Questions tagged [spinors]
The spinors tag has no usage guidance.
566
questions
2
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1answer
45 views
Why are twistors commuting?
In his book, Srednicki introduces the notion of twistor in chapter 50. It is described as a simply commuting spinor, as opposed to anti-commuting. How do we know that this object is simply commuting? ...
2
votes
1answer
49 views
Co-spinors and contra-spinors
As i was reading my teacher's notes on $SU(2)$ and $SO(3)$, i have had this question. Why do co-spinors transform differently under a rotation than contra-spinors?
0
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0answers
28 views
Computation with bispinors for a compatible pair of pure spinors in N=1 supersymmetric vacua compactified down to 4-dimension
I ask this question basically because I need help, just an hint, for a computation with bispinors. The background is string theory / supergravity, where many data of a supersymmetric vacuum can be ...
2
votes
1answer
60 views
Weinberg Volume II: Abelian Anomaly Function
The following is from page 363 of Weinberg volume II.
We wish to evaluate the RHS of
\begin{align}\label{EQbbvbv}
[d \psi][d \bar{\psi}] \rightarrow(\operatorname{Det} \mathscr{U} \operatorname{Det}...
1
vote
0answers
44 views
Gamma Matrices as nonstandard numbers, and Grassman Numbers
I'm in the process of exploring the Dirac equation and its forms and consequences, and as such have just been initiated into the theory of spinors and their accompanying formalism.
One of the things ...
0
votes
1answer
32 views
What's the field strength and energy density of a spinor field?
The field strength and energy density of a vector field $A_\mu$ can be described using the field strength tensor $F_{\mu \nu}$.
What is the field strength and energy density associated with a (Dirac) ...
1
vote
1answer
38 views
$CP$-transformation for spinor field. $C$ and $P$ do not commute?
I am bothered by an exercise about CP transformations where I get the result that CP acting on a Dirac spinor field is not the same as the PC transformation. The exercise states the following ...
0
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0answers
27 views
How do time reversal and parity inversion act on a Majorana spinor in QFT?
Dirac particles are not the same Majorana particles. However, in the simple Lorentz group (boost and rotations, but no parity or time flips), they transform the same way. Particles in QFT were defined ...
2
votes
1answer
51 views
Correct transformation of left-handed Weyl spinor
In the book "Matthew D. Schwartz, Quantum Field Theory and the Standard Model", page 164, it says that a left-handed spinor transforms as
$$\psi_L \rightarrow e^{\frac{1}{2}(i\vec{\theta} - \vec{\...
0
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0answers
19 views
Spin Projection operators in “scalar” or “covariant” form?
At page 34 of "Relativistic Quantum Mechanics" by Bjorken and Drell the following spin projection operator for spinors "at rest" is considered:
$$\frac{1+\sigma_{z}}{2}\quad ,$$
where $\sigma_{z}$ is ...
0
votes
1answer
41 views
Spinor Understanding: QFT vs pure Representation Theory
I have some questions on terminology used in QM & QFT and (pure mathematical) representation theory treating the concept of "spinor".
Let us focus on Dirac spinor as described in https://en....
3
votes
1answer
93 views
How can we interpret the components of a polarization four-vector?
The four components of a Dirac spinor can be interpreted in terms of left-chiral and right-chiral spin up and spin down states.
How can we interpret the four components of a polarization four vector $...
0
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0answers
51 views
Can we derive $\vec{S}=\frac{1}{2}\vec{\Sigma}$ in a representation independent way in terms $\vec{\alpha},\vec{\beta}$?
For the Hamiltonian $H=(\vec{\alpha}\cdot \vec{p}+\vec{\beta}m)$ of the Dirac equation $i\frac{\partial \psi}{\partial t}=H\psi$, it can be shown that $[H,\vec{L}]=-i(\vec{\alpha}\times\vec{p})$. Now, ...
1
vote
1answer
48 views
Sign-Conventions for Spinor Transformations
In the literature one encounters a lot of different conventions for how left-handed spinor transforms (rotation angle $\phi$, rapidity $\beta$), among them
$M_L = M_{(\frac{1}{2}, 0)} = e^{-i \frac{1}...
2
votes
1answer
105 views
Supercurrent conservation for super-Yang-Mills in D=3,4,6,10 dimensions
I am following the book by Freedman and Van-Proeyen and this question is related to exercise 6.3.
The supercurrent of a super Yang-Mills theory is given by $\mathcal{J}^{\mu} = \gamma^{\nu \rho} F^...
1
vote
1answer
47 views
Some counting of field degrees of freedom for a classical spin-1/2 Dirac field
A classical real scalar field admits a decomposition $$\phi(x)\sim a_pe^{-ip\cdot x}+a_p^*e^{+ip\cdot x}$$ which tells that at each $x$, there exists a real number i.e., one degree of freedom at each ...
6
votes
1answer
163 views
The spinor metric, basic spinor calculations and spinor indices
I'm currently reading the textbook "Finite Quantum Electrodynamics" by Günter Scharf, but I find myself stuck already on page 24.
Background
Scharf introduces the index-raising symbol (spinor metric)...
2
votes
1answer
123 views
Almost complex structure $J_i^j$ from covariantly constant spinor $\eta$
In the context of superstring compactification on a 6-manifold which admits a covariantly conserved spinor $\eta$, which we normalize so that $\eta^{\dagger} \eta = 1$, I am trying to show that the ...
0
votes
0answers
30 views
Why does the majorana equation preserve handedness?
In the "QFT Nutshell" by A. Zee, it is stated that
The Majorana equation is
$$i\not\partial\psi=m\psi_c$$
where $\psi_c$ is the charge conjugated spinor $\psi_c = \left(C\gamma^0\right)\psi^*$....
8
votes
3answers
416 views
Spin vs orbital angular momenta in QFT
I couldn't find past answers that quite match what I wanted, so I will try to ask in slightly different manner. In QM, we have total angular momentum operator $\vec{J}$ (I dropped the hat for ...
1
vote
0answers
19 views
Spin Sums & Conservation of Angular Momentum
External fermions and bosons have momentum and spin (polarization) degrees of freedom.
E.g. decay rate of $t\rightarrow W^{+}+\bar{b}$ with an unpolarized t beam.
Peskin & Schröder sum over t, $\...
0
votes
1answer
69 views
A Question About $4$-Spinor Contractions
Let $f_{abc}$ be a constant which is totally anti-symmetric with respect to indices $a$, $b$ and $c$. Let $\psi^{a}$, $\psi^{b}$, $\psi^{c}$ and $\epsilon$ be Grassmann-valued Majorana fermions. How ...
0
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0answers
49 views
Relativistic quantum field theory
Let $\psi(x)$ be solution of Dirac equation $$ (\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
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0answers
40 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(...
4
votes
2answers
125 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
66 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
93 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
392 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
50 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
53 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
121 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
34 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
112 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
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0answers
43 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
130 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
94 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
50 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
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0answers
46 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
56 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
80 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
25 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
181 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
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0answers
42 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
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0answers
93 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
88 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
$$
by ...
0
votes
0answers
40 views
Generally Covariant Dirac equation: The spin connection
Wikipedia, an answer on stackexchange and a few papers in the Arxiv I've found all have different definitions of the spin connection found in the Dirac equation. Can anyone please tell me what the ...
1
vote
0answers
39 views
Periodic Killing spinor on $S^1$?
In Cartesian coordinates on $R^2$ we have a constant two component Killing spinor $\epsilon_0$. If we use polar coordinates
$x = R \cos t$
$y = R \sin t$
we have the vielbein
$e^R = dR$
$e^t = ...
0
votes
2answers
137 views
Lorentz transformation of a Weyl Spinor?
A left handed Weyl Spinor belongs to the $(\frac{1}{2},0)$ representation of the Lorentz group. So given the Spinor, the unitary representation of the Lorentz transformation should look like $\exp{iA\...
4
votes
2answers
299 views
Dirac equation in 1+1D spacetime compared to “standard” 3+1D Dirac equation
In the past couple of weeks I've been studying the Dirac equation and its solutions. During a discussion with a tutor it was pointed out to me that one could formulate something similar to the Dirac ...
0
votes
1answer
57 views
A question about Lorentz transformations in spinor representation
For
$$\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}...
