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

ICMAT, Madrid, july 3-7, 2017

Celebrating D. Holm’s 70th Birthday

Index

• 1. Introduction.
• 2. The setting: multisymplectic formalism
• 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).

• 3. The bracket: canonical forms
• 4. Jacobi brackets in particle dinamics.

• 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).

• M. Forger and S. V. Romero; Covariant Poisson Brackets in Geometric Field Theory
• Commun. Math. Phys., 256, 375 (2005).
• J. E. Marsden, R. Montgomery, P.J. Morrison, W.B. Thompson; Covariant Poisson brackets for

classical Fields, Ann. of Phys., 167, 29-47 (1986).

• I. Khavkine; Covariant Phase Space, constraints and the Peierls formula,
• Int. J. Mod. Phys. A 29, 1430009 (2014).
• C. Crnkovic, E. Witten; Covariant description of canonical formalism in geometrical theories.

Three hundred years of gravitation, 676-684 (1987).

• 2. The Setting

Covariant Hamiltonian First-Order Field Theories

π0

1 : J1E → E

(xµ, ua; ua

µ)

The multisymplectic setting:

ua

µ 7! ρµ aua µ + ρ

Covariant phase space:

P(E) = Aff(J1E)/R

Vector bundle over E modelled on π∗(TM) ⊗E V E∗

τ 0

1 : P(E) → E

(xµ, ua; ρµ

a)

volM = dmx

π: E → M

(xµ, ua), a = 1, . . . , r

µ = 0, 1, . . . , d m = 1 + d

• 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)

Θ = ρµ

a dua ∧ volµ + ρ volM

volµ = i∂/∂xµvolM ρ = −H(xµ, ua, ρµ

a)

ΘH = ρµ

adua ∧ volµ − H(xµ, ua, ρµ a) volM

0 → Vm

0 E ,

→ Vm

1 E → P(E) → 0

(xµ, ua; ρµ

a)

π∗(TM) ⊗E V E∗

Covariant phase space:

P(E) = Aff(J1E)/R τ 0

1 : P(E) → E

= J1E∗ M(E) = Vm

1 E

Ω = dΘ (xµ, ua; ρ, ρν

a)

Multisymplectic model

• 2. The Setting (III)

The action

= Z

M

(P µ

a (x)∂µΦa(x) − H(x, Φ(x), P(x))) volM

S(χ) = Z

M

χ∗ΘH

M(E) J1E∗ E M ∂M E∂M = i∗E i∗(J1E∗) π π1

π0

1

µ π∂M Φ P χ Θ ΘH = h∗θ h ϕ (p, β)

τ1

τ 0

1

H

H∗Θ

i

Sections, fields, forms and all that…

FP (E)

χ = (Φ, P) ∈ FP (E)

“Double sections”

Φ ∈ FM Φ: M ! E , π Φ = idM P : E ! P(E) , τ 0

1 P = idE

P Φ = χ χ: M ! P(E) , π τ 0

1 P = idM

• 2. The Setting (IV)

M ∂M U✏ ∼ = (−✏, 0] × @M

x0 = t

xk volM = dt ∧ vol∂M

t = −✏

t = 0

ϕt(xk) = Φ(t, xk)

∂M 6= ; xk k = 1, 2, 3

Boundaries

(ϕ, p) ∈ T ∗F∂M

α∂M = paδϕa

Canonical 1-form

α(ϕ,p)(δϕ, δp) = R

∂M pa(x)δϕa(x)vol∂M

ϕ = Φ i, pa = P 0

a i

Π: FP (E) → T ∗F∂M

Π(Φ, P) = (ϕ, p)

ω∂M = −dα∂M

Canonical symplectic form

Z

∂M

(χ i)∗ (i ˜

UΘH) = (Π∗α)χ(U)

The boundary term

• 2. The Setting (V)

dSχ = ELχ + Π∗αχ

The fundamental formula

dS(χ)(U) = Z

M

χ∗ i e

UdΘH

• +

Z

∂M

(χ i)∗ i e

UΘH

• χ ∈ FP (E)

U = (δΦ, δP) ∈ TχFP (E)

ΘH = ρµ

adua ∧ volµ − H(xµ, ua, ρµ a) volM

˜ U = δΦa ∂ ∂ua + δP µ

a

∂ ∂ρµ

a

• 2. The Setting (VI)

= {(Φ, P) | ∂Φa ∂xµ = ∂H ∂P µ

a , ∂P µ a

∂xµ = − ∂H ∂Φa }

ELM = {χ = (Φ, P) | ELχ = 0}

The space of solutions of Euler-Lagrange equations

ELχ(U) = Z

M

χ∗ (i ˜

UdΘH)

The Euler-Lagrange 1-form = Z

M

✓∂Φa ∂xµ − ∂H ∂P µ

a

◆ δP µ

a +

✓∂P µ

a

∂xµ + ∂H ∂Φa ◆ δΦa

• volM

• 3. The Bracket

0-form: action S(χ) = Z

M

χ∗ΘH 1-form: Euler-Lagrange form ELχ(U) = Z

M

χ∗ (i ˜

UdΘH)

Beyond: Σ M M+ M− Σ , → M = Π∗

Σ ωΣ = −d (Π∗ ΣαΣ)

ΩΣ

χ(U, V ) =

Z

Σ

i∗(χ∗(iUiV dΘH)) = Z

Σ

(δUϕaδV pa − δUpaδV ϕa) volΣ

U = (δUΦ, δUP) ∈ TχFP (E)

Canonical forms on the space of fields FP (E)

• 3. The Bracket (II)

M Σ1

Σ2 M12 ∂M12 = Σ2 t Σ1

S12(χ) = Z

M12

χ∗ΘH dS12 = EL + Π∗

Σ2αΣ2 − Π∗ Σ1αΣ1

Π∗

Σ1ωΣ1 − Π∗ Σ2ωΣ2 = −d

• Π∗

Σ2αΣ2 − Π∗ Σ1αΣ1

= d(EL) ◆: EL , → FP (E) The pull-back of the 2-forms ΩΣ1 ΩΣ2 along the map is such that ι∗(ΩΣ1) − ι∗(ΩΣ1) = d(ι∗EL) = 0 Ω = ι∗(ΩΣ)

Canonical closed 2-form on the space of solutions EL

ΩΣ1 − ΩΣ1 = d(EL)

• 3. The Bracket (III)

In general the canonical 2-form on the space of solutions is just presymplectic ker Ω 6= 0 If the canonical 2-form is symplectic, then we may define a covariant Poisson bracket on the space

• f solutions

Ω

{F, G} = Ω(XF , XG) iXF Ω = dF

DeWitt’s formula

G is the causal Green’s function of the linearisation

• f the equations of motion along the solution

{F1, F2}(χ) = Z

M×M

δF1 δχ(x)G(x, y) δF2 δχ(y)dxdy χ

• 4. Jacobi brackets

Space of parametrized time-like geodesics such that

pµpµ + m2 = 0

Cm Cm

is a contact manifold of dimension 2m-1

(M, η)

Space-time

xµ µ = 0, 1, . . . , d (− + · · · +)

(Globally hyperbolic)

ωγ(J1, J2) = hJ1, J0

2i hJ2, J0 1i

ω = dΘ E = M × R → R

h˙ γ, ˙ γi = 1

L = m 2 ηµν ˙ xµ ˙ xν

γ ∈ Cm J ∈ TγCm Θγ(J) = h˙ γ, Ji

Jacobi field Contact 1-form

J00 + R(˙ γ, J)˙ γ = 0

Reeb field −˙

γ

• 4. Jacobi brackets (II)

Thus the canonical covariant 2-form of the theory defines a Jacobi bracket (not Poisson)

[f, g] = Λ(d f, dg) + fX(g) − gX(f) (Λ, X) , [Λ, Λ] = 2X ∧ Λ , LXΛ = 0

Jacobi manifold

i ˙

γΩ = 0

ker Θ = h˙ γi⊥

Theorem: The 2-form defined by the contact structure is the canonical covariant 2-form of the 1+0 field theory

• n with Lagrangian L

Ω

E = M × R → R ω [f, g] θ ∧ dθn = (n − 1)d f ∧ dg ∧ θ ∧ (dθ)n−1 + (fdg − gd f) ∧ dθn iXθ ∧ (dθ)n = (dθ)n

iΛθ ∧ dθn = nθ ∧ dθn−1 .

Jacobi structure of contact manifolds

• 4. Jacobi brackets (III)

Minkowski space-time Mm

[F1 , F2] (γ) = Z dsds0 ✓δF1 δγµ (s)Gµν(s − s0)δF2 δγν (s0) + F1(s)˙ γµ(s0)δF2 δγµ (s0) − F2(s)˙ γµ(s0)δF1 δγµ (s0) ◆

G(s-s’) is the causal Green function of Jacobi’s equation

Gµν(s, s0) = P µν(s − s0) , kµ = ηµν ˙ xν Pµν = ηµν − kµkν m F = Z

R

xµδ(s − s1)ds F(γ) = γµ(s1) F = xµ(s1)

[xµ(s1) , xν(s2)] (γ) = P µν(s1 − s2) + xµ(s1)kν(s2) − xν(s2)kµ(s1)

[xµ , xν] = xµkν − xνkµ

Equal-time bracket

Congratulations!