Covariant Brackets in Field Theories and Particle Dynamics A. - PowerPoint PPT Presentation

Covariant Brackets in Field Theories and Particle Dynamics A. Ibort ICMAT & Department of Mathematics Univ. Carlos III de Madrid Celebrating D. Holm’s 70th Birthday ICMAT, Madrid, july 3-7, 2017

Index 1. Introduction. 2. The setting: multisymplectic formalism 3. The bracket: canonical forms 4. Jacobi brackets in particle dinamics. M. Asorey, F. Ciaglia, F. Di Cosmo, A. Ibort. Covariant brackets for particles and fields. M. Phys. Lett. A 32 (19) 1750100, 16 pages (2017); ibid. , Covariant Jacobi bracket for test particles. To appear M. Phys. Lett. A 32 (2017).

1. Introduction R. E. Peierls; The Commutation Laws of Relativistic Field Theories , Proc. Roy. Soc. A, 214 , 1117 (1952). B. S. DeWitt; Dynamical Theory of Groups and Fields , Documents on Modern Physics (Gordon and Breach 1965). J. E. Marsden, R. Montgomery, P.J. Morrison, W.B. Thompson; Covariant Poisson brackets for classical Fields, Ann. of Phys., 167 , 29-47 (1986). C. Crnkovic, E. Witten; Covariant description of canonical formalism in geometrical theories . Three hundred years of gravitation , 676-684 (1987). M. Forger and S. V. Romero; Covariant Poisson Brackets in Geometric Field Theory Commun. Math. Phys., 256 , 375 (2005). I. Khavkine; Covariant Phase Space, constraints and the Peierls formula, Int. J. Mod. Phys. A 29 , 1430009 (2014).

2. The Setting Covariant Hamiltonian First-Order Field Theories The multisymplectic setting: π : E → M ( x µ , u a ) , a = 1 , . . . , r µ = 0 , 1 , . . . , d m = 1 + d vol M = d m x π 0 1 : J 1 E → E ( x µ , u a ; u a µ ) P ( E ) = A ff ( J 1 E ) / R Covariant phase space: u a µ 7! ρ µ a u a µ + ρ Vector bundle over E modelled on π ∗ ( TM ) ⊗ E V E ∗ τ 0 1 : P ( E ) → E ( x µ , u a ; ρ µ a ) A. Ibort and A. Spivak, Covariant Hamiltonian field theories on manifolds with boundary: Yang–Mills theories, J. Geom. Mech. 9(1) (2017) 47–82.

2. The Setting (II) Covariant phase space: P ( E ) = A ff ( J 1 E ) / R = J 1 E ∗ τ 0 ( x µ , u a ; ρ µ a ) 1 : P ( E ) → E π ∗ ( TM ) ⊗ E V E ∗ Multisymplectic model M ( E ) = V m 1 E Ω = d Θ ( x µ , u a ; ρ , ρ ν a ) a du a ∧ vol µ + ρ vol M Θ = ρ µ vol µ = i ∂ / ∂ x µ vol M 0 → V m → V m 1 E → P ( E ) → 0 0 E , ρ = − H ( x µ , u a , ρ µ a ) a du a ∧ vol µ − H ( x µ , u a , ρ µ Θ H = ρ µ a ) vol M

2. The Setting (III) The action Φ ∈ F M Φ : M ! E , π � Φ = id M τ 0 P : E ! P ( E ) , 1 � P = id E “Double sections” F P ( E ) π � τ 0 χ : M ! P ( E ) , 1 � P = id M P � Φ = χ χ = ( Φ , P ) ∈ F P ( E ) Z Z ( P µ a ( x ) ∂ µ Φ a ( x ) − H ( x, Φ ( x ) , P ( x ))) vol M = S ( χ ) = χ ∗ Θ H M M M ( E ) Θ H µ h Sections, fields, Θ H = h ∗ θ H ∗ Θ i ∗ ( J 1 E ∗ ) J 1 E ∗ forms and all that… τ 0 π 0 ( p, β ) τ 1 1 π 1 P 1 E ∂ M = i ∗ E χ E Φ π ϕ π ∂ M i ∂ M M

2. The Setting (IV) Boundaries x k ∂ M 6 = ; k = 1 , 2 , 3 Π : F P ( E ) → T ∗ F ∂ M Π ( Φ , P ) = ( ϕ , p ) p a = P 0 ( ϕ , p ) ∈ T ∗ F ∂ M ϕ = Φ � i, a � i ϕ t ( x k ) = Φ ( t, x k ) Canonical 1-form x 0 = t M ∂ M p a ( x ) δϕ a ( x )vol ∂ M R α ( ϕ ,p ) ( δϕ , δ p ) = x k ∂ M α ∂ M = p a δϕ a t = − ✏ t = 0 Canonical symplectic form U ✏ ∼ = ( − ✏ , 0] × @ M ω ∂ M = − d α ∂ M vol M = dt ∧ vol ∂ M

2. The Setting (V) The fundamental formula d S χ = EL χ + Π ∗ α χ χ ∈ F P ( E ) Z Z χ ∗ � � ( χ � i ) ∗ � � d S ( χ )( U ) = + i e U d Θ H i e U Θ H ∂ M M U = ( δ Φ , δ P ) ∈ T χ F P ( E ) Z ( χ � i ) ∗ ( i ˜ U Θ H ) = ( Π ∗ α ) χ ( U ) The boundary term ∂ M U = δ Φ a ∂ ∂ a du a ∧ vol µ − H ( x µ , u a , ρ µ Θ H = ρ µ ˜ a ) vol M ∂ u a + δ P µ ∂ρ µ a a

2. The Setting (VI) The Euler-Lagrange 1-form Z χ ∗ ( i ˜ EL χ ( U ) = U d Θ H ) M ✓ ∂ Φ a ✓ ∂ P µ ◆ ◆ � Z ∂ x µ − ∂ H ∂ x µ + ∂ H a δ P µ δ Φ a = a + vol M ∂ P µ ∂ Φ a a M The space of solutions of Euler-Lagrange equations EL M = { χ = ( Φ , P ) | EL χ = 0 } = { ( Φ , P ) | ∂ Φ a ∂ x µ = ∂ H a , ∂ P µ ∂ x µ = − ∂ H ∂ Φ a } a ∂ P µ

3. The Bracket Canonical forms on the space of fields F P ( E ) Z S ( χ ) = χ ∗ Θ H 0-form: action M 1-form: Euler-Lagrange form Z χ ∗ ( i ˜ EL χ ( U ) = U d Θ H ) M Beyond: U = ( δ U Φ , δ U P ) ∈ T χ F P ( E ) Z Ω Σ χ ( U, V ) = i ∗ ( χ ∗ ( i U i V d Θ H )) Σ Σ M Z ( δ U ϕ a δ V p a − δ U p a δ V ϕ a ) vol Σ = Σ M + = Π ∗ Σ ω Σ = − d ( Π ∗ Σ α Σ ) M − Σ , → M

3. The Bracket (II) Σ 1 Σ 2 Z S 12 ( χ ) = χ ∗ Θ H M 12 M M 12 dS 12 = EL + Π ∗ Σ 2 α Σ 2 − Π ∗ Σ 1 α Σ 1 ∂ M 12 = Σ 2 t Σ 1 � = d (EL) � Π ∗ Σ 1 ω Σ 1 − Π ∗ Π ∗ Σ 2 α Σ 2 − Π ∗ Σ 2 ω Σ 2 = − d Σ 1 α Σ 1 Ω Σ 1 − Ω Σ 1 = d(EL) The pull-back of the 2-forms Ω Σ 1 Ω Σ 2 along the map ◆ : EL , → F P ( E ) is such that ι ∗ ( Ω Σ 1 ) − ι ∗ ( Ω Σ 1 ) = d ( ι ∗ EL) = 0 Canonical closed 2-form on the space of solutions EL Ω = ι ∗ ( Ω Σ )

3. The Bracket (III) In general the canonical 2-form on the space of solutions is just presymplectic ker Ω 6 = 0 Ω If the canonical 2-form is symplectic, then we may define a covariant Poisson bracket on the space of solutions { F, G } = Ω ( X F , X G ) i X F Ω = dF DeWitt’s formula δχ ( x ) G ( x, y ) δ F 2 δ F 1 Z { F 1 , F 2 } ( χ ) = δχ ( y ) dxdy M × M G is the causal Green’s function of the linearisation of the equations of motion along the solution χ

4. Jacobi brackets ( M, η ) ( − + · · · +) (Globally hyperbolic) Space-time x µ µ = 0 , 1 , . . . , d E = M × R → R C m Space of parametrized time-like geodesics such that p µ p µ + m 2 = 0 L = m x µ ˙ h ˙ γ i = � 1 x ν γ , ˙ 2 η µ ν ˙ C m is a contact manifold of dimension 2m-1 J 00 + R (˙ γ , J )˙ γ = 0 Jacobi field γ ∈ C m J ∈ T γ C m Contact 1-form Θ γ ( J ) = h ˙ γ , J i ω = d Θ Reeb field − ˙ ω γ ( J 1 , J 2 ) = h J 1 , J 0 2 i � h J 2 , J 0 1 i γ

4. Jacobi brackets (II) Theorem: The 2-form defined by the contact structure is ω Ω the canonical covariant 2-form of the 1+0 field theory on with Lagrangian L E = M × R → R γ i ⊥ ker Θ = h ˙ γ Ω = 0 i ˙ Thus the canonical covariant 2-form of the theory defines a Jacobi bracket (not Poisson) Jacobi manifold ( Λ , X ) , [ Λ , Λ ] = 2 X ∧ Λ , L X Λ = 0 [ f, g ] = Λ ( d f, dg ) + fX ( g ) − gX ( f ) Jacobi structure of contact manifolds i X θ ∧ ( d θ ) n = ( d θ ) n i Λ θ ∧ d θ n = n θ ∧ d θ n − 1 . [ f, g ] θ ∧ d θ n = ( n − 1) d f ∧ dg ∧ θ ∧ ( d θ ) n − 1 + ( fdg − gd f ) ∧ d θ n

4. Jacobi brackets (III) ✓ δ F 1 Z δγ µ ( s ) G µ ν ( s − s 0 ) δ F 2 dsds 0 δγ ν ( s 0 ) [ F 1 , F 2 ] ( γ ) = ◆ γ µ ( s 0 ) δ F 2 γ µ ( s 0 ) δ F 1 δγ µ ( s 0 ) − F 2 ( s )˙ δγ µ ( s 0 ) + F 1 ( s )˙ G(s-s’) is the causal Green function of Jacobi’s equation Minkowski space-time M m P µ ν = η µ ν − k µ k ν G µ ν ( s, s 0 ) = P µ ν ( s − s 0 ) , m x ν k µ = η µ ν ˙ Z F = x µ δ ( s − s 1 ) ds F ( γ ) = γ µ ( s 1 ) F = x µ ( s 1 ) R [ x µ ( s 1 ) , x ν ( s 2 )] ( γ ) = P µ ν ( s 1 − s 2 ) + x µ ( s 1 ) k ν ( s 2 ) − x ν ( s 2 ) k µ ( s 1 ) [ x µ , x ν ] = x µ k ν − x ν k µ Equal-time bracket

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