Questions tagged [clifford-algebra]

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Rotor of a rotor in Geometric Algebra

I was reading Geometric Algebra for Physicists, by Doran and Lasenby, and, in section 5.5.2, they calculate the Thomas Precession. However, at a certain point, they have the exponential of an ...
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41 views

Intuition about analogy of spinors in Clifford Algebra

I've been studying the application of Clifford Algebra in quantum mechanics, more specifically in spin, but I'm stuck in a basic analogy that is part of the basics before starting to do things using ...
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1answer
54 views

Symplectic Majorana spinors in 5D

According to the book "Supergravity" written by Freedman & van Proeyen in 5D for the existence of Majorana spinors it is necessary to introduce so called sympletic ones which requires ...
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1answer
34 views

Confusion in linear independence of matrices argument in Bjorken and Drell

In Bjorken and Drell, Relativistic Quantum Mechanics, the following argument is constructed to show that a set of matrices derived from the $\gamma$ matrices are linearly independent. The matrices of ...
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62 views

$\displaystyle{\not}{a}\displaystyle{\not}{a} = a^2$ or $-a^2$ in Srednicki

I'm confused: In Srednickis Book (Equation 37.26), he has: $$\displaystyle{\not}{a}\displaystyle{\not}{a} = -a^2$$ However, every other source I found (for example this SE question says that it's: $$\...
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Calculate Connection Coefficients in Geometric Algebra

I'm trying to calculate the connection coefficients of a set of coordinates with Geometric Algebra. However, following Curvature Calculations with Spacetime Algebra and Spacetime Geometry with ...
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44 views

Respresentation Dirac group and Lie algebra

I am reading Peskin and Schroeder's book on QFT and have some difficulties with representation groups. Let's start with the Lorentz group since it is easier. let $\Lambda$ be a Lorentz transformation, ...
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1answer
45 views

Are the $\alpha_i$ and $\beta$ matrices in Dirac equation unique in the Dirac representation?

Related question is in Are the $\alpha$ and $\beta$ matrices of the Dirac equation unique? . However, it does not solve my problem. My professor asked us to prove that in the Dirac representation, $$\...
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336 views

Conflicting definitions of a spinor

I'm moving my question from math.stackexchange over here because it got no attention over there, even after 3 weeks and a 50-point bounty, and also because this is a very physics-oriented math ...
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43 views

What is the definition of a Majorana fermion in conformal field theory?

Majorana spinors background According to Eq. (4.84) and (4.85) of these notes, charge conjugation of the spinor $\Psi$ is defined as $$ \Psi^{(c)} = C \Psi^*,$$ where $C$ is the unitary charge ...
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36 views

Majorana fermions in Euclidean and Minkowski signatures - contradiction with Wikipedia Table

In this wonderful lecture note on Clifford Algebra and Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf Somehow I find some inconsistency with his Tables of Euclidean and ...
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1answer
33 views

The 'mutual' and the 'self' in terms of the 'conjugacy' of Euclidean and Minkowski Weyl fermions

Euclidean and Minkowski fermions are shown in the Table of Wikipedia. (see the bottom https://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions) My question is that what does the conjugacy mean ...
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65 views

Does $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}\mathbb1$ determine the hermiticity of the gamma matrices?

If I remember correctly, the derivation of the Dirac equation requires that $\gamma^0$ is Hermitian while $\gamma^i$ for $i=1,2,3$ is anti-Hermitian. This is clearly true for the standard Dirac ...
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52 views

Equivalence of Dirac matrix representations

I'm currently having fun with the Dirac matrices $\gamma^\mu$ which appear in the relativistic Fermion field equation (aka the Dirac equation). They need to satisfy the anticommutation relation $$ \{\...
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1answer
137 views

Spinors and spin group

It seems to me that spinors (pinors) are loosely defined as representations of the spin (pin) group $Spin(p,q)$ ($Pin(p,q)$), which double covers the spacetime symmetry group $SO(p,q)$ ($O(p,q)$). $\...
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94 views

Solution to Dirac equation

We take the solution of Dirac equation as 4 component wave function (Dirac Spinor). But how do we know that it can't be a square or rectangular matrix like 4x2 or 4x4 matrix?
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72 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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33 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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129 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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66 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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96 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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109 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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56 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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89 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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92 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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1answer
72 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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84 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
54 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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29 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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108 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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65 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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485 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
82 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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87 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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492 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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66 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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236 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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225 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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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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107 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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55 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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62 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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418 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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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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935 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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456 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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74 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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85 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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58 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
400 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^{\...