Questions tagged [spinors]

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170 views

Complexifying Lie algebras confusion

I have been studying a course on Lie algebras in particle physics and I could never understand how complexifying helps us understand the original Lie algebra. For example, consider $\mathfrak{su}(2)$...
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1answer
150 views

Coupling a spinor field to a preexisting scalar field?

So I'm not a physicist, but I'm thinking about a mathematical problem where I think physical insight might be useful. We're working on a Riemannian manifold $(M,g)$ (positive definite metric) with a ...
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149 views

Has hep-th/0312070 forgotten to fix $s_{0} = 1/2$ for the fermionic states in the second table on page 52?

Link to the original paper: The Gauge/String Correspondence Towards Realistic Gauge Theories (arXiv paper) On page 52 we see that, for a theory of Dp-branes placed at an orbifold (orbifold = $C_{2}$/$...
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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)...
4
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1answer
74 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 ...
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114 views

Physical/geometrical interpretations of spinors?

Physically, a scalar is a quantity invariant with reference frame, a vector is a quantity associated with a direction, tensors are higher relationships between vectors - what are spinors? I thought I ...
4
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2answers
415 views

Relation between spinors and anticommutation relation of fermions

I read that the state of a pair of particles is the tensor product of the single states of both, and you will get a wavefunction with the parameters of both, if you swap the parameters you will get a ...
4
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262 views

Spin Connection derivation in dirac equation

I'm learning about the spin connection, more specifically how it is derived in the dirac equation through this notes: http://web.phys.ntnu.no/~mika/CPP/ch15.pdf In the third page of the document it ...
4
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120 views

Interpretation of inequivalent spin structures (on the circle)

I was wondering about the physical interpretation of inequivalent spin structures on a given configuration space. For simplicity, I'd be satisified by only discussing the case of the circle. There ...
4
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108 views

How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
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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 - ...
3
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1answer
272 views

Identities of Pauli matrices in two-component spinor formalism

I'm reading the review by H. K. Dreiner, H. E. Haber and S. P. Martin (arXiv:0812.1594) about the two-component spinor formalism. There are some identities and notational conventions which lead to ...
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47 views

Mapping from spinor to tetrad

I am reading the journel by Patrick l. Nash: mapping from tetrad to Dirac spinor. While reading this ,I came across the term concrete real 4*4 irreducible representation of SO(3,3). I know SO(3) is ...
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138 views

Why is a spinor not a tensor?

The title says it. why is a spinor not a tensor? I know the transformation rules for a spinor but I cant see why it is not a tensor?
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235 views

Geometric meaning of killing spinor?

I know about killing vectors- They forms the basis for the isometries of a metric space. I am not clear about the definition of killing spinors, in the sense the one there is for killing vectors, ...
3
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358 views

4-vector from a spinor

Currently reading Aitchison's book on SUSY, and on page 35 (section 2.2) he asks the reader to prove that $\bar{\Psi}\gamma^\mu\Psi=\psi^\dagger\sigma^\mu\psi+\chi^\dagger\bar{\sigma}^\mu\chi$ ...
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63 views

How similar are the spin states and the matter/antimatter states?

Within the Dirac formalism, we have bispinors that represent both if a particle is spin up or spin down, and if a particle is an electron or a positron. And these representations are very similar. (...
3
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1answer
388 views

Schwartz's book: Spinor-helicity formalism

I'm trying to learn the spinor-helicity formalism from Schwartz's QFT book. His equation 27.44 is describes the annihilation of an electron(1)-positron(2) pair to a muon(3)-antimuon(4) pair. He ...
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275 views

Angular momentum of the vacuum

I'm studying quantum field theory from "An introduction to Quantum field theory" by Peskin and Schroeder and from "A modern introduction to quantum field theory" by Maggiore. I've read from "An ...
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78 views

Conformal compactification and the use of spinors (Twistor theory)

I was reading the book from Huggett and Tod "An introduction to twistor theory" and as the book evolves they reach to the necessity to "found" a Lie derivative of a spinor respect to a conformal ...
3
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292 views

Transformations of gamma-matrices through Pauli matrices transformations

I have the transformation law of the Lorentz group for Pauli matrices: $$ \tag 0 (\sigma^{\mu})_{a \dot {a}}{'} = \Lambda^{\mu}_{\quad \nu} N_{a}^{\quad c}(\sigma^{\nu})_{c \dot {c}}(N^{-1})^{\dot {c}}...
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82 views

Spinors on algebraic plane curves

I'm interested in parameterizing spinors on Riemann surfaces. For my purposes, it's best to represent the Riemann surfaces as immersed in $\mathbb{C}P^2$, i.e. as algebraic plane curves. Apparently, ...
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204 views

Some more questions on conformal spinors of $SO(n,2)$

This is somewhat of a continuation of my previous question. I had stated there that a conformal spinor ($V$) of $SO(n,2)$ can be created by taking a direct sum of two $SO(n-1,1)$ spinors $Q$ and $S$ ...
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24 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^...
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77 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 ...
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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{\...
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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^\...
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70 views

Are there supersymmetry algebras with higher spinor representations?

The super-Poincare algebra contains supersymmetry generators $Q^I$ which satisfy fermionic anticommutation relations. By the higher-dimensional analogue of the spin-statistics theorem, they must ...
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63 views

Motivating the Unintuitive Properties of Spinors

In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
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33 views

Would Left and Right Weyl spinor components mix to become massive in an expanding space?

Sorry, this might be a dumb question. I was just reading a very old paper by Schrodinger where he's talking about different frequency modes mixing in an expanding universe. Basically he says if the ...
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29 views

Vertex with spinorial structure and scattering amplitude

Consider the Lagrangian $$\mathcal{L}=\bar\psi_1\left(i\partial\!\!\!/-m_1\right)\psi_1 + \bar\psi_2\left(i\partial\!\!\!/-m_2\right)\psi_2 - g\bar\psi_1\gamma_\mu\psi_1\bar\psi_2\gamma^\mu\psi_2.$$ ...
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65 views

Feynman rule for a vertex with spinorial structure

Consider the interaction term $$\mathcal{L}_{\rm{int}}=-g\bar\psi_1\gamma_\mu\psi_1 \bar\psi_2\gamma^\mu\psi_2,$$ where $\psi_i$ are fermions. I would like to calculate the Feynman rule for the vertex....
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128 views

Spinors in Classical Mechanics and Geometry?

I'm trying to deepen my understanding of spinors by looking at applications in simple problems, preferably unrelated to quantum mechanics. For this purpose I'd like to refrain from discussing ...
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129 views

Einstein-Cartan Theory with modern Cartan structural equations?

I cannot find real derivations and analysis of the Einstein-Cartan Theory. This can probably be a neat up-climb to a Cartan structural summit, or very close. I am not asking this because I want it ...
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54 views

Let's put a $\mathrm{Pin}$ in it$.$

As it turns out, the low-energy behaviour of (gapped) systems with fermionic degrees of freedom can be described by a so-called Spin TQFT. Such models require the introduction of a (S)pin structure. ...
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116 views

Why are antiparticles associated with spin-flipped spinors?

In section 2.2 of Elvang and Huang's Scattering Amplitudes in Gauge Theory and Gravity (http://arXiv.org/abs/1308.1697), beneath equation (2.9), it is mentioned that $u^{\pm}=v^{\mp}$, where $u^\pm$ ...
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360 views

Spin covariant derivative of gamma matrices?

Where can I find a general expression (on curved manifolds) in local coordinates, for the following: $$\nabla^S_{\mu}\gamma^{\nu} = ?$$ $\nabla^S_{\mu} = \partial_{\mu} + \omega^S_{\mu}$ is the spin ...
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144 views

Transformation of Weyl spinors

I usually see Weyl spinor and Weyl equations as derived from Dirac equation, like in Peskin. Now, I'm following a course where the professor actually builds Weyl spinor lagrangians BEFORE talking ...
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65 views

Spinors, punctured plane and principle frame bundle

I am reading Applied Conformal Field Theory by Ginsparg. On page 72, while describing different boundary conditions on fermion he states the following. We shall choose to consider periodic $(P)$ ...
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291 views

Charge conjugation in chiral representation

I'm reading Maggiore's book and I got to the part of charge conjugation symmetry for Dirac spinor. I get that the definition of charge conjugation is representation-dependent, however I couldn't find ...
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357 views

Matching Dirac/Majorana/Weyl Spinor Degrees of Freedom in Minkowski signature

Question: How do we match the real degrees of freedom (DOF) of Dirac/Majorana/Weyl Spinor in terms of their quantum numbers (spin, momentum, etc) in any dimensions [1+1 to 9+1] in Minkowski signature?...
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102 views

Parity transformation of spin-3/2 field

In conventional quantum field theory textbook, we can find the expression of parity transformation of spin-0, 1/2 or 1 fields. For example, for spin-1/2 fields, we have $$U^{-1}(\mathcal{P})\Psi(x) U(...
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149 views

Formal definition of gauge field and spinors in QFT

I am trying to pin down what spaces a spinor and gluon gauge field exactly occupy. I know that the spinor is a quantity $\psi_{i\alpha f}(\vec x, t)$ where $i$ is a color index in the fundamental ...
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94 views

Dirac operator with torsion

On a spin manifold $M$, (in local coordinates) the Dirac operator $D_M$ is of the form $$-i\gamma^{\mu}(\partial_{\mu}-\frac{1}{4}\tilde{\Gamma}^b_{\mu a}\gamma^a\gamma_b),$$ where the (torsion-free) ...
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132 views

How this spinor identity is shown?

In one QFT problem it is asked to prove the following identity: $$\overline{u}_\sigma(p)\gamma^\mu u_{\sigma'}(p)=2\delta_{\sigma\sigma'}p^\mu.$$ Considering $u_\sigma$ the basis solutions to the ...
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519 views

Variation of the Spin Connection

I am trying to get an explicit expression for the variation $\delta \omega_{\mu}^{\ ab} / \delta e_\mu^a$, but when doing the actual variation I end up with a series of 16 terms that I cannot simplify ...
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134 views

How to transform Dirac fermion coupled with $SU(2)/Z_2$ gauge field to the real fermion coupled with $SO(3)$?

For instance, I am considering a free (3+1)-D Dirac fermion coupled with $SU(2)$ (fundamental representation) gauge field. Then I do a "gauging" the center $Z_2$ to the associated principal bundle by ...
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312 views

Why parity exchanges right handed and left handed spinors

Reading through David Tong lecture notes on QFT. On pages 94, he shows the action of parity on spinors. See below link: QFT notes by Tong In (4.75) he confirms that parity exchanges right handed ...
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154 views

Supercovariant Derivative action

My query is with Weinberg Vol3 equation just above 26.7.22 Weinberg follows Majorana Superfield formalism. Where, covariant derivative is defined as, $$D_{R\alpha}=-\epsilon_{\alpha \beta}\frac{\...
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660 views

Spin 1/2 wavefunction transformation under inversion and mirror symmetry

I'm considering group-theory applications to condensed matter physics now. In particular I work with the following paper: http://journals.aps.org/pr/pdf/10.1103/PhysRev.100.580 and try to understand ...