Questions tagged [poincare-symmetry]
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232
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Casimir of $SO(3)$, $SO(2)$, $IO(1,3)$, $T(4)$
It is known that $SO(3)$, a semisimple group of rank 1, has one Casimir $J^2$, and one can use this information to classify its irreps with the eigenvalues of $J^2$ and $J_3$: $(j,m)$. Now, only $j$ ...
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Question for ${\cal N}=1$ supersymmetry representations
Please see this lecture note: https://arxiv.org/abs/1011.1491.
In section "2.2.5 Massless supermultiplet"
the author defines a Casimir and says it is zero.
How can we confirm it?
We take the ...
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1answer
83 views
On Poincare group’s Casimir operators
We’ve defined Casimir operator for a group as an operator which commutes with all generators of that group. For the Poincare group we’ve found two Casimir operators: $p_\mu p^\mu$ and $W_\mu W^\mu$ ...
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Symmetry and Coordinate invariance
How are spacetime symmetries different from simple general coordinate invariance?
Physical laws should be coordinate independent. Are Poincare invariances not simply changing coordinates?
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Lorentz or restricted Lorentz group?
We say (or we observe empirically) that the laws of physics are Lorentz invariant, i.e their form does not change under transformations of the Lorentz group.
The weak interactions are not invariant ...
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138 views
Proof of Poincaré algebra with Poisson bracket
I'm trying to prove that the generators of Poincaré group in Poisson bracket close the well-known Poincaré algebra. In particular, I don't know how to prove that $\{P_\mu,P_\nu\}=0$.
Let's take an ...
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128 views
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 ...
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Poisson-Bracket representation of the Poincaré group and symmetries of dynamical systems
In canonical formalism we know that a symmetry for the dynamical system can be expressed by $\{H,f\}=0$, where $H$ is the hamiltonian of the system and $f$ is the smooth function associated to the ...
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Do 2d CFTs define healthy 4d QFTs?
When doing 2d CFTs we typically complexify coordinates and formally consider $\mathbb C^2$, with the understanding that, in the end, we are to restrict to the real slice $\bar z=z^*$. If we do not ...
asked Nov 16 '20 at 14:42
AccidentalFourierTransform
38.8k15 gold badges88 silver badges173 bronze badges
2
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1answer
39 views
Invariance of inner product under Poincare transformation
The Poincare transformation reads,
$$x\rightarrow x^\prime=\Lambda x +a $$
The scalar product is preserved under Lorentz transformation. However I do not see how it is preserved under the more general ...
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Does potential energy break Poincare invariance down to invariance under Lorentz boosts?
Landau and Lifshitz (Mechanics, Vol. 1) derive the form of the Lagrangian for a free particle by requiring invariance of action under Galilean transformation and assuming homogeneity and isotropy of ...
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Relations between the spin of representations of Lorentz group and Poincare group
It is known that
Finite dimensional irreducible representations of Lorentz group can be indexed by two half integers $(s_1,s_2)$ and the sum $s_1+s_2$ is called the spin.
Infinite dimensional unitary ...
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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", &...
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50 views
Mass as a coupling and mass as a Casimir operator
In Poincare group, we consider mass as a Casimir of the group. Hence it is a constant in various frames (I do not mean old fashion Lorentz transformation). But, in the quantum field theory mass is the ...
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Infinite dimensional representation of generators of the Lorentz group
I was reading Schwichtenberg's "Physics from Symmetry 2nd Edition". In Section 3.7.11, there is the discussion on the infinite dimensional representations.
If we consider a transformation of ...
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1answer
108 views
Why the little group of a massless spin-1 particle is ISO(2) rather than SO(2)?
As the title suggests, after going through a lot of Wikipedia, and references of books, I have learned that: The little group of a massive spin-1 particle is SO(3), while the little group of a ...
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63 views
Commutator of tetrad and Lorentz generators
While reading "Four lectures on Poincaré gauge field theory" (available at RG) the authors present a relationship between a tetrad $e^i_{\;\gamma}$ (with Latin indices coordinates, Greek ...
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Commutator of gauge Poincaré group
I'm reading along with Hehl, von der Heyde, and Kerlick's General relativity with spin and torsion on gauging the Poincaré group to include spin into GR.
They introduce the covariant derivative
$$D_\...
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47 views
Why does the Hamiltonian density transform as a scalar field and not as the zeroth component of a vector field?
Page 191 of Weinberg Vol.1. Break the Hamiltonian operator up into two parts $H=H_0+V$, where $H_0$ is the free Hamiltonian and $V$ is the interaction. Write the interaction operator $V(t)$ as a local ...
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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 ...
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0answers
163 views
Is there a formal distinction between Lorentz boosts and the others types of transformations of the Poincaré group?
The doubt arises as space translations can be associated to homogeneity of space, time translations to homogeneity of time and classical rotations to isotropy of space. These properties of space leads ...
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1answer
56 views
How many elementary particles are predicted by Wigner's classification?
My understanding is that when it comes to the correspondence between representation theory and particle physics, every irreducible representation of the Poincare group has a corresponding fundamental ...
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How is it possible for quantum fields in different irreps of the Poincaré group to interact?
In QFT, elementary particles correspond to irreps of the universal cover of the Poincaré group, and the full quantum field is then the direct sum of fields living in different irreps. So the unitary $...
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143 views
Inönü-Wigner contraction of Poincaré $\oplus$ $\mathfrak{u}$(1)
Metric = (-+++), complex $i$'s are ignored.
Using the following decompositions of the Poincaré generators,
I can write the Poincaré algebra as
I can get the Galilei algebra using the following ...
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Masses at finite density
I am reading this paper, where they make the statement:
Second, and less importantly, given our high-energy upbringing, we might be tempted to refer to an excitation’s zero-momentum gap as its “...
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1answer
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Commutativity and Associativity of Poincare Transformations
Commutativity and Associativity of Poincare Transformations:
For commutativity I showed that $2$ successive transformations does not commute with the same transformations reversed.
$$(Λ_2 Λ_1, Λ_2 ...
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1answer
100 views
Representation of the Poincaré group by means of exponential
Let $(\Lambda,a)$ be a Poincaré coordinate transformation. Let $U$ be an unitary representation of the Poincaré group on some vector space.
Is it always possible to express $U(\Lambda,a)$ in the ...
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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. ...
asked May 5 '20 at 13:26
AccidentalFourierTransform
38.8k15 gold badges88 silver badges173 bronze badges
0
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1answer
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Does $\partial^\mu X^\nu = \partial_\mu X_\nu$ (Neumann Boundary Conditions)?
Problem
I am trying to prove that the Neumann boundary condition , $\partial_\sigma X_\mu=0$ , implies that no momentum flows out of the end of an open string.
I'm told that the associated conserved ...
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1answer
627 views
Why does having a representation of the Poincaré algebra imply conservation of energy, momentum and angular momentum?
Considering that Poincaré algebra is given by the following relations
$$i[J^{\mu\nu},J^{\rho\sigma}]=\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}-\eta^{\sigma\mu}J^{\rho\nu}+\eta^{\sigma\...
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Finite Poincare transformation in momentum space?
In position space, a finite Poincare transformation changes the coordinate four-vector $x^\mu$ as
$$x^\mu\to x'^\mu=\Lambda^\mu{}_\nu x^\nu+a^\mu$$
where $\Lambda^\mu{}_\nu$ is a Lorentz ...
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1answer
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Relationship between the Klein-Gordon equation and Poincaré invariance
Derivations of the Klein-Gordon equation such as the one given by Phoenix in here, are based on studying the wave equation of the wave function of a relativistic particle. In this case, the Klein-...
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118 views
Does Fabri-Picasso theorem imply non-conservation of charge?
The vacuum expectation value of the square of the Noether charge operator $$\langle 0|\hat{Q}^2|0\rangle=\int_{\rm all space} d^3\textbf{x}\langle0|\hat{j}_0(0)\hat{Q}|0\rangle$$ diverges in case of a ...
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1answer
170 views
Commutator of the Pauli-Lubanski vector operator and the generator of translations $P^\alpha$
I'm trying to obtain the commutation relation between the Pauli-Lubanski vector operator and the generators of the Lorentz Group:
$$[W^\mu,P_\sigma]=[\frac{1}{2}\epsilon^{\mu\nu\lambda\rho} P_\nu M_\...
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2answers
83 views
Casimir operators for Poincare algebra
I have seen at various places the comment that the operator $P_\mu P^\mu$ is a Casimir operator of Lorentz algebra and thus it satisfies a on-shell condition like $P_\mu P^\mu=m^2$. Given the Poincare ...
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1answer
59 views
Proof that the commutator of angular momentum and 4-momentum is 0
I have this commutator $[P^2,J_{\mu\nu}]$ that I'm supposed to prove is zero. If we expand it (given that $[P_{\alpha}, J_{\mu\nu}] = i(g_{\mu\alpha}P_{\nu} - g_{\nu\alpha}P_{\mu})$ and $[P_{\alpha}, ...
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How to derive the general expression for eigenvalue for the square of Pauli Lubanski operator?
After some trials, I managed to get the correct eingenvalue $(\frac{-3}{4}m^2)$ for $W^2$, where $W$ is the Pauli Lubanski pseudo vector.
The expression for each $J^{\mu \nu}$ is a sum of a 4x4 ...
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1answer
44 views
Application of a Poincaré group element to a scalar function
Let $f(x)$ be a scalar function and let's say that we want to know how it transforms when it's subjected to a translation (by a vector $a^{\mu}$), rotation and a Lorentz boost. Thus we can write an ...
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Intuition for 2 commutation relations related to the Poincare group
For some commutation relations related to the Poincare group, there are intuitive way of understanding it. For example, $[J_i,H]=0$ can be understood as the conservation of angular momentum along i ...
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1answer
310 views
General relativity as a gauge theory of the Poincaré algebra
Let the Poincaré algebra be given without any factors of i as
$[P_\mu,P_\nu]=0$,
$[M_{\rho \sigma},P_\mu]=\eta_{\sigma\mu}P_\rho-\eta_{\rho\mu}P_\sigma$,
$[M_{\mu\nu},M_{\rho\sigma}]=\eta_{\nu\rho}...
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What is the problem with a generalized kinetic term as $K^{\mu\nu}(x)\partial_\mu\phi\partial_\nu\phi$?
For field theory in flat spacetime, the most general kinetic term that I can think of for a field is $$K^{\mu\nu}(x)\partial_\mu\phi\partial_\nu\phi$$ where $K^{\mu\nu}(x)$ is an arbitrary second rank ...
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48 views
Why do we use affine groups in gauge theory? What is the purpose?
When we study General Relativity in the frame of gauge theory, what's the importance of affine group?
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Poincaré Group element of pure spacetime translation
If we make a spacetime translation of the coordinates of a event $ x^{\mu}$ such that $x' ^{\mu} = x ^{\mu} + a^{\mu}$, the element $\eta _{\mu \nu} x'^\mu x' ^\nu $. Must be invariant :
\begin{...
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Can local rotations lead to a gauge theory?
I was reading about the relation between electromagnetism and the complex Klein-Gordon field. The KG field had a global $U(1)$ symmetry and upon demanding that even a local phase transformation must ...
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1answer
134 views
What is the value of $W_\mu W^\mu$ for massless particles?
What is the value of the quantity $W_\mu W^\mu$ for massless particles where $W^\mu$ is called Pauli-Lubanski vector defined as $W^\mu=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}P_\nu J_{\alpha\beta}$.
...
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Generators of $su(2) \oplus su(2)$ in the $(A, B)$ representation
Let us call the generators of $su(2)$ in the spin $A$ or spin $B$ representation $J^A_i$ and $J^B_i$ respectively. What are the generators of $su(2) \oplus su(2)$ in the $(A, B)$ representation ?
...
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Doubt in Weinberg's book on QFT
In chapter 3 of his book on QFT (volume 1), while discussing the symmetries of the S-matrix, Weinberg makes the following statement
For any proper orthochronous Lorentz transformation $x\rightarrow ...
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3answers
359 views
The role of Lorentz tranformations
My questions concern the role of Lorentz transformations in Special Relativity and General Relativity, as described in the following fragment of the series of GR lectures: https://www.youtube.com/...
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As $SL(2,\mathbb{C})$ is a double cover of the Lorentz group, is $SL(2,\mathbb{Z})$ a discrete subgroup of the Lorentz group?
The group $SL(2,\mathbb{C})$, the group of $2 \times 2$ complex matrices with determinent $1$, is a double cover of the Lorentz group. (These transformations can be understood as Mobius ...
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Commutation of position four-vector with spacetime derivatives
I am trying to understand a simple demonstration in Ashok Das' Lectures in QFT. He does the following on p. 134
$$[P_\mu,M_{\nu\lambda}]=[\partial_\mu ,x_\nu\partial_\lambda-x_\lambda\partial_\nu]=\...
