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: pˆ = i~ ∂/∂x] Schr¨odinger eq. → − → ∂S ∂S ∂ ∂ + H0(x, ) = 0 i~ + H0(x, i~ ) ψ(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. ∞ → 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 → { | } [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) → ∈ xµ, φa ... partial observables = q ... configuration space – N + D-dimensional C { } 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) ∀ ∈ 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 | | − 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 + εv(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) ∀ 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 {eµ}µ=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 ) | | 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 − | | 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 − · · {γ } ... orthogonal spacetime basis {Aµ} ... 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 | | 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 · · · − | | 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. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus, Springer (1987)]
V´aclavZatloukal Symmetries and gauge fields 16 / 17 The last slide
Similarities between Gravity and Yang-Mills (gauge theory structure) Seek a unified theory of a single gauge field → Unify spacetime and field space+ Hamiltonian constraint formulation + Requirement of invariance under generic transformations static gauge field h →
Q: Find kinetic term for the unified gauge field h. Q: What about fermions? Q: What about quantization?
Thank you for your attention.
V´aclavZatloukal Symmetries and gauge fields 17 / 17