Questions tagged [spinors]
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783
questions
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QFT - Allowed transitions for a given vertex
I'm given the following Lagrangian :
$$ L = L_{f1 }(\psi_1, \bar\psi_1) + L_{f2}(\psi_2, \bar\psi_2) + \lambda (\bar\psi_1 \gamma^{\mu} \psi_2) (\bar\psi_2 \gamma_{\mu} \psi_1) $$
where $L_{fi}$ are ...
0
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61 views
Feynman rules to spinor helicity
The following amplitudes of the 3-gluon vertex are obtained only from momentum conservation (shown in Quantum Field Theory and the Standard Model, Schwartz):
$$M^{+++}=C^{abc}\frac{1}{\langle12\rangle\...
0
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0answers
48 views
Adjoint spinors in Feynman diagrams
I am reading through Griffiths textbook "Introduction to Elementary Particles" and I am in the middle of the chapter on Quantum Electrodynamics and Feynman Rules for his diagrams.
There is ...
1
vote
2answers
63 views
How is $j=1/2$ representation, $U(R(\theta,\hat{\bf n}))=e^{i{\sigma}\cdot{\hat {\bf n}}\theta/2}$, is a projective representation of ${\rm SO}(3)$?
A projective unitary representation of ${\rm SO(3)}$ satisfies $$U(R_1)U(R_2)=e^{i\phi(R_1,R_2)}U(R_1R_2)\tag{1}$$ where $R_1,R_2\in {\rm SO(3)}$. How to show that the $j=1/2$ representation, $U(R(\...
2
votes
2answers
72 views
Are Dirac/Weyl/Majorana fermions exclusive?
I think we can be pretty sure that fermions exist. We have several ways to describe them (Dirac, Weyl, Majorana, maybe someone I'm missing?), with different equations and number of components. My ...
0
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1answer
55 views
Deriving the Generalized Fierz Transformation from Schroeder's Textbook
I am self studying QFT from the textbook An Introduction of Quantum Field Theory and the corresponding solutions from Zhong-Zhi Xianyu. The generalized Fierz Transformation is derived in problem 3.6. ...
4
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0answers
51 views
Are there non-lagrangian field theories for massive Weyl spinors?
It is a well-known fact that a chiral fermion is massless, since the mass term involves both a right-handed and a left-handed field. If you have one chirality only there are no mass terms one can ...
asked May 27 at 15:54
AccidentalFourierTransform
42.3k20 gold badges92 silver badges186 bronze badges
9
votes
2answers
253 views
Conflicting definitions of a spinor
I'm moving my question from math.stackexchange over here because it got no attention over there, even after 3 weeks and a 50-point bounty, and also because this is a very physics-oriented math ...
1
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0answers
38 views
In what meaningful way can we talk about the generators of the super-Poincaré algebra being “spinors” or “4-vectors” because of their indices?
It is often presented without much justification that the generators of the super-Poincaré algebra carry indices that imply they are elements of a representation space of the Poincaré algebra. In ...
1
vote
2answers
61 views
Levi-Civita symbol in 2-spinor notation
I'm reading An Introduction to Twistor Theory, by Huggett and Tod, and I don’t get the result we're being given page 17: the 2-spinor form of the 4 dimensional Levi-Civita symbol.
\begin{equation}
\...
0
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0answers
20 views
Flipped sign in solution of Dirac bispinor
I’ve been following David Tong’s lecture notes on QFT, and I came across something that I couldn’t work out for the general solution of the Dirac Spinors. On page 100 in the link here Tong Notes Part ...
1
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0answers
34 views
What is the definition of a Majorana fermion in conformal field theory?
Majorana spinors background
According to Eq. (4.84) and (4.85) of these notes, charge conjugation of the spinor $\Psi$ is defined as
$$ \Psi^{(c)} = C \Psi^*,$$
where $C$ is the unitary charge ...
4
votes
1answer
110 views
Spin $1/2$ motivation for twistor theory in general relativity?
I was watching this Youtube video (I have linked it at the relevant timestamp) and to paraphrase Dr. Woit' s motivation for twistor theory:
Within the standard way of thinking about general ...
12
votes
4answers
1k views
Do spinors form a vector space?
In contradiction to a number of other authors (sample ref below), Gerrit Coddens, at France’s prestigious Ecole Poytechnique, asserts that:
2.2 Preliminary caveat: Spinors do not build a vector space
...
1
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0answers
33 views
Majorana fermions in Euclidean and Minkowski signatures - contradiction with Wikipedia Table
In this wonderful lecture note on Clifford Algebra and
Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf
Somehow I find some inconsistency with his Tables of Euclidean and ...
2
votes
1answer
33 views
The 'mutual' and the 'self' in terms of the 'conjugacy' of Euclidean and Minkowski Weyl fermions
Euclidean and Minkowski fermions are shown in the Table of Wikipedia. (see the bottom https://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions)
My question is that what does the conjugacy mean ...
6
votes
2answers
484 views
Trying to understand spin
My question is fairly simple and straightforward. I'm studying Quantum Mechanics, specifically the spinor formalism. I understand that one can define a generator of rotations, say around axis $z$ by ...
0
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0answers
51 views
Accounts on the solutions of the Dirac equation
Consider the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$. As it is well known, there are different representations for the matrices $\gamma^{\mu}$, $\mu = 0,1,2,3$, the most famous ones ...
0
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0answers
49 views
How would the “belt trick” analogy of a pion look like?
Since the (charged) pion has a spin of 0, this by itself would suggest that it has spherical symmetry. However, because if it's non-zero iso-spin property it does not have spherical symmetry (is this ...
0
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0answers
65 views
Why there is a minus sign before $p^0$?
In scattering amplitudes, page 244, I am trying to verify that $$
p_{a\dot{b}} = \left( \begin{matrix} -p_0+p_3 & p_1-ip_2 \\ p_1+ip_2 & -p_0-p_3 \end{matrix} \right), \quad (1)
$$ i.e, ...
1
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0answers
34 views
Riemannian and Weyl tensors as spinor representation
There is the way of converting vector indices to spinor indices, for example, Maxwell stress tensor $F_{[\mu\nu]}$ can be decomposed to $(1,0) \oplus (0,1)$ irreducible representations of $\mathfrak{...
0
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0answers
33 views
Plane waves solutions for Dirac equation in terms of eigenstates of helicity
Suppose $\sigma_{1},\sigma_{2}$ and $\sigma_{3}$ are the Pauli matrices. Given a momentum ${\bf{p}}$, we define the helicity operator:
$$ h = \frac{1}{2}\begin{pmatrix} {\bf{\sigma}}\cdot {\bf{\hat{p}}...
1
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0answers
43 views
Coleman–Mandula theorem and Ward Identity
I was reading a paper on Coleman–Mandula theorem and Ward Identity [The Coleman-Mandula Theorem by Sascha Leonhardt]1, where I saw it says that-
Let a higher spin current $\hat{B}_{\mu\nu}$ is non ...
1
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0answers
41 views
Summing over spins in QED and calculating the square of Feynman amplitudes
I'm trying to compute the differential decay rate given by the following amplitude:
$$M = i g \bar{u}(q,s)\gamma^\mu \displaystyle{\not}\epsilon^\mu_r(p)v(\tilde{q},s')$$
which concerns the ...
0
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0answers
54 views
What is the physical meaning of the components of a spin 1/2 spinor in matrix representation?
If the spin operators for spin $1/2$ can be represented in matrix form using the Pauli Matrices,
e.g $S_x = \frac{1}{2}\hbar \sigma_x = \frac{1}{2}\hbar \begin{bmatrix}0&1\\1&0\end{bmatrix}$, ...
2
votes
1answer
45 views
Sign error when deriving Weyl spinor transformation laws (3.37) in Peskin Schroesder
I am having some trouble deriving the transormation laws for the weyl spinors, equation (3.37) in the Peskin Schroesder book on quantum field theory.
Beginning with the relation $\psi\to(1-\frac{i}{2}\...
0
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0answers
47 views
Quantum Field theory, coupling term, Electroweak Unification
In Anthony Zee, Quantum Field Theory in a Nutshell(2nd edition), Zee writes on Page 381, when $\phi$ - the Higgs field, acquires the vacuum expectation value $\begin{bmatrix} 0 \\ v \end{bmatrix}$, a ...
1
vote
1answer
111 views
Spinors and spin group
It seems to me that spinors (pinors) are loosely defined as representations of the spin (pin) group $Spin(p,q)$ ($Pin(p,q)$), which double covers the spacetime symmetry group $SO(p,q)$ ($O(p,q)$). $\...
0
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1answer
63 views
A question involving chiral transformations and gamma matrices
I'm looking at a calculation that involves an infinitesimal transformation of a Dirac fermion field:
$$\Psi \rightarrow e^{i \beta \gamma^5} \Psi.$$
Then the conjugate field $\bar{\Psi} = \Psi^{\...
0
votes
1answer
64 views
Solution to Dirac equation
We take the solution of Dirac equation as 4 component wave function (Dirac Spinor). But how do we know that it can't be a square or rectangular matrix like 4x2 or 4x4 matrix?
1
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0answers
53 views
Confused about chirality of antiparticles and QED interaction
In the middle of doing some calculations and I seem to have forgotten some basics from QFT. I would be very grateful if someone could help me out!
Taking, for example, an interaction proportional to $\...
0
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0answers
38 views
Another form of the solutions of the Dirac equation
Consider the Dirac equation $[i\gamma^\mu\partial_\mu-m]\psi=0$ and let me focus in particular on the positive-energy solutions by the ansatz
$$
\psi(x)=e^{-ipx}u(\mathbf p,r).
$$
Making this ...
0
votes
1answer
66 views
Dirac spinor in the chiral basis
In the chiral basis, the gamma matrices take the form
$$
\gamma^0=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \quad \gamma^j=\begin{bmatrix}0 & -\sigma^j \\ \sigma^j & 0\end{bmatrix}
$$...
0
votes
1answer
54 views
How to show that $\sigma^2\psi_L^*$ transforms as a right-handed spinor? (Peskin&Schroeder)
In Peskin & Schroeder, it is written that the quantity $\sigma^2\psi_L^*$ transforms as a right-handed spinor. What confuses me is that I only get the correct result when considering the following:...
0
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0answers
29 views
Dirac spinors completeness relation for $d>4$
In $d=4$ and in the rest mass frame, the four Dirac spinors $u_s(0)$ and $v_s(0)$ satisfy the completeness relation
$$
\sum_{s=1}^{2} u_s(0) \overline{u}_s(0) - \sum_{s=1}^{2} v_s(0) \overline{v}_s(0)...
1
vote
1answer
105 views
What is Dirac indices?
In Maggiore A Modern Introduction to Quantum Field Theory Eq. 4.31
$$\{\Psi_a(\vec x,t),\Psi_b(\vec x,t)\}=\delta^{(3)}(\vec x-\vec y))
\delta_{ab}$$
where "$a,b=1,2,3,4$ are the Dirac ...
0
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0answers
28 views
Helicity operator for particles and antiparticles
I´ve come across the fact that the helicity operator for particle solutions u(p) is not the same as the helicity operator for anti-particle solutions v(p).
For particle solutions:
$\begin{equation}
h =...
0
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0answers
22 views
Confusion with helicity eigenstates of massless spinors
As presented in Schwartz´s QFT and SM one can solve the free Dirac Equation in the Weyl basis.
If $p^{\mu} = (E,0,0,p_z)^T$ the four solutions are:
\begin{equation}
u_1^p = \begin{bmatrix}
\...
0
votes
1answer
42 views
Are the spinors one can find in the Feynman rules always solutions of the free Dirac equation?
For a given Feynman diagram one can calculate the matrix Element by translating the diagram into math using Feynman rules.
In these calculations one will encounter incoming and outgoing particles (and ...
0
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0answers
17 views
What's the spin state in the rest frame of an electron which is in a helicity eigenstate?
Helicity is the spin component in the direction of momentum: $\mathbf{\Sigma \cdot \hat{p}}$.
For a free electron, the helicity commutes with the free Dirac Hamiltonian $H = c\boldsymbol{\alpha}\cdot\...
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0answers
27 views
Problem with constructing a bispinor in the spinor helicity formalism
The $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group is constructed as $(\frac{1}{2},0) \otimes (0,\frac{1}{2})$.
To get an element of the vector space this specific representation acts ...
0
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0answers
31 views
Under which representation (and how exactly) transforms the given bispinor? [duplicate]
I am currently reading through Chapter 27 (Spinor helicity formalism) in Schwartz´s "QFT and the SM".
In this chapter it says that since 4-momenta transform in the (1/2,1/2) representation ...
0
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0answers
30 views
Parity of a fermion bilinear
I'm assuming that the parity transformation of a 4-vector field is:
$$x^\mu = (t,\mathbf{x}) \rightarrow x'^\mu = (t',\mathbf{x'}) = (t,-\mathbf{x})$$
$$V^\mu(t,\mathbf{x}) = (V^0(t,\mathbf{x}), V^i(t,...
2
votes
1answer
120 views
Factoring the Laplace operator $\Delta$ in dimensions $D \geq 3$
Consider the Laplace operator in 2 dimensions
\begin{equation}
\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} = \partial^2_x + \partial^2_y
\end{equation}
By defining the ...
2
votes
1answer
68 views
Transformation between left-handed spinors and right-handed spinors
I am learning (Weyl) spinor formalism from Müller-Kirsten and Wiedemann's Introduction to Supersymmetry (2nd Ed., WS, 2010, here). I am quite confused about the transformation between left-handed ...
1
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0answers
18 views
Homogeneous (projective) coordinates and spinors
When a complex number is considered as the stereographic projection from a sphere to the Argand plane, and then is represented by two “homogeneous coordinates” (in order to allow for a “point at ...
0
votes
1answer
95 views
Weyl and Dirac spinors
I know that the Dirac spinor is composed of two Weyl spinors and each of the Weyl spinors also has two components. Can I see its two components as two different wave functions? Can I see four ...
1
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0answers
30 views
Why does the Dirac beta matrix commute with the angular momentum operator?
This is the Dirac Hamiltonian,
and Beta is
The question says it all, I don't understand why Beta would commute with $ \hat L$
0
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0answers
38 views
Chiral Fierz identity
I am having trouble with proving the following:
$$(\bar{\psi}_1 P_R \psi_2) (\bar{\psi}_3 P_R \psi_4)=-1/2 (\bar{\psi}_1 P_R \psi_4)(\bar{\psi}_3 P_R \psi_2)-\dfrac{1}{8}(\bar{\psi}_1\sigma_{\mu\nu} ...
1
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0answers
50 views
Weyl Spinor Representation and Single Particle States
I'm trying to study representation theory for quantum field theory. Let me first summarize my current state of (hopefully correct, please correct me if I'm wrong about something) knowledge:
Single ...
