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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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 ...
