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Derivation of the full generator of the lorentz transformations

Let us study the subgroup of the Poincare group that leaves the point $x=0$ invariant, that is the Lorentz group. The action of an infinitesimal Lorentz transformation on a field $\Phi(0)$ is $L_{\mu ...
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Five-component field

Recently I was reading about 5-component field $(\varphi , \psi_{\mu})$, for which $$ \hat {p}^{\mu} \varphi = mc\psi^{\mu}, \quad \hat {p}_{\mu}\psi^{\mu} = mc\varphi . $$ This field refers to the ...
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121 views

Reducing massive representation of the Poincare group to the massless one

I want to ask about the connection for massive and massless representation of the Poincare group. Sorry for the awkwardness. First I must to represent the formalism for both of cases. Massive ...
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68 views

Spectrum of a quantum relativistic “distance squared” operator

This question disusses the same concepts as that question (this time in quantum context). Consider a relativistic system in spacetime dimension $D$. Poincare symmetry yields the conserved charges $M$ ...
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Notation for Translation Group Generators

The generators of the translation group $T(4)$ are given below: $P_0 \equiv -i \begin{pmatrix} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 ...
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One question about Weinberg's derivation of arbitrary spin fields expressions

In his book "QFT" (vol. 1) Weinberg writes the expression for an arbitrary spin massive field: $$ \hat {\Psi}_{a}(x) = \sum_{\sigma = -s}^{s} \int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi)^{3}2 ...
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43 views

General covariance and global Poincaré algebras

Reading an article (page 7) I read this: Just as ordinary general covariance may be regarded as the local gauge symmetry corresponding to the global Poincare algebra and local gauge invariance ...
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209 views

Unitary Lorentz transformation on quantized Dirac spinor

I am stuck again on page 59 of Peskin and Schroeder. In particular, I do not know how they get equation (3.110). Let me first give some background in the way that I understand it (but I might be ...
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216 views

How does spin appear in QFT?

In QFT, as I read, it appears naturally. It is connected with Poincare algebra, doesn't it? __ As explanation of the main part of the question. Operator of relativistic orbital angular momentum ...
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47 views

Verification of the Poincare Algebra

The generators of the Poincare group $P(1;3)$ are supposed to obey the following commutation relation to be verified: $$\left[ M^{\mu\nu}, P^{\rho} \right] = i \left(g^{\nu\rho} P^{\mu} - g^{\mu\rho} ...
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51 views

Two pairs of projection operators of the Dirac equation

The Dirac equation may be interpreted as the action of projection operator $\frac{1 - \Delta}{2}\Psi = 0$, where $$ \Delta = \begin{pmatrix} 0 & \Delta_{b \dot {a}} \\ \Delta^{\dot {b}a} & 0 ...
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82 views

How to show that higher derivative theories (mostly) breaks unitarity

How to show that higher derivative theories (mostly) breaks unitarity? Spinor field $\psi_{a_{1}...a_{n}\dot {b}_{1}..\dot {b}_{m}} $, which refer to the $\left( \frac{n}{2}, \frac{m}{2} \right)$ ...
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124 views

Relations between fields transforming by Lorentz and Poincare groups

We can analyze fields transforming by the Lorentz group as $(m, n)$ representations, where $m,n$ are the max eigenvalues of two SU(2) operators, which generate the irreducible representation of the ...
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233 views

Eigenvalues of the square of Pauli-Lubanski operator

Let's have Pauli-Lunanski 4-operator: $$ \hat {W}^{\nu} = \frac{1}{2}\varepsilon^{\nu \alpha \beta \gamma}\hat {J}_{\alpha \beta}\hat {P}_{\gamma}, $$ which easy transforms to $$ \hat {W}^{\nu} = ...
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21 views

Ultrarelativistic limit of helicity values

Let's have helicity operator: $$ \hat {h} = \frac{(\hat {\mathbf S} \cdot \hat {\mathbf p})}{|\mathbf p|}. $$ For massive case spin $s$ representation it has $2s + 1$ eigenvalues $s, s - 1,..., -s$, ...