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This is almost correct, you have just made a mistake in the beginning in the index summation : $$\begin{align}1-\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu} &= 1-\frac{i}{2}\omega_{0\nu}S^{0\nu} \color{red}{-} \frac{i}{2}\omega_{i\nu}S^{i\nu}=1 \color{red}{-} \frac{1}{2}\beta_i\begin{pmatrix}\sigma^i & 0 \\ 0 & -\sigma^i\end{pmatrix} + \frac{1}{2}\theta^...


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The spin group ${\rm Spin}(p,q)\cong {\rm Spin}(q,p)$ is connected if $\max(p,q)\geq 2$. If we exclude multiple temporal dimensions, i.e. if we consider only Minkowskian and Euclidean signatures, then the component of ${\rm Spin}(p,q)$ that is connected to the identity is simply connected; except in the cases 2+0D and 2+1D where the fundamental group is $\...


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It is a question of degrees of freedom. A solution to Dirac equation with 1$\times$4 components has precisely enough freedom for describing a single (without any "internal" quantum numbers) Dirac fermion which has only two states with different spin orientations. However, if your fermion field has additional quantum numbers (like color charge in ...


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Let us evaluate the sum in the exponential: \begin{equation} -\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}=-\frac{i}{2}(\omega_{03}S^{03}+\omega_{30}S^{30}). \end{equation} Due to $S^{\mu\nu}$ and $\omega_{\mu\nu}$ being antisymmetric, we get: \begin{equation} -\frac{i}{2}(\omega_{03}S^{03}+\omega_{30}S^{30})=-\frac{i}{2}(\omega_{03}S^{03}+(-\omega_{03})(-S^{03})) = ...


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Left handed spinors $\psi_{\text{L}}$ transform as $$ \psi_{\text{L}} \mapsto \exp\left((-\mathrm{i}\boldsymbol{\theta} +\boldsymbol{\beta})\cdot\frac{\boldsymbol{\sigma}}{2} \right)\psi_{\text{L}}\tag{1} $$ and right handed spinors $\psi_{\text{R}}$ transform as $$ \psi_{\text{R}} \mapsto \exp\left((-\mathrm{i}\boldsymbol{\theta} -\boldsymbol{\beta})\cdot\...


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The projectors $P_R, P_L$ project $\psi \in \mathcal{H}\cong \mathbb{R}^4$ onto the right- and left-handed sectors of the representation of the Lorentz algebra, which are each a two-dimensional vector space, hence (locally) isomorphic to $\mathbb{R}^2$. What "sum" is meant here is the direct sum $\oplus$: $$\begin{pmatrix} \psi_1 \\ \psi_2 \end{...


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A spinor bundle is a vector bundle associated to a principal bundle whose structure group is a spin group. There is a constraint for the spacetime manifold to support such a principal bundle. It must be a spinnable manifold. Actually, as a point of physical nomenclature, the usual phrase is: the manifold is spin. But as a point of use of the English language,...


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Feynman diagrams are a pictorial representation of a mathematical integral in Quantum Field Theory (QFT). QFTs by consruction have creation and annihilation operators operating on fields. The fields are plane wave solutions of a quantum mechanical equation, and are like a coordinate syste, occupying all space time. On the fields creation and annihilation ...


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