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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Upgrading to geometric algebra my proof of energy conservation for a rotating rigid body in $D$ dimensions, and solving the Sylvester equation
I wrote a proof from first principles that energy is conserved in a $D$-dimensional rotating rigid body without external forces, and I'd like to ask for some feedback on improving my math with more ...
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Why is SUSY algebra expressed as such?
In SUSY, many books describe its SUSY algebra like this.
$$\{Q_a, \bar Q_ \dot a\} = 2 \sigma ^\mu_{a \dot a} P_\mu,$$
and all other supercommutators vanish. But I don't understand why physicists ...
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What is the motivation for the Self-Dual Canonical Anticommutation Relation (CAR) algebra?
What exactly is the motivation for the use of the Self-Dual Canonical Anticommutation Relation (CAR) algebra in the context of infinite lattice systems? Why not remain with the CAR algebra, as both ...
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Proving that commutator algebra of Dirac matrices are isomorphic to that of matrix generators of lorentz group
In the exercise 2.6 for Supergravity by Freedman and van Proyen I am asked to Use (2.4) to show that the commutator algebras of $\sigma_{\mu \nu}$ and $\bar{\sigma}_{\mu \nu}$ are isomorphic to (1.34)
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Is Dirac theory just a real Clifford algebra?
The gamma matrices $\gamma^\mu$ appearing in the Dirac equation span the Clifford algebra ${\cal Cl}_{1,3}$ over real numbers. They are generators of Clifford algebra in that sense that their products:...
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Explicit form of the unitary representation of Lorentz transformations for spinorial field operator
When constricting quantum field theory, one studies Lorentz symmetry and find the corresponding projective representation. Just like $SU(2)$ is the universal covering group of $SO(3)$, $SL(2)$ is the ...
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Why is $( \alpha_i r_i) (\alpha_j r_j ) = \frac{1}{2} \{ \alpha_i , \alpha_j\}r_i r_j$?
Where $\alpha_i= \left(
\begin{matrix}
0 & \sigma_i \\
\sigma_i & 0
\end{matrix}
\right)$.
To me it should just be $( \alpha_i r_i) (\alpha_j r_j ) = \alpha_i \alpha_j r_i r_j$, but it is not. ...
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How to prove formula for contraction of a vector with a Multivector?
I am currently reading "Space-Time Algebra" by David Hestenes and the following proof is given for the formula of contraction of a vector $a$ and multivector $b_1 \wedge b_2 \wedge b_3 \...
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Understanding conclusion of Proof that a matrix that commutes with all gamma matrices is proportional to the identity matrix
I am trying to understand why a matrix $M$ that commutes with all gamma matrices $\gamma^\mu $ is proportional to the Identity matrix. I am following example 3.18 in Voja Radovanovic's book on solved ...
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Is there a way to rewrite all of circular motions in exterior algebra?
The torque is defined as the cross product between the position vector and the applied force:
$$\tau = \vec{r} \times \vec{F}. $$
The cross product only works in 2 and 3 dimensions and the ...
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How to define the inverse of Dirac Gamma Matrices in QFT?
The Dirac gamma matrices are a set defined by the 16 following matrices:
$$\Gamma^{(a)}=\{I_{4x4},\gamma^\mu,\sigma^{\mu\nu},\gamma_5\gamma^\mu,\gamma_5\}.\tag{2.122}$$
Now, I wish to determine the ...
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Can the QED action be expressed using geometric algebra?
Hestenes et al. have been able to rewrite the Dirac equation in terms of the "spacetime algebra" (Clifford algebra over Minkowski space), as laid out here. I was wondering if the same can ...
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Quaternions as rotation generators
The following exercise appears in Geometric Algebra for Physicists by Chris Doran and Anthony Lasenby in section 1.8.
The unit quaternions $i, j, k$ are generators of rotations about their ...
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Where can I find a real representation for 8 dimensional gamma matrices?
I understand that gamma matrices can be real in $d = 8$ and with Euclidean signature, with minimal dimension $16\times16$. Does anybody know where I can find such a representation explicitly written?
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Are all fields on spacetime spinor-valued?
I'm trying to understand the values that fields can take. For fermions, my understanding is that fields on spacetime take values as Dirac Spinors, which are $\mathbb{C}^4$ vectors. The vector space of ...
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Why is this form of writing the six antisymmetric gamma matrices correct?
I encountered the following expression in Ashok Das' QFT Lectures:
$$\sigma_{\mu \nu } =\frac{i}{2}[\gamma ^\mu,\gamma^\nu]=i(\eta ^{\mu \nu}-\gamma^\nu \gamma^\mu)=-i(\eta ^{\mu \nu}-\gamma^\mu \...
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Propagation of vectorial electric field with Fourier Optics
In Fourier Optics one can propagate in free space the electric field $E(x,y)$ of a monochromatic light beam along the z-axis by decomposing it into plane waves via the Fourier transform, then ...
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Can the complete information about a system be encoded in a Set of operators that is not tomographically complete?
For a clifford algebra of 2N or 2N -1 operators (I hope this is the right terminology):
Operators which satisfy:
\begin{align}
\{O_i, O_j \} = \delta_{ij}
\end{align}
we know that we can represent ...
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What is the smallest possible dimension to represent $N$ operators $O_i$ with $\{O_i, O_j \} = 2 \delta_{ij}$?
The question is in the title. I have an algebra of Hermitian operators that satisfy:
\begin{align}
\{O_i, O_j\} = 2 \delta_{ij}
\end{align}
that means all of those operators have eigenvalues $\pm 1$, ...
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What is the meaning of the differential in Doran and Lasenby's discussion of Noether's theorem for spacetime transformations?
Doran and Lasenby (Geometric Algebra for Physicists, pg. 450) state that if a transformation involves spacetime dependence (this brings to my mind common examples: translation and rotation), then ...
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Can a non-dimensionless physical quantity ever be continuously exponentiated?
I was really surprised and somewhat skeptical when I first learned that you can exponentiate complex numbers and matrices and such. But then it makes sense once you consider (natural) exponentiation ...
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Weyl Tensor and conformal transformations in Geometric Algebra
Geometric Algebra seems to provide numerous insights and simplify many calculations. However, I've seen very little about the Weyl tensor.
I was wondering if anybody knows if it provides a more ...
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Weyl Tensor in Geometric Algebra
In Gravity, Gauge theories and Geometric Algebra, p.39, they derive the Weyl Tensor in the following manner:
Six of the degrees of freedom in $\mathcal{R}(B)$ can be removed by
arbitrary gauge ...
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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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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 ...
- 1,397
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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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