Multi-Hamiltonian formulations of the extended Chern-Simons theory - PowerPoint PPT Presentation

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 Victoria A. Abakumova Tomsk State University 1 / 10

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 [ q i ( t )] = dt L ( q i , ˙ q i ) , (1) the system is both classically and quantum stable. If its canonical energy q i ∂L E ( q, ˙ q ) = ˙ q i − L (2) ∂ ˙ is bounded, it determines both Lyapunov function and Hamiltonian. Victoria A. Abakumova Tomsk State University 2 / 10

Theories with higher derivatives Once the Lagrangian of the theory involves higher-derivatives, � S [ q i ( t )] = dt L ( q i , ˙ q i , ¨ q i ) , (3) the canonical energy of the model q i � ∂L q, ... q i ∂L q i − d ∂L � q ) = ¨ E ( q, ˙ q, ¨ q i + ˙ − L (4) ∂ ¨ ∂ ˙ dt ∂ ¨ q i 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. Victoria A. Abakumova Tomsk State University 3 / 10

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 oscillator, 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. Victoria A. Abakumova Tomsk State University 4 / 10

The extended Chern-Simons theory of third order A class of theories of vector field A = A µ dx µ on 3 d Minkowski space with the action functional S [ A ] = 1 � α 1 m ∗ dA + α 2 ∗ d ∗ dA + m − 1 ∗ d ∗ d ∗ dA � � ∗ A ∧ . (5) 2 Here, d is the de-Rham differential, ∗ is the Hodge star operator, m is a dimen- sional constant, α 1 , α 2 are dimensionless constant real parameters. The theory admits a two-parameter series of conserved quantities H ( β ) = 1 � d 2 − → � β 2 m − 2 G µ G µ + 2 m − 1 β 1 G µ F µ + ( β 1 α 2 − β 2 α 1 ) F µ F µ � x , (6) 2 where β 1 , β 2 are parameters of the series, F = ∗ dA , G = ∗ d ∗ dA . The quantity (6) is bounded if − β 2 1 + α 2 β 1 β 2 − α 1 β 2 β 2 > 0 , 2 > 0 . (7) The conditions can be satisfied if α 1 = 0 , α 2 � = 0 or α 2 2 − 4 α 1 = 0 , α 1 � = 0 . Victoria A. Abakumova Tomsk State University 5 / 10

First-order equations The Lagrange equations for the extended Chern-Simons theory δS δA ≡ ( α 1 m ∗ d + α 2 ∗ d ∗ d + m − 1 ∗ d ∗ d ∗ d ) A = 0 (8) involve the third time derivatives of A . New variables F i , G i that absorb the time derivatives of A : F i = ε ij ( ˙ G i = − ¨ A i + ∂ i ˙ A j − ∂ j A 0 ) , A 0 + ∂ j ( ∂ j A i − ∂ i A j ) , i, j = 1 , 2 , (9) with ε ij being the 2 d Levi-Civita symbol. The first-order equations of motion in terms of the fields A µ , F i , G i : ˙ ˙ � � A i = ∂ i A 0 − ε ij F j , F i = ε ij ∂ k ( ∂ k A j − ∂ j A k ) − G j , (10) ˙ � � G i = ε ij ∂ k ( ∂ k F j − ∂ j F k ) + m ( α 2 G j + α 1 mF j ) , and one constraint � m − 1 G j + α 2 F j + α 1 mA j � Θ ≡ ε ij ∂ i = 0 . (11) Victoria A. Abakumova Tomsk State University 6 / 10

Poisson brackets The Poisson brackets y ) } β,γ = ( α 1 − α 2 2 ) β 1 + α 1 α 2 β 2 m 3 ε ij δ ( � { G i ( � x ) , G j ( � x − � y ) , β 2 1 − α 2 β 1 β 2 + α 1 β 2 2 α 2 β 1 − α 1 β 2 m 2 ε ij δ ( � { F i ( � x ) , G j ( � y ) } β,γ = x − � y ) , β 2 1 − α 2 β 1 β 2 + α 1 β 2 2 β 1 { F i ( � x ) , F j ( � y ) } β,γ = − mε ij δ ( � x − � y ) , β 2 1 − α 2 β 1 β 2 + α 1 β 2 2 (12) β 1 { A i ( � x ) , G j ( � y ) } β,γ = − mε ij δ ( � x − � y ) , β 2 1 − α 2 β 1 β 2 + α 1 β 2 2 β 2 { A i ( � x ) , F j ( � y ) } β,γ = ε ij δ ( � x − � y ) , β 2 1 − α 2 β 1 β 2 + α 1 β 2 2 γ m − 1 ε ij δ ( � { A i ( � x ) , A j ( � y ) } β,γ = x − � y ) β 2 1 − α 2 β 1 β 2 + α 1 β 2 2 are well-defined for all the parameters α, β, γ that satisfy the condition β 2 1 − α 2 β 1 β 2 + α 1 β 2 2 � = 0 . (13) Victoria A. Abakumova Tomsk State University 7 / 10

Alternative Hamiltonian formulation The first-order equations take the constrained Hamiltonian form Z i = { Z i , H T ( β, γ ) } β,γ , Z = { A, F, G } . (14) Here, � β 2 1 − α 2 β 1 β 2 + α 1 β 2 � d 2 − → 2 H T ( β 1 , β 2 , γ ) = H ( β ) + x Θ A 0 + β 1 − α 2 β 2 − α 1 γ (15) β 2 β 1 β 2 + α 1 β 2 γ − α 2 β 1 γ 2 + β 1 γ � m − 1 ε ij ∂ i A j + β 1 − α 2 β 2 − α 1 γ m − 2 ε ij ∂ i F j , β 1 − α 2 β 2 − α 1 γ and β 1 − α 2 β 2 − α 1 γ � = 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. Victoria A. Abakumova Tomsk State University 8 / 10

Alternative first-order action The non-degenerate Poisson brackets define the series of Hamiltonian action functionals � � β 2 1 − α 2 β 1 β 2 + α 2 β 2 ( α 1 mA i + 2 α 2 F i + 2 m − 1 G i ) ε ij ˙ 2 S ( β, γ ) = A j + β 1 − α 2 β 2 − α 1 γ + β 2 1 + (( α 2 2 − α 1 ) β 1 − α 1 α 2 β 2 ) γ m − 1 ε ij F i ˙ F j + β 1 − α 2 β 2 − α 1 γ +2 β 1 β 2 + ( α 2 β 1 − α 1 β 2 ) γ m − 2 ε ij G i ˙ F j + β 1 − α 2 β 2 − α 1 γ β 2 2 + β 1 γ � β 1 − α 2 β 2 − α 1 γ m − 3 ε ij G i ˙ d 3 x . + G j − H T ( β, γ ) The series includes canonical Ostrogradski action ( β 1 = 1 , β 2 = γ = 0), which is always unbounded, � � � ( α 1 mA i +2 α 2 F i +2 m − 1 G i ) ε ij ˙ A j − m − 1 ε ij F i ˙ d 3 x . S Ost. = F j − A 0 Θ − H Ost. Victoria A. Abakumova Tomsk State University 9 / 10

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 3 d 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) Victoria A. Abakumova Tomsk State University 10 / 10