# 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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### 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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### 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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### 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. ...
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 ...