Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [spinors]

The tag has no usage guidance.

25
votes
4answers
5k views

Dimension of Dirac $\gamma$ matrices

While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices: $$\...
5
votes
5answers
2k views

Total spin of two spin-$1/2$ particles

On my book I read: $S_{z-tot}\chi_+(1)\chi_+(2)=[S_{1z}+S_{2z}]\chi_+(1)\chi_+(2)=[S_{1z}\chi_+(1)]\chi_+(2)+[S_2\chi_+(2)]\chi_+(1)=...$ Now, I have two questions: What's $\chi_+(1)\chi_+(2)$ ? I ...
25
votes
4answers
10k views

What is the difference between a spinor and a vector or a tensor?

Why do we call a 1/2 spin particle satisfying the Dirac equation a spinor, and not a vector or a tensor?
25
votes
2answers
4k views

What's the relationship between $SL(2,\mathbb{C})$, $SU(2)\times SU(2)$ and $SO(1,3)$?

I'm a beginner of QFT. Ref. 1 states that [...] The Lorentz group $SO(1,3)$ is then essentially $SU(2)\times SU(2)$. But how is it possible, because $SU(2)\times SU(2)$ is a compact Lie group ...
23
votes
2answers
2k views

Introduction to spinors in physics, and their relation to representations

First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms "up to a sign" when acted on by the rotation group. I have even rotated a ...
22
votes
2answers
1k views

Wick rotation and spinors

I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; ...
5
votes
1answer
684 views

Confusion with chirality eigenstates

In the Weyl/chiral basis, the four components of the Dirac spinor represent left-chirality spin up, left-chirality spin down, right-chirality spin up, and right-chirality spin down, respectively. When ...
10
votes
1answer
5k views

What is a spinor? [closed]

In a youtube video, sir Michael Atiyah mentioned that even after working during the most of his life on spinors, he doesn't know what a spinor is. Now surely that was part of his humorous introduction ...
12
votes
2answers
1k views

If the mass of neutrino is not zero, why we cannot find right-handed neutrinos and left-handed anti-neutrinos?

I am learning P&S's Introduction of quantum field theory. My teacher said that if the mass of neutrino is exactly 0, then we should not observe any right-handed neutrinos and left-handed anti-...
5
votes
3answers
352 views

Spinor field normalisation from poles in the propagator

In the theory of free scalar bosons (KG field) it is a basic result that the propagator $\Delta(p)$ has poles at $p^2=m^2$, with residue $1$ (or any other constant, depending on conventions). Thinking ...
21
votes
1answer
3k views

What're the relations and differences between slave-fermion and slave-boson formalism?

As we know, in condensed matter theory, especially in dealing with strongly correlated systems, physicists have constructed various "peculiar" slave-fermion and slave-boson theories. For example, For ...
5
votes
1answer
843 views

How can a left-handed fermion field create a right-handed antifermion?

My question - which is likely stupid or appears due to some confusion - stems from the following considerations: when quantizing canonically we are told (see any book on QFT) that a Dirac fermion ...
4
votes
2answers
604 views

Why is the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group realized as the vector space of complex $2\times 2$ matrices?

Why can we write an arbitrary object $v_{a \dot{b} }$ our transformations in this basis act on as $$ v_{a \dot{b} } = v_{\nu} \sigma^{ \nu}_{a \dot{b} } = v^0 \begin{pmatrix} 1&0 \\ 0&1 \...
2
votes
1answer
247 views

Where do the quantum fields encode the spin information?

I know basically the difference between Klein-Gordon and Dirac field is spin. But I am not sure where we need to implement this info. The solutions of both equations are the wave packets which ...
2
votes
2answers
276 views

Can Dirac equation be reformulated in an equivalent tensor form?

1) Can Dirac equation (including bispinors) be represented by a tensor formalism? 2) If yes, what kind of tensors could be the components of the wave function in Dirac equation in such formulation? ...
0
votes
3answers
180 views

Spin Up with Indefinite Helicity

Imagine we are studying the spin quantization along the same axis as the momentum. What if I have a Dirac spinor with a spin up but no definite helicity ($\psi_L,\psi_R\neq0$): $$ u(p)= \left(\begin{...
34
votes
2answers
1k views

Is there an elegant proof of the existence of Majorana spinors?

Almost all standard sources on the existence of Majorana spinors (e.g. Appendix B.1 to Polchinski's "String Theory", Vol. 2) do so in a way I consider inherently ugly: A priori, we are dealing with ...
14
votes
6answers
1k views

Does spin have anything to do with a rate of change?

The orbital angular momentum of a particle can be related to the revolution of that particle about some external axis. But in quantum mechanics, the spin angular momentum of a particle can't really ...
19
votes
2answers
5k views

Dirac, Weyl and Majorana Spinors

To get to the point - what's the defining differences between them? Alas, my current understanding of a spinor is limited. All I know is that they are used to describe fermions (?), but I'm not sure ...
11
votes
4answers
3k views

Covariant derivative for spinor fields

scalars (spin-0) derivatives is expressed as: $$\nabla_{i} \phi = \frac{\partial \phi}{ \partial x_{i}}.$$ vector (spin-1) derivatives are expressed as: $$\nabla_{i} V^{k} = \frac{\partial V^{k}}{ \...
8
votes
3answers
1k views

Is there any relationship between gauge field and spin connection?

For a spinor on curved spacetime, $D_\mu$ is the covariant derivative for fermionic fields is $$D_\mu = \partial_\mu - \frac{i}{4} \omega_{\mu}^{ab} \sigma_{ab}$$ where $\omega_\mu^{ab}$ are the spin ...
6
votes
2answers
1k views

Number of Components of a Spinor

I'm trying to develop my understanding of spinors. In quantum field theory I've learned that a spinor is a 4 component complex vector field on Minkowski space which transforms under the chiral ...
4
votes
2answers
362 views

What spinor field corresponds to a forwards moving positron?

When we search for spinor solutions to the Dirac equation, we consider the 'positive' and 'negative' frequency ansatzes $$ u(p)\, e^{-ip\cdot x} \quad \text{and} \quad v(p)\, e^{ip\cdot x} \,,$$ ...
7
votes
3answers
3k views

Difference between spinor and vector field [duplicate]

How do we distinguish spinors and vector fields? I want to know it in terms of physics with mathematical argument.
5
votes
1answer
523 views

Regarding the Weyl spinor and its transformation properties

I am trying to prove the Lorentz invariance of the (left-handed) Weyl Lagrangian: $$\mathcal L=i\psi^\dagger\bar\sigma^\mu\partial_\mu\psi$$ A Lorentz transformation is realized as $\psi\to M\psi$, ...
4
votes
1answer
268 views

Two conflicting definitions of chirality

Consider a Majorana fermion embedded in a Dirac spinor, $$\psi = \begin{pmatrix} \psi_L \\ i \sigma_2 \psi_L^* \end{pmatrix}.$$ The Majorana fermion $\psi_L$ is left-chiral, i.e. it transforms in the ...
4
votes
2answers
287 views

A whole lot of doubts on Lorentz representation

Can someone tell me in layman's language how the $(1/2,1/2)$ represents a vector field and $(0,1/2)$ or $(1/2,0)$ represents spinors and $(0,0)$ represents scalar field. Please don't be pedantic on ...
3
votes
2answers
487 views

What are the actual transformation properties of Dirac spinors $u_\sigma(p)$?

Let $u_\sigma(p)$ be a Dirac spinor. As far as I know, it transforms under changes of reference frame according to $$ u_\sigma(p)=S(\Lambda)u_\sigma(\Lambda p)\tag{1} $$ where the $\sigma$ label doesn'...
10
votes
3answers
4k views

Is there a reason why the spin of particles is integer or half integer instead of, say, even and odd?

It seems to me that we could change all the current spin values of particles by multiplying them by two. Then we could describe Bosons as even spin particles and Fermions as odd spin particles. Is ...
0
votes
1answer
437 views

Gordon Identity confusion

For the Gordon identity $$2m \bar{u}_{s'}(\textbf{p}')\gamma^{\mu}u_{s}(\textbf{p}) = \bar{u}_{s'}(\textbf{p}')[(p'+p)^{\mu} -2iS^{\mu\nu} (p'-p)_{\nu}]u_{s}(\textbf{p}) $$ If I plug in $\mu$=5, ...
14
votes
1answer
853 views

Could this model have soliton solutions?

We consider a theory described by the Lagrangian, $$\mathcal{L}=i\bar{\Psi}\gamma^\mu\partial_\mu\Psi-m\bar{\Psi}\Psi+\frac{1}{2}g(\bar{\Psi}\Psi)^2$$ The corresponding field equations are, $$(i\...
4
votes
1answer
380 views

What is the role of the spacetime algebra?

For Minkowski space $M^4=\mathbb{R}^{1,3}$ the Clifford algebra $Cl_{1,3}$ (Dirac algebra) with $\{\gamma^\mu, \gamma^\nu \}=2 g^{\mu \nu}$ is sometimes called "spacetime algebra". What is its ...
16
votes
1answer
852 views

Chirality, helicity and the weak interaction

From what I'm understanding about Dirac spinors, using the Weyl basis for the $\gamma$ matrices the first two components behave as a left handed Weyl spinor, while the third and the fourth form a ...
1
vote
1answer
897 views

General definition of vector, spinor, and spin

I am looking for basic and exact definitions of fundamental physical concepts in graduate level. I reach this following definitions. Could you please help to improve these definitions. Spin: ...
5
votes
2answers
515 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 ...
9
votes
3answers
5k views

Dirac spinor and Weyl spinor

How can it be shown that the Dirac spinor is the direct sum of a right-handed Weyl spinor and a left-handed Weyl spinor? EDIT: - Let $\psi_L$ and $\psi_R$ be 2 component left-handed and right-handed ...
9
votes
1answer
1k views

Is the Dirac equation equivalent to the Klein-Gordon equation for its left handed component?

The Dirac equation $$(i\gamma^a\partial_a - m)\psi=0\tag{0}$$ is given by a first order operator acting on a Dirac spinor, which is the direct sum of a left handed spinor and a right handed spinor. ...
8
votes
1answer
1k views

The correspondence between Grassmann number and 4-spinor

In canonical quantization, we view the Dirac field $\psi$ as a $4\times1$ matrix of complex number. While in path integral quantization, we view the Dirac field $\psi$ as a Grassmann number. For two ...
8
votes
1answer
749 views

2 Component Spinor Formalism

In Chapters 34-36 of the Srednicki QFT book, 2 component spinors and their combinations in Dirac and Majorana spinors are carefully constructed. Specifically, in equations 36.14 and 36.15 the ...
7
votes
1answer
870 views

Does the concept of both helicity and chirality make sense for a massive Dirac spinor?

Does the concept of both helicity and chirality make sense for a massive Dirac spinor? A massive electron in the chiral basis is written as a column made up of $\psi_L$ and $\psi_R$. What is the ...
4
votes
1answer
627 views

Conceptual interpretation of the left- and right-handed spinor representations of the Lorentz group

I understand mathematically that the Lorentz group's Lie algrebra $\mathfrak{so(3,1)}$ (given by eqns. (33.11)-(33.13) in Srednicki's QFT book) is isomorphic to $\mathfrak{su(2) \times su(2)}$ (given ...
7
votes
1answer
2k views

Antiparticles, charge conjugation and chirality

(Why/how) are antiparticles and charge-conjugates different things? I am trying to understand the effect of discrete symmetries on spinor fields (neutrinos in particular). In the article, Dirac, ...
5
votes
2answers
390 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 ...
5
votes
1answer
224 views

How is the invariant speed of light encoded in $SL(2, \mathbb C)$?

In quantum field theory, we use the universal cover of the Lorentz group: $SL(2, \mathbb C)$, instead of $SO(3,1)$. The reason for this is, of course, that $SO(3,1)$ representations aren't able to ...
5
votes
1answer
458 views

The relationship between spin and spinor curvature

The identity, $$ -\gamma^b{\mathcal{R}}_{ab} = {\mathcal{R}}_{ab}\gamma^b = \frac{1}{2}\gamma^b R_{ab}$$ is presented in the answer to the question Dirac Equation in General Relativity. How does ...
3
votes
1answer
237 views

How does the Lorentz group act on a 4-vector in the spinor-helicity formalism $p_{\alpha\dot{\alpha}}$?

Given a 4-vector $p^\mu$ the Lorentz group acts on it in the vector representation: $$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$ However, I ...
2
votes
0answers
191 views

Transformation law for spinor functions multiplication

Let's have Dirac spinor $\Psi (x)$, which formally corresponds to $$ \left( 0, \frac{1}{2} \right) \oplus \left( \frac{1}{2}, 0 \right) $$ representation of the Lorentz group. What representation is ...
8
votes
2answers
326 views

Are there projective representations of the Lorentz Group NOT coming from a Clifford algebra?

Let $\mathrm{SO}(1,d-1)_{+}$ be the restricted Lorentz Group in $d$ dimensions. Are there projective irreducible representations of this group that do not descend from a representation of $\mathrm{C}\...
4
votes
5answers
235 views

Do spins have spatial directions?

When we consider a spin-1/2 particle and try to write down it's wave function, we have $$|\psi\rangle = a|+\rangle + b|-\rangle,$$ where in a reference about two-level system, the author wrote ...
3
votes
2answers
538 views

Coupling fermions with gravity

Einstein's gravity does not incorporate the "spinor" nature of fermions. The tetrad formulation or Cartan's theory is suggested as the way to go around this problem - by allowing the spin connection ...