# 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.

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1. ### 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. ### 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. ### 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. ### 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...

13. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### Are spinors just wavefunctions in the dirac field?

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