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Physics

Representation theory of the Poincare group

• Poincaré group is the group of Minkowski spacetime isometries. A Minkowski spacetime isometry has the property that the interval between events is left invariant.
• The representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics.
• The space of physical states is typically a representation of the Poincaré group. (More generally, it may be a projective representation, which amounts to a representation of the double cover of the group.)
• In a classical field theory, the physical states are sections of a Poincaré-equivariant vector bundle over Minkowski space. The equivariance condition means that the group acts on the total space of the vector bundle, and the projection to Minkowski space is an equivariant map. Therefore, the Poincaré group also acts on the space of sections. Representations arising in this way (and their subquotients) are called covariant field representations, and are not usually unitary. For a discussion of such unitary representations, see Wigner's classification.
• Quantum field theory is the relativistic extension of quantum mechanics, where relativistic (Lorentz/Poincaré invariant) wave equations are solved, "quantized", and act on a Hilbert space composed of Fock states; eigenstates of the theory's Hamiltonian which are states with a definite number of particles with individual 4-momentum.

Poincare group

• 4+6=10 AdS/CFT correspondence There are field theories in four dimensions which at strong coupling become quantum gravity theories in ten dimensions! The strong coupling effects cause the excitations to act as if they’re gravitons moving in higher dimensions. This is quite extraordinary and still poorly understood.
• 4+6 The conservation laws for 4-momentum {$p_a$} and 6-angular momentum {$M^{ab}$} arise respectively from the 4 translational symmetries and 6 (Lorentz) rotational symmetries of Minkowski space. The tensor quantity {$M^{ab}$} describes angular momentum in special relativity. This is what I need to quantize. And {$M^{ab}=x^ap^b-x^bp^a$} relates six pairs of four dimensions, including time. Compare with pairs of levels as given by the six qualities of signs. Thus "the thing" is given by time and the dimensions of space are simply signs of time of three different kinds: icon, index, symbol.

Poincare group - how does it relate to structure preservation

• orthogonal - distances, center
• conformal - angles
• symplectic - oriented area
• Heisenberg group (with terms like px-xp) has no finite dimensional representation, needs infinite dimensional. Similarly with the Poincare group.
• How are Kravchuk polynomials related to the Poincare group and the Heisenberg group and how are they not a finite dimensional representation? Is it because they are shifting?
• Quantum field theory is the field theory which is symmetric under the Poincare group. To be symmetric need to be unitary or anti-unitary transformation.
• Representation theory of the Lorentz group
• Particle physics and representation theory
• The different quantum states of an elementary particle give rise to an irreducible representation of the Poincaré group.
• The properties of the various particles, including their spectra, can be related to representations of Lie algebras, corresponding to "approximate symmetries" of the universe.
• OP1 and Lorentzian geometry
• The algebras {$\mathbb{R},\mathbb{C},\mathbb{H}$} and {$\mathbb{O}$} are related to Lorentzian geometry in 3, 4, 6, and 10 dimensions
• Super-Poincaré algebra. A related observation is that the representations of the Lorentz group include a pair of inequivalent two-dimensional complex spinor representations 2 {\displaystyle 2} 2 and 2 ¯ {\displaystyle {\overline {2}}} {\overline {2}} whose tensor product 2 ⊗ 2 ¯ = 3 ⊕ 1 {\displaystyle 2\otimes {\overline {2}}=3\oplus 1} {\displaystyle 2\otimes {\overline {2}}=3\oplus 1} is the adjoint representation. The experimental issue can roughly be stated as the question: if we live in the adjoint representation (Minkowski spacetime), then where is the fundamental representation hiding?
• https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group
  • Richard Brauer was 1935–38 largely responsible for the development of the Weyl-Brauer matrices describing how spin representations of the Lorentz Lie algebra can be embedded in Clifford algebras.
• https://en.wikipedia.org/wiki/Representation_theory_of_the_Poincar%C3%A9_group
• Understand the relationship between the Lorentz group and SL(2,C).
• https://en.m.wikipedia.org/wiki/Lorentz_group

Because the restricted Lorentz group SO+(1, 3) is isomorphic to the Möbius group PSL(2,C), its conjugacy classes also fall into five classes:

• Elliptic transformations
• Hyperbolic transformations
• Loxodromic transformations
• Parabolic transformations
• The trivial identity transformation
• In the article on Möbius transformations, it is explained how this classification arises by considering the fixed points of Möbius transformations in their action on the Riemann sphere, which corresponds here to null eigenspaces of restricted Lorentz transformations in their action on Minkowski spacetime. An example of each type is given in the subsections below, along with the effect of the one-parameter subgroup it generates (e.g., on the appearance of the night sky).