# Questions tagged [clifford-algebra]

The tag has no usage guidance.

101 questions
Filter by
Sorted by
Tagged with
91 views

### How to vary fermion action in the index-free Clifford notation with respect to spin-connection?

In ref (1), it is claimed that the Dirac action (2.30) $$S_D \sim \int ( \overline \psi \star eee D \psi + \overline {D\psi} \star eee \psi)$$ becomes \...
266 views

### Charge conjugation in arbitrary basis

Consider the matrix $C = \gamma^{0}\gamma^{2}$. It is easy to prove the relations $$C^{2}=1$$ $$C\gamma^{\mu}C = -(\gamma^{\mu})^{T}$$ in the chiral basis of the gamma matrices. Do the two ...
2k 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 ...
100 views

### Clifford Algebra in 3D [duplicate]

Why the gamma matrices are taken 2 by 2 (Pauli matrices) in 3 dimensional Clifford Algebra. As in 4D Clifford Algebra the matrices are 4 by 4, in 3D Algebra why are they not 3 by 3 matrices? The ...
360 views

318 views

### How to derive the form of the invariant spinor inner product?

So we have gamma matrices that satisfy the spacetime algebra relations, $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu}$. We know that if we set $\sigma^{\mu\nu} = \frac{1}{4}[\gamma^\mu, \gamma^\nu]$ ...
376 views

### Showing that a bilinear variation is Lorentz invariant

Let $\psi, \chi$ be a spinor (say Dirac). Then the infinitesimal Lorentz variation is given by $$\delta \psi = -\frac{1}{4}\lambda^{\mu \nu} \gamma_{\mu \nu}\psi$$ then I think that the conjugate is ...
298 views

### In what sense is the chiral decomposition of spinors unique?

We may decompose a spinor field $\psi = \psi_L + \psi_R$ where $\psi_L = \frac12 (1 - \gamma^5) \psi$ and $\psi_R = \frac12 (1 + \gamma^5) \psi$. (I believe this is because the clifford algebra has ...
157 views

I am reading that there exists a unitary matrix $C$ (the charge conjugation) matrix such that each matrix $C\Gamma^{A}$ is either symmetric or anti-symmetric. Now, $\Gamma^{A} = \{ {\bf 1}, \gamma^{\... 2answers 399 views ### Higher rank$\gamma$-matrix question I read that the higher rank$\gamma\$ matrices can be written as alternate commutators and anti-commutators. For example, the rank 3 gamma matrix can be written as $$\gamma^{123} = \frac{1}{2}\{\gamma^{... 1answer 989 views ### Is there a textbook which covers QM via Geometric Algebra (GA)? There is at least one good book on classical mechanics using Geometric Algebra (GA): New Foundations in Classical Mechanics by David Hestenes. Likewise there is at least one good book on classical E&... 4answers 13k views ### Commutator of Dirac gamma matrices Quick question...For some reason I'm having trouble finding an identity or discussion for the commutator of the gamma matrices at the moment...i.e$$\gamma^u\gamma^v-\gamma^v \gamma^u$$but I am not ... 2answers 635 views ### Dirac group representation I am currently taking a representation theory class (from a physicist), and I am very confused about the Dirac groups' irreducible representations. First of all, all the Dirac matrices in the ... 2answers 731 views ### QFT Hilbert spaces over other rings than the complex numbers \mathbb{C} I would like some help evaluating a physics theory recently proposed by a physics professor at the College of Dupage. I think the theory is utterly wrong, for very simple reasons. If an amateur ... 2answers 1k views ### Mathematica package for supergravity and string theory [duplicate] I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra ... 0answers 202 views ### Subgroups of the Clifford Group We recall the definition of a Clifford group (over n qubits) is the set of unitary transformations:$$\{U: UPU^\dagger\in\mathcal{P}\}$$where \mathcal{P} denotes the corresponding Pauli group (... 2answers 5k views ### Dirac, Weyl and Majorana Spinors To get to the point - what's the defining differences between them? Alas, my current understanding of a spinor is limited. All I know is that they are used to describe fermions (?), but I'm not sure ... 1answer 1k views ### Gamma Matrices in Dimensional Regularization Prove that tr\left(\gamma_\mu\gamma_\nu\gamma_\rho\gamma_\sigma\gamma_5\right)=0 when the spacetime dimension is not 4. What I have tried: We know that \gamma_\alpha\gamma^\alpha=d\mathbb{1}, so ... 1answer 91 views ### Problem evaluating C^{-1}M^\dagger C How can I show the following?$$\overline{\psi_L}M^\dagger (\psi_L)^c=\overline{\psi_L}CM^\dagger\overline{\psi_L}^T$$where \psi^c=C\overline{\psi}^T and C=i\gamma^2\gamma^0=-C^T=-C^\dagger=-C^{-... 3answers 2k views ### Do gamma matrices form a basis? Do the four gamma matrices form a basis for the set of matrices GL(4,\mathbb{C})? I was actually trying to evaluate a term like \gamma^0 M^\dagger \gamma^0 in a representation independent way, ... 4answers 5k views ### Dimension of Dirac \gamma matrices While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the \gamma matrices:$$\...
Generally, Gamma matrices could be constructed based on the Clifford algebra. $$\gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2h^{ij},$$ My question is how to generally ...