Dirac field Definition and 28 Threads

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.

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  1. Pouramat

    Green’s function of Dirac operator

    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 >...
  2. Pouramat

    Weyl Spinors Transformation, QFT1, Peskin, Chapter 3

    \begin{align} \psi_L \rightarrow (1-i \vec{\theta} . \frac{{\vec\sigma}}{2} - \vec\beta . \frac{\vec\sigma}{2}) \psi_L \\ \psi_R \rightarrow (1-i \vec{\theta} . \frac{{\vec\sigma}}{2} + \vec\beta . \frac{\vec\sigma}{2}) \psi_R \end{align} I really cannot evaluate these from boost and rotation...
  3. JD_PM

    I Understanding the wrong way to quantize the Dirac Field | Part 1

    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 $$\mathscr{L}=\bar{\psi}(x)(i\not{\!\partial}-m)\psi(x) \tag{5.1}$$...
  4. sakh1012

    A Dirac Field quantization and anti-commutator relation

    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...
  5. Q

    A Explicit form of annihilation and creation operators for Dirac field

    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} =...
  6. TAKEDA Hiroki

    I Double sided arrow notation in Dirac Field Lagrangian

    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...
  7. Nod

    A Quantized Dirac field calculations

    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...
  8. T

    How do Peskin/Schroeder derive 2-component Fierz identities?

    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...
  9. D

    What is the Hamiltonian density for a massive Dirac field?

    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...
  10. VintageGuy

    Advanced Dirac field propagator (spacelike separation)

    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 =...
  11. I

    Expectation value of a real scalar field in p state

    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>...
  12. Luca_Mantani

    What Is the General Solution to the Dirac Field Theory Equation?

    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...
  13. pellman

    Heisenberg equation of motion for the Dirac field?

    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...
  14. P

    Hermitian conjugate of Dirac field bilinear

    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?
  15. J

    Source of Dirac Field: Classical & Quantum Explanation

    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...
  16. F

    Spin of single particle state of free Dirac Field

    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}...
  17. M

    Quantum mechanics and Minimal coupling of Dirac field

    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...
  18. K

    Lagrangian, Hamiltonian and Legendre transform of Dirac field.

    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...
  19. maverick280857

    Quantized Dirac Field Interacting with a Classical Potential

    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) =...
  20. J

    Intrinsic angular momentum of Dirac field

    (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 (...
  21. maverick280857

    How Does the Charge Conjugate Dirac Field Transform in Quantum Field Theory?

    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...
  22. R

    A Dirac field can be written as two Weyl fields

    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}...
  23. maverick280857

    Lorentz Algebra in Boosts for the spin-1/2 Dirac Field

    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...
  24. P

    Integration on the way to Generating Functional for the free Dirac Field

    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...
  25. P

    Uniqueness of quantization of Dirac field

    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...
  26. C

    Are spinors just wavefunctions in the dirac field?

    are spinors just wavefunctions in the dirac field?
  27. C

    Angular Momentum vs Hamiltonian in Dirac Field Theory (Canonical)

    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...
  28. M

    Exploring the Conservation Laws of the Dirac Field

    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...
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