Questions tagged [clifford-algebra]

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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 ...
25
votes
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: $$\...
19
votes
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 ...
11
votes
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, ...
10
votes
3answers
1k views

Does Hestenes Zitterbewegung Explain why complex numbers appear in QM?

This question may fit better in the discussion of "Why Complex variables are required by QM?", but it also relates to the extent to which arguments by Hestenes are accepted in mainstream physics and ...
10
votes
1answer
983 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&...
10
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0answers
192 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 ...
8
votes
2answers
972 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-...
8
votes
2answers
381 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}\...
8
votes
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 ...
7
votes
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 ...
7
votes
2answers
730 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 ...
6
votes
2answers
344 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 ...
6
votes
2answers
3k views

How to construct the charge conjugation matrix for any given spacetime dimension?

Generally, Gamma matrices could be constructed based on the Clifford algebra. \begin{equation} \gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2h^{ij}, \end{equation} My question is how to generally ...
6
votes
2answers
282 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 ...
5
votes
1answer
360 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,\...
4
votes
1answer
906 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 ...
4
votes
2answers
179 views

What is the relationship between the Lorentz group and the $CL(1,3)$ algebra?

In my classes the dirac equation is always presented as the "square root" of the Klein Gordon equation, then from this you can demand certain properties from the Matrices (anticommutation relations, ...
4
votes
2answers
260 views

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 ...
4
votes
3answers
111 views

Missing identity element in the Clifford relation

While studying the Dirac equation, $$\left(i\gamma^{\mu} \partial_{\mu} - m\right)\psi = 0.$$ I have been finding difficulty understanding the following summarisation of the algebra that the $\gamma$-...
4
votes
2answers
298 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 ...
4
votes
1answer
177 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}...
4
votes
1answer
376 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 ...
4
votes
0answers
112 views

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 ...
4
votes
0answers
201 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 (...
3
votes
1answer
883 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 ...
3
votes
4answers
2k 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 ...
3
votes
2answers
270 views

A useful identity for Gell-Mann $su(3)$ matrices?

We have the following beautiful result for Pauli $su(2)$ matrices $$(\vec{\sigma}\cdot\vec{a})(\vec{\sigma}\cdot\vec{b}) = \mathbb{I} ~\vec{a}\cdot\vec{b} + i (\vec{a} \times \vec{b}) \cdot \vec{\...
3
votes
2answers
396 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 ...
3
votes
2answers
837 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 ...
3
votes
2answers
398 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^{...
3
votes
2answers
201 views

Is there a Geometric Algebra for gravity?

I have been reading a lot on geometric algebra. I came to ask whether we had a formula for gravity under this algebra? - it turns out that an electromagnetic geometric algebra does exist but I could ...
3
votes
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 ...
3
votes
2answers
629 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 ...
3
votes
2answers
207 views

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 ...
3
votes
1answer
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 ...
3
votes
0answers
180 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 ...
2
votes
3answers
3k views

Should linear algebra and vector calculus from traditional courses be replaced with `geometric algebra`? [closed]

geometric algebra gives geometric meaning to linear algebra and much more. it can provide a coordinate free geometric interpretation of spaces. those who learn of ...
2
votes
1answer
136 views

Gamma matrices in (2+1)

I am sure that is very well-known question and see on this site several similar questions but I would like to specify the answer 1) I know that in $(2+1)$-dimensions one can construct $\gamma$-...
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\...
2
votes
1answer
81 views

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 ...
2
votes
1answer
72 views

Maximal anticommuting sets of Dirac matrices

At the end of this webpage, it is said that there exist 6 maximal anticommuting sets each consisting of 5 Dirac $\Gamma$-matrices. I couldn't find anything more in the book cited there, either. I ...
2
votes
1answer
105 views

Can one find Dirac matrices for any spacetime metric?

For any metric $$g_{μν}$$ is there always a linearly independant spacetime algebra satisfying $$\{\bar{γ}_μ,\bar{γ}_ν\} = 2 g_{μν} I?$$ For a diagonal metric I was able to work out that $$\bar{γ}_μ=\...
2
votes
1answer
231 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 $\...
2
votes
3answers
167 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 ...
2
votes
1answer
297 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 ...
2
votes
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 ...
2
votes
2answers
64 views

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 ...
2
votes
1answer
71 views

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 ...
2
votes
1answer
899 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 ...