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Quantization, the Orbit Method, and Unitary Representations Quantization, the orbit method, and unitary representations David Vogan Quantization, the orbit method, and Physics Representations unitary representations Orbit method Hyperbolic orbits Elliptic orbits David Vogan Department of Mathematics Massachusetts Institute of Technology Representation Theory, Geometry, and Quantization: May 28–June 1 2018 Quantization, the Outline orbit method, and unitary representations David Vogan Physics: a view from a neighboring galaxy Physics Representations Orbit method Classical representation theory Hyperbolic orbits Elliptic orbits History of the orbit method in two slides Hyperbolic coadjoint orbits for reductive groups Elliptic coadjoint orbits for reductive groups Quantization, the Quantum mechanics orbit method, and unitary representations David Vogan ! H Physical system complex Hilbert space Physics States ! lines in H Representations Orbit method Observables ! linear operators fAj g on H Hyperbolic orbits Elliptic orbits Expected value of obs A !h Av; vi Energy ! special skew-adjoint operator A0 Time evolution ! unitary group t 7! exp(tA0) Observable A conserved ! [A0; A] = 0 Moral of the story: quantum mechanics is about Hilbert spaces and Lie algebras. Quantization, the Quantum mechanics orbit method, and unitary representations David Vogan ! H Physical system complex Hilbert space Physics States ! lines in H Representations Orbit method Observables ! linear operators fAj g on H Hyperbolic orbits Elliptic orbits Expected value of obs A !h Av; vi Energy ! special skew-adjoint operator A0 Time evolution ! unitary group t 7! exp(tA0) Observable A conserved ! [A0; A] = 0 Moral of the story: quantum mechanics is about Hilbert spaces and Lie algebras. Quantization, the Quantum mechanics orbit method, and unitary representations David Vogan ! H Physical system complex Hilbert space Physics States ! lines in H Representations Orbit method Observables ! linear operators fAj g on H Hyperbolic orbits Elliptic orbits Expected value of obs A !h Av; vi Energy ! special skew-adjoint operator A0 Time evolution ! unitary group t 7! exp(tA0) Observable A conserved ! [A0; A] = 0 Moral of the story: quantum mechanics is about Hilbert spaces and Lie algebras. Quantization, the Quantum mechanics orbit method, and unitary representations David Vogan ! H Physical system complex Hilbert space Physics States ! lines in H Representations Orbit method Observables ! linear operators fAj g on H Hyperbolic orbits Elliptic orbits Expected value of obs A !h Av; vi Energy ! special skew-adjoint operator A0 Time evolution ! unitary group t 7! exp(tA0) Observable A conserved ! [A0; A] = 0 Moral of the story: quantum mechanics is about Hilbert spaces and Lie algebras. Quantization, the Quantum mechanics orbit method, and unitary representations David Vogan ! H Physical system complex Hilbert space Physics States ! lines in H Representations Orbit method Observables ! linear operators fAj g on H Hyperbolic orbits Elliptic orbits Expected value of obs A !h Av; vi Energy ! special skew-adjoint operator A0 Time evolution ! unitary group t 7! exp(tA0) Observable A conserved ! [A0; A] = 0 Moral of the story: quantum mechanics is about Hilbert spaces and Lie algebras. Quantization, the Quantum mechanics orbit method, and unitary representations David Vogan ! H Physical system complex Hilbert space Physics States ! lines in H Representations Orbit method Observables ! linear operators fAj g on H Hyperbolic orbits Elliptic orbits Expected value of obs A !h Av; vi Energy ! special skew-adjoint operator A0 Time evolution ! unitary group t 7! exp(tA0) Observable A conserved ! [A0; A] = 0 Moral of the story: quantum mechanics is about Hilbert spaces and Lie algebras. Quantization, the Quantum mechanics orbit method, and unitary representations David Vogan ! H Physical system complex Hilbert space Physics States ! lines in H Representations Orbit method Observables ! linear operators fAj g on H Hyperbolic orbits Elliptic orbits Expected value of obs A !h Av; vi Energy ! special skew-adjoint operator A0 Time evolution ! unitary group t 7! exp(tA0) Observable A conserved ! [A0; A] = 0 Moral of the story: quantum mechanics is about Hilbert spaces and Lie algebras. Quantization, the Quantum mechanics orbit method, and unitary representations David Vogan ! H Physical system complex Hilbert space Physics States ! lines in H Representations Orbit method Observables ! linear operators fAj g on H Hyperbolic orbits Elliptic orbits Expected value of obs A !h Av; vi Energy ! special skew-adjoint operator A0 Time evolution ! unitary group t 7! exp(tA0) Observable A conserved ! [A0; A] = 0 Moral of the story: quantum mechanics is about Hilbert spaces and Lie algebras. Quantization, the Unitary representations of a Lie group G orbit method, and unitary representations David Vogan Physics Unitary repn is Hilbert space Hπ with action Representations G × Hπ !Hπ; (g; v) 7! π(g)v Orbit method respecting inner product: hv; wi = hπ(g)v; π(g)wi. Hyperbolic orbits Elliptic orbits π is irreducible if has exactly two invt subspaces. Unitary dual problem: find Gbu = unitary irreps of G. X 2 Lie(G) skew-adjoint operator dπ(X): π(tX) = exp(tdπ(X)): Moral of the story: unitary representations are about Hilbert spaces and Lie algebras. Quantization, the Unitary representations of a Lie group G orbit method, and unitary representations David Vogan Physics Unitary repn is Hilbert space Hπ with action Representations G × Hπ !Hπ; (g; v) 7! π(g)v Orbit method respecting inner product: hv; wi = hπ(g)v; π(g)wi. Hyperbolic orbits Elliptic orbits π is irreducible if has exactly two invt subspaces. Unitary dual problem: find Gbu = unitary irreps of G. X 2 Lie(G) skew-adjoint operator dπ(X): π(tX) = exp(tdπ(X)): Moral of the story: unitary representations are about Hilbert spaces and Lie algebras. Quantization, the Unitary representations of a Lie group G orbit method, and unitary representations David Vogan Physics Unitary repn is Hilbert space Hπ with action Representations G × Hπ !Hπ; (g; v) 7! π(g)v Orbit method respecting inner product: hv; wi = hπ(g)v; π(g)wi. Hyperbolic orbits Elliptic orbits π is irreducible if has exactly two invt subspaces. Unitary dual problem: find Gbu = unitary irreps of G. X 2 Lie(G) skew-adjoint operator dπ(X): π(tX) = exp(tdπ(X)): Moral of the story: unitary representations are about Hilbert spaces and Lie algebras. Quantization, the Unitary representations of a Lie group G orbit method, and unitary representations David Vogan Physics Unitary repn is Hilbert space Hπ with action Representations G × Hπ !Hπ; (g; v) 7! π(g)v Orbit method respecting inner product: hv; wi = hπ(g)v; π(g)wi. Hyperbolic orbits Elliptic orbits π is irreducible if has exactly two invt subspaces. Unitary dual problem: find Gbu = unitary irreps of G. X 2 Lie(G) skew-adjoint operator dπ(X): π(tX) = exp(tdπ(X)): Moral of the story: unitary representations are about Hilbert spaces and Lie algebras. Quantization, the Unitary representations of a Lie group G orbit method, and unitary representations David Vogan Physics Unitary repn is Hilbert space Hπ with action Representations G × Hπ !Hπ; (g; v) 7! π(g)v Orbit method respecting inner product: hv; wi = hπ(g)v; π(g)wi. Hyperbolic orbits Elliptic orbits π is irreducible if has exactly two invt subspaces. Unitary dual problem: find Gbu = unitary irreps of G. X 2 Lie(G) skew-adjoint operator dπ(X): π(tX) = exp(tdπ(X)): Moral of the story: unitary representations are about Hilbert spaces and Lie algebras. Quantization, the Here’s the big idea orbit method, and unitary representations One of Kostant’s greatest contributions was David Vogan understanding the power of the analogy Physics unitary repns quantum mech systems ! Representations Hilb space, Lie alg of ops Hilb space, Lie alg of ops Orbit method Hyperbolic orbits Unitary repns are hard, but quantum mech is hard Elliptic orbits too. How does an analogy help? Physicists have a cheat sheet! There is an easier version of quantum mechanics called classical mechanics. Theories related by quantization classical mech −! quantum mech classical limit − Quantization, the Here’s the big idea orbit method, and unitary representations One of Kostant’s greatest contributions was David Vogan understanding the power of the analogy Physics unitary repns quantum mech systems ! Representations Hilb space, Lie alg of ops Hilb space, Lie alg of ops Orbit method Hyperbolic orbits Unitary repns are hard, but quantum mech is hard Elliptic orbits too. How does an analogy help? Physicists have a cheat sheet! There is an easier version of quantum mechanics called classical mechanics. Theories related by quantization classical mech −! quantum mech classical limit − Quantization, the Here’s the big idea orbit method, and unitary representations One of Kostant’s greatest contributions was David Vogan understanding the power of the analogy Physics unitary repns quantum mech systems ! Representations Hilb space, Lie alg of ops Hilb space, Lie alg of ops Orbit method Hyperbolic orbits Unitary repns are hard, but quantum mech is hard Elliptic orbits too. How does an analogy help? Physicists have a cheat sheet! There is an easier version of quantum mechanics called classical mechanics. Theories related by quantization classical mech −! quantum mech classical limit − Quantization, the Here’s the big idea orbit method, and unitary representations One of Kostant’s greatest contributions was David Vogan understanding the power of the analogy Physics unitary repns quantum mech systems ! Representations Hilb
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