1
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1answer
73 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} ...
2
votes
1answer
101 views

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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1answer
31 views

Action of the Poincare Group on a Scalar Function

Let $F(x^\mu)$ is a scalar function; i.e. $F(x^\mu): \mathbb{R}^{1,3} \rightarrow \mathbb{R}$. How the Poincare Group $P(1,3)$ will act on it; i.e., by which formula I can calculate it for a specific ...
0
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0answers
22 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 $s, s - 1,..., -s$, ...
6
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1answer
170 views

Representations of the Poincare group

Which type of states carry the irreducible unitary representations of the Poincare group? Multi-particle states or Single-particle states?
6
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1answer
427 views

Generators of Poincare Groups

How can I determine the generators of the Poincare Group, $P(1,3)$ explicitly? Here $P(1,3)$ means a matrix Lie group.
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0answers
224 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 ...
1
vote
1answer
236 views

Trying to rhyme Peskin and Schroeder with Weinberg

This is a follow up question of this one. In the Vol 1, Weinberg derives how a unitary operator $U(\Lambda)$ acts on one-particle states, which is given by equation (2.5.2): \begin{equation} ...
1
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1answer
117 views

Question about Weinberg's derivation of a one-particle states under the Poincare group

I'm reading QFT: Vol 1 by Weinberg and I have a (perhaps trivial) question about a statement he makes on page 63. I can follow him to his derivation of equation (2.5.2): \begin{equation} P^\mu ...
1
vote
1answer
196 views

Spinor formalism in QFT

We can describe fields by two formalisms: vector and spinor. This is the result of possibility of representation of the Lorentz's group irreducible rep as straight cross product of two $SU(2)$ or two ...
1
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0answers
127 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 ...
3
votes
1answer
175 views

Connection between particles and fields and spinor representation of the Poincare group

Let's have a definition of massive particle as an irreucible representation of the Poincare group. Then, let's have a spinor field $\psi_{\alpha \alpha_{1}...\alpha_{n - 1}\dot {\beta} \dot ...
4
votes
2answers
451 views

Why do we say that irreducible representation of Poincare group represents the one-particle state?

Only because Rep is unitary, so saves positive-definite norm (for possibility density), Casimir operators of the group have eigenvalues $m^{2}$ and $m^2s(s + 1)$, so characterizes mass and spin, and ...
4
votes
2answers
267 views

Why do single particle states furnish a rep. of the inhomogeneous Lorentz group?

Following up on this question: Weinberg says In general, it may be possible by using suitable linear combinations of the $\psi_{p,\sigma}$ to choose the $\sigma$ labels in such a way that ...
0
votes
1answer
250 views

Lorentz transformation problem

In the equation (1.18) they omitted the translation vector $a^\mu$, but why?
8
votes
1answer
567 views

Identification of the state of particle types with representations of Poincare group

In the second chapter of the first volume of his books on QFT, Weinberg writes in the last paragraph of page 63: In general, it may be possible by using suitable linear combinations of the ...
3
votes
0answers
69 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 $M$ ...
8
votes
1answer
397 views

Relativistic center of mass

Recently I realized the concept of center of mass makes sense in special relativity. Maybe it's explained in the textbooks, but I missed it. However, there's a puzzle regarding the zero mass case ...
9
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2answers
1k views

Poincare group vs Galilean group

One can define the Poincare group as the group of isometries of the Minkowski space. Is its Lie algebra given either by the equations 2.4.12 to 2.4.14 (..as also given in this page - ...