Questions tagged [clifford-algebra]

Clifford algebras are associative algebras constructed from quadratic forms on vector spaces. They can be viewed as generalizations of real numbers, complex numbers and quaternions. When constrained to real numbers, the algebra are often referred to as "geometric algebra" and has use in theoretical physics.

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Lorentz invariance and Dirac Slash Notation

I have the following factor inside a scattering amplitude $$\Big[\bar{u}(p')\not{l}\gamma^{\mu}\not{l}u(p)\Big] \ \Big[\bar{u}(q')\gamma_{\mu}u(q)\Big]$$ where $p$ and $p'$ are the initial and final ...
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Pauli matrices from anticommutator

I want construct the Pauli matrices starting from $$\sigma_i=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$$ by using only the anti-commutation relation $$ \sigma^i\sigma^h+\sigma^h\sigma^i=\{\...
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Degrees of freedom in a spinor in $d$ dimensions (following Polchinski & Lounesto)

I am working through several texts on spinors and trying to deepen my understanding of this fascinating concept. In many ways I have found Polchinski's great Appendix B of String Theory, volume 2 to ...
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How to dimensionally reduce the 3+1 D Dirac equation into the 1+1D Dirac equation?

In 3+1D the Dirac equation looks like $$i\partial_\mu \gamma^\mu \Psi -m\Psi=0.$$ If we only consider $x$-direction, then it should reduce to $$i\partial_t\gamma^0\Psi =(-i\partial_x\gamma^1+m)\Psi.$$ ...
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What is the relationship between spinors and rotating motion geometrically?

Spinors are famously like spinning tops, but not actually like spinning tops since they are point particles and thus cannot rotate around their axis. It is easy to show algebraically how spinors must ...
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Charge conjugation of symplectic Majorana spinors in 4+1 dimensions

In the book "Supergravity" written by Freedman & van Proeyen, a symplectic Majorana spinor is defined in eq. (3.86) $$ \chi^i = \varepsilon^{ij} (\chi^j)^C, \tag{3.86}$$ where the upper ...
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Relation between the commutator of commutators in Dirac algebra

In an attemption to obtain the curvature tensor related to the spin connection of the fermionic fields I came across this expression with the commutator of the gamma matrices commutators. My question ...
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Why does Tong uses Euclidean Gamma matrices in this step of deriving the Chiral Anomaly?

In David Tong's GT notes on page 137, he uses the trace identity for Euclidean gamma matrices given by $$\text{Tr}(\gamma^5\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma})=4\epsilon^{\mu\nu\rho\...
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Relationship between spin and wave-function (in geometric algebra)

I am interested in determining which 'feature' of the wave-function are responsible for fixing the spin of the matter is relates to; or more precisely, an "algorithm" or "procedure"...
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Connection between Dirac-Matrices and Clifford Algebra

I have to give a quick talk at our University about the Dirac Equation and the Clifford Algebra. Since I am still at the beginning of my studies, my knowledge in this regard is still very limited. ...
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Can the $SU(3)$ gauge field be put in geometric algebra terms?

According to this article on the spacetime algebra, we know the Dirac spinor can be thought of as an even element of the Clifford algebra over spacetime, which in turn can be thought of as a general ...
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Problem analyzing Dirac equation in an arbitrary coordinate system using geometric calculus

This question was partially inspired from the Dirac equation in spherical coordinates. For simplicity, let’s suppose I have a two variable partial differential equation initially written in a ...
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Inertia tensor for rotors

For vectors we can use Inertia tensor. But if I want to use bivectors (Rotors), what should I use for the inertia tensor? I want to make a 2d game and progressively to 4d.
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The irreducible representation of rank $n$ spinor in 3D

I was reading Ref. 1, where it is asserted that the irreducible resolution of a rank $n$ spinor can be written as totally symmetric spinors. $$\Psi^{n}=\Psi^{\{n\}}+\zeta \Psi^{\{n-2\}}+\zeta \zeta \...
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Is there a fast way to simplify gamma matrix tensors?

I’m doing QFT homework, and our professor said Fermions require a lot of algebra. And I’m finding he’s right. I want to either get a mastery of these operations or find useful shortcuts. Since it’s ...
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Relation between equations of the form "Derivative" $f=0$

I'm currently taking an introductory course in QFT, and I've noticed that lots of equations in physics take the form of "Derivative" of a funcition equal 0. Some examples being the wave ...
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Scalar geometric product ambiguity involving derivatives

I'm working on deriving the Euler equations for a perfect fluid from the conservation of the energy-momentum tensor and I've found some difficulties with the notation. Given two vector fields, $v(x)$, ...
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Self or complex Weyl representation in Polchinski

In p.433 of Polchinski String theory volume 2, he said that: In $d$ spacetime dimension, For $d = 2 \mod 4$, each Weyl representation is its own conjugate. ($B \Gamma B^{-1} =-\Gamma $) For $d = 0 \...
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Is this how spinors transform and is it the square root of a vector?

I have an expression and the transformation rules, and I wonder if this qualifies as a spinor. Can the following expression written with complex Clifford algebra be seen as a spinor? In any case, it ...
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Decomposing product of spinor representation of orthogonal group

I am reading A. Zee's group theory book. In the chapter of spinor representation, page 416, he was trying to describe how to decompose of the product of spinor representation into irreducible ...
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Is Dirac equation valid only for spin-$\frac{1}{2}$ particles?

It is usually said that Dirac got his equation by looking for the square root of the 4-momentum norm (see Dirac’s coop here). The relativistic 4-momentum norm is $$(E)^2-(\mathbf{p}c)^2=(mc^2)^2 \tag{...
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How to show that $\sigma^\mu_{\alpha \dot{\alpha}} \bar\sigma_\mu^{\dot{\beta} \beta} = 2\delta_\alpha ^\beta \delta_\dot{\alpha}^\dot\beta$?

Consider the usual definitions $\sigma^\mu = (1, \sigma^i)$ and $\bar\sigma^\mu = (1, -\sigma^i)$, is it possible to show $$\sigma^\mu_{\alpha \dot{\alpha}} \bar\sigma_\mu^{\dot{\beta} \beta} = 2\...
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Understanding the degrees of freedom of the bispinors vs. the Lorentz group

Okay, I think I can say that a Lorentz transformation has 6 degrees of freedom, whereas a bispinor (not necessarily normalized) has 8. But then, I also believe I've read that bispinors are in some ...
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Space-time split in Geometric Algebra

I was performing some calculations with Geometric Algebra today and I've found myself stuck with a simple operation that I don't know how to answer. Considering that you have the usual vector ...
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Can spinors be written this way?

I've used Geometric Algebra to rewrite Quantum Mechanics in a slightly different way than what I've seen in text books so far. Some equations seem to become neater. I wonder if a particular expression ...
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Why use a spacetime indices when introducing supersymmetry for a point particle

In chapter $7$ of introduction to AdS/CFT correspondence Nastase states following: ... we generalize the first order action for the particle. The generalisation is done by introducing objects $\theta^...
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Grade operations in Geometric Algebra

I'm going through the book Geometric Algebra for Physicists by Doran and Lasenby, and I have found myself lost when the authors use the grade operator to change between products and switch the order. ...
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Help operating with differentials in Geometric Algebra

I am trying to learn General Relativity with Geometric Algebra. Following the article Spacetime Geometry with Geometric Calculus but I am finding some algebraic problems when dealing with expressions ...
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How to determine the irreducible representations of a product of Clifford algebra generators

Suppose $\psi^a$, $\psi_{a}$ ($a=1,2,3,4$) are lowering and raising spinorial oscillators, respectively, and generate a Clifford algebra $\{\psi^a, \psi_{b} \} = \delta^a_b$, where an upper $a$ index ...
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Pauli matrices in general relativity

Just as in a tetrad formalism one brings the gamma matrices from the local Lorentz frame to the manifold through: $$\gamma^\mu = \gamma^a e^\mu_a.$$ Can one do that for the Pauli spin matrices?
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Fierz identity and its application

I know there are very good explanations for Fierz identity in this platform. But I have a question about its application. Consider a matrix element $M$ and we can write $M$ in the basis of 16 Dirac ...
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Biot-Savart law in 2D using Geometric Algebra

The Biot-Savart law can be derived from Maxwell's equations for 3D using the vector product (via the curl). On the other hand, in a purely two-dimensional (2D) world, the vector product is not well ...
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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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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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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 ...
1 vote
1 answer
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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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$\displaystyle{\not}{a}\displaystyle{\not}{a} = a^2$ 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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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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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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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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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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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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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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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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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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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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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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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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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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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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