Questions tagged [clifford-algebra]

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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 ...
0
votes
0answers
32 views

About elements “factorization” in Clifford Algebras

the article linked below is very instructive and advanced about real Clifford Algebras, and their relationship with Lorentz group. After a general introduction of a Clifford algebra, $\mathcal{Cl}(V,\...
1
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0answers
28 views

Completeness relation of spin matrices

I was reading Hugh Osborne's notes on Conformal Field theory and came across a completeness relation which seems easy to prove but I am unable to do it. ${(s_{\mu\nu})}_{\alpha}^{\beta}{(s^{\mu\nu})}...
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 ...
3
votes
2answers
210 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 ...
0
votes
1answer
53 views

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 ...
0
votes
1answer
96 views

From relativistic equation to find Dirac matrices

Is this possible and then how? $$((\gamma \otimes \mathbf\sigma)\bullet\mathbf p)(\gamma^\prime\otimes\mathbf 1_2) = \gamma\gamma^\prime\otimes\sigma \bullet \mathbf p $$ where $\gamma$ and $\gamma^\...
2
votes
1answer
73 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 ...
0
votes
2answers
244 views

Do I need Gamma matrices in Majorana representation in the Lagrangian of a Majorana fermion?

I understand that the Majorana representation of the Gamma matrices are the real representations of the associated Clifford algebra. A Majorana fermion is defined as a fermion that equals to its ...
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=\...
1
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0answers
37 views

Question about Pauli Matrices

I found the following identities about Pauli matrices from the lecture notes of Supersymmetry. $$((\sigma^{\mu})^{\alpha\dot{\alpha}})^{\ast}=(\bar{\sigma}^{\mu})^{\dot{\alpha}\alpha}$$ $$((\sigma_{\...
3
votes
2answers
211 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 ...
1
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1answer
43 views

Identity Involving Grassmann Variables and Pauli Matrices

I am trying to prove the following identity: $$\theta\sigma^{\mu}\bar{\theta}\theta\sigma^{\nu}\bar{\theta}=\frac{1}{2}g^{\mu\nu}\theta\theta\bar{\theta}\bar{\theta}$$ Where $\theta$ and $\bar{\...
2
votes
2answers
65 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 ...
0
votes
1answer
37 views

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 ...
0
votes
1answer
71 views

How do I construct a Palantini action within Clifford algebra?

I want to define the following two object with spinor-type indices: $${\hat{e}}^{\alpha\beta}(x)\equiv e_n^\mu(x) \gamma_n^{\alpha\beta}\partial_\mu$$ $$\omega^{\alpha\beta}(x) \equiv e^\mu_p(x)W_\...
0
votes
1answer
52 views

Proving identity $\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu}$

In the lecture notes accompanying a course I'm following, it is stated that $$\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu} $$ Yet when I try to prove this,...
2
votes
1answer
232 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
votes
1answer
106 views

Dirac matrices in 1+1 dimensions

Given $\gamma^\mu$ in 1+3 dimensions with signature $(+,-,-,-)$, how can I obtain Dirac matrices in 1+1 dimensions expressed in terms of the $\gamma^\mu$?
2
votes
1answer
138 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 $\gamma$-...
1
vote
1answer
48 views

Confusion with trace of gamma matrices

Using $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu} \mathbf{1}$, it is easy to show that: \begin{align*} \operatorname{tr} \gamma^\mu \gamma^\nu = 4\eta^{\mu\nu} \end{align*} Now, it is also true that ...
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 ...
0
votes
1answer
105 views

Why do the $\gamma$ matrices behave like vectors (tensors)?

In the study of Quantum Field Theory and Group Theory for the spinor representation of $SO$ groups, we know the following correspondence: $\chi C\psi$ scalar $\chi C\gamma^\mu\psi$ vector $\chi C\...
3
votes
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 ...
2
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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 ...
4
votes
2answers
180 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, ...
0
votes
3answers
407 views

Is spinor the sum of scalar, vector, bi-vector, pseudo-vector, and pseudo-scalar?

Is spinor $\psi$ actually the sum of scalar, vector, bi-vector, ..., pseudo-scalar? Before talking about spinors, we have to differentiate two kinds of spacetime, demonstrated with the example of ...
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 ...
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 $\gamma$-...
1
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0answers
121 views

Relation between Dirac spinors, quaternions, and bicomplex numbers

Superficially Dirac spinor resp. Dirac gamma matrices and quaternions and bicomplex numbers seems to be very similar objects. all can be expressed by unitary 4x4 matrices so they seem to represent ...
3
votes
2answers
272 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{\...
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 ...
0
votes
1answer
204 views

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-...
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 ...
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 ...
1
vote
1answer
290 views

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 ...
4
votes
0answers
113 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 ...
1
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1answer
67 views

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)....
-1
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1answer
97 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 ...
1
vote
2answers
738 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,...
2
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0answers
211 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{...
2
votes
1answer
905 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 ...
8
votes
2answers
977 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
204 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 ...
1
vote
1answer
493 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 \...
7
votes
2answers
732 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 ...
0
votes
1answer
48 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
votes
2answers
347 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
vote
1answer
127 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
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 ...