Questions tagged [clifford-algebra]

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1answer
132 views

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 ...
3
votes
4answers
3k views

Proof of the Anti-Commutation Relation for Gamma Matrices from Dirac Equation

My textbook on QFT says that the Dirac equation can be used to show the following relation: $$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$$ I have searched around and unable to find how to prove this ...
2
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0answers
247 views

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{...
8
votes
2answers
1k views

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-...
1
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0answers
240 views

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 ...
0
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1answer
49 views

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 $v=0$? ...
6
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2answers
388 views

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 ...
1
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1answer
167 views

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 ...
3
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1answer
1k views

Transformation between Weyl and Dirac representation of Gamma matrices

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 ...
8
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2answers
429 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}\...
0
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0answers
109 views

How are the covariant Pauli matrices defined?

When doing calculations with Weyl spinors, terms like $\theta\sigma^\mu\theta^\dagger$ appear. I know that for 3+1 spacetime dimensions, $\sigma^\mu = (\textbf{1}, \sigma^i)$ with $i=1,2,3$ the usual ...
1
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0answers
69 views

Matrix representation of Clifford algebra - steps [closed]

In (Vaz and da Rocha, 2016;pg108) the following two step process is given for finding the matrix representation of a Clifford algebra: (verbatim; except for notation) (1) Choose a set of $N$ ...
4
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1answer
185 views

Fierz like identity for $\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}$

It is known that contracting over the vector indices of two Pauli matrices (the 3d γ-matrices) can be simplified to a bunch of δ functions. This is done via the Fierz formula $$\delta_{ab}\sigma^a_{ij}...
0
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1answer
286 views

Trace technology with polarisation vectors

Consider $d$-dimensional gamma matrix structures. I have an expression like $$ \sum_{h_2=\pm}\text{Tr}(\not{\xi}_2\not{p}_3\bar{\not{\xi}}_2\not{p}_1), $$ where $\not p=p^\mu \eta_{\mu\nu}\gamma^\nu$ ...
2
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1answer
282 views

Link between the Grassmann algebra and spinors

What is the exact link between spinors and the Grassmann algebra? I'm pretty sure there's one, based on the following: The Berezin integral in path integrals is done over the Grassmann algebra of $\...
0
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1answer
76 views

What is the $4\times 2\times 2$ matrix $\sigma_{A \dot B}^{\mu}$ explicitly?

In Tales of 1001 Gluons by Stefan Weinzierl, in the end of page 36, (163), $\sigma_{A \dot B}^{\mu} = (1, -\sigma_x, -\sigma_y, -\sigma_z)$. It seems that $\sigma_{A \dot B}^{\mu} = (1, -\sigma_x, -\...
2
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0answers
187 views

How this spinor identity is shown?

In one QFT problem it is asked to prove the following identity: $$\overline{u}_\sigma(p)\gamma^\mu u_{\sigma'}(p)=2\delta_{\sigma\sigma'}p^\mu.$$ Considering $u_\sigma$ the basis solutions to the ...
2
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1answer
1k views

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 ...
1
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0answers
169 views

Dimension of Representation of Majorana Fermions with Euclidean Metric?

It is possible to represent the Dirac matrices in the Majorana basis using $N= 2^{⌊d/2⌋}$-dimensional matrices, as shown here. This source uses a Minkowski metric. It would then be possible to move to ...
1
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2answers
95 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) \begin{equation} S_D \sim \int ( \overline \psi \star eee D \psi + \overline {D\psi} \star eee \psi) \end{equation} becomes \begin{equation} \...
3
votes
1answer
332 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 ...
34
votes
2answers
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 ...
0
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0answers
137 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 ...
5
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1answer
441 views

How to prove $tr\{\sigma_1\sigma_2\sigma_3\}=\pm 2i$ with only using $\{\sigma_i,\sigma_j\}=2\delta_{ij}$

How to prove $tr\{\sigma_1\sigma_2\sigma_3\}=\pm 2i$ with only using $\{\sigma_i,\sigma_j\}=2\delta_{ij}$ and we cannot use the explict representation of Pauli matrices and cannot use $[\sigma_i,\...
3
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1answer
1k views

Covariant gamma matrices

Covariant gamma matrices are defined by $$\gamma_{\mu}=\eta_{\mu\nu}\gamma^{\nu}=\{\gamma^{0},-\gamma^{1},-\gamma^{2},-\gamma^{3}\}.$$ The gamma matrix $\gamma^{5}$ is defined by $$\gamma^{5}\equiv ...
1
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2answers
1k views

Hermitian properties of the gamma matrices

The gamma matrices $\gamma^{\mu}$ are defined by $$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}.$$ There exist representations of the gamma matrices such as the Dirac basis and the Weyl basis. Is it ...
2
votes
1answer
433 views

Representations of the Dirac algebra, hermitian adjoint and traces

Strictly speaking this is a math question, but since the Dirac algebra is much more important in physics than in math I thought I'd have a better chance of getting an answer here. The Dirac algebra ...
10
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0answers
210 views

Can change in position due to acceleration be expressed using dual quaternions?

Dual quaternions seem like an appealing way to model 6DOF motion since they linearize rotation. I've reviewed what literature I can find on then, and found expressions for translation and change in ...
1
vote
2answers
847 views

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,...
3
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2answers
449 views

How to express $\gamma^{\mu} \gamma^{\nu}$ as a linear combination of {1, $\gamma^5, \gamma^{\mu}, \gamma^{u} \gamma^5, \sigma^{\mu \nu}$}?

** EDIT: I think I have completely missed the mark on asking my question. Here is another try. I do not understand what a linear combination means in this situation. My naive desire is to have an ...
0
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1answer
382 views

Dimensionality of Gamma Matrices

If I express the Dirac equation in the form of $$i\hbar \frac{\partial}{\partial t} \psi_a(x) = \left(-i\hbar c(\alpha^j)_{ab}\partial _j + mc^2(\beta)_{ab}\right)\psi_b(x),$$ with the constraints $...
0
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0answers
311 views

Spinors in dimensions greater than $4$

The Dirac equation describes the behaviour of non-interacting spin-$1/2$ fermions in a quantum-field-theoretic framework and is given by $$i\gamma^{\mu}\partial_{\mu}\psi=-m\psi,$$ where $\gamma^{\...
5
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2answers
353 views

What has “quantisation” to do with associated graded algebras?

I was currently reading an introduction into spin geometry by José Figueroa-O’Farrill. The first chapter handles Clifford algebras. When discussing the connection of the Clifford algebra to the ...
0
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1answer
261 views

Decomposition of gamma matrices into sigma matrices and their equvialence [closed]

Considering even dimension. From the definition of $\gamma^{(d+1)}$ (all products of gamma matrices) and its anti commutation, $\{ \gamma^\mu, \gamma^{(d+1)}\}=0$, if we choose $\gamma^{(d+1)}$ as ...
1
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1answer
124 views

Is there a geometric object analagous to a spinor that encodes projections onto bivectors?

The most sensible geometric interpretation of spinors that I've come across is that they encode projections in the Clifford algebra. So if $\mathbf A$ is a vector with components $A_i$ and $\psi$ is ...
1
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1answer
675 views

How to treat charge conjugation and time reversal operators for Dirac Field in representation invariant way?

Since manipulations with charge conjugation and time reversal operators involve taking complex conjugate of bispinors, most formulas are not invariant under change of representation of $\gamma$ ...
2
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0answers
79 views

From $\gamma$ to $\sigma$, finding proper basis

For $d$ dimensional case, usual gamma matrices have basis $\Gamma^A = \{1, \gamma^a, \cdots \gamma^{a_1 \cdots a_d}\}$ (Let's think about even case only for simplicity, I know for odd case only up to ...
2
votes
3answers
196 views

Notation about basis of gamma matrices in $4d$

In Quantum Field theories, we encounter gamma matrices a lot. Reading from various textbook, i encountered some textbook use different basis for their gamma matrices. Gamma matrices are defined such ...
1
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1answer
578 views

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 \...
4
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1answer
1k views

How are Clifford algebras related to Dirac Equation

Given a vector space $V$ and a quadratic form $q$ for the vector space. The tensor algebra is defined as $\mathcal{T}(V)=\sum_{i=1}^{\infty} V^{\otimes i}$. The set $\mathcal{I}=\{x\otimes x-q(x)\cdot ...
2
votes
1answer
62 views

Sign choice for sigma-matrices

I'm trying to figure out the consequences of the sign choice $$ \sigma^\mu = (\mathbf{1},\vec\sigma)\qquad\text{vs.}\qquad \sigma^\mu = (-\mathbf{1},\vec\sigma) \,. $$ This choice does not affect the ...
1
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0answers
339 views

Are Fock spaces just a special type of tensor algebra?

Are Fock spaces just a special type of tensor algebra? The definitions I am using: http://en.wikipedia.org/wiki/Fock_space http://en.wikipedia.org/wiki/Tensor_algebra From what I can tell, the ...
2
votes
1answer
254 views

Proof of two Lorentz-algebra identities

I am currently working through the QFT introduction text by Peskin and Schroeder and try to fill in two identities that I wasn't able to prove (it should be fairly simple, but my experience with this ...
1
vote
1answer
537 views

Symmetry properties of gamma matrices

While reading a paper on supersymmetry i faced the following problem. Its about the symmetry property of charge conjugation matrix in different space time dimension. The charge conjugation matrix is ...
3
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0answers
195 views

Geometric Algebra formulation of EM in nonuniform dielectric media

Normally, there is a straightforward formulation of Maxwell's equations in free space under the GA framework, which reduces to (picking the right units): $$ \nabla F = \mu_0 J $$ with some ...
4
votes
1answer
403 views

The fifth gamma matrix and fermion fields

I am aware of the various relations with Dirac spinors and chirality but how does the fifth gamma matrix $\gamma^5$ behave with fermion fields, $\psi$? Does the fifth gamma matrix have any particular ...
1
vote
1answer
795 views

Gamma matrices relations (Dirac Spinors: QFT) [closed]

The entry question in an exam paper: I think I have made an elementary error in the transpose somewhere invoked by a conceptual misunderstanding of how spinors behave with gamma matrices under a ...
7
votes
2answers
326 views

Is there some physical intuition behind Clifford Algebras?

The mathematically rigorous definition of a Clifford Algebra is as follows: Let $V$ be a vector space over a field $\mathbb{K}$ and let $Q : V\to \mathbb{K}$ be a quadratic form on $V$. A Clifford ...
3
votes
2answers
914 views

Different definitions of spinors

Recently I've read a little about the description of particles with spin in the book Quantum Mechanics by Cohen-Tannoudji. Although I yet didn't fully study the subject, I've read one interesting part ...
2
votes
1answer
2k views

Derivation of Gordon identity from Srednicki [closed]

On srednicki page 240 (print) there is a derivation of the Gordon identity, and it starts with stating that $$ \require{cancel} \gamma^{\mu}\cancel{p} = \frac{1}{2} \big\{\gamma^{\mu},\cancel{p} \big\...