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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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Why are twistors commuting?

In his book, Srednicki introduces the notion of twistor in chapter 50. It is described as a simply commuting spinor, as opposed to anti-commuting. How do we know that this object is simply commuting? ...
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How does spin influence the dynamics of quantum mechanical systems?

I have just been introduced to the Klein-Gordon Equation and the Dirac Equation for the first time. The way they were explained to me, these equations govern the (relativistic) evolution of spin-0 and ...
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What is the correct form of Dirac equation?

Usually the Dirac equation in curved space is written as $$i\Gamma^{\mu}D _{\mu}\Psi-m\Psi=0,$$ where $\Gamma_{\mu}$ are curved space gamma matrices and $D_{\mu}$ is covariante derivative. This ...
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How did we know that the Dirac equation describes the electron but not the proton?

I'm suddenly getting confused on what should be a very simple point. Recall that the $g$-factor of a particle is defined as $$\mu = \frac{ge}{2m} L$$ where $L$ is the spin angular momentum. For any ...
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$CP$-transformation for spinor field. $C$ and $P$ do not commute?

I am bothered by an exercise about CP transformations where I get the result that CP acting on a Dirac spinor field is not the same as the PC transformation. The exercise states the following ...
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Dirac equation boundary conditions

In Schroedinger equation, which is second order differential equation, one normally, equates both $\psi(x)$ and $\psi'(x)$ across the boundary, as boundary conditions. However, the dirac equation ...
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How can we interpret the components of a polarization four-vector?

The four components of a Dirac spinor can be interpreted in terms of left-chiral and right-chiral spin up and spin down states. How can we interpret the four components of a polarization four vector $...
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Commutation relation for Dirac field

In "Quantum Field Theory" by Peskin and Schroeder, I couldn't understand the commutation relation calculation for Dirac field (pg. 53): $$ \begin{align} \psi(x) &= \int \frac{d^3p}{(2\pi)^3} ...
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Is there a coordinate-free Dirac equation?

Dirac equation is always written with indices. Is there any way to write it down without any indices ABSTRACT or not, and without coordinates,basis vectors etc..?
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Can we derive $\vec{S}=\frac{1}{2}\vec{\Sigma}$ in a representation independent way in terms $\vec{\alpha},\vec{\beta}$?

For the Hamiltonian $H=(\vec{\alpha}\cdot \vec{p}+\vec{\beta}m)$ of the Dirac equation $i\frac{\partial \psi}{\partial t}=H\psi$, it can be shown that $[H,\vec{L}]=-i(\vec{\alpha}\times\vec{p})$. Now, ...
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Electron spin g-factor

I am reading Schwartz's book on Quantum field theory. In the chapter anomalous magnetic moment (chapter 17) he mentions that Dirac equation naturally implies that the electron spin $g$-factor is 2. ...
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Some counting of field degrees of freedom for a classical spin-1/2 Dirac field

A classical real scalar field admits a decomposition $$\phi(x)\sim a_pe^{-ip\cdot x}+a_p^*e^{+ip\cdot x}$$ which tells that at each $x$, there exists a real number i.e., one degree of freedom at each ...
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Why can we use the equation of motion to calculate the amplitude in “Quantum Field Theory”?

I am reading the chapter on electron-proton scattering from "Quantum Field Theory in a Nutshell". The author calculates the amplitude of the electron-proton scattering (up to the second order). The ...
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Lorentz invariant probability from the Dirac equation

My question is regarding a proof given in Greiner's "Relatavistic Quantum Mechanics", 3rd Edition textbook. On pg 148, he proves that the current density $j^{\nu}(x)$ is invariant to a Lorentz ...
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Dirac matrices for generalized metric tensors

The Dirac matrices are defined by the relations $$\left [\gamma^{i},\gamma^{j}\right]_{+}=2\eta^{i,j}\mathbb{1}$$ where $[\cdot,\cdot]_{+}$ is the anti-commutator. What happens if I replace $\eta^{i,...
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Why does the majorana equation preserve handedness?

In the "QFT Nutshell" by A. Zee, it is stated that The Majorana equation is $$i\not\partial\psi=m\psi_c$$ where $\psi_c$ is the charge conjugated spinor $\psi_c = \left(C\gamma^0\right)\psi^*$....
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Why does the charge conjugation of the spinor transform as a spinor?

I have come across (in QFT Nutshell, A. Zee) how the charge conjugation of the spinor, $\psi_c \equiv \gamma^2 \psi^*$, transform (where $\gamma^2=\sigma^2\otimes i\tau^2$ is the component of the ...
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A question about Lorentz invariant argument when writing down the Dirac equation [duplicate]

According to the chapter II.1 in "Quantum Field Theory in a Nutshell" by A. Zee, Dirac was trying to write down the relativistic wave equation linear in spacetime derivative. The author stated that ...
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Limit on speed of expansion of the bounded support interval of a position wave function in relativistic quantum mechanics

If the support of a quantum mechanical position wave function is a bounded interval, and that interval is expanding or contracting, then I think it cannot change in any direction faster than $c$. To ...
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Commutation relations in QFT [duplicate]

So I have just started learning QFT. So you take a classical field and turn the degrees of freedom into operators. All fine, just like normal quantum. However I am confused about the commutation ...
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Relativistic quantum field theory

Let $\psi(x)$ be solution of Dirac equation $$ (\gamma^\mu\Pi_\mu-mc) \psi(x)=0 $$ where $\Pi_\mu=i\partial_\mu-eA_\mu$ is momentum operator in present electromagnetic field . We consider tow ...
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Did Dirac derive the correct equation for the wrong reasons? [closed]

Did Dirac derive the correct equation for the wrong reasons? This is a question about the historical discovery of the Dirac equation and how it was deduced. Looking back at that discovery with our ...
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Removing “electric dipole moment” from non-relativistic Dirac equation

I have found many sources (c.f. Schwartz's QFT book section 10.4) that try to obtain the non-relativistic limit of the Dirac equation by first "squaring it" so that it looks somewhat like the Klein-...
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Is total mass conserved for free Dirac fermions?

I am studying quantum field theory and stumbled across the following problem: Is the total mass conserved for free Dirac fermions? I.e., does the total mass operator commute with the Dirac ...
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Why would a spinor transform under Lorentz transformations?

From my understanding of spinors, they arise as projective representations of $SO_0(1,3)$ that do not correspond to representations of $SO_0(1,3)$. But still one says here - and virtually everywhere - ...
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Non-relativistic limit of the coupled Dirac equation

In Greiner's relativistic quantum mechanics textbook he has a derivation of the Pauli equation as a nonrelativistic limit of the coupled Dirac equation. Just below Eq. (2.81) he makes the following ...
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Parity operator expression in relativistic quantum mechanics

I was reading Schwabl's Advanced quantum mechanics. In that book it is written in the Spatial reflection part that the parity operator is $P=e^{i\phi}\gamma^0$.But after some lines it is written as $P=...
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Does the Schrodinger Equation care about spin?

I have taken the non-relativistic limit of the Klein-Gordon and Dirac equation, and both have brought me to the Schrodinger equation. The Klein-Gordon equation describes spin 0 particles, and the ...
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Dirac equation derivation

I am working through a set of lecture notes containing a derivation of the Dirac equation following the historical route of Dirac. It states that Dirac postulated a hermitian first-order differential ...
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What “manifold in band parameters ” means?

I was reading an article https://arxiv.org/abs/0907.0500 in which they write about manifold in band parameter ,like in first line in my picture , and then they call it band parameter . can some ...
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Why one can swap the product of a Lorentz transformation and a Dirac $\gamma^\mu$ matrix?

Ashok tries to prove Lorentz invariance of the Dirac equation. If the spinor follows the transformation rule $\Psi' = S\Psi$, then $$ (i\gamma^\mu\partial_\mu-m)\Psi = 0\to (i\gamma^\mu\Lambda^\nu_{\;...
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Dirac spinor as null vectors

In this paper, on page 9, the authors show that a spinor is equivalent to a null vector with a bit of extra structure (just one real parameter I think?): https://arxiv.org/abs/1312.3824 They then go ...
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Dirac equation in curved spacetime

As we know, the laws of physics in curved spacetime are obtained to lowest order by upgrading the flat space laws by substituting partial derivatives with the appropriate covariant derivatives. In the ...
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Solutions of relativistic wave equations compared to classical wave functions

In classical quantum mechanics, absolute square of the wave function (i.e. $|\psi|²$) means probability density of particle's location, so when we integrate this over certain volume we get the ...
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Why is Dirac equation a matrix equation?

According to Wikipedia's Dirac equation article, the Dirac equation can be written in form $$ i\hbar\gamma^{\mu}\partial_{\mu}\psi-mc\psi=0, $$ where $\gamma^{\mu}$ are gamma matrices which are $4 \...
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Demonstration of identities appearing in Dirac spinors in the chiral representation

Using the chiral representation of the gamma matrices, Peskin and Schroeder arrive in some expressions for the 4-component spinors $u(p)$ and $v(p)$ in terms of a square root of the Pauli matrices ...
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Dirac spinor for arbitrary momentum

In many treatments of the Dirac equation (e.g. Peskin and Schroder, pages 45-46) after subbing in $\psi(x) = e^{-ix_\mu p^\mu}u(\vec p)$, with $u$ a constant spinor, into the Dirac equation, we ...
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Hamilton's equations for Dirac Hamiltonian [duplicate]

The Dirac Lagrangian $$\mathcal{L} = i\bar{\psi}\gamma^{\mu}\partial_\mu \psi - m \bar{\psi}\psi$$ gives a Hamiltonian $$\mathcal{H}(\Pi,\bar{\Pi},\psi,\bar{\psi})=\Pi \dot{\psi}-\mathcal{L}=-\bar{\...
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Why does the Dirac equation work for the hydrogen atom?

The Dirac equation works well for predicting the spectrum of the hydrogen atom, famously incorporating relativistic effects like fine structure. Yet, there seems to be a sense in which this is ...
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Legal values of spin-1/2 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, .. (Grassmann)?

For the spin-1/2 fermion field $\psi$, we may choose it to be a spinor which needs to be Grassmann variable but can also be complex $\mathbb{C}$ Grassmann. (Dirac or Weyl spinor/fermion) We can ...
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single chirality electron and photon interaction

I asked a similar question about QED Lagrangian but I guess the question wasn't clear enough since I didn't get any correct answers. So, I'll try to ask the question in a different way: how does one ...
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QED Lagrangian in terms of Weyl spinors

Let's say the electron field can be written in term of its left and right handed Weyl spinors: \begin{equation} \psi_{e} = \begin{pmatrix} \chi \\ \eta^{\dagger} \end{pmatrix} \end{equation} In ...
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Varying the Dirac action with differential forms

The Dirac action in a curved spacetime can be written in terms of the vierbein $\{ e^a \}$ and spin connection $\{ \omega^{ab} \}$ differential forms. Let the spinor field $\psi$ be interpreted as a ...
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From relativistic equation to find Dirac matrices

Is this possible and then how? $$((\gamma \otimes \mathbf\sigma)\bullet\mathbf p)(\gamma^\prime\otimes\mathbf 1_2) = \gamma\gamma^\prime\otimes\sigma \bullet \mathbf p $$ where $\gamma$ and $\gamma^\...
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Negative probability distribution function for Dirac equation

People say that the probability density function of the continuity equation for the Dirac equation is definite positive. I wanted to see it myself and followed the same path as Bjorken & Drell's ...
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Derivation of Klein Gordon equation from Dirac equation; what does it mean?

In Dirac field (Peskin and Schroeder), there is one equation in which it multiples the Dirac operator $$(-i\gamma^{\mu}\partial_{\mu}-m )$$ by $$(i\gamma^{\nu}\partial_{\nu}-m ),$$ obtaining $\...
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Time evolution using the Dirac equation

In non-relativistic qantum mechanics, the energy eigenstates (i.e.e eigensattes of the hamiltonian) evolve in phase according to their eigenenergies $\phi_(t) = e^{-iE_nt}\phi_n(0)$ using natural ...
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Second order relativistic corrections to Pauli equation from Dirac equation

I'm trying to derive the full and correct Hamiltonian for spin$\frac{1}{2}$ particles from Dirac equation up to second order in $v/c$. For a potential and magnetic field constant in time. In ...
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Conservation of $\int |\psi|^2$ for Dirac wave

When $\psi$ be Schrodinger wave $\int |\psi|^2$ is conserved even when this wave interact whit another wave say electromagnetic wave. and this is very necessary for one particle interpretation of this ...
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Microcausality for Dirac's current

I`m supposed to show as an exercises that for the Dirac field's associated current: $$j^\mu=\bar{\Psi}\gamma^\mu\Psi$$ The microcausality relation holds: $$ [j^\mu(x),j^\nu(y)]=0 \text{ for } (x-y)^...