Questions tagged [poincare-symmetry]
335 questions
60
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Why are particles thought of as irreducible representations, in plain English?
I'm a PhD student in mathematics and I have no problem in understanding what irreducible representation are. I mean that the mathematical side is not a particular problem. Nevertheless I have some problems in understanding why and in which sense…
Dac0
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32
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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
It is the representation of the global group of…
user8817
28
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3 answers
What does it mean for particles to "be" the irreducible unitary representations of the Poincare group?
I am studying QFT. My question is as the title says. I have read Weinberg and Schwartz about this topic and I am still confused. I do understand the meanings of the words "Poincaré group", "representation", "unitary", and "irreducible",…
UrsaCalli79
- 633
28
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4 answers
What is CPT, really?
The naive statement for the "CPT theorem" one usually finds in the literature is "relativistic theories should be CPT invariant". It is clear that this statement is not true as written, e.g. topological theories are typically not invariant under…
AccidentalFourierTransform
- 56,647
20
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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 $\Psi_{p,\sigma}$ to choose the $\sigma$ labels in such a…
user4235
19
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1 answer
Do the Wightman axioms uniquely fix the representation of the Poincaré group on the one-particle states given the representation on the fields?
Let $P := \mathrm{SL}(2,\mathbb{C})\ltimes \mathbb{R}^4$ be the universal cover of the connected component of the identity of the Poincaré group.
Given a classical field $\phi : \mathbb{R}^{1,3}\to V$ where $V$ carries a finite-dimensional…
ACuriousMind
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Can Poincare representations be embedded in non-standard Lorentz representations?
My impression for how Poincare and Lorentz representations are linked in $3+1$ dimensions is:
Assuming positive mass for simplicity, irreducible representations of the Poincare group are indexed by their mass $M$ and spin $s$.
Irreducible…
knzhou
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3 answers
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 - https://en.wikipedia.org/wiki/Poincar%C3%A9_group) or equations…
Student
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17
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What is a general definition of the spin of a particle?
In quantum field theory, one defines a particle as a unitary irreducible representations of the Poincaré group. The study of these representations allows to define the mass and the spin of the particle. However, the spin is not defined the same way…
Georg Sievelson
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Irreducible Representations Of Lorentz Group
In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64.
He define states as $P^{\mu} |p,\sigma\rangle = p^{\mu} |p,\sigma\rangle…
hans
- 163
16
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1 answer
Klein-Gordon inner product
Studying the scalar field and Klein-Gordon equation in quantum field theory I came across this definition for the inner product in the space of the solutions of the K.G. equation:
$$\langle \Phi_1 | \Phi_2 \rangle = i\int \mathrm{d}\vec{x}(\Phi_1 ^*…
justmyfault
- 271
15
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Rank of the Poincare group
There are two Casimirs of the Poincare group:
$$ C_1 = P^\mu P_\mu, \quad C_2 = W^\mu W_\mu $$
with the Pauli-Lubanski vector $W_\mu$. This implies the Poincare group has rank 2.
Is there a way to show that there really are no other Casimir…
Neuneck
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14
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3 answers
In what sense is SUSY a spacetime symmetry?
Clearly the SUSY anti-commutation relations involve momentum, and thus the generator of translations in spacetime:
$$\{ Q_\alpha, \bar{Q}_{\dot{\beta}} \} = 2 (\sigma^\mu)_{\alpha \dot{\beta}} P_\mu . $$
So I would say that naively SUSY has…
DJBunk
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12
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Are particles in curved spacetime still classified by irreducible representations of the Poincare group?
For QFT in Minkowski space, the usual story is that particles lie in irreps of the Poincare group. Wigner's classification labels particles by their momentum and by their transformation properties under the little group, which comprises the Poincare…
soundofthepolice
- 595
12
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Which particles do not fit into Wigner's picture?
In his accepted and highly upvoted answer to Why particles are thought as irreducible representation in plain English? @Valter Moretti finishes his ADDENDUM with "Finally not all particles fit into Wigner's picture".
Although @Kai subsequently…
iSeeker
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