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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Does the spacetime algebra (STA) formulation of the Dirac equation remove the "double cover" under Lorentz transformations?
As per this section, spacetime algebra allows the Dirac equation to be written entirely in terms of geometric objects, i.e. multivectors. So I would imagine that these multivectors are invariant ...
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References on a variant of Geometric Calculus
Geometric algebra and (standard) calculus, when synthesized, give rise to geometric calculus, a very powerful formalism.
I have read a bit about fractional calculus and time-scale calculus, both very ...
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Characterization of Pauli Spin Matrices
Say I have a finite collection $\{O_i\}_{i=1}^{d}$ of traceless, mutually orthogonal, dichotomic (meaning eigenvalues are $\pm 1$) observables satisfying the anti-commutation relations $\{O_i,O_j\}=2\...
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Spin 1 Clifford Algebra
For starters I'm not sure if what I'm looking for actually falls within Clifford Algebra,the name comes really from trying to find analogies with spin 1/2.
The system that I study (Trilayer Graphene) ...
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Mass gauge boson?
Note for the mod: Read the question. This is not about "correctness of unpublished personal theories". This is an attempt to identify the keywords to search for any papers on the subject.
...
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Pauli matrix relation of dot and cross products with complex numbers
I saw this Identity in "QFT for the gifted amateur" when explaining non-Abelian gauge theory and I just don't get any intuition on why it's true, perhaps some geometric algebra intuition ...
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Matrix representation of Clifford algebra elements [closed]
Hello,I am learning Clifford algebra as a preliminary step to studying the Dirac equation. The gamma matrices obey specific rules as elements of Clifford algebra: they square to either the identity ...
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Solving the wave equation of a tensor $h_{\mu\nu} = (1/2) (e_\mu e_\nu + e_\nu e_\mu)$
It is known that the solution to the wave equation for a tensor
$$
\square h_{\mu\nu} = 0
$$
is
$$
h_{\mu\nu}(\vec{x}, t) = \int \frac{d^3k}{(2\pi)^3} \sum_{\lambda=+,\times} \left( \epsilon_{\mu\nu}^{...
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How to compute inertia tensor off the center of mass with geometric algebra?
I was reading Doran & Lasenby's Geometric Algebra for Physicists when I stumbled upon this,
where $\mathbf{a}$ is a vector taken form the centre of mass. Returning to the definition of equation (...
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Doran Geo Algebra for Physicists Exercise 2.9 [closed]
In the question says
The Cayley-Klein parameters are a set of four real numbers $\alpha$, $\beta$, $\gamma$ and $\delta$ subject to the normalisation condition $\alpha^2+\beta^2+\gamma^2+\delta^2=1$
...
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How do you apply a transformation, or a superposition in David Hetens' geometric formulation of the wavefunction?
In David Hestenes' formulation of the wavefunction in geometric algebra, we have:
$$
\psi(x) = \sqrt{\rho(x)} R(x) e^{-ib(x)/2}
$$
where R(x) is a rotor.
For simplicity, let us now consider a two-...
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Fierz Identity in 2+1d
Wikipedia states an example of Fierz Identity, under the assumption of commuting spinors, the $\mathrm{V} \times \mathrm{V}$ product that can be expanded as,
$$
\left(\bar{\chi} \gamma^\mu \psi\right)\...
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Why are spinors members of minimal ideals?
Why do we require that spinors live in minimal left ideals of Clifford algebras and not just left ideals? I assume that it has something to do with irreps but a Dirac spinor also lives in an minimal ...
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How to motivate spinors from the Dirac equation? [closed]
I am trying to motivate spinors by making sure the Dirac equation is relativistically invariant (and it suffices to discuss just the Dirac operator).
Let $\{ e_i \}$ be an orthonormal frame and $x^i$ ...
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Proof Majorana spinors exists if maximal commutant of Clifford algebra is $\mathbb{R}$
I am searching for a proof of the claim made in this post. It states that Majorana spinors (I refer to both complex pinor and spinor representations which are restricted to the real Spin group and ...
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Is there a $\gamma^{5}$ in $d$-dimensional Clifford Algebra?
I was trying to compute the EW vacuum polarization
$$i\Pi_{LL}^{\mu\nu}=(-1)\mu^{\frac{\epsilon}{2}}e^{2}∫\frac{d^{d}k}{(2\pi)^{d}}\frac{Tr\bigl(i\gamma^{\mu}(iP_{L})i(\not{k}+m_{i})(i\gamma^{\nu})P_{...
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The dimension of the Clifford algebra for the Dirac equation
The Dirac algebra contains sixteen linearly independent elements. In general, a Clifford algebra $\mathcal{C}\!\ell(V,Q)$ generated from a vector space $V$ equipped with a quadratic form $Q$ has ...
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Obtaining the 16 elements of the Clifford algebra from the $\gamma^\mu$ generators
In my study of the Dirac equation, I have fully understood the "linearization" of the relativistic energy to obtain a matrix-valued equation that reduces to the Klein-Gordon equation if the ...
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Doubts on the Majorana & Weyl conditions compatibility
In the appendix B of Polchinski's book there is a discussion on the compatibility between Majorana and Weyl condition.
My doubts are trough this passages:
He starts constructing the basis in $(2k+2)$ ...
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Conflicting definitions of vector conjugate in QM
Let $e$ be a finitely matrix representable operator.
In physics, specially in quantum mechanics (QM), it is customary to define the conjugate operator $e^{\dagger}$, as the adjoint or the Hermitian ...
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Real representation of smallest dimension of Clifford Algebra with $d$ generators
I'm trying to understand the model described in this paper. I have a question about a claim they make. From page 2:
To describe the fermionic degrees of freedom let, as a preliminary
\begin{align*}
...
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The Dirac-Hestenes equation as an eigenvalue equation: Interpretation of the $m \psi \gamma_0$ term and the wavevector
This is somewhat of a follow-up question to my previous question on the Dirac-Hestenes equation. In that question, I asked whether the equation could be written in a form that omits the dangling ...
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Product of Dirac $\gamma^0$ and $\gamma^\mu$ generate a representation of some algebra?
I need your help with an issue about Dirac gamma matrices. Precisely, I need to know if $\gamma^0\gamma^\mu$ generates an irreducible representation of some algebra. This problem has come out in the ...
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Transformation of spinor reps and why the Dirac rep is its own conjugate
In Polchinski's String Theory volume 2, appendix B, on page 433 (in the section on Spinors and SUSY in various dimensions, specifically the subsection on Majorana spinors) he says:
"It follows ...
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What justifies the statement that a Dirac spinor can be written as two Weyl spinors?
I've cross listed this post on math SE in case it is more appropiate there. That post can be found here: https://math.stackexchange.com/q/4833722/.
I am approaching this from a Clifford algebra point ...
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Computing the Free Parafermion Spectrum
In Fendley's paper on Free Parafermion (https://arxiv.org/abs/1310.6049), Fendley used some operator techniques to show that $$Q_{2 L}\left(\epsilon_k^n\right)=0$$ which is the formal derivation of ...
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Understanding Wikipedia's definition of a spinor
I originally asked this question on math SE but I'm asking it again here due to the lack of responses. I should note that I come from a mathematical background and not a physics one so I am not ...
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On the derivation of Dirac matrices
Is it possible to retrieve the matrix elements of the $\gamma$s by simply knowing their anti-commutation relation:
$$
\{\gamma^\mu, \gamma^\nu\}=2\,g^{\mu\nu}\,\mathbb{I}_{4}
$$
I'm just trying to ...
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Dirac Equation and the Klein-Gordon Equation
I am trying to solve an exercise in Halzen and Martin's Quarks and Leptons book and got stuck on doing some math. The Dirac equation reads $$i \gamma^{\mu} \partial_{\mu} \psi - m\psi = 0.$$ Now, I ...
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Pairwise product of Pauli Matrices using Levi-Civita
Just a quick question. The professor used $$ \sigma_i \sigma_j = i \sum_k \epsilon_{ijk} \sigma_k$$
to define products of Pauli matrices. This works fine to get $$\sigma_x \sigma_y = i \sigma_z$$ and ...
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How do you show that Einstein's tensor simplifies to the unitary form in geometric algebra?
In David Hestenes' Gauge Theory Gravity paper (RG), he claims that
$$G^\beta = \tfrac{1}{2}(g^\beta \wedge g^\mu \wedge g^\nu) \cdot R(g_\mu \wedge g_\nu)$$ in geometric algebra (specifically, ...
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What does the geometric product between displacement (or maybe position) and force vectors mean?
So work is measured in joules, which is newton * meter, and it is calculated by taking a scalar product between displacement and force vectors.
And torque is very similar, it is newton * meter too, ...
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Uniqueness of Dirac matrices
I am trying to understand the motivation behind the Dirac equation for a free particle
$$
i\gamma^\mu\partial_\mu \psi-m\psi=0 \tag{1}
$$
I am wondering how to get the concrete form of the matrices $\...
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How to derive and understand the covariance property of the curvature bivector in Gauge Theory Gravity?
In David Hestenes' Gauge Theory Gravity paper (RG), the curvature tensor is defined via
$$ [D_\mu, D_\nu] M = R(g_\mu \wedge g_\nu) \times M $$
therefore
$$ R(g_\mu \wedge g_\nu) = D_\mu \omega_\nu - ...
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Rotor wave fomulation of quantum mechanics and the associated canonical commutation relations of position and momentum operators
I've been trying to determine whether it would be possible to formulate non-relativistic quantum mechanics entirely in the algebra of physical space (APS) by using rotor waves instead of complex-...
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How does the divergence change under a change of frame (with geometric algebra)?
I'm trying to prove equations (85) and (86) from Hestenes' paper Gauge Theory Gravity with Geometric Calculus (ResearchGate version).
$$
\dot{\nabla}^\prime \cdot \dot{\underline{f}}(A) = J_f^{-1}[ (\...
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Normalizers and Clifford group, how are they connected? (Nielsen and Chuang Ex 10.40)
I was studying stabilizer formalism and came across with normalizers and Clifford group and I was trying to prove that any unitary operators that map Pauli group to Pauli group may be composed of $O(n^...
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Dirac/Weyl/Majorana Basis and their Importance
I have just begun studying Dirac equations and was confused by the physical significance of Dirac Basis. In principle, we can have as many representations of Clifford Algebra as we wish by conducting ...
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Representing the Dirac equation in spacetime algebra without leftover indices
The Dirac equation as derived by Hestenes is
$$
\hbar \nabla \psi I \sigma_3 = m \psi \gamma_0
$$
where $I \sigma_3 = \gamma_2 \gamma_1$. The equation is claimed to be Lorentz invariant, because the ...
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Why can we choose a gamma matrix to be diagonal?
I am reading the textbook "Lectures on Quantum Field Theory", second edition by Ashok Das. On page 21, in equation (1.83), he writes the properties of gamma matrices:
$$
( \gamma^0)^2 = {...
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2-dimensional spin 1/2 representation of Lorentz Lie Algebra
In Peskin & Schroeder section $3.2$ they begin by telling us that they want to construct spin $1/2$ representations of the Lorentz Lie algebra. One way to do that, they say, is to first find a ...
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What isomorphisms does $SO(3,1)$ have?
The double cover of $SO(4)$ is isomorphic to $SU(2)\times SU(2)$, which might be related to $SO(3,1)$.
Also, I have seen that $so(3,1)=Cl^2(3,1)$. However, the notation confused me a bit, does the ...
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Component-free computation of Poisson bracket of Laplace-Runge-Lenz vector
How can the Poisson bracket $\{A,H\}$ be computed directly without components, where $H$ is the Hamiltonian for the inverse square force, $$H=\frac{p^2}{2m} - \frac{k}{|r|}\ ,$$
and $A$ is the ...
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Anti-persymmetric Spinorial Matrices?
An antipersymmetric matrix is one that is antisymmetric about its anti-diagonal.
The simplest example comes from the Pauli Matrices:
{σ[0],σ[1],σ[2]} are not antipersymmetric
σ[3] is antipersymmetric....
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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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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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2
answers
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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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answer
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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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