Quantum Mechanics of Degenerate Dynamical Systems · Quantum Mechanics of Degenerate Dynamical...

Quantum Mechanics

of

Degenerate Dynamical Systems

Tesis entregada a la

Pontificia Universidad Catolica de Valparaıso

en cumplimiento parcial de los requisitos

para optar al grado de

Doctor en Ciencias Fısicas

Facultad de Ciencias

por

Fiorenza de Micheli

Noviembre 2013

Directores de tesis: Jorge Zanelli

Sergio del Campo

FACULTAD DE CIENCIAS

Pontificia Universidad Catolica de Valparaıso

INFORME DE APROBACION

TESIS DE DOCTORADO

Se informa a la Direccion de Estudios Avanzados que la Tesis de Doctorado presen-

tada por la candidata

Fiorenza de Micheli

ha sido aprobada por la Comision de Evaluacion de la Tesis como requisito para

optar al grado de Doctor en Ciencias Fısicas, en el examen de Defensa de Tesis

rendido el 29 de Noviembre de 2013

Directores de tesis: Dr. Jorge Zanelli .........................................

Dr. Sergio del Campo .........................................

Comision de Evaluacion de la Tesis:

Dra. Olivera Miskovic (Presidente) .........................................

Dr. Jorge Zanelli .........................................

Dr. Sergio del Campo .......................................

Dr. Fabrizio Canfora .........................................

Dr. Joel Saavedra .........................................

to my families:

my ancient roots and

my future branchees

with me in the middle

so...

walls were my prison first

and then my freedom

Vaga luna, che inargenti

queste rive e questi fiori

ed inspiri

ed inspiri agli elementi

il linguaggio

il linguaggio dell’amor;

testimonio or sei tu sola

del mio fervido desir,

ed a lei

ed a lei che m’innamora

conta i palpiti

i palpiti e i sospir.

Vincenzo Bellini

Contents

Introduction 2

1 Classical degenerate systems 6

1.1 Review of degenerate systems . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Dynamical role of the degeneracy . . . . . . . . . . . . . . . . . . . . 11

1.3 Local coordinates transformations . . . . . . . . . . . . . . . . . . . 12

1.3.1 Darboux coordinates . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.2 Time reparametrization . . . . . . . . . . . . . . . . . . . . . 13

1.3.3 Example: dynamically degenerate harmonic oscillator . . . . . 14

1.4 Reducible and irreducible degenerate systems . . . . . . . . . . . . . 20

2 The quantum problem 23

2.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 The quantum problem: simplest first order Lagrangian . . . . . . . . 24

2.3 Dealing with the degeneracy . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.1 Excluding the degeneracy: x ∈ (0, a) . . . . . . . . . . . . . . 27

2.3.2 Probability density . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.3 Example: a wave packet . . . . . . . . . . . . . . . . . . . . . 30

2.3.4 Including the degeneracy: x ∈ (a−, a+), a− < 0 < a+ . . . . . 31

2.4 Tunneling: equation of continuity . . . . . . . . . . . . . . . . . . . . 35

3 Coupled system 37

3.1 General classical solution . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Quantum coupled system . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Path integral 47

4.1 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

9

CONTENTS

5 Discussion 54

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A On the self-adjointness 58

B Integrating the continuity equation 60

B.1 The contribution of the discontinous probablity current . . . . . . . . 60

C Calculating the propagator 62

10

Abstract

Degenerate dynamical systems are characterized by symplectic structures whose

rank is not constant throughout phase space. The degenerate phase space is divided

into causally disconnected, nonoverlapping regions such that there are no classical

orbits connecting two different regions. Here the question of whether this classi-

cal disconnectedness survives quantization is addressed. Our conclusion is that in

irreducible degenerate systems –in which the degeneracy cannot be eliminated by re-

defining variables in the action–, the disconnectedness is maintained in the quantum

theory: there is no quantum tunnelling across degeneracy surfaces. This shows that

the degeneracy surfaces are boundaries separating distinct physical systems, not

only classically, but in the quantum realm as well. The relevance of this feature for

gravitation and Chern-Simons theories in higher dimensions cannot be overstated.

Introduction

Classical degenerate systems are characterized by an evolution which is globally

determined by the equations of motion, except on a certain set Σ of measure zero in

phase space Γ. On this set the evolution is indeterminate because the matrix that

multiplies the highest derivatives in the evolution equations –the Hessian matrix in

the Lagrangian formalism or the symplectic form in the Hamiltonian description–,

degenerates: its determinant vanishes there [1]. This produces an abrupt change

in the order dynamical equation, or a change of character between a hyperbolic

evolution equation and an elliptic boundary value problem [36].

Many physically relevant systems including gravitation and Chern-Simons the-

ories in dimensions greater than four [2], vortex interactions in fluids [3, 4], and

piecewise smooth systems in electronics, mechanics or engineering (the so-called

Filippov systems), exhibit this feature [5]. Degenerate systems can also be found

in the force-free plasma equations that govern the strong electromagnetic fields in

quasar magnetospheres [6].

Generically, the degeneracy sets are surfaces of codimension one in phase space,

and higher codimension at their intersections. These surfaces split the phase space

into nonoverlapping regions, causally disconnected from each other, each describing a

nondegenerate system. A classical system would lose degrees of freedom irreversibly

if its orbits reach a degeneracy surface; the Liouville current is not conserved at the

degeneracy surfaces where there is a net ingoing or outgoing flux. The sign of the

flux distinguishes between sources and endpoints of orbits. In the latter case, once

the system reaches the domain wall, it generically acquires a new gauge invariance

and one degree of freedom is frozen, while the remaining degrees of freedom evolve

regularly thereafter [1].

Degeneracies of this kind present challenges that require extending the stan-

dard treatments. For instance, in Dirac’s time-honored approach for constrained

Hamiltonian systems [8], [9], [10], the rank of the symplectic form is less than the

phase space dimension, but constant throughout Γ. In degenerate systems, instead,

2

the rank is not constant throughout phase space. For this peculiarity a quantum

formulation of these system is not available.

Some degenerate systems like, Chern-Simons theories in dimensions higher than

three, may have constraints that are not functionally independent in some region of

phase space (irregularity) constraints whose rank is not constant throughout phase

space (irregularity). This problem makes the formulation of a quantum problem

much more complex and will not be discussed here.

Although degenerate systems could be viewed as extensions of constrained sys-

tems, degeneracy is explicitly excluded from the hypotheses of the standard Dirac

approach, and this introduces conceptual difficulties that must be addressed [12].

Our aim is to clarify to what extent the difficulties in degenerate systems are insur-

mountable obstacles for their quantization, or whether they can be circumvented,

reducing the problem to one already known and solved. Our conclusion is that de-

generate systems can be quantized following the standard postulates of quantum

mechanics, although they exhibit a number of peculiar and unexpected features.

One surprising aspect of degenerate systems is that they possess different number

of degrees of freedom in different part of phase space: as they evolve some local

degrees of freedom could be frozen out at the degeneracy surface and become non

dynamical. This seems to correspond to a sort of dynamical dimensional reduction

mechanism in higher-dimensional gravity, where a reduction of degrees of freedom

would correspond, for example, to freezing out some components of the metric,

which implies that the effective geometry has fewer dimensions.

The extension of this analysis beyond system of a finite number of degrees of

freedom to field theory is an important question. A crucial issue in the quantization

of Yang-Mills theories is the Gribov ambiguity, a problem that may be relevant for

the understanding of confinement in nonabelian gauge theories [7]. This problem

manifests in the occurrence of Gribov horizons, where the matrix of Poisson brack-

ets between generators of gauge transformations and gauge fixing conditions is not

invertible. This can be understood as an indication that gauge conditions in general

fail to do the job of eliminating the gauge degrees of freedom, which is related to the

fact that the gauge generators and the gauge fixing constraints become degenerate

at the Gribov horizons. Hence, understanding quantum degenerate systems may

shed some light on the Gribov ambiguity issue. It is known that in a generic gauge,

the presence of Gribov horizons affects the quantization procedure, but this is an

independent inevitable feature of nonabelian gauge theories.

3

Introduction

The first part of this thesis is intended as a review of the properties of degenerate

sistems as presented in [1]; we then investigate the aspects that make the degeneracy

an intrinsic property and we clarify the consequences of this degeneratcy on the

dynamics of the system, both globally and locally.

Chapter 1 provides the necessary classical background in the context of the first

order formalism; the advantage of this formalism is that in principle it includes all

hamiltonian dynamical systems, giving rise to simpler equations of motion. These

equations are first order and the pre-symplectic form depends generically on the

coordinates. In Section 1.2 we outline the role of the pre-symplectic form and the

hamiltonian as the main although independent characters in the evolution of the

system.

In Section 1.3 we investigate how a change of coordinates or a reparametrization

of time can remove at least locally the presence of degeneracy and we illustrate the

results with the example of a dynamically degenerate harmonic oscillator. Finally

Section 1.4 distinguishes what we called irreducible degenerate systems as those

that cannot be handled with equivalent and simpler reformulation of their action

principle, neither globally nor locally, so a special treatment is needed to quantize

them.

Chapter 2 is devoted to the quantization of degenerate systems in the Schrodinger

approach. The method and the results are illustrated with two examples, a pure

simple degenerate system and a mixed one composed by a degenerate system cou-

pled to a regular (harmonic oscillator). In this chapter we construct the quantum

formalism: the basic fact is that using the canonical substitution, by promoting

the Dirac bracket to a commutator, the classical degeneracy surface appears as a

singular set of the Hamiltonian domain. With a simple model in Section 2.2 we

show how to manage the Hamiltonian as a singular differential operator in an ap-

propriate Hilbert space, excluding or including this singularity. The conclusion is

that the degeneracy surface appears as a barrier that does not allow transmission

of information across it.

The absence of tunnelling is confirmed in Section 2.4 by the continuity equation

for the Schrodinger equation. In order to investigate further this issue in the case of

more complex dynamics, in Chapter 3 we quantum analyze a mixed system whose

classical form had been studied in [1]. This system consists of a regular harmonic

oscillator coupled to the simple degenerate system already discussed in Section 2.2.

Following the same line of reasoning as before, no tunnelling effects are found in

this mixed case either (Section 3.3). After the canonical method, we investigate in

4

Chapter 4 the quantization of the same simple degenerate system of the Chapter 4

with the famous alternative method of the path integral: our conclusion is that also

in this approach tunneling effects are excluded. Finally, Chapter 5 summarizes the

results and presents some open questions that illustrate possible lines for further

investigation.

5

Chapter 1

Classical degenerate systems

In this chapter we analyze the classical dynamics of degenerate systems, building on

the ideas developed in [1]. Let us begin by considering a dynamical system in the

2n-dimensional phase space Γ whose state at time t is described by its coordinates

zi(t) ∈ Γ. In order to fix ideas, consider that the equations of motion of this system

are given by a first order action

I[z; 1, 2] =

∫ t2

t1

[Ai(z)zi + A0(z)]dt, with i = 1, 2, · · · 2n, (1.1)

In spite of its simplicity, a system described by the above action 1.1 captures the

problem we are interested in and, at the same time, represents any dynamical hamil-

tonian system - degenerate or not - of a finite number of degrees of freedom (see [13]

and Section 1.2).

Observe that the above general formula can be viewed as an action in Hamiltonian

form where zi = (p, q) are noncanonical coordinates in phase space Γ, if we recog-

nize the Hamiltonian in the function −A0(z) and we identify the functions Ai(z)

with the noncanonical momenta. Alternatively, the integrand L(z, z) = Aizi + A0

can be taken as the Lagrangian for a system with 2n primary constraints φi(z, p) =

pi − Ai(z) ≈ 0. In both cases, in order to set the equations of motion as an analyt-

ically well-posed problem, we require Ai(z) and A0(z) to be functions of class C1,

i.e., continuous differentiable at least.

1.1 Review of degenerate systems

In this section, we review the results reported in [1]. According to the Hamilton’s

principle of least action, the trajectory of a system coincides with the path that

6


extremizes its action. Varying the action (1.1) with respect to z(t) we find

δI =

∫ t2

t1

δzi[Fij(z)zj + Ei(z)

]+ δziAi(z)

∣∣t2t1

(1.2)

where1

Fij ≡ ∂iAj − ∂jAi, (1.3)

Ei ≡ ∂iA0. (1.4)

We are looking for the extremal curve that satisfies δI = 0 under arbitrary vari-

ation of δz, this implies that the two terms on the r.h.s. of (1.2) have to vanish

independently. So the first gives the equations of motion

Fij zj + Ei = 0, (1.5)

and the second imposes condition at the end points of the path

δzi(t2) · Ai(z(t2)

)− δzi(t1) · Ai

(z(t1)

)= 0, (1.6)

The equations 1.5 define a Hamiltonian system where the pre-symplectic two-form

is defined by the skew-symmetric 2n× 2n matrix Fij,

F = dA =1

2Fij(z) dzi ∧ dzj. (1.7)

From (1.5) it follows that ziEi ≡ 0, and therefore the orbits are contained in the

surfaces A0 = constant, which corresponds to the conservation of energy.

In order to implement (1.6) we observe that the usual Dirichlet boundary con-

ditions δzi(t2) = δzi(t1) = 0 are generically incompatible with the equations of

motion. In general, solutions that connect zi(t1) = zi1 to zi(t2) = zi2 do not exist for

arbitrary values of those coordinates because (1.5) is a first order equation. Then, if

the action is to be varied over a class of functions for which the boundary conditions

are adequate for integrating the equations of motion, a weaker compatible boundary

condition can be imposed in our case, namely, periodic boundary conditions

δzi(t2) = δzi(t1), Ai(z(t2)

)= Ai

(z(t1)

), (1.8)

or anti-periodic conditions, zi(t1) + zi(t2) = 0, provided A is odd,

δzi(t2) = −δzi(t1), Ai(− z)

= −Ai(z)

(1.9)

1Here we are interested in autonomous systems with time-independent Ai, but this could beeasily extended to include the time-dependent case, where Ei = ∂iA0 − ∂0Ai.

7


as is usually the case for fermionic variables.

The identification of the end points would restrict the functional space on which

the action is defined that can be viewed as a restriction on the topology of the

problem, but this does not alter the local character of the equations or the solutions.

As we will see in Chapter 2 this condition agrees with the same periodic boundary

condition one has to impose on the wave function in the quantum version of a

degenerate dynamical system.

Let us now summarize the basic facts about degenerate systems.

• Existence of degeneracy surfaces. Solving (1.5) for zi(t) requires inverting

Fij(z), which is clearly the problem for degenerate systems at the points where the

determinant of Fij(z) vanishes: on the set

Σ = z ∈ Γ |∆(z) = 0 , (1.10)

where ∆(z) = det[Fij(z)], the equations of motion are indeterminate. Analytically,

the degeneracy set Σ ⊂ Γ is defined by one relation among the coordinates zi,

and therefore it corresponds generically to a collection of codimension one surfaces,

which eventually divide the phase space into dynamically disconnected regions, as

shown in [1].

• Robustness of degeneracies. The degenerate character of a dynamical sys-

tem is a feature that cannot be eliminated by an appropriate change of coordinates.

To see this, it suffices to observe that the determinant of the pre-symplectic form,

∆ =det(Fij), transforms as a pseudoscalar under coordinate changes

∆→ ∆′ = J2∆ , (1.11)

where J is the Jacobian of the transformation z → z′. Hence, the zeros of detF

cannot be removed by general coordinate transformations in phase space.

• Intrinsic two-dimensionality. Darboux theorem2 states that in a symplectic

manifold M2n, equipped with a nondegenerate differential 2-form ω2, there are

always local coordinates in which the symplectic structure ω2 is represented by the

constant canonical form

ω2 =∑

dpi ∧ dqi . (1.12)

In the case of degenerate dynamical systems the invariance of the degeneracy surfaces

Σ implies that it is impossible to transform the pre-symplectic form F = 12Fij(z)dzi∧

2as fundamental references see [16] and [32], see also [33]

8


dzj globally into the canonical symplectic form,

Ωij =

0 1

−1 0

0 1

−1 0. . .

. (1.13)

However, in an open neighborhood it is possible to find Darboux coordinates, such

that F can be block-diagonalized as

Fij =

0 f1(zi)

−f1(zi) 0

0 f2(zi)

−f2(zi) 0. . .

, (1.14)

so that ∆(z) = [f1(z)f2(z) · · · fn(z)]2. Then we can assert that for degenerate sys-

tems only in an open, simply connected set where ∆(z) = [f1(z)f2(z) · · · fn(z)]2 > 0,

there exist a coordinate redefinition for which the pre-symplectic form F can be writ-

ten in the canonical symplectic way, with f1(z) = f2(z) = · · · = fn(z) = 1 (Darboux

theorem). But clearly, this is not possible for degenerate systems in an open set

containing points of degeneracy, z ∈ Σ, where at least one of the fr vanishes.

Moreover, since F is exact, the Bianchi identity dF = 0 implies that each fr(z)

of the block diagonal form is a function of only two coordinates,

F =n∑r=1

fr(z2r−1, z2r)dz2r−1 ∧ dz2r . (1.15)

Finally, the pre-symplectic form (1.3) can be block-diagonalized as

Fij(z) =

0 f1(z1, z2)

−f1(z1, z2) 0

0 f2(z3, z4)

−f2(z3, z4) 0. . .

, (1.16)

and hence, the zeros of ∆(z) = det[Fij(z)] given by∏1≤r≤n

fr(z2r−1, z2r) = 0 , (1.17)

9


describe the degeneracy sets in two-dimensional surfaces spanned by the coordinates

(z2r−1 , z2r). This means that in order to inspect the dynamical properties of de-

generate systems in the vicinity of the degenerate surfaces, it is sufficient to focus

on two-dimensional submanifolds embedded in phase space,fr(z

2r−1, zr) z2r−1 = −∂rA0(z2r−1; zr)

−fr(z2r−1, zr) zr = −∂2r−1A0(z2r−1; zr) .(1.18)

• Degenerate dynamical flow. Assuming Ai(z) and A0(z) to be of class C1,

then Fij(z) = ∂iAj − ∂jAi and his determinant ∆(z) will be continous well behaved

functions in the phase space Γ. Then, without loss in generality, we can assume fr

to be Morse functions.3 This basically means that the level curves of the functions

fr(z2r−1, z2r) are either infinitely extended lines, isolated points, or closed lines in

the (z2r−1, z2r)-plane, and the sets Σr = z ∈ Γ|fr(z2r−1, z2r) = 0, that are not

isolated points, are simple zeros. Thus, the set Σr are generically either infinitely

long lines or closed curves that separate region where fr have different signs4. This

implies that the components of z on the (z2r−1, z2r)-plane change sign at Σr (see

1.18) and then Σr acts as a generic source or sink for orbits. Moreover, the velocity

field zi has a nonvanishing divergence,

∂izi = f−2εij∂if∂jA0 6= 0 . (1.19)

Consequently, the time evolution of a degenerate system does not preserve the vol-

ume in phase space: the volume v of a small region in phase space evolves as

div v(z) = f−2f,H

, (1.20)

which blows up as the orbit approaches a degeneracy point: v → ±∞, depending

on the sign of the gradient of f along the orbit.

• From second to first class constraints along the evolution. The inte-

grand of the action (1.1) L(z, z) = Ai(z)zi + A0 is a first-order Lagrangian which

can be viewed as a Legendre transform defined on the tangent space of the phase

space Γ. This Lagrangian gives the same equations of motion as the Hamiltonian

3 In the context of the Morse theory, a smooth real valued function f defined on a manifoldf : M → R is a Morse function if its critical points (where the differential of f vanishes) are allnondegenerate (i.e. the Hessian matrix of second order partial derivatives is non singular). Formore details, see [30] and [31].

4Exceptionally, Σ may have a finite number of self intersections and isolated points or cuspswhich can be removed by a continuous deformation of A0.

10

1.2 Dynamical role of the degeneracy

H(z) = −A0 using the presymplectic form 1.3 Fij ≡ ∂iAj − ∂jAi, with 2n primary

constraints φi(z) = pi − Ai(z) ≈ 0.

From Dirac’s formalism for constrained systems, it is essential for the dynamical

interpretation to distinguish between first class constraints, related to the presence

of gauge symmetries, and second class ones, corresponding to redundant degrees of

freedom. The nature of the constraints depends on the rank of the matrix composed

by the brackets of constraints φi, φj. In our case this matrix is given by

φi, φj = Fij(z), (1.21)

where · · · , · · · stands for the Poisson bracket. Then, in the case of degenerate

systems, the constraints appear to be generically of second class φi, φj 6= 0 except

on the degenaracy set, where some of them become first class φi, φj = 0.

• Causally disconnected dynamics. As mentioned above, for degenerate sys-

tems some components5 of the velocity z2r−1 , z2r change sign across Σr. Therefore,

the orbits either start, end, or run tangent to the set Σr. It should be stressed that

the inversion of the velocity happens only in those 2-dimensional subspaces where

the degeneracy occurs, while the dynamics evolves independently in the other coor-

dinates. This means that the classical evolution in those subspaces cannot take the

system across the degeneracy surfaces: there is no causal connection between states

separated by a degeneracy surface.

The question that naturally arises is whether this condition continues to hold if

quantum mechanical effects are taken into account. Can there be tunnelling across

Σ? What happens to a wave packet prepared on one side that corresponds to a

classical trajectory that approaches Σ? This will be discussed in Chapter 2.

1.2 Dynamical role of the degeneracy

As mentioned in the previous section, degeneracies can be studied as intrinsically

two-dimensional problems. This means that in order to analyze the dynamical

properties of degenerates systems near the degeneracy, it is sufficient to focus on

two-dimensional surfaces embedded in phase space. In particular, the equations

of motion (1.5) appear as a system of n equations of two variables (z1, z2), which

5at least one pair of the components of the velocity diverges and changes sign because at leastone of the fr has to vanish in (1.17) when calculating the zero’s of the matrix (1.16) in the newcoordinates

11


depend parametrically on the remaining coordinates za, So let’s suppose that the

degeneracy is present on the (z1, z2) surface, i.e. the term ff(z1, z2) z1 = −∂2A0(z1, z2; za)

−f(z1, z2) z2 = −∂1A0(z1, z2; za) .(1.22)

The system (1.22) describes a 2-dimensional vector field, not necessarily smooth,

but mildly singular due to the unbounded velocity z → ∞ at f(z1, z2) = 0. This

system represents a continuous directional field6 given by

z2

z1:= tanα(z1, z2; za) = −∂1A0(z)

∂2A0(z), (1.23)

whose integral curves are completely determined by A0(z). This expression is in-

sensitive to the change t → −t, so it carries no information about the reversal of

orientation that takes place at the points where the orbits intersect the degenerate

surfaces Σ. More importantly, this expression is also independent of f , and therefore,

there are infinitely many dynamical systems analogous to (1.22), whose orbits have

the same “shape” but with different dynamics and different degenerate surfaces. In

particular, f(z1, z2) = 1 gives the simplest nondegenerate analogous system, corre-

sponding to a standard mechanical system, with Hamiltonian H = −A0, and z1 = p

and z2 = q.

The level curves A0(z1, z2; za) = constant implicitly define the shape of the or-

bits, while f = ∂1A2(z1, z2)−∂2A1(z1, z2) determines the dynamics, i.e. the pace at

which the orbits are traced. In other words, the Hamiltonian draws the orbits and

the pre-symplectic form determines the time evolution.

1.3 Local coordinates transformations

1.3.1 Darboux coordinates

We have seen that it is impossible to set f(z) = 1 globally by a coordinate change

but, is it possible to do it within an open nondegenerate neighborhood? Is it pos-

sible to find appropriate coordinates, within each nondegenerate domain, so the

dynamical equations look nondegenerate? The answer is yes.

6A direction field is called continuous if tg(α) depends continuously on the points (z1, z2).

12


Let us consider a degenerate system given as in (1.22),

f(z)zi = εij∂A0(z)

∂zj, ε12 = −ε21 = 1. (1.24)

In terms of new coordinates ξi(z), this equation reads

ξa =1

f

∂ξa

∂zi∂ξb

∂zjεij∂A0(ξ)

∂ξb=

1

f

∣∣∣∂ξ∂z

∣∣∣εab∂A0(ξ)

∂ξb, (1.25)

which reduces to (1.22), provided

det(∂ξi/∂zk) = f, and A0(ξ) = A0(z(ξ)) . (1.26)

There are certainly many choices of coordinates ξ that satisfy conditions (1.26).

The first condition says that the determinant of the Jacobian for this change of

coordinates vanishes at the degenerate surface. Then the fact that the change of

coordinates z → ξ is valid where det(∂ξi/∂zk) 6= 0 means that the new coordinates

ξ are valid in local regions that exclude the set f(z) = 0. Hence, there are many

regular autonomous systems that have the same dynamics as a degenerate one within

a non-degenerate region. Degenerate dynamical systems can always be reduced to

non-degenerate ones in an open neighbourhood that does not include degenerate

sets. This explains why textbooks on differential equations never discuss degenerate

systems.

1.3.2 Time reparametrization

The fact that the shape of the orbit is independent of f suggests that a change of

time parameter may yield an evolution that could be reproduced by a nondegenerate

dynamical system. If that is the case, then a new time parameter τ(t) should exist

such thatdt

dτ=

1

f(z1, z2). (1.27)

This relation could be integrated if the trajectory zi(t) is known,

τ(t; z0) =

∫ t

t0

f(z1(t′), z2(t′))dt′ . (1.28)

This relation could be integrated for each trajectory zi(t), obtained by solving (1.22)

13


for some initial conditions. The complete expression for the trajectories depends on

the initial data z0 = (z10 , z

20) and is given by z1(t; z1

0 , z20) and z2(t; z1

0 , z20)

τ(t; z0) =

∫ t

t0

f[z1(t′; z0), z2(t′; z0)

]dt′ . (1.29)

This relation, however is not a redefinition of the time parameter for the entire

dynamical system, but for each individual orbit. Moreover, the reparametrization

τ = τ(t) fails precisely at the degeneracy points, where f changes sign: at f = 0 the

integrand changes sign, its primitive fails to be monotonic, the reparametrization

τ(t; z0) is reversed and becomes singular, regardless of which orbit is considered.

This highlights the fact that the degenerate surfaces that intersect the classical

trajectories are the beginning or the endpoints of orbits.

1.3.3 Example: dynamically degenerate harmonic oscillator

We have already shown that in general is it possible to find appropriate coordinates,

within each nondegenerate domain, so the dynamical equations look nondegenerate.

It is clear that even if in principle it is always possible to find such coordinates, this

can be awkward in practice. Here we will illustrate how a change of coordinates can

set f(z) = 1 locally in an ad-hoc example. Consider a dynamical system of the form

(1.22) corresponding to a “degenerate” harmonic oscillator:p = −1

pq

q =1

pp = 1 .

(1.30)

with hamiltonian H(p, q) = 12(p2 + q2) and degeneracy function f(p, q) = p.

We look for the change of coordinates (p, q) → (P,Q) such that the system

(1.30) in (p, q)-coordinates can be written in new coordinates (P,Q) as the usual

(not degenerate) harmonic oscillator whose Hamiltonian is H(P,Q) = 1/2(P 2 +Q2)

with the standard symplectic form f(P,Q) = 1P = −Q

Q = P .

(1.31)

As discussed in Section 1.2, the orbits are circles in both (p, q) and (P,Q) phase

spaces (see Figures 1.1 and 1.2). The system (1.30) can be described in terms of the

original coordinates p(P,Q) and q(P,Q), using (1.31)

14


p

q

Figure 1.1: Classical orbits for the degenerate harmonic oscillator in the phase space

(p, q) as described in (1.30) with degeneracy set at p = 0: the orbits emerge with

infinite velocity from the negative q semi axes and in a finite time end up with

infinite velocity in the positive q semi axes.

P

Q

Figure 1.2: Classical orbits for the harmonic oscillator in the phase space (P,Q) as

described in (1.31): there is no degeneracy in this space and the orbits are traced

with constant velocity.15


p =

∂p(P,Q)

∂PP +

∂p(P,Q)

∂QQ =

∂p(P,Q)

∂P(−Q) +

∂p(P,Q)

∂QP = −q(P,Q)

p(P,Q)

q =∂q(P,Q)

∂PP +

∂q(P,Q)

∂QQ =

∂q(P,Q)

∂P(−Q) +

∂q(P,Q)

∂QP = 1

(1.32)

These equations provide a set of first-order semi-linear partial differential equations

for p(P,Q) and q(P,Q),−Q ∂p(P,Q)

∂P+ P

∂p(P,Q)

∂Q= −q(P,Q)

p(P,Q)

−Q ∂q(P,Q)

∂P+ P

∂q(P,Q)

∂Q= 1 ,

(1.33)

which can be solved for some initial or boundary conditions. Let’s observe that the

characteristic curves7 of both differential equations in (1.33) are by construction the

orbits of the non-degenerate analogous system (1.31), that is,P = −Q

Q = P

−→ P 2 +Q2 = P 20 +Q2

0 . (1.34)

Inorder for this to be a well-posed problem, it will be sufficient to give an initial

condition on a curve that intersects these characteristic curves only once. We choose

for simplicity to give initial boundary conditions on the semi positive P -axes asq(P > 0, Q = 0) = 0

p(P > 0, Q = 0) = P(1.35)

Solving the second of (1.33) for q(P,Q), with these boundary conditions, one finds

q(P,Q) = arctg

(Q

P

)(modnπ), (1.36)

which is the angle measured from the P -axis. The arctangent function is multivalued

so, in order for q(P,Q) to cover the interval [0, 2π] in the (P,Q)-plane, we include a

7The method of characteristics is the technique for solving partial differential equations reducingthem to ordinary differential equation along the characteristic curves, see for example [34], [35].

16


p

q

Figure 1.3: Coordinate lines for P in

the (p, q) space as given by (1.39).

p

q

Figure 1.4: Coordinate lines for Q in

the (p, q) space as given by (1.39).

branch cut along the semi positive P -axis. With this solution for q(P,Q), the first

equation in (1.35) becomes

−Q ∂p(P,Q)

∂P+ P

∂p(P,Q)

∂Q= −

arctg(QP

)p(P,Q)

, (1.37)

which can be integrated for the boundary condition p(P > 0, Q = 0) = P , giving

p2 = (P 2 +Q2)− arctg2(QP

). (1.38)

We are interested in the inverse functions, P (p, q) and Q(p, q),P = ±

√(p2 + q2)cos(q)

Q = ±√

(p2 + q2)sin(q)

(1.39)

whose contour lines are drawn in Fig. (1.3) and (1.4).

The Jacobian determinant for this change of coordinates is

detJ =

∣∣∣∣∣∂(P,Q)

∂(p, q)

∣∣∣∣∣ = p . (1.40)

As expected from (1.26), the Jacobian vanishes with the degeneracy function. This

means that the new set of coordinates (P,Q) is well defined everywhere in the (p, q)

17


p

q

Figure 1.5: Contour lines of the new coordinates (P,Q) in the (p, q) space. This

new set of coordinates fails at p = 0: here the lines P and Q are tangent.

18


p

q

Figure 1.6: How are transformed the degenerate set p = 0 and a degenerate orbit in

the nondegenerate phase space? These are drawn with the same color in the (p, q)

space in the Fig. (1.7).

P

Q

p=0

Figure 1.7: The blue line is the degenerate set p = 0 described by√P 2 +Q2 =

arctg(P,Q), a curve known as Archimedean spiral (see 1.38 ). The red curve repre-

sents both two branches of the degenerate orbit in the nondegenerate(P,Q) plane.

Compare with above figure (1.6).

19


space except on the set p = 0, where the coordinates lines of P and Q are tangent

(see Fig (1.5)). Thus, changing to (P,Q) coordinates in the (p, q) phase space

allows to describe the degenerate oscillator as a non degenerate one only locally:

the dynamical equations of the system (1.30) look non-degenerate only in region

that excludes the degenerate set.

Next, we consider the time reparametrization that could eliminate the degeneracy

in the (p, q) coordinates. Actually, it is easier to look for the inverse reparametriza-

tion, which produces the degeneracy in the nondegenerate coordinates. In other

words, we look for a time parameter t = t(τ) such that if it is applied to an ordinary

harmonic oscillator in coordinates (p, q) the equations becomedpdτ

= dpdt

dtdτ

= −1pq

dqdτ

= dqdt

dtdτ

= 1pp = 1

(1.41)

Solving for the nondegenerate part in term of time t one finds the solution depending

on the initial conditions (po, qo)dpdt

= −q

dqdt

= p

→

p(t; p0, q0) =

√p2

0 + q20 cos

(t+ arctan(q0/p0)

)q(t; p0, q0) =

√p2

0 + q20 sin

(t+ arctan(q0/p0)

) (1.42)

Then, the time τ is given by equation (1.29) that in this case reads

τ(t; p0, q0) =

∫ t

t0

p(t′; p0, q0) dt′ =

∫ t

t0

√p2

0 + q20 cos

[t+ arctan(q0/p0)

]dt′ =

=√p2

0 + q20 sin

[t+ arctan

(q0

p0

)]t(τ ; p0, q0) = arcsin

[τ√

p20 + q2

0

]− arctan

(q0

p0

)(1.43)

Clearly this time reparametrization t = t(τ) is valid only for τ in the interval (−1, 1)

and cover the range t in intervals −π/2 < t < π/2 modπ. These are the two regions

p < 0 and p > 0 into which the phase space (p, q) is divided by the degeneracy

function f(p, q) = p.

1.4 Reducible and irreducible degenerate systems

We have seen that coordinates can be found such that the equations of motion take a

canonical form everywhere within a nondegenerate domain, and don’t seem to have

20

1.4 Reducible and irreducible degenerate systems

any problem; but, can those equations be obtained from an action principle? Can

the dynamical system within a nondegenerate domain be described by a regular,

nondegenerate action principle? Can the action of a degenerate system like (1.1)

be replaced by a nondegenerate action that reproduces the same evolution within a

region that does not contain degeneracies?

The question is whether any of the infinitely many equivalent nondegenerate

descriptions can be obtained from an action principle of the type (1.1). As we will

see next, the answer is negative, as stated in the following

Lemma: Given a generic degenerate system obtained from the action principle

I[z] as in (1.1), none of its nondegenerate analogues can be obtained from a local

action principle I[z]. The only exceptional (non-generic) case in which an action

principle exists for both, degenerate and non-degenerate systems, occurs if the de-

generacy function f(z) is a constant of motion, or equivalently, if the orbits do not

intersect the surface f(z) = 0.

Proof : Suppose there exists an action I[z], for which the equations of motion

are nondegenerate,

zi = εij∂A0(z)

∂zj. (1.44)

Since these equations describe the same degenerate orbits as described, for instance,

in (1.24), then

∂iA0(z) = f−1∂iA0(z) . (1.45)

A fast check of the mixed second partial derivatives shows that

~∇f = ϕ~∇A0 . (1.46)

where ϕ is any scalar function, or equivalently,

εij∂if∂jA0 =f, A0

= 0 . (1.47)

Eq. (1.46) means that the level curves of f and A0 must concide, and (1.47) implies

that f(z) is a constant of motion. In other words, only the action for degenerate

systems whose orbits run tangent to the degenerate surfaces can be replaced by

an action describing a nondegenerate system. We call this type of degeneracy a

reducible one. Irreducible degenerate systems, on the other hand, will be those

whose classical orbits intersect the degenerate surfaces and therefore cannot be de-

scribed by an equivalent nondegenerate action principle. For example, a degenerate

system

f(z)εabzb = Ea(z) (1.48)

21


Figure 1.8: Example of an irreducible

degenerate system, in the case that the

degeneracy acts as a sink

Figure 1.9: Example of an irreducible

degenerate system, in the case that the

degeneracy acts as a source.

Figure 1.10: Example of an reducible degenerate system, in this case the orbits run

tangent to the degeneracy surface.

is reducible iff ∂af = ϕ(z)Ea(z); otherwise, it is irreducible. This conclusion is rel-

evant for the study of quantum degenerate systems. The point is that, in order to

discuss the quantum mechanics of a particular system, it is not sufficient to have its

dynamical equations, it is necessary to know the action principle that defines it [14]:

As is well known, systems without an action principle –like a damped harmonic

oscillator– do not have a well defined quantum mechanical description.

22

Chapter 2

The quantum problem

2.1 Formalism

As we have seen, irreducible degenerate system cannot be obtained from a non-

degenerate action principle. This means that the quantization of irreducible de-

generate systems is a problem that cannot be addressed following the standard

procedures of quantum mechanics as it applies to nondegenerate systems. The pe-

culiar feature is that the Dirac bracket not only depends on the coordinates, but

moreover, it is undefined on the degenerate surface. When the symplectic structure

degenerates and is no longer invertible, what is the correct approach to define the

quantum theory?

There are two standard constructions of a quantum theory starting from a clas-

sical one: the canonical (Schrodinger) and the path integral. In this chapter, we

analyze the simplest irreducible degenerate system following the first approach and

we will analyze the second one in the Chapter 4.

Let us consider a generic 2-dimensional first order Lagrangian of the form

L(x, y) = Axx+ Ayy + A0 . (2.1)

The Dirac bracket is given by the inverse of symplectic form,

x, y∗ =1

f(x, y), (2.2)

where f(x, y) = ∂xAy − ∂yAx.The phase space coordinates have a noncanonical symplectic structure, and there

is no metric and no preferred coordinate system in the problem. However, since for

irreducible degenerate systems, f has been assumed to be a smooth Morse function

23

The quantum problem

whose level curves do not coincide with the level curves of A0, a natural option would

be to take the value of f as a coordinate, which may be called “x”. The level curves

of Morse functions are either closed or infinitely extended and, in a local patch, the

coordinate lines for x can be identified with the gradient of f .

Quantization still requires finding an adequate prescription of operators such

that the Dirac bracket (2.2) becomes the commutator at the quantum level,

[x, y] = i~1

x. (2.3)

The operators x and y that satisfy this relation can be chosen as

x : = x (2.4)

y : = −i~1

x∂x . (2.5)

The quantum operator H = H(x, y) that replaces the classical Hamiltonian Hc =

−A0(x, y), is a singular differential operator with a leading coefficient 1/x. Still,

given the fact that classically the energy is conserved, (Hc = 0), we expect the quan-

tum Hamiltonian H to have observable real eigenvalues. Consequently H should be

self-adjoint, but this eventually depends on the choice of boundary conditions that

define the Hilbert space.

2.2 The quantum problem: simplest first order

Lagrangian

We illustrate the procedure by analyzing the simplest Lagrangian for which Fij(x) =

xεij. This system, discussed in [1], is obtained for Ax = xy, Ay = 0, A0 = −νy, so

that

L = xyx− νy , (2.6)

H = νy = −A0 , (2.7)

whose degeneracy at x = 0 can be thought of as an approximation near the degen-

erate surface of a generic system. In spite of its simplicity –and possibly unrealistic

nature–, this problem has some interesting features that help to understand more

general cases.

24

2.2 The quantum problem: simplest first order Lagrangian

x

y

infinite velocity

-xo +xo

Figure 2.1: Classical orbits x2(t) = 2νt+x20 in the phase space (x, y) for the system

described by L = xyx− νy with ν < 0. For ν > 0 the orbits are time-reversed.

The classical solution is given by [1]

x2(t) = 2νt+ x20 . (2.8)

For ν < 0, the system presents an attractive surface of degeneracy at x = 0 (repulsive

for ν > 0). Note that the orbits flow towards this surface, reaching the degeneracy

with infinite velocity in a finite time. Conversely, for ν < 0, the orbits emerge from

this surface with infinite velocity and go to ±∞ with a velocity that approaches

zero at infinity.

Using the prescription (2.4,2.5) the Hamiltonian operator in this case is

H = νy = −i~ν 1

x∂x . (2.9)

The domain of this singular differential operator must be chosen so that the corre-

sponding Hilbert space will be equipped with a well-defined weighted scalar product.

In general the weighted Hilbert space L2(Ω ⊂ R, w(x) dx)1 consists of (all equiv-

alence classes of) complex-valued functions, defined on a subset Ω of R, that are

1Also denoted as L2(Ω ⊂ R, w(x)) or L2w(Ω ⊂ R).

25

The quantum problem

square-integrable with a weight w(x),

‖ϕ(x)‖ =(∫|ϕ(x)|2w(x) dx

) 12. (2.10)

The weight w(x) is chosen in such a way that the Hamiltonian (2.9) is symmetric,∫ϕ∗1(x)

[Hϕ2(x)

]w(x)dx =

∫ [Hϕ1(x)

]∗ϕ2(x)w(x)dx , (2.11)

up to boundary terms. The symmetry condition together with the positivity of

the scalar product require w(x) = |Fij| = |x|. This is the measure implied by

the noncanonical Dirac bracket (2.2), and is consistent with the presence of the

degenerate surface at x = 0. Hence, the domain where H defines a proper scalar

product is

Do(H) = ψ ∈ L2(R, |x|dx) : H(ψ(x)) ∈ L2(R, |x| dx). (2.12)

The corresponding scalar product and norm in the Hilbert space are

< ϕ1, ϕ2 >=

∫ϕ∗1|x|ϕ2 dx, and ||ϕ|| =

(∫|ϕ|2|x| dx

) 12, (2.13)

respectively.

In this case, the Schrodinger equation reads

− i~ν 1

x

∂

∂xΨ(x, t) = i~

∂

∂tΨ(x, t), (2.14)

which is a singular differential equation with indefinite weight w(x) = x that can

change sign and vanish. 2

The general solution of equation (2.14) is Ψ = ϕ(x2 − 2νt), where ϕ is any

differentiable function. Since the classical system is conservative, the quantum states

ψ can be spanned in a basis of eigenstates of the Hamiltonian (2.9). Hence, a

stationary solution ψ(x, t) is also an eigenstate of H of the form

ΨE(x, t) = ψE(x)αE(t) (2.15)

2As an example, an equation of the Sturm-Liouville type, − ddx

[p(x) dydx

]+ q(x) · y = λw(x) · y,

is a (second order) singular equation with weight w(x). This operator is self-adjoint in the Hilbertspace L2(Ω ⊂ R, w(x) dx), but usually considering only regions for which w(x) is positive. Thetypical example of a differential equation of mixed type studied in intervals where a coefficient canchange sign is given by the Tricomi equation uxx + xuyy = 0: depending on on the sign of x thisequation changes between elliptic, hyperbolic or parabolic type [36].

26

2.3 Dealing with the degeneracy

with

ψE(x) = ψ0 ei~E2νx2

and αE(t) = α0 e− i

~Et, (2.16)

where E= constant is an eigenvalue of H.

The crucial point now is the choice of the domain where the Hamiltonian op-

erator is self-adjoint. Let’s stress that for unbounded (linear) operator, as H, self-

adjointness and symmetry may not coincide depending on the domain. In practice,

the process to establish the self-adjointness requires the symmetry condition (see

Appendix B). In our case, H = −i~ν x−1∂x, is self-adjoint provided the functions

in the Hilbert space Do(H) satisfy appropriate boundary conditions, depending on

whether the domain includes the degeneracy or not.


In the presence of the divergence at x = 0, the Schrodinger equation (2.14) can be

solved by either restricting the domain so as to exclude the origin, or by imposing

some additional boundary conditions involving the values of ψ at x = 0±. The latter

option is a subtle issue in view of the first order nature of equation (2.14).

2.3.1 Excluding the degeneracy: x ∈ (0, a)

A simple possibility is to consider the domain (0, a), in which case, the normalized

stationary states are

ΨE(x, t) =

√2

aexp

[iE

2~ν(x2 − 2νt)

]. (2.17)

This solution is even in x and never vanishes in the range, although its domain

of definition does not include x ≤ 0. The equation is separable and therefore, the

solution can be factorized as Ψ(x, t) = ψ(x)α(t). In order to evaluate the self-

adjointness we first consider the symmetry condition,

< Hψ, φ >=< ψ, Hφ > ∀ψ, φ ∈ D(H), (2.18)

that reduces to

ψ∗(a)φ(a)− ψ∗(0+)φ(0+) = 0. (2.19)

It then follows that the operator H is self-adjoint in the space of functions which

differ by an arbitrary but fixed phase θ at the end points of the domain, ψ(a) =

27

The quantum problem

eiθψ(0+),

< ψ, Hφ >=< H†ψ, φ >,

∀ψ, φ ∈ D(0,a)(H) ≡ D(0,a)(H†) , (2.20)

where

D(0,a)(H) = L2((0, a), |x|dx) : ψ(a) = eiθψ(0+) 6= 0. (2.21)

Hence, the eigenfunctions

ψE(x) =

√2

aexp

[iE

2~νx2

](2.22)

form a complete orthonormal basis spanning the space D(0,a)(H).

The parameter θ must be the same for all the functions in the Hilbert space.

Different choices of θ give rise to different Hilbert spaces which, however, describe

equivalent physical systems. Hence, changing the value of θ has no effect on the

energy differences between states, or on the matrix elements < Ψ1(x, t)MΨ2(x, t) >

for any operator M (e.g., in the probability amplitude). Without loss of generality,

θ can be set to zero, which implies the added symmetry Ψ∗n = Ψ−n among the energy

eigenstates.

The boundary condition in (2.21) implies a discrete energy spectrum,

En :=2ν~a2

(2nπ + θ) =4πν~na2

+2ν~a2

θ, n ∈ Z and θ ∈ [0, 2π] (2.23)

En − Em := ∆E =4πν~a2

(n−m) (2.24)

The parameter θ produces a shift of energy levels by the constant ∆E = 2ν~θ/a2,

which can be seen as the effect of putting the system in an environment at a constant

potential [15]. Thus, the energy eigenstates are described by the wave functions

Ψn,θ(x, t) =

√2

aexp

[i2nπ + θ

a2(x2 − 2νt)

], (2.25)

and the general solution in the interval (0, a) is Ψ(x, t) =∑cnΨn,θ(x, t), with the

coefficients cn given by

cn =< Ψn,θ(x, t)|Ψ(x, t) >=

∫ a

0

Ψ∗n,θ(x, t)Ψ(x, t)x dx (2.26)

which are determined by the initial condition Ψ(x, 0) = ψ0(x).

28


The energy spectrum En is unbounded below because the are no restrictions on

the values of n ∈ Z as seen in (2.23). This is due to the first-order character of the

Schrodinger operator in this case. Thus, the spectrum of this Schrodinger equation

is analogous to the Dirac case, where the negative energy states are interpreted

as anti-particles states going backwards in time. Here, the energy eigenvalues En

remain unchanged under simultaneous reversal of n and ν,

En,ν = −E−n,ν = −En,−ν = E−n,−ν . (2.27)

Hence, the negative values of E correspond to the states of a system where ν has

the opposite sign, passing from a system of attractive character to a repulsive one

or vice-versa, which are precisely the time-reversal of each other.

2.3.2 Probability density

The probability of finding the state in a configuration around x is given by

P (x < x′ < x+ dx, t) = |Ψ(x, t)|2|x|dx, (2.28)

which is the same for all n and any value of θ,

Pn,θ(x < x′ < x+ dx, t) = |Ψn,θ(x, t)|2|x|dx =2

a2|x|dx . (2.29)

The probability density is ρ(x, t) = |Ψ(x, t)|2|x|. The conservation of probability

of finding a particle anywhere in the interval 0 < x < a is ensured by the continuity

condition obtained multiplying the Shcrodinger equation by xΨ∗(x, t),

∂tρ+ ∂xJ = 0 , (2.30)

where J = ν|Ψ(x, t)|2. The behaviour of the quantum probability density follows

the classical pattern: for a particle moving according to x2(t) = 2νt + x20, the

probability P (x)dx ∝ dt = dx/|v| (where v is the velocity) can be calculated using

the fact that v = ν/x and requiring the probability to be normalized in the interval,∫ a0P (x)dx = 1. The result is

Pclass(x) =2

a2|x| , (2.31)

which indicates that the probability of finding a particle decreases linearly with x.

Additionally, for ν < 0 J = ν|Ψ(x, t)|2 represents a constant flux towards x = 0.

29

The quantum problem

2.3.3 Example: a wave packet

Ideally one can prepare the system in an initial state that describes a particle local-

ized in an interval and study its evolution: how long does it take for a generic state

to be absorbed? At which rate does the source absorb particle-like waves? In order

to address these issues, one can consider a state can prepared as

ψ(x) =

0 0 < x < x1

k x1 ≤ x ≤ x2

0 x2 < x < a

, (2.32)

and normalized,

||ψ(x)||2 =

∫|ψ(x)|2|x|dx = k2 x

22 − x2

1

2= 1 , k =

√2

x22 − x2

1

. (2.33)

This state can be spanned in the basis (2.25)

ψ(x) =n=∞∑n=−∞

cnψn(x) , (2.34)

where cn =< ψn(x)|ψ(x) >=∫ a

0

√2ae−i

2nπa2

x2

|x|ψ(x)dx. substituting (2.32), yields

cn = iak√2nπ

[e−i

2nπa2

x22 − e−i

2nπa2

x21

]and therefore,

ψ(x) =n=∞∑n=−∞

ik

2nπ

[ei

2nπa2

(x2−x21) − ei

2nπa2

(x2−x22)]. (2.35)

This state evolves as

ψ(x, t) =n=∞∑n=−∞

ik

2nπ

[ei

2nπa2

(x2−x21−2νt) − ei

2nπa2

(x2−x22−2νt)

](2.36)

It is clear that this is a wave packet with constant profile rigidly moving towards

the origin (for ν < 0) or towards the x = a value (for ν > 0).

ψ(x) =

0 0 < x <

√x2

1 + 2νt

k√x2

1 + 2ν ≤ x ≤√x2

2 + 2νt

0√x2

2 + 2νt < x < a

(2.37)

30


x

Wave Function

ax1 x2

Figure 2.2: Evolution of a sharply wave packet localized as in (2.32), with ν < 0.

One can say that this wave translates rigidly with the law

x(t) :=√x2(0) + 2νt (2.38)

After a maximum time t = a2/2|ν| the wave is totally absorbed at x = 0 (or at

x = a in the repulsive case).

2.3.4 Including the degeneracy: x ∈ (a−, a+), a− < 0 < a+

We now examine the Schrodinger equation (2.14) in an interval that extends across

the surface of degeneracy. The idea is to describe a situation in which both the initial

and the final states can be on either side of x = 0, in order to explore the possibility

of tunnelling across the degeneracy surface. The difficulty is not so much to find

the space of solutions in the interval (a−, a+), with a− < 0 < a+, –in other words,

a Hilbert space L2((a−, a+), |x|dx)– but to make sure that H is self-adjoint in that

space of solutions. In fact, the solution analogous to the previous case, normalized

in this domain with measure |x|dx, is

ψE(x) =

√2

(a−)2 + (a+)2exp

[iE

2~νx2

], (2.39)

31

The quantum problem

which reduces to (2.17) for a− = 0, a+ = a. The condition (2.19) for the symmetry

of H, however, is replaced by the requirement,3

ψ∗(a−)φ(a−)− ψ∗(0−)φ(0−)− ψ∗(0+)φ(0+) + ψ∗(a+)φ(a+) = 0. (2.40)

The Hamiltonian operator H = −i~x−1∂x is singular at x = 0 and hence, the

wave function need not be defined there. The correct definition of the domain is not

the continuous interval (a−, a+), but rather (a−, 0) ∪ (0, a+), and the wavefunction

ψ(x) is allowed to be discontinuous at x = 0. Therefore, we look for solutions

that are everywhere bounded but not necessarily continuous at x = 0, where they

can present a (finite) discontinuity. This kind of functions belong to the class of

piecewise continuous function and they are not almost everywhere continuous. This

means, in particular, that (2.40) must be interpreted as two separate statements,

ψ∗(a−)φ(a−) = ψ∗(0−)φ(0−) and ψ∗(a+)φ(a+) = ψ∗(0+)φ(0+), (2.41)

and the results of the previous section be can expected to hold for both intervals

(a−, 0) and (0, a+) separately.

The self-adjoint condition for H must be respected in both domains, so the

wavefunctions must satisfy the following boundary conditions

ψ(a−) = eiθ−ψ(0−), and ψ(a+) = eiθ

+

ψ(0+) . (2.42)

This in turn implies that the Hilbert space splits into two subspaces H±

H− = φ(x) ∈ L2((a−, 0), |x|dx) : φ(a−) = eiθ−φ(0−) (2.43)

H+ = φ(x) ∈ L2((0, a+), |x|dx) : φ(a+) = eiθ+

φ(0+) , (2.44)

where in general φ(0−) 6= φ(0+). This shows that the quantum problem in a region

that extends across a degenerate surface reduces to the previous case on the disjoint

sets (a−, 0) and (0, a+), and the Hilbert space splits into a direct sum

H = H− ⊕H+, (2.45)

where H− = L2((a−, 0), |x|dx) and H+ = L2((0, a+), |x|dx) are mutually orthog-

onal projections of L2((a−, a+), |x|dx) on the intervals (a−, 0) and (0, a+). These

3This condition appears because the weight |x| in the scalar product (2.13) splits the integral,

< ψ, Hφ >=∫ a+

a−ψ∗|x| (Hφ) dx = −

∫ 0

a−ψ∗x(Hφ)dx+

∫ a+

0ψ∗x(Hφ)dx.

32


projections can be implemented through the action of the operator P , defined as

P : f(x) 7→ sgn[x] · f(x), and

H± =1

2(P ± 1) · L2((a−, a+), |x|dx) . (2.46)

In this splitting, the support of each function space is restricted to either one side

or the other, and the wave functions are

ψ =

(ψ+(x)

ψ−(x)

)= ψ+(x)⊕ ψ−(x), (2.47)

where

ψ−(x) =

√2

a−exp

[i

2~νE−n x

2], a− < x < 0

0 , 0 < x < a+(2.48)

ψ+(x) =

0 , a− < x < 0√

2a+ exp

[i

2~νE+n x

2], 0 < x < a+

(2.49)

These are admissible solutions of the Schrodinger equation,

Hψ± = E±ψ± (2.50)

where the eigenvalues E± are found to be

E±n = (2nπ + θ±)2~ν

(a±)2. (2.51)

The Hamiltonian takes a block-diagonal form, each block having its own spectrum

E±n . The complete energy spectrum is the union of the two spectra

En = E+n ∪ E−n . (2.52)

Some eigenvalues could have a matching one on the other side, i.e. E+n = E−m, then

n =

(a+

a−

)2

m+ κ , (2.53)

where κ = [(a+)2θ− − (a−)2θ+]/[2π(a−)2]. As in the previous case, changing the

energy spectrum by a constant corresponds to an equal shift in the phases of all

wave functions, θ± → θ± + δθ with no observable effects. This freedom can be used

to set κ = 0, so that the ground states on both sides (n = 0 = m) have the same

energy. In that case, we can distinguish three possibilities:

33

The quantum problem

x

Pro bability de ns ity fo r e ige ns tate s

a+

a-

Figure 2.3: Probability density for the energy eigenstates (2.48,2.49) in (a−, a+).

• If (a+/a−)2 is a generic irrational number, the two spectra have only one

common eigenvalue –a doubly degenerate ground state–

• If (a+/a−)2 takes a rational value, there are some doubly degenerate eigenstates

and the rest are nondegenerate

• In the extreme case a symmetric domain, a+ = a−, all states are doubly

degenerate.

The general time-dependent solution reads

Ψ(x, t) =

∑c−m√

2a−

exp [2πim(x2 − 2νt)(a−)−2] , x ∈ (a−, 0)

∑c+n

√2

a+ exp [2πin(x2 − 2νt)(a+)−2] , x ∈ (0, a+)

(2.54)

where the coefficients are given as before

c−m = < Ψn(x, t)|Ψ(x, t) >=

∫ 0

a−Ψ∗n(x, t)|x|Ψ(x, t) dx

c+n = < Ψm(x, t)|Ψ(x, t) >=

∫ a+

0

Ψ∗m(x, t)|x|Ψ(x, t) dx

34

2.4 Tunneling: equation of continuity

2.4 Tunneling: equation of continuity

Note that there is no overlap between wavefunctions with support on opposite sides

of the degeneracy surface. Consequently, a wave packet initially prepared in the

region x < 0 will never evolve into x > 0, and vice-versa. This is in complete

agreement with the classical behaviour of the system whose orbits never extend to

the other side of a degeneracy surface. In other words, there is neither classical nor

quantum flow across the degeneracy surface.

The term |x| in the probability density ρ(x, t) = |Ψ(x, t)|2|x| reflects the role

of the degeneracy as a singularity of the probability flow, where particle states are

created or annihilated. In fact, the quantum mechanical probability density satisfies

a continuity equation with a sink (or source) at the degeneracy,

∂tρ+ ∂xJ = σ , (2.55)

where

ρ(x, t) = |Ψ(x, t)|2|x| (2.56)

J(x, t) = ν sgn(x) |Ψ(x, t)|2 (2.57)

σ(x, t) = 2ν δ(x) |Ψ(x, t)|2. (2.58)

The direction of the flow is determined by the sign of ν and the degeneracy at

x = 0 acts as sink (ν < 0) or source (ν > 0) of states. As seen in (2.58), there is a

delta function supported at the discontinuity of the wave function that acts as this

source or sink of particles (see Appendix).

Let us analyze the continuity equation, both in the case in which the degeneracy

is excluded, 0 < x < a, and included, x ∈ (a−, a+) with a− < 0 < a+. Integrating

the continuity equation (2.55) over the interval 0 < x < a, the source term does not

contribute and therefore∫ a

0

∂

∂tρ(x, t) dx+

∫ a

0

∂

∂xJ(x, t) dx = 0 (2.59)

and denoting with P (t) the total probability∫ρ(x, t) dx we have

d

dtP (t) + J(x, t)

∣∣∣a0

= 0 (2.60)

d

dtP (t) + ν sgn(x) |Ψ(x, t)|2

∣∣∣a0

= 0 (2.61)

d

dtP (t) + ν

[|Ψ(a, t)|2 − |Ψ(0, t)|2

]= 0 (2.62)

35

The quantum problem

finally, imposing the boundary condition Ψ(a, t) = Ψ(0, t)eiθ one gets

d

dtP (t) = 0 for x ∈ (0, a) , (2.63)

which means that the probability of finding the particle inside the interval that

excludes the degeneracy is constant.

On the other hand, integrating the continuity condition over the interval (a−, a+)

contributions from the source (sink) cannot be ignored,∫ a+

a−

∂

∂tρ(x, t) dx+

∫ a+

a−

∂

∂xJ(x, t) dx =

∫ a+

a−σ(x, t) dx . (2.64)

The current density is a continuous function except for a jump ∆ at x = 0,

∆ = J(0+, t)− J(0−, t) = ν|ψ(0+, t)|2 + ν|ψ(0−, t)|2 . (2.65)

Its derivative gives a delta at the jump that exactly matches that of the source.

Then the results are (see Appendix)∫ a+

a−

∂

∂xJ(x, t) dx =

∫ a+

a−

∂

∂x

(ν sgn(x) |Ψ(x, t)|2

)x6=0

+

∫ a+

a−∆ · δ(x) =

= ν[|Ψ(a−)|2 + |Ψ(a+)|2

]=

∫ a+

a−σ(x, t) dx . (2.66)

So the continuity equation (2.64) integrated over the interval (a−, a+) reads

d

dtP (t) = 0 (2.67)

For the interval (a−, a+) the total probability density is conserved: given the bound-

ary conditions (2.41), the flux entering (leaving) from the endpoints a− and a+ is

absorbed (suplied) by the degeneracy surface. Once more this shows that imposing

boundary condition that make H self-adjoint results in conservation probability and

there is no net flux of the wave packet across the degeneracy in any case.

36

Chapter 3

Coupled system

An important feature of a degenerate system is that the degeneracy generically

affects a two-dimensional submanifold of the phase space. This means that in a

degenerate system there usually coexist coupled degenerate and a non-degenerate

subsystems. Next, we analyze quantum-mechanically this situation, which had been

previously discussed in a classical context in [1].

3.1 General classical solution

The next step is to study the quantum behaviour of a coupled system: what is the

fate of two systems, a regular one and a degenerate one, when they interact? Some

important questions arise: can the two systems influence each other to the point of

cancelling or amplifying the effects or characteristics of the degeneracy? Could it

happen that once the degenerate system collapses on the surface of degeneracy, the

regular system continues to evolve in a form that certain information can overcome

the ”barrier”? Would this tunnelling effect depend on the type of coupling?

We first summarize what is known at classical level for a simple coupled system

with a Lagrangian of the form [1],

L(xα, za, xα, za) = LDEG(xα, , xα) + LREG(za, za)− Vλ(xα, za) α = 1, 2, a = 1, 2

(3.1)

where the coordinates xα refer to the degenerate system, while za refer to the regular

system. For our purposes the Lagrangians are written in their first order version,

where

LDEG = Aα(x)xα −HDEG(x), (3.2)

37

Coupled system

describes a 2-dimensional system that degenerates at a surface described by f(xα) =

0 (see Chapter 1.2), that is f(x, y) = ∂xAy(x, y)−∂yAx(x, y) = 0. On the other hand

LREG describes a 2n-dimensional regular system with a corresponding Hamiltonian

HREG(za)

LREG(z) = paza −HREG(z), a = 1, ..., 2n (3.3)

The interacting term plays the central role, a possible simple choice is to write a

coupling that does not change the character of the degeneracy surface, i.e. a coupling

that vanishes at the degeneracy as in

Vλ(xα, za) = λf(xα)HREG(za) (3.4)

Finally the complete Lagrangian reads

L = Aαxα −HDEG(x) + pza −HREG(z) + λf(xα)HREG(za) α = 1, 2, a = 1, 2 (3.5)

Here Vλ depends on a constant parameter λ, it vanishes at the degeneracy surface

and does not change the flux density there (note that this coupling would be trivial

in case of nondegenerate systems). Classically such coupling allows the degenerate

system to evolve until it reaches the degeneracy surface, while the regular system

evolves as in the decoupled case but with a reparametrized time τ(t). Once the

degeneracy is reached, both time coordinates become identical (τ = t), all traces

of the degenerate system disappear, the regular system evolves undisturbed and

the solution thereafter contains no traces of the initial conditions of the degenerate

subsystem.

The equations of motion areza =

[1 + λf(x)

]F ab∂bHREG(za)

εαβf(x)xβ = ∂α

[HDEG(xα) + λf(x)HREG(z)

] , (3.6)

where F ab is the inverse of the regular (canonical) symplectic form. The first equa-

tion in (3.6) implies that the regular Hamiltonian HREG is independently conserved

HREG = za ∂aHREG(z) = 0.

This permits to solve the equations of motion for the degenerate part in the xα(t)

coordinates, independently from the regular subsystem,

εαβf(x)xβ =[∂αHDEG(x) + λHREG(z)∂αf(x)

]. (3.7)

38

3.1 General classical solution

The solution xα(t) can be put back in the first equation (3.6) to confirm that the

regular system describes the same orbit as in the decoupled case, with a reparametrized

time

za(t) =dza

dτ

dτ

dt= F ab ∂bHREG(z)

[1 + λf

(x(t)

)], (3.8)

dτ

dt= 1 + λf

(x(t)

), (3.9)

and finally the equation of motion for the regular part becomes

za(t) = zaλ=0

(τ(t)

). (3.10)

In particular, the coupling between the simplest degenerate system L = xyx−νywith f(x, y) = x, as discussed in (2.6), and a harmonic oscillator gives [1]

L = LDEG(x, y) + LREG(p, q) + λf(x, y)HREG(p, q)

= xyx− νy + pq − 1

2(1 + λx)(p2 + q2) (3.11)

with ν < 0. In this case the degenerate subsystem collapses to x = 0 in a finite

time. The four equations of motion read

xy = −1

2λ(p2 + q2) (3.12)

xx = ν (3.13)

p = −(1 + λx) (3.14)

q = (1 + λx) . (3.15)

From equations (3.14) and (3.15) it is clear that p2 + q2 = constant. Hence from

(3.12) and (3.13) the orbits in the (x, y) plane are straight lines

The solutions are

x2(t) = 2νt+ x02, 0 < t < x2

o

2|ν| (ν < 0)

x(t) = 0, t < to

y(x) = −12λν(p2

0 + q20)x+ y0

y(x) = yo, t < to

p(τ) = p0cos (τ + φ0)

q(τ) = q0sin (τ + φ0)

, (3.16)

39

Coupled system

where τ = t+ λ3ν

[2νt+ x2o]

3/2.

This means in particularx(t) = ±

√2νt+ x2

o for t < x2o

2|ν|

x(t) = 0 for t > x2o

2|ν|

(3.17)

with

τ(t) = t+λ

3ν(2νt+ xo

2)3/2 for 0 < t < .x2o

2|ν|= to (3.18)

This shows that the orbits in the (q, p) plane are circles that are traced at a speed

that changes in time for 0 < t < to = x2o/2|ν|. Once the degenerate system collapses

for t ≥ xo2/2ν, the harmonic oscillator evolves with time τ = t.

3.2 Quantum coupled system

Let us now study the quantum behaviour of the same simple degenerate system

discussed in Section 2.2, coupled to a harmonic oscillator,

L = xyx− νy + pq − 1

2(p2 + q2)− 1

2λx(p2 + q2) (3.19)

The corresponding Hamiltonian operator is

Hc = νy +1

2(1 + λx)(p2 + q2) , (3.20)

The proposed quantization prescription is

q : = q (3.21)

p : = −i~∂q (3.22)

x : = x (3.23)

y : = −i~1

x∂x . (3.24)

These operators are defined on the Hilbert space of function of two variables

(x, q), whose scalar product, according to previous Chapter 2 should be

< f(x, q)|g(x, q) >:=

∫f(x, q) g(x, q) |x| dx dq . (3.25)

The Hamiltonian operator is time independent and reads

Hc = −i~ν 1

x

∂

∂x+

1

2(1 + λx)(−~2 ∂

2

∂q2+ q2) . (3.26)

40


The Schrodinger equation

Hcψ(x, q, t) = i~∂

∂tψ(x, q, t) (3.27)

is separable and the solutions can be factorized as ψ(x, q, t) = u(x, q)φT(t), and

1

u(x, q)H(u(x, q)

)= i~

1

φT(t)

d

dtφT(t) . (3.28)

Equating both sides to a constant W gives

i~1

φT(t)

d

dtφT(t) = W (3.29)

1

u(x, q)H(u(x, q)

)= W . (3.30)

As usual, the constant W plays the role of the total energy of the system. The first

equation has the typical solution φT(t) ∝ exp(− i~Wt). The second is the eigenvalue

equation for H

H(u(x, q)

)= Wu(x, q) , (3.31)

where W represents allowed values of total energy of the coupled system. According

to (3.26), (3.31) is the differential equation

− i~ν 1

x

∂

∂xu− 1

2~2 ∂

2

∂q2u− 1

2~2λx

∂2

∂q2u+

1

2q2 u+

1

2λx q2 u = Wu (3.32)

We look for factorized solutions u(x, q) = φD(x) · φR(q), where the labels D and

R stand for “degenerate” and “regular”

φR(q)[− i~ν 1

x

∂

∂xφD(x)

]+

1

2(1 + λx)φD(x)

[− ~2 ∂

2

∂q2+ q2

]φR(q) = WφD(x)φR(q) .

(3.33)

Dividing by (1 + λx) · φD(x) · φR(q) one obtains1

1

φR(q)

1

2

[− ~2 d2

dq2+ q2

]φR(q) = i~ν

1

x(1 + λx)

1

φD(x)

d

dxφD(x) +

W

1 + λx. (3.34)

Here each side should be equal to a constant, ER,12

[− ~2 d2

dq2+ q2

]φR(q) = ER φR(q)

i~ν 1x(1+λx)

1φD(x)

ddxφD(x) + W

1+λx= ER .

(3.35)

1The condition (1 + λx) 6= 0 is assumed here because otherwise the Hamiltonian (3.20) wouldmerely describe the purely degenerate system already discussed in previous sections.

41

Coupled system

The first is clearly the eigenvalue equation for a normal harmonic oscillator2 whose

solutions are

φRn (q) = NnHn(q)e−12q2 , ER

n = ~ω(n+1

2) , (3.36)

with q ∈ (−∞,∞), φRn (q) → 0 for q → ±∞. Here Nn is a normalization constant

and Hn is the nth Hermite polynomial.

The second equation in (3.35) is still separable and can be written as

1

φD(x)

d

dxφD(x) =

i

~ν[Wx− ERx(1 + λx)

], (3.37)

which is similar to the simplest degenerate case we have already studied, corrected

by the term −ERx(1 + λx). Its integral is

φD(x) = φ0exp

[i

2~ν

[(W − ER

n

)x2 − 2

3λERx3

]]. (3.38)

The main difference between this solution and that of the quantum degenerate

system discussed before is the term x3, that apparently has no classical counterpart,

but as we will show, it can be interpreted in the context of the harmonic oscillator.

For λ = 0, the Hamiltonian (3.20) describes a decoupled system, whose total

energy W is simply the sum of the energy of its two subsystems, W = ER + ED.

Then W − ERn = ED is the energy of a degenerate system in the presence of a

(decoupled) harmonic oscillator .

In order to write down completely the factorized solution ψW(x, q) = φD(x)φR(q),

the differential equation (3.30) should be solved with some boundary condition that

makes the Hamiltonian operator self-adjoint. The symmetry condition

< ϕ1, Hϕ2 >=< ϕ2, Hϕ1 >∗,

plus the condition of self-adjointness for H = HDEG +(1+λx)HOSC require boundary

conditions for ψW(x, q) that are the intersection of the boundary conditions that

make self-adjoint separately HDEG and HOSC, each one in its proper domain.

This happens because (1 + λx) term is real

< g |H(f) >=< g |HDEG(f) > + < g |(1 + λx)HOSC(f) > (3.39)

< H†(g)|f >=< H†DEG(g)|f > + < (1 + λx)H†OSC(g)|f > (3.40)

2In the coupled Hamiltonian (3.20) we assumed ω = 1.

42


So, using the usual harmonic oscillator’s boundary condition and recovering previous

results (2.42) for the degenerate system (for a− < 0 < a+)

q ∈ (−∞,∞) with φRn (q)→ 0 for q → ±∞ (3.41)

x ∈ (a−, a+) with φD

W (a−) = φD

W (0−) and φD

W (a+) = φD

W (0+) (3.42)

the Hamiltonian H = HDEG + (1 + λx)HOSC is self-adjoint. The energy spectrum is

given by

Wk,n =4πk~νa2

+ ERn +

2

3λaER

n , (3.43)

where ERn = ~ω(n + 1

2). This spectrum coincides with the perturbative correction

for the Hamiltonian HDEG + HOSC produced by a perturbation given by λ · xHOSC.

Here 4πk~ν/a2 +ERn is the energy of the decoupled system composed by degenerate

system and a harmonic oscillator, and 23λER

n a is the first and only correction to the

energy (the same for every level).

We observe that for λ = − 32a

the coupling energy and the harmonic oscillator

energy cancel out and the total energy does not depend on n: regardless of the

state of harmonic oscillator the total energy is exactly the energy of the degenerate

system.

The complete wave function ψ(x, q, t) = φR(q)φD(x)φT(t) reads

ψk,n(x, q, t) = ψ0 ei~

12ν

[(Wk,n−ERn

)x2− 2

3λERn x

3

]φRn (q) e−

i~Wk,nt , (3.44)

that can also be written as

ψk,n(x, q, t) = ψ0 exp

i

~1

2ν

[4kπ~νa2

+2

3λa ~ω

(n+

1

2

)](x2 − 2νt)

(3.45)

×NnHn(q)exp

− 1

2q2 − iω

(n+

1

2

)[t+

1

3

λ

νx3

]This describes a degenerate system whose energy levels are corrected by the en-

ergy levels of harmonic oscillator, 23λa~ω(n+ 1

2), and the harmonic oscillator has a

reparametrized time τ(t) = t+ 13λνx3. As usual, ψk,n(x, q) must be properly normal-

ized, ∫ ∫|ψ(x, q, t)|2 |x| dx dq = 1 . (3.46)

The most general solution is

ψk,n(x, q, t) =∑n≥0,∀k

ck,n φD

k,n(x, t)φR

n(q, t) , (3.47)

43

Coupled system

and as the sum runs over n and k, this general wave function cannot be expressed

as the product of a degenerate and a regular wave function.

44

3.3 Continuity equation

3.3 Continuity equation

Until now we have shown that the dynamics of the degenerate system is not altered

by the coupling, only its energy levels are displaced by a constant gap, while the

harmonic oscillator is more conditioned in his evolution, with reparametrization in

time: what are the consequences in the continuity equation? To answer this it is

natural to define first the following quantities

• total probability density: ρ(x, q, t) := |ψ(x, q, t)|2|x|

• degenerate probability density: ρD(x, t) :=∫|ψ(x, q, t)|2|x|dq

• regular probability density: ρR(q, t) :=∫|ψ(x, q, t)|2|x|dx

where ψ stands always for the most general wave function ψ(x, q, t) .

Even if the general wave function can not be factorized into a degenerate and a

regular part, the densities ρD(x, t) and ρR(q, t) can be related as usual to some

square amplitudes. So, let’s tentatively define

ρD(x, t) =

∫|ψ(x, q, t)|2|x|dq = |x|

∫|ψ(x, q, t)|2dq =: |x||ϕD(x, t)|2 (3.48)

ρR(q, t) =

∫|ψ(x, q, t)|2|x|dx =: |ϕR(q, t)|2 (3.49)

As expected from the definition previously given in (2.28), the so-called degenerate

part |ϕD(x, t)|2 of the wave is not dimensionally function a probability amplitude

density by itself, it misses the |x| factor to be the proper probability amplitude in

the degenerate Hilbert space.

From the Schrodinger equation in (3.26) and in (3.27), the continuity equation is

∂

∂tρ(x, q, t) +

∂

∂xJx(x, q, t) +

∂

∂qJq(x, q, t) = 2νδ(x)|ψ|2 , (3.50)

where

• total current density:−→J (x, q, t) =

(Jx(x, q, t), Jq(x, q, t)

)• Jx(x, q, t) = νsgn(x)|ψ|2

• Jq(x, q, t) = (1 + λx)|x| i~2m

[ψ ∂∂qψ∗ − ψ∗ ∂

∂qψ]

45

Coupled system

Moreover let’s define

• degenerate current density: JD(x, t) =∫Jx(x, q, t)dq

• regular current density: JR(q, t) =∫Jq(x, q, t)dx

Integration over x in the interval (0, a) or (a−, a+) gives (for the integration involving

the terms Jx(x, q, t) and 2ν δ(x)|ψ| see the Appendix B)

∫∂

∂tρ(x, q, t)dx+

∫∂

∂xJx(x, q, t)dx+

∫∂

∂qJq(x, q, t)dx =

∫2νδ(x)|ψ|2dx ,

(3.51)

∂

∂tρR(q, t) +

∂

∂qJR(q, t) = 0 . (3.52)

This equation nearly recover the typical harmonic oscillator result with the quanti-

ties ρR(q, t) and JR(q, t) defined above

∂

∂tρR(q, t) +

∂

∂q

i~2m

∫ a

0

(1 + λx)[ψ∂

∂qψ∗ − ψ∗ ∂

∂qψ]|x|dx = 0 . (3.53)

For an eigenstate of the system as written in (3.44) this regular part reads

∂

∂tρR(q, t) +

(a2

2+ λ

a3

3

)∂

∂q

i~2m

[φR

n

∂

∂qφR ∗n − φR

n

∂

∂qφR∗n

]= 0 , (3.54)

ρR(q, t) = |φR

n(q, t)|2 =(NnHn(q)e−

12q2)2. (3.55)

On the other hand integration over q ∈ (−∞,∞) gives

∫∂

∂tρ(x, q, t)dq +

∂

∂x

∫Jx(x, q, t)dq +

1

2(1 + λx)|x|

∫∂

∂qJq(x, q, t)dq =

2νδ(x)

∫|ψ(x, q, t)|2dq ,

∂

∂tρD(x, t) +

∂

∂xJD(x, t) = 2νδ(x)|ϕD(x, t)|2 . (3.56)

As one can easily compare with first equations of Section 2.4, this is the same result

obtained for a singular degenerate system described by a wave function ϕD(x, t) with

density ρD(x, t) = |ϕD(x, t)|2|x| and with current JDx (x, t) = νsgn(x)|ϕD(x, t)|2. The

dynamics of the degenerate systems is as before (up to a constant in the energy),

so for ν < 0 (or ν > 0 ) the wave evolves towards x = 0 (or away from x = 0)

independently from the evolution of harmonic oscillator: for this coupled system

described by (3.20) there is no net flux across the degeneracy.

46

Chapter 4

Path integral

In the previous chapters we have followed the more conventional way to quantize a

classical system: the canonical quantization method. In this method the Hamilto-

nian governs the dynamics through a differential point of view: once the quantum

Hamiltonian operator is constructed the differential Schrodinger’s equation dictates

the time development of a system. This is in contrast with the global point of view

based on the action principle: among all the possible evolution (paths) between to

points at two distinct times, classically the system follows the path that extremizes

the action S =∫L(z, z, t) dt. The quantum counterpart of this approach resides

in the famous method developed by Feynman [24] [25], based on the Dirac’s ideas

that Lagrangians have a fundamental role in quantum formulation [26]. Feynman

constructed for quantum mechanics a generalization of the action principle via the

so-called path integral formulation: if in classical mechanics only matters the par-

ticular trajectory that extremizes the action functional, in the quantum version all

the paths contribute.

In this chapter we will investigate the quantization through the path integral

formulation of the same simple example L = xyx− νy we have already analised in

Chapter 2 with canonical quantization.

4.1 Propagator

In this approach a system classically governed by the generic action S[x; t1, t2] =∫ t2t1L (x(t), x(t), t) dt has a probability amplitude to evolve between two prescribed

points at two distinct instants of time given by

47

Path integral

K(x2, t2;x1, t1) = A

∫Dx(t) e

i~S[x;t1,t2] , (4.1)

where the measure of integration Dx(t) is over all possible continuous trajectories in

configuration space joining initial and final point x1 and x2, and A is a normalization

factor. The above expression K(x2, t2;x1, t1) is called path integral.

Eventually the measure can be realized by dividing the possible path in N in-

finitesimal time intervals and then realizing the integration as a product of integra-

tion over each individual i-th interval keeping fix the initial (i = 0) and final (i = N)

points

Dx = limN→∞

N−1∏i=1

dx(ti) . (4.2)

Considering the action in Hamiltonian form, S[p, q; t1, t2] =∫ t2t1

[pq −H(p, q)] dt,

the path integral is a sum over all possible trajectories in phase space, this permits

to consider a wider class of dynamical systems, e.g. those with constraints without

standard symplectic structures ( [37], [38], [39]). With general coordinates (p, q) = zi

with i = 1, ...2n the action functional reads

K(z2, t2; z1, t1) =

∫Γ

Dω(z) exp

i

~S[z; t1, t2]

, (4.3)

where the integral measure takes into account the symplectic structure F ij = [zi, zj]

of the phase space Γ

Dω(z) =∏

(det[F (z)])1/2 dz(ti) . (4.4)

For the simple degenerate system we have studied in Chapter 2.2 described by

the Lagrangian L = xyx− νy , we can explicitly write the propagator through (4.3)

K(xf , tf ;xi, ti) =

∫Γ

DyDxei~∫

(xyx−νy)dt . (4.5)

We discretize the Lagrangian slicing the time interval (ti, tf ) into N intervals

of magnitude ε, then we take the limits N → ∞ and ε → 0. Accordingly the

x (position) and y (momentum) coordinates will be discretized as x → xn−xn−1

2,

x→ xn−xn−1

ε, y → yn with x0 as initial point and xN as final point.

The Hamiltonian action will read

48

4.1 Propagator

S = limε→0N→∞

i

~ε

N∑n=1

(xn − xn−1

2· yn ·

xn − xn−1

ε− νyn

)=

= limε→0N→∞

i

~

N∑n=1

(x2n − x2

n−1

2· yn − νεyn

), (4.6)

and the propagator reads


= limε→0N→∞

∫dx1 · · · dxN−1

dy12π~ · · ·

dyN2π~ e

i~∑Nn=1

(x2n−x

2n−1

2yn−νεyn

)=

= limε→0N→∞

∫dy12π~ · · ·

dyN2π~ e

− i~∑Nn=1 νεyn

∫dx1 · · · dxN−1e

i2~∑Nn=1(x2

n−x2n−1)yn . (4.7)

All the calculation for the above expression is performed in the Appendix C, here

we outline the principal steps. The integration in x results in the following infinite

product

∫dx1 · · · dxN−1exp

[i

2~

N∑n=1

(x2n − x2

n−1

)yn

]=

= ei

2~(−y1x20+yNx

2N) ·

j=N−1∏j=1

∫dxje

i2~ (yj−yj+1)x2

j , (4.8)

whose generic element is1

∫ ∞−∞

dxjeiyj−yj+1

2~ x2j = (4.9)

=

∫ ∞−∞

cos

(yj − yj+1

2~x2

1

)dxj + i

∫ ∞−∞

sin

(yj − yj+1

2~x2j

)dxj = (4.10)

=

√2π~

|yj − yj+1|ei[sgn(yj−yj+1)]π

4 . (4.11)

After completing the x integration the action functional will read

1up to a phase exp(2nπ)

49

Path integral

K (xf , tf ;xi, ti) =

= limε→0N→∞

∫dy1

2π~· · · dyN

2π~e−

i~∑Nn=1 νεyne

i2~(−y1x2

0+yNx2N) ·

·j=N−1∏j=1

∫dxje

i2~ (yj−yj+1)x2

j =

= lim

∫dy1

2π~· · · dyN

2π~ei(−

x202~ y1−

νε~ y1−

νε~ y2−···−

νε~ yN−1− νε2~yN+

x2N2~ yN )

×j=N−1∏j=1

√2π~


4 . (4.12)

For the integration over y1 we use the substitution

k1 =x2o

2~+νε

~, z1 =

(x2o

2~+νε

~

)(y1 − y2) = k1(y1 − y2) . (4.13)

So the first term, the integral en y1 reads

∫dy1

2π~

√2π~|y1 − y2|

ei[−(x2o2~ + νε

~ )y1+sgn(y1−y2)π4

] = (4.14)

=2√π√

2π~1√k1

e−ik1y2 . (4.15)

For the integration over y2 one obtains

2√π√

2π~√k1

∫dy2

2π~e−ik1y2e−i

νε~ y2

√2π~|y2 − y3|

ei[sgn(y2−y3)]π4 =

= · · · = 2√π√

2π~1√k1

2√π√

2π~1√k2

e−ik2y3 , (4.16)

with the coefficient k2 = k1 + νε~ , and so on, until the integration on yN−1 with the

coefficient kN−1 = kN−2 + νε~ = x2

o

2~ + (N − 1)νε~ . Then the propagator assumes the

expression

50

4.1 Propagator


= limε→0N→∞

∫dyN2π~

j=N−1∏j=1

(2√π√

2π~1√kj

)e(−i kN−1·yN)ei(−

νε~ yN+

x2N2~ yN )) =

= limε→0N→∞

j=N−1∏j=1

(2√π√

2π~1√kj

)∫dyN2π~

ei(−x2o2~−(N−1) νε~ −

νε~ +

x2N2~ )yN =

= limε→0N→∞

j=N−1∏j=1

(2√π√

2π~1√kj

)∫dyN2π~

ei(−x2o2~−N

νε~ +

x2N2~ )yN =

= limε→0N→∞

j=N−1∏j=1

(2√π√

2π~1√kj

)δ

(1

2(−x2

o − 2νNε+ x2N)

)=

= limε→0N→∞

j=N−1∏j=1

(2√π√

2π~1√kj

)2 · δ

(x2N − x2

o − 2νNε))

= A · 2δ(x2N − x2

o − 2νt) . (4.17)

The infinite product in front of the δ is related to the normalization factor, then let’s

observe that the presence of the δ-function makes the propagator vanishing along

any path that does not match the classical one given by the equation (2.8) with

x′ = x(0)

x2(t) = x′2

+ 2νt . (4.18)

This confirms the fact that tunneling effect through the degeneracy is not allowed.

The connection with the Schrodinger’s picture is realized using K(x2, t2;x1, t1)

as a propagator

ψ(x2, t2) =

∫K(x2, t2;x1, t1)ψ(x1, t1) dx1 , (4.19)

where ψ(x2, t2) is the wave function of the system at (x2, t2) and the integration

characterizes the state as a sum of all the probabilities to reach this state from

everywhere (from every previously accessible states). In this context the propagator

K(x2, t2;x1, t1) is given in terms of the eigenfunctions un(x) and eigenenergies Wn

of the Schrodinger’s corresponding problem

K(x2, t2;x1, t1) =∑n

u∗n(x1)un(x2) exp[− i~Wn(t2 − t1)] . (4.20)

51

Path integral

For our simple degenerate system described in Chapter 2.2, first we recall that

the eigenenergies (2.23) are

En :=2ν~a2

(2nπ + θ) =4πν~na2

+2ν~a2

θ, n ∈ Z and θ ∈ [0, 2π] , (4.21)

and the eigenfunctions (2.25) are

un,θ(x) =

√2

aexp

[i2nπ + θ

a2x2

], (4.22)

Then the propagator that takes an initial state ψ(x1 = x′, t1 = 0) to a generic state

ψ(x, t) will be

K(x, t;x′, 0) =∑n

2

a2exp

[i2nπ + θ

a2(x2 − x′2)

]exp

[−i2ν

a2(2nπ + θ)t

]=

=2

a2exp

[iθ

a2(x2 − x′2 − 2νt)

]∑n

exp

[in

2π

a2(x2 − x′2 − 2νt)

]=

= 2

exp

[iθ

a2(x2 − x′2 − 2νt)

]· δ(x2 − x′2 − 2νt) =

= 2 · δ(x2 − x′2 − 2νt) . (4.23)

The above δ is the same found in (4.17), so also with this calculation we re-

covered the same functional dependence of the propagator on the coordinates: the

propagator is a δ-function of the classical trajectory (2.8).

x2(t) = x′2

+ 2νt . (4.24)

Note that we have use the fact that 2

δ(x2) =∑n

cnun(x) =∑n

cn

√2

aexp

[in

2π

a2x2

]=

1

a2

∑n

exp

[in

2π

a2x2

](4.25)

cn := < un(x), δ(x2) >=

∫ a

0

√2

aexp

[−i2nπ

a2x2

]δ(x2)|x|dx =

=

√2

a

∫exp

[−i2nπ

a2x2

]δ(x)

2|x||x|dx =

1

2

√2

a∀n . (4.26)

2The Dirac Delta of a function is given by δ[F (x)] =∑iδ(x−xi)|F ′(xi)| where the sum is over all the

simple zero’s of F (x), i.e. F (xk) = 0 and F ′(xk) 6= 0, to integrate δ[F (x)] is sufficient that thesimple zero’s fall inside the integration interval, as in our case.

52

4.1 Propagator

We can now check how an eigenstate taken as initial state

ψk(x′, t′ = 0) =

√2

aexp

[i2kπ + θ

a2x′2]

(4.27)

propagates through the above calculated propagator 4.23 with the correct integral

measure given by the symplectic structure on the (x, y) phase space according to

(4.4)

det[F (x, y)] = x2 . (4.28)

In principle in the expression of (4.27) the initial value x′ can be either positive

or negative, then3

ψ(x, t) =

∫ a

−aK(x, t;x′, 0)ψ(x′, 0)|x′| dx′ =

=

∫ a

−a2 δ(x2 − x′2 − 2νt) ·

√2

aexp

[i2kπ + θ

a2x′2]|x′|dx′ =

=2√

2

a

∫ a

−aδ(x2 − x′2 − 2νt) exp

[i2kπ + θ

a2x′2]|x′|dx′ = (4.29)

=2√

2

a

−∫ 0

−a

δ(x′ +√x2 + 2νt)

| − 2√x2 + 2νt|

exp

[i2kπ + θ

a2x′2]x′dx′ +

∫ a

0

δ(x′ −√x2 + 2νt)

|2√x2 + 2νt|

exp

[i2kπ + θ

a2x′2]x′dx′

=

=

√2

aexp

[i2kπ + θ

a2(−√x2 + 2νt)2

]ifx < 0

√2

aexp

[i2kπ + θ

a2(√x2 + 2νt)2

]ifx > 0.

(4.30)

Even if we integrate over all possible (negative and positive) initial values of x′,

the above calculated result illustrates how only initial x′ positive (negative) states

contribute to a final x positive (negative) state ψ(x, t).

Therefore in path integral approach all paths that in principle could be contribute

to a tunneling effects are excluded: also this formalism confirms that for a simple

degenerate system there is no net flux across the degeneracy.

3The integration is broken according to δ(x2 − x′2 − 2νt) = δ(x′+√x2+2νt)

|−2√x2+2νt| + δ(x′−

√x2+2νt)

|2√x2+2νt|

53

Chapter 5

Discussion

1. Although the discussion here has been restricted to a rather simple case in which

the configuration space corresponds to the entire real axis, it is easy to see that the

conclusions do not change radically if the configuration space is compact (x ∈ S1).

The only modifications introduced by the topology of the phase space are essentially

two:

i) The restrictions imposed by the Poincare-Hopf theorem that relates the degree of

the singularities in the Hamiltonian flow and the Euler characteristic of the phase

space manifold [23]

ii) The fact that the orbits reverse orientation at the degeneracy surfaces. This

restrict the functional form for the expression of degeneration surface to be a Jordan

curve.

2. The orthogonality of the Hilbert spaces on different sides of a degeneracy sur-

face is not affected by the fact that some eigenvalues might accidentally match –e.g.,

if (2.53) holds. Even in the extreme case, in which all eigenvalues are identical (for

a− = a+), the eigenstates supported on different sides are still orthogonal. In that

case, the states could also be arranged into a basis of symmetric and antisymmetric

wavefunctions, and not supported only on one side or the other. Such a basis of

parity eigenstates may be convenient since the parity operator commutes with the

Hamiltonian.

3. The conclusion of no quantum tunnelling obtained in the canonical quantiza-

tion approach is consistent with other methods like Dirac’s Hamiltonian approach,

Feynman’s path integral, or the old Bohr-Sommerfeld quantization condition.

4. Lovelock Lagrangians that generalize the Einstein-Hilbert theory for dimen-

sions higher than four, as well as Chern-Simons theories in dimensions five or more,

54

5.1 Summary

are widely studied models for the potential description of spacetime and fundamen-

tal interactions at high energies. However, these actions are well known to present

degeneracies [2, 19–22], and should be regarded, therefore, as describing a host of

different physical systems. Depending on the initial conditions where the system

starts, the evolution may take the system towards a degeneracy surface, where the

number of degrees of freedom is reduced and the resulting field theory has an effec-

tive dynamics that corresponds to fewer dimensions, a sort of dynamical dimensional

reduction as discussed in [20].

5.1 Summary

Let us summarize our results:

• Two types of degenerate systems can be distinguished: those whose orbits never

intersect the surfaces of degeneracy (reducible systems) and the rest (irreducible

ones). In the first case, there exists an action principle which yields the same

dynamics everywhere in phase space (with the possible exception of the degenerate

surfaces, where the orbits may not exist). Irreducible systems on the other hand

cannot be globally described by a nondegenerate action principle.

• The quantum mechanical description is obtained by the canonical substitution

as in the standard Schrodinger picture, c.f., Eqs. (2.3, 2.4,2.5). The only difference is

that the degeneracy of the symplectic form becomes the singular set of the quantum

Hamiltonian operator. Since the singular points must be removed from the domain

of the Hamiltonian operator, for consistency they should also be removed from the

domain of the wave functions. This means that the Hilbert space must allow for

wave functions that can be discontinuous at the degenerate surfaces, where they are

allowed to present a finite jump.

• The proper Hilbert space is equipped with a weighted scalar product defined as

the absolute value of the symplectic form w(x) = |F ij| (see 2.10) due to the required

symmetry condition and the positivity of scalar product

• Allowing discontinuous wave functions implies that the solutions have support

restricted to a single region bounded by a degenerate surface Σ. This is realized by

a Hilbert space that splits into a direct sum of orthogonal subspaces. Generalizing

this conclusion, we see that since Σ defines a collection of Jordan curves, the Hilbert

space that describes a quantum degenerate system must be of the form

H = ⊕Ni=1Hi, (5.1)

55

Discussion

where each Hilbert subspace describes the dynamics of a subsystem supported in

only one nondegenerate domain.

• The physical consequence of this is that there is no overlap between wavefunc-

tions on different nondegenerate regions, and therefore no tunnelling across surfaces

of degeneracy. This is in complete analogy with the classical picture of degenerate

systems.

• The degeneracy surface does not permit tunneling effects, then it is analo-

gous to an event horizon: an observer outside knows only what happen before the

degeneracy surface and beyond there is region of information loss.

• For the case of mixed system, here we have presented one composed of the

simple degenerate system and a harmonic oscillator. We found that there is no

evidence for tunneling effect, not even if the nondegenerate system survives once the

degenerate one colapsed: the harmonic oscillator continues undisturbed its evolution

without producing any excitation of the degenerate one.

5.2 Open questions

•What we have shown in this thesis is a first step in quantize degenerate systems by

means of simple examples. Even in the context of finite number degrees of freedom

there are still many problems for different toy model including relativistic or spin

effects, considering different geometries or boundary conditions. Finally there could

be many examples of practical interest in physics or other disciplines. First of all

we

• In the simple coupled model, we discussed a particular coupling, so a question

that needs more attention is about other couplings and possible tunneling effects in

those cases.

• A particularly interesting question is to understand how the present discussion

extends to field theories. In particular, this would allow deciding whether the Chern-

Simons systems in five spacetime dimensions or more are reducible or not. Although

the notion of orbit in a field theory is only formally defined, some of the essential

features of the distinction between reducible and irreducible systems can be applied.

It might be conjectured that CS systems forD ≥ 5 are generically irreducible (they

cannot be replaced by a non-degenerate action principle), it is far from obvious how

to settle this question.

Another, even more difficult question is how does a CS system behave quantum

mechanically.

56

5.2 Open questions

• In nonabelian theories the existence of the Gribov copies that comes from

the non-uniqueness of the gauge condition forces one to restrict the domain of the

integration in field space. It can be profitable to study the connection between the

existence of the so-called Gribov horizon and the existence of the degeneracy. This

imposes an accurate analysis of the constraints at the Gribov horizon to distinguish

between regular and irregular constraints. Preliminary results on this issue suggest

the possibility of interpreting Gribov horizon as degeneracy surface and viceversa.

We can expect that a deeper understanding of this interconnection would be relevant

in order to face field theory issues given the results in the degenerate context.

• Loss of information about initial data and irreversibility of evolution are the

relevant characteristics of the degeneracy. Then the degeneracy surface can be

interpeted as a sort of dynamical dimensional reduction mechanism: a degenerate

system can start from an initial configuration and evolve irreversibly towards another

with few degrees of freedom. In order to complete this interpretation one should be

able to answer some additional questions: is it possible to distinguish if a system

is living on a degenerate surface, i.e. it comes from higher dimensions and it has

already colapsed? Would it be possible to interpret a gauge freedom as a colapsed

dimension?

57

Appendix A

On the self-adjointness

With regard to properties of operators1, it is worth to clarify and repeat some already

known results (see [27] and [28]), emphasizing once more that defining an operator

means to specify its action and at the same time its domain.

First we remember that an operator A on a Hilbert space A : D(A) ⊂ E → F is

called symmetric (or Hermitian) if

< Aψ, φ >=< ψ,Aφ > ∀ψ, φ ∈ D(A) (A.1)

The adjoint of A is denoted by A† and defined as

< A†ψ, φ >=< ψ,Aφ > ∀ψ ∈ D(A†),∀φ ∈ D(A) (A.2)

It means that in principle the domain D(A†) of the adjoint operator A† is not

necessarily identical to the domain D(A) of the operator A, it can be larger: if the

operator is simmetric then D(A) ⊆ D(A†).

If the domains of A† and A coincide, then the operator is self-adjoint

A = A† and D(A) = D(A†) (A.3)

Clearly for operators defined in all the space, symmetry and self-adjointntess coin-

cide. For bounded operators symmetry and self-adjointness coincide, too. For un-

bounded operator, as the differential operator are, the simmetry is a less-demanding

condition to respect the adjointness, but it ensures that the eigenvalues of H are real

and that eigenvectors corresponding to different eigenvalues are orthogonal. Beside

note that the definition of the adjoint operator, like the self-adjointness, depend

1Remember that byoperators one usually refers to linear operators

58

on how the scalar product is defined: changing the scalar product a self-adjoint

operator will no longer be self-adjoint.

Proof: If v is an eigenvector of H with α as eigenvalue, then Hv = αv. If H is

symmetric, then

< Hv, v >=< v, Hv >→ α < v, v >= α∗ < v, v >→ α = α∗ , (A.4)

the eigenvalue α has to be real.

If w is an other eigenvector of H with a different eigenvalue β, then

< Hv, w >=< v, Hw >→ α < v,w >= β∗ < v,w >→< v,w >= 0 . (A.5)

So eigenvectors corresponding to different eigenvalues are orthogonal.

Further more, the adjointness of H ensures the completeness of its eigenvectors,

so the eigenvectors of a self-adjoint operator form a complete othonormal system,

a basis with which one can construct all the element in the space. This in turn

ensures the unitarity of the time evolution operator, i.e. the time conservation of

probability in the evolution of the system described by H. The evolution operator

is connected to the hamiltonian by

U(t) = e−i~ Ht . (A.6)

If H is self-adjoint, U is unitary (isometric with inverse), i.e. U † = U−1. If H is

not self-adjoint, but only simmetric it is because the domains of H and H† do not

concide, D(H) ⊆ D(H†), so the eigenvectors of H can not describe (or span) all

the space. Then as a consequence U can be isometric but its inverse is not defined:

U preserves the probability amplitude, but is not guaranteed the existence of the

inverse U−1, because the domain of the inverse could fall outside the domain of

U . i.e. of H. This means a temporal evolution is possible, but its reverse may

not be well defined evolution. If the Hamiltonian operator is not self-adjoint the

time-reversibility of the system that H describes is not guaranteed to be physical

any more.

59

Appendix B

Integrating the continuity

equation

Here we outline some details about the integration of the continuity equation.

B.1 The contribution of the discontinous proba-

blity current

Integrating the continuity equation for the simple degenerate system, one has to

evaluate the contribution of the current density J(x, t) as written in (2.58)

J = ν sgn(x) |Ψ(x, t)|2 . (B.1)

Let’s stress that the current density J(x, t) is a continous function except at x = 0

where it presents a step

∆ = J(0+, t)− J(0−, t) = ν|ψ(0+, t)|2 + ν|ψ(0−, t)|2

so its derivative is

∂

∂xJ(x, t) =

∂

∂x


)x6=0

+ ∆ · δ(x) .

60

B.1 The contribution of the discontinous probablity current

Then we can integrate this last expression over the entire domain∫ a+

a−

∂

∂xJ(x, t) dx =

∫ a+

a−

∂

∂x


)x 6=0

+

∫ a+

a−∆ · δ(x) =

=

∫ 0−

a−

∂

∂x


)+

∫ a+

0+

∂

∂x


)+ ∆ =

= ν sgn(x) |Ψ(x, t)|2∣∣∣0−a−

+ ν sgn(x) |Ψ(x, t)|2∣∣∣a+

0++ ∆ =

= ν− |Ψ(0−)|2 + |Ψ(a−)|2 + |Ψ(a+)|2 − |Ψ(0+)|2 + (B.2)

+|Ψ(0−)|2 + |Ψ(0+)|2

= (B.3)

= ν|Ψ(a+)|2 + |Ψ(a−)|2

. (B.4)

61

Appendix C

Calculating the propagator

We show here the details in calculating the propagator in Chapter 4, through 4.3,

using our lagrangian L = xyx− νy


∫Γ

DyDxei~∫

(xyx−νy)dt . (C.1)

We discretize the Lagrangian discretizing the time interval (ti, tf ) in N intervals

of magnitude ε, then we will take the limit N → ∞ and ε → 0. Accordingly the

x (position) and y (momentum) coordinates will be discretized as x → xn−xn−1

2,

x→ xn−xn−1

ε, y → yn with x0 as initial point and xN as final point.

The Hamiltonian action becomes

S = limε→0N→∞

i

~ε

N∑n=1

(xn − xn−1

2· yn ·

xn − xn−1

ε− νyn

)

= limi

~

N∑n=1

(x2n − x2

n−1

2· yn − νεyn

), (C.2)

and the propagator reads


= limε→0N→∞

∫dx1 · · · dxN−1

dy1

2π~· · · dyN

2π~ei~∑Nn=1

(x2n−x

2n−1

2yn−νεyn

)=

= limε→0

∫dy1

2π~· · · dyN

2π~e−

i~∑Nn=1 νεyn

∫dx1 · · · dxN−1e

i2~∑Nn=1(x2

n−x2n−1)yn .(C.3)

The integration in x results in the following infinite product

62

∫dx1 · · · dxN−1exp

[i

2~

N∑n=1

(x2n − x2

n−1

)yn

]=

=

∫dx1 · · · dxN−1 ·

·exp

i

2~[− y1x

20 + (y1 − y2)x2

1 + (y2 − y3)x22 + · · ·

· · ·+ (yN−1 − yN)x2N−1 + yNx

2N

]=

= ei

2~(−y1x20+yNx

2N)∫

dx1 · · · dxN−1e( i

2~∑N−1j=1 (yj−yj+1)x2

j ] =

= ei

2~(−y1x20+yNx

2N) ·

j=N−1∏j=1

∫dxje

i2~ (yj−yj+1)x2

j . (C.4)

The generic element of the infinite product is1

∫ ∞−∞

dxjeiyj−yj+1

2~ x2j = (C.5)

=

∫ ∞−∞

cos

(yj − yj+1

2~x2

1

)dxj + i

∫ ∞−∞

sin

(yj − yj+1

2~x2j

)dxj = (C.6)

=

√π

2

2~|yj − yj+1|

+ i sgn(yj − yj+1)

√π

2

2~|yj − yj+1|

= (C.7)

=

√~

|yj − yj+1|+ i sgn(yj − yj+1)

√~

|yj − yj+1|= (C.8)

=

√~

|yj − yj+1|(1 + i sgn(yj − yj+1)) =

=

√2π~


4 . (C.9)

After completing the x integration the action functional will read

1Up to a phase exp(2nπi)

63



= limε→0N→∞

∫dy1

2π~· · · dyN

2π~e−

i~∑Nn=1 νεyne

i2~(−y1x2

0+yNx2N) ·

·j=N−1∏j=1

∫dxje

i2~ (yj−yj+1)x2

j =

= lim

∫dy1

2π~· · · dyN

2π~ei(−

x202~ y1−

νε~ y1−

νε~ y2−···−


x2N2~ yN ) ·

·j=N−1∏j=1

√2π~


4 =

= lim

∫dy1

2π~· · · dyN

2π~ei(−

x202~ y1−

νε~ y1−

νε~ y2−···−


x2N2~ yN ) ·

·

√2π~|y1 − y2|

ei[sgn(y1−y2)]π4 ·

√2π~|y2 − y3|

ei[sgn(y2−y3)]π4 · · ·

·

√2π~

|yN−2 − yN−1|ei[sgn(yN−2−yN−1)]π

4 ·

√2π~

|yN−1 − yN |ei[sgn(yN−1−yN )]π

4 .(C.10)

For th integration over y1 we use the substitution

k1 =x2o

2~+νε

~, z1 =

(x2o

2~+νε

~

)(y1 − y2) = k1(y1 − y2) (C.11)

64

then

∫dy1

2π~

√2π~|y1 − y2|

ei[−(x2o2~ + νε

~ )y1+sgn(y1−y2)π4

] = (C.12)

=1√2π~

∫dy1

√1

|y1 − y2|ei[−(

x2o2~ + νε

~ )y1+sgn(y1−y2)π4

] = (C.13)

=1√2π~

∫dz1

k1

√k1

|z1|e−ik1y2ei[−z1+sgn(z1)π

4] = (C.14)

=1√2π~

e−ik1y2√k1

∫dz1

1√|z1|

ei[−z1+sgn(z1)π4

] = (C.15)

=1√2π~

e−ik1y2√k1

∫ 0

−∞dz1

1√|z1|

ei[−z1−π4

] +

∫ ∞0

dz11√|z1|

ei[−z1+π4

]

= (C.16)

=1√2π~

e−ik1y2√k1

(2√π) = (C.17)

=2√π√

2π~1√k1

e−ik1y2 . (C.18)

Similarly, integration over y2 yelds

2√π√

2π~√k1

∫dy2

2π~e−ik1y2e−i

νε~ y2

√2π~|y2 − y3|

ei[sgn(y2−y3)]π4 =

=2√π√

2π~1√k1

∫dy2

2π~

√2π~|y2 − y3|

ei[−(k1+ νε~ )y2+sgn(y2−y3)π

4] =

= · · · = 2√π√

2π~1√k1

2√π√

2π~1√k2

e−ik2y3 , (C.19)

where k2 = k1 + νε~

And so on, until the integration on yN−1 with the coefficient kN−1 = kN−2 + νε~ =

x2o

2~ + (N − 1)νε~ . Then the propagator assumes the expression

65



= limε→0N→∞

∫dyN2π~

j=N−1∏j=1

(2√π√

2π~1√kj

)e(−i kN−1·yN)ei(−

νε~ yN+

x2N2~ yN )) =

= limε→0N→∞

j=N−1∏j=1

(2√π√

2π~1√kj

)∫dyN2π~

ei(−x2o2~−(N−1) νε~ −

νε~ +

x2N2~ )yN =

= limε→0N→∞

j=N−1∏j=1

(2√π√

2π~1√kj

)∫dyN2π~

ei(−x2o2~−N

νε~ +

x2N2~ )yN =

= limε→0N→∞

j=N−1∏j=1

(2√π√

2π~1√kj

)δ

(1

2(−x2

o − 2νNε+ x2N)

)=

= limε→0N→∞

j=N−1∏j=1

(2√π√

2π~1√kj

)2 · δ

(x2N − x2

o − 2νNε)). (C.20)

66

Acknowledgements

Vorrei poter ringraziare tutte le persone incontrate in questo lungo tempo, che non

e stato solo il tempo del dottorato, e stato anche un pezzetto non piccolo della mia

vita. Credo di poter ricordare nitidamente ogni frase, commento o sguardo o mano

di chi da vicino o da lontano mi ha accompagnata, vorrei poter pronunciare i nomi

di tutte e tutte, ma la carta non mi da’ lo spazio, gli occhi si chiudono e ringrazio

in silenzio col cuore e con la mente...

I would like to thank all the people encountered in this long time, that is was

not just the time of the my PhD, but it also was a not so small piece of my life. I

think I can clearly remember every sentence, comment or look or hand of who from

near or far accompanied me, I would be able to pronounce the names of all, but the

paper does not give me the space, so with closed eyes I thank in silence with heart

and mind ...

67

This work was supported by Fondecyt grants # 1110102, 1100328, 1100755,

1100328, and by Conicyt grant Southern Theoretical Physics Laboratory, ACT-91.

The Centro de Estudios Cientıficos (CECS) is funded by the Chilean Government

through the Centers of Excellence Base Financing Program of Conicyt.

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