Questions tagged [clifford-algebra]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0
votes
0answers
62 views

Product of an odd number of Dirac $\gamma$ matrices

Suppose $n$ is an odd number. Why can we write $a\llap{/}_1 a\llap{/}_2 ... a\llap{/}_n$ as $$a\llap{/}_1 a\llap{/}_2 ... a\llap{/}_n = V_\mu \gamma^\mu + A_\mu \gamma^\mu \gamma_5$$ for some $V_\mu, ...
1
vote
1answer
37 views

(Anti)commutation relations for higher-dimensional anti-symmetrized Gamma matrices

Suppose $a,b,c...=0,....,D-1$ are Lorentz indices of $SO(1,D-1)$ tangent space and consider $D$-dimensional Clifford algebra defined by the usual anticommutation relation $$\{\Gamma^a,\Gamma^b\}=2\eta^...
0
votes
0answers
52 views

The Lagrangian density of electromagnetism, expressed in geometric algebra such that $\nabla \mathbf{F}=0$ is the equation of motion?

Geometric algebra admits a very short and sweet definition of Maxwell's laws of electromagnetism: $$ \nabla \mathbf{F}=0 $$ where $$ \mathbf{F}=\mathbf{E}+i\mathbf{B} $$ and where $$ \nabla \mathbf{F}=...
2
votes
1answer
64 views

Are the $\alpha$ and $\beta$ matrices of the Dirac equation unique?

Assume we are dealing wth three spatial dimensions $d=3$ which requires 3 $\alpha$ matrices. Furthermore assume that we are looking for them in the space of 4-dimensional matrices, not in higher ...
2
votes
0answers
49 views

The “basic hamiltonian” of topological systems

I am currently studying topological insulators and repeatedly found the claim (e.g. here), that the "basic hamiltonian" of a topological system in $d$ spatial dimensions can be written using ...
0
votes
2answers
85 views

How do we know there are only 16 Dirac bilinears?

We know there are 5 types of bilinears in 4 dimensions, all of them add up to contribute with 16 independent DoF (degrees of freedom). Namely, these bilinears are known as: scalar (1DoF), pseudoscalar(...
1
vote
0answers
143 views

An $SL(2,C)$ representation and Dirac Spinor

In PCT, spin and statistics, and all that book, the following example is given: Let $S(A)$ be a representation of $SL(2,C)$ given as : $$S(A)=\frac{1}{2}\left(a^{0} \mathbf{1}+\mathbf{a} \cdot \...
1
vote
0answers
59 views

Is there a bosonic representation of Clifford algebra in (1,3) spacetime?

By a suitable combination of Dirac's $\gamma_\mu$ matrices one can define creation and destruction operators satisfying fermionic anticommutators. Is there a similar result for bosons in the context ...
0
votes
1answer
56 views

How does $\{\gamma^\mu , \gamma^\nu\}= 2g^{\mu\nu}I$ imply that $\gamma^\mu$ is traceless?

How does $\{\gamma^\mu , \gamma^\nu\}= 2g^{\mu\nu}I$ imply that $\gamma^\mu$ is traceless? where I represents the identity matrix I know that $$\{\gamma^\mu , \gamma^\nu\}=\gamma^\mu\gamma^\nu + \...
0
votes
0answers
35 views

Spinor tensor multiplication rules

I’m wondering how to contract a product of gamma matrices like $$(\gamma^5)_b^{~~d}(\gamma^5\gamma^{\mu})_{ad}$$ Is this just $$(\gamma^5)_b^{~~d}(\gamma^5\gamma^{\mu})_{ad} = (\gamma^5\gamma^5\...
0
votes
0answers
22 views

Equivalence of the Dirac representation of the Lorentz algebra and its conjugate in even dimensions - Polchinski

In Polchinski's String Theory, Appendix B.1 we look at the smallest irreps of the Clifford algebra in even-dimensional spacetimes $d=2k+2$. In (B.1.16) he defines two operators from the gamma matrices ...
1
vote
3answers
51 views

Trace of 3 gamma matrices in 3 dimensions

I know that for a Lorentzian metric with signature $(-,+,+,..,+)$, in an even number of dimensions the trace of any odd number of gamma matrices is zero. This can be proven by defining $\gamma_\ast=\...
1
vote
0answers
32 views

Defining a velocity in projective geometry

In spacetime geometric algebra, Hestenes and Doran define a 4-velocity for a particle moving in spacetime $\mathcal{V}^{3, 1}$ by simply differentiating w.r.t. the parameter defining the curve to get $...
4
votes
3answers
184 views

Is there an identity for anti-commutator $\{ A B, \, C D \}$ in terms of commutators $[\, , \,]$ only?

I'm looking for an identity that could express the anti-commutator $$\tag{1} \{ A B , \, C D \} \equiv A B C D + C D A B $$ expressed as a combination of commutators only: $[A,\, C]$, $[A, \, D]$, etc....
2
votes
1answer
118 views

Going from the Dirac Lagrangian to the adjoint Dirac equation

I am comfortable doing the following calculation, Derivation of the adjoint of Dirac equation, notably — going from the standard Dirac equation to the adjoint Dirac equation via using Dirac ...
1
vote
0answers
71 views

Generalization of Dirac matrices

Is it possible to have a set of 16-dimensional matrices $\gamma_{\mu}^{a}$ such that $$\{\gamma_{\mu}^{a},\gamma_{\nu}^{b}\} = 2\delta^{ab}\eta_{\mu\nu}$$ where $\eta_{\mu\nu}$ is the Minkowski metric ...
2
votes
0answers
64 views

Dirac matrix algebra from bosonic creation/anihilation operators?

Using some generic fermionic creation/anihilation operators $a_i$ and $a_i^{\dagger}$ ($i, j = 1, 2, 3, \dots, N$) such that \begin{align} \{ a_i, \, a_j^{\dagger} \} &= \delta_{ij}, \tag{1} \\[...
2
votes
1answer
39 views

What transformation gives a Weyl-like representation by flipping $\gamma^0$ and $\gamma^5$?

The usual Weyl representation of the Dirac matrices is defined like this: $$\tag{1}\gamma_W^a = T_W \, \gamma^a \, T_W^{-1},$$ where \begin{align}\tag{2} T_W &= \frac{1}{\sqrt{2}} (1 + \gamma^5 \, ...
0
votes
3answers
118 views

Proving identity $tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{5})=-4i\epsilon^{\mu\nu\rho\sigma}$

Im trying to proof the following identity: $tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{5})=-4i\epsilon^{\mu\nu\rho\sigma}$ when $\gamma^{\mu},\gamma^{\nu},\gamma^{\rho},\gamma^{\...
1
vote
1answer
100 views

Lie algebra/group/basis of the four gamma matrices along with the identity?

Do the four gamma matrices along with the identity element constitute a lie algebra? With real coefficients we have $$ \mathbf{v}_{\mathbb{R}}=aI+t\gamma_0+x\gamma_1+y\gamma_2+z\gamma_3 \tag{real ...
2
votes
0answers
54 views

How to demonstrate that Fierz-like identity for 2-components Weyl spinors? [duplicate]

Consider the 2-components Weyl spinors with the following scalar product \begin{equation}\tag{1} \langle \, \phi, \, \psi \, \rangle = \phi^{\top} \, \sigma_y \: \phi, \end{equation} where $\sigma_y$ ...
3
votes
1answer
84 views

Metric in 2-components spinor space

Consider the two components spinors (usually Weyl spinors): $\psi = (c_1, c_2)^{\top}$, where $c_i$ are complex numbers. The metric in spinor space is usually defined with the second Pauli matrix $\...
6
votes
1answer
211 views

Time reversal for Dirac particles in 2+1 or 6+1 (mod 8) spacetime dimensions

It is my understanding that time reversal invariance for Dirac fermions is usually (in 3+1 dimensions at least) implemented by an antiunitary operator ${\mathfrak T}$ that acts on the Dirac field ...
2
votes
2answers
194 views

Contracting gamma matrices with explicit indices

So I was calulating the matrix element of an interaction and arrived at the following contraction $$\gamma^\mu_{ab}\gamma_{\mu\,cd}$$ With $a,b,c,d$ spinor indices that are never contracted with ...
1
vote
1answer
103 views

Why do vectors appear to transform like rank-2 tensors in the Clifford algebra representation?

In Euclidean space with metric $\delta_{ij}$ a Galilean vector rotates via $\hat{V} = \Lambda(\theta) V$ where $\Lambda$ is a member of $O(3)$. This can be represented by having the $V$ be column ...
0
votes
1answer
50 views

Spinorial representation of Lorentz group for solution to Dirac equation

In my relativistic quantum mechanics course, we found plane wave solution to the Dirac equation by first studying it the reference frame of the particle. Using a plane wave solution for both positive ...
5
votes
1answer
159 views

Spinors, Spacetime and Clifford algebra

I'm looking to understand the intrinsic connection that Clifford algebra allows one to make between spin space and spacetime. For a while now I've trying to wrap my head around how the Clifford ...
0
votes
1answer
87 views

Is $\gamma_\mu \gamma^\mu$ a unit operator?

Is the term: $$γ^μ γ_μ$$ An identity matrix? Since,if we start with both the Dirac equation, $$(iγ^μ ∂_μ-m)Ѱ=0$$ We find that, $$iγ^μ ∂_μ=m$$ If we square both sides, we get, $$-γ^μ γ_μ∂^μ ∂_μ=m^{2}$$...
2
votes
1answer
117 views

Does the geometric algebra of curved space have a matrix representation?

Suppose the geometric algebra defined by $$ \frac{1}{2}(e_\mu e_\nu +e_\nu e_\mu)=g_{\mu\nu} $$ where $e_\mu,e_\nu$ are generators of the algebra, and where $g_{\mu\nu}$ are elements of the reals. I ...
4
votes
3answers
757 views

Physical interpretation of gamma matrices

Just out of plain curiosity, I want to ask: What are/is the physical interpretation(s) of the gamma matrices? If there is none, is it right to assume that it is just a mathematical fudge-factor?
0
votes
0answers
202 views

Traces of gamma matrices in $d$ dimensions

For $d=4$, some identities of the traces of gamma matrices are: $tr[\gamma_\mu] = 0$ $tr[\gamma_\mu \gamma_\nu ] = 4g_{\mu\nu}$ $tr[\gamma_\mu\gamma_\alpha\gamma_\nu] = 0$ $tr[\gamma_\mu\gamma_\alpha\...
1
vote
0answers
38 views

What Is a Geometric Algebra Model For Compound Gyroscopes? [closed]

Since geometric algebra (quaternion, clifford algebra, etc.) intrinsically represents 3D rotations, I was wondering how one would model systems involving two or more gyroscopes. For example, ...
0
votes
0answers
23 views

Is there a Cauchy integeal formula in Geometric algebra (GA) or space-time algebra or clifford algebra>

I am curious about Cauchy integral formula which is generalized to Clifford algebra especially Cl(1,3). If you are familiar with paper or theorems about this subject and share them. It will be very ...
1
vote
1answer
53 views

Matrix representation of the CAR for the fermionic degrees of freedom

The canonical anticommutation relations (CAR) for a fermionic degree of freedom can be written as follows: $$ a^2 = \left( a^{\dagger} \right) ^2 = 0, $$ $$ a a^{\dagger} + a^{\dagger} a = 1. $$ ...
2
votes
1answer
150 views

Spinor Understanding: QFT vs pure Representation Theory

I have some questions on terminology used in QM & QFT and (pure mathematical) representation theory treating the concept of "spinor". Let us focus on Dirac spinor as described in https://en....
1
vote
2answers
432 views

Anticommutator of spin-1 matrices

We know that in the spin-1/2 representation the anticommutation relation of the Pauli matrices is $\{\sigma_{a},\sigma_{b}\}=2\delta_{ab}I$. Does a similar relation hold for the spin-1 representation?
0
votes
1answer
89 views

Clifford algebra formulation of the Nambu-goto action

Using the wedge product one can pair the generators the Clifford algebra $Cl_{1,3}(\mathbb{R})$ to produce 2-vectors (area elements). The Nambu-Goto action is a statement on the evolution of ...
0
votes
0answers
29 views

Why is there a negative sign in the (non-relativistic) bivector formulation of the Lorentz force?

I'm currently trying to update my understanding of basic (Newtonian, non-relativistic) physics to use bivectors and Clifford products instead of pseudovectors and cross products. And I've come up ...
1
vote
0answers
100 views

Dirac matrices for generalized metric tensors

The Dirac matrices are defined by the relations $$\left [\gamma^{i},\gamma^{j}\right]_{+}=2\eta^{i,j}\mathbb{1}$$ where $[\cdot,\cdot]_{+}$ is the anti-commutator. What happens if I replace $\eta^{i,...
0
votes
0answers
43 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
vote
0answers
112 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
82 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 ...
0
votes
1answer
66 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
170 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
111 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 ...
4
votes
2answers
579 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 ...
1
vote
0answers
45 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
364 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
vote
1answer
94 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
132 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 ...