Universit`
a degli Studi di Pisa
Facolt`a di Fisica
Corso di Laurea Magistrale in Fisica Teorica
Is negative kinetic energy meta-stable in classical mechanics
and field theory?
Candidato: Matteo Zirilli Relatore: Prof.Alessandro Strumia Correlatore: Dr.Daniele Teresi
"Considerate la vostra semenza: fatti non foste a viver come bruti, ma per seguir virtute e canoscenza"
Acknowledgments
I would like to thank for their precious support and help Prof. Alessandro Strumia and Dr.Daniele Teresi. Moreover I desire to thank my family for having supported me during this period, in particular my brother Alessio who was always there when I needed him.
Contents
1 Introduction 5
2 Unitarity, gravity renormalizability and Ostrogradsky's theorem 8
1 Unitarity of a theory . . . 8
1.1 The optical theorem . . . 8
2 General relativity unitarity and non-renormalizability. . . 10
3 Higher derivative quantum gravity . . . 11
4 Ostrogradsky's theorem . . . 12
3 Classical mechanics 16 1 The Hamilton Jacobi equation. . . 17
2 Action-angle variables . . . 20
3 Canonical pertubation theory . . . 22
4 Ghost toy model: pertubative expansion . . . 25
5 Birkho series . . . 26
6 KAM theorem. . . 28
7 Ghost toy model: Birkho pertubative expansion . . . 29
7.1 First order in the coupling . . . 29
7.2 Generic order in the coupling . . . 30
8 Stability estimates . . . 31
8.1 Stability at lowest order . . . 32
8.2 Stability at generic order. . . 32
9 Resonances . . . 33
9.1 Example: stable ghost close to resonance . . . 34
9.2 Example: ghost that undergoes run-away close to resonance . . . 35
4 Classical eld theory 38 1 Classical equations of motion in momentum space. . . 40
2 Classical lattice equations . . . 40
3 Analytic study of one ghost resonance in eld theory . . . 43
5 The ghost run-away rate . . . 48
6 Lattice simulations . . . 49
Chapter 1
Introduction
Particle physics has experienced a great development in the last sixty years, contributing to a better understanding of how things work at fundamental level. Three of the four known interac-tions (electromagnetic, strong and weak interaction) are well-described by a consistent theory, the Standard Model, which enjoys of the fundamental properties of locality, renormalizability, unitarity. The gravitational interactions cannot be described by such a theory, because, when trying to include gravity, some of the fundamental consistence properties are lost, in particular renormalizability and unitarity.
The Hilbert-Einstein action (see section 2 of chapter 2) is not renormalizable. A good way to renormalize gravity is to consider the so called "higher derivative theories". Stelle introduced a trial higher-derivative quantum gravity action [2]. He included other terms in the classical gravity action. S = Z d4xpdet g R2 6f02 + 1 3R 2− R2 µν f22 − 1 2M 2 P lR+ Lmatter , (1.1)
where gµν is the graviton eld with mass-dimension 0, Rµν is the Ricci tensor, R is the scalar
curvature, Lmattercontains all the interactions with the other physical elds i.e scalars, fermions,
vectors, f0, f2 are the gravitational couplings. The rst two terms are graviton kinetic terms with
4-derivatives. However, higher derivative theories have problems with unitarity. The two princi-ples seemed to mutually exclude each other when gravity is considered. As it will be explained in section1of chapter 2, sacri cing this principle means, loosely speaking, admitting states with negative kinetic energy or equivalently stating the existence of negative probabilities, allowing a "ghost" degree of freedom, which seems to be a nonsense.
This thesis moves its steps from this point. The questions which it tries to answer are: consid-ering higher derivative theories, does the presence of negative energy states invalidate the whole theory? Does some metastable state exist, in the same way as with as unbounded-from-below potential energy where a meta-stable state can exist in a false vacuum? If the answer to the previous question is positive, how long does this state remain meta-stable? These questions are investigated in their classical physics aspects i.e classical mechanics and classical eld theory. To
have an idea of what happens in the quantum mechanics case and in the quantum eld theory case see [1]. Ostrogradsky proved that is always possible to switch from a classical degree of free-dom with 4-derivatives to two degrees of freefree-dom with 2-derivatives, one of them being a ghost with negative kinetic energy (see section4of chapter 2 and [5] for completeness). The 4-derivative graviton splits into the massless graviton and a ghost-graviton with mass Mghost = f2MP l/
√ 2. The full action (see [3]) is
S ∝ Z d4x√−g R+ 1 6m20R 2− 1 2m22C 2 , (1.2) where C2 = RλµνρRλµνρ− 2RµνRµν+1 3R 2.
When a classical degree of freedom with positive kinetic energy interacts with a negative kinetic energy degree of freedom, troubles may arise since the individual energies can diverge, while the total energy is conserved. To know whether a metastable state can exist, a simpler system is studied that contains physical issues equivalent to the more complicated higher-derivative gravity: L= m1 ˙q12 2 − ω 2 1 q12 2 ! − ˙q 2 1 2 − ω 2 2 q22 2 ! − λ 2q 2 1q22. (1.3)
The positive-energy degree of freedom q1 interacts with the negative-energy degree of freedom q2
by means of a quartic interaction λq2 1q
2
2. In the case of relativistic eld theory, scalar elds φ1,2
are considered with Lagrangian density L = (∂µφ1) 2− m2 1φ21 2 − (∂µφ2)2− m22φ22 2 − λφ21φ22 2 . (1.4)
The thesis is composed by the following parts:
In the second chapter the concept of unitarity [6] and renormalizability of gravity will be introduced. The implications of the optical theorem will be analyzed. Unitarity and non renormalizability of the Einstein-Hilbert action will be shown [14]. Subsequently the focus will be moved to the general case of higher derivative quantum gravity, whose describing action is renormalizable but not unitary [14]. The last part of this chapter regards the Ostrogradsky's theorem, of fundamental importance for understanding where the negative energies emerge in a higher derivative theory ( [4], [5]) and what are the implications; The third chapter will focus on the analysis of the system described by eq. (1.3). All
the mathematical tools necessary to analitically understand the classical physical aspects will be developed in detail, i.e Hamilton Jacobi equations, action-angle variables, Birkho series expansion, Kolmogorov-Arnold-Moser theorem [7]. Analitycal understanding will be compared to numerical simulation in terms of region of stability and stability time. Varying the value of the coupling interaction λ, the time at which run-aways take place is computed. During the evolution, the system can hit some resonance (such as ω1 = ω2), which can lead
to run-away solutions. The results show that a ghost system is stable if the interactions are small, while it goes through run-away solutions if the interactions are large;
Finally the classical eld theory case is considered. In this case the number of possible res-onances which could lead to run-away solutions even at small coupling is in nite. Therefore again analytical and numerical analysis is necessary to better understand the issues. Local eld theories can give resonances of benign type, while the in nite number of resonances re-moves the hidden quasi constants of motion that "protect" the system. Therefore, assuming no protection, a statistical analysis is performed to show that in this case a thermal state cannot exist: heat keeps owing from ghost elds to positive-energy elds, so that entropy increases. The Boltzamm equations will be necessary to compute the rate of instability. In the case of 4-derivative gravity, the interactions between graviton and ghost are so small that the ghost run-away rate is not problematic in cosmology.
Conclusions are presented in the last chapter. This thesis will follow the work done in [1].
Chapter 2
Unitarity, gravity renormalizability and
Ostrogradsky's theorem
1 Unitarity of a theory
Unitarity is one of the fundamental features a quantum eld theory must have to well de ne a scattering theory. Unformally, it states the conservation of probability. Unitarity constrains the states of a Hilbert space, which de ne a system. It also constrains the form that interactions can have in the system since the S-matrix must be unitary. The next subsection treats the implications of unitarity focusing on the optical theorem which gives a non pertubative relation between the cross sections and the imaginary part of scattering amplitudes.
1.1 The optical theorem
Conservation of probability in quantum theory implies the norm of a state to be stationary, so given the state |ψ; ti the following must be true:
hψ; t|ψ; ti = hψ; 0|ψ; 0i , ∀t. (2.1)
Since |ψ; ti = e−iHt
|ψ; 0i, in order to have a unitary theory the Hamiltonian operator Hb must be hermitian so Hb
†
= bH. The S-matrix is S = e−iHt, so, with the hermiticity property of the Hamiltonian operator, the S-matrix must be unitarian:
SS†= 1. (2.2)
Writing the S-matrix as S = 1 + iT , the S-matrix elements can be written as:
hf |T |ii = (2π)4δ4 pf − pi M (i → f). (2.3)
where the kets |ii and |fi indicate some initial and nal states. Unitarity implies that 1 = SS†
= (1 + iT )1 − iT†, which leads to the relation:
Sandwiching the left hand side of equation (2.4) between the initial and nal states i and f and using the equation (2.3) the relation is:
hf |i(T†− T )|ii = i hi|T |f i∗− i hf |T |ii = i(2π)4δ pf− pi
M∗(f → i) − M (i → f ) . (2.5) Inserting a completness relation it produces:
hf |T†T)|ii = X X Z dΠXhf |T†|Xi hX|T |ii =X X (2π)4δ4(pf − X)δ4(pi− X) Z dΠXM(i → X)M∗(f → X). (2.6)
Substituting the right hand side of equations (2.5) and (2.6) in (2.4) the following equivalance must be true: M(i → f ) − M∗(f → i) = iX X Z dΠX(2π)4δ4(pf − pi)M (i → X)M ∗ (f → X). (2.7) Equation (2.7) represents the generalized optical theorem, which must hold in pertubation theory for every order. It is important to notice that the LHS of the above equation contains matrix elements, instead the RHS contains matrix elements squared. This fundamental relation tells that for the order λ2 in pertubation theory the left hand side must be a loop in order to be
consistent with the right hand side, which is a tree-level computation. So to keep unitarity a classical interacting theory must have loops. In the particular case where the initial and nal states coincide (|ii = |fi = |Ai), equation (2.7) becomes:
2iImM (A → A) = iX X Z dΠx(2π) 4 δ4(pf − pi)|M (A → X)| 2 . (2.8)
The decay rate of the one-particle state |Ai is: Γ(A → X) = 1
2mA
Z
X
dΠx(2π)4δ4(pf − pi)|M (A → X)|2, (2.9)
relating it to equation (2.8) it becomes:
ImM(A → A) = mAX
X
Γ(A → X) = mAΓT OT. (2.10) Therefore, the imaginary part associated with the exact propagator is related to the total decay rate(which is the lifetime's inverse of a particle) times the mass. Similarly if |Ai is a two-particle state then there is a direct relation between the amplitude and the cross section in the center of mass frame. This is because the cross section is:
σ(A → X) = 1 4ECM|−→pi|
Z
X
dΠx(2π)4δ4(pf− pi)|M (A → X)|2, (2.11) which leads to the optical theorem:
Finally it is important to estabilish whether the propagators of some elds of a theory propagates physical degrees of freedom or not. Considering the simplest case of the imaginary part of a scalar propagator (which is the meaningful one in computing physical processes):
Im 1 p2− m2+ i = 1 2i 1 p2− m2+ i− 1 p2− m2− i = − p2− m22+ 2 . (2.13)
The above expression vanishes as → 0, except near p2
= m2. The integration over p2 results in: Z +∞ 0 dp2 − p2− m22+ 2 = −π (2.14)
which implies that
Im 1 p2− m2+ i = −πδ(p2− m2) (2.15)
stating that the imaginary part of the propagator is non-null only if the on-shellness condition is satis ed.
2 General relativity unitarity and non-renormalizability
Gravity is well described at low energies by general relativity, which moves its steps from the Hilbert-Einstein action:
SHE(g) = − 1 2κ2
Z
d4x√−g (R + 2ΛC) (2.16)
where ΛC is the cosmological constant with [ΛC] = 2, R is the scalar curvature, function of the
metric gµν. This is a dieomorphisms invariant action (xµ(x) → x
0µ(x)). A dieomorphism is
a function between two dierentiable manifolds with the property of being dierentiable itself, invertible and having the inverse dierentiable. Coupling gravity to matter through a matter action Sm, the equation of motion is derived:
Rµν− 1 2gµνR − gµνΛC = κ 2 Tµν (2.17) with Tµν de ned as Tµν = − 2 √ −g δSm δgµν
, i.e the energy momentum tensor of the matter sector Sm.
Therefore, the local transformation de nes a gauge trajectory in which a gauge must be xed in order to overcome the usual problem of degrees of freedom's redundance of a quantum eld theory. This is why the distinct con guarations of a theory are equivalance classes of eld con gurations. Two descriptions of the theory, which are related to one another by a gauge transformation (in the gravity case a dieomorphism), are two equivalent physical con gurations. The solutions of a QFT exist not in a space of elds with values at every point of the spacetime but in a quotient space whose elements are equivalence classes of eld con gurations. Therefore, the action must
be gauge- xed: SHE → SGF. The gauge usually used is Gµ = ∂νφµν − ω2∂µφ, where ω is a
gauge- xed parameter and φ is the graviton eld. SGF = SHE+
1 2λ
Z
d4xGµηµνGν+ Sghost. (2.18)
Changing the action arbitrarily is not possible because the physics cannot be changed. This is why another term has been added to SGF which is Sghost. The ghost action involves two new
elds: cµ and cµ, which is the Faddeev-Popov ghost.
Sghost = Z d4xd4ycµ(x)δGµ(x) δφρσ(y) δcφρσ(y) = Z d4xcµ∂νδcφµν−ω 2∂µδcφ (2.19) where by δcφρσ(y) it is meant an in nitesimal variation of the eld under dieomorphism. The
variation under dieomorphism of the φ- eld is directly related to the variation of the metric gµν. The physical results are completely independent of Gµ, λ, ω, these are gauge-independent,
no dependance on the gauge- xing parameters. Isolating the quadratic parts in the action SGF
of the graviton and ghost elds their propagators come out: -GRAVITON→ Pµνρσ = i k2 ηµρηνσ+ ηµσηνρ− ηµνηρσ 2 -GHOST∝ −→1 k2.
It can be immediately noticed that the graviton propagator has a pole at zero momentum (the graviton is a massless particle) and the ghost propagator has no poles. This is consistent with the equation (2.15), telling which degrees of freedom are physical or not. It is important to note that this is not the same "ghost" de ned in the introduction, with negative kinetic energy (which is a physical degree of freedom). This comes from the gauge- xing procedure.
Focussing on renormalizability, it must be noticed that general relativity is non renormalizable. From the expansion of the metric gµν = ηµν+ 2κφµν where the energy dimension of the metric
is [gµν] = 0and from the quadratic term of the graviton action it comes out that [φ] = 1, which
leads to the negative dimension of the coupling constant [κ] = −1. This is why an in nite number of counter terms can be built, with arbitrarily large dimensions each one multiplying a suitable power of κ.
3 Higher derivative quantum gravity
There are ways to make the theory of gravity renormalizable. For example, considering the following action:
SHD= − 1 2κ2
Z √
−g2Λ + ζR + αRµνRµν+ βR2. (2.20) The theory previously described (general relativity) is an eective theory, predictive in the low energy approximation. In order to extend it to describe a quantum theory of gravity, it must be multiplied by the extra quadratic Ricci terms with small parameters (in this case α and β). In the case of the higher derivative quantum gravity, the parameters α and β are not small.
The expansion must be done pertubatively around the entire SHD. This changes completely
the theory, because expanding around the atspace (gµν = ηµν + 2κφµν), and ignoring the
cosmological constant, it comes out that the quadratic part contains: ζ(∂φ)2
+ αφ∂4φ+ βφ∂4φ. The last two terms derive from the action terms proportional to RµνRµν and R2. In this case, the
propagator for large momentum decreases with the fourth power: ∝ 1
p4−p2. In 4 dimension, by
power counting, this is su cient to make the theory renormalizable. If energy dimension of the parameters are chosen so that [α] = [β] = 0, then consequently [φ] = 0 ⇒ [ζ] = 2, [Λ] = 4 and [κ] = 0. Therefore, the theory does not contain any coupling of negative dimension, which means that it is a strictly renormalizable theory. By adding new terms to the action more renormalizable theories can be constructed (superrenormalizable theory):
SHD = − 1 2κ2
Z √
−g2Λ + ζR + αRµνRµν+ βR2+ γR2R + δRµν2Rµν+ O(R3). (2.21) The parameters and eld dimension are [δ] = [γ] = 0,[φ] = −1, [α] = [β] = 2,[ζ] = 4,[κ] = 1. The number of divergent terms quickly disappear after a certain order of pertubative expansion. Therefore, the huge advantage of these theories is that they are renormalizable, the great problem is that they violate unitarity. The propagator of equation (2.20) can be found as usual and written proportional to: − m 2 p2(p2− m2) = 1 p2 − 1 p2− m2 . (2.22)
The minus sign between the two propagators is what makes the whole structure fail because, following the same arguments of the previous sections, it comes out that the imaginary part of the propagator is proportional to −πδ(p2) − δ(p2− m2). The theory has an eective physical degree of freedom which propagates with a negative normalization, and this is a drama for the physics. It is a ghost, but not a Faddev-Popov ghost, this is the real problem. Higher derivative theories have problems also at the classical level. Instead of having a lagrangian L(q, ˙q), they have a lagrangian L(q, ˙q, ¨q, ...). The minus sign will be seen as an instability in the energy. The energy seems to be not bounded from below. In the next section this topic will be analyzed in more detail.
4 Ostrogradsky's theorem
In the previous section, the problems that emerge when an higher derivative theory is considered were brie y discussed. In this section a more speci c discussion will be developed. Ostrogradsky was a Russian mathematician and physicist whose credit was having generalized the Hamilton's construction to the higher derivative case. This construction turned out to be remarkable for fundamental theories.
Lagrangian L = L(x, ˙x). The Euler-Lagrange equations are: ∂L ∂x − d dt ∂L ∂˙x = 0. (2.23)
Assuming a non-degenerate lagrangian, which means in this case that ∂
2
L
∂x2 6= 0, the Euler-Lagrange equations can be written in the usual Newton form:
¨
x= F (x, ˙x) =⇒ x(t) = χ(t, x0,˙x0) (2.24) where x0 = x(0)and ˙x0 = ˙x(0)are the initial conditions.
Considering two canonical coordinates X and P de ned as X ≡ x, P ≡ ∂L
∂˙x. These relations can be inverted because of the non-degeneracy assumption. Therefore, a function V (X, P ) must exist such that ˙x = V (X, P ) and ∂L
∂˙x|x=X, ˙x=V (X,P ). By Legendre-transforming in the ˙x coordinate the Lagrangian L the Hamiltonian is obtained:
H(X, P ) = P ˙x − L = P V (X, P ) − L(X, V (X, P )) (2.25) from which the Hamilton equations can be found:
˙ X= ∂H ∂P = V + P ∂V ∂P − ∂L ∂˙x ∂V ∂P = V (2.26) ˙ P = −∂H ∂X = −P ∂V ∂X + ∂L ∂x + ∂L ∂˙x ∂V ∂X = ∂L ∂x. (2.27)
Furthermore, when the Lagrangian has no explicit time dependence the Hamiltonian is a con-served quantity. Considering a Lagrangian of the functional form L(x, ˙x, ¨x) the Euler-Lagrange equation is: ∂L ∂x − d dt ∂L ∂˙x + d dt2 ∂L ∂x¨ = 0. (2.28)
The non-degeneracy condition is ∂
2
L
∂¨x2 6= 0, which implies that the Euler-Lagrange equation can be written in the form:
....x
= F (x, ˙x, ¨x,...x) =⇒ x(t) = χ(t, x0,˙x0,x¨0,...x0). (2.29)
Solutions now depends upon four pieces of initial values, four canonical coordinates can be de ned: X1 ≡ x; (2.30) P1 ≡ ∂L ∂˙x − d dt ∂L ∂¨x; (2.31) X2 ≡ ˙x; (2.32) P2 ≡ ∂L ∂x¨. (2.33)
Even in this case non-degeneracy allows to invert the previous relations. Therefore, a function A(X1, X2, P2) must exists such that
∂L
∂¨x|x=X1, ˙x=X1,¨x=A= P2. (2.34)
The Hamiltonian becomes: H(X1, X2, P1, P2) = 2 X i=1 Pix(i)− L = (2.35) = P1X2+ P2A(X1, X2, P2) − L(X1, X2, A(X1, X2, P2)).
The time evolution equations are: ˙ Xi= ∂H ∂Pi , ˙Pi = −∂H ∂Xi (2.36) The Hamiltonian (2.35) is linear in the canonical momentum P1, which means that no system
of this form can be stable. A stable system is the harmonic oscillator which is quadratic in the canonical coordinate P .
The results just obtained are general, the only assumption is non-degeneracy. It is useful to look at an example to better clarify this issue. Considering the lagrangian:
L= −m 2ω2x¨ 2 +m 2 ˙x 2−mω2 2 x 2 (2.37)
The Euler-Lagrange equation is:
0 = −m ω2 ....x2 + ¨x+ ω2x (2.38) whose solution is:
x(t) = C+cos(k+t) + S+sin(k+t) + C−cos(k−t) + S−sin(k−t) (2.39)
where the frequencies k± are de ned as:
k±≡ ω
s
1 ∓√4
2 , (2.40)
the constants C± and S± are de ned as:
C±= k2∓x0+ ¨x0 k∓2 − k2± ; (2.41) S±= k∓2 ˙x0+...x0 k± k∓2 − k±2 ; (2.42)
the momenta are:
P1 = m ˙x + m ω2
...x ⇔ ...x = ω2P1− mω2X2
P2 = −
m
ω2x ⇔¨ x ≡ A¨ = − ω2P2
m (2.44)
It is interesting to express the Hamiltonian in terms of the canonical variables: H = P1X2− ω 2 2mP 2 2 − m 2X 2 2 + mω2 2 X 2 1 = = m 2 √ 1 − 4k+2 C+2 + S+2−m 2 √ 1 − 4k−2 C−2 + S−2 . (2.45)
If the hamiltonian is written in this form it is clear that there are positive energy modes (+) and negative energy modes (−) that generates the instabilities. Similar issues happen for the gravity case.
The argument can be extended also to N derivatives. The procedure is analogous to the ones made above. The Hamiltonian will be linear in P1, P2, ..., PN −1. Taking into account of all the
interactions, such a system will suer of such instabilities, which lead to a collection of positive and negative energy excitations. In a continuum eld theory the in nite number of degrees of freedom, and so the great entropy, will make the decay instantaneous. The kind of instability which develops from these arguments is a kinetic one, though potential energy instabilities are more familiar. There is a problem with kinetic energy, this is why the dynamical variables de-velop a special time dependence. The examples show that instable Hamiltonians are linear in several canonical coordinates. Therefore, moving in the phase space one can arbitrarily increase or decrease the energy of the system. If the system interacts, there will be interactions between positive-energy particles with the negative ones. Depending on the interaction the empty state can decay into a collection of such states. In the next chapters the implications of such theorem will be discussed in the classical case. It will be argued whether in the case of K-instability some metastable state can emerge (as in the V-instability case). The analysis of a simpler model will be developed, compared with the gravity case, with one degree of freedom in 0 + 1D with 4-derivative kinetic term and quartic interaction. The system's Lagrangian will be rewritten as a system with two degrees of freedom, one of them being a ghost (the cause of K-instability).
Chapter 3
Classical mechanics
The Lagrangian to be studied is L= −1 2q ∂ ∂t2 + ω1 ∂ ∂t2 + ω2 q − VI(q, ¨q). (3.1) Adding zero as a perfect square containing an auxiliary degree of freedom ˜q so that
L= 1 2 − ¨q2+ (ω21+ ω22) ˙q2− ω21ω22q2 +1 2 ¨ q+ (ω12+ ω22) q 2− ˜ q 2 2 − VI, (3.2) the Lagrangian can be expanded to become
L= −q˜q¨ 2 + (ω 2 1 − ω 2 2) 2q2 8 − (ω 2 1 + ω 2 2) ˜ qq 4 + ˜ q2 8 − VI. (3.3)
The previous formula can be diagonalized in the kinetic and mass term by performing a eld-rede nition ˜ q = q ω22− ω12(q1− q2) q = (q1+ q2)/ q ω22− ω12 (3.4)
Rewriting the Lagrangian in terms of the new variables q1,2 the equivalent system to be studied
becomes L= ˙q 2 1− ω 2 1q 2 1 2 − ˙q22− ω22q22 2 − VI q1+ q2 q ω22− ω12 , −ω 2 1q1+ ω 2 2q2 q ω22− ω12 . (3.5)
In the case of a quartic interaction i.e VI= λ2q 2 1q
2
2 the classical equations of motion are
¨
q1+ ω12q1+ λq1q22 = 0, q¨2+ ω22q2− λq2q21 = 0. (3.6) Up to rescalings they have one free parameter ω1/ω2. The total energy E = E1− E2+ VI is the
unique constant of motion where
Ei = ˙q 2 i 2 + ω 2 i q2i 2 >0 (3.7)
Numerical evolution shows that solutions starting from |E1− E2| & VI quickly undergo run-away
i.e the system is unstable. On the other hand, for solutions starting from small enough initial energies E1, E2 VI, E1(t) and E2(t) evolve remaining con ned to a small range, for a time
longer than what can be numerically computed. To understand analitically why such behaviour happens some mathematical tools are used. The next section will describe such tools in a brief digression. It will present a recap of the Hamilton-Jacobi theory and some useful issues of the canonical pertubation theory will be described.
1 The Hamilton Jacobi equation
The problem of nding the solutions to the equations of motion of an Hamiltonian system with Hamiltonian H(−→p , −→q , t)can be reduced to nding a canonical transformation from the variables (−→p , −→q) to new variables (−→P ,−→Q)generated by a function S(−→q ,−→P , t)in such a way that the new Hamiltonian K(−→P ,−→Q , t) = 0. In this case the integration of the equations of motion is simple and for every t the following conditions are valid:
Pj(t) = ηj Qj(t) = ξj, j = 1, ..., l. (3.8) (−→η ,−→ξ) are constant vectors which are determined from the initial conditions. Through the inverse transformation the original variables can be reobtained:
−
→p = −→p(−→η ,−→ξ , t) →−q = −→q(−→η ,−→ξ , t). (3.9) To realise such a trasformation the generating function must be found:
S = S(−→q , −→η , t) (3.10)
The equation to be solved, consistently with the theory of canonical transformation, is: H(∇−→qS, −→q , t) +∂S
∂t = 0. (3.11)
The following condition must be valid to ensure the invertibility of the transformation: det ∂ 2 S ∂qi∂ηj ! 6= 0, (3.12)
Equation (3.11) is known as the Hamilton-Jacobi equation. The independent variables are q1, ..., ql,
t. It is not necessary to nd a general integral of the equation (i.e a solution depending on an arbitrary function), it is su cient to nd a complete integral, i.e a solution depending on as many constants as the number of independent variables (l + 1). Since in the equation (3.11) the generating function S appears with its rst derivatives, S(−→q , −→η , t) + η0 surely will be a solution
equation (3.11) and satis es the condition (3.12) then, because of dealing with canonical trans-formations and canonical coordinates (−→η ,−→ξ), whose Hamiltonian is identical zero, the remaining coordinates are: pj = ∂S ∂qj ξj = ∂S ∂ηj j= 1, ..., l (3.13)
It is worthwhile noticing an interesting physical remark. Computing the time derivative along the motion, the Lagrangian emerges:
dS dt = l X j=1 ∂S ∂qj ˙qj+ ∂S ∂t = l X j=1 pj˙qj− H = L. (3.14) It is evident that S|t1 t0 = Rt1
t0 L is the classical action. If the Hamiltonian H does not depend
explicitly on time, a solution in the following form can be found
S = −E(−→α)t + W (−→q , −→α) (3.15) where −→α = (α1, ..., αl) denotes the vector of l arbitrary constants and E(−→α) is a function of
class at least C2 such that ∇
αE 6= 0. The Hamilton Jacobi equation (3.11) reduces to
H ∇−→qW, −→q = E(−→α) (3.16)
From the previous equation it is evident that the function E is the total energy. The function W is called Hamilton's characteristic function, which satis es the relation
∂2S ∂qi∂αj =
∂2W
∂qi∂αj. (3.17)
Thus W is the generating function of a canonical transformation in the new variables (−→α ,−→β) with the new Hamiltonian E(−→α). The coordinates β1, ...βl are cyclic, so that the new kinetic momenta α1, ..., αl are rst integral of the motion. The βj coordinates satisfy
˙
βj = ∂E ∂αj
= γj(−→α) → βj(t) = γj(−→α)t + βj(0). (3.18)
The invertibility condition, which relates the new coordinates (−→α ,−→β) to the old coordinates (−→p , −→q), is assured by the condition on the generating function det ∂
2 W ∂βj∂αj ! 6= 0. The relations βj = ∂W ∂αj, pj = ∂W ∂qj , j = 1, ..., l (3.19)
yield the solutions
qj(t) = qj(α1, ..., αl, γ1t+ β1(0), ..., γlt+ βl(0)), j= 1, ..., l. (3.21) It is useful to look at an example in order to better clarify the procedure. An easy case is the one-dimensional harmonic oscillator. The Hamiltonian is
H(p, q) = 1 2m
p2+ m2ω2q2. (3.22)
The Hamilton-Jacobi equation (3.11) takes the form 1 2m " ∂S ∂q 2 + m2ω2q2 # +∂S ∂t = 0. (3.23)
Setting S(q, E.t) = W (q, E) − Et, the equation (3.23) then becomes 1 2m " ∂W ∂q 2 + m2ω2q2 # = E. (3.24) Solving for W W(q, E) =√2mE Z q q0 s 1 −mω 2 x2 2E dx. (3.25)
Choosing q0 = 0 the result of the previous integration is
W(q, E) = 1 2 √ 2mE q s 1 −mω 2 x2 2E + r 2E mω2 arcsin s mω2 2E q . (3.26) The β coordinate is found by dierentiating W with respect to E:
β = 1 ωarcsin s mω2 2E q . (3.27)
Inverting the previous formula and dierentianting again W with respect to q the old coordinates (p, q) can be found in terms of (E, β)
p= ∂W ∂q = √ 2mE cos(ωβ); (3.28) q= r 2E mω2sin(ωβ). (3.29)
In this case β = t+β(0) and by imposing the initial conditions the well-known solution (p(t), q(t)) is found. For more details about the integrability methods see [7].
2 Action-angle variables
In this section a system of canonical coordinates will be introduced. This is very useful to study a wide range of systems in which the one described by the Lagrangian (3.5) is included.
Consider an Hamiltonian of a system with one degree of freedom H(p, q) = E. This relation is satis ed by a closed curve in the phase space γ = γE. In this case the motion is called libration
or oscillatory motion. The area enclosed by the curve is proportional to the energy E. The question is: does a canonical transformation exist, which leads to a pair of canonical coordinates (J, χ) ∈R ⊗ S1, that allows to write the Hamiltonian in terms of the variable J? The coordinate χ is an angle. It satis es
I
γE
dχ= 2π, (3.30)
E = H (p(J, χ), q(J, χ)) = K(J). (3.31)
The system is said to be completely canonically integrable and the new variables are the action-angle variables. The Hamilton equations become
˙ J = −∂K ∂χ = 0, χ˙ = ∂K ∂J . (3.32) Setting ω = ω(J) = dK dJ (3.33) this yields J(t) = J(0), χ(t) = χ(0) + ω(J(0))t, ∀t ∈R. (3.34) The action variable is a constant of the motion. The motion is periodic with period T = 2π
ω(J ).
Taking again as example the harmonic oscillator, the canonical trasformation to action-angle variables is
p=√2mωJ cos(χ), q =r 2J
mωsin(χ). (3.35)
The new Hamiltonian is
K(J) = ωJ (3.36)
A general de nition of the action variable is J = 1 2π I γE pdq = A(E) 2π , (3.37)
where A(E) is the area enclosed by the curve γE and the integration is made over a period.
In general every Hamiltonian system with one degree of freedom and with periodic motion is canonically integrable. This is true also in the case of l degrees of freedom if a completely canonical transformation exists
−
→p = ˜p(−→J , −→χ); (3.38)
−
→q = ˜q(−→J , −→χ). (3.39)
The dependence of ˜p and ˜q on each variable χi is 2π-pediodic.
The new variables are the action angle variables: (−→J , −→χ) ∈ Rl ⊗Tl. Tl represents the l-dimensional torus where the χi coordinates live in.
Arnold proved that considering an Hamiltonian H(−→p , −→q) with l degrees of freedom and which has l independent rst integrals of the motion, the manifold M of the rst integrals is compact and connected. Therefore it is dieomorphic to an l-dimensional torus (analogously in the 1-dimensional case the closed curve γE is dieomorphic to a circumference).
The new Hamiltonian K is only function of the actions−→J. Then let Ma be the manifold of the rst integrals of the motion−→f, compact and connected, it can be paramatrized by l angles. For xed−→f = −→a and varying between [0, 2π] only one of the angles, a cycle γi⊂ Ma is obtained for i= 1, ..., l. It is now possible to introduce the action variables de ned in the l-dimensional case as Ji = 1 2π I γi l X j=1 pjdqj (3.40)
In summary in this section a new useful system of canonical coordinates has been introduced. This consists in the action-angle variables, which concerns with completely integrable hamiltonian systems for which the phase space can be foliated in invariant tori. If l = 1 the motion is periodic. This latter property is not assured in the higher dimensional case.
− →
˙
χ = −→ω (3.41)
In the higher dimensional case the following theorem helps to distinguish between periodic and quasi-periodic motion. De ned M−→ω as
M−→ω =
h−→
m ∈Zl|−→m · −→ω = 0i, (3.42) of dimension varying between 0 and l − 1 (the trivial case −→ω = −→0 is excluded), the motion's orbit is:
dense on the whole torus Tl if dim M−→ ω = 0;
dense on a torus of dimension l − d in the d-dimensional case with 0 < d < l − 1. If the last two conditions hold the motion is called quasi-periodic.
Going back to the toy ghost model described by the Lagrangian (3.5) the transformation to action-angle variables leads to the Hamiltonian
H = ω1J1− ω2J2+ εJ1J2sin2χ1sin2χ2, ε= 2λ ω1ω2
. (3.43)
The system described by the previous Hamiltonian will be discussed in section (4) of chapter 3.
3 Canonical pertubation theory
This section will focus on the study of complete canonically integrable system h0 pertubed by a
small pertubation. Such system is called quasi-integrable system
h(−→p , −→q , ε) = h0(−→p , −→q ) + εf (−→p , −→q), (3.44) where h0 is the Hamiltonian of the complete canonically integrable system,(−→p , −→q ) ∈ R
2l and
0 < ε 1 is the pertubation of the previous unperturbed Hamiltonian. Since h0 is completely canonically integrable it is possible to change variables and write it in terms of action-angle variables (−→J , −→χ). After the transformation the new Hamiltonian is
H(−→J , −→χ , ε) = H0(→−J) + εF (−→J , −→χ). (3.45) The function F is periodic in all the angles χ1, ..., χl and assumed regular of class C
∞
, as well as H and H0, in each argument. The procedure to follow in order to solve the system
is to nd a new completely canonical transformation, which eliminates the dependence of the Hamiltonian on the angular variables at rst order in ε. Then the iteration must be continued until the dependence on the χi variables is eliminated at the desired order. The goal is to nd a
generating function W (−→J0, −→χ , ε) that allows the transformation from the action-angle variables (−→J , −→χ) to new variables (−→J0, −→χ0), where the new Hamiltonian H0(−→J0, −→χ0, ε) is independent of the angular variables, at least up to order O(ε2
) H0(−→J0, −→χ0, ε) = H00( − → J0) + εH10( − → J0) + ε2F0(→−J0, −→χ0, ε). (3.46) Noticing that when ε = 0 the Hamiltonian (3.46) is independent of the angle variables, it can be seen that the trasformation to look for is ε-near the identity. Then the generating function W can be expanded into a power series in ε whose zero-order term is the generating function of the identity transformation.
with W(1)
(−→J0, −→χ , ε) unknown function. The transformation it generates is Ji= Ji0+ ε∂W (1) ∂χi (−→J0, −→χ) + O(ε2), i= 1, ..., l, (3.48) χ0i = χi+ ε∂W (1) ∂Ji0 (−→J0, −→χ) + O(ε2), i= 1, ..., l. (3.49) Substituting equation (3.48) into (3.45) and using equation (3.46), the following equation is found
H0(
− →
J0+ ε∇−→χW(1)) + εF (→−J0, −→χ) + O(ε2) = H00(−→J0) + εH10(−→J0) + O(ε2) (3.50)
At zero-order in ε the previous equation gives H00( − → J0) = H0( − → J0) (3.51)
therefore the two Hamiltonians coincide at the lowest order. At the rst order in ε the equation reads
− →ω(−→J0
) · ∇χW(1)(−→J0, −→χ) + F (−→J0, −→χ) = H10(−→J0), (3.52) where W(1)
(−→J0, −→χ) and H10(→−J0) are unknown functions and −→ω(−→J0) = ∇−→ J0H
0
0 is the vector of
frequencies of the new Hamiltonian. Equation (3.52) is a linear partial dierential equation of rst order on the torus Tl. If equation (3.52) admits a solution, with W(1)
(−→J0, −→χ) a 2π-periodic function with respect to −→χ, the action variables satisfy the equation of motion
˙
Ji0 = −∂H
0
∂χi
(−→J0, ε) = O(ε2), i= 1, ..., l. (3.53) In the l = 1 case the equation (3.52) has solution
H10(J0) = 1 2π Z 2π 0 F(J0, χ)dχ, (3.54) W(1)(J0, χ) = 1 ω(J0) Z χ 0 H1(J0) − F (J0, x) dx. (3.55) which is unique if 1 2π Z 2π 0 W(1)(J0, χ)dχ = 0. (3.56)
For this case it is possible to formally solve the following Hamilton-Jacobi equation to all orders in ε
H(∇−→χW, −→χ , ε) = H0(∇−→χW) + εF (∇−→χW, −→χ) = H0(−→J0, ε), (3.57)
obtained by having substituted the solutions of the equations of motion of the new Hamiltonian into equation (3.52). The formal solution for the l = 1 and ω(J0
H0(J0, ε) = ∞ X n=0 εnHn0(J0), (3.58) W(J0, χ, ε) = J0χ+ ∞ X n=0 εnW(n)(J0, χ), (3.59) which is unique if it is assumed that W(n) has zero average with respect to χ for every n ≥ 1.
The previous series expansions are uniformally convergent, under the assumptions that H0 and
F are analytic functions.
Going back to the l-dimensional case, it is possible to show that a formal solution W(1) of (3.52)
exists in the non-resonant case. It is also unique if it is required that the mean of W(1) on the
torus Tl is zero. Then a Fourier series expansion is possible. In order to show the uniqueness of
the solution the Fourier expansion is performed, ignoring the convergence of the series problem: F(−→J0, −→χ) = X − → m∈Zl b Fm( − → J0)ei−→m·−→χ (3.60) W(1)(−→J0, −→χ) = X − →m∈Zl c Wm(1)(−→J0)ei−→m·−→χ. (3.61) Substituting the previous equations in (3.52) the following expression is found:
i−→m · −→ω(−→J0)cW−→m(1)(
− →
J0) + bF−→m(−→J0) = 0, (3.62) for every −→m ∈Zl with −→m =−→0 excluded. The previous equation is solved by
c W−→m(1)( − → J0) = Fb−→m( − → J0) −i−→m · −→ω. (3.63)
The denominator must not vanish, therefore the above expression is valid only in the non-resonant case i.e −→m · −→ω 6= 0. The frequencies −→ω depend on the action variables in general, therefore they are not constant. Under the assumption of non-degeneracy of the integrable Hamiltonian (i.e ∀−→J ∈ A ⊂ Rl, det ∂ 2 H0 ∂Ji∂Jk ! (−→J)
≥ c) the map ω : A −→ Rl is a local dieomorphism i.e − → J −→ −→ω(−→J) = ∇−→ JH0( − →
J). It is possible to perform an expansion of the rst integrals I in terms of the parameter ε to nd a solution for the equation of the rst integrals I, H = 0:
I(−→J , −→χ , ε) =
∞
X
n=0
εnI(n)(−→J , −→χ). (3.64) This expansion has to be substituted in the equation for the rst integrals, in order to obtain an in nite system of equations for the coe cients of the expansion
Poincaré proved that if H(−→J , −→χ , ε) is a non-degenerated Hamiltonian of a quasi-integrable sys-tem, an analytic rst integral of motion I(−→J , −→χ , ε), for which the previous expansion is well de ned and convergent with ε su ciently small, does not exist.
4 Ghost toy model: pertubative expansion
In the previous sections the utility of a convenient change of variables (action-angle variables) and the ε-expansion has been discussed (see equations (3.48) and (3.49)). In this section the system described by the Hamiltonian (3.43) will be studied with the formalism of pertubation theory. The energies Ei are: Ei = ωiJi ≥ 0. It can be noticed that the Hamiltonian (3.43) is
linear in the action variables. Since the action variables are positively de ned, the minus sign in front of the action variable J2 signals a ghost, which could lead to an instability as Ostrogradsky
proved. For small ε the equations of motion (3.53) can be analytically solved as power series in ε. At 0th order the equations of motion are solved by
Ji(t) = Ji0, χ1(t) = χ1(0) + ω1t, χ2(t) = χ1(0) − ω2t. (3.66) Ji0is constant for i = 1, 2. At rst order the equations of motion for the action variables become J10 = −εJ10J20sin(2ω1t) sin(ω2t)2, J20 = εJ10J20sin(2ω2t) sin(ω1t)2, (3.67) which are solved by
J1(t) = J10+ εJ10J20ω1ω2−1cos(2tω1+2) + ω1+2(2ω1−2cos(2tω1) − ω1cos(2tω1−2)) + 2ω
2 2
8ω1(ω1− ω2)(ω1+ ω2)
, (3.68) having de ned ω1±2= ω1± ω2. At the rst order of pertubation a resonance could emerge, since
this order diverges if ω1 = ±ω2. At higher orders the same problem could arise if for some integers
N1,2 the equation N1ω1+ N2ω2 = 0 is satis ed. Furthermore, as Poincarè pointed out, there is
the big problem of the non-convergence of the pertubative series, which at most is asymptotic. Therefore absolute stability is not guaranteed.
Fig.3.1 shows that, for small λ, J1 and J2 remain con ned in a well-de ned region for long
times. Above a critical value of λ, suddenly ghost run-away appears. Therefore, adding small interactions to unperturbed systems (as systems of n independent oscillators), a near ordered behaviour persists. The unperturbed system trajectories evolve along tori in the phase space. Including small interactions these tori are slightly deformed (see section (6) of chapter 3). For large coupling the pertubative expansion fails and the system evolves chaotically. In the next three sections a dierent approach will be followed in order to study the system described by the Hamiltonian (3.43). The goal is to diagonalize the Hamiltonian in such a way that it will depend only by the action variables. A general discussion about the Birkho method, the theorem of Kolmogorov-Arnold-Moser will be developed. This explains in which condition a trajectory can
Figure 3.1. Time evolution of the two quasi-conserved energies (J0 1, J
0
2) computed at the rst
three orders. For small λ (left plot) time evolutions remain in a con ned region that get smaller at higher orders. As λ increases meta-stability is lost (right plot). The pertubative series stops converging.
remain stable in presence of an interaction. Subsequently the discussion will be focused on the system (3.43).
5 Birkho series
Since it is not possible to nd a solution to the fundamental equation (3.52) in the non-degenerate Hamiltonian case, the discussion will now be restricted to the degenerate case. Considering a system described by the Hamiltonian
H(−→J , −→χ , ε) = −→ω ·→−J + εF (−→J , −→χ), (3.69) the frequencies are now xed constants and independent of the actions, therefore there is no restriction on the action variables, as in the previous case. If the frequencies satisfy a condition such that |−→m · −→ω | ≥ γ |−→m|−µ, with γ > 0, µ ≥ l − 1, −→m ∈Zl, −→m 6= 0, i.e a diophantine condition, then the Fourier series is convergent and a regular solution of the fundamental equation (3.52) exists. The solution is
W(1)(−→J0, −→χ) = X − →m∈Zl ,−→m6=0 −Fb−→m( − → J0) i−→m · −→ω e i−→m·−→χ , (3.70)
which, as said, is a convergent expansion. Starting to generalize, the following Hamiltonian is considered H(−→J , −→χ) = −→ω ·−→J + ∞ X r=1 εrFr( − → J , −→χ). (3.71)
The previous expression will be transformed into the new Hamiltonian H0, depending only on
the action variables, through the transformation generated by the function W H0(−→J0, ε) =
∞
X
r=0
εrHr0(−→J0). (3.72) The generating function W is expanded in power series of ε near the identity generating function
W(−→J0, −→χ , ε) =−→J0· −→χ +
∞
X
r=1
εrW(r)(−→J0, −→χ), (3.73) as well as the action variables
− → J =−→J0+ ∞ X r=1 εr∇−→χW(r)(−→J0, −→χ). (3.74) The series (3.72) and (3.73) are called Birkho series. Substituting everything in equation (3.72) the following relation is found
− →ω · " − → J0+ ∞ X r=1 εr∇−→χW(r)( − → J0, −→χ) # + ∞ X r=1 εrFr −→J0+ ∞ X r=1 εr∇−→χW(r)( − → J0, −→χ) ! = ∞ X r=0 εrHr0(−→J0). (3.75) Expanding in Taylor series the second term:
Fr −→J0+ ∞ X r=1 εr∇−→χW(r)( − → J0, −→χ) ! = Fr+ ∇−→ JFr· ∞ X r=1 εr∇−→χW(r)+ ... (3.76) +1 k! l X m1,...,mk=1 ∂kFr ∂Jm1...∂Jmk ∞ X n=k εn X j1+...+jk=n ∂W(j1) ∂χm1 ...∂W (jk) ∂χmk , equation (3.75) becomes (−→ω ·−→J0− H00) + ε(−→ω · ∇−→χW(1)+ F1− H10) + (3.77) +ε2(−→ω · ∇−→χW(2)+ F2+ ∇−→ JF1· ∇−→χW (1)− H0 2) + ... εr(−→ω · ∇−→χW(r)+ Fr+ ∇−→ JFr−1· ∇−→χW (1) + ... − Hr0) + ... = 0. Therefore the fundamental equation, to solve to all orders in ε, is
− →ω · ∇−→ χW (r) (−→J0, −→χ) + G(r)(−→J0, −→χ) = Hr0( − → J0), (3.78) where G(r)= Fr+ r−1 X n X 1 k! l X ∂kFr−n ∂J ...∂J X ∂W(j1) ∂χ ... ∂W(jk) ∂χ . (3.79)
The series (3.72) and (3.73) converge for |ε| < ε0 in the domain A ⊗ Tl, where A is an open set
of Rl. The Hamiltonian would be completely canonically integrable. Since in general these series
diverge, it seems impossible to admit the existence of quasi-periodic motions for Hamiltonian quasi-integrable systems. The idea behind the next section, which is about the KAM theory, is, in the ε 6= 0 case, requiring that some of the quasi-periodic motions of the unperturbed system keep their property of being bounded. This means that it will not be necessary to seek a foliation of the phase spece into invariant tori anymore.
6 KAM theorem
The Kolmogorov-Arnol'd-Moser theorem states that for su ciently small values of ε the majority of invariant tori corresponding to diophantine frequencies ω are conserved and they are slightly deformed by the pertubation. The motions on these tori are quasi-periodic. What does it mean that the tori are slightly deformed by the pertubation? Given ε0 >0 and a torus τ0 =
− →
J0⊗Tl, a submanifold τε of R
2l, with ε ∈ (−ε
0, ε0), is an analytic deformation of the torus τ0 if ∀ε ∈
(−ε0, ε0), τε has parametric equations
− → J =−→J0+ ε−→A(ψ, ε), (3.80) − →χ = ψ + ε−→B(ψ, ε), where ψ ∈ Tl ,−→A : Tl⊗ [−ε0, ε0] −→ Rl and − →
B : Tl⊗ [ε0, ε0] −→ Tl are analytic functions.
Under the same previous assumptions the deformation τε of τ0 is a deformation into invariant
tori for the quasi integrable system
H(−→J , −→χ , ε) = H0(→−J) + εF (−→J , −→χ) (3.81) if the following parametrization is valid:
− → J(t, ψ0) = − → J0+ ε − → A(ψ0+ −→ω( − → J0), ε), (3.82) − →χ(t, ψ 0) = ψ0+ −→ω( − → J0)t + ε − → B(ψ0+ −→ω( − → J0), ε).
(−→J(t, ψ0), −→χ(t, ψ0))belongs to τε, ∀t ∈R. At this point the goal is to compute pertubatively the functions−→A and −→B, which can be expanded in power series of ε (Lindstedt series):
− → A(ψ, ε) = ∞ X k=0 εk−→A(k)(ψ), (3.83) − → B(ψ, ε) = ∞ X k=0 εk−→B(k)(ψ).
Expanding also −→ω(−→J0+ε−→A(ψ, ε))around−→J0, substituting everything in the equation of motion (−→J , ˙˙ −→χ)it is possible to nd equations at every order in ε. If H0 is non-degenerate in the open set
A, it is always possible to determine formally the functions −→A and−→B, for any xed−→J0∈ Aand non-resonant −→ω0 = −→ω(
− →
J0) and the deformation can be parametrized into invariant tori of the
initial torus of frequency −→ω0. But are the series (3.83) convergent? The answer is given by the KAM theorem and it is aermative. Under the usual assumptions of analytic and non-degenerate Hamiltonian, having µ > l − 1, γ > 0 xed, there exists a constant εc>0 depending on γ such
that for every −→J0 ∈ Aγ,µ there exists a deformation τε∈(−ε
c,εc) of the torus τ0 =
− →
J0⊗Tl into invariant tori for the quasi integrable system. The theorem protects from the divergence of the series expansion.
7 Ghost toy model: Birkho pertubative expansion
As anticipated at the end of section (4) a generic canonical transformation Ji → J0
i and χi → χ 0 i
is performed to write the Hamiltonian (3.43) in terms of the action variables only:
H(Ji, χi) = H0(Ji0), (3.84) Let W (J0
, χ)be the generator of the canonical transformation such that J = J0+ ∂χ
iW, χ
0
= χ + ∂J0W. (3.85)
De ning f = sin2
χ1sin2χ2, the new Hamiltonian becomes H0(J0) = ω1(J10 + ∂χ 1W) − ω2(J 0 2+ ∂χ2W) + εf (J 0 1+ ∂χ1W)(J 0 2+ ∂χ2W). (3.86)
If this equation was solvable all J0
i would be exact constants of motion. Instead, due to the
asymptotic character of the Birkho series, J0
iare the approximated constants of motion. However
the only procedure that can be adopted is to expand pertubatively in powers of ε (see equations (3.58) and (3.59))
W = εW(1)+ ε2W(2)+ ..., H0 = H + εH(1)+ ε2H(2)+ .... (3.87) The terms of the generating function W(n)are periodic in the angle-variables χ
1,2, then they can
be expanded in Fourier series i.e W(n)(χ1, χ2) = −i +∞ X N1,N2=−∞ ei − → N ·−→χ W−→(n) N , − → N = (N1, N2). (3.88) The non-zero Fourier coe cient of the function f are easily computed using Mathematica and they are f00= 1/4, f±2,±2= f±2,∓2 = 1/16, f±2,0= f0,±2= −1/8.
7.1 First order in the coupling
The rst order expansion of equation (3.86) gives H(1) = ω ∂W
(1)
− ω ∂W
(1)
Remembering that the terms W(n) are periodic function with period 2π, the average over a
period is zero. Averaging over χi the rst term of the Hamiltonian's expansion becomes
H(1)= J 0 1J 0 2 4 H 0 = ω1J10 − ω2J20 + εJ 0 1J 0 2 4 + O(ε 2 ). (3.90)
Therefore the nal Hamiltonian does not depend on the angle variables anymore, at least at this order of expansion. Now the terms W(1)
N1,N2 can be easily computed substituting the Fourier
expansions of W(1) and f in equation (3.89):
H(1) = J 0 1J 0 2 4 = −i ω1 +∞ X N1=N2=−∞ iN1W (1) N1,N2e i(N1χ1+N2χ2)− ω 2 +∞ X N1,N2=−∞ iN2W (1) N1,N2e i(N1χ1+N2χ2) + +J10J 0 2 1 4 + +∞ X 06=N1,N2=−∞ fN1,N2e i(N1χ1+N2χ2) . (3.91)
Matching term to term one gets that W(1)
0,0 = 0 and in the case N1 6= 0, N2 6= 0
WN(1) 1,N2 = fN 1,N2J 0 1J 0 2 N2ω2− N1ω1. (3.92)
Remembering which are the non-zero Fourier terms of f and summing over the non-vanishing − → N, W(1) is computed: W(1)= J 0 1J 0 2 8(ω22− ω21) " ω1cos(2χ2) − ω1+ ω22 ω1) sin(2χ1) + (ω2cos(2χ1) − ω2+ ω12 ω2 ! sin(2χ2) # . (3.93) The new action variable J0
i is computed substituting eq. (3.93) into eq. (3.48), getting
J10 = J1+ J1J2 4ω1(ω12− ω22)
h
cos 2χ1ω22− ω12+ ω12cos 2χ2− ω1ω2sin 2χ1sin 2χ2i+O(2) (3.94) It is evident that the resonance, which has been introduced in section (5) of chapter 3, is ω1 = ±ω2
at this order. Getting closer to the resonance the pertubative expansion fails.
7.2 Generic order in the coupling
As already found in equation (3.78), the expansion to generic order reads H(n)(J10, J 0 2) = ω1 ∂W(n) ∂χ1 − ω2∂W (n) ∂χ2 + f (χ1, χ2) × × " J10∂W (n−1) ∂χ2 + J 0 2 ∂W(n−1) ∂χ1 + n−2 X m=1 ∂W(m) ∂χ1 ∂W(n−1−m) ∂χ2 # . (3.95) The expansion of the previous equation in Fourier series gives
H(n)(J10, J 0 2) = ω1 +∞ X − → N =−∞ N1W (n) − → N e i−→N ·−→χ − ω 2 +∞ X − → N =−∞ N2W (n) − → N e i−→N ·−→χ +
+ +∞ X − →a =−∞ f−→aei−→a ·−→χ[J10 +∞ X − → b =−∞ b2W−→(n−1) b e i−→b ·−→χ + +J20 +∞ X − → b =−∞ b1W−→(n−1) b e i−→b ·−→χ + + n−2 X m=1 +∞ X − → b =−∞ +∞ X − →c =−∞ b1c2W (m) − → b W (n−1−m) − →c ei( − → b +−→c )·−→χ ] (3.96)
In order to nd the expression for H(n)
(J10, J 0
2)it is necessary to isolate from the previous
expres-sion all the terms which do not depend on the angle-variables. Then it is clear that the terms in the rst square brackets of eq.(3.96) do not contribute to H(n)
(J10, J 0
2) since the only term that
does not depend on the angle-variables is the one with N1 = N2 = 0. Regarding the remaining
part of (3.96) it can be easily checked that
H(n)(J10, J20) = X − →a +−→b =−→0 f−→aW−→(n−1) b b2J 0 1+ b1J 0 2 + n−2 X m=1 X − →a +−→b +−→c =−→0 f−→ab1c2W−→(m) b W (n−1−m) − →c (3.97) By performing a χ-only transformation, W(n)
00 can be set to zero. Factorizing e
i−→N ·−→χ, W(n) − → N 6=−→0 is nally found: (ω1N1− ω2N2) W−→(n) N + X − →a +−→b =−→N f−→a J10b2W−→(n−1) b + J 0 2b1W (n−1) − → b + + n−2 X m=1 X − →a +−→b +−→c =−→N f−→ab1c2W−→(m) b W (n−1−m) − →c = 0, (3.98) which leads to W−→(n) N = 1 ω2N2− ω1N1 X − →a +−→b =−→N f−→a J10b2W−→(n−1) b + J 0 2b1W (n−1) − → b + + 1 ω2N2− ω1N1 n−2 X m=1 X − →a +−→b +−→c =−→N f−→ab1c2W−→(m) b W (n−1−m) − →c . (3.99)
8 Stability estimates
The Birkho series guarantees stability but, depending on the initial state from which the inter-acting system starts to evolve, the frequencies can vary, hitting resonances which invalidate the expansion. However Kolmogorov proved that the initial conditions that lead to instabilities are as rare as rational numbers within real numbers. Now the goal is to compute the meta-stability time for the toy ghost model, nding the optimal order of expansion, given the value of the coupling λ i.e the time τn(Jmaxin → Jmax > Jmaxin ) for which any evolution starting from J
0
i ≤ Jmaxin remains
within J0
i ≤ Jmax (see [13]). The stability time must depend only on the initial conditions, so
n. Clearly this last point depends on the power of the computer. Stability will be guaranteed up to the largest τn. J
0
i are approximated constants of motion. Infact thanks to the Birkho
series, interaction are eliminated up to an arbitrary order, therefore the evolution of this pseudo constants of motion are characterized by the remainder of the Hamiltonian
H(Ji, χi) = H(≤n)(J 0 i) + δH(J 0 i, χ 0 i), J˙ 0 i = − ∂δH ∂χ0i (3.100)
8.1 Stability at lowest order
At lowest order the remainder in the Birkho series is the whole interaction δH(0)= 2λJ1J2
ω1ω2sin
2
χ1sin2χ2. (3.101)
Using the inequality
|Ji0(t) − Ji0(0)| ≤ t max Ji0≤Jmax ˙ Ji0 ≤ t 2λJmax2 ω1ω2 , (3.102)
one nds that the time to leave the region of stability is
t ≥ τ0(Jmaxin → Jmax) = ω1ω2Jmax− J
in max
2λJmax2 , (3.103)
where the right handed side is maximum when Jmax = 2Jmaxin , therefore the stability time at
lowest order is
τ0(Jmaxin ) = ω1ω2
8λJmaxin , (3.104)
which depends only on the initial region where the system starts to evolve and the parameters of the system.
8.2 Stability at generic order
The leading order contribution to rhe residual is δH = −εn+1 " ω1∂W (n+1) ∂χ1 − ω2∂W (n+1) ∂χ2 # + O(εn+2). (3.105)
The Fourier coe cient are easily computed: δH(n+1)~ N = ( H(n+1) for ~n = ~0 (N2ω2− N1ω1)W(n+1)~ N for ~n 6= ~0 . (3.106)
Starting from eq. (3.100) one can get the following relation using the triangular inequality ∂ ∂χ0iδH ≤X ~ p ∂ ∂χ0iδH (n) ~ p = X N1,N2 Ni(N2ω2− N1ω1)W (n) ~ N . (3.107)
Figure 3.2. Ghost model with n = 2 degrees of freedom and quartic coupling λ(eq. (3.43)). The dots are numerical results of observed ghost instability. The black curve is the analytic lower bound on the ghost stability time, computed up to 20th order in λ.
From the previous expression it follows max i,Ji0≤Jmax ∂ ∂χiδH ≤ 2n+1 max i,Ji0≤Jmax X N1,N2 Ni(N2ω2− N1ω1)W (n+1) ~ N ≡ n+1 Jmaxn+2βn, (3.108) where the factor 2 is due to higher orders, the function βn(ω1, ω2) can be computed numerically
and diverges close to resonances:
τn(Jmaxin → Jmax) = Jmax− J
in max
n+1Jmaxn+2βn. (3.109) Maximising over Jmax the stability time is
τn(Jmaxin ) = 1 βn (n + 1)n+1 (n + 2)n+2 ω1ω2 2λJmaxin !n+1 . (3.110)
The stability time must still be maximised over the order n which can be found numerically, testing dierent orders. In Figure (3.2) the curve outlined after having computed the optimal order is shown. Therefore for small coupling ghost meta-stability is proved up to cosmological times, compatibly with numerical eort.
9 Resonances
Close to resonances the Birkho expansion fails. The most dangerous one corresponds to ω1 = ω2
way of studying this problem is to modify the Birkho normal form into a "resonant normal form", which put aside the goal of cancelling all dependece on the angle variables, to avoid the problematic terms which would make the entire structure fall. Below there is an example to clarify the procedure.
9.1 Example: stable ghost close to resonance
As previously said, the resonant limit case ω1 → ω2 is considered for the model of eq.(3.43).
Expanding in Fourier series eq.(3.89) one gets H(1) = X − → N ei − → N ·−→χ H−→(1) N = X N16=N2 ei(N1χ1+N2χ2)H(1) N1,N2 + X N1=N2=N eiN (χ1+χ2)H(1) N ,N = = ω1 X N16=N2 N1ei(N1χ1+N2χ2)W(1) N1,N2 + ω1 X N1=N2=N N eiN (χ1+χ2)W(1) N ,N − −ω2 X N16=N2 N2ei(N1χ1+N2χ2)W(1) N1,N2 − ω2 X N1=N2=N N eiN (χ1+χ2)W(1) N ,N + +J10J 0 2 X N16=N2 ei(N1χ1+N2χ2)f N1,N2 + X N1=N2=N eiN (χ1+χ2)f N ,N (3.111)
Having separated all the summations, the term-to-term match is trivial. In the case N1 6= N2
the following reltion comes out HN(1) 1,N2 = − (ω2N2− ω1N1) WN1,N2+ J 0 1J 0 2fN1,N2. (3.112)
Using eq.(3.92), eq. (3.112) gives H(1)
N1,N2 = 0. Considering the case N1 = N2 = N, from eq.
(3.111) the following is obtained H(1) N ,N = ω1N W (1) N ,N − ω2N W (1) N ,N + J1J 0 2fN ,N. (3.113)
Since the resonant limit case ω1 ' ω2 is considered, the rst two terms of the right handed side
of the previous expression cancel out and one gets H(1) N ,N = J 0 1J 0 2fN ,N, H (1) =X N eiN (χ1+χ2)J0 1J 0 2fN ,N (3.114)
Summing over N, H(1) becomes
H0 = ω1J 0 1− ω2J 0 2+ ε J10J 0 2 4 1 +1 2cos 2(χ 0 1+ χ 0 2) + ..., (3.115)
taking into account that χi' χ 0 i.
The system is now bounded, since in the interaction term there is 1
2 that multiplies a cosine,
which means that the most negative value it can assume is −1
2. Therefore the interaction term
is strictly positive. This can be better seen by performing a canonical transformation: Q ≡ χ 0 1+ χ 0 2 2 , J ≡ J 0 1+ J 0 2, E ≡ J 0 1− J 0 2 (3.116)
Figure 3.3. Phase portrait of the auxiliary system close to the resonance. The shaded gray region in phase space cannot be accessed with J0
1,2≥ 0.
such that the Hamiltonian becomes H0 ' ωE + ∆ωJ 2 + ε 16(J 2− E2 ) 1 +1 2cos 4Q , (3.117) where ω ≡ (ω1 + ω2)/2, ∆ω ≡ ω1 − ω2. H 0
and E are constants of motion, while J forms, together with Q, a system of 1 degree of freedom, which means that J is no longer conserved. But now the key point is that the Hamiltonian is bounded, it is not a ghost system. Numerical simulation shows that the energies E1,2grow by O(1) factor but remains con ned (Fig.(3.4)). The
possible motions are show in Fig. (3.3) where the two typical motions of libration and rotation are separeted by the black line. As seen before stability is guaranteed by the Birkho expansion and the KAM theorem away from resonances. Close to resonances the equivalent system is not a ghost system, the motion is bounded: the leading order non-resonant terms prevail on the higher order resonances. Therefore the ghost system with quartic interaction q2
1q 2
2 is stable.
9.2 Example: ghost that undergoes run-away close to resonance
There are systems that are not stable when they are in a resonance condition. This is the case of two oscillators which interact by a cubic interaction as q2
1q2. The most important resonace
is again the one which comes out from the rst order of expansion in the pertubative series i.e ω2 = 2ω1. The procedure is the same as the one followed in the previous section. Starting from
the Hamiltonian of the system in action-angle variables:
H= ω1J1− ω2J2+ J1pJ2sin2χ1sin χ2, (3.118) the goal is to put eq. (3.118) in resonant form. Therefore the usual canonical transformation to the variables (J0
i, χ 0
i) is performed so that the rst order of expansion gives
H0(1)= ω ∂ 0W(1)− ω ∂ 0W(1)+ J0
q
Figure 3.4. Resonant case, system described by eq.(3.117): energies variation that remains con- ned in a nite region
where f ≡ sin χ2
1sin χ2. Since now a resonant case (ω2= 2ω1) is under consideration the expanded
Hamiltonian will depend also on a combination of the angle variables χi. The Fourier coe cient
of f = P−→ Ne i−→N ·−→χ f−→ N are: f±2,1 = i 8, f±2,−1 = − i 8, f0,±1 = ∓ i
4. Out of resonance averaging eq.
(3.119) over the two angles gives H(1)
= 0, since in the non resonant case the Hamiltonian must not depend on the angle variables. This helps to nd the general Fourier coe cients of the generating function W(1) =P−→ Ne i−→N ·−→χ W−→(1) N which are WN(1) 1,N2 = J10 q J20fN 1,N2 ω2N2− ω1N1. (3.120)
Going back to the resonant case and expanding eq. (3.119) in Fourier series it comes out
H(1) = X N1,N2 ei(N1χ 0 1+N2χ 0 2)H(1) N1,N2 = ω1 X N1,N2 N1ei(N1χ 0 1+N2χ 0 2)W(1) (N1,N2)− −ω2 X N1,N2 N2ei(N1χ 0 1+N2χ 0 2)W(1) N1,N2+ J 0 1 q J20 X N1,N2 ei(N1χ 0 1+N2χ 0 2)f N1,N2. (3.121)
Matching term to term the Fourier coe cients one gets HN(1) 1,N2 = ω1N1W (1) N1,N2 − ω2N2W (1) N1,N2 + J 0 1 q J20fN1,N2. (3.122)
