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Symmetries and Gauge Fields in the Hamiltonian Constraint Formulation

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Symmetries and Gauge Fields in the Hamiltonian Constraint Formulation Symmetries and gauge fields in the Hamiltonian constraint formulation of classical field theories V´aclavZatloukal (www.zatlovac.eu) Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague talk given at \ICCA11" Ghent, 8 August 2017 V´aclavZatloukal Symmetries and gauge fields 1 / 17 Motivation I: Unification of fundamental interactions Faraday-Maxwell's field theory of electromagnetism Einstein's field theory of gravity (general relativity) (interactions at distance replaced by interactions mediated by fields) [H. Goenner, On the History of Unified Field Theories, Living Rev. Relativity 7, 2 (2004)] Nowadays we have: Electromagnetism weak strong interaction ⊕ ⊕ (Yang-Mills theories with gauge groups U(1), SU(2) and SU(3), resp.) Gravity (see Gauge Theory Gravity) ⊕ [A. Lasenby, C. Doran, S. Gull, Gravity, gauge theories and geometric algebra, Phil. Trans. Roy. Soc. Lond. A 356 (1998) 487-582, arXiv:gr-qc/0405033] My aim: Understand the relation between Gravity and Yang-Mills on the level of classical field theory. V´aclavZatloukal Symmetries and gauge fields 2 / 17 Motivation II: Quantization in Hamiltonian formalism Non-relativistic mechanical system with Hamiltonian H0(x; p): Hamilton-Jacobi [quantization: p^ = i~ @=@x] Schr¨odinger eq. ! − ! @S @S @ @ + H0(x; ) = 0 i~ + H0(x; i~ ) (x; t) = 0 (1) @t @x ! − @t − @x Field theoretic Hamiltonian formalism: [M. J. Gotay et. al, arXiv:physics/9801019v2, arXiv:math-ph/0411032] 1) canonical: -dim. space of field configurations functional Schr. eq. 1 ! 2) covariant: finite-dim. configuration space, generalized momentum, classical De Donder-Weyl theory precanonical Schr. eq. ! [I. V. Kanatchikov, Rep. Math. Phys. 43 (1999) 157, arXiv:9810165, arXiv:1312.4518] My talk: Hamiltonian constraint formalism ( covariant) ∼ V´aclavZatloukal Symmetries and gauge fields 3 / 17 Outline Unifying spacetime and the space of fields Hamiltonian constraint formulation of field theories Canonical equations of motion Local Hamilton-Jacobi theory Symmetries and Hamiltonian Noether theorem Local symmetries and the gauge field Example: Scalar field coupled to gravity and Yang-Mills field [V.Z., Int. J. Geom. Methods Mod. Phys. 13, 1650072 (2016), arXiv:1504.08344] [V.Z., Adv. Appl. Clifford Algebras 27, 829-851 (2017), arXiv:1602.00468] [V.Z., arXiv:1611.02906 (2016)] V´aclavZatloukal Symmetries and gauge fields 4 / 17 Mathematical language: Geometric algebra Geometric algebra and calculus [D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus, Springer (1987)] [C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge Univ. P. (2007)] ( Clifford algebra, Dirac algebra of γ-matrices) , Geometric product: ab = a b + a b · ^ { associative, invertible, non-commutative ( ) inner product: a b = b a ( ) outer product a b = b a · · · ^ ^ − ^ a2 a1 =a1 a2 = ∧ ∧ Many useful tools: multivector derivative, rotors, bivector algebras, ... ! Coordinate-free differential geometry ! V´aclavZatloukal Symmetries and gauge fields 5 / 17 Unifying spacetime and the field space Fields as functionsΦ( x) surfaces γ = q = (x; y) g(x; y) = 0 ! f j g [x = (x µ); y = (φa)] φa φa P γ a µ φ (x ) b (xµ, φa)= q x1 x1 x0 x0 (cf. special relativity: t; x xµ Minkowski space) ! 2 xµ; φa ::: partial observables = q ::: configuration space { N + D-dimensional C f g γ ::: motions { D-dim. surfaces (correlations among partial observ.) ⊂ C [Chap. 3 in C. Rovelli, Quantum Gravity, Cambridge Univ. Press (2004)] V´aclavZatloukal Symmetries and gauge fields 6 / 17 Variational principle with Hamiltonian constraint Variational principle RD+N Extremize P ′ C ≃ dΓ′ b q′ f [γ; P] = P(q) dΓ(q) (2) P γ′ A γ · Z Iγ dΓ γ b under Hamiltonian constraint q dΣ ∂γ = ∂γ′ H(q; P(q)) = 0 q γ: (3) 8 2 dΓ ::: oriented surface element of γ P ::: multivector of grade D (H is a generic function of q and P, not energy or its density!) Non-relativistic mechanics ::: H = p et + H0(q; px ) 1 · N 2 Scalar field theory ::: H = P Ix + Ix (P ea) + V (y) · 2 a=1 · · String theory ::: H = 1 ( P 2 Λ2) 2 j j − P V´aclavZatloukal Symmetries and gauge fields 7 / 17 Classical field theory from Hamiltonian constraint R Lagrange multiplier of Hamiltonian constraint: A[γ; P; λ] = γ [P · dΓ − λ(q)H(q; P)] Canonical equations of motion: ! λ @P H(q; P) = dΓ (4a) D dΓ @qP for D = 1 (particles) ( 1) λ @_qH(_q; P) = · (4b) − ((dΓ @q) P for D > 1 (fields) · · H(q; P) = 0 (4c) Local Hamilton-Jacobi theory: ! H(q;@q S) = 0 (5) ^ (Partial diff. eq., S(q) is not classical action cl [@γ]) A [V.Z., Int. J. Geom. Meth. Mod. Phys. 13, 1650072 (2016), arXiv:1504.08344] [F. H´eleinand J. Kouneiher, J. Math. Phys. 43, 2306 (2002), arXiv:0010036] V´aclavZatloukal Symmetries and gauge fields 8 / 17 Symmetries and conservation laws (Active) Transformation q0 = f (q): RD+N C ≃ 1 P ′ = f − (P ) γ′ P b q′ dΓ′ = f(dΓ) dΓ f γ b q −1 −1 f (a) = a · @qf (q) (push-forward), f (b) = @qf (q) · b (pull-back) Symmetry if: H(q0; P0) = H(q; P)( [γ0; P0; λ0] = [γ; P; λ]) !A A Conservation law: (corresp. to infsm. sym. f (q) = q + "v(q)) ! D > 1 : (dΓ @q) (P v) = 0 ; D = 1 : dΓ @q(P v) = 0 (6) · · · · · P v ::: conserved multivector of grade D 1( Noether current) [V.Z.,· Adv. Appl. Cliff. Alg. 27, 829-851 (2017),− arXiv:1602.00468,∼ arXiv:1604.03974] V´aclavZatloukal Symmetries and gauge fields 9 / 17 Local transformations Assume H(q0; P) = H(q; P) and define the Gauged Hamiltonian Hh(q; P) = H(q; h(P; q)) (7) where h is the adjoint of a q-dependent linear mapping h (cf. vielbein) Transformation rule for the (static) gauge field h: h0(b; q0) = h f (b; q); q ( b) (8) 8 0 0 Then: Hh0 (q ; P ) = Hh(q; P) classical motions of Hh are transformed to classical motions of Hh0 ) [V.Z., arXiv:1611.02906 (2016)] V´aclavZatloukal Symmetries and gauge fields 10 / 17 Gauge field strength Canonical equations of motion for Hh(q; P): (denote P¯ = h(P)) ¯ −1 λ @P¯H(q; P) = h (dΓ) (9a) D −1 −_ 1 ( 1) λ h(@_q)H(_q; P¯) h (dΓ) h @_q h (P¯) − − · ^ dΓ @qP¯ for D = 1 = · (9b) −1 ¯ (h (dΓ @q) P for D > 1 · · H(q; P¯) = 0 (9c) Field strength: linear mapping from n to n + 1-vectors −_ 1 F (P¯) = h @_q h (P¯) (10) − ^ 0 0 Gauge invariance: F (P¯ ) = F (P¯) D+N µ µ j j −1 Coordinate frame feµgµ=1 : hj = γj · h(e ) ; h µ = γ · h (eµ) ρ ρ j ρ j −1 ρ Field strength as torsion: T µν = hj @µh ν − hj @ν h µ = h (eµ ^ eν ) · F (h(e )) V´aclavZatloukal Symmetries and gauge fields 11 / 17 Example: Scalar field D-dim. spacetime: x = (x µ) P (q) e φ = y N-dim. field space: y = (φ1; : : : ; φN ) a a a P b q Lagrangian formulation: γ dΓ= g(dX) Iy SF [y(x)] = g A P(x) 1 D µ b x d x (@µy) (@ y) V (y) dX 2 · − Ω Z Ix Hamiltonian constraint for scalar field coupled to a gauge field: N 1 2 HSF ;h(q; P) = h(P) Ix + Ix (h(P) ea) + V (y) (11) · 2 · · a=1 X Eliminating momentum P (using the first canonical equation (9a)) N 1 −1 −1 2 −1 −1 SF ;h[γ] = I (h (dΓ) ea) λV (y) ; λ = I h (dΓ) A 2λ x · · − x · Zγ " a=1 # X V´aclavZatloukal Symmetries and gauge fields 12 / 17 Scalar field and gravity 0 Spacetime diffeomorphisms: q = fGr (q) = fx (x) + y Gravitational gauge field: hGr (b; q) = hx (bx ; x)+ by 0 0 Transformation rule: hx (bx ; x ) = hx f x (bx ; x); x −1 We calculate: λGr = dX det(h ) j j Gr Scalar field in gravitational background: N −1 1 2 SF ;h [y(x)] = det(h ) hGr (@x φa) V (y) dX (12) A Gr Gr 2 − j j ZΩ " a=1 # X V´aclavZatloukal Symmetries and gauge fields 13 / 17 Scalar field and Yang-Mills 0 −By (x)=2 Local field-space rotations: q = fYM (q) = R(x)qR(x), R(x) = e (Assuming V (y 0) = V (y)) e µ Yang-Mills gauge field: hYM (b; q) = S(x)bS(x) γ y Aµ(x) b µ − · · fγ g ::: orthogonal spacetime basis fAµg ::: field-space bivectors (, skew-symm.e mapA µ : y 7! y · Aµ) S ::: field-space rotor 0 0 Transformation rules: S = RS , A = RAµR 2R@µR µ − We calculate: λYM = dX j j e e Scalar field in Yang-Mills background: 1 µ µ SF ;YM [y(x)] = (@µy + y Aµ) (@ y + y A ) V (y) dX A Ω 2 · · · − j j Z (13) V´aclavZatloukal Symmetries and gauge fields 14 / 17 Dynamics of the gauge field −_ 1 Field strength corresponding to gauge field h: F (b) = h @_q h (b) − ^ Gravitational field strength (cf. teleparallel gravity) −_ 1 FGr (b) = hx @_x hx (bx ) (14) − ^ Yang-Mills field strength 1 µ ν FYM (b) = γ γ [y (@νAµ @µAν Aµ Aν)] (SbS) 2 ^ · − − × · µ + γ (SAµS 2S@µS) b (15) ^ − · e e e Q: Find a kinetic term for the unified gauge field h that reduces to −1 1) Gravitational kinetic term: det(hGr ) R µν 2) Yang-Mills kinetic term: Tr(FµνF ) V´aclavZatloukal Symmetries and gauge fields 15 / 17 Detour: Geometry of embedded surfaces n N Embedded manifold R with pseudoscalar IM(x) M ⊂ −1 Shape tensor: S(a) = I a @IM , a is tangent vector M · Covariant derivative: a DM = a @M M S(a) · · − × (Shape S ! ChristoffelsΓ and Gauge potentials A?) Curvature: ! S(a) S(b) = R(a b) + F (a b) (16) × ^ ^ R ::: intrinsic, F ::: external [D.
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