Poincaré gauge theory and its deformed Lie algebra mass-spin classification of elementary particles
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1 Poincaré gauge theory and its deformed Lie algebra mass-spin classification of elementary particles Jens Boos Perimeter Institute for Theoretical Physics Friday, Dec 4, 2015 PSI Student Seminar Perimeter Institute for Theoretical Physics
2 Introduction and outline Question: What do quantum field theory and gravity have in common? Quantum field theory: Gravity: states in a Hilbert space, field operators on Minkowski space dynamic spacetime fields on curved background 1/8
3 Introduction and outline Question: What do quantum field theory and gravity have in common? Quantum field theory: Gravity: states in a Hilbert space, field operators on Minkowski space dynamic spacetime fields on curved background but both are intrinsically tied to the Poincaré group 1/8
4 Introduction and outline Question: What do quantum field theory and gravity have in common? Quantum field theory: Gravity: states in a Hilbert space, field operators on Minkowski space dynamic spacetime fields on curved background but both are intrinsically tied to the Poincaré group Outline: 0. The Poincaré group 1. Quantum field theory 2. Poincaré gauge theory of gravity 3. Putting it all together 1/8
5 0. The Poincaré group Isometry group of Minkowski space. Noether's theorem predicts conserved energymomentum, angular momentum, and orbital angular momentum. Poincaré group = {n translations} { Lorentz transformations} 2/8
6 0. The Poincaré group Isometry group of Minkowski space. Noether's theorem predicts conserved energymomentum, angular momentum, and orbital angular momentum. Poincaré group = {n translations} { Lorentz transformations} Lie algebra: semidirect product 2/8
7 0. The Poincaré group Isometry group of Minkowski space. Noether's theorem predicts conserved energymomentum, angular momentum, and orbital angular momentum. Poincaré group = {n translations} { Lorentz transformations} Lie algebra: semidirect product Casimir operators: values are independent of representation (scalars) Pauli Lubanski pseudovector (in 4D only): 2/8
8 1. QFT: Wigner's little group Why is the Poincaré group important in particle physics? Wigner (1939): it allows us to invariantly classify one-particle states (momentum eigenvalues given by ) How do we find additional good quantum numbers (little group procedure)? 3/8
9 1. QFT: Wigner's little group Why is the Poincaré group important in particle physics? Wigner (1939): it allows us to invariantly classify one-particle states (momentum eigenvalues given by ) How do we find additional good quantum numbers (little group procedure)? 1. Lorentz transformed momentum unitary transformation on quantum state: 3/8
10 1. QFT: Wigner's little group Why is the Poincaré group important in particle physics? Wigner (1939): it allows us to invariantly classify one-particle states (momentum eigenvalues given by ) How do we find additional good quantum numbers (little group procedure)? 1. Lorentz transformed momentum unitary transformation on quantum state: 2. Find Wigner little group elements W with, k = reference vector 3/8
11 1. QFT: Wigner's little group Why is the Poincaré group important in particle physics? Wigner (1939): it allows us to invariantly classify one-particle states (momentum eigenvalues given by ) How do we find additional good quantum numbers (little group procedure)? 1. Lorentz transformed momentum unitary transformation on quantum state: 2. Find Wigner little group elements W with, k = reference vector 3. Find invariants quantities ( Casimir operators ) of W 3/8
12 1. QFT: Wigner's little group Why is the Poincaré group important in particle physics? Wigner (1939): it allows us to invariantly classify one-particle states (momentum eigenvalues given by ) How do we find additional good quantum numbers (little group procedure)? 1. Lorentz transformed momentum unitary transformation on quantum state: 2. Find Wigner little group elements W with, k = reference vector 3. Find invariants quantities ( Casimir operators ) of W Example: p² = -m² (massive particle): W = SO(3) spin as quantum number p² = 0 (massless particle): W = SE(2) helicity as quantum number (or continuous spin particles) 3/8
13 1. QFT: Relativistic wave equations Suitable real-space representation of the Poincaré algebra: How is that related to relativistic wave equations? 4/8
14 1. QFT: Relativistic wave equations Suitable real-space representation of the Poincaré algebra: How is that related to relativistic wave equations? Klein Gordon equation as an eigenvalue equation: can be generalized for arbitrary spin values using Wigner's little group technique 4/8
15 1. QFT: Relativistic wave equations Suitable real-space representation of the Poincaré algebra: How is that related to relativistic wave equations? Klein Gordon equation as an eigenvalue equation: can be generalized for arbitrary spin values using Wigner's little group technique How can that all be related to gravity? Poincaré group is the isometry group of Minkowski space cannot simply be extended to a curved background let us consider Poincaré gauge theory as a viable theory of gravity 4/8
16 2. Poincaré gauge theory of gravity (1/2) Newton successfully wrote apple = moon, but you cannot write apple = neutron. J. L. Synge Consider a matter field (e.g. Dirac spinor) on Minkowski background: 5/8
17 2. Poincaré gauge theory of gravity (1/2) Newton successfully wrote apple = moon, but you cannot write apple = neutron. J. L. Synge Consider a matter field (e.g. Dirac spinor) on Minkowski background: Make this theory invariant under local Poincaré transformations: 5/8
18 2. Poincaré gauge theory of gravity (1/2) Newton successfully wrote apple = moon, but you cannot write apple = neutron. J. L. Synge Consider a matter field (e.g. Dirac spinor) on Minkowski background: Make this theory invariant under local Poincaré transformations: Compared to Minkowski space, we are forced to introduce gauge potentials: = 4 translational gauge potentials = 6 rotational gauge potentials 5/8
19 2. Poincaré gauge theory of gravity (2/2) Field strengths related to the potentials and : = translational curvature = rotational curvature 6/8
20 2. Poincaré gauge theory of gravity (2/2) Field strengths related to the potentials and : = translational curvature = rotational curvature This class of theories indeed describes gravity (e.g. Einstein Cartan theory) and includes General Relativity in the limit with the Lagrangian. 6/8
21 2. Poincaré gauge theory of gravity (2/2) Field strengths related to the potentials and : = translational curvature = rotational curvature This class of theories indeed describes gravity (e.g. Einstein Cartan theory) and includes General Relativity in the limit with the Lagrangian. Translation into differential geometry: Latin indices coordinate indices, Greek indices orthonormal frame indices gauge potentials and correspond to tetrad and connection gauge curvatures and correspond to torsion and curvature 6/8
22 2. Deformed Lie algebra Field strengths related to the potentials and : = translational curvature = rotational curvature Note that the Poincaré transformations act on space itself (not some internal space): 7/8
23 2. Deformed Lie algebra Field strengths related to the potentials and : = translational curvature = rotational curvature Note that the Poincaré transformations act on space itself (not some internal space): This deformed Lie algebra is unique to external gauge theories. 7/8
24 3. Putting it all together? Deformed Lie algebra: Questions: What is the resulting group? How do the Casimir operators look like? What does that tell us about particles on a curved background?... Thank you for your attention. 8/8
25 References [1] F. W. Hehl, P. von der Heyde, G. D. Kerlick and J. M. Nester, "General Relativity with Spin and Torsion: Foundations and Prospects," Rev. Mod. Phys. 48 (1976) 393. [2] F. W. Hehl, Four Lectures on Poincaré Gauge Field Theory, Erice, [3] E. P. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group," Annals Math. 40 (1939) 149 [Nucl. Phys. Proc. Suppl. 6 (1989) 9]. [4] W. K. Tung, "Relativistic Wave Equations and Field Theory for Arbitrary Spin," Phys. Rev. 156 (1967) [5] X. Bekaert and N. Boulanger, "The Unitary representations of the Poincaré group in any spacetime dimension," arxiv:hep-th/ i/ii
26 Appendix: conserved Noether currents for the Poincaré group Conserved Noether currents: energy-momentum: total angular momentum: Note: the semidirect product structure is everywhere. ii/ii
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