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Questions tagged [spinors]

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1answer
114 views

Why two different spinors are Grassmann quantities?

In Rydberg Quantum Field Theory page 441 (this edition, unfortunately page 441 is not in the link) it says If $\xi$ and $\eta$ are Majorana spinors [...] and since $\xi$ and $\eta$ are Grassmann ...
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82 views

Metrics and Spinors

this might be better posed in mathematics but I'll ask here anyway. So the Lagrangian for the spinor field can be viewed as follows. Let $(M,g,\nabla)$ denote a locally Minkowskian spacetime, Where $\...
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81 views

Covariantly constant 2 Component Spinor

Note: For this question I am using the conventions of "Ideas and Methods of Supersymmetry and Supergravity" by Ioseph Buchbinder and Sergei Kuzenko (mostly p16 & p44). Let our space be equipped ...
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71 views

Relation between Dirac spinors, quaternions, and bicomplex numbers

Superficially Dirac spinor resp. Dirac gamma matrices and quaternions and bicomplex numbers seems to be very similar objects. all can be expressed by unitary 4x4 matrices so they seem to represent ...
1
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1answer
56 views

Newman-Penrose formalism: Can a normalized spin-frame be found for every null tetrad?

I have a question within the framework of spin-coefficient formalism / Newman-Penrose formalism. Suppose I have a space-time metric whose components with respect to some coordinate basis are known. I ...
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1answer
110 views

Breaking of a commutator involving Dirac spinors and gamma matrices

I'm trying to understand a particular step in the solution to problem 27 in THIS solution sheet. By the middle of the page, they start with the simplification of this expression $$\left[s^{\mu}\left(...
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55 views

Spinor Lorentz Transformation

Why should the transformation between the solutions of the Dirac equation for different inertial observers be linear?
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107 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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57 views

Lorentz transformation of vector of sigma matrices

$\newcommand{\vec}[1]{\mathbf{#1}}$I'm trying to show that the relations \begin{equation} \rho_L(\Lambda)^\dagger\bar{\sigma}^\mu\rho_L(\Lambda)={\Lambda^\mu}_\nu\bar{\sigma}^\nu\\ \rho_R(\Lambda)^\...
3
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1answer
225 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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128 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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24 views

How is a half-odd integer spin field defined in the Wightman axioms?

Can anyone give me a proper definition of 'half-odd integer spin fields' in terms of the Wightman axioms? I'm trying to prove the anti-unitarity of the PCT operator in 'PCT, Spin and Statistics, and ...
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2answers
145 views

Commuting Dirac spinor with scalar field

I have two Lagrangians which only differ in their interaction term - one of them has $L_{Yuk_{1}}$ while the other has $L_{Yuk_{2}}$. I want to know if this two interaction Lagrangians are equivalent. ...
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2answers
155 views

Dirac Lagrangian after decomposing the Dirac spinor into Weyl spinors

Consider the Dirac Lagrangian, $$L=\overline{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi$$ and take the Dirac spinor chiral decomposition with $\psi_{L}=\frac{1}{2}\left(1-\gamma^{5}\right)\...
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2answers
128 views

What does it mean to act a Dirac field operator on the vacuum?

The usual interpretation from my QFT courses is that when acting the scalar field operator onto the vacuum, we create particle: $$ |x\rangle = \phi(x)|0\rangle. $$ If I have a multi-component field ...
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28 views

Gamma matrices and a vector

Let $\theta$ be an anticommuting Majorana spinor and $A^\mu$ a vector. Is there an easy way to show then that $\bar{\theta} \gamma^{\mu} \gamma_5 \theta \bar{\theta} \gamma^{\nu} \gamma_5 \theta A_{\...
2
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1answer
91 views

How to understand spinors in 1+1 spacetime?

I am struggling to understand spinors in 1+1 spacetime. I know in this case the Clifford algebra is realized by two by two matrices so the spinors have two components. Then what do we mean by spin or ...
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107 views

spinor indices and Weyl spinors

This is a follow up to Spinor dotted and undotted indices, to see if I understand things correctly. So a Dirac spinor $\psi_D$ can be written as a direct sum of a left handed Weyl spinor and a right ...
3
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1answer
237 views

Spinor dotted and undotted indices

I have had an introduction to QFT following the book of Mandl and Shaw. However, I have been asked to write a report on the CPT theorem. For this, the main reference I'm using is PCT, spin and ...
0
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1answer
144 views

Spacetime dimension and the dimension of Clifford algebra

The dimension of the Clifford algebra $C_p$ generated by a vector space $V^p$ is given by $2^p$, where $p$ is the dimension of the vector space (T. Frankel, the geometry of physics). Based on the top-...
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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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1answer
26 views

Questions regarding the elements of vector space spin representations act on

Elements of vector space spin-$1/2$ representations act on are spinors. What about half-integers in general? And what about integer spins? Do spin-$0$,$1$ reps always act on vectors?
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1answer
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Do massless spin-1/2 particles have to be Weyl spinors?

Weyl spinors are massless. Is the converse also true? Does any massless spin-1/2 fermion have to be a two-component Weyl spinor? In the Standard model, before symmetry breaking, the electron (for ...
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61 views

Weyl Semimetal Hamiltonian and spinor

This is what I've known The Hamiltonian for right and left-handed weyl fermion can be written as $H = \pm v\ \textbfσ\cdot \textbf k$ So the the wave function derived from above should be a 2 ...
8
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1answer
184 views

A question on the transformation law of the spin connection

The covariant derivative may be defined as: $$\nabla_{a}=e_{a}^{~\mu}(x)\partial_{\mu}+\frac{1}{2}e_{a}^{~\mu}(x)\omega_{\mu bc}(x)M^{bc}\tag{1}$$ where $e_{a}^{~\mu}(x)$ is the vielbein, $\omega_{\...
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What is the problem of non-unitarity in RQM?

In Peskin and Schroeder, section 3.2, it is stated that Lorentz group being non-compact it does not have any finite dimensional, faithful unitary representation. But it has also been said that one ...
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0answers
109 views

Adjoint of Weyl Spinor

Given a (Dirac), spinor in the Weyl basis, $\psi = \begin{pmatrix} \psi_{L}\\ \psi_{R} \end{pmatrix} $ , where $\psi_{L}$ and $\psi_{R}$ are Weyl spinors we define the adjoint of the Dirac spinor as; ...
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1answer
193 views

Helium ground state spatial component of the wave function symmetric

I do not understand why the spatial component of the wave function in the ground state for the helium is necessarily symmetric. In that case the spin component is antisymmetric according to pauli ...
1
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1answer
94 views

Where did the square root come from in this spin 1/2 spinor equation?

This is coming from the spin-$1\over 2$ section of David J. Griffith's textbook Introduction to Quantum Mechanics. My textbook gives the generic expression for a spinor as $$\chi= \begin{pmatrix} ...
3
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0answers
43 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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0answers
184 views

Gamma matrices invariant under lorentz transformation

I know this has been asked before but I just can't seem to get my head around it based on the answers I've read. So the idea is that we have the gamma matrices $\gamma^{\mu}$. Now from my ...
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59 views

On the Pauli-Lubansky vector and spin

Lahiri's A First Book on Quantum Field Theory states on problem 4.24 that from the Pauli-Lubansky vector $$W_\mu=-\frac{1}{2}\epsilon_{\mu\nu\lambda\rho}P^\nu J^{\lambda\rho}$$ one can prove that for ...
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29 views

Do one-half spin phenomena breakdown equivalence between active and passive transformations?

If i rotate a physical system by say 10 degrees it is equivalent to my self rotate -10 degrees. but if i rotate an electron 360 degrees it spin state have a detectable minus sing. now if instead of ...
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99 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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38 views

What is the most generic spin mixed state?

For a spin generic mixed state $\rho$. How should I write $\rho$? $\rho_1 = q_1 |\uparrow\rangle \langle \uparrow| + q_2 |\downarrow\rangle \langle \downarrow|$ or $\rho_2 = q_1 |\uparrow\rangle \...
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102 views

A few doubts with showing Lorentz invariance of Dirac equation and probability current

Trying to understand some about Lorentz invariance and representation theory, I thought that the best way is with an example of application: Show the Lorentz invariance of the Dirac Equation $$(i \...
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103 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 ...
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39 views

When I construct SUSY multiplet how can I distinguish a Majorana fermion from a Weyl one?

When I construct a SUSY multiplet (massive or massless) I can act with the supercharges on a Clifford vacuum and construct the states in the multiplet, that I could interpret in terms of “ordinary” ...
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Can we say the origin of spin is due to mixing of various components of field under Lorentz transformation?

when x,y,z,t are transformed under Lorentz transformation all the components get mixed up and gives the angular momentum conservation of the field. On the other hand if the field has some components ...
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28 views

Why is spinor along x-axis biased in terms of probability?

If we use the Paulispin matrices of x-axis to find its Eigenvalues and eigenstates and use those to represent a spin state, we get the equation: when we look at spinor in terms of along x-axis, the ...
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1answer
58 views

Spin of an electron (What is the meaning of spinor in terms of Hilbert space and Euclidian space?)

In quantum mechanics, electron has a spin of 1/2 either up or down. As shown by the Stern-Gerlach experiment, the spin is quantized so it could only be either up or down. The spinor matrices, for ...
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71 views

Some clarification on Dirac and Weyl spinor terminology

I came across a seemingly trivial exercise in Schwartz's "QFT and the Standard Model" that I am just a little confused about. The problem is 11.6, "The physics of spin, helicity and chirality". (a) ...
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1answer
86 views

Why a 2-state photon is interpreted as spin 1?

Both Ising spin and photon's polarization degree of freedom are used in quantum information as Qbit implementation. They both have 2 level state systems, which means mathematically their state could ...
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2answers
258 views

A simple question about the scattering amplitude $\mathcal{M}$ in QFT

Every scattering amplitude that I see have all the tensor indices contracted but spinor indices floating around, whose only disappear after you square the amplitude and do the sum and average over ...
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2answers
357 views

Choice of Dirac gamma matrix representation and definition of adjoint spinor

Is the definition of the adjoint spinor $\bar{\psi}=\psi^\dagger \gamma^0$ forcing a particular choice of representation of the Dirac matrices (or a subset of the possible choices)? More precisely, I ...
2
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1answer
161 views

Dirac spinor parity

I'm not sure I understand the effect of a parity transform on a Dirac spinor $\left( \begin{array}{c} \psi_R\\ \psi_L\\ \end{array} \right)$. I've been given the definitions $P\psi=\gamma_0\psi$, ...
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2answers
415 views

spinor vs vector indices of Dirac gamma matrices

I am struggling to understand the nature of the components of the Dirac matrices. If we view the four components of a Dirac spinor as $\psi^a$ with $a$ being a 'spinor' index, then if a gamma matrix ...
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1answer
157 views

What's the reasoning behind propagators definitions (specifically fermionic propagators)

I'm studying QFT by David Tong's lecture notes. When he discusses causility with real scalar fields, he defines the propagator as $$D(x-y)=\left\langle0\right|\phi(x)\phi(y)\left|0\right\rangle=\int\...
3
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1answer
488 views

How to understand the Dirac Lagrangian?

I am having some basic questions about how to interpret Lagrangians, lets start with Dirac: $L = \bar{\Psi} (i \gamma^{\mu} \partial_{\mu} -m) \Psi$, where $\Psi$ is a Dirac-Spinor, $m$ is the mass,...
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0answers
116 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?