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Multi-Hamiltonian formulations of the extended Chern-Simons theory with higher derivatives

VICTORIA A. ABAKUMOVA Tomsk State University

Faculty of Physics, Quantum Field Theory Department Tomsk April 01, 2018

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Classical and quantum stability

Stability (classical or quantum) is an important characteristic of the dynamics. Classical stability. Classical solutions are bounded at any period of time (Lyapunov stability). Quantum stability. Quantum system has a well-defined vacuum state with the lowest possible energy. Once the classical equations of motion come from a variational principle for the action functional without higher derivatives, S[qi(t)] =

• dt L(qi, ˙

qi) , (1) the system is both classically and quantum stable. If its canonical energy E(q, ˙ q) = ˙ qi ∂L ∂ ˙ qi − L (2) is bounded, it determines both Lyapunov function and Hamiltonian.

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Theories with higher derivatives

Once the Lagrangian of the theory involves higher-derivatives, S[qi(t)] =

• dt L(qi, ˙

qi, ¨ qi) , (3) the canonical energy of the model E(q, ˙ q, ¨ q, ... q ) = ¨ qi ∂L ∂¨ qi + ˙ qi ∂L ∂ ˙ qi − d dt ∂L ∂¨ qi

• − L

(4) is linear in ... q i. Thus, it is unbounded. The unbounded canonical energy is not an admissible Lyapunov function of the

• system. Thus, the energy cannot ensure the classical stability of the model.

The canonical (Ostrogradski) Hamiltonian, being the phase-space equivalent of energy, is also unbounded. It is a quantum instability. There can exist an another bounded conserved quantity. It ensures classical and quantum stability of the model.

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Previous work

The alternative Hamiltonian formulations (i.e. formulations, where the Hamil- tonian is not equivalent to canonical energy) are studied since 2005.

• The alternative Hamiltonian formulations are introduced for the Pais-

Uhlenbeck oscillator of fourth order.

• K. Bolonek, P. Kosinski, Hamiltonian structures for Pais-Uhlenbeck
• scillator, Acta Phys.

Polon. B 36 (2005) 2115.

• The alternative Hamiltonian formulations are introduced for the general one-

dimensional Pais-Uhlenbeck oscillator.

• E. V. Damaskinsky and M. A. Sokolov, Remarks on quantization of

Pais-Uhlenbeck oscillators, J. Phys.

• A. 39 (2006) 10499.
• The general multi-dimensional Pais-Uhlenbeck oscillator is considered.
• I. Masterov, An alternative Hamiltonian formulation for the Pais-

Uhlenbeck oscillator, Nucl. Phys. B 902 (2016) 95. In this work, we first consider the alternative Hamiltonian formulation in the higher-derivative field theory.

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The extended Chern-Simons theory of third order

A class of theories of vector field A = Aµdxµ on 3d Minkowski space with the action functional S[A] = 1 2

• ∗A ∧
• α1m ∗ dA + α2 ∗ d ∗ dA + m−1 ∗ d ∗ d ∗ dA
• .

(5) Here, d is the de-Rham differential, ∗ is the Hodge star operator, m is a dimen- sional constant, α1, α2 are dimensionless constant real parameters. The theory admits a two-parameter series of conserved quantities H(β) = 1 2

• d2−

→ x

• β2m−2GµGµ + 2m−1β1GµFµ + (β1α2 − β2α1)FµFµ
• , (6)

where β1, β2 are parameters of the series, F = ∗dA , G = ∗d ∗ dA . The quantity (6) is bounded if β2 > 0 , −β2

1 + α2β1β2 − α1β2 2 > 0 .

(7) The conditions can be satisfied if α1 = 0 , α2 = 0 or α2

2 − 4α1 = 0 , α1 = 0 .

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First-order equations

The Lagrange equations for the extended Chern-Simons theory δS δA ≡ (α1m ∗ d + α2 ∗ d ∗ d + m−1 ∗ d ∗ d ∗ d)A = 0 (8) involve the third time derivatives of A. New variables Fi, Gi that absorb the time derivatives of A : Fi = εij( ˙ Aj − ∂jA0) , Gi = − ¨ Ai + ∂i ˙ A0 + ∂j(∂jAi − ∂iAj) , i, j = 1, 2 , (9) with εij being the 2d Levi-Civita symbol. The first-order equations of motion in terms of the fields Aµ, Fi, Gi : ˙ Ai = ∂iA0 − εijFj , ˙ Fi = εij

• ∂k(∂kAj − ∂jAk) − Gj
• ,

˙ Gi = εij

• ∂k(∂kFj − ∂jFk) + m(α2Gj + α1mFj)
• ,

(10) and one constraint Θ ≡ εij∂i

• m−1Gj + α2Fj + α1mAj
• = 0 .

(11)

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Poisson brackets

The Poisson brackets {Gi( x), Gj( y)}β,γ = (α1 − α2

2)β1 + α1α2β2

β2

1 − α2β1β2 + α1β2 2

m3εijδ( x − y) , {Fi( x), Gj( y)}β,γ = α2β1 − α1β2 β2

1 − α2β1β2 + α1β2 2

m2εijδ( x − y) , {Fi( x), Fj( y)}β,γ = − β1 β2

1 − α2β1β2 + α1β2 2

mεijδ( x − y) , {Ai( x), Gj( y)}β,γ = − β1 β2

1 − α2β1β2 + α1β2 2

mεijδ( x − y) , {Ai( x), Fj( y)}β,γ = β2 β2

1 − α2β1β2 + α1β2 2

εijδ( x − y) , {Ai( x), Aj( y)}β,γ = γ β2

1 − α2β1β2 + α1β2 2

m−1εijδ( x − y) (12) are well-defined for all the parameters α, β, γ that satisfy the condition β2

1 − α2β1β2 + α1β2 2 = 0 .

(13)

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Alternative Hamiltonian formulation

The first-order equations take the constrained Hamiltonian form Zi = {Zi, HT (β, γ)}β,γ, Z = {A, F, G} . (14) Here, HT (β1, β2, γ) = H(β) +

• d2−

→ x Θ β2

1 − α2β1β2 + α1β2 2

β1 − α2β2 − α1γ A0 + β1β2 + α1β2γ − α2β1γ β1 − α2β2 − α1γ m−1εij∂iAj + β2

2 + β1γ

β1 − α2β2 − α1γ m−2εij∂iFj

• ,

(15) and β1 − α2β2 − α1γ = 0 . (16)

• The constrained Hamiltonian formulation exists for almost all α, β, γ. The

theory is multi-Hamiltonian.

• The parameter γ is an accessory one. The series of Hamiltonian formulations

is two-parameter.

• An alternative Hamiltonian can be bounded from below, and it can ensure

the quantum stability of the theory.

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Alternative first-order action

The non-degenerate Poisson brackets define the series of Hamiltonian action functionals S(β, γ) = β2

1 − α2β1β2 + α2β2 2

β1 − α2β2 − α1γ (α1mAi + 2α2Fi + 2m−1Gi)εij ˙ Aj + + β2

1 + ((α2 2 − α1)β1 − α1α2β2)γ

β1 − α2β2 − α1γ m−1εijFi ˙ Fj+ +2β1β2 + (α2β1 − α1β2)γ β1 − α2β2 − α1γ m−2εijGi ˙ Fj + + β2

2 + β1γ

β1 − α2β2 − α1γ m−3εijGi ˙ Gj − HT (β, γ)

• d3x .

The series includes canonical Ostrogradski action (β1 = 1, β2 = γ = 0), which is always unbounded,

• SOst. =

(α1mAi+2α2Fi+2m−1Gi)εij ˙ Aj −m−1εijFi ˙ Fj −A0Θ−HOst.

• d3x .

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Summary

The theory admits a two-parameter series of conserved quantities. These quantities can be bounded, and they ensure the classical stability of the model. The theory admits a two-parameter series of Hamiltonian formulations. The Hamiltonians can be bounded, and they ensure the quantum stability. The series of Hamiltonians includes the canonical Ostrogradski Hamilto- nian, which is unbounded in all the instances. References: [1] V. A. Abakumova, D. S. Kaparulin, S. L. Lyakhovich, Multi-Hamiltonain formulations and stability of higher-derivative extensions of 3d Chern-Simons. Eur. Phys.

• J. C 78, 115 (2018)

[2] V. A. Abakumova, D. S. Kaparulin, S. L. Lyakhovich, A bounded Hamiltonian in the extended Chern-Simons theory of fourth order. Russ. Phys.

• J. 60 (12), 40-47 (2017)(in Russian)

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