What is Dirac field: Definition and 28 Discussions
In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields.
The most prominent example of a fermionic field is the Dirac field, which describes fermions with spin-1/2: electrons, protons, quarks, etc. The Dirac field can be described as either a 4-component spinor or as a pair of 2-component Weyl spinors. Spin-1/2 Majorana fermions, such as the hypothetical neutralino, can be described as either a dependent 4-component Majorana spinor or a single 2-component Weyl spinor. It is not known whether the neutrino is a Majorana fermion or a Dirac fermion; observing neutrinoless double-beta decay experimentally would settle this question.
I started from eq(3.113) and (3.114) of Peskin and merge them with upper relation for $S_F$, as following:
\begin{align}
S_F(x-y) &=
\theta(x^0-y^0)(i \partial_x +m) D(x-y) -\theta(y^0-x^0)(i \partial_x -m) D(y-x) \\
&= \theta(x^0-y^0)(i \partial_x +m) < 0| \phi(x) \phi(y)|0 >...
I've been studying Tong's beautiful chapter (pages 106-109; See also Peskin and Schroeder pages 52-58), together with his great lectures at Perimeter Institute, on how to quantize the following Dirac Lagrangian in the wrong way
$$\mathscr{L}=\bar{\psi}(x)(i\not{\!\partial}-m)\psi(x) \tag{5.1}$$...
Can anyone explain while calculating $$\left \{ \Psi, \Psi^\dagger \right \} $$, set of equation 5.4 in david tong notes lead us to
$$Σ_s Σ_r [b_p^s u^s(p)e^{ipx} b_q^r†u^r†(q)e^{-iqy}+ b_q^r †u^r†(q)e^{-iqy} b_p^s u^s(p)e^{ipx}].$$
My question is how the above mentioned terms can be written as...
I'm unclear on what exactly an annihilation or creation operator looks like in QFT. In QM these operators for the simple harmonic oscillator had an explicit form in terms of
$$
\hat{a}^\dagger = \frac{1}{\sqrt{2}}\left(- \frac{\mathrm{d}}{\mathrm{d}q} + q \right),\;\;\;\hat{a} =...
In a thesis, I found double sided arrow notation in the lagrangian of a Dirac field (lepton, quark etc) as follows.
\begin{equation}
L=\frac{1}{2}i\overline{\psi}\gamma^{\mu}\overset{\leftrightarrow}{D_{\mu}}\psi
\end{equation}
In the thesis, Double sided arrow is defined as follows...
Hi everyone!
I'm having a problem with calculating the fermionic propagator for the quantized Dirac field as in the attached pdf. The step that puzzles me is the one performed at 5.27 to get 5.28. Why can I take outside (iγ⋅∂+m) if the second term in 5.27 has (iγ⋅∂-m)? And why there's a...
On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity...
Hey guys,
So here's the deal. Consider the Lagrangian
\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi
where \bar{\psi}=\psi^{\dagger}\gamma^{0} .
I need to find the Hamiltonian density from this, using
\mathcal{H}=\pi_{i}(\partial_{0}\psi_{i})-\mathcal{L}
So I get the following...
Homework Statement
[/B]
I'm supposed to calculate the advanced propagator for the Dirac field, and I have no problem with that. Then I'm supposed to show it vanishes for spacelike separation (that is (x-y)^2<0).
Homework Equations
For the advanced propagator I get something like:
S_A =...
Hello,
I've been trying to find <p'|φ(x)|p> for a free scalar field. and integral of <p'|φ(x)φ(x)|p> over 3d in doing the space
In writing φ(x) as
In doing the first, I get the creation and annihilation operators acting on |p> giving |p+1> and |p-1> which are different from the bra state |p>...
Homework Statement
[/B]
This is an excercise that was given by my professor in a previous test:
Consider the equation:
$$
\displaystyle{\not} p
=\gamma^\mu p_\mu= m$$
where the identity matrix has been omitted in the second member.
Find its most general solution.
Homework Equations
The...
I would expect that the Heisenberg equation of motion for the Dirac field would yield the Dirac equation. Indeed, these lecture notes claim it as a fact in eq 7.7 but without proof. My trouble is that I know the anti-commutation rules for the Dirac field but I don't know how to calculate the...
In the standard QFT textbook, the Hermitian conjugate of a Dirac field bilinear
\bar\psi_1\gamma^\mu \psi_2 is \bar\psi_2\gamma^\mu \psi_1.
Here is the question, why there is not an extra minus sign coming from the anti-symmetry of fermion fields?
Classically as well as quantum-mechanically, the source of the Maxwell field is the electron/four-current (Dirac field), so the use of the Green Function propagator for the Maxwell field makes perfect sense: the Maxwell field is inhomogenous in the presence of matter.
But what about the source...
Homework Statement
Show that the state d^{\dagger}_{\alpha}(0)\mid 0\rangle describes a postrion at rest by showing that it is an eigenstate of the operators P^{\mu}, Q, J^z . Homework Equations
The Fourier expansion of \psi, \psi^{\dagger}:
\psi = \int \frac{d^3k}{(2\pi)^3} \frac{m}{k_0}...
Hi
I have a simple question:
We know from non-relativistic quantum mechanics that the spin of an electron couples only to the magnetic field, i.e. it processes around the magnetic field. How is this resolved in the relativistic context where it would seem that the spin should couple to...
In most of the physical systems, if we have a Lagrangian L(q,\dot{q}), we can define conjugate momentum p=\frac{\partial L}{\partial{\dot{q}}}, then we can obtain the Hamiltonian via Legendre transform H(p,q)=p\dot{q}-L. A important point is to write \dot{q} as a function of p.
However, for the...
Hi,
I'm working through Section 4-3 of Itzykzon and Zuber's QFT textbook, but I am a bit stuck while trying to understand some of the quantities and equations.
First of all, what is this "one-body scattering operator \mathcal{F}(A)"? It is defined (eqn 4-89, page 188) as
\mathcal{F}(A) =...
(I'm sorry about my pool English..)
I have a question about some exercise for intrinsic angular momentum part of quantized Dirac field.
S_3 = \frac{1}{2}\int d^3 x :\Psi^\dagger \Sigma_3 \Psi :
\Psi = \int \frac{d^3 k}{\left ( 2\pi \right )^3} \frac{m}{k_0}
\left ( b_\alpha \left (...
Hi,
I'm trying to work my way through Halzen and Martin's section 5.4. I'd appreciate if someone could answer the following question:
How does
j^{\mu}_{C} = -e\psi^{T}(\gamma^{\mu})^{T}\overline{\psi}^{T}
become
j^{\mu}_{C} = -(-)e\overline{\psi}\gamma^{\mu}\psi
? Is there some...
A Dirac field can be written as two Weyl fields stacked on top of each other: \Psi= \left( \begin{array}{cc} \psi \\ \zeta^{\dagger} \end{array}\right) , where the particle field is \psi and the antiparticle field is \zeta.
So a term like P_L\Psi=.5(1-\gamma^5)\Psi=\left( \begin{array}{cc}...
Hi,
What is the origin of the following commutation relation in Lorentz Algebra:
[J^{\mu\nu}, J^{\alpha\beta}] = i(g^{\nu\alpha}J^{\mu\beta}-g^{\mu\alpha}J^{\nu\beta}-g^{\nu\beta}J^{\mu\alpha}+g^{\mu\beta}J^{\nu\alpha})
This looks a whole lot similar to the commutation algebra of...
Hi,
if I want to calculate the generating functional for the free Dirac Field, I have to evaluate a general Gaussian Grassmann integral. The Matrix in the argument of the exponential function is (according to a book) given by:
I don't understand the comment with the minus-sign and the...
Let's have a theory involving Dirac field \psi. This theory is decribed by some Lagrangian density \mathcal{L}(\psi,\partial_\mu\psi). Taking \psi as the canonical dynamical variable, its conjugate momentum is defined as
\pi=\frac{\partial\mathcal{L}}{\partial(\partial_0\psi)}
Than the...
I need some suggestions and/or corrections if I understand this correct? My questions are based on the book by Mandl and Shaw.
Conserved currents are based on Noethers theorem and directly connected to spacetime and field transformations (rotations, translations, phase, ...). One can...
I have a question about the Dirac field. If as quantum field theory states , every point in the Universe is filled with "virtual" photons , and if these "virtual" photons in turn give rise to electron-positron pairs , which being components of matter and anti-matter collide and annihilate each...