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Questions tagged [clifford-algebra]

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Is there a Cauchy integeal formula in Geometric algebra (GA) or space-time algebra or clifford algebra>

I am curious about Cauchy integral formula which is generalized to Clifford algebra especially Cl(1,3). If you are familiar with paper or theorems about this subject and share them. It will be very ...
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Matrix representation of the CAR for the fermionic degrees of freedom

The canonical anticommutation relations (CAR) for a fermionic degree of freedom can be written as follows: $$a^2 = \left( a^{\dagger} \right) ^2 = 0,$$ $$a a^{\dagger} + a^{\dagger} a = 1.$$ ...
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Spinor Understanding: QFT vs pure Representation Theory

I have some questions on terminology used in QM & QFT and (pure mathematical) representation theory treating the concept of "spinor". Let us focus on Dirac spinor as described in https://en....
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Anticommutator of spin-1 matrices

We know that in the spin-1/2 representation the anticommutation relation of the Pauli matrices is $\{\sigma_{a},\sigma_{b}\}=2\delta_{ab}I$. Does a similar relation hold for the spin-1 representation?
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Clifford algebra formulation of the Nambu-goto action

Using the wedge product one can pair the generators the Clifford algebra $Cl_{1,3}(\mathbb{R})$ to produce 2-vectors (area elements). The Nambu-Goto action is a statement on the evolution of ...
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Why is there a negative sign in the (non-relativistic) bivector formulation of the Lorentz force?

I'm currently trying to update my understanding of basic (Newtonian, non-relativistic) physics to use bivectors and Clifford products instead of pseudovectors and cross products. And I've come up ...
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Maximal anticommuting sets of Dirac matrices

At the end of this webpage, it is said that there exist 6 maximal anticommuting sets each consisting of 5 Dirac $\Gamma$-matrices. I couldn't find anything more in the book cited there, either. I ...
304 views

Dirac equation in 1+1D spacetime compared to “standard” 3+1D Dirac equation

In the past couple of weeks I've been studying the Dirac equation and its solutions. During a discussion with a tutor it was pointed out to me that one could formulate something similar to the Dirac ...
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Proving identity $\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu}$

In the lecture notes accompanying a course I'm following, it is stated that $$\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu}$$ Yet when I try to prove this,...
134 views

Dirac matrices in 1+1 dimensions

Given $\gamma^\mu$ in 1+3 dimensions with signature $(+,-,-,-)$, how can I obtain Dirac matrices in 1+1 dimensions expressed in terms of the $\gamma^\mu$?
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Do I need Gamma matrices in Majorana representation in the Lagrangian of a Majorana fermion?

I understand that the Majorana representation of the Gamma matrices are the real representations of the associated Clifford algebra. A Majorana fermion is defined as a fermion that equals to its ...
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Confusion with trace of gamma matrices

Using $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu} \mathbf{1}$, it is easy to show that: \begin{align*} \operatorname{tr} \gamma^\mu \gamma^\nu = 4\eta^{\mu\nu} \end{align*} Now, it is also true that ...
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Gamma matrices in (2+1)

I am sure that is very well-known question and see on this site several similar questions but I would like to specify the answer 1) I know that in $(2+1)$-dimensions one can construct $\gamma$-...
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In the study of Quantum Field Theory and Group Theory for the spinor representation of $SO$ groups, we know the following correspondence: $\chi C\psi$ scalar $\chi C\gamma^\mu\psi$ vector $\chi C\... 1answer 108 views Can one find Dirac matrices for any spacetime metric? For any metric $$g_{μν}$$ is there always a linearly independant spacetime algebra satisfying $$\{\bar{γ}_μ,\bar{γ}_ν\} = 2 g_{μν} I?$$ For a diagonal metric I was able to work out that $$\bar{γ}_μ=\... 2answers 219 views What is the relationship between the Lorentz group and the CL(1,3) algebra? In my classes the dirac equation is always presented as the "square root" of the Klein Gordon equation, then from this you can demand certain properties from the Matrices (anticommutation relations, ... 3answers 427 views Is spinor the sum of scalar, vector, bi-vector, pseudo-vector, and pseudo-scalar? Is spinor \psi actually the sum of scalar, vector, bi-vector, ..., pseudo-scalar? Before talking about spinors, we have to differentiate two kinds of spacetime, demonstrated with the example of ... 3answers 114 views Missing identity element in the Clifford relation While studying the Dirac equation,$$\left(i\gamma^{\mu} \partial_{\mu} - m\right)\psi = 0.$$I have been finding difficulty understanding the following summarisation of the algebra that the \gamma-... 2answers 225 views What are Clifford fragments? In his article/lecture on "What quantum physics can learn from Egyptian hieroglyphs"", researcher Robert Spekkens talks about Clifford fragments. He describes them as "containing only a subset of the ... 0answers 136 views Relation between Dirac spinors, quaternions, and bicomplex numbers Superficially Dirac spinor resp. Dirac gamma matrices and quaternions and bicomplex numbers seems to be very similar objects. all can be expressed by unitary 4x4 matrices so they seem to represent ... 2answers 293 views A useful identity for Gell-Mann su(3) matrices? We have the following beautiful result for Pauli su(2) matrices$$(\vec{\sigma}\cdot\vec{a})(\vec{\sigma}\cdot\vec{b}) = \mathbb{I} ~\vec{a}\cdot\vec{b} + i (\vec{a} \times \vec{b}) \cdot \vec{\... 1answer 232 views Spacetime dimension and the dimension of Clifford algebra The dimension of the Clifford algebra$C_p$generated by a vector space$V^p$is given by$2^p$, where$p$is the dimension of the vector space (T. Frankel, the geometry of physics). Based on the top-... 1answer 74 views A question about the decoupling of Dirac equation in 1+1 dimension It is said that in 1+1 dimension, if we take$\gamma^0=i\sigma^2$and$\gamma^1=\sigma^1$, then the two components of dirac spinor$\psi_L$(upper component) and$\psi_R$(lower component) decouple in ... 1answer 301 views How to prove$\{\gamma^{\mu}, \gamma^{\nu}\}$? (notation problem) I want to prove that$ \{\gamma^{\mu}, \gamma^{\nu} \}=2g^{\mu \nu} $what are the indices$ \mu$and$ \nu$here? because I know the gamma matrices from 0 to 5 and I need to verify the anti ... 0answers 117 views Physical/geometrical interpretations of spinors? Physically, a scalar is a quantity invariant with reference frame, a vector is a quantity associated with a direction, tensors are higher relationships between vectors - what are spinors? I thought I ... 1answer 68 views How to count states in SUSY multiplets? There is an easy proof of the structure of multiplet that I don't reproduce here (it can be found in Bertolini, Lecture on Supersymmetry, pp.40-41 for the massless case and p.47 for the massive one).... 1answer 107 views How is$\gamma^{\mu}$defined in the anti commutation relation$\{{\gamma_{5},\gamma^{\mu}}\}$? how is$\gamma^{\mu}$defined in the anti commutation relation$\{\gamma_{5},\gamma^{\mu}\}$? does it make a difference if you write the index${^\mu}lower? what does usually change if the index is ... 4answers 2k views Proof of the Anti-Commutation Relation for Gamma Matrices from Dirac Equation My textbook on QFT says that the Dirac equation can be used to show the following relation: $$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$$ I have searched around and unable to find how to prove this ... 0answers 237 views Gamma matrices in higher (even) spacetime dimensions Suppose we write the gamma matrices in this following representation: \begin{align*} \gamma^{0}=\begin{pmatrix} \,\,0 & \mathbb{1}_{2}\,\,\\ \,\,\mathbb{1}_{2} & 0\,\, \end{... 2answers 1k views Relation between the Dirac Algebra and the Lorentz group In their book Introduction to Quantum Field Theory, Peskin and Schroeder talk about a trick to form the generators for the Lorentz group from the commutators of the gamma matrices, using their anti-... 0answers 225 views Weyl and Majorana-Weyl spinors why need commutation? Let\psi$denote a Dirac spinor then Weyl spinors are defined by: $$\psi_{L,R}=\frac{1}{2} (I\pm \gamma)\psi$$ on even dimensions$\gamma$commutes with$\sigma_{\mu \nu}$(generators used to define ... 1answer 49 views Is the Heighest weight vector in the Spinor rep of$SO(1,d-1)$zero? Consider the highest weight vector of the Spinor rep of$SO(1,d-1)$where$d=2m+1$. It can be shown that: $$\gamma_i \gamma_{m+i}v=v\tag{*}$$ I cannot see why this relation does not imply that$v=0$? ... 2answers 362 views Clifford Algebra: Wedge product, cross product, and Hodge duality I've been reading some papers related to Bell's Theorem which involve Clifford Algebra. I am investigating it for an undergrad project but none of my professors seem to know anything about Clifford ... 1answer 143 views How to prove$\gamma^0=(\gamma^0)^T$? The Dirac gamma matrix$\gamma^0$is symmetric in Dirac, Weyl and Majorana representation. Is it in general true that$\gamma^0=(\gamma^0)^T$? Can it be proved that$\gamma^0=(\gamma^0)^T$in a ... 1answer 1k views Transformation between Weyl and Dirac representation of Gamma matrices I want to find a similarity transformation$T$between the Weyl representation and the Dirac representation of the gamma matrices:$\gamma_W^\mu=T \gamma_D^\mu T^{-1}$. It turns out that I can look at ... 2answers 406 views Are there projective representations of the Lorentz Group NOT coming from a Clifford algebra? Let$\mathrm{SO}(1,d-1)_{+}$be the restricted Lorentz Group in$d$dimensions. Are there projective irreducible representations of this group that do not descend from a representation of$\mathrm{C}\...
When doing calculations with Weyl spinors, terms like $\theta\sigma^\mu\theta^\dagger$ appear. I know that for 3+1 spacetime dimensions, $\sigma^\mu = (\textbf{1}, \sigma^i)$ with $i=1,2,3$ the usual ...
In (Vaz and da Rocha, 2016;pg108) the following two step process is given for finding the matrix representation of a Clifford algebra: (verbatim; except for notation) (1) Choose a set of $N$ ...