Questions tagged [clifford-algebra]

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68 views

How to invert this matrix?

Is there a smarter method for finding the determinant and inverse of the following matrix, without using the brute force procedure? (When I say brute force, it is to write the matrix with each term ...
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19 views

Majorana fermions algebra confusion

This is a quiet embarrasing question. Consider the Majorana fermion fields $\psi_i(x)$ and $\psi_j(x)$, where $i$ and $j$ denote lattice sites and $x$ is a spatial coordinate, which satisfy the ...
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1answer
66 views

Proof of $\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=0$

Using $\gamma^{5}\gamma^\mu=-\gamma^\mu\gamma^{5}$ and $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ I obtain \begin{equation}\tag{1} T_{\mu\nu}:=\mathrm{tr}(\gamma^{5}\gamma^\mu\gamma^\nu)=-\mathrm{tr}(\gamma^\...
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1answer
46 views

Reducibility of Lorentz group generators, contrasted with irreducible gamma-matrix representation

After introducing the gamma matrices as $$\gamma^0=-i \pmatrix{\begin{matrix} 0 & \Bbb I_{2x2} \\ \Bbb I_{2x2} & 0 \\ \end{matrix}} , \qquad \gamma^i=-i\pmatrix{\begin{matrix} ...
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57 views

Geometric algebra from the vector space of $N\times N$ Hermitian matrices [closed]

My research in quantum mechanics has led me to geometric algebra several times, so I finally decided to delve into its study. Particularly, I'm interested in geometric algebras which can be ...
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1answer
69 views

Help understanding Clifford/Geometric Algebra on curved spacetimes

Geometric Algebra on curved spacetimes. I seem to come across a lot of mentions of “spacetime algebra” (especially by Hestenes). As I understand it, this is simply the Clifford algebra $Cl(3,1)$. I'm ...
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1answer
102 views

How do you compute the successive action of two angular momentum operators on a multivector in geometric algebra?

I'm trying to compute the action of two angular momentum operators $J_i$ on some multivector $\psi$ in geometric algebra as in page 290 of Doran & Lasenby (Geometric Algebra for Physicists). ...
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1answer
38 views

How do I normalize the relativistic wavefunction in general?

https://en.wikipedia.org/wiki/Spacetime_algebra#Relativistic_quantum_mechanics claims that the wave-function can be written using geometric algebra as follows: $$ \psi=R(\rho e^{i\beta})^{1/2} $$ ...
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78 views

A calculation on the relation between gamma-matrices and total antisymmetric tensor

In some papers I found the following relations: $$\left[\gamma^N,\left[\gamma^I,\gamma^J\right]\right]=4\left(\eta^{NI}\gamma^J-\eta^{NJ}\gamma^I\right)$$ $$\left\{\gamma^N,\left[\gamma^I,\gamma^J\...
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48 views

Weyl's spinor and Dirac spinor [duplicate]

Weyl's spinor and Dirac's spinor What is the difference between the two from a mathematical point of view? So are there different mathematical definitions of spinor? Is it correct to say that the Weyl ...
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66 views

Why do physicists use wave functions with more than two components?

For $n$-body systems you just need a single component wavefunction. For example, for a two-body system you would need a wavefunction of 6 variables. $\psi(x_1,x_2,y_1,y_2,z_1,z_2)$ That satisfies the ...
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1answer
62 views

How do spinors arise in systems of less than 3 effective spatial dimensions as representations of the Lorentz group?

In 3+1 Minkowski spacetime, we can use the fact that the Lie algebra of the Lorentz group decomposes (I am omitting some details here to keep it short) into $su(2) \oplus su(2)$ and then use the fact ...
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1answer
64 views

Cyclic identity of $\gamma$-matrices in 3D and 4D

Reading the book on Supergravity from Freedman & van Proeyen I was very bewildered by the so called cyclic identity of $\gamma$-matrices of the Clifford-algebra(eq. 3.67) important in string ...
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1answer
50 views

Spinor matrix representation of a spacetime vector in 2+1D

Let $a^\mu$ be a spacetime vector in 2+1D: $(t,x,y)$. What would be the spinor-matrix representation ${A^\alpha}_\beta$ of the spacetime vector $a^\mu$? I have not been able to even find examples or ...
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26 views

SUSY and free Majorana field theory

Given a free Majorana field theory in $D$ spacetime dimension. Say $D=2$ to $10$. Question: Which $D$ spacetime dimension,do we have a free Majorana field theory with Majorana spinor in the real ...
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79 views

Unitary transformation of Dirac equation

Dirac equation is given by $$(i\gamma^\mu\partial_\mu-m)\psi=0.$$ The matrices $\gamma^\mu$ satisfy the relation $$\{\gamma^\mu,\gamma^\nu\}=\gamma^\mu\gamma^n+\gamma^\nu\gamma^\mu=2g^{\mu\nu},$$ ...
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62 views

Quantum Field Theory Identity

We are asked to show that $$\not A\not B = A\cdot B - i\sigma_{\mu\nu}A^\mu B^\nu $$ I know that $$\not A = A_\mu \gamma^\nu $$by definition, and: $$\sigma_{\mu \nu}=\frac{i}{2} [\gamma_\mu, \gamma_\...
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2answers
337 views

Product of an odd number of Dirac $\gamma$ matrices [closed]

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, ...
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1answer
65 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^...
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73 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}=...
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1answer
233 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 ...
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64 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 ...
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2answers
162 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(...
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207 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 \...
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75 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 ...
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1answer
80 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 + \...
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48 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\...
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1answer
56 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 ...
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3answers
260 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=\...
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36 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 $...
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3answers
653 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....
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1answer
314 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 ...
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73 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 ...
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76 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} \\[...
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1answer
51 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 \, ...
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3answers
300 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^{\...
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1answer
124 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 ...
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0answers
135 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 $\sigma_y$ ...
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1answer
201 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 $\...
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1answer
383 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
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2answers
503 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 ...
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1answer
130 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 ...
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1answer
61 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 ...
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1answer
267 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 ...
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1answer
120 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, $$-γ^μ γ_μ∂^μ ∂_μ=m^{2}$$...
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1answer
161 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 ...
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3answers
1k 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?
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447 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\...
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0answers
50 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, ...
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1answer
81 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: $$ a^2 = \left( a^{\dagger} \right) ^2 = 0, $$ $$ a a^{\dagger} + a^{\dagger} a = 1. $$ ...