A fully relativistic (Lorentz covariant) description, first put forward by Paul Dirac in 1928, of the first quantized, spin one half fermion with nonzero mass. Physical notions to do with this equation include the Dirac sea, Dirac hole theory, the Klein Paradox and the fine structure of the Hydrogen ...

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5k views

What is the difference between a spinor and a vector or a tensor?

Why do we call a 1/2 spin particle satisfying the Dirac equation a spinor, and not a vector or a tensor?
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3answers
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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: $$\...
16
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2answers
1k views

Introduction to spinors in physics, and their relation to representations

First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms "up to a sign" when acted on by the rotation group. I have even rotated a ...
6
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4answers
1k views

Dirac equation as Hamiltonian system

Let us consider Dirac equation $$(i\gamma^\mu\partial_\mu -m)\psi ~=~0$$ as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that ...
11
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2answers
4k views

Charge conjugation in Dirac equation

According to Dirac equation we can write, \begin{equation} \left(i\gamma^\mu( \partial_\mu +ie A_\mu)- m \right)\psi(x,t) = 0 \end{equation} We seek an equation where $e\rightarrow -e $ and which ...
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Where is spin in the Schroedinger equation of an electron in the hydrogen atom?

In my current quantum mechanics, course, we have derived in full (I believe?) the wave equations for the time-independent stationary states of the hydrogen atom. We are told that the Pauli Exclusion ...
13
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2answers
2k views

Is Zitterbewegung an artefact of single-particle theory?

I have seen a number of articles on Zitterbewegung claiming searches for it such as this one: http://arxiv.org/abs/0810.2186. Others such as the so-called ZBW interpretation by Hestenes seemingly ...
3
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1answer
596 views

Why must the Dirac equation multiplied by its complex conjugate give the KG equation?

This may be a simple question. I can show this is the case mathematically but cannot explain why it happens. It was only when asked why this happens when I realised I couldn't explain it intuitively/...
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4answers
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Why would Klein-Gordon describe spin-0 scalar field while Dirac describe spin-1/2?

The derivation of both Klein-Gordon equation and Dirac equation is due the need of quantum mechanics (or to say more correctly, quantum field theory) to adhere to special relativity. However, excpet ...
7
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1answer
346 views

Sign in front of QFT kinetic terms

I'd like to know if the sign in front of a kinetic term in QFT important. For the scalar field we conventionally write (in the $ + --- $ metric), \begin{equation} {\cal L} _{ kin} = \frac{1}{2} \...
8
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2answers
247 views

Can we treat $\psi^{c}$ as a field independent from $\psi$?

When we derive the Dirac equation from the Lagrangian, $$ \mathcal{L}=\overline{\psi}i\gamma^{\mu}\partial_{\mu}\psi-m\overline{\psi}\psi, $$ we assume $\psi$ and $\overline{\psi}=\psi^{*^{T}}\gamma^{...
3
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1answer
182 views

Wrong sign anticommutation relation for the Dirac field?

Consider the Dirac Lagrangian $$\mathcal{L}=\psi ^{\dagger }\gamma ^{0}\left( \mathrm{i}\gamma ^{\rho }\partial _{\rho }-m\right) \psi .$$ The conjugate momenta to $\psi ^{a}$ are defined, as usual, ...
3
votes
1answer
655 views

Matrix order in Dirac equations

The trace of matrix is always sum of its eigen values , which can be seen if $\hat{U}$ transforms the matrix $\alpha_i$ into it's diagonal form . $$ \begin{pmatrix} A_1 & 0 & \cdots & 0 \...
11
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3answers
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What was missing in Dirac's argument to come up with the modern interpretation of the positron?

When Dirac found his equation for the electron $(-i\gamma^\mu\partial_\mu+m)\psi=0$ he famously discovered that it had negative energy solutions. In order to solve the problem of the stability of the ...
2
votes
2answers
367 views

CPT invariance of Dirac equation

We know that Dirac equation is \begin{equation} ( i \partial _\mu \gamma ^\mu - m ) \psi ~=~0. \end{equation} How can we show that Dirac equation is invariant under CPT transformation?
3
votes
1answer
160 views

Equation for relativistic electron and two-component spinor

Recently I heard that there is some "alternate" equation for the Dirac one. It can be introduced if we refuse some properties of the theory describes the electron, which Dirac used in his original ...
1
vote
1answer
126 views

Nonlinear Dirac's Equation?

Are there any nonlinear variations of Dirac's Equation analogous to the Nonlinear Schrodinger Equation, that have been studied and published in any mainstream journals or books? Perhaps such a ...
29
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5answers
1k views

Why fermions have a first order (Dirac) equation and bosons a second order one?

Is there a deep reason for a fermion to have a first order equation in the derivative while the bosons have a second order one? Does this imply deep theoretical differences (like space phase dimesion ...
14
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2answers
2k views

How to prove $(\gamma^\mu)^\dagger=\gamma^0\gamma^\mu\gamma^0$?

Studying the basics of spin-$\frac{1}{2}$ QFT, I encountered the gamma matrices. One important property is $(\gamma^5)^\dagger=\gamma^5$, the hermicity of $\gamma^5$. After some searching, I stumbled ...
6
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1answer
4k views

Majorana mass vs Dirac Mass

Why is it said that the Dirac mass term conserves the fermion number but the Majorana mass term does not? Can someone explain this mathematically? Which breakdown of symmetry is responsible for ...
10
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2answers
217 views

Is there a 2D manifold on which the Dirac equation has a zero mode?

The two-dimensional (2D) Dirac equation $(\sigma_1iD_1+\sigma_2 iD_2)\psi=E\psi$ admits zero mode ($E=0$) solutions on a non-trivial gauge background, such as the zero mode at the core of a U(1) gauge ...
3
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1answer
1k views

What happens to the Lagrangian of the Dirac theory under charge conjugation?

Consider a charge conjugation operator which acts on the Dirac field($\psi$) as $$\psi_{C} \equiv \mathcal{C}\psi\mathcal{C}^{-1} = C\gamma_{0}^{T}\psi^{*}$$ Just as we can operate the parity operator ...
8
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1answer
535 views

Dirac equation as canonical quantization?

First of all, I'm not a physicist, I'm mathematics phd student, but I have one elementary physical question and was not able to find answer in standard textbooks. Motivation is quite simple: let me ...
3
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0answers
80 views

Substitution $\partial_\mu \to D_\mu \equiv \partial_\mu + ieA_\mu$ allows the introduction of electromagnetic interactions [closed]

I want to show that the substitution $\partial_u \to D_\mu \equiv \partial_\mu + ieA_\mu$, or equivalently $p_\mu \to p_\mu - eA_\mu$ allows the introduction of electromagnetic interactions. Here $e$ ...
4
votes
3answers
241 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 \...
8
votes
2answers
611 views

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

Why the lowest order of matrices in Dirac equation (Relativistic Quantums) are 4x4 matrices (and can not be 2x2 matrices)? How to prove it?
4
votes
2answers
82 views

Bilinears in adjoint representation

Below are two statements from my notes and I am trying to verify them explicitly. In both cases the fields are assumed to transform under the fundamental representation of $O(N)$ - --'The kinetic ...
4
votes
1answer
630 views

Components of the Weyl spinor field

In the Weyl basis we can separate the spinor field into 2 components: the right-chiral spinor and the left-chiral spinor. Each of these fields has again 2 components which are coupled. What is the ...
4
votes
4answers
170 views

Is time an observable in Relativistic Quantum Mechanics?

Relativistic Quantum Mechanic is based, as far as I know, in the Dirac Equation. Now, the Schrödinger equation, in the abstract state space takes the form: $$i\hbar \dfrac{d|\psi(t)\rangle}{dt}=H|\...
3
votes
1answer
438 views

Dirac adjoint of a matrix

The Dirac adjoint for Dirac spinors is defined as, $$ \bar{u} = u^{\dagger} \gamma^{0} \, . $$ However I have come across this, $$ \overline{\gamma^{\mu}} = \gamma^{\mu} \, , \tag{1} $$ (where $\gamma^...
3
votes
1answer
594 views

Explanation of equation that shows a failed approach to relativize Schrodinger equation

I'm reading the Wikipedia page for the Dirac equation: $\rho=\phi^*\phi\,$ ...... $J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$ with the conservation of probability ...
3
votes
1answer
350 views

Dirac operator in curved spacetime in 2 dimensions – hermitian?

I'm currently trying to learn about the Dirac equation in curved spacetime and have come across an odd remark in Nakahara's well-known textbook "Geometry, Topology and Physics" that I would like to ...
1
vote
1answer
1k views

Dirac field and stress-energy tensor density

I read somewhere that stress-energy tensor density is a symmetric tensor. But if I take the Dirac Field tensor: $$T^{\mu \nu}=i \psi^\dagger \gamma^0 \gamma^\mu \partial^\nu \psi $$ How could I ...
1
vote
1answer
323 views

Complete set and Klein-Gordon equation

In http://www.physics.ucdavis.edu/~cheng/teaching/230A-s07/rqm2_rev.pdf, it says that when there is some external potential, the Klein-Gordon equation is altered, and it says the following: The ...
6
votes
4answers
704 views

Why is the Dirac equation not used for calculations?

From what I understand the Dirac equation is supposed to be an improvement on the Schrödinger equation in that it is consistent with relativity theory. Yet all methods I have encountered for doing ...
3
votes
0answers
100 views

Tunneling from Dirac material into Schrodinger material?

When a Dirac material, like graphene or TI, has a connection with a normal metal which Schrodinger equation govern on their carriers, how could we manipulate the tunneling of electron from Dirac side ...
2
votes
1answer
221 views

Spin tensor and Lorentz group operator in bispinor case

For infinisesimal bispinor transformations we have $$ \delta \Psi = \frac{1}{2}\omega^{\mu \nu}\eta_{\mu \nu}\Psi , \quad \delta \bar {\Psi} = -\frac{1}{2}\omega^{\mu \nu}\bar {\Psi}\eta_{\mu \nu}, \...
0
votes
0answers
31 views

Variables in the Dirac Equation Lagrangian [duplicate]

(Warning: I'm a student of mathematics with no training in physics.) In derivations of the Dirac equation from an action principle, one encounters the action $$S= \displaystyle\int\,d^4x \,\bar\psi(x)...
0
votes
0answers
79 views

Four-momentum and Dirac equation in curved spacetime

Norm of four momentum in Minkowski spacetime is proportional to the square of rest mass as \begin{equation}|P|^2= P^\alpha \eta_{\alpha\beta}P^\beta= (E/c)^2 - p^2 = (mc)^2 \end{equation} While in ...