page 53 of P&S, they postulate the commutation relations for the annihilation and creation operators of spin 1/2 particles. From there, they start computing the commutation relations of $\psi$ ...
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32 views

How to decompose the spinor kinetic term of $D=10$ SYM in terms of lower dimensional parts?

The kinetic term for the spinors in $D=10$ SYM is $\lambda \gamma^\mu \partial_\mu \lambda$, where $\lambda$ is a 16 component Majorana-Weyl spinor and $\gamma^\mu$ is a 16 by 16 matrix satisfying $\{\...
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1answer
69 views

Parity transformations and Dirac Spinor

I'm reading "No-Nonsense quantum field theory" and I have some doubts about the transformation law for Dirac Spinors as explained by the author. In the book the left chiral spinors $\chi$ ...
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1answer
48 views

Spin 1/2 as belt trick in a smooth field

In the (English) Wikipedia article on Spinor, there is an animation, demonstrating the Dirac belt trick as a model for Spin 1/2. My interpretation of that animation goes like this: If you rotate an ...
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1answer
43 views

Orthogonality of positron/electron Dirac spinors with different momenta?

Consider the Dirac spinor for a positron at rest, and the spinor for an electron with non-zero 3-momentum. In the Dirac basis it is clear that these are not orthogonal, as I would expect. Does this ...
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1answer
57 views

Proof of the Fierz Identity

In p.51 of the Peskin's QFT book, the author derives the Fierz identity as the following: $$(\bar{u}_{1R}\,\sigma^\mu\, u_{2R})(\bar{u}_{3R}\,\sigma_\mu\, u_{4R}) = 2\epsilon_{\alpha\gamma}\,\bar{u}_{...
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1answer
25 views

Can you define a spinor in relation to a mesh?

Taking a mesh of points which approximates a manifold $M$, one can define a vector field at each point by assigning each point A to another point B, which creates a vector AB. Similarly one can define ...
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59 views

$\rm SU(2)$ transformation of spinors

In the book QFT by Ryder on the topic $\rm SU(2)$ and the rotation group, it is stated that, A spinor $\xi$ transforms under $\rm SU(2)$ as, $$\xi \rightarrow \xi' = U \xi, \quad \xi^\dagger \...
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Weyl's spinor and Dirac spinor [duplicate]

Weyl's spinor and Dirac's spinor What is the difference between the two from a mathematical point of view? So are there different mathematical definitions of spinor? Is it correct to say that the Weyl ...
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1answer
94 views

Spin representations of Lorentz group

In the context of Classical Field Theory, we know that irreducible representation are labelled by the values of the two Casimir operators of the Poincaré group: we can have massive fields $P^2=m^2>...
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47 views

The objects that the $(\frac{1}{2}, \frac{1}{2})$ representation of the Lorentz group act on

In the Physics from Symmetry book (pg. 80-83), the author uses the convention of dotted and undotted spinor indices such that the $(\frac{1}{2},\frac{1}{2})$ representation of the $SO(1,3)$ Lorentz ...
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38 views

Why does $(\frac{1}{2}, \frac{1}{2})$ representation of $SO(1,3)$ Lorentz group act on $2\times2$ Hermitian matrices? [duplicate]

In pg. 81 of the Physics from Symmetry book, the author deduced that the $(\frac{1}{2}, \frac{1}{2})$ representation of the Lorentz $SO(1,3)$ group acts on $2\times2$ Hermitain matrices using the ...
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1answer
94 views

Transforming a Left-Handed Spinor into a Right-Handed Spinor

In pg. 125-126 of Symmetry and the Standard Model book, it was stated that for a spinor $\psi_L$ in the left-handed representation and a spinor $\psi_R$ in the right-handed representation, we can ...
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1answer
71 views

What is the best way to solve an half spin particle exercise?

This question is strongly related to this other question regarding the definition of spinor. Let's take for example the following exercise: Given a particle with mass $m$ and spin $1/2$, described by ...
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118 views

Problem regarding the definition of spinor

I am trying to build a simple understanding of what a spinor is, in doing so I have stumbled across two different definitions of spinor: First definition: If $|\psi\rangle$ is a generic state of a ...
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90 views

How to know if a spinor $\Psi\left(x\right)$ is the ground state of the system?

Suppose we have time independent one-dimensional single particle Schrödinger-like equation$$-\frac{d}{dx}\left(A\left(x\right)\frac{d}{dx}\psi\left(x\right)\right)+V\left(x\right)\psi\left(x\right)=E\...
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61 views

Why do physicists use wave functions with more than two components?

For $n$-body systems you just need a single component wavefunction. For example, for a two-body system you would need a wavefunction of 6 variables. $\psi(x_1,x_2,y_1,y_2,z_1,z_2)$ That satisfies the ...
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35 views

Ordering ambiguity in the spinor helicity formalism of scattering amplitudes

I am following Nima's course on "Quantum mechanics and spacetime, total positivity and motives", and I wonder if there's a direct way to understand the ordering ambiguity when we are summing ...
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1answer
60 views

How do spinors arise in systems of less than 3 effective spatial dimensions as representations of the Lorentz group?

In 3+1 Minkowski spacetime, we can use the fact that the Lie algebra of the Lorentz group decomposes (I am omitting some details here to keep it short) into $su(2) \oplus su(2)$ and then use the fact ...
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2answers
105 views

Notations related to identities for spinors in the rest frame

This question pertains to some notation in Zee's QFT book, Section II.2. The Dirac equation is $$ (i\gamma^\mu\partial_\mu-m)\psi(x)=0, $$ which we can write in momentum space with the Fourier ...
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1answer
105 views

Are spinors representations of the Lorentz group or its associated algebra?

I got confused about the true origin of spinors with regards to their connection to the proper orthochronous Lorentz group $SO^+(3,1)$. In the book on QFT by Maggiore that I'm following, we reached ...

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