# Questions tagged [clifford-algebra]

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### Dirac equation in 1+1D spacetime compared to “standard” 3+1D Dirac equation

In the past couple of weeks I've been studying the Dirac equation and its solutions. During a discussion with a tutor it was pointed out to me that one could formulate something similar to the Dirac ...
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### Doubt about mathematical construction underlying physical systems

Consider the first and second videos of this playlist $[1]$. It seems the professor tried to discuss some heuristic approach between number theory abstract algebra and physics; Classical Physics is ...
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### What are Clifford fragments?

In his article/lecture on "What quantum physics can learn from Egyptian hieroglyphs"", researcher Robert Spekkens talks about Clifford fragments. He describes them as "containing only a subset of the ...
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### Legal values of spin-1/2 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, .. (Grassmann)?

For the spin-1/2 fermion field $\psi$, we may choose it to be a spinor which needs to be Grassmann variable but can also be complex $\mathbb{C}$ Grassmann. (Dirac or Weyl spinor/fermion) We can ...
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### Proving an identity relating the gamma matrices

I'm looking to prove the following identity: $$k_a \gamma^a \gamma^\nu K_b \gamma^b p_c \gamma^c \gamma_\nu P_d \gamma^d = 4(p\cdot K)(P\cdot k)$$ I tried this many times but always seem to be stuck ...
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### General formulation of time reversal symmetry action on fermions

I'm wondering about a general way to define the action of time reversal on a fermion field $\psi$. From a few sources I've read (e.g. appendix A of Witten's paper on fermion path integrals), it seems ...
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### How to construct the charge conjugation matrix for any given spacetime dimension?

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 ...
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### 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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### 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 ...
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### A question about the decoupling of Dirac equation in 1+1 dimension

It is said that in 1+1 dimension, if we take $\gamma^0=i\sigma^2$ and $\gamma^1=\sigma^1$, then the two components of dirac spinor $\psi_L$(upper component) and $\psi_R$(lower component) decouple in ...
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### How to prove $\{\gamma^{\mu}, \gamma^{\nu}\}$ ? (notation problem)

I want to prove that $\{\gamma^{\mu}, \gamma^{\nu} \}=2g^{\mu \nu}$ what are the indices $\mu$ and $\nu$ here? because I know the gamma matrices from 0 to 5 and I need to verify the anti ...
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### 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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### How to count states in SUSY multiplets?

There is an easy proof of the structure of multiplet that I don't reproduce here (it can be found in Bertolini, Lecture on Supersymmetry, pp.40-41 for the massless case and p.47 for the massive one)....
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### How is $\gamma^{\mu}$ defined in the anti commutation relation $\{{\gamma_{5},\gamma^{\mu}}\}$?

how is $\gamma^{\mu}$ defined in the anti commutation relation $\{\gamma_{5},\gamma^{\mu}\}$? does it make a difference if you write the index ${^\mu}$ lower? what does usually change if the index is ...
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### Dirac spinors in 2+1 dimensions

In 3+1 dimensions, Dirac spinors have four complex components. In 2+1 dimensions, the representation of the Clifford algebra by $\sigma^3$ and $-i\sigma^3\sigma^i$, with $i\in\{1,2\}$ is 2-dimensional,...
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### Gamma matrices in higher (even) spacetime dimensions

Suppose we write the gamma matrices in this following representation: \begin{align*} \gamma^{0}=\begin{pmatrix} \,\,0 & \mathbb{1}_{2}\,\,\\ \,\,\mathbb{1}_{2} & 0\,\, \end{...
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### What is the difference between a Pauli spinor, a Weyl spinor, and a Cartan spinor?

I know that a spinor is a complex two components "vector", which is acted on by the $SU(2)$ group under a rotation. In the physics litterature, I often read "Weyl spinors", "Pauli spinors", "Cartan ...
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### Relation between the Dirac Algebra and the Lorentz group

In their book Introduction to Quantum Field Theory, Peskin and Schroeder talk about a trick to form the generators for the Lorentz group from the commutators of the gamma matrices, using their anti-...
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### Weyl and Majorana-Weyl spinors why need commutation?

Let $\psi$ denote a Dirac spinor then Weyl spinors are defined by: $$\psi_{L,R}=\frac{1}{2} (I\pm \gamma)\psi$$ on even dimensions $\gamma$ commutes with $\sigma_{\mu \nu}$ (generators used to define ...
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### Some general formula with trace of gamma matrices relating $\gamma^{(d+1)}$

I want to figure out the trace of gamma matrices relating with $\gamma^{(d+1)}$ for even $d$ dimensional case. First define $\gamma^{(d+1)}$ as \begin{align} \gamma^{(d+1)} = \gamma^1 \gamma^2 \...
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### 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 ...
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### Is the Heighest weight vector in the Spinor rep of $SO(1,d-1)$ zero?

Consider the highest weight vector of the Spinor rep of $SO(1,d-1)$ where $d=2m+1$. It can be shown that: $$\gamma_i \gamma_{m+i}v=v\tag{*}$$ I cannot see why this relation does not imply that $v=0$? ...
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### Clifford Algebra: Wedge product, cross product, and Hodge duality

I've been reading some papers related to Bell's Theorem which involve Clifford Algebra. I am investigating it for an undergrad project but none of my professors seem to know anything about Clifford ...
### How to prove $\gamma^0=(\gamma^0)^T$?
The Dirac gamma matrix $\gamma^0$ is symmetric in Dirac, Weyl and Majorana representation. Is it in general true that $\gamma^0=(\gamma^0)^T$? Can it be proved that $\gamma^0=(\gamma^0)^T$ in a ...
I want to find a similarity transformation $T$ between the Weyl representation and the Dirac representation of the gamma matrices: $\gamma_W^\mu=T \gamma_D^\mu T^{-1}$. It turns out that I can look at ...