# Exploring Left & Right Spinor Fields in Srednicki

• LAHLH
In summary, the conversation discusses the differences between left and right spinor fields, how they are distinguished by a dot over them, and how they transform under Lorentz and infinitesimal transformations. The conversation also touches on the Hermitian conjugate of a left spinor and its relation to the Lorentz group. The conversation also briefly discusses the Poincare group and the concept of SL(2,C) as a representation of the Lorentz group.
LAHLH
Hi,

I'm just looking at the stuff on left and right handed spinor fields in Srednicki. Srednciki distinguishes fields in the left rep from those in the right rep by putting a dot over them. Since hermitian conjugation swaps the two SU(2) algebras, the hermitian conj of a left spinor is a right spinor etc. e.g. $$\left[\psi_a(x)\right]^{\dag}=\psi^{\dag}_{\dot{a}}(x)$$

Under a Lorentz transformation we have, $$U(\Lambda)^{-1}\psi^{\dag}_{\dot{a}}(x)U(\Lambda)=R^{\dot{b}}_{\dot{a}}(\Lambda)\psi^{\dag}_{\dot{b}}(\Lambda^{-1}x)$$.

Under infinitesimal transformations we have, $$R^{\dot{b}}_{\dot{a}}(1+\delta\omega)=\delta^{\dot{b}}_{\dot{a}}+\frac{i}{2}\delta\omega_{\mu\nu}\left(S^{\mu\nu}_{R}\right)^{\dot{b}}_{\dot{a}}$$. $$R^{\dot{b}}_{\dot{a}}(1+\delta\omega)=\delta^{\dot{b}}_{\dot{a}}+\frac{i}{2}\delta\omega_{\mu\nu}\left(S^{\mu\nu}_{R}\right)^{\dot{b}}_{\dot{a}}$$.
From this one finds,

$$\left[\psi^{\dag}_{\dot{a}}(0),M^{\mu\nu}\right]=(S^{\mu\nu}_{R})^{\dot{b}}_{\dot{a}}\psi^{\dag}_{\dot{b}}(0)$$Now I want to take the Hermitian conjugate of this to get Srednicki's 34.16. I presume this conjugate is w.r.t. the a,b, internal indices not the spacetime indices,

$$\psi^{\dag}_{\dot{a}}(0)M^{\mu\nu}-M^{\mu\nu}\psi^{\dag}_{\dot{a}}(0)=(S^{\mu\nu}_{R})^{\dot{b}}_{\dot{a}}\psi^{\dag}_{\dot{b}}(0)$$

OK, so taking the h.c.:

$$(M^{\mu\nu})^{\dag}\psi_{a}(0)-\psi_{a}(0)(M^{\mu\nu})^{\dag}=\left[(S^{\mu\nu}_{R})^{..\dot{b}}_{\dot{a}}\psi^{\dag}_{\dot{b}}(0)\right]^{\dag}$$

$$(M^{\mu\nu})^{\dag}\psi_{a}(0)-\psi_{a}(0)(M^{\mu\nu})^{\dag}=\psi_{b}(0)\left[(S^{\mu\nu}_{R})^{..\dot{b}}_{\dot{a}}\right]^{\dag}$$
But given that the M's have no a,b type indices I don't really see what this would mean.

Could anyone point me in the right direction on how to derive this 34.16, thanks

Last edited:
Hi,

"the hermitian conj of a left spinor is a right spinor"

To my understanding this is not correct. If $$\chi$$ is a left-spinor, then let

$$\varepsilon$$ =
( 0 -1 )
( 1 0 )
Then $$\varepsilon \chi^c$$ transforms as a right-spinor, where the superscript-c denotes complex conjugation (without transpose). Left- and right-handed spinors transform as $$R$$ and $$(R^{-1})^\dagger$$ respectively, where $$R \in SL(2)$$ and the dagger denotes hermitian conjugate. The reason it works is because
$$\varepsilon R = (R^{-1})^T \varepsilon$$
which you can show to be true by writing it out longhand and using det R = 1.

I'm sorry I can't post in more detail because I have to run off urgently... I hope I got all the details right but at least in broad sweep I think what I've said is correct. Hope it helps get you part way towards the rest of the question which I haven't had a chance to get to,

Cheers
Dave

Hi,

thanks for the reply schieghoven. I see, it's strange then that Srednicki should say on p210 "the Hermitian conjugate of a field in the (2,1) representation should be a field in the (1,2) representation"

Is this an error then, or is he just being sloppy? How does one get from his 34.15 to 34.16 with either definition of a RH spinor?

cheers

After reading my nearest group theory book (Jones). I wonder if what you are describing here is the transformations of upper and lower spinors under SU(2), where $$C_{ab}=\epsilon_{ab}$$ is the (invariant analogue of the GR metric) and takes upper spinors that transform under $$U*$$ of SU(2) to lower spinors that transform under U of SU(2).

But I am under the impression that these spinors are not the same as a Weyl spinor? since Weyl LH/RH spinors (2,1) or (1,2) transform under the Poincare group which can be thought of as a product of SU(2)XSU(2). So it's kind of like, one "component" (probably not the correct word here) of the Weyl spinor transforms under U and the other under U* (i.e. we have two SU(2) algebras that are complex conjugates of each other)

hmm, somewhat confused.

Hi,

I disagree with books that say SU(2) x SU(2) is a rep of the Lorentz group. It's transparently not the case, because SU(2) x SU(2) is compact; the Lorentz group SO(1,3) is not. I kind of understand why authors try to do it; it is because if J_j, K_k are generators for rotations and boosts respectively, these authors construct the quantities $J_j \pm iK_j$ and show that they obey SU(2) commutation relations. Doing so belies the fact that J_j and K_j are generators of the real (i.e. not complex) Lie group SO(1,3). $J_j \pm iK_j$ is not a real linear combination of the J_j, K_j, so it's not correct to change basis this way.

(Note that you referred to the Poincare group, which generally speaking refers to something different again:
Lorentz = rotations and boosts = 6 generators
Poincare = rotations, boosts, and translations = 10 generators)

I think of spinors (= Weyl spinors I think??) as transforming under the group SL(2,C) = 2-by-2 complex matrices with unit determinant. This group has six real dimensions and the generators obey the commutation relations for the Lorentz group, given in e.g. Weinberg's text or Foldy (PR 102, 568, 1956). There are two distinct representations which you can switch between using $U \mapsto U^{-1}^*$. I don't think you can use U* because it wouldn't preserve the order of operation: $U^*V^* \neq (UV)^*$, so it's not a representation.

I don't have a copy of Srednicki, so I can't comment on the other statements. I'm afraid we seem to get a step further away from your original question with each reply :-) .

Cheers

Dave

Hi,

OK admittedley you kind of lost me, as all I know is what I've read in Srednicki/Jones, who both do exactly the process you described to argue we can split the Lorentz algebra into two SU(2) algebras etc etc. I'm aware the Poincare and Lorentz group are different in the way you stated, however I think if I remember correctly from Jones, using the Poincare group he still drew the same conclusions about this splitting into su(2)xsu(2), his algebra including not just the J,K generators but also P and H.

I've heard people saying SL(2,C) is a rep of Lorentz, but don't really understand this idea, could you perhaps explain it at all?

As for my original question a free copy of Srednicki is available here: http://www.physics.ucsb.edu/~mark/qft.html if anyone is interested

thanks

Cool, I hadn't seen either of these papers before.

I like Bilal's construction more than Srednicki's... Still I don't know why Bilal didn't simply state

$$J_j = \sigma_j, \qquad K_j = i\sigma_j, \qquad (j=1,2,3)$$

from which it can be verified directly that they satisfy the Lie bracket (Bilal eq 2.1) for the Lorentz group. To construct the Lie group from the Lie algebra, take the exponential
$$\exp( a_j J_j + b_j K_j )$$
(Bilal eq 2.8) for arbitrary $a_j, b_j \in \mathbb{R}$. The exponential of a traceless matrix is a matrix with unit determinant --- which is why we get to $SL(2,\mathbb{C})$.

My take on the spin-half representation of the Lorentz group is that you need to find a group whose elements can be mapped into the Lorentz group, such that the group operation is preserved. This group is called a covering group. In other words, if U_1, U_2 are elements of the covering group, then you have a map P into the Lorentz group such that
P(U_1) P(U_2) = P(U_1 U_2)
where on the RHS the product is the group operation of the Lorentz group, on the LHS the product is the group operation of the covering group. So while U_1, U_2 represent transformations on some other space (the space of spinors), they can be mapped to a unique Lorentz transformation, and the effect of multiple transformations is reflected consistently under that mapping. The map is not one-to-one, so I think that Srednicki's (33.8) (a kind of "inverse" to the above) is misleading at best.

The group and the covering group are locally isomorphic, so they have isomorphic Lie algebras. So you first satisfy the Lie bracket, then exponentiate to construct the group from the Lie algebra.

I'm afraid I can't remember a good reference off the top of my head, which is why I wrote out this fairly hashed-up overview. Perhaps you can find an topology text with a chapter on topological groups... but depends how much time you have to throw at it.

Hope that helps
Dave

## 1. What is a spinor field?

A spinor field is a mathematical construct used to describe the properties and behavior of spin particles in quantum field theory. It is a complex-valued field that transforms under rotations in a specific way, depending on the spin of the particle it represents.

## 2. What is the significance of left and right spinor fields?

In quantum field theory, particles can have either left-handed or right-handed spin, which corresponds to their direction of spin relative to their direction of motion. Left and right spinor fields represent these two types of spin and play a crucial role in understanding the behavior of particles in the universe.

## 3. How are spinor fields explored in Srednicki's book?

In Srednicki's book, spinor fields are explored through the use of mathematical equations and diagrams that illustrate how they transform under rotations and interactions with other particles. The book also discusses their role in various physical phenomena, such as the Standard Model of particle physics.

## 4. What are some real-world applications of studying spinor fields?

The study of spinor fields has many practical applications, such as in the development of new technologies, understanding the behavior of particles in high-energy collisions, and predicting the properties of new particles. They are also used in various areas of physics, including particle physics, condensed matter physics, and cosmology.

## 5. Are there any controversies or open questions surrounding spinor fields?

While spinor fields are a well-established concept in quantum field theory, there are still ongoing debates and unanswered questions surrounding their properties and behavior. For example, some researchers are still trying to reconcile the fundamental differences between left and right spinor fields, and there is ongoing research into the role of spinors in the unification of fundamental forces.

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