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We all know that in addition to scalars and vectors, there are pseudoscalars and pseudovectors, which have an additional sign flip under parity. These are useful and necessary when constructing theories.

It seems logically possible to have a pseudospinor, which is simply a Dirac spinor with an additional sign flip upon parity. However, I have never seen any textbooks even mention this possibility.

Since every term in a Lorentz-invariant Lagrangian requires an even number of spinors, it can be argued that we can always globally replace spinors with pseudospinors, so it is ambiguous whether any specific spinor field can be called a pseudospinor. However, pseudospinors are still necessary to define parity in some cases. For example, if we have $$\mathcal{L} \supset \bar{\psi}_1 \psi_2 \varphi$$ where $\varphi$ is a pseudoscalar, then one of $\psi_1$ and $\psi_2$ must be a pseudospinor if we want the theory to conserve parity, though it's ambiguous which one.

Are pseudospinors valid? If they aren't, what's wrong with them? If they are, why don't textbooks seem to mention them?

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  • $\begingroup$ Chiral $\psi_L,\psi_R$ more practical? $\endgroup$ – Cosmas Zachos Feb 8 at 10:33
  • $\begingroup$ @CosmasZachos I mean a sign flip in addition to exchanging left/right chirality. Or in terms of Dirac spinors, $\psi \to - \gamma_0 \psi$. $\endgroup$ – knzhou Feb 8 at 11:50
  • $\begingroup$ Of course, but it is not hard to be thinking of even and odd combinations thereof for parity eigenstates... $\endgroup$ – Cosmas Zachos Feb 8 at 14:51
  • $\begingroup$ @CosmasZachos Can you please be more explicit? Suppose I have a Lagrangian that is only parity-invariant if I define $\psi_1 \to - \gamma_0 \psi_1$ and $\psi_2 \to \gamma_0 \psi_2$. How exactly do you propose to get rid of the minus sign? $\endgroup$ – knzhou Feb 8 at 15:36
  • $\begingroup$ I am lost here. What prevents you from defining a pseudospinor as a spinor with $\gamma^5 \psi$? That is a $\psi_R -\psi_L$? $\endgroup$ – Cosmas Zachos Feb 8 at 15:52
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  1. Careful, the word "pseudo-" prefixed to various types of spinors (e.g. "pseudo-Majorana spinor") already denotes certain representations of the Clifford algebra, see e.g. this question of mine and its answer.

  2. "Pseudovector" (and pseudoscalar") are properly not "vectors (scalars) with a sign flip", but (d-1)- and d-forms/elements of the exterior algebra on a vector space, cf. this answer of mine

  3. There is no such thing as a pseudo-spinor in your sense. The action of parity on the Dirac spinor representation $(1/2,0)\oplus(0,1/2)$ is simply to exchange the representations, i.e. parity acting on a Dirac spinor simply maps the left-chiral part of a Dirac spinor to its right-chiral part and vice versa. See also this answer by Nephente.

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As stated by Cosmas Zachos in the comments, pseudospinors are perfectly valid but not ever necessary, because if $\psi$ is a pseudospinor, then $\gamma_5 \psi$ is an ordinary spinor and we can work with that instead. That's why they're never mentioned.

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  1. Note that only the double cover $SL(2,\mathbb{C})$ of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$ [or the double cover $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$ of the complexified proper Lorentz group $SO(1,3;\mathbb{C})$] act within a single Weyl representation, cf. e.g. this Phys.SE post. In particular, the improper Lorentz transformations $P$ and $T$ do not act within a single Weyl representation. Therefore the notion of pseudospinor does not make sense for a single Weyl spinor.

  2. Instead the parity $P$ turns left-handed Weyl spinors into right-handed Weyl spinors $\psi_L\leftrightarrow \psi_R$, and vice-versa. Alternatively, the parity $P$ acts on a Dirac spinor as $\psi_D\rightarrow \gamma^0\psi_D$ (for Dirac matrices in a Weyl basis). NB: The above is not all: Additional sign factors arise from intrinsic parity.

References:

  1. M.E. Peskin & D.V. Schroeder, An Intro to QFT, 1995; eq. (3.126).

  2. M.D. Schwartz, QFT & the standard model, 2014; p. 197.

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