Dedicated to

Alexei Sisakian
co-author $\&$ friend

On the connection between quantum and classical descriptions
J.Manjavidze

Joint Institute for Nuclear Research

(VBLHEP & LNP, Dubna, Russia)

$\&$

Andronikashvili Institute of Physics

(LThP, Tbilisi State University, Georgia),
Tel: (09621) 6 35 17, Fax: (09621) 6 66 66, E-mail<EMAIL_ADDRESS>
###### Abstract

The review paper presents generalization of d’Alembert’s variational
principle: the dynamics of a quantum system for an external observer is
defined by the exact equilibrium of all acting in the system forces, including
the random quantum force $\hbar j$, $\forall\hbar$. Spatial attention is
dedicated to the systems with (hidden) symmetries. It is shown how the
symmetry reduces the number of quantum degrees of freedom down to the
independent ones. The sin-Gordon model is considered as an example of such
field theory with symmetry. It is shown why the particles $S$-matrix is
trivial in that model.

###### Contents

  1. 1 Introduction
  2. 2 Simplest examples
    1. 2.1 Introduction
    2. 2.2 The generalized stationary-phase method
    3. 2.3 Complex trajectories
    4. 2.4 Conclusions
  3. 3 Path integrals on Dirac measure
    1. 3.1 Introduction
    2. 3.2 Canonical transformation
    3. 3.3 Selection rule
    4. 3.4 Coordinate transformation
    5. 3.5 Conclusions
  4. 4 Reduction of quantum degrees of freedom
    1. 4.1 Introduction
    2. 4.2 Unitary definition of the path-integral measure
    3. 4.3 Perturbation theory on the cotangent manifold
    4. 4.4 Cancelation of angular perturbations
    5. 4.5 Conclusions
  5. 5 Example: H-atom
    1. 5.1 Introduction
    2. 5.2 Mapping
    3. 5.3 Reduction
    4. 5.4 Perturbations
    5. 5.5 Conclusions
  6. 6 Example: sin-Gordon model
    1. 6.1 Introduction
    2. 6.2 Reduction procedure
    3. 6.3 Quantum corrections
  7. 7 Summary
  8. 8 Conclusion

Preface

Present paper is the review of the works which was done after my first paper
[1]. I returned from time to time to the idea [1] that it seems interesting to
embed the total probability conservation condition into the quantum field
theory formalism and discuss it with Alexei Sissakian during our team-work. It
seems that this suggestion is unnecessary noting that the $S$-matrix is the
unitary operator and it is not evident why this attempt can give something
new. But it turns out that exist the correspondence among quantum theory and
classics which is independent from the value of quantum corrections. Besides
this new quantum field theory is free from divergences and the value of
quantum corrections ingenuously depend on the topology of classical field. All
that is new from the point of view of ordinary theory and at last Alexei
Sissakian propose to write on paper all result in details. Present
introductory paper devoted to simplest examples and more interesting field
theory models will be published later.

##  1 Introduction

The basis of the method of calculations is the following [1]. The $S$-matrix
unitarity condition, $S^{\dagger}S=1$, in terms of amplitudes, $S=1+iA$, looks
as follows:

$iA^{\dagger}A=(A-A^{\dagger}).$ (1\. 1)

The nonlinearity of this equality points on existence of the cancelations
mechanism (of the real part of amplitude) which reduces quadratic form down
the linear one. Our purpose is to show how this reduction removes the
”unwanted” contributions111This means that the theory must be formulated
directly in terms of probability. But notice that it is the frequently used
method of particle physics. For example, one must integrate over unobserved
final state in the inclusive approach to the multiple production phenomena.
Another example: describing the very high multiplicity (VHM) processes the
number of produced particles $n$ must be considered as the dynamical
parameter. In the frame of $S$-matrix thermodynamics, where the ”rough”
description of final state is used, one must also integrate over final
particles momenta. In all cases one must consider quantities $\sim|A|^{2}$
directly, where $A$ is the corresponding amplitude..

One may consider the simplest vacuum-into-vacuum transition probability,
$|Z|^{2}$, as the main quantity, where $Z$ is the functional integral over
fields,

$Z=\int D\varphi~{}e^{iS(\varphi)},~{}D\varphi=\prod_{x}d\varphi(x).$ (1\. 2)

One may include into the action, $S$, also the linear over field $\varphi$
term,

$\int dxJ(x)\varphi(x)$ (1\. 3)

to describe production of particles. We will assume on the early stages that
$J=0$. Then the vacuum-into-vacuum transition probability

$|Z|^{2}=\int
D\varphi_{+}D\varphi^{*}_{-}~{}e^{iS(\varphi_{+})-iS^{*}(\varphi_{-})},$ (1\.
4)

where $\varphi_{+}$ and $\varphi_{-}$ are completely independent fields.

It can be shown that Eq. (1\. 1) means that a reduced form must also exist
[1]:

$|Z|^{2}=\lim_{j=e=0}e^{i\hat{\mathbb{K}}(j,e)}\int DMe^{iU(\varphi,e)},$ (1\.
5)

where $\hat{\mathbb{K}}=\hat{\mathbb{K}}(j,e)$ is a definite differential
operator over $j(x)$ and $e(x)$, see the examples (2\. 42), (6\. 7). The
expansion of $\exp\\{i\hat{\mathbb{K}}\\}$ generates perturbation series. The
functional $U(\varphi,e)$ introduces interaction among quantum degrees of
freedom and the integral measure is $\delta$-functional:

$DM=\prod_{x}d\varphi(x)\delta\left(\frac{\delta
S(\varphi)}{\delta\varphi(x)}+\hbar j(x)\right).$ (1\. 6)

Sometimes the $\delta$-like measure [2] is called in mathematical literature
as the ”Dirac measure”. It follows from (1\. 6) that

— the quantum system for an externa observer looks like classical which is
excited by the external random force $\hbar j,~{}\forall\hbar$.
The established generalized correspondence principle222This formulation of the
principle was offered by A.Sisakian. is the main consequence of Eq. (1\. 1).
Therefore the complete set of acceptable field states for external
observer333Since the ”probability” is considered is known having (1\. 6).

It is important that the restricted problem is considered. We will calculate
the imaginary part of amplitude believing that it will be sufficient for us.
In this case the unmeasurable phase of amplitude stay undefined444 Therewith
why must the calculations of unnecessary, i.e. unmeasurable, phase be
performed? Just in this sense the unitarity condition (1\. 1) is a necessary
one. It says that the real part is the ”unwanted” part of the amplitude.. A
main mathematical problem in the searching representation (1\. 5) is to find
the way how to find the imaginary part from the modulo square of amplitude. To
be more precise, we will find the imaginary part as a result of cancelation of
”unwanted” contribution in the modulo square of amplitude.

The $\delta$-function (1\. 6) solves the problem of definition of
contributions into the path integral but can not solve the problem completely
since the action of operator $\hat{\mathbb{K}}$ remains unknown. It must be
noted that $\exp\\{i\hat{\mathbb{K}}\\}$ generates the asymptotic series
ordinary in quantum theories [3] and it seems that $\delta$-like measure gives
nothing new555Looking at the approach from the point of view of the stationary
phase methods. In other words, one can think that the present approach gives
nothing new to the Bohr’s correspondence principle.. But this is not entirely
so. I would like draw attention to the appearance of source of quantum
excitations $\hbar j$ in the r.h.s. of classical Lagrange equation, i.e. the
changes of l.h.s. in equation of motion leads to the change of $j$. It is
crucially important that (1\. 6) is rightful independently from the value of
$\hbar$.

The theory defined on the Dirac measure (1\. 6) for this reason has quite
unexpected properties, e.g. allows to perform transformation of the path
integral variables. So, it will be shown that in theories with symmetry the
reduced form of representation (1\. 5) exists:

$|Z|^{2}=\lim_{j=e=0}e^{i\hat{\mathbb{K}}(j,e)}\int
DM(j)e^{iU(\varphi_{c},e)},$ (1\. 7)

where $\hat{\mathbb{K}}$ is again the perturbations generating operator and
$U$ introduces interactions. Note that $\hat{\mathbb{K}}$ and $U$ in (1\. 7)
depends on the sets $\\{j_{\xi_{k}},j_{\eta_{k}}\\}$,
$\\{e_{\xi_{k}},e_{\xi_{k}}\\}$ of new variables. One must take this auxiliary
variables equal to zero at the very end of calculations. At the same time the
transformed measure $DM$ is again $\delta$-like:

$DM=\prod_{k}\prod_{t}d\xi_{k}(t)d\eta_{k}(t)\times$
$\times\delta\left(\dot{\xi}_{k}(t)-\frac{\delta
h}{\delta\eta_{k}(t)}-j_{\xi_{k}}(t)\right)\delta\left(\dot{\eta}_{k}(t)+\frac{\delta
h}{\delta\xi_{k}(t)}+j_{\eta_{k}}(t)\right),$ (1\. 8)

where $t$ is the time variable and $h=h(\eta)$ is the transformed Hamiltonian:

$h(\eta)=H(\varphi_{c}),$ (1\. 9)

where $\varphi_{c}=\varphi_{c}({\mathbf{x}};\xi,\eta)$ is given solution of
Lagrange equation at $j=0$.

The formulae (1\. 8) is the main result. Therefore, as it follows from it the
problem of the quantum field theory with symmetry is reduced down to quantum
mechanical one, with potential defined by $\varphi_{c}$.

(A) The Dirac measure (1\. 6) prescribes that $|Z|^{2}$ is defined by the
$sum$ of $strict$ solutions of equation of motion:

$\frac{\delta S(\varphi)}{\delta\varphi(x)}=\hbar j(x),$ (1\. 10)

in vicinity of $j=0$, i.e. by definition Eq. (1\. 10) must be solved expanding
the solution over $j$ 666It should be noted that it may be that the limit
$j=0$ is absent. For example it may happen if the system is unstable against
symmetry breaking. This important possibility will not be considered in
present paper.. Following the ordinary rule we obviously leave the
contribution which ensures the minimal vacuum energy. On the other hand,
having theory on Dirac measure, which calls for the complete set of
contributions, we have offered another selection rule in our dynamic theory of
$S$-matrix. Namely, we simply propose777This selection rule is used widely in
classical mechanics, see e.g. formulation of Kolmogorov-Arnold-Mozer (KAM)
theorem [4].  that
— the largest terms in the sum over solutions of (1\. 10) are significant from
the physics point of view.
To be more precise, this selection rule means that if $G$ is the symmetry of
action and $TG^{*}$ is the symmetry of the extremum of the action, then in the
situation of a general position only the trajectories with the highest
dimension factor group, $(G/TG^{*})$, are sufficient.

It will be seen that this kind of definition of the ”ground state” extracts
the maximally ”feeling” symmetry contributions since other ones will be
realized on a zero measure, or, more precisely, only maximal symmetry breaking
field configurations, $\varphi_{c}$, are most probable. We will call such
solution of the problem the field theory with symmetry. It is the main formal
distinction of present approach.

It is important here that the zero width of $\delta$-function excludes the
interference among contributions from various trajectories. Therefore the
formalism naturally takes into account the orthogonality of Hilbert spaces
built on various trajectories. This is achieved through the special boundary
conditions in the frame of which the total action of the product

$Z\cdot Z^{*}=<in|out>\cdot<out|in>$

always describes a closed path, i.e. the necessary for d’Alembert variational
principle time reversible motion. It points to the necessity to be careful
with boundary conditions in a considered formalism888 The necessity to count
all possible boundary conditions of a given problem was mention to author by
L.Lipatov..

(B) The Dowker’s theorem [5] insists that the semiclassical approximation to
be exact for path integrals on the simple Lie group manifolds. For this reason
one can expect that the quantum-mechanical problems, as well as the field-
theoretical ones, may be at least transparent to the symmetry manifolds.

However we know how to construct correctly the path integral formalism only in
the restricted case of canonical variables [6]. At first sight the path
integrals in terms of generalized coordinates can be defined through the
corresponding transformation. But there is an opinion that it is impossible to
perform the transformation of path-integral variables: the naive
transformation of coordinates give wrong results because of their stochastic
nature in quantum theories999One can find corresponding examples in [6, 7].
The mostly popular method of transformation of the path-integral variables is
a ”time-sliced” method [8], which induces corrections to the interaction
Lagrangian proportional at least to $\hbar^{2}$ [9], i.e. the problem of
transformation is of a quantum nature. For this reason the usage of the ”time-
sliced” method in general case is cumbersome, see also [10].. That is why such
general principle as the conservation of total probability (1\. 1) should play
important role. Indeed, it is evident that $\delta$-like Dirac measure (1\. 6)
allows to perform arbitrary transformation [1] just as in the classical
mechanics.

Therefore, the theory on Dirac measure straight away leads to the new for
quantum field theory selection rule and latter one gives the theory with
symmetry. All this is attained by transition to the appropriate variables,
$(\xi,\eta)\in W$ in our notations. The last circumstance means that we go
away from ordinary spectral analysis of quantum fluctuations to the
description of the classical trajectories topology conserving deformations,
since $\varphi_{c}=\varphi_{c}({\mathbf{x}};\xi,\eta)$ is given, of symmetry
manifold, $W$ 101010It will be seen from our selection rule that the measure
on which particles mechanics is realized is equal to zero in the field
theories with symmetry.. It must be underlined that our method of
transformations is rightful for arbitrary case, i.e. not only for simple Lie
group manifolds, where the semiclassical approximation is exact.

Next, the dimensions of initial phase space of field and of the transformed
space of independent degrees of freedom, i.e. of the symmetry manifold, will
not coincide. That means that the mapping to the independent degrees of
freedom, $(\xi,\eta)$, will be singular. For this reason the transformation

$\varphi_{c}:\varphi\rightarrow(\xi,\eta)$

will be irreversible and the notion of particle should be considered as the
wrong idea of quantum field theory with symmetry 111111Considering gluon
production in the frame of Yang-Mills field theory with symmetry the
conclusion that gluons can not be created should be confirmed by direct
calculations, taking into account also the quark fields. That was mentioned to
the author by P.Culish and will be shown in later publications. It is
noticeable that the mapping in quantum mechanics is not singular and for this
reason both representations before and after transformation have the equal
status..

(C) It will be shown that the result of action of the operator
$\exp\\{i\hat{\mathbb{K}}\\}$ for transformed theories may be expressed as the
sum of contributions on all boundaries $\partial W$:

$|Z|^{2}=|Z|^{2}_{sc}+\sum_{k}\int
d\xi_{k}(0)\frac{\partial}{\partial\xi_{k}(0)}C_{\xi}+\sum_{k}\int
d\eta_{k}(0)\frac{\partial}{\partial\eta_{k}(0)}C_{\eta}$ (1\. 11)

where the first term presents a semiclassical contribution and $C_{\xi}$,
$C_{\eta}$ contains quantum corrections. This result shows that the quantum
corrections greatly depend on the topology of classical trajectory.

This important observation solves a number of problems. For instance, it is
known that the Coulomb trajectory is closed because of Bargman-Fock symmetry,
independent from the initial conditions. For this reason the corrections on
$\partial W$ of Coulomb problem are canceled and the H-atom problem is pure
semiclassical. We will find the same for sine-Gordon model [11] as the
consequence of mapping on the Arnold’s hypertorus [12].

It is extremely important to keep in mind that the symmetry constraints can
not be taken into account perturbatively over the interaction constant, $g$.
Indeed, we will see below that the expansion in $polinomial$ theories with
symmetry is performed in terms of the inverse interaction constant, $1/g$. It
points to the absence of the weak-coupling limit in such theories.

In the end our present aim is

— to find representation (1\. 5);

— to investigate the main properties of theory defined on the Dirac measure
(1\. 6);

— to investigate the structure of perturbation theory generated by operator
$\hat{\mathbb{K}}$ on the measure (1\. 8);

— to find particles production probabilities for theories with symmetry.

I understand that the perturbations scheme in terms of new variables,
especially in theories with symmetry, is outside the habitual one121212See
[13, 14, 15] and for this reason the approach will be describe in more
details, giving step-by-step the properties of a new quantization scheme by
appropriate examples. I think that such non-formal scheme of the description
is much more transparent, although the text may contain reiterations with the
used method of description far from completeness.

##  2 Simplest examples

###  2.1 Introduction

It it has mentioned above a technical aspect of our idea is the suggestion to
calculate the probability without the intermediate step of calculations of the
amplitudes. In present Section we restrict ourselves to the simplest problem -
to the motion of a particle in a potential $V(x)$.

Let the amplitude $A(x_{2},T;x_{1},0)$ describes the motion of the particle
from the point $x_{1}$ to the point $x_{2}$ during the time $T$. Using the
spectral representation:

$A(x_{2},T;x_{1},0)=\sum_{n}\psi_{n}(x_{2})\psi_{n}^{*}(x_{1})e^{iE_{n}T},$
(2\. 1)

we have for probability:

$W(x_{2},T;x_{1},0)=\sum_{n_{1},n_{2}}\psi_{n_{1}}(x_{2})\psi_{n_{1}}^{*}(x_{1})\psi_{n_{2}}^{*}(x_{2})\psi_{n_{2}}(x_{1})e^{i(E_{n_{1}}-E_{n_{2}})T}.$
(2\. 2)

Taking into account the ortho-normalizability condition:

$\int dx\psi_{n}(x)\psi_{m}^{*}(x)=\delta_{n,m},$ (2\. 3)

the total probability:

$\int dx_{2}dx_{1}W(x_{2},T;x_{1},0)=\sum_{n}\delta_{n,n}=\Omega$ (2\. 4)

is the time independent quantity which coincides with the number of existing
physics states. Therefore, the amplitude (2\. 1) is time dependent, but the
total probability (2\. 4) is not. This means that the time is the unwanted
parameter from the point of view of experiment described by the probability
(2\. 4). Notice also the role of boundary condition (2\. 3).

The quantity (2\. 4) is of no interest to experiment. Much more interesting
the probability $\rho(E)$, where $E$ is the energy experimentally measured.
The Fourier transform of $A(x_{2},T;x_{1},0)$ with respect to $T$

$a(x_{2},x_{1};E)=\sum_{n}\frac{\psi_{n}(x_{2})\psi_{n}^{*}(x_{1})}{E-(E_{n}+i\varepsilon)}$
(2\. 5)

leads to the probability

$\omega(x_{2},x_{1};E)=|a(x_{2},x_{1};E)|^{2}=\sum_{n_{1},n_{2}}\frac{\psi_{n_{1}}(x_{2})\psi_{n_{1}}^{*}(x_{1})}{E-(E_{n_{1}}+i\varepsilon)}\frac{\psi_{n_{2}}^{*}(x_{2})\psi_{n_{2}}(x_{1})}{E-(E_{n_{2}}-i\varepsilon)}$
(2\. 6)

and the total probability:

$\rho(E)=\int
dx_{1}dx_{2}\omega(x_{2},x_{1};E)=\sum_{n}\left|\frac{1}{E-E_{n}-i\varepsilon}\right|^{2}=$
$=\frac{1}{\varepsilon}\sum_{n}{\rm
Im}\frac{1}{E-E_{n}-i\varepsilon}=\frac{\pi}{\varepsilon}\sum_{n}\delta(E-E_{n}).$
(2\. 7)

The total probability $\rho(E)$ again coincides with number of existing states
but for all that it is seen that the unphysical, i.e. needless, states from
the point of view of measurement with $E\neq E_{n}$ was canseled131313Such
contributions enter into the real part of $a(x_{2},x_{1};E)$..

Let us use now the proper-time representation:

$a(x_{1},x_{2};E)=\sum_{n}\Psi_{n}(x_{1})\Psi^{*}_{n}(x_{2})i\int^{\infty}_{0}dTe^{i(E-E_{n}+i\varepsilon)T}$
(2\. 8)

to see the integral form of cancelation of unwanted contributions and insert
it into definition of total probability ($\varepsilon\rightarrow+0$):

$\rho(E)=\int
dx_{1}dx_{2}|a(x_{1},x_{2};E)|^{2}=\sum_{n}\int^{\infty}_{0}dT_{+}dT_{-}e^{-(T_{+}+T_{-})\varepsilon}e^{i(E-E_{n})(T_{+}-T_{-})}.$
(2\. 9)

We will introduce new time variables instead of $T_{\pm}$:

$T_{\pm}=T\pm\tau,$ (2\. 10)

where, as it follows from Jacobian of transformation, $|\tau|\leq T,~{}0\leq
T\leq\infty$. But we can put $|\tau|\leq\infty$ since $T\sim
1/\varepsilon\rightarrow\infty$ is essential in integral over $T$. As a
result,

$\rho(E)=4\pi\sum_{n}\int^{\infty}_{0}dTe^{-2\varepsilon
T}\int^{+\infty}_{-\infty}\frac{d\tau}{\pi}e^{2i(E-E_{n})\tau}=\frac{\pi}{\varepsilon}\sum_{n}\delta(E-E_{n}).$
(2\. 11)

In the last integral all contributions with $E\neq E_{n}$ has been canceled
and only the acceptable from physics point of view contributions with
$E=E_{n}$ has survived. This peculiarity of considered interference phenomena
which is the consequence of unitarity condition, i.e. its ability to extract
only physics states, would have the significant applications.

Note also that the product of amplitudes $a\cdot a^{*}$ was ”linearized” after
introduction of ”virtual” time $\tau=(T_{+}-T_{-})/2$, i.e. after
transformation (2\. 10) we start calculation of the imaginary part. The
meaning of such variables will be discussed also in Sec.2.2.

###  2.2 The generalized stationary-phase method

1. 0-dimensional model

Let us practise considering the ”$0$-dimensional” integral:

$A=\int^{+\infty}_{-\infty}\frac{dx}{(2\pi)^{1/2}}e^{i(\frac{1}{2}ax^{2}+\frac{1}{3}bx^{3})},$
(2\. 12)

with ${\rm Im}a\rightarrow+0$ and $b>0$. This example is useful since it
allows to illustrate practically all technical tricks of the approach.

We want to compute the ”probability”

$R=|A|^{2}=\int^{+\infty}_{-\infty}\frac{dx_{+}dx_{-}}{2\pi}e^{i(\frac{1}{2}ax_{+}^{2}+\frac{1}{3}bx_{+}^{3})-i(\frac{1}{2}a^{*}x_{-}^{2}+\frac{1}{3}bx_{-}^{3})}.$
(2\. 13)

New variables:

$x_{\pm}=x\pm e$ (2\. 14)

will be introduced to find out the cancelation phenomenon. In result:

$R=\int^{+\infty}_{-\infty}\frac{dxde}{\pi}e^{-2(x^{2}+e^{2}){\rm
Im}a}e^{2i({\rm Re}a\;x+2bx^{2})e}e^{2i\frac{b}{3}e^{3}},$ (2\. 15)

where the prescription that ${\rm Im}a\rightarrow+0$ has been used. Note that
integrations are performed along the real axis.

We will compute the integral over $e$ perturbatively. For this purpose the
transformation:

$F(e)=\lim_{j=e^{\prime}=0}e^{\frac{1}{2i}\hat{j}\hat{e}^{\prime}}e^{2ije}F(e^{\prime}),$
(2\. 16)

which is valid for any differentiable function, will be used. In (2\. 16) two
auxiliary variables $j$ and $e^{\prime}$ has been introduced and the ”hat”
symbol means the differential over corresponding quantity:

$\hat{j}=\frac{\partial}{\partial
j},\;\;\;\hat{e^{\prime}}=\frac{\partial}{\partial e^{\prime}}.$ (2\. 17)

The auxiliary variables must be taken equal to zero at the very end of
calculations.

Choosing

$\ln F(e)=-2e^{2}{\rm Im}a+2i\frac{b}{3}e^{3}$ (2\. 18)

we will find:

$R=\lim_{j=e=0}e^{\frac{1}{2i}\hat{j}\hat{e}}\int^{+\infty}_{-\infty}dxe^{-2(x^{2}+e^{2}){\rm
Im}a}e^{2i\frac{b}{3}e^{3}}\delta({\rm Re}a~{}x+bx^{2}+j).$ (2\. 19)

Therefore, the destructive interference among two exponents in the product
$a\cdot a^{*}$ unambiguously determines both integrals, over $x$ and over $e$.
The integral over difference $e=(x_{+}-x_{-})/2$ gives $\delta$-function and
then this $\delta$-function defines the contributions in the last integral
over $x=(x_{+}+x_{-})/2$. Following the definition of $\delta$-function only a
strict solutions of equation

${\rm Re}a\;x+bx^{2}+j=0$ (2\. 20)

gives the contribution into $R$.

But one can note that this is not the complete solution of the problem: the
expansion of operator exponent $\exp\\{\frac{1}{2i}\hat{j}\hat{e}\\}$
generates the asymptotic series. Note also that it is impossible to remove the
source, $j$, dependence (only harmonic case, $b=0$, is free from $j$).

The equation (2\. 20) at $j=0$ has the solutions, at $x_{1}=0$ and at
$x_{2}=-a/b$. Performing trivial transformation $e\rightarrow ie$,
$\hat{e}\rightarrow-i\hat{e}$ of auxiliary variable we find at the limit ${\rm
Im}a=0$ that the contribution from $x_{1}$ extremum (minimum) has the
expression141414The contribution of $x_{2}$ leads to divergent series.:

$R=\frac{1}{a}e^{-\frac{1}{2}\hat{j}\hat{e}}(1-4bj/a^{2})^{-1/2}e^{2\frac{b}{3}e^{3}}$
(2\. 21)

and the expansion of an operator exponent gives the asymptotic series:

$R=\frac{1}{a}\sum^{\infty}_{n=0}(-1)^{n}\frac{(6n-1)!!}{n!}\left(\frac{2b^{4}}{3a^{6}}\right)^{n},\;\;\;\;(-1)!!=0!!=1.$
(2\. 22)

This series is convergent in Borel’s sense. Therefore the described
destructive interference has not an action upon the value of perturbation
series convergence radii.

Let us calculate now $R$ using stationary phase method. The contribution from
the minimum $x_{1}$ gives $({\rm Im}a=0)$:

$A=e^{-i\hat{j}\hat{x}}e^{-\frac{i}{2a}j^{2}}e^{i\frac{b}{3}x^{3}}({i}/{a})^{1/2}.$
(2\. 23)

The corresponding “probability” is

$R=\frac{1}{a}e^{-i(\hat{j}_{+}\hat{x}_{+}-\hat{j}_{-}\hat{x}_{-})}e^{-\frac{i}{2a}(j_{+}^{2}-j_{-}^{2})}e^{i\frac{b}{3}(x_{+}^{3}-x_{-}^{3})}.$
(2\. 24)

Introducing new auxiliary variables:

$j_{\pm}=j\pm j_{1},\;\;\;\;x_{\pm}=x\pm e$ (2\. 25)

and, correspondingly,

$\hat{j}_{\pm}=(\hat{j}\pm\hat{j}_{1})/2,\;\;\;\;\hat{x}_{\pm}=(\hat{x}\pm\hat{e})/2$
(2\. 26)

we find from (2\. 24):

$R=\frac{1}{a}e^{-\frac{1}{2}\hat{j}\hat{e}}e^{2\frac{b}{3}e^{3}}e^{\frac{2b}{a^{2}}ej^{2}}$
(2\. 27)

This expression does not coincide with (2\. 21) but it leads to the same
asymptotic series (2\. 22). We may conclude that both considered methods of
calculation of $R$ are equivalent since the Borel’s regularization scheme of
asymptotic series gives a unique result.

The difference between this two methods of calculation is in different
organization of perturbations. So, if $F(e)$, instead of (2\. 18), is chosen
in the form:

$\ln F(e)=-2e^{2}{\rm Im}a+2i\frac{b}{3}e^{3}+2ibx^{2}e,$ (2\. 28)

we may find (2\. 27) straightforwardly.

Therefore, our method has the freedom in choice of (quantum) source
$j$151515This freedom was mentioned firstly by A.Ushveridze.. Indeed, the
transition from perturbation theory with Eq.(2\. 18) to the theory with
Eq.(2\. 28) formally looks like following transformation of the argument of
$\delta$-function:

$\delta(ax+bx^{2}+j)=\lim_{e^{\prime}=j^{\prime}=0}e^{-i\hat{j}^{\prime}\hat{e}^{\prime}}e^{i(bx^{2}+j)e^{\prime}}\delta(ax+j^{\prime}).$
(2\. 29)

Here the transformation (2\. 16) of the Fourier image of $\delta$-function was
used. Inserting Eq.(2\. 29) into (2\. 19) we easily find (2\. 27).

During analytic calculations it will be useful to have a corresponding quantum
sources of the new dynamical variables. Formally this will be done using
transformation (2\. 29). Note that this transformation will not lead to
changing of the Borel’s regularization procedure.

2\. 1-dimensional model

Let us calculate now the probability using the path-integral definition of
amplitudes [1]. Calculating the quantity

$|A|^{2}=\rm<in|out><in|out>^{*}=<in|out><out|in>,$ (2\. 30)

the converging and diverging waves in the product $A\cdot A^{*}$ interfere in
such a way that the continuum of contributions cancel each other. Indeed, the
amplitude

$A(x_{2},T;x_{1},0)=\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{C_{T}}e^{-iS_{T}(x)},~{}Dx=\prod_{t=0}^{T}\frac{dx(t)}{(2\pi)^{1/2}},$
(2\. 31)

where the action $S_{T}$ is given by the expression:

$S_{T}(x)=\int^{T}_{0}dt\left(\frac{1}{2}~{}\dot{x}^{2}-v(x)\right)$ (2\. 32)

and $C_{T}$ is the standard normalization coefficient:

$C_{T}=\int^{x(T)=x_{2}}_{x(0)=x_{1}}Dxe^{\frac{i}{2}\int^{T}_{0}dt~{}\dot{x}^{2}}$
(2\. 33)

Let us calculate the quantity

$R(x_{2},T;x_{1},0)=\int^{x_{\pm}(T)=x_{2}}_{x_{\pm}(0)=x_{1}}\frac{Dx_{+}}{C_{T}}\frac{Dx_{-}}{C_{T}^{*}}e^{-iS_{T}(x_{+})+iS_{T}(x_{-})}$
(2\. 34)

We assume for simplicity that the integration in (2\. 31) is performed over
real trajectories. Later a general case of complex trajectories will be
considered.

The convergence of functional integral at that is not important. One may
restrict the range of integration for better confidence, or introduce into the
Lagrangian $i\varepsilon$ term, and later remove the restriction in the
expression (2\. 40). It is interesting that the interference phenomena
naturally regularize divergent integrals of (2\. 31) type, accumulating
divergence into $\delta$-function.

In order to take into account explicitly the interference between
contributions of the trajectories $x_{+}(t)$ and $x_{-}(t)$ we shall go over
from the integration over two independent trajectories $x_{+}$ and $x_{-}$ to
the pair $(x,e)$:

$x_{\pm}(t)=x(t)\pm e(t).$ (2\. 35)

It must be stressed that the transformation (2\. 35) is linear and for this
reason may be done in the path integral. Substituting (2\. 35) into (2\. 34)
the argument of the exponent takes the form

$S_{T}(x+e)-S_{T}(x-e)=2\int_{0}^{T}dte(\ddot{x}+v^{\prime}(x))-U_{T}(x,e),$
(2\. 36)

where $U_{T}(x,e)$ is the remainder of the expansion in powers of $e(t)$
($U_{T}=O(e^{3})$). Note that in (2\. 36) we have discarded the ”surface” term

$\int_{0}^{T}dt\partial_{t}(e\dot{x})=e(T)\dot{x}(T)-e(0)\dot{x}(0)=0,$ (2\.
37)

since the boundary points of the trajectories $x_{+}(0)=x_{-}(0)=x_{1}$ and
$x_{+}(T)=x_{-}(T)=x_{2}$ are not varied, i.e.

$e(0)=e(T)=0.$ (2\. 38)

Next,

$Dx_{+}Dx_{-}=JDxDe=2\pi J\prod_{t=0}^{T}dx(t)\prod_{t\neq
0,T}\frac{de(t)}{2\pi},$ (2\. 39)

where $J$ is an unimportant Jacobian of the transformation.

As a result of the replacement (2\. 35) we have

$R(x_{2},T;x_{1},0)=2\pi
J\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}\int^{e(T)=0}_{e(0)=0}De~{}e^{2i\int_{0}^{T}dte(\ddot{x}+v^{\prime}(x))+U_{T}(x,e)}.$
(2\. 40)

One can make use of the formulae

$e^{iU_{T}(x,e)}=e^{\hat{\mathbb{K}}(e^{\prime},j)}e^{iU_{T}(x,e^{\prime})}e^{-2i\int_{0}^{T}e(t)j(t)dt},$
(2\. 41)

where we have introduced the operator

$\hat{\mathbb{K}}(e,j)=\lim_{e=j=0}\exp\left\\{-\frac{1}{2i}\int_{0}^{T}\frac{\delta}{\delta
j(t)}\frac{\delta}{\delta e(t)}\right\\},$ (2\. 42)

after which from (2\. 40) we have found that

$R(x_{2},T;x_{1},0)=2\pi
Je^{\hat{\mathbb{K}}(e^{\prime},j)}\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}e^{iU_{T}(x,e^{\prime})}\times$
$\times\int^{e(T)=0}_{e(0)=0}De~{}\exp\left\\{2i\int_{0}^{T}dt(\ddot{x}+v^{\prime}(x)-j)e\right\\}=$
$=2\pi
Je^{\hat{\mathbb{K}}(e,j)}\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}e^{iU_{T}(x,e)}\prod_{t\neq
0,T}\delta(\ddot{x}+v^{\prime}(x)-j),$ (2\. 43)

where the functional $\delta$-function

$\prod_{t\neq
0,T}\delta(\ddot{x}+v^{\prime}(x)-j)=\int^{e(T)=0}_{e(0)=0}De~{}\exp\left\\{2i\int_{0}^{T}dt(\ddot{x}+v^{\prime}(x)-j)e\right\\}$
(2\. 44)

has arisen as a result of total reduction of unnecessary contributions from
the point of view of equation of motion

$\ddot{x}(t)+V^{\prime}(x)=j(t).$ (2\. 45)

The operator (2\. 42) is Gaussian so that the system is perturbed by the
random force $j(t)$.

If $x(t)$ is the ”true” trajectory and the virtual deviation is $e(t)$ then
the quantity $e(\ddot{x}+v^{\prime}(x)-j)$ coincides with the virtual work. It
must be equal to zero in classical mechanics since only the time reversible
motion is considered. In result we came to equation of motion since $e$ is
arbitrary in classics.

The difference $S_{T}(x_{+})-S_{T}(x_{-})$ in (2\. 34) with boundary
conditions (2\. 38) coincides with the action of reversible motion. Upon the
substitution (2\. 35) we have identified the mean trajectory, $x(t)$, and the
deviation from it, $e(t)$. One must integrate over $e(t)$ in quantum case, in
contrast to classical one. In result the measure of the remaining path
integral over mean trajectory $x(t)$ takes the Dirac $\delta$-function form
which unambiguously chooses the ”true” trajectory.

In other words, the proposed definition of the measure of the path integral is
generalization of classical d’Alambert’s principle on the quantum case. The
theory in the frame of this principle can take into account any external
perturbations, $j(t)$ in our case, if the time reversibility of motion is
conserved. In quantum case the reversibility is established through the
boundary conditions (2\. 38). Next, one may generalize the approach adding
also the probe force which can lead to dynamical symmetry breaking
[16]161616It is important that if the expectation value of the probe force is
not equal to zero then the symmetry is broken. This important possibility will
not be considered in present work..

In the semiclassical approximation $\hat{\mathbb{K}}(e,j)=1$ and taking the
limit $e=j=0$ we find that

$R(x_{2},T;x_{1},0)=2\pi
J\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}\prod_{t\neq
0,T}\delta(\ddot{x}+v^{\prime}(x)),$ (2\. 46)

Let the solution of the homogeneous equation

$\ddot{x}+v^{\prime}(x)=0$ (2\. 47)

be $x_{c}(t)$, with $x_{c}(0)=x_{1}$ and $x_{c}(T)=x_{2}$. Then

$R(x_{2},T;x_{1},0)=2\pi
J\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}\prod_{t\neq
0,T}\delta(\ddot{x}+v^{\prime\prime}(x_{c})x),$ (2\. 48)

The remaining integral is calculated by the standard methods171717Here it is
more convenient to represent (2\. 48) as a production of two Gauss integrals;
later on more effective method of calculation of the functional determinant
will be offered.. As a result we find

$R(x_{2},T;x_{1},0)=\frac{1}{2\pi}\left|\frac{\partial^{2}S_{T}(x_{c})}{\partial
x_{c}(0)\partial x_{c}(T)}\right|_{x_{c}(0)=x_{1},x_{c}(T)=x_{2}}.$ (2\. 49)

Next, let us recall that the full derivative of the classical action is

$dS=p_{2}dx_{2}-p_{1}dx_{1},$ (2\. 50)

where $p_{2}$ and $p_{1}$ are, respectively, the final and initial momentum.
Noting this definition,

$\left|\frac{\partial^{2}S_{T}}{\partial x_{1}\partial
x_{2}}\right|dx_{2}=dp_{1},$ (2\. 51)

and in result we find that

$\int
dx_{1}dx_{2}R(x_{2},T;x_{1},0)=\int\frac{dx_{1}dp_{1}}{2\pi}=\Omega^{2},$ (2\.
52)

which coincides with (2\. 4), i.e. it agree with conservation of total
probability since (2\. 52) again coincides with the total number of physical
states.

Deriving (2\. 52) we somewhat simplify the problem considering a unique
solution of Eq.(2\. 47). A more complicate and important examples will be
considered in the next Sections.

###  2.3 Complex trajectories

Let us consider the one dimensional motion with fixed energy $E$ on the
complex trajectory181818The necessity to extend the formalism on the case of
complex trajectories was mention to the author by A.Slavnov.. The
corresponding amplitude has the form:

$A(x_{1},x_{2};E)=i\int^{\infty}_{0}dTe^{iET}\int_{x_{1}=x(0)}^{x_{2}=x(T)}D_{C_{+}}xe^{iS_{C_{+}}(x)},$
(2\. 53)

where the action

$S_{C_{+}}(x)=\int_{C_{+}}dt(\frac{1}{2}\dot{x}^{2}-v(x))$ (2\. 54)

and the measure

$D_{C_{+}}x=\prod_{t\in C_{+}}\frac{dx(t)}{(2\pi)^{1/2}}$ (2\. 55)

are defined on the shifted in the upper half time plane Mills’ contour
$C_{+}=C_{+}(T)$ [17]:

$t\rightarrow t+i\varepsilon,\;\;\;\varepsilon\rightarrow+0,\;\;\;0\leq t\leq
T.$ (2\. 56)

Therefore, we will consider integration over real functions of complex
variables:

$x^{*}(t)=x(t^{*}).$ (2\. 57)

It must be underlined also that the boundary conditions in (2\. 53) have the
classical meaning, i.e. they do not vary, and $x_{1}$, $x_{2}$ are the real
quantities.

The probability looks as follows:

$R(E)=\int^{\infty}_{0}e^{iE(T_{+}-T_{-})}\int^{x_{\pm}(T_{\pm})=x_{2}}_{x_{\pm}(0)=x_{1}}D_{C_{+}}x_{+}D_{C_{-}}x_{-}\times$
$\times e^{iS_{C_{+}(T_{+})}(x_{+})-iS_{C_{-}(T_{-})}(x_{-})},$ (2\. 58)

where $C_{-}(T)=C^{*}_{+}(T)$ is the time contour in the lower half of complex
time plane.

New time variables

$T_{\pm}=T\pm\tau$ (2\. 59)

will be used. Considering ${\rm Im}E\rightarrow+0$ we can consider $T$ and
$\tau$ as the independent variables:

$0\leq T\leq\infty,\;\;\;-\infty\leq\tau\leq\infty.$ (2\. 60)

We will apply the boundary conditions, see (2\. 58):

$x_{1}=x_{+}(0)=x_{-}(0),~{}~{}x_{2}=x_{+}(T_{+})=x_{-}(T_{-}).$ (2\. 61)

Inserting (2\. 59) one can find in zero order over $\tau$ from (2\. 61) that

$x_{+}(0)=x_{-}(0),~{}~{}x_{+}(T)=x_{-}(T),$ (2\. 62)

Now we will introduce also the mean trajectory $x(t)=(x_{+}(t)+x_{-}(t))/2$
and the deviation $e(t)$ from $x(t)$:

$x_{\pm}(t)=x(t)\pm e(t).$ (2\. 63)

We have consider $e(t)$ and $\tau$ as the virtual quantities. The integrals
over $e$ and $\tau$ will be calculated perturbatively. In zero order over $e$
and $\tau$, i.e. in the semiclassical approximation, $x$ is the classical path
and $T$ is the total time of classical motion. Note that one can do surely the
linear transformations (2\. 63) in the path integrals.

The higher terms over $\tau$ put a unphysical constrains on the trajectory
$x(t)$:

$\frac{d^{(2n+1)}x(T)}{dT^{(2n+1)}}=0,~{}n=0,1,2,...,$

since $e(t)$ must be arbitrary. Therefore, to avoid this constraints and since
the boundaries have classical unvaried meaning we will use the minimal
boundary conditions:

$e(0)=e(T)=0,$ (2\. 64)

which ensures the time reversibility. Note that it is sufficient to have (2\.
64) if the integrals over $e(t)$ are calculated perturbatively. At the same
time

$x(0)=x_{1},~{}x(T)=x_{2}.$ (2\. 65)

Let us extract now the linear over $e$ and $\tau$ terms from the closed-path
action:

$S_{C_{+}(T_{+})}(x_{+})-S_{C_{-}(T_{-})}(x_{-})=$ $=-2\tau
H_{T}(x)-\int_{C^{(+)}(T)}dte(\ddot{x}+v^{\prime}(x))-\tilde{H}_{T}(x;\tau)-U_{T}(x,e),$
(2\. 66)

where

$C^{(+)}(T)=C_{+}(T)+C_{-}(T)$ (2\. 67)

is the total-time path, $H_{T}$ is the Hamiltonian:

$2H_{T}(x)=-\frac{\partial}{\partial T}(S_{C_{+}(T)}(x)+S_{C_{-}(T)}(x)),$
(2\. 68)

and

$-\tilde{H}_{T}(x;\tau)=S_{C_{+}(T+\tau)}(x)-S_{C_{-}(T-\tau)}(x)+2\tau
H_{T}(x),$ (2\. 69)
$-U_{T}(x,e)=S_{C_{+}(T)}(x+e)-S_{C_{-}(T)}(x-e)+\int_{C^{(+)}}dte(\ddot{x}+v^{\prime}(x))$
(2\. 70)

are the remainder terms, where $v^{\prime}(x)=\partial v(x)/\partial x$.
Deriving the decomposition (2\. 66) the definition

$C_{-}(T)=C^{*}_{+}(T)$ (2\. 71)

and the boundary conditions (2\. 64) was used.

One can find the compact form of expansion of

$e^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)}$

over $\tau$ and $e$ using formulae (2\. 16):

$\exp\\{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)\\}=\exp\left\\{\frac{1}{2i}\hat{\omega}\hat{\tau}^{\prime}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}^{\prime}(t)\right\\}\times$
$\times\exp\left\\{2i\omega\tau+i\int_{C^{(+)}(T)}dtj(t)e(t)\right\\}\exp\\{-i\tilde{H}_{T}(x;\tau^{\prime})-iU_{T}(x,e^{\prime})\\}.$
(2\. 72)

At the end of calculations the auxiliary variables
$(\omega,\tau^{\prime},j,e^{\prime})$ should be taken equal to zero.

Using (2\. 66) and (2\. 72) we find from (2\. 58) that

$R(E)=2\pi\int^{\infty}_{0}dT\exp\left\\{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)\right\\}\times$
$\times\int Dx\exp\\{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)\\}\delta(E+\omega-
H_{T}(x))\prod_{C^{(+)}}\delta(\ddot{x}+v^{\prime}(x)-j).$ (2\. 73)

The expansion over the differential operators:

$\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)=\frac{1}{2i}\left(\frac{\partial}{\partial\omega}\frac{\partial}{\partial\tau}+{\rm
Re}\int_{C+}dt\frac{\delta}{\delta j(t)}\frac{\delta}{\delta e(t)}\right)$
(2\. 74)

will generate the perturbation series. We propose that it is summable in Borel
sense.

The first $\delta$-function in (5\. 33) fixes the conservation of energy:

$E+\omega=H_{T}(x)$ (2\. 75)

where $E$ is the observed energy, $H_{T}(x)$ is the energy at the mean
trajectory at the time moment $T$ and $\omega$ is the energy of quantum
fluctuations. The second $\delta$-function191919Following shorthand entry of
$\delta$-function of the complex argument:
$\prod_{C^{(+)}}\delta(f(t))=\prod_{C_{+}}\delta(f(t))\prod_{C_{-}}\delta(f(t))=\prod_{C_{+}}\delta({\rm
Re}f(t)+i{\rm Im}f(t))\delta({\rm Re}f(t)-i{\rm
Im}f(t))=\prod_{C_{+}}\delta({\rm Re}f(t))\cdot\\\ \delta({\rm Im}f(t))$ will
be useful during calculations. The condition (2\. 57) is important here. The
inessential constant can be canceled by normalization. So, in the result of
analytical continuation of $C_{\pm}$ on the real axis the product of two
$\delta$-functions reduces to single one since $\delta^{2}({\rm
Re}f(x))=\delta(0)\delta({\rm Re}f(x))=\delta(0)\delta(f(x))$ and $\delta(0)$
must be canceled by normalization. Offered abbreviated notation will allow to
consider $\delta$-function on the complex time contour as the ordinary one.

$\prod_{t\in
C^{(+)}}\delta(\ddot{x}+v^{\prime}(x)-j)=(2\pi)^{2}\int\prod_{t\in
C^{(+)}}\frac{de(t)}{\pi}\delta(e(0))\delta(e(T))\times$ $\times e^{-2i{\rm
Re}\int_{C_{+}}dte(\ddot{x}+v^{\prime}(x)-j)}=\prod_{t\in C_{+}(T)}\delta({\rm
Re}(\ddot{x}+v^{\prime}(x)-j))\delta({\rm Im}(\ddot{x}+v^{\prime}(x)-j))$ (2\.
76)

fixes the function $x(t)$ of complex argument on $C^{(+)}$ completely by the
equation

$\ddot{x}+v^{\prime}(x)=j.$ (2\. 77)

The physics meaning of $\delta$-function (2\. 76) was discussed in Sec.2.3
noting that the unitarity condition of quantum theories plays the same role as
the d’Alambert’s variational principle in classical mechanics.

In (2\. 77) $j(t)$ describes the external quantum force. The solution
$x_{j}(t)$ of this equation will be found expanding it over $j(t)$:

$x_{j}(t)=x_{c}(t)+\int dt_{1}G(t,t_{1})j(t_{1})+...$ (2\. 78)

This is sufficient since $j(t)$ is the auxiliary variable202020See also
footnote 15.. In this decomposition $x_{c}(t)$ is the strict solution of
unperturbed equation:

$\ddot{x}+v^{\prime}(x)=0$ (2\. 79)

Note that the functional $\delta$-function in (2\. 76) does not contain the
end-point values of $x(t)$, at $t=0$ and $t=T$. This means that if we
integrate over $x_{1}$ and $x_{2}$ then the initial conditions to the Eq.(2\.
79) are not fixed and the integration over them must be performed.

Inserting (2\. 78) into (2\. 77) we find the equation for Green function:

$(\partial^{2}+v^{\prime\prime}(x_{c}))_{t}G(t,t^{\prime};x_{c})=\delta(t-t^{\prime}).$
(2\. 80)

It is too hard to find the exact solution of this equation if $x_{c}(t)$ is
the nontrivial function of $t$. We will see that the canonical transformation
to the (action-angle)-type variables can help to avoid this problem, see
following Section.

###  2.4 Conclusions

1\. The path integral must be defined on the Mills time contour. This
condition will be important in the field theories with high space-time
symmetries (such as the Yang-Mills type theory) since it seems that for such
theories with symmetry one can not perform surely the analytic continuation
over time variable212121The fact that a theory must satisfy certain conditions
upon analytic continuation over time variable is clear from [18]..

2\. The quantization can be performed without transition to the canonical
formalism, using only the Lagrange one which is more natural for relativistic
field theories.

3\. Only the exact solutions of the equation of motion must be taken into
account defining the contributions into the functional integral.

##  3 Path integrals on Dirac measure

###  3.1 Introduction

In present Section we will offer two methods which may simplify calculation of
path integrals on Dirac measure. They are based on the possibility to perform
transformation of the path-integral variables.

We will consider two examples. In the first example the transformation to the
(action,angle)-type variables will be considered. This example shows how much
the calculations of path integrals may be simplified.

In the second part of present Section the coordinate transformation will be
described. For the sake of definiteness the transformation to cylindrical
coordinates will be considered.

###  3.2 Canonical transformation

Let us introduce the first-order formalism. We will insert in (2\. 73)

$1=\int Dp\prod_{t}\delta(p-\dot{x}).$ (3\. 1)

As a result,

$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}+{\rm
Re}\int_{C_{+}(T)}dt\hat{j}(t)\hat{e}(t))}\int
DxDpe^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)}\times$ $\times\delta(E+\omega-
H_{T}(x))\prod_{t}\delta\left(\dot{x}-\frac{\partial H_{j}}{\partial
p}\right)\delta\left(\dot{p}+\frac{\partial H_{j}}{\partial x}\right),$ (3\.
2)

where

$H_{j}=\frac{1}{2}p^{2}+v(x)-jx$ (3\. 3)

may be considered as the total Hamiltonian which is time dependent through
$j(t)$. Notice that in present simplest case $x$ and $p$ are independent
parameters and therefore (3\. 3) define the Hamiltonian.

Instead of pare $(x(t),p(t))$ we introduce new pare $(\theta(t),h(t))$
inserting in (3\. 2)

$1=\int\prod_{t}d\theta
dh\delta\left(h-\frac{1}{2}p^{2}-v(x)\right)\delta\left(\theta-\int^{x}dx(2(h-v(x)))^{-1/2}\right).$
(3\. 4)

Note that the integral measures in (3\. 2) and (3\. 4) are both $\delta$-like,
i.e. have the equal power. It allows to change the order of integration and
firstly integrate over $(x,p)$. We find that

$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}+{\rm
Re}\int_{C_{+}(T)}dt\hat{j}(t)\hat{e}(t))}\int D\theta
Dhe^{-i\tilde{H}_{T}(x_{c};\tau)-iU_{T}(x_{c},e)}\times$
$\times\delta(E+\omega-h(T))\prod_{t}\delta\left(\dot{\theta}-\frac{\partial
H_{c}}{\partial h}\right)\delta\left(\dot{h}+\frac{\partial
H_{c}}{\partial\theta}\right),$ (3\. 5)

where

$H_{c}=h-jx_{c}(h,\theta)$ (3\. 6)

is the transformed Hamiltonian and $x_{c}(\theta,h)$ is the given solution of
algebraic equation:

$\theta=\int^{x}dx(2(h-v(x)))^{-1/2},$ (3\. 7)

i.e. $x_{c}$ is the classical trajectory parametrized in terms of $h(t)$ and
$\theta(t)$.

As it follows from (3\. 5) new variables, $h(t)$ and $\theta(t)$, are
subjected to the action of quantum force $j(t)$ and the topology of classical
trajectory $x_{c}$ remains unchanged.

So, instead of Eq.(2\. 77) we must solve the equations:

$\dot{h}=j\frac{\partial
x_{c}}{\partial\theta},\;\;\;\;\;\dot{\theta}=1-j\frac{\partial
x_{c}}{\partial h},$ (3\. 8)

which have a simpler structure. Expanding the solutions over $j$ we will find
the infinite set of recursive equations. This is the important peculiarity of
used quantization scheme.

Note now that $j\partial x_{c}/\partial\theta$ and $j\partial x_{c}/\partial
h$ in the r.h.s. can be considered as the new sources. We will use this
property of Eqs.(3\. 8) and introduce in the perturbation theory new
”renormalized” sources:

$j_{h}=j\frac{\partial
x_{c}}{\partial\theta},\;\;\;\;\;j_{\theta}=j\frac{\partial x_{c}}{\partial
h},$ (3\. 9)

i.e. $j_{\xi}$ and $j_{\eta}$ are the forces on the cotangent bundle. We will
use transformation (2\. 29):

$\prod_{t}\delta(\dot{h}-j\frac{\partial
x_{c}}{\partial\theta})=e^{\frac{1}{2i}{\rm
Re}\int_{C_{+}}dt\hat{j}_{h}(t)\hat{e}_{h}(t)}e^{2i{\rm
Re}\int_{C_{+}}e_{h}j\frac{\partial
x_{c}}{\partial\theta}}\prod_{t}\delta(\dot{h}-j_{h})$ (3\. 10)

and

$\prod_{t}\delta(\dot{\theta}-1+j\frac{\partial x_{c}}{\partial
h})=e^{\frac{1}{2i}{\rm
Re}\int_{C_{+}}dt\hat{j}_{\theta}(t)\hat{e}_{\theta}(t)}e^{2i{\rm
Re}\int_{C_{+}}e_{\theta}j\frac{\partial x_{c}}{\partial
h}}\prod_{t}\delta(\dot{\theta}-1-j_{\theta})$ (3\. 11)

to introduce them. The re-scaling of source $j$ lead to the re-scaling of
auxiliary field $e$. In the new perturbation theory we will have two sources
$j_{h}$, $j_{\theta}$ and two auxiliary fields $e_{h}$, $e_{\theta}$. Notice
that the momentum $p$ never arose.

Inserting (3\. 10), (3\. 11) into (3\. 5) we find:

$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}-i\int_{C^{(+)}}dt(\hat{j}_{h}(t)\hat{e}_{h}(t)+\hat{j}_{\theta}(t)\hat{e}_{\theta}(t)))}\times$
$\times\int DhD\theta
e^{-i\tilde{H}_{T}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\times$
$\times\delta(E+\omega-h(T))\prod_{t}\delta(\dot{\theta}-1-j_{\theta})\delta(\dot{h}-j_{h}),$
(3\. 12)

where

$e_{c}=e_{h}\frac{\partial x_{c}}{\partial\theta}-e_{\theta}\frac{\partial
x_{c}}{\partial h}$ (3\. 13)

carry the simplectic structure of Hamilton equations of motion and the ”hat”
symbol means differential operator over corresponding quantity. At the very
end one should take all auxiliary variables,
$(e_{h},j_{h},e_{\theta},j_{\theta})$, equal to zero.

Hiding the $x_{c}(t)$ dependence into $e_{c}$ we solve the problem of the
functional determinants, see (3\. 12), and simplify the Hamilton equations of
motion as much as possible:

$\dot{h}(t)=j_{h}(t),\;\;\;\;\;\dot{\theta}(t)=1+j_{\theta}(t)$ (3\. 14)

We will use the boundary conditions

$h(0)=h_{0},\;\;\;\;\theta(0)=\theta_{0},$ (3\. 15)

as the extension of boundary conditions in (2\. 58). This lead to the
following Green function of transformed perturbation theory:

$g(t-t^{\prime})=\Theta(t-t^{\prime}),$ (3\. 16)

with the properties of projection operator:

$\displaystyle\int dtdt^{\prime}g^{2}(t-t^{\prime})=\int
dtdt^{\prime}g(t-t^{\prime}),$ $\displaystyle\int
dtdt^{\prime}g(t-t^{\prime})g(t^{\prime}-t)=0$ (3\. 17)

and, at the same time, we will assume that

$g(0)=1.$ (3\. 18)

It is important to note that ${\rm Im}g(t)$ is regular on the real time axis.
This is the very simplification of the perturbation theory since it eliminates
the doubling of degrees of freedom. One may use here the analytical
continuation to the real time axis.

In result, shifting $C_{+}$ and $C_{-}$ contours on the real time axis we
find:

$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}+\int^{\infty}_{0}dt_{1}dt_{2}\Theta(t_{1}-t_{2})(\hat{e}_{h}(t_{1})\hat{h}(t_{2})+\hat{e}_{\theta}(t_{1})\hat{\theta}(t_{2})))}\times$
$\times\int
dh_{0}d\theta_{0}e^{-i\tilde{H}_{T}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega-
h_{0}+h(T)),$ (3\. 19)

where the solutions of eqs.(3\. 14) was used. In this expression
$x_{c}(t)=x_{c}(h_{0}-h(t),t+\theta_{0}-\theta(t))$ and
$(h(t),e_{h}(t),\theta(t),e_{\theta}(t))$ are the auxiliary fields. At the
very end one must take them equal to zero.

###  3.3 Selection rule

Let us consider the theory with Lagrangian

$L(x)=\frac{1}{2}\dot{x}^{2}-\frac{1}{2}\omega^{2}x^{2}-\frac{g}{4}x^{4}.$
(3\. 20)

The Dirac measure gives the equation (of motion):

$\ddot{x}+\omega^{2}x+gx^{3}=j.$ (3\. 21)

It has two solutions:

$x_{1}(t)=x_{c}(t)+O(j),~{}~{}x_{2}(t)=O(j).$ (3\. 22)

For this reason

$R(E)=R_{1}(E;x_{1})+R_{2}(E;x_{2})$ (3\. 23)

and which one defines $R(E)$ is a question. Following to our selection rule
just $R_{1}$. This will be shown.

Let us return now to the example with Lagrangian (3\. 20). In the
semiclassical approximation

$R_{1}(E;x_{1})=\int_{0}^{\infty}dT\int_{0}^{\infty}dh_{0}\int_{-\infty}^{+\infty}d\theta_{0}e^{-iU_{T}(x_{c},0)}\delta(E-h_{0}).$
(3\. 24)

Therefore,

$R_{1}(E;x_{1})\sim\int_{-\infty}^{+\infty}d\theta_{0}\equiv\Omega,$ (3\. 25)

i.e. it is proportional to the volume of group of time translations.

At the same time

$R_{2}(E;x_{2})=O(1)$ (3\. 26)

in the semiclassical approximation. Therefore,

$R=R_{1}(1+O(1/\Omega)).$ (3\. 27)

This result explains the source of chosen selection rule.

###  3.4 Coordinate transformation

In this section the coordinate transformation of two dimensional quantum
mechanical model with potential

$v=v((x^{2}_{1}+x^{2}_{2})^{1/2})$ (3\. 28)

will be considered. Repeating calculations of previous sections,

$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{\vec{j}}(t)\hat{\vec{e}}(t)}\int
D^{(2)}M(x)e^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)},$ (3\. 29)

where the $\delta$-like Dirac measure

$D^{(2)}M(x)=\delta(E+\omega-
H_{T}(x))\prod_{t}d^{2}x(t)\delta^{(2)}(\ddot{x}+v^{\prime}(x)-j).$ (3\. 30)

In the classical mechanics the problem with potential (3\. 28) is solved in
the cylindrical coordinates:

$x_{1}=r\cos\phi,\;\;\;\;\;x_{2}=r\sin\phi.$ (3\. 31)

We insert into (3\. 29)

$1=\int
DrD\phi\prod_{t}\delta(r-(x^{2}_{1}+x^{2}_{2})^{1/2})\delta(\phi-\arctan\frac{x_{2}}{x_{1}}).$
(3\. 32)

to perform the transformation. Note that the transformation (3\. 31) is not
canonical. In result we will find a new measure:

$D^{(2)}M(r,\phi)=\delta(E+\omega-H_{T}(x))\prod_{t}drd\phi J(r,\phi),$ (3\.
33)

where the Jacobian of transformation

$J(r,\phi)=\int\prod
d^{2}x\delta^{(2)}(\ddot{x}+v^{\prime}(x)-j)\delta(\phi-\arctan\frac{x_{2}}{x_{1}})\delta(r-(x^{2}_{1}+x^{2}_{2})^{1/2})$
(3\. 34)

is the product of two $\delta$-functions:

$J(r,\phi)=\prod_{t}r^{2}(t)\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j_{r})\delta(\partial_{t}(\dot{\phi}r^{2})-rj_{\phi}),$
(3\. 35)

where $v^{\prime}(r)=\partial v(r)/\partial r$ and

$j_{r}=j_{1}\cos\phi+j_{2}\sin\phi,\;\;\;\;j_{\phi}=-j_{1}\sin\phi+j_{2}\cos\phi$
(3\. 36)

are the components of $\vec{j}$ in the cylindrical coordinates.

It is useful to organize the perturbation theory in terms of $j_{r}$ and
$j_{\phi}$. For this purpose following transformation of arguments of
$\delta$-functions will be used:

$\prod_{t}\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j_{r})=e^{-i\int_{C^{(+)}}dt\hat{j}^{\prime}_{r}\hat{e}_{r}}e^{i\int_{C^{(+)}}dtj_{r}e_{r}}\prod_{t}\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j^{\prime}_{r})$
(3\. 37)

and

$\prod_{t}\delta(\partial_{t}(\dot{\phi}r^{2})-rj_{\phi})=e^{-i\int_{C^{(+)}}dt\hat{j}^{\prime}_{\phi}\hat{e}_{\phi}}e^{i\int_{C^{(+)}}dtj_{\phi}re_{\phi}}\prod_{t}r(t)\delta(\partial_{t}(\dot{\phi}r^{2})-j^{\prime}_{\phi}).$
(3\. 38)

Here $j_{r}$ and $j_{\phi}$ was defined in (3\. 36). In result, we get to the
path integral formalism written in terms of cylindrical coordinates. This is a
very simplification which will help to solve a lot of mechanical problems. One
can note that in result of mapping our problem reduced to the description of
quantum fluctuations of the surface of cylinder:

$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt(\hat{j}_{r}(t)\hat{e}_{r}(t)+\hat{j}_{\phi}(t)\hat{e}_{\phi}(t))}\times$
$\times\int D^{(2)}M(r,\phi)e^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e_{C})},$ (3\.
39)

where

$\displaystyle D^{(2)}M(r,\phi)=\delta(E+\omega-
H_{T}(r,\phi))\prod_{t}r^{2}(t)dr(t)d\phi(t)\times$
$\displaystyle\times\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j_{r})\delta(\partial_{t}(\dot{\phi}r^{2})-j_{\phi})$
(3\. 40)

and

$e_{C,1}=e_{r}\cos\phi-
re_{\phi}\sin\phi,\;\;\;\;e_{C,2}=e_{r}\sin\phi+re_{\phi}\cos\phi.$ (3\. 41)

This is the final result. The transformation looks quite classically but (3\.
39) can not be deduced from naive coordinate transformation of initial path
integral for amplitude.

Inserting

$1=\int DpDl\prod_{t}\delta(p-\dot{r})\delta(l-\dot{\phi}r^{2})$ (3\. 42)

into (3\. 39) we can introduce the motion in the phase space with Hamiltonian

$H_{j}=\frac{1}{2}p^{2}+\frac{l^{2}}{2r^{2}}+v(r)-j_{r}r-j_{\phi}\phi.$ (3\.
43)

The Dirac’s measure becomes four dimensional:

$D^{(4)}M(r,\phi,p,l)=\delta(E+\omega-
H_{T}(r,\phi,p,l))\prod_{t}dr(t)d\phi(t)dp(t)dl(t)\times$
$\times\delta\left(\dot{r}-\frac{\partial H_{j}}{\partial
p}\right)\delta\left(\dot{\phi}-\frac{\partial H_{j}}{\partial
l}\right)\delta\left(\dot{p}+\frac{\partial H_{j}}{\partial
r}\right)\delta\left(\dot{l}+\frac{\partial H_{j}}{\partial\phi}\right)$ (3\.
44)

Note absence of the coefficient $r^{2}$ in this expression. This is the result
of special choice of transformation (3\. 38).

Since the Hamilton’s group manifolds are more rich then Lagrange ones the
measure (3\. 44) can be considered as the starting point of farther
transformations. One must to note that the $(action,angle)$ variables are
mostly useful [12]. Note also that to avoid the technical problems with
equations of motion and with functional determinants it is useful to linearize
the argument of $\delta$-functions in (3\. 44) hiding nonlinear terms in the
corresponding auxiliary variables $e_{c}$.

###  3.5 Conclusions

1\. Our perturbation theory describes the quantum fluctuations of the
parameters $(h,\theta)$ of classical trajectory $x_{c}$. It is more
complicated than canonical one, over an interaction constant [19], since
demands investigation of analytic properties of $4N$-dimensional integrals,
where $2N$ is the phase space dimension. Indeed, in the considered case with
$N=1$ the perturbations generating operator, $\hat{\mathbb{K}}$, see (3\. 12),
contain derivatives over four auxiliary parameters,
$(j_{h},e_{h},j_{\theta},e_{\theta})$.

Our transformed theory describes the ”direct” deformations of classical
trajectory $x_{c}=x_{c}(h,\theta)$, i.e. just $h$ and $\theta$ are the objects
of quantization in the considered example. In another words, the quantum
deformations of the invariant hypersurface, $(h,\theta)$, is described in the
new quantum theory. This possibility is the consequence of $\delta$-likeness
of measure, i.e. it based on the conservation of total probability.

Dirac measure allows to perform classical transformations of the measure and
to use high resources of classical mechanics. For example, the interesting
possibility may arise in connection with Kolmogorov-Arnold-Mozer (KAM) theorem
[4]: the system which is not strictly integrable can show the stable motion
peculiar to integrable systems. This is the argument in favor of the idea that
there may be another, non-topological, mechanism of suppression of the quantum
excitations.

2\. One can note that the transformed perturbation theory describes only the
retarded quantum fluctuations, see definition of Green function (3\. 16). This
feature of the theory can lead to the imaginary time irreversibility of
quantum processes and it must be explained.

The starting expression (2\. 58) describes the reversible in time motion since
total action $S_{C_{+}(T_{+})}(x_{+})-S_{C_{-}(T_{-})}(x_{-})$ is time
reversible. But the unitarity condition forced us to consider the interference
picture between expanding and converging waves. This is fixed by the boundary
conditions $e(0)=e(T)=0$. The quantum theory remain time reversible up to
canonical transformation to the invariant hypersurface of the constant energy.
The causal Green function $G(t,t^{\prime})$ , see (2\. 80), is able to
describe both advanced and retarded perturbations and the theory contains the
doubling of degrees of freedom. It means that the theory ”keeps in mind” the
time reversibility. But after the canonical transformation, using above
mentioned boundary conditions, and continuing the theory to the real time, the
quantum perturbations were transferred on the inner degrees of freedom of
classical trajectory. In result the memory of doubling of the degrees of
freedom was disappeared and the theory becomes ”time irreversible”.

The key step in this calculations was an extraction of the classical
trajectory $x_{c}$ which can not be defined without definition of boundary
conditions. Just $x_{c}$ introduces the direction of motion and the order of
quantum perturbations of trajectories inner degrees of freedom play no role,
i.e. the mechanical motion is time reversible while the corrections to energy
of trajectory, $h$, and to the phase, $\theta$, can not be time reversible.
Therefore, the considered irreversibility of the quantum mechanics in terms of
$(h,\theta)$ seems to be imaginary.

##  4 Reduction of quantum degrees of freedom

###  4.1 Introduction

It will be shown in this Section that the quantum fluctuations of angular
variables may be removed if the classical motion is periodic. This cancelation
mechanism can be used for path-integral explanation of integrability of the
quantum-mechanical problems, for example of H-atom problem where the classical
trajectories is closed independently from the initial conditions222222The
approach may be extended on the case of rigid rotator problem [20]. Last one
is isomorphic to the Pocshle-Teller problem [21]. The main result of present
Section is based on the statement that the topology properties of classical
trajectory takes special significance232323Since the action of perturbations
generating operator of transformedf theory, $\hat{\mathbb{K}}$, maps quantum
corrections on the boundaries of cotangent foliation, $\partial W$, see (4\.
41)..

Our technical problem consist in necessity to extract the quantum angular
degrees of freedom. For this purpose we will define path integral in the phase
space of action-angle variables. For simplicity the effect of cancelations we
will demonstrate on the one-dimensional $\lambda x^{4}$ model. In the
following subsection the brief description of unitary definition of the path-
integral measure will be given. The perturbation theory in terms of action-
angle variables will be contracted in Sec.4.3 (the scheme of transformed
perturbation theory was given firstly in [1]). In Sec.4.4 the cancelation
mechanism will be demonstrated.

###  4.2 Unitary definition of the path-integral measure

We will calculate the probability

$R(E)=\int dx_{1}dx_{2}|A(x_{1},x_{2};E)|^{2},$ (4\. 1)

to introduce the unitary definition of path-integral measure [1]. Here

$A(x_{1},x_{2};E)=i\int^{\infty}_{0}dTe^{iET}\int_{x(0)=x_{1}}^{x(T)=x_{2}}Dxe^{iS_{C_{+}(T)}(x)}$
(4\. 2)

is the amplitude of the particle with energy $E$ moving from $x_{1}$ to
$x_{2}$. The action

$S_{C_{+}(T)}(x)=\int_{C_{+}(T)}dt(\frac{1}{2}\dot{x}^{2}-\frac{\omega_{0}^{2}}{2}x^{2}-\frac{\lambda}{4}x^{4})$
(4\. 3)

is defined on the Mills’ contour [17]:

$C_{\pm}(T):t\rightarrow t\pm
i\epsilon,\;\;\;\epsilon\rightarrow+0,\;\;\;0\leq t\leq T.$ (4\. 4)

So, we will omit the calculation of the amplitude.

Inserting (4\. 2) into (4\. 1) we find, see previous Section, that

$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)}\int
Dxe^{-i\tilde{H}(x;\tau)-iU_{T}(x,e)}\times$ $\times\delta(E+\omega-
H_{T}(x))\prod_{t}\delta(\ddot{x}+\omega_{0}^{2}x+\lambda x^{3}-j).$ (4\. 5)

The ”hat” symbol means differentiation over corresponding auxiliary quantity.
For instance,

$\hat{\omega}\equiv\frac{\partial}{\partial\omega},~{}~{}~{}\hat{j}(t)=\frac{\delta}{\delta
j(t)}.$ (4\. 6)

It will be assumed that

$\hat{j}(t\in C_{\pm})j(t^{\prime}\in C_{\pm})=\delta(t-t^{\prime}),$
$\hat{j}(t\in C_{\pm})j(t^{\prime}\in C_{\mp})=0.$ (4\. 7)

The time integral over contour $C^{(\pm)}(T)$ means that

$\int_{C^{(\pm)}(T)}=\int_{C_{+}(T)}\pm\int_{C_{-}(T)}.$ (4\. 8)

At the end of calculations the limit $(\omega,\tau,j,e)=0$ must be calculated.
The explicit form of $\tilde{H}(x;\tau)$ $U_{T}(x,e)$ will be given later;
$H_{T}(x)$ is the Hamiltonian at the time moment $t=T$.

The functional $\delta$-function unambiguously determines the contributions in
the path integral. For this purpose we must find the strict solution
$x_{j}(t)$ of the equation of motion:

$\ddot{x}+\omega_{0}^{2}x+\lambda x^{3}-j=0,$ (4\. 9)

expanding it over $j$. In zero order over $j$ we have the classical trajectory
$x_{c}$ which is defined by the equation of motion:

$\ddot{x}+\omega_{0}^{2}x+\lambda x^{3}=0.$ (4\. 10)

This equation is equivalent to the following one:

$t+\theta_{0}=\int^{x}dx\\{2(h_{0}-\omega_{0}^{2}x^{2}-\lambda
x^{4})\\}^{-1/2}.$ (4\. 11)

The solution of this equation is the periodic elliptic function.

Here $(h_{0},\theta_{0})$ are the constants of integration of Eq.(4\. 10),
i.e. $(h_{0},\theta_{0})$ are the coordinates of point on the surface defined
by elliptic function. The integration over $(h_{0},\theta_{0})$ is assumed
since the integration over all trajectories in (4\. 2) must be performed, i.e.
$(h_{0},\theta_{0})$ takes on all values available by elliptic function. Let
$W$ be the corresponding manyfold. One can say therefore that classical
trajectory belongs $W$ completely.

The mapping of our problem on the action-angle phase space will be performed
using representation (4\. 5) [22]. Using the obvious definition of the action:

$I=\frac{1}{2\pi}\oint\\{2(h-\omega_{0}^{2}x^{2}-\lambda x^{4})\\}^{1/2},$
(4\. 12)

and of the angle

$\phi=\frac{\partial h}{\partial
I}\int^{x_{c}}\\{2(h-\omega_{0}^{2}x^{2}-\lambda x^{4})\\}^{-1/2}$ (4\. 13)

variables [12] we easily find from (4\. 5) that

$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)}\int
DID\phi e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e)}\times$
$\times\delta(E+\omega-h_{T}(I))\prod_{t}\delta(\dot{I}-j\frac{\partial
x_{c}}{\partial\phi})\delta(\dot{\phi}-\Omega(I)+j\frac{\partial
x_{c}}{\partial I}),$ (4\. 14)

where $x_{c}=x_{c}(I,\phi)$ is the solution of Eq.(4\. 13) with $h=h(I)$ as
the solution of Eq.(4\. 12) and the frequency

$\Omega(I)=\frac{\partial h}{\partial I}.$ (4\. 15)

Representation (4\. 14) is not the full solution of our problem: the action
and angle variables are still interdependent since they both are exited by the
same source $j(t)$. This reflects the Lagrange nature of the path-integral
description of phase-space motion. The true Hamilton’s description must
contain independent quantum sources of action and angle variables.

###  4.3 Perturbation theory on the cotangent manifold

The structure of source terms, $j\partial x_{c}/\partial\phi$ and $j\partial
x_{c}/\partial I$, show that the source of quantum fluctuations is the
classical trajectories perturbation and $j$ is the auxiliary variable. It
allows to regroup the perturbation series in a following manner. Let us
consider the action of the perturbation-generating operators on
$\delta$-functions:

$e^{-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)}e^{-iU_{T}(x,e)}\prod_{t}\delta\left(\dot{I}+j\frac{\partial
x_{c}}{\partial\phi}\right)\delta\left(\dot{\phi}-\Omega(I)-j\frac{\partial
x_{c}}{\partial I}\right)=$ $=\int
D_{C^{(+)}}e_{I}D_{C^{(+)}}e_{\phi}e^{i\int_{C^{(+)}}dt(e_{I}\dot{I}+e_{\phi}(\dot{\phi}-\Omega(I)))}e^{-iU_{T}(x,e_{c})},$
(4\. 16)

where

$e_{c}(e_{I},e_{\phi})=e_{I}\frac{\partial
x_{c}}{\partial\phi}-e_{\phi}\frac{\partial x_{c}}{\partial I}.$ (4\. 17)

The integrals over $(e_{I},e_{\phi})$ will be calculated perturbatively:

$e^{-iU_{T}(x,e_{c})}=\sum^{\infty}_{n_{I},n_{\phi}=0}\frac{1}{n_{I}!n_{\phi}!}\int\prod^{n_{I}}_{k=1}(dt_{k}e_{I}(t_{k}))\prod^{n_{\phi}}_{k=1}(dt^{\prime}_{k}e_{\phi}(t^{\prime}_{k}))\times$
$\times
P_{n_{I},n_{\phi}}(x_{c},t_{1},...,t_{n_{I}},t^{\prime}_{1},...,t_{n_{\phi}}),$
(4\. 18)

where

$P_{n_{I},n_{\phi}}(x_{c},t_{1},...,t_{n_{I}},t^{\prime}_{1},...,t_{n_{\phi}})=\prod^{n_{I}}_{k=1}\hat{e}^{\prime}_{I}(t_{k})\prod^{n_{\phi}}_{k=1}\hat{e}^{\prime}_{\phi}(t^{\prime}_{k})e^{-iU_{T}(x,e^{\prime}_{c})},$
(4\. 19)

where $e^{\prime}_{c}\equiv e_{c}(e^{\prime}_{I},e^{\prime}_{\phi})$ and the
derivatives in (4\. 19) are calculated at $e^{\prime}_{I}=0$,
$e^{\prime}_{\phi}=0$. At the same time,

$\prod^{n_{I}}_{k=1}e_{I}(t_{k})\prod^{n_{\phi}}_{k=1}e_{\phi}(t^{\prime}_{k})=\prod^{n_{I}}_{k=1}(i\hat{j}_{I}(t_{k}))\prod^{n_{\phi}}_{k=1}(i\hat{j}_{\phi}(t^{\prime}_{k}))e^{-i\int_{C^{(+)}}dt(j_{I}(t)e_{I}(t)+j_{\phi}(t)e_{\phi}(t))}.$
(4\. 20)

The limit $(j_{I},j_{\phi})=0$ is assumed. Inserting (4\. 19), (4\. 20) into
(4\. 16) we will find new representation for $R(E)$:

$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt(\hat{j}_{I}(t)\hat{e}_{I}(t)+\hat{j}_{\phi}(t)\hat{e}_{\phi}(t))}\times$
$\times\int DID\phi e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\times$
$\times\delta(E+\omega-
h_{T}(I))\prod_{t}\delta(\dot{I}-j_{I})\delta(\dot{\phi}-\Omega(I)-j_{\phi}),$
(4\. 21)

in which the action and the angle are the decoupled degrees of freedom.

Solving the canonical equations of motion:

$\dot{I}=j_{I},\;\;\;\dot{\phi}=\Omega(I)+j_{\phi}$ (4\. 22)

the boundary conditions:

$I_{j}(0)=I_{0},\;\;\;\phi_{j}(0)=\phi_{0}$ (4\. 23)

will be used. This will lead to the following Green function:

$g(t-t^{\prime})=\Theta(t-t^{\prime}),$ (4\. 24)

with boundary condition: $\Theta(0)=1$. The solutions of eqs.(4\. 22) have the
form:

$\displaystyle I_{j}(t)=I_{0}+\int
dt^{\prime}g(t-t^{\prime})j_{I}(t^{\prime})\equiv I_{0}+I^{\prime}(t),$
$\displaystyle\phi_{j}(t)=\phi_{0}+\tilde{\Omega}(I_{j})t+\int
dt^{\prime}g(t-t^{\prime})j_{\phi}(t^{\prime})\equiv\phi_{0}+\tilde{\Omega}(I_{0}+I^{\prime})t+\phi^{\prime}(t),$
(4\. 25)

where

$\tilde{\Omega}(I_{j})=\frac{1}{t}\int
dt^{\prime}g(t-t^{\prime})\Omega(I_{0}+I^{\prime}(t^{\prime})).$ (4\. 26)

Inserting (4\. 25) into (4\. 21) we find:

$R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt(\hat{j}{{}_{I}}(t)\hat{e}_{I}(t)+\hat{j}_{\phi}(t)\hat{e}_{\phi}(t))}\times$
$\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}d\phi_{0}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega-
h_{T}(I_{j})),$ (4\. 27)

where

$x_{c}=x_{c}(I_{j},\phi_{j})=x_{c}(I_{0}+I(t),\phi_{0}+\tilde{\Omega}(I_{0}+I)t+\phi(t))$
(4\. 28)

and $e_{c}$ was defined in (4\. 17). Note that the measure of the integrals
over $(I_{0},\phi_{0})$ was defined without of the Faddeev-Popov’s ansatz and
there is not any “hosts” since the Jacobian of transformation is equal to one.

We can extract the Green function into the perturbation-generating operator
using the equalities:

$\hat{j}_{I}(t)=\int dt^{\prime}g(t-t^{\prime})\hat{I}(t),\hat{j}_{\phi}=\int
dt^{\prime}g(t-t^{\prime})\hat{\phi}(t),$ (4\. 29)

which evidently follows from (4\. 25). In result,

$R(E)=2\pi\int^{\infty}_{0}dTe^{\\{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dtdt^{\prime}g(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))\\}}\times$
$\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}d\phi_{0}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega-
h_{T}(I_{0}+I)),$ (4\. 30)

where $x_{c}$ was defined in (4\. 28).

We can define the formalism without doubling of the degrees of freedom. One
can use fact that the action of perturbation-generating operators and the
analytical continuation to the real times are commuting operations. This can
be seen easily using the definition (4\. 7). In result the expression:

$R(E)=2\pi\int^{\infty}_{0}dTe^{\\{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))\\}}\times$
$\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}d\phi_{0}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega-
h_{T}(I_{0}+I(T)),$ (4\. 31)

where

$\tilde{H}_{T}(x_{c};\tau)=2\sum^{\infty}_{n=1}\frac{\tau^{2n+1}}{(2n+1)!}\frac{d^{2n}}{dT^{2n}}h(I_{0}+I(T))$
(4\. 32)

and

$-U_{T}(x_{c},e_{c})=S(x_{c}+e_{c})-S(x_{c}-e_{c})-2\int_{0}^{T}dte_{c}\frac{\delta
S(x_{c})}{\delta x_{c}}$ (4\. 33)

defines quantum theory on the cotangent manifold $W$.

Now we can use the last $\delta$-function:

$R(E)=2\pi\int^{\infty}_{0}dTe^{\\{\frac{1}{2i}(\hat{\omega}\hat{\tau}+\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))\\}}\times$
$\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E+\omega)}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}.$
(4\. 34)

Here

$x_{c}(t)=x_{c}(I_{0}(E+\omega)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t+\phi(t)).$
(4\. 35)

Eq.(4\. 34) contains unnecessary contributions: the action of the operator

$\int^{T}_{0}dtdt^{\prime}\Theta(t-t^{\prime})\hat{e}_{I}(t)\hat{I}(t^{\prime})$
(4\. 36)

on $\tilde{H}_{T}$, defined in (4\. 32), leads to the time integrals with zero
integration range:

$\int^{T}_{0}dt\Theta(T-t)\Theta(t-T)=0.$

Using this fact,

$\displaystyle
R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))}\times$
$\displaystyle\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E)}e^{-iU_{T}(x_{c},e_{c})},$
(4\. 37)

where

$x_{c}(t)=x_{c}(I_{0}(E)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t+\phi(t)).$ (4\.
38)

is the periodic function:

$x_{c}(I_{0}(E)+I(t)-I(T),(\phi_{0}+2\pi)+\tilde{\Omega}t+\phi(t))=$
$=x_{c}(I_{0}(E)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t+\phi(t)).$ (4\. 39)

Now we can consider the cancelation of angular perturbations.

###  4.4 Cancelation of angular perturbations

1\. Simples example

Introducing the perturbation-generating operator into the integral over
$\phi_{0}$:

$\displaystyle
R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)\hat{I}(t)\hat{e}_{I}(t^{\prime})}\times$
$\displaystyle\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E)}e^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime})}e^{-iU_{T}(x_{c},e_{c})},$
(4\. 40)

the mechanism of cancelations of the angular perturbations becomes evident.
One can formulate the statement:

(i) if

$e^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime})}e^{-iU_{T}(x_{c},e_{c})}=e^{-iU_{T}(x_{c},e_{c})}|_{e_{\phi}=\phi=0}+dF(\phi_{0})/d\phi_{0},$
(4\. 41)

and

(ii) if

$F(\phi_{0}+2\pi)=F(\phi_{0}),$ (4\. 42)

then:

$R(E)=2\pi\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E)}\int^{\infty}_{0}dTdI_{0}e^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})}$
$\times e^{S(x_{c}+e\partial x_{c}/\partial\phi_{0})-S(x_{c}-e\partial
x_{c}/\partial\phi_{0})},$ (4\. 43)

i.e. we find the expression in which the angular corrections was canceled. In
this case the problem becomes semiclassical over the angular degrees of
freedom.

For the $(\lambda x^{4})_{1}$-model

$S(x_{c}+e\partial x_{c}/\partial\phi_{0})-S(x_{c}-e\partial
x_{c}/\partial\phi_{0})=S_{0}(x_{c})-2\lambda\int^{T}_{0}dtx_{c}(t)\\{e\partial
x_{c}/\partial\phi_{0}\\}^{3},$ (4\. 44)

where [1]

$S_{0}(x_{c})=\oint_{T}dt\left(\frac{1}{2}\dot{x}_{c}^{2}-\frac{\omega_{0}^{2}}{2}x_{c}^{2}-\frac{\lambda}{4}x_{c}^{4}\right)$
(4\. 45)

is the closed time-path action and

$x_{c}(t)=x_{c}(I_{0}(E)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t).$ (4\. 46)

Here $I(t)$ and $I(T)$ are the auxiliary variables.

The condition (4\. 42) requires that the classical trajectory $x_{c}$ with all
derivatives over $I_{0}$, $\phi_{0}$ is the periodic function. In the
considered case of $(\lambda x^{4})_{1}$-model $x_{c}$ is periodic function
with period $1/\Omega$, see (4\. 39). Therefore, we can concentrate the
attention on the condition (4\. 41) only.

Expanding $F(\phi_{0})$ over $\lambda$:

$F(\phi_{0})=\lambda F_{1}(\phi_{0})+\lambda^{2}F_{2}(\phi_{0})+...$ (4\. 47)

we find that

$\frac{d}{d\phi_{0}}F_{1}(\phi_{0})=$
$=\int^{T}_{0}\prod^{3}_{k=1}dt^{\prime}_{k}\hat{\phi}(t^{\prime}_{k})\left(\left(-\frac{6}{(2i)^{3}}\right)\int^{T}_{0}dt\prod^{3}_{k=1}\Theta(t-t^{\prime}_{k})x_{c}(t)(\partial
x_{c}/\partial I_{0})^{3}e^{iS_{0}(x_{c})}_{k}\right)=$
$=\int^{T}_{0}dt^{\prime}\hat{\phi}(t^{\prime})B_{1}(\phi),$ (4\. 48)

where

$B_{1}(\phi)=\left\\{-\frac{6}{(2i)^{3}}\int^{T}_{0}dt\Theta(t-t^{\prime})\right.\times$
$\times\left.\prod^{2}_{k=1}(\Theta(t-t^{\prime}_{k})\hat{\phi}(t^{\prime}_{k}))x_{c}(t)(\partial
x_{c}/\partial I_{0})^{3}e^{iS_{0}(x_{c})}\right\\}$ (4\. 49)

This example shows that the sum over all powers of $\lambda$ can be written in
the form:

$\frac{d}{d\phi_{0}}F(\phi_{0})=\int^{T}_{0}dt^{\prime}\hat{\phi}(t^{\prime})B(\phi),$
(4\. 50)

where, using the definition (4\. 35),

$B(\phi)=\int^{T}_{0}dt\tilde{B}(\phi_{0}+\phi(t)).$ (4\. 51)

Therefore,

$\hat{\phi}(t^{\prime})B(\phi)=\frac{d}{d\phi_{0}}\int^{T}_{0}dt\delta(t-t^{\prime})\tilde{B}(\phi_{0}+\phi(t))$
(4\. 52)

coincides with the total derivative over initial phase $\phi_{0}$, and

$F(\phi_{0})=\tilde{B}(\phi_{0}+\phi(t))|_{\phi=0}.$ (4\. 53)

This result ends the prove of (4\. 41).

2\. General case

Now we will offer following important statement:

— each order of perturbation theory in the invariant subspace can be
represented as the sum of total derivative over the subspace coordinate.
This statement directly follows from structure of perturbations generating
operator $\hat{\mathbb{K}}$ and the assumption (3\. 18). It explains the
statement, offered in Preface.

Let us remind that integration with last $\delta$-function gives the result of
action of operator $\hat{\mathbb{K}}$ written in the form:

$R(E)=2\pi\int_{0}^{\infty}dT\int_{0}^{2\pi}\frac{d\varphi_{0}}{\Omega(E)}:e^{-iU_{(}x_{c},\hat{e}/2i)}:,$
(4\. 54)

where the colons mean normal product,

$\hat{e}=\hat{j}_{\varphi}\frac{\partial x_{c}}{\partial
I}-\hat{j}_{I}\frac{\partial x_{c}}{\partial\varphi},$ (4\. 55)

and by definition $U_{T}$ is the odd over $\hat{e}_{c}$ functional:

$U_{T}(x_{c},e_{c})=2\int_{0}^{T}\sum_{n=1}(\hat{e}_{c}(t)/2i)^{2n+1}u_{n}(x_{c}),$
(4\. 56)

where $u_{n}$ is the function of only $x_{c}$ at the time $t$. Inserting (4\.
55) one can write:

$:e^{-iU_{(}x_{c},\hat{e}/2i)}:=\prod_{n=1}^{\infty}\prod_{k=0}^{2n+1}:e^{-iU_{k,n}(j,x_{c})}:,$
(4\. 57)

where

$U_{k,n}(j,x_{c})=\int_{0}^{T}dt(\hat{j}_{\varphi}(t))^{2n-k+1}(\hat{j}_{I}(t))^{k}b_{k,n}(x_{c}(t))$
(4\. 58)

and the explicit form of $b_{k,n}(x_{c})$ is not important.

Using the evident definition:

$\hat{j}_{X}=\int_{0}^{T}dt^{\prime}\Theta(t-t^{\prime})\hat{X}(t^{\prime}),~{}~{}X=\varphi,I,$

it is easy to find that

$j_{X}(t_{1})b_{k,n}(x_{c}(t_{2}))=\Theta(t_{1}-t_{2})\partial
b_{k,n}(x_{c}(t_{2}))/\partial X_{0},$

since $x_{c}=x_{c}(X+X_{0})$, or shortly:

$j_{1}b_{2}=\Theta_{12}\partial_{X_{0}}b_{2}=\partial_{X_{0}}(\Theta_{12}b_{2})$
(4\. 59)

since the indexes $(k,n)$ are not important.

Let us start consideration from the first term with $k=0$. In this case we
describe only the angular fluctuations. Noting that $\partial_{X_{0}}$ and
$\hat{j}$ commute we can consider the lowest order over $\hat{j}$. The typical
term looks as follows (omitting the index $X_{0}$):

$\hat{j}_{1}\hat{j}_{2}\cdots\hat{j}_{m}b_{1}b_{2}\cdots b_{m}.$

It is sufficient to show that this expression is the total derivative over
$X_{0}$.

Case $m=1$. In this approximation we have, see (4\. 59):

$\hat{j}_{1}b_{1}=\Theta_{11}\partial_{0}b_{1}\neq 0.$ (4\. 60)

Here (3\. 18) was used.

Case $m=2$. This order is less trivial:

$\hat{j}_{1}\hat{j}_{2}b_{1}b_{2}=\Theta_{21}b_{1}^{2}b_{2}+b_{1}^{1}b_{2}^{1}+\Theta_{12}b_{1}b_{2}^{2},$
(4\. 61)

where

$b_{i}^{n}\equiv\partial^{n}b_{i}.$ (4\. 62)

At first glance (4\. 61) is not the total derivative. But inserting

$1=\Theta_{12}+\Theta_{21}$

we can symmetrize it:

$\hat{j}_{1}\hat{j}_{2}b_{1}b_{2}=\Theta_{21}(b_{1}^{2}b_{2}+b_{1}^{1}b_{2}^{1})+\Theta_{12}(b_{1}b_{2}^{2}+b_{1}^{1}b_{2}^{1})=$
$=\partial_{0}(\Theta_{21}b_{1}^{1}b_{2}+\Theta_{12}b_{1}b_{2}^{1})\equiv$
$\equiv\partial_{0}(b_{1}^{1}\rightarrow b_{2}+b_{2}^{1}\rightarrow b_{1})$
(4\. 63)

since the explicit form of the function is not important. Therefore, the
second order term can be also reduced to the total derivative. Notice that
(4\. 63) shows time reversibility.

Case $m=3$. In this order one can find that

$\hat{j}_{1}\hat{j}_{2}\hat{j}_{3}b_{1}b_{2}b_{3}=\partial_{0}\left\\{\sum_{i\neq
j\neq k=1}^{3}(i^{2}\rightarrow j\rightarrow k+i^{1}\rightarrow
j^{1}\rightarrow k)\right\\}$ (4\. 64)

The $m$-th order contribution is also total derivative:

$\hat{j}_{1}\hat{j}_{2}\cdots\hat{j}_{m}b_{1}b_{2}\cdots
b_{m}=\partial_{0}\\{\sum_{i_{1}\neq i_{2}\neq i_{3}\neq\cdots\neq
i_{m}=1}^{m}(i_{1}^{m}\rightarrow i_{2}\rightarrow
i_{3}\rightarrow\cdots\rightarrow i_{m}+$ $+i_{1}^{m-1}\rightarrow
i_{2}^{1}\rightarrow i_{3}\rightarrow\cdots\rightarrow
i_{m}+i_{1}^{m-2}\rightarrow i_{2}^{1}\rightarrow
i_{3}^{1}\rightarrow\cdots\rightarrow i_{m}+\cdots$
$\cdots+i_{1}^{1}\rightarrow i_{2}^{1}\rightarrow
i_{3}^{1}\rightarrow\cdots\rightarrow i_{m-1}^{1}\rightarrow i_{m})\\}$ (4\.
65)

Let us consider now the case with $k\neq 0$. The typical term looks as
follows:

$\hat{j}_{1}^{1}\hat{j}_{2}^{1}\cdots\hat{j}_{l}^{1}\hat{j}_{l+1}^{2}\hat{j}_{l+2}^{2}\cdots\hat{j}_{m}^{2}b_{1}b_{2}\cdots
b_{m},~{}0<l<m,$ (4\. 66)

where, for instance

$\hat{j}_{k}^{1}\equiv\hat{j}_{I}(t_{k}),~{}~{}\hat{j}_{k}^{2}\equiv\hat{j}_{\varphi}(t_{k})$
(4\. 67)

and

$\hat{j}_{1}^{i}b_{2}=\Theta_{12}\partial_{0}^{i}b_{2}.$ (4\. 68)

Case $m=2,l=1$.In this case:

$\hat{j}_{1}^{1}\hat{j}_{2}^{2}b_{1}b_{2}=\Theta_{21}(b_{2}\partial_{0}^{1}\partial_{0}^{2}b_{1}+(\partial_{0}^{2}b_{2})(\partial_{0}^{1}\partial_{0}^{2}b_{1}))+\Theta_{12}(b_{1}\partial_{0}^{1}\partial_{0}^{2}b_{2}+(\partial_{0}^{2}b_{2})(\partial_{0}^{1}\partial_{0}^{2}b_{1}))=$
$=\partial_{0}^{1}(\Theta_{21}b_{2}\partial_{0}^{2}b_{1}+\Theta_{12}b_{1}\partial_{0}^{2}b_{2})+\partial_{0}^{2}(\Theta_{21}b_{2}\partial_{0}^{1}b_{1}+\Theta_{12}b_{1}\partial_{0}^{1}b_{2}).$
(4\. 69)

Therefore we have the total-derivative structure yet. This property is
conserved in arbitrary order over $m$ and $l$ since the time-ordered structure
does not depends from upper index of $\hat{j}$, see (4\. 68).

One can conclude that the contribution are defined by topology properties of
classical trajectory $x_{c}$. We will see that this important property of
perturbation theory remains unchanged also for field theories with symmetry.

###  4.5 Conclusions

1\. It was shown that the real-time quantum problem can be semiclassical over
the part of the degrees of freedom and quantum over another ones. Following to
the result of this Section one may introduce the (probably naive)
interpretation of the quantum systems integrability (we suppose that the
classical system is integrable and can be mapped on the compact hypersurface
in the phase space [12]): the quantum system is strictly integrable in result
of cancelation of all quantum degrees of freedom. The mechanism of cancelation
of the quantum corrections is varied from case to case.

For some problems (as the rigid rotator, or the Pocshle-Teller) the
cancelation of angular degrees of the freedom is enough since they carry only
the angular ones. In an another case (as in the Coulomb problem, or in the
one-dimensional models) the problem may be partly integrable since the quantum
fluctuations of action degrees of freedom just survive. Theirs absence in the
Coulomb problem needs special discussion (one must take into account the
dynamical (hidden) symmetry of Coulomb problem [23]).

The transformation to the action-angle variables maps the $N$-dimensional
Lagrange problem on the $2N$-dimensional phase-space torus. If the winding
number on this hypertorus is a constant (i.e. the topological charge is
conserved) one can expect the same cancelations. This is important for the
field-theoretical problems (for instance, for sin-Gordon model [24]).

2\. In the classical mechanics following approximated method of calculations
is used [12]. The canonical equations of motion:

$\dot{I}=a(I,\phi),~{}~{}\dot{\phi}=b(I,\phi)$ (4\. 70)

are changed on the averaged equations:

$\dot{J}=\frac{1}{2\pi}\int^{2\pi}_{0}d\phi
a(J,\phi),~{}~{}\dot{\phi}=b(J,\phi),$ (4\. 71)

It is possible if the oscillations can be extracted from the systematic
evolution of the degrees of freedoms.

In our case

$a(I,\phi)=j\partial x_{c}/\partial\phi,~{}~{}b(I,\phi)=\Omega(I)-j\partial
x_{c}/\partial I.$ (4\. 72)

Inserting this definitions into (4\. 71) we find evidently wrong result since
in this approximation the problem looks like pure semiclassical for the case
of periodic motion:

$\dot{J}=0,~{}~{}\dot{\phi}=\Omega(J).$ (4\. 73)

The result of this Section was used here. This shows that the procedure of
extraction of the oscillations from the systematic evolution is not trivial
and this method should be used carefully in the quantum theories. (This
approximation of dynamics is ”good” on the time intervals $\sim 1/|a|$ [12].)

##  5 Example: H-atom

###  5.1 Introduction

The mapping

$J:T\rightarrow W,$ (5\. 1)

where $T$ is the $2N$-dimensional phase space and $W$ is a linear space solves
the mechanical problem iff

$J=\otimes^{N}_{1}J_{i},$ (5\. 2)

where $J_{i}$ are the first integrals in involution, see e.g. [12]242424The
formalism of reduction (5\. 1) in classical mechanics is described also in
[25]. The aim of this Section is to adopt this procedure for H-atom.

The mapping (5\. 1) introduces integral $manifold$ $J_{\omega}=J^{-1}(\omega)$
in such a way that the $classical$ phase space flaw belongs to $J_{\omega}$
$completely$. We wish quantize the $J_{\omega}$ manifold instead of flow in
$T$ noting that the quantum trajectory also should belong to $J_{\omega}$
completely. This important conclusion was demonstrated in previous Section by
transformation of the path-integral measure to the canonical variables
$(\xi,\eta)$. New perturbation theory is extremely simple since $W$ is the
linear space.

The ”direct” mapping (5\. 1) used in [26] assumes that $J$ is known. But it
seems inconvenient having in mind the general problem of nonlinear waves
quantization, when the number of degrees of freedom $N=\infty$, or if the
transformation is not canonical. We will consider by this reason the ”inverse”
approach assuming that just the classical flow is known. Then, since the flow
belongs to $J_{\omega}$ completely [26], we would be able to find the quantum
motion in $W$. It is the main technical result illustrated in this Section.

The manifold $J_{\omega}$ is invariant relatively to some subgroup
$G_{\omega}$ [27] in accordance to topological class of classical flaw. This
introduces the $J_{\omega}$ classification and summation over all (homotopic)
classes should be performed. Note, the classes are separated by the boundary
bifurcation lines in $W$ [27]. If the quantum perturbations switched on
adiabatically then the homotopic group should stay unbroken. It is the
ordinary statement for quantum mechanics, but, generally speaking, this is not
true for field theories.

We will calculate the bound state energies in the Coulomb potential252525 We
will restrict ourselves by the plane problem. Corresponding phase space
$T=(p,l,r,\varphi)$ is 4-dimensional.. This popular problem was considered by
many authors, using various methods, see e.g. [23]. The path-integral solution
of this problem was offered firstly in [28].

The classical flaw of this problem can be parameterized by the angular
momentum $l$, corresponding angle $\varphi$ and by the normalized on total
Hamiltonian Runge-Lentz vector length $n$. So, we will consider the mapping
($p$ is the conjugate to $r$ radial momentum in the cylindrical coordinates):

$J_{l,n}:(p,l,r,\varphi)\rightarrow(l,n,\varphi)$ (5\. 3)

to construct the perturbation theory in the $W=(l,n,\varphi)$ space. I.e. $W$
is not considered as the cotangent foliation on $T$.

The mapping (5\. 3) assumes additional reduction of the four-dimensional
incident phase space up to three-dimensional linear subspace262626$W$ would
not have the simplectic structure. Actually in considered case $W=R+TW$, where
$R$ is the zero-modes space and $TW$ is the simplectic subspace.. Just this
reduction phenomena leads to corresponding stability of $n$ concerning quantum
perturbations and will allow to solve our H-atom problem completely272727 In
other words, we would demonstrate that the hidden Bargman-Fock [23] $O(4)$
symmetry is stay unbroken concerning quantum perturbations..

In Subsec. 5.2 we will show how the mapping (5\. 3) can be performed for path-
integral differential measure. In Subsec. 5.3 the consequence of reduction
will be derived and in Subsec. 5.4 the perturbation theory in the $W$ space
will be analyzed. The calculations are based on the formalism offered in
previous Sections.

###  5.2 Mapping

We will calculate the integral [26]:

$\rho(E)=\int^{\infty}_{0}dTe^{-i{\hat{\mathbb{K}}}(j,e)}\int
DM(p,l,r,\varphi)e^{-iU(r,e)},$ (5\. 4)

where $\rho(E)$ is the $probability$ to find a particle with energy $E$, i.e.
we should find [22] that normalized on the zero-modes volume

$\rho(E)=\pi\sum_{n}\delta(E-E_{n}),$ (5\. 5)

where $E_{n}$ are the bound states energies. For $H$-atom problem $E_{n}\leq
0$. This condition will define considered homotopy class.

Expansion over operator

${\hat{\mathbb{K}}}(j,e)=\frac{1}{2}\int^{T}_{0}dt(\hat{j}_{r}\hat{e}_{r}+\hat{j}_{\varphi}\hat{e}_{\varphi}),~{}~{}~{}\hat{X}(t)\equiv\delta/\delta
X(t),$ (5\. 6)

generates the perturbation series. It will be seen that in our case we may
omit the question of perturbation theories convergence.

The differential measure

$DM(p,l,r,\varphi)=\delta(E-H_{0})\prod_{t}dr(t)dp(t)dl(t)d\varphi(t)\times$
$\delta\left(\dot{r}-\frac{\partial H_{j}}{\partial
p}\right)\delta\left(\dot{p}+\frac{\partial H_{j}}{\partial
r}\right)\delta\left(\dot{\varphi}-\frac{\partial H_{j}}{\partial
l}\right)\delta\left(\dot{l}+\frac{\partial H_{j}}{\partial\varphi}\right),$
(5\. 7)

with total Hamiltonian ($H_{0}=H_{j}|_{j=0}$)

$H_{j}=\frac{1}{2}p^{2}-\frac{l^{2}}{2r^{2}}-\frac{1}{r}-j_{r}r-j_{\varphi}\varphi$
(5\. 8)

allows perform arbitrary transformation of variables because of its
$\delta$-likeness. Notice that $H_{j}$ contains only the ”Lagrange forces”
$j_{r}$ and $j_{\varphi}$.

The functional

$U(r,e)=-s_{0}(r)+$
$+\int^{T}_{0}dt\left[\frac{1}{((r+e_{r})^{2}+r^{2}e_{\varphi}^{2})^{1/2}}-\frac{1}{((r-e_{r})^{2}+r^{2}e_{\varphi}^{2})^{1/2}}+2\frac{e_{r}}{r}\right]$
(5\. 9)

describes the interaction between various quantum modes and $s_{0}(r)$ defines
the non-integrable phase factor [22]. The quantization of this factor
determines the bound state energy. Such factor will appear if the phase of
amplitude can not be fixed 282828As, for instance, in the Aharonov-Bohm case..
Note that the Hamiltonian (5\. 8) contains the energy of radial $j_{r}r$ and
angular $j_{\varphi}\varphi$ excitation independently.

Let us introduce the functional

$\Delta=\int\prod_{t}d^{2}\xi d^{2}\eta\times$
$\times\delta(r(t)-r_{c}(\xi,\eta))\delta(p(t)-p_{c}(\xi,\eta))\delta(l(t)-l_{c}(\xi,\eta))\delta(\varphi(t)-\varphi_{c}(\xi,\eta))$
(5\. 10)

which is defined by given functions
$(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$. If given functions $(\xi,\eta)$
zeroes argument of $\delta$-functions in (5\. 10) then it is assumed that the
functional determinant

$\displaystyle\Delta_{c}=\int\prod_{t}d^{2}\bar{\xi}d^{2}\bar{\eta}\delta\left(\frac{\partial
r_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
r_{c}}{\partial\eta}\cdot\bar{\eta}\right)\delta\left(\frac{\partial
p_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
p_{c}}{\partial\eta}\cdot\bar{\eta}\right)\times$
$\displaystyle\times\delta\left(\frac{\partial\varphi_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial\varphi_{c}}{\partial\eta}\cdot\bar{\eta}\right)\delta\left(\frac{\partial
l_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
l_{c}}{\partial\eta}\cdot\bar{\eta}\right)\neq 0.$ (5\. 11)

Note that this is the condition only for
$(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$.

To perform the mapping we will insert

$1=\Delta/\Delta_{c}$ (5\. 12)

into (5\. 4) and integrate over $r(t)$, $p(t)$, $\varphi(t)$ and $l(t)$. In
result we find the measure:

$\displaystyle
DM(\xi,\eta)=\frac{1}{\Delta_{c}}\delta(E-H_{0})\prod_{t}d^{2}\xi
d^{2}\eta\delta\left(\dot{r_{c}}-\frac{\partial H_{j}}{\partial
p_{c}}\right)\times$
$\displaystyle\times\delta\left(\dot{p_{c}}+\frac{\partial H_{j}}{\partial
r_{c}}\right)\delta\left(\dot{\varphi_{c}}-\frac{\partial H_{j}}{\partial
l_{c}}\right)\delta\left(\dot{l_{c}}+\frac{\partial
H_{j}}{\partial\varphi_{c}}\right),$ (5\. 13)

Note that the functions $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$ must obey
only one condition (5\. 11).

A simple algebra gives:

$\displaystyle
DM(\xi,\eta)=\frac{\delta(E-H_{0})}{\Delta_{c}}\prod_{t}d^{2}\xi
d^{2}\eta\int\prod_{t}d^{2}\bar{\xi}d^{2}\bar{\eta}$
$\displaystyle\times\delta^{2}\left(\bar{\xi}-\left(\dot{\xi}-\frac{\partial
h_{j}}{\partial\eta}\right)\right)\delta^{2}\left(\bar{\eta}-\left(\dot{\eta}+\frac{\partial
h_{j}}{\partial\xi}\right)\right)$
$\displaystyle\times\delta\left(\frac{\partial
r_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
r_{c}}{\partial\eta}\cdot\bar{\eta}+\\{r_{c},h_{j}\\}-\frac{\partial
H_{j}}{\partial p_{c}}\right)$ $\displaystyle\times\delta\left(\frac{\partial
p_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
p_{c}}{\partial\eta}\cdot\bar{\eta}+\\{p_{c},h_{j}\\}+\frac{\partial
H_{j}}{\partial r_{c}}\right)$
$\displaystyle\times\delta\left(\frac{\partial\varphi_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial\varphi_{c}}{\partial\eta}\cdot\bar{\eta}+\\{\varphi_{c},h_{j}\\}-\frac{\partial
H_{j}}{\partial l_{c}}\right)$ $\displaystyle\times\delta\left(\frac{\partial
l_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial
l_{c}}{\partial\eta}\cdot\bar{\eta}+\\{l_{c},h_{j}\\}+\frac{\partial
H_{j}}{\partial\varphi_{c}}\right).$ (5\. 14)

The Poisson notation:

$\\{X,h_{j}\\}=\frac{\partial X}{\partial\xi}\frac{\partial
h_{j}}{\partial\eta}-\frac{\partial X}{\partial\eta}\frac{\partial
h_{j}}{\partial\xi}$

was introduced in (5\. 14).

Next, the ”auxiliary” quantity $h_{j}$ have been introduced by following
equalities:

$\displaystyle\\{r_{c},h_{j}\\}-\frac{\partial H_{j}}{\partial
p_{c}}=0,~{}\\{p_{c},h_{j}\\}+\frac{\partial H_{j}}{\partial r_{c}}=0,$
$\displaystyle\\{\varphi_{c},h_{j}\\}-\frac{\partial H_{j}}{\partial
l_{c}}=0,~{}\\{l_{c},h_{j}\\}+\frac{\partial H_{j}}{\partial\varphi_{c}}=0.$
(5\. 15)

Then the functional determinant $\Delta_{c}$ is canceled and

$DM(\xi,\eta)=\delta(E-H_{0})\prod_{t}d^{2}\xi
d^{2}\eta\delta^{2}(\dot{\xi}-\frac{\partial
h_{j}}{\partial\eta})\delta^{2}(\dot{\eta}+\frac{\partial
h_{j}}{\partial\xi}),$ (5\. 16)

It is the desired result of transformation of the measure for given generating
functions $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$. In this case the
”Hamiltonian” $h_{j}(\xi,\eta)$ is defined by four equations (5\. 15).

But there is another possibility. Let us assume that

$h_{j}(\xi,\eta)=H_{j}(r_{c},p_{c},\varphi_{c},l_{c})$ (5\. 17)

and the functions $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$ are unknown.
Then eqs.(5\. 15) are the equations for this functions. It is not hard to see
that the eqs.(5\. 15) simultaneously with equations fixed by
$\delta$-functions in (5\. 16) are equivalent of incident equations if the
equality (5\. 17) is hold. Indeed, for example,

$\dot{r}_{c}=\frac{\partial r_{c}}{\partial\xi}\cdot\dot{\xi}+\frac{\partial
r_{c}}{\partial\eta}\cdot\dot{\eta}=\\{r_{c},h_{j}\\}=\frac{\partial
H_{j}}{\partial p_{c}},$ (5\. 18)

where (5\. 16) and (5\. 15) was used successively.

So, incident dynamical problem was divided on two parts. First one defines the
trajectory in the $W$ space through eqs.(5\. 15). Second one defines the
dynamics, i.e. the time dependence, through the equations fixed by
$\delta$-functions in the measure (5\. 16).

Therefore, we should consider $r_{c},~{}p_{c},~{}\varphi_{c},~{}l_{c}$ as the
solutions in the $\xi,~{}\eta$ parametrization. The desired parametrization of
classical orbits has the form (one can find it in arbitrary textbook of
classical mechanics):

$r_{c}=\frac{\eta_{1}^{2}(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}}{(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}+\eta_{2}\cos\xi_{1}},~{}p_{c}=\frac{\eta_{2}\sin\xi_{1}}{\eta_{1}(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}},~{}\varphi_{c}=\xi_{1},~{}l_{c}=\eta_{1},$
(5\. 19)

i.e. $r_{c}$ and $p_{c}$ are $\xi_{2}$ independent. At the same time,

$h_{j}=\frac{1}{2(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}}-j_{r}r_{c}-j_{\varphi}\xi_{1}\equiv
h(\eta)-j_{r}r_{c}-j_{\varphi}\xi_{1}.$ (5\. 20)

Noting that the derivatives of $h_{j}$ over $\xi_{2}$ are equal to
zero292929To have the condition (5\. 11) we should assume that $\partial
r_{c}/\partial\xi_{2}\sim\epsilon\neq 0$. We put $\epsilon=0$ completing the
transformation. we find that

$DM(\xi,\eta)=\delta(E-h(T))\prod_{t}d^{2}\xi
d^{2}\eta\delta\left(\dot{\xi}_{1}-\omega_{1}+j_{r}\frac{r_{c}}{\partial\eta_{1}}\right)$
$\times\delta\left(\dot{\xi}_{2}-\omega_{2}+j_{r}\frac{r_{c}}{\partial\eta_{2}}\right)\delta\left(\dot{\eta}_{1}-j_{r}\frac{\partial
r_{c}}{\partial\xi_{1}}-j_{\varphi}\right)\delta(\dot{\eta}_{2}),$ (5\. 21)

where

$\omega_{i}=\partial h/\partial\eta_{i}$ (5\. 22)

are the conserved in classical limit $j_{r}=j_{\varphi}=0$ ”velocities” in the
$W$ space.

###  5.3 Reduction

We see from (5\. 21) that the length of Runge-Lentz vector is not perturbated
by the quantum forces $j_{r}$ and $j_{\varphi}$. To investigate the
consequence of this fact it is useful to project this forces on the axis of
$W$ space. This means splitting of $j_{r},~{}j_{\varphi}$ on
$j_{\xi},~{}j_{\eta}$. The equality

$\prod_{t}\delta\left(\dot{\xi}_{1}-\omega_{1}+j_{r}\frac{r_{c}}{\partial\eta_{1}}\right)=e^{\frac{1}{2i}\int^{T}_{0}dt\hat{j}_{\xi_{1}}\hat{e}_{\xi_{1}}}e^{2i\int^{T}_{0}dtj_{r}e_{\xi_{1}}\partial
r_{c}/\partial\eta_{1}}\prod_{t}\delta(\dot{\xi}_{1}-\omega_{1}+j_{\xi_{1}})$

becomes evident if the Fourier representation of $\delta$-function is used
(see also [26]). The same transformation of arguments of other
$\delta$-functions in (5\. 21) can be applied. Then, noting that the last
$\delta$-function in (5\. 21) is source-free, we find the same representation
as (5\. 4) with

$\hat{\mathbb{K}}(j,e)=\int^{T}_{0}dt(\hat{j}_{\xi_{1}}\hat{e}_{\xi_{1}}+\hat{j}_{\xi_{2}}\hat{e}_{\xi_{2}}+\hat{j}_{\eta_{1}}\hat{e}_{\eta_{1}}),$
(5\. 23)

where the operators $\hat{j}$ are defined by the equality:

$\hat{j}_{X}(t)=\int^{T}_{0}dt^{\prime}\Theta(t-t^{\prime})\hat{X}(t^{\prime})$
(5\. 24)

and $\Theta(t-t^{\prime})$ is the Green function of our perturbation theory
[26].

We should change also

$e_{r}\rightarrow e_{c}=e_{\eta_{1}}\frac{\partial
r_{c}}{\partial\xi_{1}}-e_{\xi_{1}}\frac{\partial
r_{c}}{\partial\eta_{1}}-e_{\xi_{2}}\frac{\partial
r_{c}}{\partial\eta_{2}},~{}~{}e_{\varphi}\rightarrow e_{\xi_{1}}$ (5\. 25)

in the Eq.(5\. 9). The differential measure takes the simplest form:

$DM(\xi,\eta)=\delta(E-h(T))\prod_{t}d^{2}\xi
d^{2}\eta\delta(\dot{\xi}_{1}-\omega_{1}-j_{\xi_{1}})\delta(\dot{\xi}_{2}-\omega_{2}-j_{\xi_{2}})$
$\times\delta(\dot{\eta}_{1}-j_{\eta_{1}})\delta(\dot{\eta}_{2}).$ (5\. 26)

Note now that the $\xi,\eta$ variables are contained in $r_{c}$ only:

$r_{c}=r_{c}(\xi_{1},\eta_{1},\eta_{2}).$

This means that the action of the operator $\hat{j}_{\xi_{2}}$ gives identical
to zero contributions into perturbation theory series. And, since
$\hat{e}_{\xi_{2}}$ and $\hat{j}_{\xi_{2}}$ are conjugate operators, see (5\.
23), we can put

$j_{\xi_{2}}=e_{\xi_{2}}=0.$

This conclusion ends the reduction:

$\hat{\mathbb{K}}(j,e)=\int^{T}_{0}dt(\hat{j}_{\xi_{1}}\hat{e}_{\xi_{1}}+\hat{j}_{\eta_{1}}\hat{e}_{\eta_{1}}),$
(5\. 27) $e_{c}=e_{\eta_{1}}\frac{\partial
r_{c}}{\partial\xi_{1}}-e_{\xi_{1}}\frac{\partial r_{c}}{\partial\eta_{1}}.$
(5\. 28)

The measure has the form:

$DM(\xi,\eta)=\delta(E-h(T))d\xi_{2}(0)d\eta_{2}(0)\prod_{t}d\xi_{1}d\eta_{1}\delta(\dot{\xi}_{1}-\omega_{1}-j_{\xi_{1}})\delta(\dot{\eta}_{1}-j_{\eta_{1}})$
(5\. 29)

since $V=V(r_{c},e_{c},\xi_{1})$ is $\xi_{2}$ independent and

$\int\prod_{t}dX(t)\delta(\dot{X})=\int dX(0).$

###  5.4 Perturbations

One can see from (5\. 29) that the reduction can not solve the H-atom problem
completely: there are nontrivial corrections to the orbital degrees of freedom
$\xi_{1},\eta_{1}$. By this reason we should consider the expansion over
$\hat{\mathbb{K}}$.

Using last $\delta$-functions in (5\. 29) we find, see also [26] (normalizing
$\rho(E)$ on the integral over $\xi_{2}(0)\eta_{2}(0)$):

$\rho(E)=\int^{\infty}_{0}dTe^{-i\hat{\mathbb{K}}(j,e)}\int
dMe^{-iU(r_{c},e)},$ (5\. 30)

where

$dM=\frac{d\xi_{1}d\eta_{1}}{\omega_{2}(E)}.$ (5\. 31)

The operator $\hat{\mathbb{K}}(j,e)$ was defined in (5\. 27) and

$U(r_{c},e_{c})=-s_{0}(r)+$
$+\int^{T}_{0}dt[\frac{1}{((r_{c}+e_{c})^{2}+r_{c}^{2}e_{\xi_{1}}^{2})^{1/2}}-\frac{1}{((r_{c}-e_{c})^{2}+r_{c}^{2}e_{\xi_{1}}^{2})^{1/2}}+2\frac{e_{c}}{r_{c}}]$
(5\. 32)

with $e_{c},~{}e_{\xi_{1}}$ was defined in (5\. 28), (5\. 25) and

$r_{c}(t)=r_{c}(\eta_{1}+\eta(t),\bar{\eta}_{2}(E,T),\xi_{1}+\omega_{1}(t)+\xi(t)),~{}~{}E\equiv
h(\eta_{1}+\eta(T),\bar{\eta}_{2}),$ (5\. 33)

where $\bar{\eta}_{2}(E,T)$ is the solution of equation $E=h$.

The integration range over $\xi_{1}$ and $\eta_{1}$ is as follows:

$0\leq\xi_{1}\leq 2\pi,~{}~{}-\infty\leq\eta_{1}\leq+\infty.$ (5\. 34)

First inequality defines the principal domain of the angular variable
$\varphi$ and second ones take into account the clockwise and anticlockwise
motions of particle on the Kepler orbits.

We can write:

$\rho(E)=\int^{\infty}_{0}dT\int dM:e^{-iV(r_{c},\hat{e})}:$ (5\. 35)

since the operator $\ln\hat{\mathbb{K}}$ is linear over
$\hat{e}_{\xi_{1}},\hat{e}_{\eta_{1}}$. The colons means ”normal product” with
differential operators staying to the left of functions and $U(r_{c},\hat{e})$
is the functional of operators:

$2i\hat{e}_{c}=\hat{j}_{\eta_{1}}\frac{\partial
r_{c}}{\partial\xi_{1}}-\hat{j}_{\xi_{1}}\frac{\partial
r_{c}}{\partial\eta_{1}},~{}~{}2i\hat{e}_{\xi_{1}}=\hat{j}_{\xi_{1}}.$ (5\.
36)

Expanding $U(r_{c},\hat{e})$ over $\hat{e}_{c}$ and $\hat{e}_{\eta_{1}}$ we
find:

$U(r_{c},\hat{e})=-s_{0}(r_{c})+2\sum_{n+m\geq
1}C_{n,m}\int^{T}_{0}dt{\hat{e}_{c}^{2n+1}\hat{e}_{\eta_{1}}^{m}}\frac{1}{r_{c}^{2n+2}},$
(5\. 37)

where $C_{n,m}$ are the numerical constants. We see that the interaction part
presents expansion over $1/r_{c}$ and, therefore, the expansion over $U$
generates an expansion over $1/r_{c}$.

In result, see Sec.4.5,

$\rho(E)=\int^{\infty}_{0}dT\int
dM\\{e^{is_{0}(r_{c})}+B_{\xi_{1}}(\xi_{1},\eta_{1})+B_{\eta_{1}}(\xi_{1},\eta_{1})\\}.$
(5\. 38)

The first term is the pure semiclassical contribution and last ones are the
quantum corrections. The functionals $B$ are the total derivatives:

$B_{\xi_{1}}(\xi_{1},\eta_{1})=\frac{\partial}{\partial\xi_{1}}b_{\xi_{1}}(\xi_{1},\eta_{1}),~{}~{}B_{\eta_{1}}(\xi_{1},\eta_{1})=\frac{\partial}{\partial\eta_{1}}b_{\eta_{1}}(\xi_{1},\eta_{1}).$
(5\. 39)

This means that the mean value of quantum corrections in the $\xi_{1}$
direction are equal to zero:

$\int^{2\pi}_{0}d\xi_{1}\frac{\partial}{\partial\xi_{1}}b_{\xi_{1}}(\xi_{1},\eta_{1})=0$
(5\. 40)

since $r_{c}$ is the closed trajectory independently from initial conditions,
see (5\. 19).

In the $\eta_{1}$ direction the motion is classical:

$\int^{+\infty}_{-\infty}d\eta_{1}\frac{\partial}{\partial\eta_{1}}b_{\eta_{1}}(\xi_{1},\eta_{1})=0$
(5\. 41)

since (i) $b_{\eta_{1}}$ is the series over $1/r_{c}^{2}$ and (ii)
$r_{c}\rightarrow\infty$ when $|\eta_{1}|\rightarrow\infty$. Therefore,

$\rho(E)=\int^{\infty}_{0}dT\int dMe^{is_{0}(r_{c})}.$ (5\. 42)

This is the desired result.

Noting that

$s_{0}(r_{c})=kS_{1}(E),~{}~{}k=\pm 1,\pm 2,...,$

where $S_{1}(E)$ is the action over one classical period $T_{1}$:

$\frac{\partial S_{1}(E)}{\partial E}=T_{1}(E),$

and using the identity [22]:

$\sum^{+\infty}_{-\infty}e^{inS_{1}(E)}=2\pi\sum^{+\infty}_{-\infty}\delta(S_{1}(E)-2\pi
n),$

we find:

$\rho(E)=\pi\Omega\sum_{n}\delta(E+1/2n^{2})$ (5\. 43)

where $\Omega$ is the zero-modes volume.

###  5.5 Conclusions

The demonstrated above mechanism of reduction is universal: one can introduce
from the very beginning the arbitrary number of coordinates $(\xi,\eta)$. But
later on the formalism automatically, through dependence of classical
trajectory on coordinates of $W$, will extract the necessary set of variables
$(\xi,\eta)$. At the same time $\dim(\xi,\eta)=\dim W$ and the integrals over
other ones will give the volume

$V_{0}=\int\prod d\xi(0)d\eta(0),$

see (5\. 29) where $\dim V_{0}=2$.

Notice that appearance of the ”0-dimensional” integral measure

$d\xi_{2}(0)d\eta_{2}(0)$

in (5\. 29) reflects the hidden $O(4)$ symmetry of H-atom problem [23].
Therefore, following our selection rule, we must consider in a first place the
classical trajectory which leads to the maximal value of $\dim V_{0}$, i.e. we
must consider the contributions with maximal number of zero modes.

##  6 Example: sin-Gordon model

###  6.1 Introduction

First of all we will describe ”canonical” transformation in the path-integral
formalism. The method of canonical transformations in spite of its expected
effectiveness is unpopular in quantum theories since on this way exist the
problem: it is necessary to find the transformation from Lagrangian to
Hamiltonian descriptions. This transition in general is very difficult if
$\varphi(x)$ and $\dot{\varphi}(x)=p(x)$ are not the independent quantities
[13]. But we may use following trick. We start from the simplest verse of the
canonical formalism introducing the ”first-order” description303030In other
words, we will still stay in the frame of Lagrangian formalism. and after
transformation come to independent canonically conjugate pares, $(\xi,\eta)$,
i.e. come to Hamiltonian description. It is evident that in general the
transformation

$\varphi_{c}:(\varphi,p)\rightarrow(\xi,\eta)$

will not be canonical. The formalism of present Section is the same as in the
H-atom problem but there is some distinction.

We will continue in this Section description of influence of the phase-space
structure on the result of quantum-mechanical measurements started in previous
Sections. Now we will calculate the expectation value of the ”order parameter”
(mass-shell particles production vertex) $\Gamma(q;u)$ [29]:

$\rho(q)=<|\Gamma(q;u)|^{2}>_{u},$

where $q$ is the mass-shell ($q^{2}=m^{2}$) particles momentum and $<>_{u}$
means averaging over the field $u(x,t)$. Just the procedure of averaging would
be the object of our interest considering the quantum Hamiltonian system with
symmetry $G$. By definition, $\rho$ is the $probability$ to find one mass-
shell particle. Certainly, $\rho(q)=0$ on the sourceless vacuum but, generally
speaking, $\rho(q)\neq 0$ in a field with nonzero energy density.

Calculations will be illustrated by the integrable (1+1)-dimensional model
with non-polynomial Lagrangian

$L=\frac{1}{2}(\partial_{\mu}u)^{2}+\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda
u)-1],$ (6\. 1)

We will consider following formulation of the problem. Formally nothing
prevents to linearize partly our problem considering the Lagrangian

$L=\frac{1}{2}[(\partial_{\mu}u)^{2}-m_{h}^{2}u^{2}]+\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda
u)-1+\frac{\lambda^{2}}{2}u^{2}]\equiv L_{0}(u)-v(u)$ (6\. 2)

to describe creation (and absorption) of the mass $m_{h}$ particles. Then the
last term in (6\. 2),

$v(u)=-\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda
u)-1+\frac{\lambda^{2}}{2}u^{2}],$ (6\. 3)

describes interactions. The corresponding to this theory order parameter is

$\Gamma(q;u)=\int
dxdte^{iqx}(\partial^{2}+m_{h}^{2})u(x,t),~{}~{}~{}q^{2}=m_{h}^{2}.$ (6\. 4)

It will be shown by explicit calculations that

$\rho(q)=0$ (6\. 5)

as the consequence of unbroken $\tilde{sl}(2,C)$ Kac-Moody algebra on which
the solitons of theory (6\. 1) live313131Trivialness of soliton $S$-matrix was
shown in [30], see e.g. [31] and references cited therein323232 It may be
useful at this point to compare our approach with ordinary thermodynamics of
ferromagnetic. The external magnetic field is $\sim<{\mu}>$, where the order
parameter $<\mu>$ is the mean value of the spin, and the phase transition
means that $<\mu>\neq 0$, i.e. $<\mu>=0$ means that corresponding symmetry
stay unbroken. We will suppose that the mean value of $|\Gamma(q,u)|^{2}$,
which is the function of external fields parameter $q$, play the same role for
field theories with symmetry, i.e. $<|\Gamma(q,u)|^{2}>_{u}=0$ means that
corresponding symmetry stay unbroken. Therefore in our approach only the
”external” display of symmetry can be described.. The solution (6\. 5) seems
interesting since it can be interpreted as the explicit demonstration of field
$u(x,t)$ confinement. The main purpose of this paper is to investigate how the
solution (6\. 5) appears.

We will be able to find exact equality (6\. 5) since the model (6\. 1) possess
infinite number of integrals of motion. It is well known that each integral of
motion in involution allows to shrink a number of phase space $\bar{\gamma}$
variables on two units, see e.g. [12]. Resulting phase space $\gamma$ is
called as the reduced phase space [25]. The summation over all reduced phase
space topological classes [27] is assumed.

By this way the field-theoretical problem will reduced to the quantum-
mechanical one. We would consider $\eta$ as the ”particles” generalized
momentum and would introduce $\xi$ as the conjugate to $\eta$ coordinate of
soliton. The $2N$-dimensional phase space (cotangent manifold) $\gamma_{N}$
with local coordinates $(\xi,\eta)$ on it has natural simplectic structure,
and $DM(\gamma_{N})=D^{N}M(\xi,\eta)$ in practical calculations (see
Subsec.6.2). The summation over $N$ is assumed.

The quantum corrections to semiclassical approximation of transformed theory
are simply calculable since $\eta$ are conserved in the classical limit. This
is the particularity of solitons dynamics (solitons momenta is the conserved
quantities). One can consider the developed in this paper formalism as the
path-integral version of nonlinear waves (solitons in our case) quantum theory
(the canonical quantization of sin-Gordon model in the soliton sector was
described also in [14].)

In Subsec.6.3 we will demonstrate Eq.(6\. 5). It will be shown that this
solution is consequence of the previously developed proposition (we would
justify it in Subsec.6.2) that the semiclassical approximation is exact for
sin-Gordon model [11]. The semiclassical approximation in the $\gamma_{N}$
phase space will be considered in Subsec.6.2.

We would not use the complicated algebra to show the reduction procedure
explicitly noting that all solutions of model (6\. 1) are known [24]. Then,
using the $\delta$-likeness of measure $DM(\tilde{\gamma})$, we will find in
Subsec.6.2 $DM(\gamma_{N})$ considering the mapping as an ordinary
transformation to useful variables333333We will apply inverse reduction
procedure. Let $G$ be a group of canonical transformations acting on the
simplectic manifold $\tilde{\gamma}$ and let $\bar{G}$ be the Lie algebra of
$G$ with $G^{*}$ dual of it. Then the momentum [32] mapping
$J:~{}\tilde{\gamma}\rightarrow G^{*}$ introduces the integrals of motion
which reduces the $\tilde{\gamma}$ manifold. Noting that the set of levels
$J^{-1}({\eta})$ (solution of equations $J(\pi)=\eta$, $\pi\in\tilde{\gamma}$)
is a manifold then $\gamma_{\eta}=J^{-1}({\eta})/\bar{G}_{\eta}$ is the
reduced phase space, where $\bar{G}_{\eta}$ is the co-adjoint isotropy
subgroup of $G$. Therefore, the differential measure
$dM=dM(\eta,\gamma_{\eta})$ for reduced phase space. For integrable mechanical
systems (infinite dimensional as well, see e.g. [24]) $\gamma_{\eta}$ shrinks
to the point and in this case $dM=dM(\eta)$ is the measure of momentum
manifold. Just this simplest case would be considered working with Lagrangian
(6\. 1) and more general and interesting case with measure
$DM=DM(\eta,\gamma_{\eta})$, $\gamma_{\eta}\neq\emptyset$, will be considered
later. So, the reduction procedure of our Hamiltonian system with symmetry $G$
looks like canonical transformation [31]. This problem is nontrivial since,
generally speaking, $\dim\tilde{\gamma}$ and $\dim\gamma$ are not the same for
model (6\. 1).. Corresponding perturbation theory, see Subsec.6.3, in the
momentum space $J$ was described in [26]. In Subsec.6.2 the path-integral
definition of $\rho(q)$ will be given.

We would conclude (this is the main result) that a theory in the ”nonlinear
waves” sector may be nontrivial ($\rho\neq 0$) iff the manifold $\gamma$ is
not compact.

###  6.2 Reduction procedure

6.2.1. Introduction into formalism.

Our aim is to calculate the integral:

$\rho(q)=e^{-i\hat{\mathbb{K}}(j,e)}\int
DM(u,p)|\Gamma(q;u)|^{2}e^{iS_{O}(u)-iU(u,e)},$ (6\. 6)

where $\Gamma(q;u)$ was defined in (6\. 4). In this expression the expansion
over operator

$\hat{\mathbb{K}}(j,e)={\rm Re}\int_{C_{+}}dxdt\frac{\delta}{\delta
j(x,t)}\frac{\delta}{\delta e(x,t)}\equiv{\rm
Re}\int_{C_{+}}dxdt\hat{j}(x,t)\hat{e}(x,t)$ (6\. 7)

generates the perturbation theory series. We will assume that this series
exist. The functionals $U(u,e)$ and $S_{O}(u)$ are defined by the equalities:

$\displaystyle V(u+e)-V(u-e)=U(u,e)+\int dxdte(x,t)v^{\prime}(u),$
$\displaystyle S_{0}(u+e)-S_{0}(u-e)=S_{O}(u)+\int
dxdte(x,t)(\partial^{2}+m_{h}^{2})u(x,t).$ (6\. 8)

The action $S_{0}(u)$ corresponds to the free part of Lagrangian (6\. 1) and
$V(u)$ describes interactions. The quantity $S_{O}(u)$ is not equal to zero
since the soliton configurations have nontrivial topological charge (see also
[1]). All time integrals in this expressions were defined on the Mills time
contour [17]:

$2{\rm Re}\int_{C_{+}}=\int_{C_{+}}+\int_{C_{-}}$

and

$C_{\pm}:t\rightarrow t\pm
i\epsilon,~{}~{}~{}\epsilon\rightarrow+0,~{}~{}~{}-\infty\leq t\leq+\infty,$

to avoid the possible light-cone singularities of the perturbation theory. The
variational derivatives in (6\. 7) are defined by the following way:

$\frac{\delta u(x,t\in C_{i})}{\delta u(x^{\prime},t^{\prime}\in
C_{j})}=\delta_{ij}\delta(x-x^{\prime})\delta(t-t^{\prime}),~{}~{}~{}i,j=+,-.$

The auxiliary variables $(j,e)$ must be taken equal to zero at the very end of
calculations.

Considering the first order formalism with new coordinates $(u,p)$ the measure
$DM(u,p)$ has the form:

$DM(u,p)=\prod_{x,t}du(x,t)dp(x,t)\delta\left(\dot{u}-\frac{\delta
H_{j}(u,p)}{\delta p}\right)\delta\left(\dot{p}+\frac{\delta
H_{j}(u,p)}{\delta u}\right)$ (6\. 9)

with the total ”Hamiltonian”

$H_{j}(u,p)=\int
dx\left\\{\frac{1}{2}p^{2}+\frac{1}{2}(\partial_{x}u)^{2}-\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda
u)-1]-ju\right\\}.$ (6\. 10)

The problem will be considered assuming that $u(x,t)$ belongs to Schwartz
space:

$u(x,t)|_{|x|=\infty}=0~{}({\rm{mod}}\frac{2\pi}{\lambda}).$ (6\. 11)

This means that $u(x,t)$ tends to zero $(\rm{mod}\frac{2\pi}{\lambda})$ at
$|x|\rightarrow\infty$ faster then any power of $1/|x|$. Note that
$\dot{u}=p$, i.e. $u$ and $p$ are not the independent quantities.

The measure (6\. 9) allows to perform arbitrary transformations. But, as was
explained in Introduction, we will use the analog of canonical transformation
which conserves the form of equations of motion. Hence, it is sufficient on
this stage of calculations to know only the fact that this transformation
exist [24]. One may propose that in result we should find for $N$-soliton
topology:

$D^{N}M(\xi,\eta)=\prod_{t}d^{N}\xi(t)d^{N}\eta(t)\delta^{(N)}\left(\dot{\xi}-\frac{\partial
h_{j}(\xi,\eta)}{\partial\eta(t)}\right)\delta^{(N)}\left(\dot{\eta}+\frac{\partial
h_{j}(\xi,\eta)}{\partial\xi(t)}\right),$ (6\. 12)

where $h_{j}$ is the ”transformed Hamiltonian”:

$h_{j}=h_{N}(\eta)-\int dxj(x,t)u_{N}(x;\xi,\eta)$ (6\. 13)

and $u_{N}(x;\xi,\eta)$ is the $N$-soliton configuration the time dependence
of which is parameterized by $(\xi,\eta)$. Therefore, the local coordinates
$(\xi,\eta)$ are defined by the equations:

$\dot{\xi}=\frac{\partial
h_{j}}{\partial\eta},~{}~{}~{}\dot{\eta}=-\frac{\partial h_{j}}{\partial\xi},$
(6\. 14)

where $h_{j}$ must obey the Poisson conditions343434See previous Section:

$\\{u_{c}(x,t),h_{j}\\}=\frac{\delta H_{j}}{\delta
p_{c}(x,t)},~{}~{}~{}\\{p_{c}(x,t),h_{j}\\}=-\frac{\delta H_{j}}{\delta
u_{c}(x,t)}.$ (6\. 15)

One can see choosing

$h_{j}(\xi,\eta)=H_{j}(u_{c},p_{c})$ (6\. 16)

that the initial equations have been restored:

$\dot{u}_{c}=\frac{\partial u_{c}}{\partial\xi}\dot{\xi}+\frac{\partial
u}{\partial\eta}\dot{\eta}=\\{u_{c},h_{j}\\}=\frac{\delta H_{j}}{\delta
p_{c}}.$

The same we will have for $\dot{p}_{c}$. Therefore $(u_{c},p_{c})$ are
solutions of equations of motion (6\. 14), if the equality (6\. 16) is hold.

The field theory case in $(1+1)$-dimensional configuration space needs
additional explanations. First of all, the analog of (5\. 10) must be
introduced:

$\Delta(u,p)=\int\prod_{t}d^{N}\xi(t)d^{N}\eta(t)\prod_{x,t}\delta(u(x,t)-u_{c}(x;\xi,\eta))\delta(p(x,t)-p_{c}(x;\xi,\eta))$
(6\. 17)

if the $N$-soliton configuration is considered. Notice that the one-
dimensional $\delta$-functions are introduced in (6\. 17) and $u_{c}$, $p_{c}$
are the functions of sets $(\xi,\eta)$, $\dim(\xi,\eta)=2N$. Introducing (6\.
17) we make the attempt to ”hide” the time dependence entirely into the set of
$independent$ variables $(\xi,\eta)$.

Comparing (6\. 9) and (6\. 12) one can note that $x$ dependence disappeared
and the transformed measure depends on the number $N=1,2,...$ Therefore,
occurs the reduction of the quantum degrees of freedom since the power of the
coordinate set is continuum and the number of solitons $N$ is the countable
set. This means that the proposed transformation to coordinates of solitons
will be unavoidably singular.

Notice then that the $x$ dependence of $\Delta(u,p)$ remain unimportant since
last one always appear under the integrals over all $u(x,t)$ and $p(x,t)$. At
the same time it is important that introduced in previous Section $\Delta_{c}$
disappeared in the final result, if the integral form of Poisson brackets (6\.
15) are hold353535 See the transformation (5\. 12), described in previous
Section. For more confidence one can introduce the appropriate cells in the
$x$ space [24]..

One can try to propose also the local form of canonical commutators (6\. 15),
if the definition (6\. 16) is hold. Indeed, one can find inserting (6\. 16)
into (6\. 15) that:

$\\{u_{c}(x,t),H_{j}(u_{c},p_{c})\\}=\frac{\delta H_{j}(u_{c},p_{c})}{\delta
p_{c}(x,t)},~{}~{}~{}\\{p_{c}(x,t),H_{j}(u_{c},p_{c})\\}=-\frac{\delta
H_{j}(u_{c},p_{c})}{\delta u_{c}(x,t)}.$ (6\. 18)

This equalities must hold for arbitrary $j$. Making use the definition:

$H_{j}(x_{c},p_{c})=\int dy\tilde{H}_{j}(x_{c},p_{c}),$

where $\tilde{H}_{j}$ is the Hamiltonian density, one can write from (6\. 18):

$\int
dy\\{u_{c}(x;\xi,\eta),u_{c}(y;\xi,\eta)\\}\frac{\delta\tilde{H}_{j}}{\delta
u_{c}(y,t)}+$ $+\int
dy(\\{u_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}-\delta(x-y))\frac{\delta\tilde{H}_{j}}{\delta
p_{c}(y,t)}=0$

and

$\int
dy\\{p_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}\frac{\delta\tilde{H}_{j}}{\delta
p_{c}(y,t)}-$ $-\int
dy(\\{u_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}-\delta(x-y))\frac{\delta\tilde{H}_{j}}{\delta
u_{c}(y,t)}=0.$

Then one can propose the solutions of these equations:

$\\{u_{c}(x;\xi,\eta),u_{c}(y;\xi,\eta)\\}=\\{p_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}=0,$
$\\{u_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}=\delta(x-y).$ (6\. 19)

But it is interesting that the local commutators (6\. 19) are not
satisfied363636That circumstances was mentioned firstly by V.Voronyuk.. One
can see this inserting the soliton solution into (6\. 19). On the other hand
the integral form (6\. 18) is satisfied. All this means that $u_{c}$ and
$p_{c}$ are not the completely independent variables. It must be stressed that
the local relations (6\. 19) are not the necessary conditions in our
formalism.

In our terms, the quantum force $j(x,t)$ excites the $(\xi,\eta)$ manifold
only, leaving the topology of classical trajectory $(u,p)_{c}$ unchanged. We
can use them immediately since the complete set of canonical coordinates
$(\xi,\eta)$ of sin-Gordon model is known, see e.g. [24].

6.2.3. Perturbation theory on the cotangent bundle.

The classical Hamiltonian $h_{j}$ is the sum:

$h_{j}(\eta)=\int
dp\sigma(r)\sqrt{r^{2}+m_{h}^{2}}+\sum^{N}_{i=1}h(\eta_{i}),$ (6\. 20)

where $\sigma(r)$ is the continuous spectrum and $h(\eta)$ is the soliton
energy. Note absence of interaction energy among solitons.

New degrees of freedom $(\xi,\eta)(t)$ must obey the equations (6\. 14):

$\dot{\xi}_{i}=\Omega(\eta_{i})-\int dxj(x,t)\frac{\partial
u_{N}(x;\xi,\eta)}{\partial\eta_{i}},~{}~{}~{}\Omega(\eta)\equiv\frac{\partial
h(\eta)}{\partial\eta},$ $\dot{\eta}_{i}=\int dxj(x,t)\frac{\partial
u_{N}(\xi,\eta)}{\partial\xi_{i}}.$ (6\. 21)

Hence the sources of quantum perturbations are proportional to the time-local
fluctuations of soliton configurations

$\frac{\partial u_{N}(x;\xi,\eta)}{\partial\eta_{i}},~{}~{}~{}\frac{\partial
u_{N}(x;\xi,\eta)}{\partial\xi_{i}}.$

One can split the Lagrange source onto ”Hamiltonian” ones:

$j(x,t)\rightarrow(j_{\xi},j_{\eta}).$

This gives weight functional $U(u_{N};e_{\xi},e_{\eta})$ and operator
$\hat{\mathbb{K}}(e_{\xi},e_{\eta};j_{\xi},j_{\eta})$. In result:

$\displaystyle\rho(q)=\sum_{N}e^{-i\hat{K}(e_{\xi},e_{\eta};j_{\xi},j_{\eta})}\int
D^{N}M(\xi,\eta)e^{iS_{O}(u_{N})}e^{-iU(u_{N};e_{\xi},e_{\eta})}\times$
$\displaystyle\times|\Gamma(q;u_{N})|^{2}$ (6\. 22)

where, using vector notations,

$\hat{\mathbb{K}}(e_{\xi},e_{\eta};j_{\xi},j_{\eta})=\frac{1}{2}\int
dt\\{\hat{j}_{\xi}(t)\cdot\hat{e}_{\xi}(t)+\hat{j}_{\eta}(t)\cdot\hat{e}_{\eta}(t)\\}.$
(6\. 23)

The measure takes the form:

$D^{N}M(\xi,\eta)=\prod^{N}_{i=1}\prod_{t}d\xi_{i}(t)d\eta_{i}(t)\delta(\dot{\xi}_{i}-\Omega(\eta_{i})-j_{\xi,i}(t))\delta(\dot{\eta}_{i}-j_{\eta,i}(t))$
(6\. 24)

The effective interaction potential

$U(u_{N};e_{\xi},e_{\eta})=-\frac{2m^{2}}{\lambda^{2}}\int dxdt\sin\lambda
u_{N}~{}(\sin\lambda e-\lambda e)$ (6\. 25)

with

$e(x,t)=e_{\xi}(t)\cdot\frac{\partial
u_{N}(x;\xi,\eta)}{\partial\eta(t)}-e_{\eta}(t)\cdot\frac{\partial
u_{N}(x;\xi,\eta)}{\partial\xi(t)}.$ (6\. 26)

Performing the shifts:

$\displaystyle\xi_{i}(t)\rightarrow\xi_{i}(t)+\int
dt^{\prime}g(t-t^{\prime})j_{\xi,i}(t^{\prime})\equiv\xi_{i}(t)+\xi^{\prime}_{i}(t),$
$\displaystyle\eta_{i}(t)\rightarrow\eta_{i}(t)+\int
dt^{\prime}g(t-t^{\prime})j_{\eta,i}(t^{\prime})\equiv\eta_{i}(t)+\eta^{\prime}_{i}(t),$
(6\. 27)

we can move the Green function $g(t-t^{\prime})$ into the operator:

$\hat{\mathbb{K}}(e_{\xi},e_{\eta};{\xi}^{\prime},{\eta^{\prime}})=\frac{1}{2}\int
dtdt^{\prime}g(t-t^{\prime})\\{\hat{\xi}^{\prime}(t^{\prime})\cdot\hat{e}_{\xi}(t)+\hat{\eta}^{\prime}(t^{\prime})\cdot\hat{e}_{\eta}(t)\\}.$
(6\. 28)

Notice that the Green function $g(t-t^{\prime})$ of eqs.(6\. 21) is again the
step function:

$g(t-t^{\prime})=\Theta(t-t^{\prime})$ (6\. 29)

Its imaginary part is equal to zero for real times and this allows to shift
$C_{\pm}$ to the real-time axis (see [26]).

In result:

$D^{N}M(\xi,\eta)=\prod^{N}_{i=1}\prod_{t}d\xi_{i}(t)d\eta_{i}(t)\delta(\dot{\xi}_{i}-\Omega(\eta+\eta^{\prime}))\delta(\dot{\eta}_{i})$
(6\. 30)

with

$u_{N}=u_{N}(x;\xi+\xi^{\prime},\eta+\eta^{\prime}).$ (6\. 31)

The equations:

$\dot{\xi}_{i}=\Omega(\eta_{i}+\eta^{\prime}_{i})$ (6\. 32)

are trivially integrable. In quantum case $\eta^{\prime}_{i}\neq 0$ this
equation describes the motion on nonhomogeneous and anisotropic manifold. So,
the expansion over
$(\hat{\xi^{\prime}},~{}\hat{e}_{\xi},~{}\hat{\eta}^{\prime},~{}\hat{e}_{\eta})$
generates the local in time deformations of $\gamma_{N}$ manifold,
$(\xi,\eta)\in\gamma_{N}$ completely. The weight of this deformations is
defined by $U(u_{N};e_{\xi},e_{\eta})$.

Using the definition:

$\int Dx\delta(\dot{x})=\int dx(0)=\int dx_{0}$

functional integrals are reduced to the ordinary integrals over initial data
$(\xi,\eta)_{0}$. This integrals define the zero modes volume.

###  6.3 Quantum corrections

The proof of (6\. 5) we would divide on two parts. First of all we would
consider the semiclassical approximation (Sec.6.3.1) and in Sec.6.3.2. we will
show that this approximation is exact.

6.1. Introduction and definitions.

The $N$-soliton solution $u_{N}$ depends from $2N$ parameters. Half of them
$N$ can be considered as the position of solitons and other $N$ as the
solitons momentum. Generally at $|t|\rightarrow\infty$ the $u_{N}$ solution
decomposed on the single solitons $u_{s}$ and on the double soliton bound
states $u_{b}$ [24]:

$u_{N}(x,t)=\sum^{n_{1}}_{j=1}u_{s,j}(x,t)+\sum^{n_{2}}_{k=1}u_{b,k}(x,t)+O(e^{-|t|})$

We will see later that main elements of our formalism are the one soliton
$u_{s}$ and two-soliton bound state $u_{b}$ configurations. Its $(\xi,\eta)$
parameterizations, confirmed to eqs.(6\. 15), have the form:

$u_{s}(x;\xi,\eta)=-\frac{4}{\lambda}\arctan\\{\exp(m_{h}x\cosh\beta\eta-\xi)\\},~{}~{}~{}\beta=\frac{\lambda^{2}}{8}$
(6\. 33)

and

$u_{b}(x;\xi,\eta)=-\frac{4}{\lambda}\arctan\\{\tan\frac{\beta\eta_{2}}{2}\frac{m_{h}x\sinh\frac{\beta\eta_{1}}{2}\cos\frac{\beta\eta_{2}}{2}-\xi_{2}}{m_{h}x\cosh\frac{\beta\eta_{1}}{2}\sin\frac{\beta\eta_{2}}{2}-\xi_{1}}\\}.$
(6\. 34)

The $(\xi,\eta)$ parametrization of solitons individual energies $h(\eta)$
takes the form:

$h_{s}(\eta)=\frac{m_{h}}{\beta}\cosh\beta\eta,~{}~{}~{}h_{b}(\eta)=\frac{2m_{h}}{\beta}\cosh\frac{\beta\eta_{1}}{2}\sin\frac{\beta\eta_{2}}{2}\geq
0.$

The bound-states energy $h_{b}$ depends from $\eta_{1}$ and $\eta_{2}$. First
one defines inner motion of two bounded solitons and second one the bound
states center of mass motion. Correspondingly we will call this parameters as
the internal and external ones. Note that the inner motion is periodic, see
(6\. 24).

Performing last integration in (6\. 22) with measure (6\. 30) we find:

$\rho(q)=\sum_{N}\int\prod^{N}_{i=1}\\{d\xi_{0}d\eta_{0}\\}_{i}e^{-i\hat{\mathbb{K}}}e^{iS_{O}(u_{N})}e^{-iU(u_{N};e_{\xi},e_{\eta})}|\Gamma(q;u_{N})|^{2}$
(6\. 35)

where

$u_{N}=u_{N}(\eta_{0}+\eta^{\prime},\xi_{0}+\Omega(t)+\xi^{\prime}).$ (6\. 36)

and

$\Omega(t)=\int
dt^{\prime}\Theta(t-t^{\prime})\Omega(\eta_{0}+\eta^{\prime}(t^{\prime}))$
(6\. 37)

In the semiclassical approximation $\xi^{\prime}=\eta^{\prime}=0$ we have:

$u_{N}=u_{N}(x;\eta_{0},\xi_{0}+\Omega(\eta_{0})t).$ (6\. 38)

Note now that if the surface term

$\int\partial_{\mu}(e^{iqx}\partial^{\mu}u_{N})=0$ (6\. 39)

then

$\int d^{2}xe^{iqx}(\partial^{2}+m_{h}^{2})u_{N}(x,t)=-(q^{2}-m_{h}^{2})\int
d^{2}xe^{iqx}u_{N}(x,t)=0$ (6\. 40)

since $q^{2}$ belongs to mass shell by definition. The condition (6\. 39) is
satisfied since $u_{N}$ belong to Schwartz space (the periodic boundary
condition for $u(x,t)$ do not alter this conclusion). Therefore, in the
semiclassical approximation (6\. 5) is hold.

Expending the operator exponent in (6\. 35) we will find the expansion over

$\rho_{n,m}(q)=\frac{(1/2i)^{n}}{n!}\frac{(1/2i)^{m}}{m!}\lim_{(\xi^{\prime},\eta^{\prime},e_{\xi},e_{\eta})=0}\sum_{N}\int
d^{N}\xi_{0}d^{N}\eta_{0}\times$
$\times\int\prod^{n}_{i=1}\\{dt_{i}dt^{\prime}_{i}\theta(t_{i}-t^{\prime}_{i})\hat{\xi}^{\prime}(t^{\prime}_{i})$
$\times\int\prod^{m}_{i=1}\\{dt_{i}dt^{\prime}_{i}\theta(t_{i}-t^{\prime}_{i})\hat{\eta}^{\prime}(t^{\prime}_{i})\\}e^{iS_{O}(u_{N})}|\Gamma(q;u_{N})|^{2}$
$\times\\{\prod^{n}_{i=1}\hat{e}_{\xi}(t_{i})\prod^{m}_{j=1}\hat{e}_{\eta}(t_{j})e^{-iU(u_{N};e_{\xi},e_{\eta})}\\}|_{e=0},$
(6\. 41)

where $U(u_{N};e_{\xi},e_{\eta})$ was defined in (6\. 25), (6\. 26). Notice
that the action of operators $\hat{\xi}^{\prime}$, $\hat{\eta}^{\prime}$
create terms

$\int d^{2}xe^{iqx}\theta(t-t^{\prime})(\partial^{2}+m^{2})u_{N}(x,t)\neq 0.$
(6\. 42)

6.2. Quantum corrections

Now we will show that The semiclassical approximation is exact in the soliton
sector of (6\. 1), (6\. 11) theory.

The structure of the perturbation theory is readily seen in the ”normal-
product” form:

$\rho(q)=\sum_{N}\int\prod^{N}_{i=1}\\{d\xi_{0}d\eta_{0}\\}_{i}:e^{-iU(u_{N};\hat{j}/2i)}e^{iS_{O}(u_{N})}|\Gamma(q;u_{N})|^{2}:,$
(6\. 43)

where

$\hat{j}=\hat{j}_{\xi}\cdot\frac{\partial
u_{N}}{\partial\eta}-\hat{j}_{\eta}\cdot\frac{\partial
u_{N}}{\partial\xi}=\omega\hat{j}_{X}\frac{\partial u_{N}}{\partial X}$ (6\.
44)

and

$\hat{j}_{X}=\int dt^{\prime}\Theta(t-t^{\prime})\hat{X}(t^{\prime})$ (6\. 45)

with $2N$-dimensional vector $X=(\xi,\eta)$. In Eq. (6\. 44) $\omega$ is the
ordinary simplectic matrix.

The colons in (6\. 43) mean that the operator $\hat{j}$ should stay to the
left of all functions. The structure (6\. 44) shows that each order over
$\hat{j}_{X_{i}}$ is proportional at least to the first order derivative of
$u_{N}$ over conjugate to $X_{i}$ variable.

The expansion of (6\. 43) over $\hat{j}_{X}$ can be written [26] in the form
of total derivatives (omitting the semiclassical approximation):

$\rho(q)=\sum_{N}\int\prod^{N}_{i=1}\\{d\xi_{0}d\eta_{0}\\}_{i}\left\\{\sum^{2n}_{i=1}\frac{\partial}{\partial
X_{0i}}P_{X_{i}}(u_{N})\right\\},$ (6\. 46)

where $P_{X_{i}}(u_{N})$ is the infinite sum of ”time-ordered” polynomials
(see [26]) over $u_{N}$ and its derivatives. The explicit form of
$P_{X_{i}}(u_{N})$ is complicated since the interaction potential is non-
polynomial. But it is enough to know, see (6\. 44), that

$P_{X_{i}}(u_{N})\sim\omega_{ij}\frac{\partial u_{N}}{\partial X_{0j}}.$ (6\.
47)

Therefore,

$\rho(q)=0$ (6\. 48)

since (i) each term in (6\. 46) is the total derivative, (ii) we have (6\. 47)
and (iii) $u_{N}$ belongs to Schwartz space.