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2answers
94 views

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 - ...
0
votes
1answer
45 views

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_{\;...
2
votes
1answer
104 views

Could there be a pseudovector kinetic term for fermions?

Could there be a kinetic term of the form $\bar{\Psi} \gamma_5 \gamma^\mu \partial_\mu \Psi $ in addition to the usual one? Or is this forbidden by a symmetry?
1
vote
1answer
150 views

Lorentz covariant propagator

So the Feynman propagator for a Klein Gordon is manifestly Lorentz invariant clearly by looking at the momentum space representation written in terms of Lorentz scalars. But in the case of the Dirac/ ...
1
vote
0answers
68 views

Spinor Lorentz Transformation

Why should the transformation between the solutions of the Dirac equation for different inertial observers be linear?
0
votes
1answer
125 views

Why is the Lagrangian $-\partial^u\overline\Psi\partial_u\Psi-m^2\overline\Psi\Psi$ not Lorentz invariant?

Let $-\partial^u\overline\Psi\partial_u\Psi-m^2\overline\Psi\Psi$ be a Lagrangian density. Here $\Psi$ is the Dirac spinor and $\overline \Psi$ is defined to be $\Psi^\dagger \gamma^0$. It is said ...
1
vote
3answers
345 views

Mathematical proof on helicity of a massive fermion is not Lorentz invariant

What is the mathematical proof that the helicity of a massive spin-$1/2$ fermion is not Lorentz invariant? Something is Lorentz invariant (e.g., $P_\mu P^\mu$) if it commutes with all the generators ...
0
votes
0answers
65 views

Non-unitarity of finite dimensional Lorentz group and its implications

In Peskin and Schroeder, section 3.2, it is stated that Lorentz group being non-compact it does not have any finite dimensional, faithful unitary representation. But it has also been said that one ...
1
vote
0answers
183 views

A few doubts with showing Lorentz invariance of Dirac equation and probability current

Trying to understand some about Lorentz invariance and representation theory, I thought that the best way is with an example of application: Show the Lorentz invariance of the Dirac Equation $$(i \...
2
votes
2answers
227 views

Can we generate the Proca action (spin-1) from the Dirac action (spin-1/2)?

The Lagrangian for a spin-$\frac{1}{2}$ particle is the Dirac Lagrangian, while for a spin-$1$ particle is the Proca Lagrangian. But $1$ should just be $\frac{1}{2} \bigoplus \frac{1}{2}$, so is it ...
2
votes
0answers
142 views

Transformation of Weyl spinors

I usually see Weyl spinor and Weyl equations as derived from Dirac equation, like in Peskin. Now, I'm following a course where the professor actually builds Weyl spinor lagrangians BEFORE talking ...
0
votes
1answer
49 views

Dirac vs KG propagation amplitude

Can someone explain to me the physical meaning of $\bar{\psi}=\psi^\dagger\gamma^0$ in the Dirac equation? I understand it is obtained as one of the solutions of Dirac equation and it is used to build ...
1
vote
1answer
571 views

Pauli matrices and Lorentz transformations

Consider the Weyl equations: \begin{align} i\sigma^{\mu} \partial_{\mu} \psi_{L} & = 0 \\ i\overline{\sigma}^{\mu} \partial_{\mu} \psi_{R} & = 0, \end{align} where $\sigma^{\mu} = \left ( \...
3
votes
1answer
745 views

How to prove that Weyl spinors equations are Lorentz invariant? [duplicate]

The Dirac equation is given by: $[iγ^μ ∂_μ − m] ψ(x) = 0$ . We can prove that it's Lorentz invariant when: $ψ(x) \to S^{-1} \psi'(x')$ and $\partial_\mu \to \Lambda^\nu_\mu \partial'_\nu$, where ...
0
votes
1answer
400 views

Covariance of the Dirac equation

In the Ashok Das book "Lectures on quantum field theory" , it's written that in page 76 : therefore, the matrix $ S~ \gamma^0~ S^\dagger ~ \gamma^0$ must be proportional to the identity matrix (this ...
2
votes
1answer
252 views

Thinking about spin triplet and singlet states in QFT

In the case of quantum mechanics, we can think of $SU(2)$'s 2-dimensional representation, which describes spin-1/2 space. This allows us to understand the spin state the pair of spin-1/2 particles by ...
1
vote
0answers
225 views

Lorentz transformation and causality of the Dirac field

Consider the Dirac quantum field $$\psi(x) = \int \frac{d^{3}p}{(2\pi)^3} \frac{1}{\sqrt{2E_{\mathbb{p}}}} \sum_{s} [a^{s}_{\mathbb{p}}u^{s}(\mathbb{p})e^{-\mathrm{i}p \cdot x} + b^{s\dagger}_{\mathbb{...
3
votes
2answers
511 views

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'...
9
votes
1answer
648 views

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 ...
3
votes
2answers
980 views

Where does the Lorentz boost for a Dirac spinor come from?

I have read, that if you have a Dirac spinor \begin{equation} \psi = \begin{pmatrix} \phi_R\\ \phi_L \end{pmatrix} \end{equation} that you can apply a Lorentz boost along the $z$-direction with ...
2
votes
1answer
736 views

Lorentz transformations and gamma matrices

I am reading Zee's QFT in a nutshell, 2nd ed. On pg. 97 below eq. 14 he writes: $$ S \gamma^{\lambda } S^{-1} = \omega_{\,\, \mu }^{\lambda } \gamma ^{\mu }+\gamma ^{\lambda }. $$ Building up ...
0
votes
1answer
1k views

Showing Dirac equation's Lorentz invariance and use of unitary matrix $U$

Dirac equation is $i \hbar \gamma^\mu \partial_\mu \psi - m c \psi = 0 $ To show its Lorentz invariance, we convert spacetime into $x'$ and $t'$ from $x$ and $t$ and then $( iU^\dagger \gamma^\mu U\...
1
vote
0answers
28 views

How to find full energy of field of an arbitrary half-integer spin?

Let's have arbitrary half-integer spin $n + \frac{1}{2}$ representation: $$ \Psi_{\mu_{1}...\mu_{n}} = \begin{pmatrix} \psi_{a, \mu_{1}...\mu_{n}} \\ \kappa^{\dot {a}}_{\quad \mu_{1}...\mu_{n}}\end{...