# Questions tagged [clifford-algebra]

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### Why do physicists use wave functions with more than two components?

For $n$-body systems you just need a single component wavefunction. For example, for a two-body system you would need a wavefunction of 6 variables. $\psi(x_1,x_2,y_1,y_2,z_1,z_2)$ That satisfies the ...
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### How do spinors arise in systems of less than 3 effective spatial dimensions as representations of the Lorentz group?

In 3+1 Minkowski spacetime, we can use the fact that the Lie algebra of the Lorentz group decomposes (I am omitting some details here to keep it short) into $su(2) \oplus su(2)$ and then use the fact ...
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### Cyclic identity of $\gamma$-matrices in 3D and 4D

Reading the book on Supergravity from Freedman & van Proeyen I was very bewildered by the so called cyclic identity of $\gamma$-matrices of the Clifford-algebra(eq. 3.67) important in string ...
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### Spinor matrix representation of a spacetime vector in 2+1D

Let $a^\mu$ be a spacetime vector in 2+1D: $(t,x,y)$. What would be the spinor-matrix representation ${A^\alpha}_\beta$ of the spacetime vector $a^\mu$? I have not been able to even find examples or ...
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### SUSY and free Majorana field theory

Given a free Majorana field theory in $D$ spacetime dimension. Say $D=2$ to $10$. Question: Which $D$ spacetime dimension,do we have a free Majorana field theory with Majorana spinor in the real ...
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### Unitary transformation of Dirac equation

Dirac equation is given by $$(i\gamma^\mu\partial_\mu-m)\psi=0.$$ The matrices $\gamma^\mu$ satisfy the relation $$\{\gamma^\mu,\gamma^\nu\}=\gamma^\mu\gamma^n+\gamma^\nu\gamma^\mu=2g^{\mu\nu},$$ ...
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### Is there a bosonic representation of Clifford algebra in (1,3) spacetime?

By a suitable combination of Dirac's $\gamma_\mu$ matrices one can define creation and destruction operators satisfying fermionic anticommutators. Is there a similar result for bosons in the context ...
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### Equivalence of the Dirac representation of the Lorentz algebra and its conjugate in even dimensions - Polchinski

In Polchinski's String Theory, Appendix B.1 we look at the smallest irreps of the Clifford algebra in even-dimensional spacetimes $d=2k+2$. In (B.1.16) he defines two operators from the gamma matrices ...
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### Is there an identity for anti-commutator $\{ A B, \, C D \}$ in terms of commutators $[\, , \,]$ only?

I'm looking for an identity that could express the anti-commutator $$\tag{1} \{ A B , \, C D \} \equiv A B C D + C D A B$$ expressed as a combination of commutators only: $[A,\, C]$, $[A, \, D]$, etc....
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### Going from the Dirac Lagrangian to the adjoint Dirac equation

I am comfortable doing the following calculation, Derivation of the adjoint of Dirac equation, notably — going from the standard Dirac equation to the adjoint Dirac equation via using Dirac ...
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### Generalization of Dirac matrices

Is it possible to have a set of 16-dimensional matrices $\gamma_{\mu}^{a}$ such that $$\{\gamma_{\mu}^{a},\gamma_{\nu}^{b}\} = 2\delta^{ab}\eta_{\mu\nu}$$ where $\eta_{\mu\nu}$ is the Minkowski metric ...
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### Dirac matrix algebra from bosonic creation/anihilation operators?

Using some generic fermionic creation/anihilation operators $a_i$ and $a_i^{\dagger}$ ($i, j = 1, 2, 3, \dots, N$) such that \begin{align} \{ a_i, \, a_j^{\dagger} \} &= \delta_{ij}, \tag{1} \\[...
### What transformation gives a Weyl-like representation by flipping $\gamma^0$ and $\gamma^5$?
The usual Weyl representation of the Dirac matrices is defined like this: $$\tag{1}\gamma_W^a = T_W \, \gamma^a \, T_W^{-1},$$ where \begin{align}\tag{2} T_W &= \frac{1}{\sqrt{2}} (1 + \gamma^5 \, ...