Poincare algebra Definition and 17 Threads

The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group, which is of importance as a model in our understanding the most basic fundamentals of physics. For example, in one way of rigorously defining exactly what a subatomic particle is, Sheldon Lee Glashow has expressed that "Particles are at a very minimum described by irreducible representations of the Poincaré group."

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  1. redtree

    I The Pauli-Lubanski vector and angular momentum

    Given $$M_{\rho \sigma} = i (x^{\rho} \partial_{\sigma} - x^{\sigma} \partial_{\rho})$$ and $$W^{\mu} = \frac{1}{2} \epsilon^{\mu \nu \rho \sigma} P_{\nu} M_{\rho \sigma}$$ Why does ##W^{\mu}## pick up only the spin part of the total angular momentum?
  2. redtree

    I Generators of translations

    I just want to make sure I understand this correctly. For an infinite-dimensional representation, the generators of translation can be written as ##i \frac{\partial}{\partial_{\mu}}= i \partial_{\mu}##, where the generators of the Lorentz group can be written as ##i (x^{\mu}\partial_{\nu} -...
  3. nicholas_eng

    I Boost Generators: Physical Meaning & Observable Quantities

    So, all of the generators of the Poincare group are associated with pretty well-known physical quantities. Time translation is associated with energy, space translations with momentum, rotations with angular momentum, and boosts... well, boosts are generated by the "generators of boosts". Do...
  4. C

    Proving Poincare Algebra Using Differential Expression of Generator

    Using differential expressions for the generator, verify the commutator expression for ##[J_{\mu\nu},P_{\rho}]=i(\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu})## in Poincare group Generator of translation: ##P_{\rho}=-i\partial_{\rho}## Generator of rotation...
  5. LCSphysicist

    I Poincaré algebra and quotient group

    I see that the first four equations are definitions. The problem is about the dimensions of the quotient. Why does the set Kx forms a six dimensional Lie algebra?
  6. RicardoMP

    Proof of the commutator ## [P^2,P_\mu]=0 ##

    I want to make certain that my proof is correct: Since ## P^2 = P_\nu P^\nu=P^\nu P_\nu ##, then ## [P^2,P_\mu]=[P^\nu P_\nu,P_\mu]=P^\nu[P_\nu,P_\mu]+[P^\nu,P_\mu]P_\nu=[P^\nu,P_\mu]P_\nu=g^{\nu\alpha}[P_\alpha,P_\mu]P_\nu=0 ##, since ## g^{\nu\alpha} ## is just a number, I can bring it...
  7. M

    Poincare algebra and its eigenvalues for spinors

    Homework Statement Show that for $$W^\mu = -\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}M^{\nu\rho}P^{\sigma},$$ where ##M^{\mu\nu}## satisfies the commutation relations of the Lorentz group and ##\Psi## is a bispinor that transforms according to the ##(\frac{1}{2},0)\oplus(0,\frac{1}{2})##...
  8. JuanC97

    I Minimum requisite to generalize Proca action

    Hello guys, In 90% of the papers I've read about diferent ways to achieve generalizations of the Proca action I've found there's a common condition that has to be satisfied, i.e: The number of degrees of freedom allowed to be propagated by the theory has to be three at most (two if the fields...
  9. binbagsss

    Is the Expression Valid in Demonstrating a Poincare Algebra Commutator?

    Homework Statement Does ##x_p\partial_v\partial_u-x_v\partial_p\partial_u=0## Homework Equations I need this to be true to show a poincare algebra commutator. We have just shown that ##[P_u, P_v] =0 ##, i.e. simply because partial derivatives commute. Where ##P_u=\partial_u## The Attempt...
  10. B

    A 3dim Poincare Algebra - isl(2,R)

    The Poincare algebra is given by isl(2, R) ~ sl(2,R) + R^3. What exactly does the i stand for? Thanks a lot in advance!
  11. Muratani

    I Relation between Poincare matrix and electromagnetic field t

    We know that Poincare matrix which is 0 Kx Ky Kz ( -Kx 0 Jz -Jy ) describes the boost and rotation is very similar to -Ky -Jz 0 Jx...
  12. S

    Manipulating Tensor Expressions to Derive the Poincare Algebra

    Hey guys, as this is a basic QFT question, I wasn't sure to put it in the relativity or quantum section. Since this question specifically is about manipulating tensor expressions, i figured here would be appropriate. My question is about equating coefficients in tensor expressions...
  13. jfy4

    Poincare Algebra from Poisson Bracket with KG Action

    Homework Statement Consider the Klein-Gordan action. Show that the Noether charges of the Poincare Group generate the Poincare Algebra in the Poisson brackets. There will be 10 generators.Homework Equations \{ A,B \}=\frac{\delta A}{\delta \phi}\frac{\delta B}{\delta \pi}-\frac{\delta...
  14. A

    Question on the representation of Poincare algebra generators on fields

    Hi, I am working through Maggiore's QFT book and have a small problem that is really bothering me. It involves finding the representation of the Poincare algebra generators on fields. I always end up with a minus sign for my representation of a translation on fields compared to Maggiore...
  15. maverick280857

    Galilean Algebra in the low velocity limit of Poincare Algebra (Weinberg vol 1)

    Hi, Can someone please explain the following statement on page 62 of Weinberg's Vol 1 on QFT: (I understand the part for P ~ mv, so the "quote" is slightly distorted, intentionally). Also how is ? Thanks in advance!
  16. N

    Understanding the Dilemma: Differential Operators and the Poincare Algebra

    I am facing a dilemma which leaves me quite puzzled. I hope someone can straighten this out. The short version is : when we use representations of the charges as differential operators to calculate their commutators, we always get -1 times the correct result. So, doe sthat mean that we...
  17. P

    Deriving the Poincare algebra in scalar field theory

    Homework Statement Find the commutators [P^\sigma,J^{\mu \nu}] The answer is part of the Poincare algebra [P^\sigma,J^{\mu \nu}]=i(g^{\mu \sigma}P^\nu-g^{\nu \sigma}P^\mu) If someone can convince me that \partial_i T^{0\mu} = 0, (i.e. the energy-momentum tensor has no explicit spatial...
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