Questions tagged [dirac-matrices]

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26
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4answers
6k views

Dimension of Dirac $\gamma$ matrices

While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices: $$\...
20
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3answers
5k views

Lorentz transformation of Gamma matrices $\gamma^{\mu}$

From my understanding, gamma matrices transforms under Lorentz transformation $\Lambda$ as \begin{equation} \gamma^{\mu} \rightarrow S[\Lambda]\gamma^{\mu}S[\Lambda]^{-1} = \Lambda^{\mu}_{\nu}\gamma^{\...
11
votes
3answers
2k views

Do gamma matrices form a basis?

Do the four gamma matrices form a basis for the set of matrices $GL(4,\mathbb{C})$? I was actually trying to evaluate a term like $\gamma^0 M^\dagger \gamma^0$ in a representation independent way, ...
11
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2answers
2k views

Why are usually 4x4 gamma matrices used? [duplicate]

As far as I understand gamma matrices are a representation of the Dirac algebra and there is a representation of the Lorentz group that can be expressed as $$S^{\mu \nu} = \frac{1}{4} \left[ \gamma^\...
10
votes
4answers
16k views

Commutator of Dirac gamma matrices

Quick question...For some reason I'm having trouble finding an identity or discussion for the commutator of the gamma matrices at the moment...i.e $$\gamma^u\gamma^v-\gamma^v \gamma^u$$ but I am not ...
9
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2answers
2k views

Do Dirac Gamma Matrices act like Tensors?

Do Dirac Gamma Matrices act like Tensors? Is it true that $$ \gamma_\mu = \eta_{\mu\nu}\gamma^\nu~? $$ Also what about $\sigma_{\mu\nu}$, where $\sigma_{\mu\nu}$ is defined to be: \begin{align*} \...
9
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2answers
4k views

Can one show that ${\gamma^5}^\dagger = \gamma^5$ directly from the anticommutation relations?

Is it possible to show that ${\gamma^5}^\dagger = \gamma^5$, where $$ \gamma^5 := i\gamma^0 \gamma^1 \gamma^2 \gamma^3,$$ using only the anticommutation relations between the $\gamma$ matrices, $$ \...
9
votes
2answers
521 views

Can we make the Dirac representation a gauge theory?

I'm looking for comments and references about an idea : gauging the Dirac representation of the Dirac matrices. What kind of field interaction would it give ? Specifically, the Dirac equation is ...
8
votes
2answers
2k views

How to find a particular representation for the gamma matrices?

I asked this question as a subquestion in another thread, but got the answer below and thought it deserved a thread of its own. Two well-known representation of the gamma matrices are the Weyl and ...
8
votes
1answer
866 views

How to show that $\bar\psi\gamma^\mu\psi$ of a Dirac spinor $\psi$ transforms as a vector?

This is part 2 of exercise II.1.1 of Zee's QFT in a Nutshell (here's part 1). This is what I have got: \begin{align} \bar\psi\gamma^\lambda\psi \mapsto \bar\psi^{\,\prime}\gamma^\lambda\psi^{\,\...
8
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2answers
2k views

Why the lowest order of matrices in Dirac equation are 4x4 matrices? [duplicate]

Why the lowest order of matrices in Dirac equation (Relativistic Quantums) are 4x4 matrices (and can not be 2x2 matrices)? How to prove it?
7
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2answers
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How does Schur's Lemma mean that the Dirac representation is reducible?

In chapter 3 of Peskin and Schroeder, when they're talking about "Dirac Matrices and Dirac Field Bilinears," they introduce $\gamma^{5}$ and give some properties of it. One of the properties is $[\...
7
votes
1answer
3k views

Parity on gamma matrices

I want to understand clearly why $ P \gamma^{\mu} P = \gamma^{\mu} $, where $ P $ is the parity operator. This result follow for example from pag. 66 of Peskin-Schroeder. The parity operator acts ...
6
votes
2answers
497 views

Zee's use of Kronecker Product in “QFT in a Nutshell” to represent Dirac matrices

In his book Quantum Theory in a Nutshell (2nd edition, p. 94), Zee describes the Dirac gamma matrices and lists a representation using Pauli matrices and the identity matrix. For example he writes $$ ...
6
votes
2answers
2k views

Hermitian adjoint of 4-gradient in Dirac equation

I'm having issues deriving the Dirac adjoint equation, $$\overline{\psi}(i\gamma^{\mu}\partial_{\mu}+m)=0.\tag{1}$$ I started by taking the Hermitian adjoint of all components of the original Dirac ...
6
votes
4answers
875 views

Representation under which Pauli matrices transform

In Peskin and Schroeder's Quantum Field Theory, there is an identity of Pauli matrices which is connected to the Fierz identity, (equation 3.77) $$(\sigma^{\mu})_{\alpha\beta}(\sigma_\mu)_{\gamma\...
6
votes
1answer
730 views

How to show that $\bar\psi\psi$ of a Dirac spinor $\psi$ transforms as a scalar?

I would like to show that for a Dirac spinor $\psi$, the scalar product $\bar\psi\psi$ transforms as a scalar under a Lorentz transformation $\Lambda$, where $\bar\psi = \psi^\dagger\gamma^0$. This is ...
6
votes
1answer
510 views

What is the role of the spacetime algebra?

For Minkowski space $M^4=\mathbb{R}^{1,3}$ the Clifford algebra $Cl_{1,3}$ (Dirac algebra) with $\{\gamma^\mu, \gamma^\nu \}=2 g^{\mu \nu}$ is sometimes called "spacetime algebra". What is its ...
6
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1answer
388 views

Simplifying a seemingly simple gamma matrix identity

When studying from the book by Wise and Manohar, Heavy Quark Physics (pg 102), I came across a seemingly simple identity that I am not able to prove. It's likely an easy problem but I can't for the ...
6
votes
1answer
195 views

Time reversal for Dirac particles in 2+1 or 6+1 (mod 8) spacetime dimensions

It is my understanding that time reversal invariance for Dirac fermions is usually (in 3+1 dimensions at least) implemented by an antiunitary operator ${\mathfrak T}$ that acts on the Dirac field ...
6
votes
1answer
983 views

Hermitian properties of Dirac operator

I am trying to understand the Hermiticity of the (massless) Dirac operator in both (flat) Minkowski space and Euclidean space. Let us define the Dirac operator as $D\!\!\!/=\gamma^\mu D_\mu$, where $...
6
votes
2answers
588 views

Weinberg QFT Chapter 5: gamma matrices consistency

Currently reading through Weinberg's QFT book (Vol. 1) [readable in parts here]. In his derivation of the causal Dirac field in Ch. 5, he chooses his gamma matrices as (5.4.17) \begin{align} \gamma^0&...
6
votes
2answers
378 views

Diffeomorphisms and the Dirac action

I have a question concerning fermions in curved space-time. Please read it to the end before suggesting the spin-connection and vierbein-based approach. I heard that there is a special way of ...
5
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2answers
1k views

spinor vs vector indices of Dirac gamma matrices

I am struggling to understand the nature of the components of the Dirac matrices. If we view the four components of a Dirac spinor as $\psi^a$ with $a$ being a 'spinor' index, then if a gamma matrix ...
5
votes
2answers
284 views

Dirac equation without $i$

In Witten's review paper "Fermion path integrals and topological phases", the Dirac equation (Eq(2.2)) is $$(\gamma^{\mu}D_{\mu}-m)\psi=0$$ which appears very strange to me. Initially I thought this ...
5
votes
2answers
673 views

Choice of Dirac gamma matrix representation and definition of adjoint spinor

Is the definition of the adjoint spinor $\bar{\psi}=\psi^\dagger \gamma^0$ forcing a particular choice of representation of the Dirac matrices (or a subset of the possible choices)? More precisely, I ...
5
votes
1answer
691 views

Spinor representation and Lorentz transformation in Peskin &Schroeder

I am a newbie in group theory. In Peskin & Schroeder's QFT P.42 (3.29) it says that, since we have $$\Lambda_{\frac{1}{2}}^{-1}\gamma ^{\mu}\Lambda_{\frac{1}{2}}~=~\Lambda^{\mu}_{~~\nu }\gamma^{\...
5
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1answer
158 views

Spinors, Spacetime and Clifford algebra

I'm looking to understand the intrinsic connection that Clifford algebra allows one to make between spin space and spacetime. For a while now I've trying to wrap my head around how the Clifford ...
5
votes
1answer
440 views

Spinor indices manipulation in the Noether's current for free fermions

I can't solve this apparent paradox; I have the free lagrangian for massive fermions $\mathscr L = i\bar\Psi\gamma^\mu\partial_\mu\Psi - m\bar\Psi\Psi$ which is invariant under the global phase ...
4
votes
3answers
755 views

Physical interpretation of gamma matrices

Just out of plain curiosity, I want to ask: What are/is the physical interpretation(s) of the gamma matrices? If there is none, is it right to assume that it is just a mathematical fudge-factor?
4
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3answers
457 views

Why do we need matrices in the Dirac equation?

Consider the following equation: \begin{equation} \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = \left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right)\left(A \...
4
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4answers
4k 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 ...
4
votes
2answers
302 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, ...
4
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3answers
125 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 $\gamma$-...
4
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2answers
1k views

Has the relative sign in the Dirac equation any meaning?

I know there are many questions about this topic and also various answers, but it's never stated explicitly, why there is a certain sign before the mass term in the Dirac Lagrangian. I'm also confused,...
4
votes
1answer
4k views

How can pseudospin be a vector? (Graphene)

In graphene science, I don't understand how one interprets pseudospin as a vector. I thought 'pseudospin' was the vector of Pauli matrices. So how can it be a vector that one can plot for example in ...
4
votes
1answer
1k views

A more general completeness relation for Dirac spinors

Assume that we have two 1/2-spin particles with four-momenta $p$ and $p'$. Particle Dirac spinors satisfy the completeness relation $$ \sum_{s=1}^2u_s(p)\overline{u}_s(p)=\not p+m $$ My goal now is to ...
4
votes
1answer
317 views

Why do we need tetrads/vierbeins/frame-fields to describe fermions in curved space?

I'm learning about the frame formalism and read that to couple fermions to gravity you need to go to the frame-formalism. As a motivation to learn more about frame-fields would someone sketch me why ...
4
votes
1answer
816 views

Normalization of Dirac bispinors

Let $u_\lambda(\vec{k})$ and $v_\lambda (\vec{k})$ be solutions of the following equations $$(\not k-m)u_\lambda(\vec{k})=0$$ $$(\not k+m)v_\lambda(\vec{k})=0$$ Suppose that $u_\lambda(\vec{k})^\...
4
votes
1answer
327 views

Should the trace of a product of gamma matrices depend on the convention I use?

I am trying to work out $$\text{Tr}[\gamma_5\gamma_\mu\gamma_\nu\gamma_\alpha\gamma_\beta]$$ using the same convention as J.J. Sakurai (Advanced Quantum Mechanics), what I get is $$\text{Tr}[\gamma_5\...
4
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1answer
4k views

Deriving the Spinor Completeness Relation without using a Representation

Reference: DAMTP problem set 3, question 5 but ignore the spinor solutions given. To preface, this has taken up 1 entire day and a further 2 afternoons of work so I will just list the most promising ...
4
votes
1answer
135 views

Some diracology in traces

Suppose I want to evaluate the trace $p_{\alpha} q_{\beta}\text{Tr}(\gamma^{\alpha} \gamma^0 \gamma^{\beta} \gamma^0)$. Using the standard trace formula for four gamma matrices I arrive at $$p_{\...
4
votes
1answer
173 views

What does $\Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$ mean?

\begin{equation} \Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu \end{equation} In P&S, p. 42: Equation (3.29) says that the $\gamma$ ...
4
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1answer
430 views

The fifth gamma matrix and fermion fields

I am aware of the various relations with Dirac spinors and chirality but how does the fifth gamma matrix $\gamma^5$ behave with fermion fields, $\psi$? Does the fifth gamma matrix have any particular ...
4
votes
0answers
125 views

How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
3
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1answer
121 views

Why does the trace show up in such expressions?

I've been studying different scattering processes (from Mandl & Shaw QFT's book, chapter 8) and there's always a purely-mathematical common step I do not understand: the showing-up of the trace. ...
3
votes
2answers
2k views

Construction of Pauli Matrices

How can we construct the Pauli matrices starting from $$\sigma_i=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$$ by using the conditions $$\sigma^2_i=1,$$$$\left [ \sigma_x,\sigma_y \right ]=2i\...
3
votes
2answers
2k views

Fierz identity with Weyl spinors

The following Fierz relation does not seem so obvious to me: \begin{equation} \bar{\psi}_1 \gamma^\mu (1+\gamma_5)\psi_2 \bar{\psi}_3 \gamma_\mu (1-\gamma_5) \psi_4 = -2 \bar{\psi}_1 (1-\gamma_5) \...
3
votes
1answer
429 views

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 $\gamma$-...
3
votes
1answer
132 views

Is $\displaystyle{\not} p$ a Lorentz Scalar?

I am a bit confused about something. $\gamma^\mu$ is a (Lorentz) vector (c.f. Pesking & Schroeder chapter 3), and so is $p^\mu$, therefore I’d expect their product $\displaystyle{\not}p \triangleq ...

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