Conservation laws and stability of higher derivative extended Chern-Simons
aa r X i v : . [ h e p - t h ] J u l Conservation laws and stability of higher derivativeextended Chern-Simons
V. A. Abakumova ∗ , D. S. Kaparulin † , and S. L. Lyakhovich ‡ Physics Faculty, Tomsk State University, Tomsk 634050, Russia
Abstract
The higher derivative field theories are notorious for the stability problems both at classical and quantum level. Classicalinstability is connected with unboundedness of the canonical energy, while the unbounded energy spectrum leads to thequantum instability. For a wide class of higher derivative theories, including the extended Chern-Simons, other boundedconserved quantities which provide the stability can exist. The most general gauge invariant extended Chern-Simons theoryof arbitrary finite order n admits ( n − -parameter series of conserved energy-momentum tensors. If the -component ofthe most general representative of this series is bounded, the theory is stable. The stability condition requires from the freeextended Chern-Simons theory to describe the unitary reducible representation of the Poincar´e group. The unstable theorycorresponds to nonunitary representation. Theories with higher derivatives are important in modern theoretical physics. The most known examples of such modelsare the Pais-Uhlenbeck (PU) oscillator [1], generalized Podolsky electrodynamics [2], modified theories of gravity [3, 4, 5, 6],and conformal higher spin theories [7]. Theories with higher derivatives have higher symmetry and better convergencyproperties at the classical and quantum level compared to their counterparts without higher derivatives, but also they areknown for classical and quantum instability.Stability is an important characteristic of dynamics. Classical stability means that the motions of theory are finite at everymoment of time. For mechanical models it can be provided by existence of the Lyapunov function. For theories withouthigher derivatives the canonical energy plays the role of Lyapunov function. So, if the canonical energy is bounded, thetheory is classically stable. Quantum stability means that quantum system has a well-defined vacuum state with the lowestpossible energy. This type of stability is provided by the bounded Hamiltonian being the phase-space equivalent of thecanonical energy.Let us consider the stability problem of higher derivative theories. The theory with second time derivatives, described bythe action functional S [ x i ( t )] = Z dt L ( x i , ˙ x i , ¨ x i ) , (1) ∗ [email protected] † [email protected] ‡ [email protected] E can = ¨ x i ∂L∂ ¨ x i + ˙ x i (cid:16) ∂L∂ ˙ x i − ddt ∂L∂ ¨ x i (cid:17) − L . (2)This expression is linear in ... x i , and so it is unbounded. The unbounded canonical energy cannot serve as a Lyapunov functionof the system, and it cannot ensure classical stability of the model. The canonical Hamiltonian is also unbounded. So, themodel is not stable at quantum level.In [8], it was shown that a wide class of theories with higher derivatives admits, except for canonical energy, anotherconserved quantities. If some of these conserved quantities are bounded, they can ensure classical and quantum stabilityof the model. First such conservation laws were introduced for the Pais-Uhlenbeck oscillator [9, 10]. Later the stability ofhigher derivative theories was studied in [11, 12, 13, 14]. We will apply this idea to study the stability problem for the higherderivative extended Chern-Simons theory [15].The rest of the paper is organized as follows. In Section 2, we consider the stability of the Pais-Uhlenbeck oscillator[1] from the viewpoint of existence of conservation law, which is different from the canonical energy. In Section 3, weanalyze the stability condition for the extended Chern-Simons of arbitrary finite order. In Section 4, the general constructionis exemplified by the third-order extension of Chern-Simons theory. In Conclusion we summarize the results. In this section, we illustrate how the idea of stabilizing dynamics by the conserved quantities works in the simplest higherderivative theory.The PU oscillator of fourth order is a theory of a single dynamical variable x ( t ) with the action functional S [ x ( t )] = Z L ( x, ˙ x, ¨ x ) dt , L ( x, ˙ x, ¨ x ) = 12 (cid:0) − ¨ x + ( ω + ω ) ˙ x − ω ω x (cid:1) . (3)Here, the parameters ω i , i = 1 , , are the frequencies of oscillations. We assume that the frequencies are different andnonzero, ω = ω , ω + ω = 0 . (4)The Euler-Lagrange equation for the model (3) has the form, δSδx ≡ x (4) + ( ω + ω )¨ x + ω ω x = 0 . (5)The solution to this equation is the bi-harmonic oscillation, x ( t ) = A sin( ω t + ϕ ) + A sin( ω t + ϕ ) , (6)where the amplitudes A i , and initial phases ϕ i , i = 1 , , are integration constants. The motion is finite, | x ( t ) | ≤ | A | + | A | . (7)2hus, the Pais-Uhlenbeck oscillator is a stable model.Let us explain the stability of the PU theory from the viewpoint of the conserved quantities of the model. There aretwo-parameter series of symmetries of the action functional (3), δ β x = β ˙ x + β ... x , (8)where β , β are infinitesimal transformation parameters, being constants. The first symmetry in the set is time translation.The second transformation in (8) is the higher symmetry. The two-parameter series of conserved quantities is associated withthese symmetries by the Noether theorem: J = β E can + β J , (9)where E can is the canonical energy of the model, and J is another independent integral of motion, E can = (cid:16) ˙ xx (3) −
12 ¨ x (cid:17) + 12 (cid:16) ( ω + ω ) ˙ x + ω ω x (cid:17) ; J = 12 (cid:16) ... x + ( ω + ω )¨ x (cid:17) + ω ω (cid:16) x ¨ x −
12 ˙ x (cid:17) . (10)Both of these quantities are unbounded quadratic forms of initial data ˙ x, ¨ x, ... x, .... x , but they can be joined in two boundedcombinations, J = J + ω E can , J = J + ω E can ,J i = 12 (cid:16) ( x (3) + ω i ˙ x ) + ( ω + ω − ω i )(¨ x + ω i x ) (cid:17) , i = 1 , . (11)Any bounded combination of these bounded quantities with positive coefficients is a positive-definite quadratic form of initialdata.The positive definite conserved quantity, being constructed from the integrals of motion (11), selects the stationarybounded surface in the phase-space of the theory. This bounded conserved quantity stabilizes the dynamics of the PU theory. Consider d Minkowski space with the local coordinates x µ , µ = 0 , , , and the metric η µν = diag (+1 , − , − . (12)The extended Chern-Simons is a gauge theory of vector field A = A µ ( x ) dx µ with the action functional S [ A ( x )] = m n X p =1 (cid:18) α p Z A µ ( x ) F ( p ) µ ( x ) d x (cid:19) . (13)3ere, m is a parameter with dimension of mass, and dimensionless real constants α , . . . , α n are parameters of the model.Without loss of generality we assume that α n = 0 , and the notation is used: F ( p ) µ = m − p ε µνρ ∂ ν F ( p − ρ , F (0) µ ≡ A µ , r = 1 , . . . , n , (14)where ε denotes the d Levi-Civita symbol with ε = 1 . The Euler-Lagrange equations for the action functional (13) havethe form δSδA ≡ m n X p =1 α p F ( p ) = 0 . (15)These equations involve the n -th time derivatives of A .The action (13) is Poincar´e-invariant. The space-time translations are symmetries of the action functional (13), δ ξ A µ = ξ ν ∂ ν A µ , (16)where ξ is the transformation parameter. The canonical energy-momentum is associated with this symmetry, T canµν ( α ) = m n X p =1 X r + s = p α p (cid:16) F ( r ) µ F ( s ) ν + F ( r ) ν F ( s ) µ − η µν η ρσ F ( r ) ρ F ( s ) σ (cid:17) . (17)The -component of the energy-momentum tensor has the form T can ( α ) = m X µ =0 n X p =1 X r + s = p α p F ( r ) µ F ( s ) µ . (18)This quantity is linear in F ( n − for n > , T can ( α ) = m α n F ( n − F (1)0 + . . . , (19)the dots denote the terms that do not include F ( n − . Thus, the energy in the extended Chern-Simons theory is unboundedwhenever the higher derivatives are included in the Lagrangian.The series of higher symmetries, which generalize (16) of the action functional (13) has the form δ ξ A µ = n − X q =1 β q ξ ν ∂ ν F ( q − µ , (20)where β q , q = 1 , . . . , n − , are the parameters of symmetry series, being real constants. The ( n − -parameter series ofconserved tensors is connected with this symmetry by the Noether theorem, T µν ( α, β ) = n − X r,s =1 C r,s ( α, β ) (cid:16) F ( r ) µ F ( s ) ν + F ( r ) ν F ( s ) µ − η µν η ρσ F ( r ) ρ F ( s ) σ (cid:17) . (21)4he Bezout matrix C r,s ( α, β ) of two polynomials is defined by the generating relation C r,s ( α, β ) = ∂ r + s ∂ r z ∂ s z ′ (cid:16) M ( z ) N ( z ′ ) − M ( z ′ ) N ( z ) z − z ′ (cid:17)(cid:12)(cid:12)(cid:12) z = z ′ =0 , (22)where z and z ′ are two independent variables, and M ( z ) = n X p =1 α p z p , N ( z ) = n − X q =1 β q z q . (23)The representatives of the series (21) are defined by the formula T ( q ) µν ( α ) = ∂T µν ( α, β ) ∂β q , q = 1 , . . . , n − . (24)By construction, T (1) µν ≡ T canµν , and other conserved tensors are independent.The -component of general conserved tensor (21) has the form T ( α, β ) = m X µ =0 n − X r,s =1 C r,s ( α, β ) F ( r ) µ F ( s ) µ . (25)This quantity is a quadratic form of the variables F ( r ) µ . So, it is bounded if C r,s ( α, β ) is a positive definite matrix . (26)This condition is a restriction on the parameters β in the series of energy-momentum tensors (21). It is consistent, iff thepolynomial M ′ ( z ) = n − X q =0 α q +1 z q (27)has simple and real roots. From the viewpoint of the representation theory, it means that the stability condition requires fromthe free extended Chern-Simons theory to describe the unitary reducible representation of the Poincar´e group. If the roots of M ′ ( z ) are multiple or complex, there is no bounded integral of motion that can stabilize the dynamics. Let us demonstrate the general construction in the case n = 3 . The action functional of the model reads S [ A ( x )] = 12 Z (cid:16) α m − ε µνρ ∂ ν G ρ + α G µ + α mF µ (cid:17) , α = 0 , (28)where the notation is used G µ ≡ F (2) µ = m − (cid:0) ∂ µ ∂ ν A ν − ∂ ν ∂ ν A µ (cid:1) , F µ ≡ F (1) µ = m − ε µρν ∂ ρ A ν . (29)5he model (28) is invariant under the following two-parameter series of symmetries: δ ξ A µ = β ξ ν ∂ ν A µ + β ξ ν ∂ ν F µ , (30)The corresponding two-parameter series of conserved tensors reads T µν ( α, β ) = β T canµν ( α ) + β T (2) µν ( α ) , (31)where T canµν ( α ) = m (cid:16) α (cid:0) G µ F ν + G ν F µ − η µν G ρ F ρ (cid:1) + α (cid:0) F µ F ν − η µν F ρ F ρ (cid:1)(cid:17) (32)is the canonical energy-momentum, and T (2) µν ( α ) = m (cid:16) α (cid:0) G µ G ν − η µν G ρ G ρ (cid:1) − α (cid:0) F µ F ν − η µν F ρ F ρ (cid:1)(cid:17) (33)is another independent conserved tensor. The canonical energy is linear in G µ and unbounded, T can ( α ) = m X µ =0 (cid:16) α G µ F µ + 12 α F µ F µ (cid:17) . (34)The -component of the general conserved tensor (31) has the form T ( α, β ) = m X µ =0 (cid:16) β α G µ G µ + 2 β α G µ F µ + ( β α − β α ) F µ F µ (cid:17) . (35)It can be bounded. The boundedness condition for the quadratic form reads α β > , − α β + α β β − α β > . (36)It is consistent if parameters of the model (28) satisfy the condition α − α α > . (37)According to the representation theory, the stable theory corresponds to one of the two cases: theory of two self-dual massivespin 1 with different masses, or theory of massless spin 1 and massive spin 1 subject to a self-duality condition [16, 17]. We considered a class of vector field models whose wave operator is a polynomial in the Chern-Simons operator. Wedemonstrated that the gauge theory of order n admits ( n − -parameter series of conserved tensors, whose -componentcan be bounded, while the canonical energy is always unbounded for n > . The bounded conservation laws ensure thestability of dynamics at classical level. At quantum level the stability is provided by bounded Hamiltonian. The constrained6amiltonian formulations with bounded Hamiltonian of the extended Chern-Simons were constructed in [18, 19]. Acknowledgments.
This research was funded by the state task of Ministry of Science and Higher Education of RussianFederation, grant number 3.9594.2017/8.9.
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