Questions tagged [spinors]
The spinors tag has no usage guidance.
553
questions
2
votes
1answer
49 views
Supercurrent conservation for super-Yang-Mills in D=3,4,6,10 dimensions
I am following the book by Freedman and Van-Proeyen and this question is related to exercise 6.3.
The supercurrent of a super Yang-Mills theory is given by $\mathcal{J}^{\mu} = \gamma^{\nu \rho} F^...
1
vote
1answer
37 views
Some counting of field degrees of freedom for a classical spin-1/2 Dirac field
A classical real scalar field admits a decomposition $$\phi(x)\sim a_pe^{-ip\cdot x}+a_p^*e^{+ip\cdot x}$$ which tells that at each $x$, there exists a real number i.e., one degree of freedom at each ...
4
votes
0answers
74 views
The spinor metric, basic spinor calculations and spinor indices
I'm currently reading the textbook "Finite Quantum Electrodynamics" by Günter Scharf, but I find myself stuck already on page 24.
Background
Scharf introduces the index-raising symbol (spinor metric)...
2
votes
0answers
79 views
Almost complex structure $J_i^j$ from covariantly constant spinor $\eta$
In the context of superstring compactification on a 6-manifold which admits a covariantly conserved spinor $\eta$, which we normalize so that $\eta^{\dagger} \eta = 1$, I am trying to show that the ...
0
votes
0answers
24 views
Why does the majorana equation preserve handedness?
In the "QFT Nutshell" by A. Zee, it is stated that
The Majorana equation is
$$i\not\partial\psi=m\psi_c$$
where $\psi_c$ is the charge conjugated spinor $\psi_c = \left(C\gamma^0\right)\psi^*$....
8
votes
3answers
366 views
Spin vs orbital angular momenta in QFT
I couldn't find past answers that quite match what I wanted, so I will try to ask in slightly different manner. In QM, we have total angular momentum operator $\vec{J}$ (I dropped the hat for ...
1
vote
0answers
19 views
Spin Sums & Conservation of Angular Momentum
External fermions and bosons have momentum and spin (polarization) degrees of freedom.
E.g. decay rate of $t\rightarrow W^{+}+\bar{b}$ with an unpolarized t beam.
Peskin & Schröder sum over t, $\...
0
votes
1answer
63 views
A Question About $4$-Spinor Contractions
Let $f_{abc}$ be a constant which is totally anti-symmetric with respect to indices $a$, $b$ and $c$. Let $\psi^{a}$, $\psi^{b}$, $\psi^{c}$ and $\epsilon$ be Grassmann-valued Majorana fermions. How ...
0
votes
0answers
45 views
Relativistic quantum field theory
Let $\psi(x)$ be solution of Dirac equation $$ (\gamma^\mu\Pi_\mu-mc) \psi(x)=0 $$ where $\Pi_\mu=i\partial_\mu-eA_\mu$ is momentum operator in present electromagnetic field . We consider tow ...
0
votes
0answers
33 views
What does it mean to take the tensor product of two reps of the Lorentz group? [duplicate]
If I reduce the Lorentz group to the representation $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$, I can write left and right-handed Weyl spinors respectively as $\left( \frac{1}{2},0 \right)$ and $\left(...
3
votes
2answers
102 views
Why would a spinor transform under Lorentz transformations?
From my understanding of spinors, they arise as projective representations of $SO_0(1,3)$ that do not correspond to representations of $SO_0(1,3)$.
But still one says here - and virtually everywhere - ...
1
vote
0answers
63 views
Derivation of the spin connection in curved spacetime
I am trying to understand the derivation of spin connection from the book "Quantum Field theory in Curved spacetime" by L. Parker and D. Toms.
In chapter 5, page 223, they have written (which I am ...
2
votes
2answers
86 views
Quantized value of spin angular momentum and underlying mysteries
I think the quantized value of the spin angular momentum is $\hbar/2 $ rather than $\hbar $ is the basic reason for the $4\pi$ rotation of a wave function to retain its initial state again? Is it true?...
7
votes
4answers
321 views
Position representation of spin states and spin operators
How can we represent a spin states
$ \lvert S_x:+\rangle, \lvert S_y:+\rangle,\lvert S_z:+\rangle ,\lvert S_x:-\rangle, \lvert S_y:-\rangle $ and $\lvert S_z:-\rangle$ in position representation ...
1
vote
0answers
31 views
Completeness relation of spin matrices
I was reading Hugh Osborne's notes on Conformal Field theory and came across a completeness relation which seems easy to prove but I am unable to do it.
${(s_{\mu\nu})}_{\alpha}^{\beta}{(s^{\mu\nu})}...
0
votes
0answers
44 views
Spinor transformations as representations of $\mathrm{SL}(2, \mathbb{C})$
Background
In the Weyl representation of the Dirac $\gamma$-matrices, the spinor transformations $S=e^{\frac{1}{2} \omega_{\alpha \beta} \Sigma^{\alpha \beta}} \in \,\mathrm{G}_{\mathrm{L}} \leq \...
2
votes
1answer
92 views
Dirac equation derivation
I am working through a set of lecture notes containing a derivation of the Dirac equation following the historical route of Dirac. It states that Dirac postulated a hermitian first-order differential ...
1
vote
0answers
31 views
What does a supercharge physically conserve? [duplicate]
What is actually being conserved? I've calculated it for the Wess-Zumino model but I still have no idea what is actually being conserved due to Noether's Theorem.
There is already a similar question, ...
1
vote
1answer
103 views
Setting spinors and $SU(2)$ representations on the same patch
I am sorry for the naivety of this question, I am a mathematician and I am trying to put together different ideas.
I am trying to understand the vocabulary of physics, in particular, I want to know:
...
1
vote
0answers
38 views
Dirac spinor as null vectors
In this paper, on page 9, the authors show that a spinor is equivalent to a null vector with a bit of extra structure (just one real parameter I think?):
https://arxiv.org/abs/1312.3824
They then go ...
2
votes
1answer
109 views
Dirac equation in curved spacetime
As we know, the laws of physics in curved spacetime are obtained to lowest order by upgrading the flat space laws by substituting partial derivatives with the appropriate covariant derivatives. In the ...
0
votes
1answer
90 views
Why is Dirac equation a matrix equation?
According to Wikipedia's Dirac equation article, the Dirac equation can be written in form
$$ i\hbar\gamma^{\mu}\partial_{\mu}\psi-mc\psi=0, $$
where $\gamma^{\mu}$ are gamma matrices which are $4 \...
0
votes
1answer
39 views
Demonstration of identities appearing in Dirac spinors in the chiral representation
Using the chiral representation of the gamma matrices, Peskin and Schroeder arrive in some expressions for the 4-component spinors $u(p)$ and $v(p)$ in terms of a square root of the Pauli matrices ...
1
vote
0answers
43 views
Dirac spinor for arbitrary momentum
In many treatments of the Dirac equation (e.g. Peskin and Schroder, pages 45-46) after subbing in $\psi(x) = e^{-ix_\mu p^\mu}u(\vec p)$, with $u$ a constant spinor, into the Dirac equation, we ...
0
votes
1answer
53 views
Legal values of spin-1/2 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, .. (Grassmann)?
For the spin-1/2 fermion field $\psi$,
we may choose it to be a spinor which needs to be
Grassmann variable
but can also be
complex $\mathbb{C}$ Grassmann. (Dirac or Weyl spinor/fermion)
We can ...
4
votes
1answer
75 views
Varying the Dirac action with differential forms
The Dirac action in a curved spacetime can be written in terms of the vierbein $\{ e^a \}$ and spin connection $\{ \omega^{ab} \}$ differential forms. Let the spinor field $\psi$ be interpreted as a ...
0
votes
0answers
22 views
Hund's Rule Coupling in an Effective Hamiltonian/Lagrangian
I am reading a book on Skyrmions, and I am at the part where the interaction of skyrmions with electrons is discussed. The chapter speaks of Spin-Transfer Torque (STT) and makes the following ...
2
votes
2answers
163 views
Can gamma matrices be real in 6 dimensions?
I'm trying to find the really real representation of 6D gamma matrices. The problem is that "do they really exist?"
If yes, then how am I supposed to construct them? Thank you!
0
votes
0answers
41 views
Transformation of Vectors
let $\Psi \in V$ be a vector and we have the action of a lorentz transformation on the object $\sigma_2 \Psi $. And $\sigma_2 \Psi $ is then in V as well. V is "Weyl or Dirac space". And the lorentz ...
1
vote
0answers
83 views
Again on Spin Operator in Dirac Field Theory (Peskin & Schroeder)
Good morning,
I've already seen that this topic has been discussed so long, but my doubts remain unchanged. At page 61 of Peskin & Schroeder, An Introduction to QFT, there is the demonstration ...
0
votes
1answer
77 views
Fierz identities and Weinberg operator
I've been told that
$$
(\bar{L_i^c}\widetilde{\phi}^{\ *})(\widetilde{\phi}^\dagger L_k) = -\frac{1}{2}(\bar{\widetilde{L}}_i \vec{\sigma}L_k)(\widetilde{\phi}^\dagger \vec{\sigma} \phi) \tag1
$$
by ...
0
votes
0answers
34 views
Generally Covariant Dirac equation: The spin connection
Wikipedia, an answer on stackexchange and a few papers in the Arxiv I've found all have different definitions of the spin connection found in the Dirac equation. Can anyone please tell me what the ...
1
vote
0answers
39 views
Periodic Killing spinor on $S^1$?
In Cartesian coordinates on $R^2$ we have a constant two component Killing spinor $\epsilon_0$. If we use polar coordinates
$x = R \cos t$
$y = R \sin t$
we have the vielbein
$e^R = dR$
$e^t = ...
0
votes
2answers
117 views
Lorentz transformation of a Weyl Spinor?
A left handed Weyl Spinor belongs to the $(\frac{1}{2},0)$ representation of the Lorentz group. So given the Spinor, the unitary representation of the Lorentz transformation should look like $\exp{iA\...
4
votes
2answers
271 views
Dirac equation in 1+1D spacetime compared to “standard” 3+1D Dirac equation
In the past couple of weeks I've been studying the Dirac equation and its solutions. During a discussion with a tutor it was pointed out to me that one could formulate something similar to the Dirac ...
0
votes
1answer
55 views
A question about Lorentz transformations in spinor representation
For
$$\Lambda^{\mu}{}_{\nu}= \frac{1}{2} Tr(\bar{\sigma}^{\mu} S \sigma_{\nu}S^{\dagger}) $$
We need to prove that
$$\Lambda (S)= \Lambda (-S)$$
Am I naive to say that by adding $-S$, $S^{\dagger}...
0
votes
0answers
27 views
Sources to learn about Killing spinors
What are some good sources to learn about Killing spinors from?
I am currently learning about Killing vectors and how they are the generators of a Lie algebra that corresponds to the isometry group ...
0
votes
0answers
55 views
Trying to prove the Wess Zumino invariance under a SUSY transformation
I have the Lagrangian density
$$L=-\partial_\mu \phi^\star \partial^\mu \phi - \bar{\chi}_R \gamma^\mu \partial_\mu \chi_L - \bar{\chi}_L\gamma^\mu\partial_\mu \chi_R.$$
where $\epsilon$ is the ...
0
votes
1answer
69 views
Infinitesimal parameter of Lorentz transformation
I'm working through the SUSY lecture notes by Lambert, and he does something which seems strange to me during the calculation of the Wess Zumino model.
He says the spinor $\psi$ has the ...
1
vote
1answer
80 views
Spinor representation
I am trying to study Special unitary group of order 2 and some textbooks mention objects transform under special unitary group are called Spinors. then How can we represent a spinor using matrix?
0
votes
1answer
56 views
Vanishing of a solution of Dirac equation
Let $\psi(x,t)$ be a solution of the free Dirac equation. Assume that $$\psi(\vec x,0)=\delta^{(3)}(\vec x) u,$$
where u is a fixed spinor. (In other words $\psi(\vec x,0)$ is assumed to be supported ...
1
vote
1answer
32 views
How does one find the parity trasformation matrix of spinors for non-free field theory?
In many QFT textbook, for example, the book of Srednicki, they use free field theory to derive the transformation matrix of the Spinors:
$$P^{-1}\Psi(x)P=D(P)\Psi(P^{-1}x)$$
Then we have a relation:
...
1
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0answers
37 views
Question about Pauli Matrices
I found the following identities about Pauli matrices from the lecture notes of Supersymmetry.
$$((\sigma^{\mu})^{\alpha\dot{\alpha}})^{\ast}=(\bar{\sigma}^{\mu})^{\dot{\alpha}\alpha}$$
$$((\sigma_{\...
0
votes
1answer
39 views
Unphysical degrees of freedom for the Weyl spinor?
I am attempting to solve the Weyl equation:
$$\bar\sigma^{\mu}\partial_{\mu}\phi=0$$ Where $\bar\sigma^{\mu}=(-1,\vec{\sigma})$ in my convention, and $\phi$ is a two component Weyl spinor. I consider ...
0
votes
0answers
22 views
I want to know the conformal weights of spinors in 2D
I want to know the conformal weights(or dimensions) of left/right-moving fermions in 2D, ${\cal N}=(2,2)$ superconformal theory.
More specifically, what is the left/right-moving conformal dimension ($...
1
vote
0answers
33 views
Naive question about particles and spinor fields [duplicate]
What is the difference between the "real" particle electron (for instance) and the spinor field of electron? I mean, which means that the electron have been described by a spinor field?My question is ...
2
votes
0answers
36 views
How does the spin connection affect the dynamics of a fermion in curved space?
Consider a massless right-handed Majorana fermion in curved spacetime. Without any other fields present, the Lagrangian density is (I believe) the following:
$$ \mathcal{L}_{\psi} = \sqrt{g}i\bar{\...
1
vote
1answer
44 views
Identity Involving Grassmann Variables and Pauli Matrices
I am trying to prove the following identity:
$$\theta\sigma^{\mu}\bar{\theta}\theta\sigma^{\nu}\bar{\theta}=\frac{1}{2}g^{\mu\nu}\theta\theta\bar{\theta}\bar{\theta}$$
Where $\theta$ and $\bar{\...
1
vote
0answers
30 views
Quantization of the massless neutrino field
If a massless neutrino or anti-neutrino is considered (in the whole post I consider neutrinos res. anti-neutrinos as mass-less), it is described by the Weyl-equation:
$$\overline{\sigma}^{\mu}\...
2
votes
0answers
46 views
Why can the spin operator be written as a product of fermions?
I was studying the Hubbard model, where we define the spin operator $\vec{S} = \frac{1}{2} c^\dagger \vec{\sigma} c$, where the creation and annihilation operators are both vectors of the form $c^\...
