Questions tagged [dirac-equation]

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 spectrum.

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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: $$\...
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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 ...
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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 ...
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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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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 ...
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Confusion with chirality eigenstates

In the Weyl/chiral basis, the four components of the Dirac spinor represent left-chirality spin up, left-chirality spin down, right-chirality spin up, and right-chirality spin down, respectively. When ...
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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^{...
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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, ...
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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 \...
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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 ...
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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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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 ...
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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^\...
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Classical Fermion and Grassmann number

In the theory of relativistic wave equations, we derive the Dirac equation and Klein-Gordon equation by using representation theory of Poincare algebra. For example, in this paper http://arxiv.org/...
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Spinor field normalisation from poles in the propagator

In the theory of free scalar bosons (KG field) it is a basic result that the propagator $\Delta(p)$ has poles at $p^2=m^2$, with residue $1$ (or any other constant, depending on conventions). Thinking ...
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Derivative with respect to a spinor of the free Dirac lagrangian

When we derived Dirac Equation starting form the lagrangian, our QFT professor said: "let's take the free lagrangian $$\mathscr L = i\bar\Psi\gamma^\mu\partial_\mu\Psi - m\bar\Psi\Psi$$ and perform $...
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Why are S-state solutions of Dirac equation for hydrogen atom allowed to be unbounded?

In this Wikipedia page there's a statement about 1S orbital as solved from Dirac equation: Note that $\gamma$ is a little less than $1$, so the top function is similar to an exponentially ...
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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, ...
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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 ...
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Is there an algebraic approach for the topological boundary (defect) states?

There are many free fermion systems that possess topological edge/boundary states. Examples include quantum Hall insulators and topological insulators. No matter chiral or non-chiral, 2D or 3D, ...
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Interpretation of Dirac equation states

In Pauli theory the components of two-component wavefunction were interpreted as probability amplitudes of finding the particle in particular spin state. This seems easy to understand. But when ...
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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} \...
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Where do the quantum fields encode the spin information?

I know basically the difference between Klein-Gordon and Dirac field is spin. But I am not sure where we need to implement this info. The solutions of both equations are the wave packets which ...
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Can Dirac equation be reformulated in an equivalent tensor form?

1) Can Dirac equation (including bispinors) be represented by a tensor formalism? 2) If yes, what kind of tensors could be the components of the wave function in Dirac equation in such formulation? ...
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Why is the Dirac mass term Hermitian if Grassmann-valued fields anticommute?

Let's have Dirac mass term in lagrangian: $$ L_{M} = \bar{\Psi}\Psi $$ Lagrangian must be real-valued, i.e., its Hermitian conjugation doesn't change it. But due to Grassmann nature of spinor fields, $...
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Spin Up with Indefinite Helicity

Imagine we are studying the spin quantization along the same axis as the momentum. What if I have a Dirac spinor with a spin up but no definite helicity ($\psi_L,\psi_R\neq0$): $$ u(p)= \left(\begin{...
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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 ...
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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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Why do we need $2^\text{nd}$ quantization of the Dirac equation

As a Mathematician reading about the Dirac equation on the internet, leaves me with a great deal of confusion about it. So let me start with its definition: The Dirac equation is given by, $$ i \hbar ...
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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 ...
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How did Paul Dirac predict the existence of antiproton?

The existence of the antiproton with -1 electric charge, opposite to the +1 electric charge of the proton, was predicted by Paul Dirac in his 1933 Nobel Prize lecture. Quotation by Wikipedia. ...
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What spinor field corresponds to a forwards moving positron?

When we search for spinor solutions to the Dirac equation, we consider the 'positive' and 'negative' frequency ansatzes $$ u(p)\, e^{-ip\cdot x} \quad \text{and} \quad v(p)\, e^{ip\cdot x} \,,$$ ...
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How to arrive at the Dirac Equation from Poincaré Algebra?

For the case of Galilean group, the time translation is given by the generator $H$. Hence, $$\mid\psi(t)\rangle\to \mid\psi(t+s)\rangle =e^{-iHs}\mid\psi(t)\rangle$$ Which immediately is the ...
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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?
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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|\...
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What are the actual transformation properties of Dirac spinors $u_\sigma(p)$?

Let $u_\sigma(p)$ be a Dirac spinor. As far as I know, it transforms under changes of reference frame according to $$ u_\sigma(p)=S(\Lambda)u_\sigma(\Lambda p)\tag{1} $$ where the $\sigma$ label doesn'...
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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?
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Why can't the Klein-Gordon equation explain the hydrogen atom but the Dirac equation does?

Why can't the Klein-Gordon equation with a Couloumb potential describe the hydrogen atom? Why can the first order Dirac equation explain it? What are the failures?
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$(c^{\mu}\partial_{\mu}-m)\phi(x)=0$ not Lorentz-invariant?

I have a question about the following passage on pg. 89 of Zee's QFT in a nutshell: At first sight, what Dirac wanted does not make sense. The equation is supposed to have the form "some linear ...
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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 ...
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Dirac Equation in RQM (as opposed to QFT) is written in which representation?

In introductory Quantum Mechanics treatments it is common to see the Schrödinger's equation being written, simply as: $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},t)+V(\mathbf{r})\Psi(\mathbf{r},t)=...
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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 ...
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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 ...
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Is the Cosmological constant part of or predicted by or supported by the Dirac Equation?

By Cosmological constant here I mean the constant as would be predicted by the vacuum. The one and the same that if compared to the actual expansion of the universe it is off by a factor of $10^{120}$....
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Need help with solution of the Dirac equation

$$\left(\vec\sigma \cdot \vec{p} \right)^2=\left(\vec\sigma \cdot \vec{p}\right) \left(\vec\sigma \cdot \vec{p} \right)=\vec{p} \cdot \vec{p}+\mathrm{i}\left(\vec\sigma \cdot \left[ \vec{p} \times \...
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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 ...
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Interpretation of Dirac equation states for moving electron

I try to understand a physical interpretation of the four components of the Dirac 4-spinor for a moving electron (in the simplest case, a plane wave). There is a very good question and answer about ...
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Dirac equation in curved space-time

I have seen the Dirac equation in curved space-time written as $$[i\bar{\gamma}^{\mu}\frac{\partial}{\partial x^{\mu}}-i\bar{\gamma}^{\mu}\Gamma_{\mu}-m]\psi=0 $$ This $-i\bar{\gamma}^{\mu}\Gamma_{\...
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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 ...
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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 ...