Questions tagged [spinors]
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936
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Physical difference between $\vert S=0, m = 0 \rangle$ and $\vert S=1, m = 0 \rangle$? [closed]
In context of a two spin $\frac{1}{2}$ particle systems, we know that,
$\vert S=0, m = 0 \rangle = \frac{1}{\sqrt2}(\vert\uparrow\downarrow\rangle - \vert\downarrow\uparrow\rangle)$
$\vert S=1, m = 0 \...
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How does the two index spinor $(v_{a\dot{b}})$ transforms?
Using the Van der Waerden Notation, we define the four-vector as:
$$v_{a\dot{b}}=v_\nu \sigma^\nu_{a\dot{b}}$$
I'm trying to see how this transforms. Defining:
$$\Lambda \equiv e^{i\vec{\theta}\cdot \...
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Derivation of covariant derivative of Spinor
I am following this derivation for the covariant derivative of spinors. I have some questions about this derivation:
On page 3 they use the fact, that
\begin{align*}
V^a(x) = \bar{\Psi}(x)\gamma^a\...
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Dirac spinor manipulation [closed]
I'm trying to simplify the following relationship as much as possible:
$$\partial_\mu\bar\psi[\gamma^\mu,\gamma^\nu]\psi$$
where $\psi$ satisfies the dirac equation.
Using the fact that $[\gamma^\mu,\...
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A question for "Scattering amplitudes for all masses and any spins" : What is the exact two-component expression of the high spin wave function?
Now I'm studying the spinor helicity formalism with several liturature. I could understand roughly how to calculate the $n$-point on-shell amplitudes for massless particles very efficiently and ...
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Spinor space and curved spacetime
I am currently thinking about the Dirac equation in curved (1+1)-dimensional spacetime. First I have tried to understand how vectors can be defined in curved space and how the covariant derivative ...
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Hermitian conjugates of Dirac equation
I am given the following Dirac equations:
$$
(\gamma^{\mu}p_{\mu} - m)u_s(p) = 0\;\;\;\;\;\;\;\;\;eqn\;(1)
$$
$$
(\gamma^{\mu}p_{\mu} + m)v_s(p) = 0\;\;\;\;\;\;\;\;\;eqn\;(2)
$$
where u are the ...
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Is there a way to transform Dirac column matrix representation to canonical operators?
In canonical quantization the components of Dirac spinor are canonical operators. But in the easiest representation Dirac spinor is a column matrix. Is there a connection between column matrix ...
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Grassmann property of Fermion field in Hilbert space and spinor space [duplicate]
If we consider the trace of two fermion field ($\psi_A \ \psi_B$), and using the cyclic property of trace (which has Grassmann property in general I thought).
$$
\text{Tr}(\psi_A \ \psi_B)=(-)\text{Tr}...
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(Srednicki) How to obtain the normalization condition for Dirac field?
I'm reading through srednicki's qft and I met a problem. In its section 41, after he make an assumption that the creation operators of free field
theory would work comparably in the interacting theory ...
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Parity transformation of spinor helicity brackets
I'm trying to figure out why a parity transformation $P: (E, \textbf{p} ) \rightarrow (E, - \textbf{p})$ implies $\langle i \ j \rangle \rightarrow - [i \ j]$ and $[i \ j]\rightarrow - \langle i \ j \...
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What's the equivalent of the polarization vector of photons for gravitons?
Spin-0, spin-1/2, and spin-1 particles have an associated quality expressing the spin structure of the field.
Spin-0 fields obviously have no extra quality. Spin-1/2 fields are expressed by spinors ...
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Fermi Spinor overbar notation
I'm doing a scattering exercise with fermions and I'm working with the expression of Dirac's spinor that arises from the field solutions of the Euler-Lagrange equation:
$$\psi(x) = \frac{1}{\sqrt{(2\...
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How to decompose Dirac spinor into plane wave solutions?
Suppose that we know Dirac spinor $\psi$ (as complex numbers) in every point in 3d space (4th dimension is time)
How do we decompose it into plane wave solutions $u^s(p) e^{ipx}$ and $v^s(p) e^{ipx}$?
...
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Connection between column matrix and Grassmann numbers in Dirac field
In canonical quantization the Dirac equation is a complex column matrix, while in path integral formulation it's Grassmann numbers.
Is there a formula to convert from complex matrix to Grassmann ...
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Charge Conjugation at Spinors
In the schwartz book, Quantum Field Theory and the Standard Model, at the chapter 11 is define the charge conjugation operator
$C:\;\;\psi\rightarrow-i\gamma^2\psi^*$
$C:\;\;\psi^*\rightarrow-i\gamma^...
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Twistor equation and Killing equation
Yesterday was the birthday of Roger Penrose. And reading again about twistors I realized that twistor equation is strikingly similar to a Killing equation. My question is, are they "equivalent&...
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Why the hermitian conjugation swaps the two $SU(2)$ Lie algebras that comprise the Lie algebra of the Lorentz group?
I've start reading the part II (spin 1/2) of srednicki's qft book and I met a problem about group theory. In the section 34, the author describes the left and right handed spinor field.
He says that ...
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What is the thermodynamic spin-vector distribution?
Background
So my understanding is the Fermi-Dirac distribution tells us about the probability of a particular energy state for spinors. Also, from Spinors and Space-Time Volume $1$ - Penrose and ...
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Generality of Rotational transformation of Berry's Formula for Berry flux
In his paper, Michael Berry has given the general expression for the calculation of Berry flux for a particular level as:
$$\vec{B}^{(n)}(R) = -\text{Im}\sum_{n'\neq n}\frac{\langle n(R)|\nabla H|n'(R)...
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Degrees of freedom in a spinor in $d$ dimensions (following Polchinski & Lounesto)
I am working through several texts on spinors and trying to deepen my understanding of this fascinating concept. In many ways I have found Polchinski's great Appendix B of String Theory, volume 2 to ...
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Derivation of the transformation law for spinors
I'm reading the book Quantum Field Theory: An Integrated Approach by Eduardo Fradkin, and I got stuck where the transformation law for spinors
$$
\psi'(x') = S(\Lambda) \psi(x)
$$
is derived.
In ...
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How the factor of $1/2$ is purely conventional?
From Zee's book on group theory, he mentioned that factor $1/2$ is conventional due to historical reasons, but I thought that it was risen to match the Lie algebra of $SO(3)$?
For $N=2$, the ...
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How do you multiply a 4x1 spinor by a $SU(3)$ matrix?
I understand multiplication by a complex number, i.e. $SU(1)$
I maybe understand multiplication of the spinor by a 2x2 matrix i.e. $SU(2)$. We probably copy the 2x2 matrix twice to form a 4x4 matrix ...
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How to dimensionally reduce the 3+1 D Dirac equation into the 1+1D Dirac equation?
In 3+1D the Dirac equation looks like
$$i\partial_\mu \gamma^\mu \Psi -m\Psi=0.$$
If we only consider $x$-direction, then it should reduce to
$$i\partial_t\gamma^0\Psi =(-i\partial_x\gamma^1+m)\Psi.$$
...
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What is the relationship between spinors and rotating motion geometrically?
Spinors are famously like spinning tops, but not actually like spinning tops since they are point particles and thus cannot rotate around their axis. It is easy to show algebraically how spinors must ...
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Doubt on transformation laws of tensors and spinors using standard tensor calculus and group theory
1) Introduction
From standard tensor calculus, here restricted to Minkowski spacetime, we learned that:
A scalar field is a object that transforms as:
$$\phi'(x^{\mu'}) = \phi(x^{\mu})\tag{1}$$
A ...
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Reflection property in traces of $SU(N)$ generators
Pure gluon amplitudes could be organized on different basis, most common ones are the trace basis and the DDM (Del Duca-Dixon-Maltoni) basis, see DDM's paper, for better comprehension and to see how ...
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Why is angular momenum related to the spin?
What I know about spin ½ particles is that they are represented by spinors, and thus, you need to apply a 720° rotation in order for the spinor to return to its original value. Spin 1 particles are ...
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Interpretation of Feynman Slash Notation
I'm self-learning Relativistic Quantum Mechanics and was playing around with the Dirac equation when I noticed something.
I was trying to interpret the meaning of ${\not} \partial$. So since I can ...
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How to contract spinor indices?
In normal vector representation, vectors can be contracted as follows:
$$v^\mu v_\mu$$
with one covariant and one contravariant index. But in spinor representation, there are 4 possible type of ...
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Is there a completely matrix representation for the Dirac Equation?
Sorry if this question makes no sense, but I've learned that you can represent spinors as vectors, vectors as hermitian matrices, and transformation matrices as unitary operators acting on both sides ...
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Lorentz boost property of gamma matrices
I was watching this video where he boosted the Dirac equation. He reached this equation:
$$S^{-1}(\Lambda)\gamma^\mu S(\Lambda)=\Lambda^\mu{}_\nu \gamma^\nu$$
My question is since $\gamma^\mu$ is a ...
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Mechanical realization of three dimensional spinors
This question asks for a mechanical device
whose configuration space is homeomorphic to the 3-sphere $\mathbb S^3$;
which contains a rigid body, which has rotation $R(q)$ (an element of $\mathrm{SO}(...
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Spin space of a photon
I'm a self learner and I've been struggling with spinors for months. However, I know how to work with them in Pauli and Dirac equations.
So back to the question. My understanding is that one of the ...
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Are Pauli matrices invariant tensors in the representation of $\frac12 \otimes \frac12 \otimes 1$?
If we raise the index of the Pauli matrices with Levi-Civita symbol $\epsilon$ we obtain the 2-index spinors $(\sigma_i)^{AB} = (\sigma_i)^A{}_C \ \epsilon^{CB}$. The textbook (Ref. 1) argued that ...
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A matrix in a matrix representation of vectors
My understanding of representing spinors is to increase the rank of tensors so that a spinor looks like a vector and a vector looks like a matrix.
My question is how would transformation matrices be ...
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Why is it that the Quantised Dirac field spin has non-half integral eigenvalues?
The z-spin operator is:
$$\sum_{\vec{p}, r} \frac{m}{E} u_r^{\dagger} (\vec{p}) \Sigma_zu_r(\vec{p})N_r(\vec{p})$$
For a state like $|\vec{p}=p_3 \vec{k}, s_z=\frac{1}{2}\rangle$, this operator ...
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Why do we normalise the spinor to $\frac{E}{m}$ and not just $E$?
The spinor, say $u(p)$, normalises to $\frac {E}{m}$. But the $m$ is at the bottom only because we defined $u(p)$ to have a $\sqrt {2m}$ in the denominator. So isn't this unnecessary? Just have a $\...
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Charge conjugation of symplectic Majorana spinors in 4+1 dimensions
In the book "Supergravity" written by Freedman & van Proeyen, a symplectic Majorana spinor is defined in eq. (3.86)
$$ \chi^i = \varepsilon^{ij} (\chi^j)^C, \tag{3.86}$$
where the upper ...
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Why does spin transform as a spinor? [duplicate]
It is often said that shortly after the binary nature of the electron spin was discovered (either plus or minus 1/2), Pauli suggested that the 2-component wave function of the electron should ...
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Explicit calculate covariant derivative for spinor field
I want to explicitly calculate the covariant derivative for spinor-fields for a given metric (to investigate the 1d dirac equation in curved spacetime):
$\begin{align*} g_{\mu\nu}=\frac{L^2}{cos^2(\...
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How would a free particle with known spin evolve?
I searched a lot for a Hamiltonian of a pauli spinor with no potential energy but got no luck, so I tried deriving one my own.
I took an overkill shortcut and used pauli's equation:
$$i\hbar \frac{\...
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IKKT matrix model: invariance of action under supersymmetry
I have been trying to check the invariance of the action of the IKKT matrix model
$$
S = N~ {\rm tr} \left( -\frac{1}{4} \left[X_\mu,X_\nu \right]^2 - \frac{1}{2} \bar{\psi} \Gamma^{\mu} \left[X_\mu, ...
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Relationship between spin and wave-function (in geometric algebra)
I am interested in determining which 'feature' of the wave-function are responsible for fixing the spin of the matter is relates to; or more precisely, an "algorithm" or "procedure"...
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Can one express the evolution of a particle with a one-parameter group of $SO(3,1)$?
Can one express the evolution of a particles using a sequence of $SO(3,1)$ transformations? If yes, how?
Is it sufficient to apply $SO(3,1)$ transformations to a spinor?
$$
\psi(t) = e^{t\mathfrak{so}(...
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Spin Connection Vanishes?
I'm trying to reproduce a result for the components of the spin connection in FRW spacetime. The formula for the spin connection $\Gamma_{\mu}$ is
$$\Gamma_{\mu} = \frac{1}{2} \Sigma^{a b} L_{a}^{\nu} ...
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Do spinor fields violate causality?
It is a theorem that spin structures on a spacetime $M$ exist iff the second Stiefel-Whitney class $w_2(M)=H^2(M, \mathbb{Z}_2)$ vanishes. I find this confusing for two reasons.
First, it implies that ...
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How can I describe, in QFT, Weyl spinors coupled to a strong version of the gluon gauge field?
Let's assume a massless Weyl-spinor field. My aim is to let them interact by a gluon-like gauge-field, which has a bigger much bigger coupling, color charge, than the standard color interaction.
This ...
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Invariants of inner product in pseudoreal representation of $SU(2)$
I am reading Peskin's and Schroeder (P&S), "An introduction to Quantum Field Theory", specifically the first paragraph on page 499 in section 15.4 "Basic Facts about Lie Algebras&...
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