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Relation between the Dirac Algebra and the Lorentz group

It's pretty annoying that P&S just give you $$S^{\mu \nu} = \frac{i}{4} [\gamma^{\mu},\gamma^{\nu}]$$ from thin air, here is a way to derive it similar to Bjorken-Drell's derivation (who start ...
bolbteppa's user avatar
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13 votes

Does Hestenes Zitterbewegung Explain why complex numbers appear in QM?

While this is an old question, I imagine many encounter geometric algebra through David Hestenes's writing on zitterbewegung, so I'd like to offer a different perspective than the existing answers. ...
Luke Burns's user avatar
13 votes

Relation between the Dirac Algebra and the Lorentz group

I want to know if we take any arbitrary metric $g_{\mu\nu}$ on some space $V$, will the generators defined as $S^{\mu\nu}$ generate a Lie group whose elements are transformations on $V$ that conserve ...
Mike's user avatar
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12 votes
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Understanding Wikipedia's definition of a spinor

The spin group has a multiple irreducible representations of dimension 4. Two of them are the left- and right-handed spin-3/2 representations. The other one is called the vector representation. "...
HTNW's user avatar
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11 votes
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Is there an elegant proof of the existence of Majorana spinors?

To answer the confusion between the three sources you list: Using the signature convention of Figueroa O'Farrill, we have Majorana pinor representations for $p - q \pmod 8 = 0,6,7$ and Majorana spinor ...
Dernier Cri's user avatar
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How are Clifford algebras related to Dirac Equation

The vector space is Minkowski spacetime. The quadratic form is the spacetime interval. The relationship to the Dirac equation is that when you start with Minkowski spacetime, the vector represent ...
Timaeus's user avatar
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9 votes

Understanding Wikipedia's definition of a spinor

Here's a purely mathematical answer. The Lie algebra $\mathfrak{so}(1,3)$ of the Lorentz group is a simple 6-dimensional real Lie algebra which is isomorphic to $\mathfrak{sl}_2(\Bbb{C})$ viewed as a ...
Chad K's user avatar
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8 votes
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Covariant gamma matrices

Indeed geometric interpretation of $\gamma_5$ is related to the volume form $$ V=\frac 1 {4!} \epsilon_{\mu\nu\alpha\beta} dx^\mu \wedge dx^\nu \wedge dx^\alpha \wedge dx^\beta = \frac 1 {4!} \sqrt{-...
Saksith Jaksri's user avatar
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Proof of the Anti-Commutation Relation for Gamma Matrices from Dirac Equation

Even if this is similar, this answer should be clearer, as it was to me. We are here. \begin{eqnarray*} (\gamma^\nu \gamma^\mu \partial_\nu \partial_\mu + m^2)\psi &=& 0\\ (\gamma^\mu \gamma^\...
MycrofD's user avatar
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8 votes

Do I need Gamma matrices in Majorana representation in the Lagrangian of a Majorana fermion?

In Euclidean $d=2k$ dimensions, we can take $\gamma^1, \gamma^2,\ldots, \gamma^{2k}$ to be hermitian. We can also arrange the usual $a_n, a^\dagger_n$ construction so that the $\gamma^i$ are ...
mike stone's user avatar
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Physical interpretation of gamma matrices

Here are interpretations for at least two gamma matrices: $\gamma_0$ is the spinor metric. It's role is analogous to the role of the Minkowski metric for four-vectors. We need the Minkowski metric to ...
jak's user avatar
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What has "quantisation" to do with associated graded algebras?

The mathematical term of "quantization" of algebras is much broader than the physical notion of quantization of a classical theory, however, a very prominent quantization in the mathematical sense ...
ACuriousMind's user avatar
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7 votes
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How to prove $tr\{\sigma_1\sigma_2\sigma_3\}=\pm 2i$ with only using $\{\sigma_i,\sigma_j\}=2\delta_{ij}$

It is impossible to prove the wanted identity (initial version) making only use of the anticommutation relation. It is because if you change the sign of all $\sigma_k$, the anticommutation relation ...
Valter Moretti's user avatar
7 votes

Are there projective representations of the Lorentz Group NOT coming from a Clifford algebra?

This answer is based on the seminal paper by Berg, DeWitt-Morette, Gwo and Kramer (BDGK) about the physics of the double covers of the Lorentz groups. Although, the article treats the general case of ...
David Bar Moshe's user avatar
7 votes

Is spinor the sum of scalar, vector, bi-vector, pseudo-vector, and pseudo-scalar?

There are no "two spacetimes". There is a single spacetime $M$, which is a four-dimensional Lorentzian manifold, and there's a bunch of bundles over it. The spin connection is formally a connection ...
ACuriousMind's user avatar
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Spinor Understanding: QFT vs pure Representation Theory

Spinors are vectors in the representation vector space, not matrices in the image of the representation map. A Dirac spinor or bispinor transforms in the (only) irreducible representation of the ...
ACuriousMind's user avatar
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7 votes

Physical interpretation of gamma matrices

They are not just mathematical definitions; there is some physics in them. Actually, they are intrinsically connected with the spin structure of the fields. You can see this from the fact that the ...
Masso's user avatar
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7 votes

Is there an identity for anti-commutator $\{ A B, \, C D \}$ in terms of commutators $[\, , \,]$ only?

Let me put on my math hat by providing a counterexample: Let's say $$ A = a_1, \\B= a_2, \\C= a_3, \\D = a_4 $$ where $a_i$ are regular bosonic anhiliation operators, therefore $$ [a_i, a_j] \equiv 0 ...
MadMax's user avatar
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Does $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}\mathbb1$ determine the hermiticity of the gamma matrices?

I don't think so. If $\gamma^\mu$ is a set of gamma matrices obeying the clifford algebra, then so is $S^{-1}\gamma^\mu S$ for any invertible $S$. But unless $S$ is unitary, $\gamma^\mu$ being ...
mike stone's user avatar
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6 votes

Dimension of Dirac $\gamma$ matrices

Thats a good question. To answer this lets start with Clifford algebra generated by $\gamma$ matrices. \begin{equation} \gamma_{\mu}\gamma_{\nu}+ \gamma_{\mu}\gamma_{\nu}=2\eta_{\mu\nu} \end{equation} ...
Mass's user avatar
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6 votes

Proof of the Anti-Commutation Relation for Gamma Matrices from Dirac Equation

To be honest I think in this case the best proof is by direct computation. The gamma matrices are $$ \begin{equation} \gamma^{0}=\begin{pmatrix} 1 & 0 & 0 & 0\newline 0 & 1 & 0 &...
Physics_Et_Al's user avatar
6 votes
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Metric in 2-components spinor space

Yes, the vector space $V\cong \mathbb{C}^2$ for the left Weyl representation $\rho:SL(2,\mathbb{C})\to {\rm End}(V)$ is a symplectic vector space with symplectic structure given by the Levi-Civita ...
Qmechanic's user avatar
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6 votes
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Going from the Dirac Lagrangian to the adjoint Dirac equation

The Euler-Lagrange-Equation is given by: $$\frac{\partial\mathcal{L}}{\partial {\psi}} - \partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\psi)} = 0$$ Let us take both derivatives ...
infinitezero's user avatar
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6 votes

How does $\{\gamma^\mu , \gamma^\nu\}= 2g^{\mu\nu}I$ imply that $\gamma^\mu$ is traceless?

Using the relation, we know that $ \frac{\gamma^b \gamma^b}{2g^{bb}} = 1 $. Thus, assuming $a\ne b$ we write \begin{align} Tr(\gamma^a) &= \frac{1}{2g^{bb}}Tr(\gamma^a \gamma^b \gamma^b) \end{...
Semoi's user avatar
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5 votes

Different definitions of spinors

You are a bit confused by the wording in Cohen-Tannoudji et al. Namely, it is not the function $[\varphi]: \mathbf R^3 \to \mathbf R^2$ that is called a spinor, it is its value at a particular point, ...
Alex Shpilkin's user avatar
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Fierz like identity for $\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}$

$$\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}= 2i \Big(\delta ^i_q \delta ^k_j\delta ^p_l - \delta ^i_l\delta ^k_q\delta ^p_j\Big ), $$ with all the right pairwise interchange symmetries. (...
Cosmas Zachos's user avatar
5 votes
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Gamma matrices in (2+1)

Gamma matrices don't have a unique representation. They only requirement is that they satisfy the axiom of the Clifford algebra $$ \{\gamma^\mu,\gamma^\nu\} = 2 \,\eta^{\mu\nu}\,. \tag{1} $$ The usual ...
MannyC's user avatar
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5 votes

Physical interpretation of gamma matrices

In natural units, the Hamiltonian your text tells you was introduced by Dirac is $$ H=\vec{\alpha}\cdot \nabla /i+ \beta m , \implies i\partial_t \psi = H \psi ,\\ \beta =\gamma_0 , \qquad \vec {\...
Cosmas Zachos's user avatar
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Does the geometric algebra of curved space have a matrix representation?

In curved space the coordinate basis will not be orthonormal, but an orthonormal basis still exists, and can be used to construct an arbitrary basis. So... Start with $\gamma_\mu$ as a matrix ...
Joe Schindler's user avatar
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Is $\gamma_\mu \gamma^\mu$ a unit operator?

You are using $\mu$ as an index too many times. When you square $\gamma^\mu \partial_\mu$, you have to use a different index for the two factors. Here is a correct derivation: Starting with $(i \...
Elias Riedel Gårding's user avatar

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