Cluster Properties and Relativistic Quantum Mechanics Wayne Polyzou [email protected] The University of Iowa Cluster Properties – p.1/45 Why is quantum field theory difficult? number of degrees of freedom. • 1 No rigorous ab-initio error bounds. • Fock-space generators not generally densely defined • H = d3xT 00(x) H2 = Z h j j i 1 regularization destroys commutation relations. No few-body problem. • Cluster Properties – p.2/45 Why is non-relativistic few-body quantum mechanics successful ? Meaningful few-body problem directly related to • experiment. Solutions with ab-initio error bounds possible. • Cluster properties relate the few- and many-body • problem. Cluster Properties – p.3/45 Relativistic few-body quantum mechanics ? Determine degrees of freedom by experiment. • Construct the most general class quantum models with • desired degrees of freedom and exact Poincaré symmetry. Constrain few-body dynamics by experiment. • Use cluster properties to relate few and many-body • problem. Cluster Properties – p.4/45 It is possible? • (Weinberg) “There have been many attempts to formulate a relativistically invariant theory that would not be a local field theory, and it is indeed possible to construct theories that are not field theories and yet yield a Lorentz invariant S-matrix for two- particle scattering, but such efforts have always run into trouble in sectors with more than two particles; either the three particle S-matrix is not Lorentz invariant, or else it violates the cluster decomposition principle.” • (Schroer) An anthology of non-local QFT and QFT on noncommutative spacetime, Bert Schroer, hep-th/0405105 “ This raised the question whether there exist consistent relativistic unitary and macro-causal particle theories are all. · · · In particular these theories fulfill the very non-trivial cluster separability properties of the associated Poincaré invariant unitary S-matrix; as a result their existence contradicts a dictum (which is ascribed to S. Weinberg) saying that a Poincaré invariant unitary S-matrix with these properties is characteristic for a (local) QFT.” Cluster Properties – p.5/45 Relativity and Quantum Theory • Inertial coordinate systems are coordinate systems where free particles move with constant velocity. • Experiments on isolated systems cannot make an absolute determination of inertial coordinate system. Cluster Properties – p.6/45 Relativity • Because solutions of the Schrödinger equation are not observable in quantum theory, relativity does not require that solutions of the Schrödinger equation transform covariantly. • This is an important difference between the classical and quantum formulation of relativistic invariance. Cluster Properties – p.7/45 Relativistic Invariance Galilean Relativity: X and X0 inertial • ) ∆~x = ∆~x 0 j ijj j ij j ∆tij = ∆tij0 Special Relativity: X and X0 inertial • ) 2 2 2 2 2 2 ∆~x c ∆t = ∆~x0 c ∆t0 j ijj − j ijj j ijj − j ijj Cluster Properties – p.8/45 Relativity • The Michelson-Morley experiment verified that special relativity gives the observed relation between inertial coordinate systems. • The most general transformation relating two inertial coordinate systems is a Poincaré transformation: µ µ µ ν µ x x0 = Λ x + a ! ν µν µ ν αβ g = Λ αΛ βg Cluster Properties – p.9/45 Quantum Theory • States are represented by unit vectors (rays), , in a complex vector space j i φ = φ ∗ h j i h j i • The predictions of a quantum theory are the probabilities P = φ 2 φψ jh j ij Cluster Properties – p.10/45 Relativity in Quantum Theory X X0 ! φ ; φ0 ; 0 j i j i ! j i j i + 2 2 φ = φ0 0 jh j ij jh j ij Must hold for all , φ and all inertial • j i j i coordinate systems X and X0 Cluster Properties – p.11/45 Wigner’s Theorem 2 2 φ = φ0 0 jh j ij jh j ij + φ0 = U φ 0 = U 0 φ0 = φ j i j i j i j i h j i h j i or φ0 = A φ 0 = A 0 φ0 = φ ∗ j i j i j i j i h j i h j i Cluster Properties – p.12/45 Wigner’s Theorem • For rotations, translations, and rotationless Lorentz transformations: U = U 1=2U 1=2 A = A1=2A1=2 = U • The correspondence between states must be unitary! + 0 = U j i j i Cluster Properties – p.13/45 Wigner’s Theorem X X0 X00 ! ! (Λ;a) (Λ0;a0) |{z}(Λ00;a|{z}00) | {z } + U(Λ0; a0)U(Λ; a) = U(Λ0Λ; Λ0a + a0) • U(Λ; a) is a unitary representation of the Poincaré group Cluster Properties – p.14/45 Elements of RQM Model Hilbert space: • H • Unitary representation of Poincaré group: U(Λ; a) : H ! H iHt Spectral condition: H 0 U(I; t) = e− • ≥ • Cluster properties: lim [U(Λ; a) U1(Λ; a) U2(Λ; a)]U1(I; d~1) U2(I; d~2) = 0 jd~1−d~2j!1 k − ⊗ ⊗ j ik Cluster Properties – p.15/45 Possible approaches Covariant wave functions quasi-Wightman functions. • ! Covariant constraint dynamics quasi-Wightman • ! functions based on first class constraints. Euclidean relativistic quantum theory - based on • reflection positive quasi-Schwinger functions. Directly interacting particles (this talk). • Cluster Properties – p.16/45 Difficulties Space Time Diagram t -4 -2 0 2 4 x Cluster Properties – p.17/45 Difficulties γ 0 0 γβ 0 0 1 0 0 1 Λ = B 0 0 1 0 C B C B γβ 0 0 γ C @ A ~a = (0; 0; 0; z) ~b = (0; 0; 0; γz) t := (γβz;~0) − 1 (Λ− ;~b)(Λ;~a) = (I; (t;~0)) Cluster Properties – p.18/45 Difficulties i λiGi U(Λ; a) = e− P G = Gy G H; P~ ; J~; K~ i i i 2 f g [Gi; Gj] = igijkGk + [Pi; Kj] = iδijH Cluster Properties – p.19/45 Dynamics Galilean Invariant Dynamics H; P~ ; J~; K~ ; M • f g [Pi; Kj] = iδijM Poincaré Invariant Dynamics H; P~ ; J~; K~ • f g [Pi; Kj] = iδij(H0 + V ) ~ ~ ~ • Puts non-linear constraints on H; P ; J; K for consistent initial value problem!f g Cluster Properties – p.20/45 The Currie-Jordan-Sudarshan Theorem X ; P = δ f i jg ij X ; J = X f i jg ijk k X ; K = X X ; H world-line condition f i jg jf i g ~ ~ ~ • Poisson brackets of H; P ; J; K satisfy Lie algebra of Poincaré gfroup. g Cluster Properties – p.21/45 The Currie-Jordan-Sudarshan Theorem + • Can only be satisfied for free particles! • The CJS theorem suggests that it might be difficult to formulate relativistic models for systems of interacting particles. • The out in relativistic quantum mechanics is that there is no position operator! Cluster Properties – p.22/45 Position Let ~0; 0 be a vector corresponding to a • j i particle at ~x = ~0 at time t = 0 p~ ~0; 0 = p~ U y(Λ(p~); 0) ~0; 0 h j i h j j i pm = p~ = 0 ~0; 0 (p~ 2 + m2)1=4 h j i N = (p~ 2 + m2)1=4 p~ ~x; t = 0 = p~ U(I; ~x ) ~0; 0 h j i h j j i Cluster Properties – p.23/45 + Position e ip~ ~x p~ ~x = N − · h j i (p~ 2 + m2)1=4 3 2 d p ip~ ~x ~0 ~x = N e− · = h j i Z (p~ 2 + m2)1=2 mc ~x −mcj~xj const D (0; ~x ) j j e ~ = 0 × + j j ∼ ~ 6 Cluster Properties – p.24/45 Cluster Properties K~ = K~ + K~ + K~ 2K~ (12)(3) (23)(1) (31)(2) − 0 [K1; K2] = iJ 3 = iJ 3 − − 0 [K1; K2] = [K1 ; K2 ] + · · · (12)(3) (23)(1) · · · Commutator of interacting generators create many-body • interactions that need to be canceled. Result is that either some cluster limits do not exist or quantities • that should survive cluster limits actually vanish. Cluster Properties – p.25/45 Cluster Properties ip~3·λ~a −ip~3·λ~a lim e V12e =? λ!1 p~3 = ~q3 + P~ F (m012; P~ ; q~) = P~ ·λ~aF (m012;P~ ;~q) −iP~·λ~aF (m012;P~ ;~q3) lim e V12e λ!1 [V ; q~ ] = [V ; P~ ] = 0; [V ; m ] = 0 12 3 12 12 120 6 + ip~3·λ~a −ip~3·λ~a lim e V12e = 0 λ!1 Cluster Properties – p.26/45 Model Hilbert space Determine a complete measurement • De®nes a complete set of commuting observables • The model Hilbert space is the space of square summable • functions of the eigenvalues of the complete set of commuting observables. + Adequate to describe results of any experiment • Cluster Properties – p.27/45 One-Particle Relativistic Quantum Mechanics U (Λ; a) H; P~ ; J~; K~ 1 ) f g + M; j; η1; ; η4; ∆η1; ; ∆η4 [ηi; ηj] = 0 f · · · · · · g − + 1 : φ = ∗(η)φ(η)dη H h j i X Z + m;j U1(Λ; a) (m; j)η = (m; j)η0 D 0 (Λ; a) j i Z j i η η X Cluster Properties – p.28/45 Two-particle Hilbert space := U (Λ; a) := U (Λ; a) U (Λ; a) H H1 ⊗ H1 0 1 ⊗ 1 + Gi = Gi I + I Gi 0 1 ⊗ 2 1 ⊗ 2 + H ; P~ ; J~ ; K~ f 0 0 0 0g + Cluster Properties – p.29/45 Two-particle Hilbert space M; j; η ; ; η ; ∆η ; ; ∆η f 1 · · · 4 1 · · · 4g U1(Λ; a) U1(Λ; a) = U0;mj(Λ; a) ⊗ Z ⊕ (m ; j )η ; (m ; j )η (m; j)η; d h 1 1 1 2 2 2j f gi m;j U0(Λ; a) (m; j)η; d = (m; j)η0; d Dη0η (Λ; a) j f gi X Z j f gi Cluster Properties – p.30/45 Two particle Hilbert space • The form of the Clebsch-Gordan coefficients depend on choice of vector labels η.