# Reduction of path integrals for interacting systems: The case of using
dependent coordinates in the description of reduced motion on the orbit space

S. N. Storchak111E-mail adress<EMAIL_ADDRESS>
Institute for High Energy Physics,
NRC “Kurchatov Institute,”
Protvino, 142284, Russian Federation

###### Abstract

We consider a reduction procedure in Wiener-type path integral for a finite-
dimensional mechanical system with a symmetry representing the motion of two
interacting scalar particles on a manifold that is the product of the total
space of the principal fiber bundle and a vector space. By analogy with what
is done in gauge theories, the local description of the reduced motion on the
space of orbits is carried out using dependent coordinates. The factorization
of the measure in the path integral, which is necessary for the reduction, is
based on the application of the stochastic differential equation of the
optimal nonlinear filtering from the theory of stochastic processes. The non-
invariance of the measure in the path integral under the reduction is shown.
The integrand of the path integral reduction Jacobian is generated by the
projection of the mean curvature vector field of the orbit onto the
submanifold, which is used to determine the adapted coordinates in the
principal fiber bundle associated with the problem under study.

##  1 Introduction

Dynamical systems with symmetry are characterized by a deep relationship with
geometry. These include, for example, dynamical systems with gauge symmetries,
especially those studied in gauge field theories, where the Yang-Mills fields
are represented by connections on the principal fiber bundles.

In gauge theories, the main problem is to describe quantum evolution in terms
of gauge invariant variables. This is the same as describing the evolution
given on the space of orbits of the group action. A satisfactory solution has
not yet been found for this problem. At present, in the path integral
quantization of the gauge systems, an approach based on the Faddeev-Popov
mechanism [1] is used. In this approach, the additional restrictions are
imposed on the gauge fields, and therefore we are forced to describe the
quantum dynamics on the orbit space in terms of constrained (or dependent)
variables.

However, the transition to constrained gauge fields, carried out in path
integrals, looks like a heuristic method of the path integral transformations.
To clarify the question of the correctness of such a transition, we studied
how this happens in path integrals for finite-dimensional dynamical systems
with symmetry[2, 3]. This was done because for finite-dimensional Wiener-type
path integrals, unlike of path integrals used in gauge theories, the
integration measure can be rigorously determined.

The model system we studied was a mechanical system describing the motion of a
scalar particle on a finite-dimensional manifold, on which the free isometric
action of a compact semisimple Lie group is given. The dynamics of this system
is largely due to the presence of the principal fiber bundle associated with
the considered dynamical system. The original configuration space of the
system can be viewed as a total space of this bundle. Therefore, as
coordinates in the original configuration space, one can use the coordinates
that are usually given in the principal fiber bundle. The possibility of using
a coordinate system including dependent coordinates follows from the existence
of a local isomorphism between the original principal bundle and the trivial
principal bundle whose base space is a local submanifold (‘gauge surface’)[4].

However, in this approach, the problem of the global description of quantum
evolution remains unsolved. From a geometric point of view, this is due to the
fact that, in the general case, the principal bundles of gauge theories are
nontrivial. Therefore, there are no global sections in the principal bundle,
and we can only deal with local gauge fixing surfaces. It also follows from
this that a global description of evolution with the help of dependent
variables defined on these surfaces is possible only when considering trivial
principal bundles.

In our paper, the path integral reduction procedure consists in transformation
of the original path integral, whose measure is generated by a stochastic
process, to the path integral that determines the diffusion (or “quantum
evolution”) of the reduced dynamical system given on the orbit space of the
principal bundle. In this transformation, the main role is played by the
factorization of the measure in the original path integral into the ‘group’
measure (given in the space of paths on a group) and a measure defined in path
space on a special surface of the principal bundle known as a ‘gauge fixing
surface’ (or ‘gauge surface’).

In our papers[2, 5, 6] we used the definition of path integrals proposed by
Daletskii and Belopolskaya in [7, 8]. According to their approach, the measure
of path integrals is determined by those stochastic processes that are
associated with local evolution semigroups. The global path integral is
defined as the limit of these semigroups upon refining the subdivision of the
time interval. Transformations of path integrals (local evolution semigroups)
can be obtained from transformations of local stochastic differential
equations whose solutions are stochastic processes that generate measures in
these path integrals. In our works, the factorization of the measure in the
path integrals was carried out using the stochastic differential equation of
the optimal nonlinear filtering from the stochastic process theory. Note that
a similar method for studying reduction in path integrals was used in [9].

The path integral reduction leads to the integral relation between the
original path integral and the path integral used to describe the “quantum
evolution” on the orbit space of the principal fiber bundle. We also note that
the Hamilton operator of the dynamical system obtained as a result of the
reduction of the path integral has an additional potential term, which is the
integrand of the path integral reduction Jacobian.

In our articles [10, 11], the path integral reduction method was applied to a
simple model system describing the interaction of two scalar particles that
are given on a manifold represented by the product of two manifolds: a smooth
compact Riemannian manifold and a vector space. The product manifold was
endowed with a free proper isometric action of a semisimple compact unimodular
Lie group. Such a group action gives rise to a principal fiber bundle, so that
the original configuration space, the product manifold, can be regarded as the
total space of this bundle. In the papers cited, we considered the case where
the gauge surface (used to obtain a local section of the principal fiber
bundle) has a parametric representation and, therefore, it is possible to
define invariant variables that can serve as coordinates in the orbit space of
this bundle.

The purpose of the present paper is to consider the case of the path integral
reduction when, as in gauge theories, dependent variables are used to describe
the motion on the orbit space. It should be noted that the geometric
properties of the considered mechanical system are similar to the properties
of an infinite-dimensional dynamical system describing the interaction of the
Yang-Mills gauge field with a scalar field. This makes it possible to use the
mechanical system under study as a finite-dimensional model for such field
theories.

The paper is organizad as follows. In Section 2 we give basic definitions that
are in fact the same as those from our previous articles. In Section 3 we
briefly describe the geometry of the problem and discuss the introduction of
the adapted coordinates. Some of the information presented in this section was
also borrowed from our previous work. In Sections 4 we define the drift
coefficients of our stochastic differential equations. Section 5 deals with
the factorization of the measure in the path integral and the calculation of
the reduction Jacobian for the case of the reduction onto a zero-momentum
level. In Conclusion, the details of the obtained result are discussed. Some
relationships between Christoffel symbols used in the article are given in
Appendix.

##  2 Definitions

In this article, we consider the path integrals that are used to represent the
solution of the backward Kolmogorov equation which is given on a smooth
compact Riemannian manifold
$\tilde{\mathcal{P}}=\mathcal{P}\times\mathcal{V}$:

$\left\\{\begin{array}[]{l}\displaystyle\left(\frac{\partial}{\partial
t_{a}}+\frac{1}{2}\mu^{2}\kappa\bigl{[}\triangle_{\cal
P}(p_{a})+\triangle_{\cal V}(v_{a})\bigr{]}+\frac{1}{\mu^{2}\kappa
m}V(p_{a},v_{a})\right){\psi}_{t_{b}}(p_{a},t_{a})=0,\\\
{\psi}_{t_{b}}(p_{b},v_{b},t_{b})=\phi_{0}(p_{b},v_{b}),\qquad\qquad\qquad\qquad\qquad(t_{b}>t_{a}),\end{array}\right.$
(1)

where $\mu^{2}=\frac{\hbar}{m}$ , $\kappa$ is a real positive parameter,
$V(p,f)$ is the group-invariant potential term: $V(pg,g^{-1}v)=V(p,v)$,
$g\in\mathcal{G}$, $\triangle_{\cal P}$ is the Laplace–Beltrami operator on
the Riemannian manifold $\cal P$. The scalar Laplacian $\triangle_{\cal V}$
acts on the space of functions defined on the finite-dimensional vector space
$\cal V$. In a chart $(U_{\cal P}\times U_{\cal
V},{\varphi}^{\tilde{\mathcal{P}}})$,
${\varphi}^{\tilde{\mathcal{P}}}=(\varphi^{A},\varphi^{a})$, from the atlas of
the manifold $\tilde{\mathcal{P}}$, where the point $(p,v)$ has the
coordinates $(Q^{A},f^{a})={\varphi}^{\tilde{\mathcal{P}}}(p,v)$,
$\triangle_{\cal P}$ is given by the following expression222In our formulas we
assume that there is sum over the repeated indices. The indices denoted by the
capital letters run from 1 to $n_{\cal P}=\dim\cal P$, and the indices of
small Latin letters, except $i,j,k,l$, run from 1 to $n_{\cal V}=\dim\cal V$.:

$\triangle_{\cal P}(Q)=G^{-1/2}(Q)\frac{\partial}{\partial
Q^{A}}G^{AB}(Q)G^{1/2}(Q)\frac{\partial}{\partial Q^{B}},$ (2)

where $G=det(G_{AB})$ and $G_{AB}(Q)$ represents the Riemannian metric on
$\cal P$: $G_{AB}(Q)=G(\frac{\partial}{\partial
Q^{A}},\frac{\partial}{\partial Q^{B}})$.

The coordinate expression of the operator $\triangle_{\cal V}$ is

$\triangle_{\cal V}(f)=G^{ab}\frac{\partial}{\partial f^{a}\partial f^{b}},$

where $G^{ab}$ is an inverse matrix to the matrix $G_{ab}$ representing the
metric on $\mathcal{V}$ in the coordinate basis $\\{\frac{\partial}{\partial
f^{a}}\\}$. We assume that the matrix $G_{ab}$ consists of fixed constant
elements.

To obtain the Schrödinger equation, one should perform the transition from the
equation (1) to the forward Kolmogorov equation and then set $\kappa=i$.

With the definition of the path integral from [7], the solution of the
equation (1) can be written as

$\displaystyle{\psi}_{t_{b}}(p_{a},v_{a},t_{a})$ $\displaystyle=$
$\displaystyle{\rm
E}\Bigl{[}\phi_{0}(\eta_{1}(t_{b}),\eta_{2}(t_{b}))\exp\\{\frac{1}{\mu^{2}\kappa
m}\int_{t_{a}}^{t_{b}}V(\eta_{1}(u),\eta_{2}(u))du\\}\Bigr{]}$ (3)
$\displaystyle=$
$\displaystyle\int_{\Omega_{-}}d\mu^{\eta}(\omega)\phi_{0}(\eta(t_{b}))\exp\\{\ldots\\},$

where ${\eta}(t)=(\eta_{1}(t),\eta_{2}(t))$ is a global stochastic process on
a manifold $\tilde{\cal P}=\cal P\times\cal V$, the measure ${\mu}^{\eta}$ in
the space of paths
$\Omega_{-}=\\{\omega(t)=(\omega^{1}(t),\omega^{2}(t)):\omega^{1,2}(t_{a})=0,\eta_{1}(t)=p_{a}+\omega^{1}(t),\eta_{2}(t)=v_{a}+\omega^{2}(t)\\}$
is determined by the probability distribution of the stochastic process
$\eta(t)$.

Note that (3) is a symbolical notation of the global semigroup, defined
according to [7, 8] as a limit (under refinement of the subdivision of the
time interval) of the superposition of the local semigroups:

$\\!\psi_{t_{b}}(p_{a},v_{a},t_{a})={\lim}_{q}\bigl{[}{\tilde{U}}_{\eta}(t_{a},t_{1})\cdot\ldots\cdot{\tilde{U}}_{\eta}(t_{n-1},t_{b})\phi_{0}\bigr{]}(p_{a},v_{a}),$
(4)

where each local evolution semigroup ${\tilde{U}}_{\eta}$ acting in the space
of functions on the manifold $\tilde{\mathcal{P}}$ is given by333In what
follows, the potential term of the Hamilton operator, as being insignificant
for our transformations of path integrals, will be temporarily omitted in the
path integrals. In the final formulae, the potential term will be recovered.

${\tilde{U}}_{\eta}(s,t)\phi(p,v)={\rm
E}_{s,p,v}[\phi(\eta_{1}(t),\eta_{2}(t))],\,\,\,\,\,\,s<t,\,\,\,\,\,\,\eta_{1}(s)=p,\;\eta_{2}(s)=v,$
(5)

in which ${\tilde{U}}_{\eta}(s,t)\phi(p,v)$ should be understand as
$[{\tilde{U}}_{\eta}(s,t)\phi](p,v)$.

These local semigroups are also expressed in terms of path integrals with the
integration measures determined by the local representatives
$\varphi^{\tilde{\cal P}}(\eta_{t})={\eta}^{\varphi^{\tilde{\cal
P}}}(t)\equiv\\{\eta^{A}_{1}(t),\eta^{a}_{2}(t)\\}\in R^{n_{\tilde{\cal P}}}$
of the global stochastic process $\eta(t)$.

The local processes $\\{\eta_{1}^{A}(t),\eta_{2}^{a}(t)\\}$ are solutions of
the stochasic differential equations

$\displaystyle d\eta_{1}^{A}(t)$ $\displaystyle=\frac{1}{2}\mu^{2}\kappa
G^{-1/2}\frac{\partial}{\partial
Q^{B}}(G^{1/2}G^{AB})dt+\mu\sqrt{\kappa}{\mathscr{X}}_{\bar{M}}^{A}(\eta_{1}(t))dw^{\bar{M}}(t),$
(6) $\displaystyle d\eta_{2}^{a}(t)$
$\displaystyle=\mu\sqrt{\kappa}{\mathscr{X}}_{\bar{b}}^{a}dw^{\bar{b}}(t),$
(7)

in which the stochastic differentials are taken in the sence of Itô and where
${\mathscr{X}}_{\bar{M}}^{A}$ and ${\mathscr{X}}_{\bar{b}}^{a}$ are defined by
the local equalities $\sum^{n_{\mathcal{P}}}_{\bar{\scriptscriptstyle
K}\scriptscriptstyle=1}{\mathscr{X}}_{\bar{K}}^{A}{\mathscr{X}}_{\bar{K}}^{B}=G^{AB}$
and $\sum^{n_{\mathcal{V}}}_{\bar{\scriptscriptstyle
b}\scriptscriptstyle=1}{\mathscr{X}}_{\bar{b}}^{a}{\mathscr{X}}_{\bar{b}}^{c}=G^{ac}$,
$dw^{\bar{M}}(t)$ and $dw^{\bar{b}}(t)$ are the independent Wiener processes.
(Here and further we denote the Euclidean indices by over-barred indices.)
Also note that equations (6) and (7) are the Stratonovich equations.

Note that with the local processes $\\{\eta_{1}^{A}(t),\eta_{2}^{a}(t)\\}$ one
can rewrite (5) as

${\tilde{U}}_{\eta}(s,t)\phi(p,v)={\rm E}_{s,\varphi^{\tilde{\cal
P}}(p,v)}[\phi(\,({\varphi}^{\tilde{\cal P}})^{-1}(\eta^{\varphi^{\tilde{\cal
P}}}(t))\,)],\,\,\,\,\,\,{\eta}^{\varphi^{\tilde{\cal
P}}}(s)=\varphi^{\tilde{\cal P}}(p,v).$ (8)

The fundamental solution, also known as the Green’s function $G_{\tilde{\cal
P}}$, is the kernel of the semigroup, with the help of which the left-hand
side of (3) is determined as follows:

$\psi_{t_{b}}(p_{a},v_{a},t_{a})=\int G_{\tilde{\cal
P}}(p_{b},v_{b},t_{b};p_{a},v_{a},t_{a})\phi_{0}(p_{b},v_{b})dv_{\tilde{\cal
P}}(p_{b},v_{b}),$

where $dv_{\tilde{\cal P}}(p,v)$ is a volume element on the manifold
$\tilde{\cal P}$. Note that in this formula the integration is carried out
using a partition of unity subordinated to the local finite covering of the
manifold $\tilde{\cal P}$.

The probabilistic representation for the kernel $G_{\tilde{\cal P}}$ can be
formally obtained by replacing $\varphi_{0}$ in (3) with the delta-function.
This means that the measure of the path integral for the Green’s function
$G_{\tilde{\cal P}}$ will be defined in the space of paths with both ends
fixed. The same can be done more correctly by considering functions
approximating the delta function (instead of $\varphi_{0}$) and then taking
the appropriate limit.

The fundamental solution to the equation (1) also satisfies the forward
Kolmogorov equation. As for the existence of a fundamental solution to the
equation (1), this depends on the smoothness constraints imposed on the
coefficients of the equation, and in this article we assume that in our
equation the coefficients are chosen in accordance with these requirements.

##  3 Adapted coordinates in the principal fiber bundle

Our main assumption in the article is that on the Riemannian manifold
$\tilde{\mathcal{P}}=\mathcal{P}\times\mathcal{V}$, which is the configuration
space of our original mechanical system, there is a smooth isometric free and
proper action of the compact semisimple Lie group $\mathcal{G}$. We consider
the right action of the group by which the point $(p,v)$ is mapped to the
point $(\tilde{p},\tilde{v})$ so that
$(\tilde{p},\tilde{v})=(p,v)g=(pg,g^{-1}v)$. If the point $(p,v)$ has the
local coordinates $(Q^{A},f^{a})$ ( in the coordinate basis
$(\frac{\partial}{\partial Q^{A}},\frac{\partial}{\partial f^{a}}$)), then
this action in coordinates can be written as

${\tilde{Q}}^{A}=F^{A}(Q,g),\;\;\;\;{\tilde{f}}^{b}=\bar{D}^{b}_{a}(g)f^{a},$

where $\bar{D}^{b}_{a}(g)\equiv D^{b}_{a}(g^{-1})$, and by $D^{b}_{a}(g)$ we
denote the matrix of the finite-dimensional representation of the group
$\mathcal{G}$ acting in the vector space $\mathcal{V}$.

For the right action, we have the compatibility relation

$F(F(Q,g_{1}),g_{2})=F(Q,\rm\hat{\Phi}(g_{1},g_{2})),$

where the function $\rm\hat{\Phi}$ determines the group multiplication law in
the space of the group parameters.

The manifold $\tilde{\mathcal{P}}$ is equipped with the following Riemannian
metric:

$ds^{2}=G_{AB}(Q)dQ^{A}dQ^{B}+G_{ab}\,df^{a}df^{b}.$ (9)

We note, that the components of the metric are not arbitrary, but must satisfy
certain relations due to the isometric action of the group $\mathcal{G}$ on
$\tilde{\mathcal{P}}$.

In particularly,

$G_{AB}(Q)=G_{DC}(F(Q,g))F^{D}_{A}(Q,g)F^{C}_{B}(Q,g),$ (10)

with $F^{B}_{A}(Q,g)\equiv\partial F^{B}(Q,g)/\partial Q^{A}$, and

$G_{pq}=G_{ab}\bar{D}^{a}_{p}(g)\bar{D}^{b}_{q}(g).$ (11)

The last relation can be derived from the linear isometrical action of the
group $\mathcal{G}$ in the vector space $\mathcal{V}$.

The Killing vector fields $K_{\alpha}$,
$\alpha=1,...,n_{\mathcal{G}},n_{\mathcal{G}}=\dim\mathcal{G}$, of the
manifold $\tilde{\mathcal{P}}$ with the metric (9) are expressed in terms of
local coordinates $(Q^{A},f^{b})$ as follows:

$K_{\alpha}=K^{A}_{\alpha}(Q)\partial/\partial
Q^{A}+K^{b}_{\alpha}(f)\partial/\partial f^{b},$

where $K^{A}_{\alpha}(Q)=\partial{\tilde{Q}}^{A}/\partial a^{\alpha}|_{a=e}$
and

$K^{b}_{\alpha}(f)=\partial{\tilde{f}}^{b}/\partial
a^{\alpha}|_{a=e}=\partial{\bar{D}}^{b}_{c}(a)/\partial
a^{\alpha}|_{a=e}f^{c}\equiv({\bar{J}}_{\alpha})^{b}_{c}f^{c}.$

The generators ${\bar{J}}_{\alpha}$ of the representation
${\bar{D}}^{b}_{c}(a)$ satisfy the commutation relation
$[{\bar{J}}_{\alpha},{\bar{J}}_{\beta}]={\bar{c}}^{\gamma}_{\alpha\beta}{\bar{J}}_{\gamma}$,
where the structure constants
${\bar{c}}^{\gamma}_{\alpha\beta}=-{c}^{\gamma}_{\alpha\beta}$.

As it follows from the general theory (see, for example, [12]) the action of
the group $\mathcal{G}$ on the manifold $\tilde{\mathcal{P}}$ leads to the
principal fiber bundle
$\pi^{\prime}:\mathcal{P}\times\mathcal{V}\to\mathcal{P}\times_{\mathcal{G}}\mathcal{V}$.444$\pi^{\prime}:(p,v)\to[p,v]$,
where $[p,v]$ is the equivalence class formed by the relation
$(p,v)\sim(pg,g^{-1}v).$ We denote this principal bundle as $\rm
P(\tilde{\mathcal{M}},\mathcal{G})$, where the orbit space
$\tilde{\mathcal{M}}=\mathcal{P}\times_{\mathcal{G}}\mathcal{V}$ is the base
space of the bundle.

The fact that the principal bundle is related to our problem means that the
original manifold $\tilde{\mathcal{P}}$ can be considered as the total space
of this bundle. It follows that we can express the local coordinates
$(Q^{A},f^{a})$ of the point $(p,v)\in\tilde{\mathcal{P}}$ in terms of the
coordinates defined in the principal fiber bundle.

As the coordinates in the principal bundle, we will use the adapted
coordinates. This choice is due to the fact that the quantization of gauge
fields is mainly carried out using such coordinates [13, 14, 15].

Adapted coordinates are defined with the help of ‘gauge fixing surfaces’,
which are local submanifolds $\tilde{\Sigma}_{i}$ in the total space of the
principal bundle. There is a certain correspondence between these local
submanifolds and the open neighborhoods $\tilde{U}_{i}$ from the atlas of the
manifold defined on $\tilde{\mathcal{M}}$. In addition, $\tilde{\Sigma}_{i}$
must have a transversal intersection with the group orbits.

In our case, to determine $\tilde{\Sigma}_{i}$ in the principal fiber bundle
$\rm P(\tilde{\mathcal{M}},\mathcal{G})$, we use local surfaces $\Sigma_{i}$
in the total space $\mathcal{P}$ of the principal bundle $\rm
P(\mathcal{M},\mathcal{G})$ with the base space
$\mathcal{M}=\mathcal{P}/\mathcal{G}$. In fact, the local surfaces
$\Sigma_{i}$ serve to define local sections of this bundle. On the other hand,
if the sections are given, then the images of these sections are the local
surfaces $\Sigma_{i}$.

We assume that local surfaces $\tilde{\Sigma}_{i}$ form a global surface
$\tilde{\Sigma}$ in the original manifold $\tilde{\cal P}$ or, which is the
same, that there exists a global section in the principal fiber bundle $\rm
P(\tilde{\mathcal{M}},\mathcal{G})$. This implies that our principal fiber
bundle is trivial and hence $\tilde{\cal
P}\sim\tilde{\mathcal{M}}\times\mathcal{G}$. Such a case corresponds to what
we have in gauge theories, where in practice one has to deal with trivial
principal bundles. Note also that triviality of $\rm
P(\tilde{\mathcal{M}},\mathcal{G})$ is interrelated with the triviality of the
principal fiber bundle $\rm P({\mathcal{M}},\mathcal{G})$.

Let’s consider how we can apply the general scheme of introducing adapted
coordinates in our case. The submanifold $\Sigma$ of $\mathcal{P}$ in this
approach is given by the system of equations
$\chi^{\alpha}=0,\,\alpha=1,...,n_{\mathcal{G}}$. And the points belonging to
$\Sigma$ have coordinates $Q^{\ast A}$ for which $\chi^{\alpha}(Q^{\ast
A})=0$. For this reason, the coordinates $Q^{\ast A}$ are called dependent
coordinates.

The group coordinates $a^{\alpha}(Q)$ of a point $p\in\mathcal{P}$ are defined
by the solution of the following equation:

$\chi^{\alpha}(F^{A}(Q,a^{-1}(Q)))=0.$

This means that

$Q^{\ast A}=F^{A}(Q,a^{-1}(Q)).$

That is, the group element $g^{-1}(p)$ with the coordinates $a^{-1}(Q)$
carries the point $p$ to the submanifold $\Sigma$.

Therefore, if we are given the (global) gauge fixing surface $\Sigma$ in the
principal fiber bundle $\pi:\cal P\to\cal M$, then the section of the bundle
can be defined as follows. Restricting the projection $\pi$ to $\Sigma$, we
have $\pi|_{\Sigma}:U_{\Sigma}\to U_{\cal M}$ for local neighborhoods on
$\Sigma$ and $\cal M$. The section ${\sigma}_{\Sigma}$ is defined as a map
${\sigma}_{\Sigma}:U_{\cal M}\to U_{\Sigma}$ such that
$\pi|_{\Sigma}\circ{\sigma}_{\Sigma}={\rm id}_{\pi|_{\Sigma}}$. It follows
from the above coordinate transformations that
${\sigma}_{\Sigma}([p])=p\,g^{-1}(p)$.

Using ${\sigma}_{\Sigma}$ one can define the section
${\sigma}_{\tilde{\Sigma}}$ of the principal fiber bundle $\rm
P(\tilde{\mathcal{M}},\mathcal{G})$ as the map
${\sigma}_{\tilde{\Sigma}}:\mathcal{P}\times_{\mathcal{G}}\mathcal{V}\to\mathcal{P}\times\mathcal{V}$.
This section is defined as
${\sigma}_{\tilde{\Sigma}}([p,v])=({\sigma}_{\Sigma}([p]),g(p)v)=(p\,g^{-1}(p),g(p)v)=(p,v)g^{-1}(p).$
The gauge fixing surface $\tilde{\Sigma}$ in $\mathcal{P}\times\mathcal{V}$ is
the image of the mapping given by ${\sigma}_{\tilde{\Sigma}}$. [14]

Having obtained $\tilde{\Sigma}$, we can define the principal bundle
$\pi^{\prime}_{(\tilde{\Sigma})}:\mathcal{P}\times\mathcal{V}\to{\tilde{\Sigma}}$
with
$\pi^{\prime}_{(\tilde{\Sigma})}={\sigma}_{\tilde{\Sigma}}\circ\pi^{\prime}$.
This principal bundle isomorphic to $\rm P(\tilde{\mathcal{M}},\mathcal{G})$.

The group element $g^{-1}(p)$ obtained above (participating in the transition
(along the orbit) from the point $p\in\mathcal{P}$ to the corresponding point
given on the gauge fixing surface $\Sigma$ in the total space of the principal
bundle ${\rm P}(\mathcal{M},\mathcal{G})$) is also used to determine adapted
coordinates on the manifold $\tilde{\cal P}$.

The coordinate functions $\tilde{\varphi}^{-1}$ that make the transition to
adapted coordinates are as follows:

$\tilde{\varphi}^{-1}:(Q^{A},f^{b})\to(Q^{\ast}{}^{A}(Q),\tilde{f}^{b}(Q),a^{\alpha}(Q)\,),$

where

$\tilde{f}^{b}(Q)=D^{b}_{c}(a(Q))\,f^{c},$

($\bar{D}^{b}_{c}(a^{-1})\equiv D^{b}_{c}(a))$ and $a(Q)$ are the coordinates
of the group element $g(p)$).

The inverse transformation is given by the coordinate functions
$\tilde{\varphi}$:

$\tilde{\varphi}:(Q^{\ast}{}^{B},\tilde{f}^{b},a^{\alpha})\to(F^{A}(Q^{\ast},a),\bar{D}^{c}_{b}(a)\tilde{f}^{b}).$

These functions, $\tilde{\varphi}$ and $\tilde{\varphi}^{-1}$, represent the
bundle maps $\tilde{\varphi}:\tilde{\Sigma}\times\cal
G\to{\mathcal{P}}\times\cal V$ and
${\tilde{\varphi}^{-1}}:{\mathcal{P}}\times{\mathcal{V}}\to{\tilde{\Sigma}}\times{\mathcal{G}}$
for the isomorphic trivial principal bundles
$\tilde{\Sigma}\times\mathcal{G}\to\tilde{\Sigma}$ and ${\rm
P}(\tilde{\mathcal{M}},\mathcal{G})$.

Note also that in the general case when ${\rm
P}(\tilde{\mathcal{M}},\mathcal{G})$ is nontrivial, the trivial principal
bundles ${\tilde{\Sigma}}_{i}\times{\mathcal{G}}\to{\tilde{\Sigma}}_{i}$ are
only locally isomorphic to the principal bundle ${\rm
P}(\tilde{\mathcal{M}},\mathcal{G})$[14, 15],[4]. In this case the adapted
coordinates can only be used to describe the local motion on the orbit space
$\tilde{\mathcal{M}}$.

Thus, we have defined the special bundle coordinates in the total space of the
principal fiber bundle
$\pi^{\prime}:\mathcal{P}\times\mathcal{V}\to\mathcal{P}\times_{\mathcal{G}}\mathcal{V}$.
They are such that

$\tilde{\varphi}^{\tilde{\cal
P}}(p,v)=(Q^{\ast}{}^{A},\tilde{f}^{b},a^{\alpha})\;\;\rm{where}\;\;\tilde{\varphi}^{\tilde{\cal
P}}:=\tilde{\varphi}^{-1}\circ\varphi^{\tilde{\cal P}}.$

Note that adapted coordinates can be obtained in the same way using a finite
set of fixing surfaces $\tilde{\Sigma}_{i}$. But in this case, it is also
required that adapted coordinates must be consistent to each other on the
overlapping of the charts.

Replacing coordinates $(Q^{A},f^{a})$ of a point $(p,v)\in\mathcal{P}\times V$
with new coordinates $(Q^{\ast}{}^{B},\tilde{f}^{b},a^{\alpha})$,

$Q^{A}=F^{A}(Q^{\ast}{}^{B},a^{\alpha}),\;\;\;f^{a}=\bar{D}^{a}_{b}(a)\tilde{f}^{b},$
(12)

leads to the following transformations of the local coordinate vector fields:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\frac{\partial}{\partial
f^{a}}=D^{b}_{a}(a)\frac{\partial}{\partial{\tilde{f}}^{b}},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\frac{\partial}{\partial
Q^{B}}=\frac{\partial Q^{\ast}{}^{A}}{\partial Q^{B}}\frac{\partial}{\partial
Q^{\ast}{}^{A}}+\frac{\partial a^{\alpha}}{\partial
Q^{B}}\frac{\partial}{\partial
a^{\alpha}}+\frac{\partial{\tilde{f}}^{b}}{\partial
Q^{B}}\frac{\partial}{\partial{\tilde{f}}^{b}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\check{F}^{C}_{B}\Biggl{(}N^{A}_{C}(Q^{\ast})\frac{\partial}{\partial
Q^{\ast}{}^{A}}+{\chi}^{\mu}_{C}({\Phi}^{-1})^{\beta}_{\mu}\bar{v}^{\alpha}_{\beta}(a)\frac{\partial}{\partial
a^{\alpha}}-{\chi}^{\mu}_{C}({\Phi}^{-1})^{\nu}_{\mu}(\bar{J}_{\nu})^{b}_{p}\tilde{f}^{p}\frac{\partial}{\partial{\tilde{f}}^{b}}\Biggr{)}\\!.$
(13)

Here $\check{F}^{C}_{B}\equiv F^{C}_{B}(F(Q^{\ast},a),a^{-1})$ is an inverse
matrix to the matrix $F^{A}_{B}(Q^{\ast},a)$,
${\chi}^{\mu}_{C}\equiv\frac{\partial{\chi}^{\mu}(Q)}{\partial
Q^{C}}|_{Q=Q^{\ast}}$,
$({\Phi}^{-1})^{\beta}_{\mu}\equiv({\Phi}^{-1})^{\beta}_{\mu}(Q^{\ast})$ – the
matrix which is inverse to the Faddeev–Popov matrix

$({\Phi})^{\beta}_{\mu}(Q)=K^{A}_{\mu}(Q)\frac{\partial{\chi}^{\beta}(Q)}{\partial
Q^{A}},$

the matrix $\bar{v}^{\alpha}_{\beta}(a)$ is inverse of the matrix
$\bar{u}^{\alpha}_{\beta}(a)$.555$\det\bar{u}^{\alpha}_{\beta}(a)$ is the
density of the right-invariant measure given on the group $\mathcal{G}$.

The operator $N^{A}_{C}$, defined as

$N^{A}_{C}(Q)=\delta^{A}_{C}-K^{A}_{\alpha}(Q)({\Phi}^{-1})^{\alpha}_{\mu}(Q){\chi}^{\mu}_{C}(Q),$

is the projection operator ($N^{A}_{B}N^{B}_{C}=N^{A}_{C}$) onto planes
perpendicular to the gauge orbits in ${\rm P}(\mathcal{M},\mathcal{G})$, since
$N^{A}_{C}K^{C}_{\alpha}=0$[13]. $N^{A}_{C}(Q^{\ast})$ is the restriction of
$N^{A}_{C}(Q)$ to the submanifold $\Sigma$:

$N^{A}_{C}(Q^{\ast})\equiv
N^{A}_{C}(F(Q^{\ast},e)),\;\;\;N^{A}_{C}(Q^{\ast})=F^{B}_{C}(Q^{\ast},a)N^{M}_{B}(F(Q^{\ast},a))\check{F}_{M}^{A}(Q^{\ast},a),$

where $e$ is the unity element of the group.

As an operator, the vector field $\frac{\partial}{\partial Q^{\ast}{}^{A}}$ is
determined by means of the following rule:

$\frac{\partial}{\partial
Q^{\ast}{}^{A}}\varphi(Q^{\ast})=(P_{\bot})^{D}_{A}(Q^{\ast})\frac{\partial\varphi(Q)}{\partial
Q^{D}}\Bigl{|}_{Q=Q^{\ast}},$

where the projection operator $(P_{\bot})^{A}_{B}$ onto the tangent plane to
the submanifold $\Sigma$ is given by

$(P_{\bot})^{A}_{B}=\delta^{A}_{B}-\chi^{\alpha}_{B}\,(\chi\chi^{\top})^{-1}{}^{\beta}_{\alpha}\,(\chi^{\top})^{A}_{\beta}.$

In this formula, $(\chi^{\top})^{A}_{\beta}$ is a transposed matrix to the
matrix $\chi^{\nu}_{B}$:

$(\chi^{\top})^{A}_{\mu}=G^{AB}{\gamma}_{\mu\nu}\chi^{\nu}_{B}\;\;\;\;{\gamma}_{\mu\nu}=K^{A}_{\mu}G_{AB}K^{B}_{\nu}.$

Using the above explicit expressions for the projection operators, it is easy
to get the following properties:

$(P_{\bot})^{A}_{B}N^{C}_{A}=(P_{\bot})^{C}_{B},\;\;\;\;\;N^{A}_{B}(P_{\bot})^{C}_{A}=N^{C}_{B}.$

In the new coordinate basis $\displaystyle(\partial/\partial Q{}^{\ast
A},\partial/\partial\tilde{f}^{a},\partial/\partial a^{\alpha})$ the metric
(9) of the original manifold $\mathcal{P}\times\mathcal{V}$ is written as
follows:

$\displaystyle{\tilde{G}}_{\cal A\cal
B}(Q{}^{\ast},\tilde{f},a)=\\!\left(\begin{array}[]{ccc}G_{CD}(P_{\bot})^{C}_{A}(P_{\bot})^{D}_{B}&0&G_{CD}(P_{\bot})^{C}_{A}K^{D}_{\nu}\bar{u}^{\nu}_{\alpha}\\\
0&G_{ab}&G_{ap}K^{p}_{\nu}\bar{u}^{\nu}_{\alpha}\\\
G_{BC}K^{C}_{\mu}\bar{u}^{\mu}_{\beta}&G_{bp}K^{p}_{\nu}\bar{u}^{\nu}_{\beta}&d_{\mu\nu}\bar{u}^{\mu}_{\alpha}\bar{u}^{\nu}_{\beta}\\\
\end{array}\right),$ (14)

where $G_{CD}(Q{}^{\ast})\equiv G_{CD}(F(Q{}^{\ast},e))$:

$G_{CD}(Q{}^{\ast})=F^{M}_{C}(Q{}^{\ast},a)F^{N}_{D}(Q{}^{\ast},a)G_{MN}(F(Q{}^{\ast},a)),$

the projection operator $P_{\bot}$ and the components $K^{A}_{\mu}$ of the
Killing vector fields depend on $Q{}^{\ast}$,
$\bar{u}^{\mu}_{\beta}=\bar{u}^{\mu}_{\beta}(a)$,
$K^{p}_{\nu}=K^{p}_{\nu}(\tilde{f})$, $d_{\mu\nu}(Q{}^{\ast},\tilde{f})$ is
the tensor used to determine the metric given on the orbits of the group
action. Its components are given by the relation

$\displaystyle d_{\mu\nu}(Q{}^{\ast},\tilde{f})$ $\displaystyle=$
$\displaystyle
K^{A}_{\mu}(Q{}^{\ast})G_{AB}(Q{}^{\ast})K^{B}_{\nu}(Q{}^{\ast})+K^{a}_{\mu}(\tilde{f})G_{ab}K^{b}_{\nu}(\tilde{f})$
$\displaystyle\equiv$
$\displaystyle\gamma_{\mu\nu}(Q^{\ast})+\gamma^{\prime}_{\mu\nu}(\tilde{f}).$

Also we note that when obtaining (14), the following transformations were
used:

$df^{a}=\bar{D}^{a}_{p}(a)d{\tilde{f}}^{p}+\frac{\partial\bar{D}^{a}_{p}(a)}{\partial
a^{\mu}}\tilde{f}^{p}da^{\mu}$

and

$\frac{\partial\bar{D}^{a}_{p}(a)}{\partial
a^{\mu}}\tilde{f}^{p}=(\bar{J}_{\beta})^{c}_{p}\bar{D}^{a}_{c}(a){\bar{u}}^{\beta}_{\mu}(a)\tilde{f}^{p}=K^{c}_{\beta}(\tilde{f})\bar{D}^{a}_{c}(a){\bar{u}}^{\beta}_{\mu}(a).$

The last equality is due to the identity
$D^{c}_{b}(a)(\bar{J}_{\alpha})^{b}_{p}\bar{D}^{p}_{e}(a)={\rho}^{\beta}_{\alpha}(a)(\bar{J}_{\beta})^{c}_{e}$,
in which
${\rho}^{\beta}_{\alpha}(a)=\bar{u}^{\beta}_{\gamma}(a)v^{\gamma}_{\alpha}(a)$
is the matrix of the adjoint representation of the group $\mathcal{G}$.

The pseudoinverse matrix ${\tilde{G}}^{\cal A\cal B}(Q{}^{\ast},\tilde{f},a)$
to the matrix (14) is given by the following expression:

$\displaystyle\left(\begin{array}[]{ccc}{G}^{EF}N_{E}^{A}N_{F}^{B}&-G^{EF}N^{A}_{E}{\Lambda}^{\nu}_{F}K^{a}_{\nu}&G^{EF}N^{A}_{E}{\Lambda}^{\beta}_{F}\bar{v}^{\alpha}_{\beta}\\\
-G^{EF}N^{B}_{F}{\Lambda}^{\nu}_{E}K^{b}_{\nu}&G^{ba}+G^{EF}{\Lambda}^{\nu}_{E}{\Lambda}^{\mu}_{F}K^{b}_{\nu}K^{a}_{\mu}&-G^{EF}{\Lambda}^{\nu}_{E}{\Lambda}^{\mu}_{F}K^{b}_{\nu}{\bar{v}}^{\alpha}_{\mu}\\\
G^{EF}N^{B}_{F}{\Lambda}^{\mu}_{E}\bar{v}_{\mu}^{\beta}&-G^{EF}{\Lambda}^{\nu}_{E}{\Lambda}^{\mu}_{F}K^{a}_{\mu}{\bar{v}}^{\beta}_{\nu}&G^{EF}{\Lambda}^{\nu}_{E}{\Lambda}^{\mu}_{F}{\bar{v}}^{\alpha}_{\nu}{\bar{v}}^{\beta}_{\mu}\\\
\end{array}\right).$ (15)

Here
${\Lambda}^{\nu}_{E}\equiv({\Phi}^{-1})^{\nu}_{\mu}(Q{}^{\ast}){\chi}^{\mu}_{E}(Q{}^{\ast})$.

The pseudoinversion of ${\tilde{G}}_{\cal A\cal B}$ means that

$\displaystyle{\tilde{G}}^{\tilde{\cal A}\tilde{\cal
D}}{\tilde{G}}_{\tilde{\cal D}\tilde{\cal
B}}=\left(\begin{array}[]{ccc}(P_{\bot})^{A}_{B}&0&0\\\
0&{\delta}^{a}_{b}&0\\\ 0&0&{\delta}^{\alpha}_{\beta}\\\ \end{array}\right).$

The determinant of the matrix (14) is defined as

$\det{\tilde{G}}_{{\cal A\cal B}}=(\det
d_{\mu\nu})\,(\det{\bar{u}}^{\mu}_{\nu})^{2}\,H,\,\;\;\;\;\;\rm{with}$
$\displaystyle
H=\displaystyle\det\left(\begin{array}[]{cc}(P_{\bot})^{A^{\prime}}_{A}\tilde{G}^{\rm
H}_{A^{\prime}B^{\prime}}(P_{\bot})^{B^{\prime}}_{B}&(P_{\bot})^{A^{\prime}}_{A}\tilde{G}^{\rm
H}_{A^{\prime}a}\\\ (P_{\bot})^{B^{\prime}}_{B}\tilde{G}^{\rm
H}_{bB^{\prime}}&\tilde{G}^{\rm H}_{ab}\\\ \end{array}\right),$ (18)

where ${\tilde{G}}^{\rm
H}_{AB}=G_{AB}-G_{AC}K^{C}_{\mu}d^{\mu\nu}K^{D}_{\nu}G_{DB},$
$\;\;\tilde{G}^{\rm H}_{Aa}=-G_{AB}K^{B}_{\mu}d^{\mu\nu}K^{b}_{\nu}G_{ba}$,
$\;\tilde{G}^{\rm
H}_{ba}=G_{ba}-G_{bc}K^{c}_{\mu}d^{\mu\nu}K_{\nu}^{p}G_{pa}$.

Note that $\det{\tilde{G}}_{{\cal A\cal B}}$ does not vanish only on the
surface $\tilde{\Sigma}$. On this surface $\det(P_{\bot})^{A}_{B}$ is equal to
unity.

It is also worth noting that the matrix (14), representing the metric on
$\mathcal{P}\times\mathcal{V}$, can also be written in terms of the components
of the mechanical connection existing in the principal fiber bundle $\rm
P(\tilde{\mathcal{M}},\mathcal{G})$. This connection is given by a Lie-algebra
valued one-form, which in local coordinates is represented as

$\omega^{\alpha}={\mathscr{A}}^{\alpha}_{D}\,dQ^{\ast}{}^{D}+{\mathscr{A}}^{\alpha}_{p}d\tilde{f}^{p}+u^{\alpha}_{\mu}da^{\mu},$

where

${\mathscr{A}}^{\alpha}_{D}(Q^{\ast},\tilde{f})=d^{\alpha\beta}K^{C}_{\beta}G_{DC},\;\;\;{\mathscr{A}}^{\alpha}_{p}(Q^{\ast},\tilde{f})=d^{\alpha\beta}K^{a}_{\beta}G_{ap}.$

In this case, ${\tilde{G}}_{\tilde{\mathcal{A}}\tilde{\mathcal{B}}}$ is given
by the following expression:

$\displaystyle\left(\begin{array}[]{ccc}(P_{\bot})^{A^{\prime}}_{A}(P_{\bot})^{B^{\prime}}_{B}\bigl{(}\tilde{G}^{\rm
H}_{A^{\prime}B^{\prime}}+d_{\mu\nu}{\mathscr{A}}_{A^{\prime}}^{\mu}{\mathscr{A}}_{B^{\prime}}^{\nu}\bigr{)}&0&(P_{\bot})^{A^{\prime}}_{A}{\mathscr{A}}_{A^{\prime}}^{\mu}d_{\mu\nu}\bar{u}^{\nu}_{\alpha}\\\
0&G_{ab}&{\mathscr{A}}^{\mu}_{a}d_{\mu\nu}{\bar{u}}^{\nu}_{\alpha}\\\
(P_{\bot})^{B^{\prime}}_{B}{\mathscr{A}}^{\mu}_{B^{\prime}}d_{\mu\nu}{\bar{u}}^{\nu}_{\beta}&{\mathscr{A}}^{\mu}_{b}d_{\mu\nu}{\bar{u}}^{\nu}_{\beta}&d_{\mu\nu}{\bar{u}}^{\mu}_{\alpha}{\bar{u}}^{\nu}_{\beta}\\\
\end{array}\\!\\!\right)\\!.$ (19)

The pseudoinverse matrix to the matrix (19) is

$\displaystyle\left(\begin{array}[]{ccc}G^{A^{\prime}B^{\prime}}N^{A}_{A^{\prime}}N^{B}_{B^{\prime}}&G^{A^{\prime}B^{\prime}}N^{C}_{A^{\prime}}N^{B}_{B^{\prime}}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\mu}_{C}}K^{a}_{\mu}&-G^{A^{\prime}B^{\prime}}N^{C}_{A^{\prime}}N^{B}_{B^{\prime}}\,\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\beta}_{C}}\bar{v}^{\alpha}_{\beta}\\\
G^{A^{\prime}B^{\prime}}N^{A}_{A^{\prime}}N^{C}_{B^{\prime}}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\mu}_{C}}K^{b}_{\mu}&G^{AB}N^{a}_{A}N^{b}_{B}+G^{ab}&-G^{EC}{\Lambda}^{\beta}_{E}{\Lambda}^{\mu}_{C}K^{b}_{\mu}\bar{v}^{\alpha}_{\beta}\\\
-G^{A^{\prime}B^{\prime}}N^{A}_{A^{\prime}}N^{C}_{B^{\prime}}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\varepsilon}_{C}}\bar{v}^{\beta}_{\varepsilon}&-G^{EC}{\Lambda}^{\varepsilon}_{E}{\Lambda}^{\mu}_{C}K^{a}_{\mu}\bar{v}^{\beta}_{\varepsilon}&G^{BC}{\Lambda}^{\alpha^{\prime}}_{B}{\Lambda}^{\beta^{\prime}}_{C}\bar{v}^{\alpha}_{\alpha^{\prime}}v^{\beta}_{\beta^{\prime}}\end{array}\\!\\!\right)\\!,$
(20)

where
$\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\mu}_{C}}=\gamma^{\mu\nu}K^{B}_{\nu}G_{BC}$,
$N^{a}_{B}=-K^{a}_{\mu}{\Lambda}^{\mu}_{B}$ is the component of the projection
operator $N^{\tilde{A}}_{\tilde{B}}=(N^{A}_{B},0,N^{a}_{B},\delta^{a}_{b})$,
(in our notation, the index $\tilde{A}=(A,a)$). Since
$N^{\tilde{A}}_{\tilde{B}}K^{\tilde{B}}_{\alpha}=0$,
$N^{\tilde{A}}_{\tilde{B}}$ projects onto planes perpendicular to the gauge
orbits in ${\rm P}(\tilde{\mathcal{M}},\mathcal{G})$.

Note that the elements of the matrix (20) can also be rewritten somewhat
differently, using the following identities:

$G^{BB^{\prime}}{\Lambda}^{\alpha}_{B}{\Lambda}^{\beta}_{B^{\prime}}=(\gamma^{\alpha\beta}+G^{A^{\prime}B^{\prime}}N^{A}_{A^{\prime}}N^{B}_{B^{\prime}}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\alpha}_{A}}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\beta}_{B}}),$
$G^{AB}N^{a}_{A}N^{b}_{B}+G^{ab}=\bigl{(}\gamma^{\alpha\beta}+G^{A^{\prime}B^{\prime}}N^{A}_{A^{\prime}}N^{B}_{B^{\prime}}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\alpha}_{A}}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\beta}_{B}}\bigr{)}K^{a}_{\alpha}K^{b}_{\beta}+G^{ab}.$

##  4 Stochastic differential equations in coordinates of the principal fiber
bundle

Transition from the original coordinates $(Q^{A},f^{a})$, given on the local
chart of the manifold $\tilde{\mathcal{P}}$, to the special coordinates
$(Q^{\ast}{}^{A},\tilde{f}^{b},a^{\alpha})$ leads to the need for a
corresponding replacement of the original stochastic process $\eta_{t}$
generating the measure in the path integral (3). The local components
$(Q^{\ast}_{t}{}^{A},\tilde{f}^{b}_{t},a^{\alpha}_{t})$ of a new process must
be solutions of those local stochastic differential equations that can be
obtained from the equations (6) and (7) as a result of the change of variables
performed in stochastic processes. These equations can be derived using the
Itô’s differentiation formula.

Consider, for example, how to obtain a stochastic differential equation for
the component $Q^{\ast}{}^{A}(t)(\equiv Q^{\ast}_{t}{}^{A})$ of a new
stochastic process. We know that the local coordinate $Q^{\ast}{}^{A}$ is
defined by the following equation $Q^{\ast}{}^{A}=F^{A}(Q,a^{-1}(Q))$ . The
same equation must hold for the local stochastic variable $Q^{\ast}_{t}{}^{A}$
and the stochastic variable $\eta^{A}_{1}(t)$:
$\eta_{1}^{A}(t)=F^{A}(Q^{\ast}{}^{A}(t),a^{\alpha}(t))$. Then, applying the
Itô’s differentiation formula to the stochastic variable $Q^{\ast A}_{t}$, we
obtain

$dQ^{\ast}_{t}{}^{A}=\Bigl{(}\frac{\partial Q^{\ast}{}^{A}}{\partial
Q^{B}}\Bigr{)}d\eta^{B}_{1}(t)+\frac{1}{2}\frac{\partial^{2}Q^{\ast}{}^{A}}{\partial
Q^{C}\partial Q^{D}}<d\eta^{C}_{1}(t),d\eta^{D}_{1}(t)>.$

Using the stochastic differential equation (6) for $\eta_{1}^{A}(t)$ in this
equation and re-expressing $\eta_{1}^{A}(t)$ through $Q^{\ast}{}^{A}(t)$ and
$a^{\alpha}(t)$ in all terms depending on $\eta_{1}^{A}(t)$ on the right-hand
side of the above equation, we obtain expressions for the drift and diffusion
coefficients of the stochastic differential equation for $Q^{\ast}_{t}{}^{A}$.
A similar approach can be used to write down the stochastic differential
equations for the processes $\tilde{f}^{b}_{t}$ and $a^{\alpha}_{t}$.

Another way to derive stochastic differential equations for transformed
variables is based on the use of the Laplace-Beltrami operator which is
obtained from the original differential operator of the equation (1) as a
result of introducing new coordinates on the manifold $\tilde{\mathcal{P}}$.
Namely, the drift coefficients of these equations will be represented by the
coefficients at the first derivatives of thus obtained Laplace-Beltrami
operator, and the diffusion coefficients can be found from the coefficients at
the second partial derivatives of this operator. Using this method, we get the
following stochastic differential equations:

$\displaystyle
dQ^{\ast}_{t}{}^{B}=\frac{1}{2}(\mu^{2}\kappa)\Bigl{[}\frac{1}{\sqrt{dH}}\frac{\partial}{\partial
Q^{\ast}{}^{A}}\Bigl{(}\sqrt{dH}G^{A^{\prime}B^{\prime}}N^{A}_{A^{\prime}}N^{B}_{B^{\prime}}\Bigr{)}$
$\displaystyle+\frac{1}{\sqrt{dH}}\frac{\partial}{\partial\tilde{f}^{a}}\Bigl{(}\sqrt{dH}G^{A^{\prime}B^{\prime}}N^{C}_{A^{\prime}}N^{B}_{B^{\prime}}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\mu}_{C}}K^{a}_{\mu}\Bigr{)}\Bigr{]}dt+\mu\sqrt{\kappa}N^{B}_{C}\mathscr{X}^{C}_{\bar{M}}dw^{\bar{M}}_{t},$
(21)

where $\mathscr{X}^{A}_{\bar{M}}(Q^{\ast}_{t})$ is determined from the local
equality
$\sum_{\bar{M}=1}^{n_{\mathcal{P}}}\mathscr{X}^{A}_{\bar{M}}\mathscr{X}^{B}_{\bar{M}}=G^{AB}$.

$\displaystyle
d\tilde{f}^{b}_{t}=\frac{1}{2}(\mu^{2}\kappa)\Bigl{[}\frac{1}{\sqrt{dH}}\frac{\partial}{\partial
Q^{\ast}{}^{A}}\Bigl{(}\sqrt{dH}G^{A^{\prime}B^{\prime}}N^{A}_{A^{\prime}}N^{B}_{B^{\prime}}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\mu}_{B}}K^{b}_{\mu}\Bigr{)}$
$\displaystyle\;\;\;\;\;\;\;\;\;+\frac{1}{\sqrt{dH}}\frac{\partial}{\partial\tilde{f}^{a}}\Bigl{(}\sqrt{dH}(G^{ab}+G^{AB}N^{a}_{A}N^{b}_{B})\Bigr{)}\Bigr{]}dt$
$\displaystyle\;\;\;\;\;\;\;\;\;+\mu\sqrt{\kappa}(N^{b}_{C}\mathscr{X}^{C}_{\bar{M}}dw^{\bar{M}}_{t}+\mathscr{X}^{b}_{\bar{c}}dw^{\bar{c}}_{t}),$
(22)

and where $\mathscr{X}^{a}_{\bar{c}}$ is defined by the local equality
$\sum_{\bar{c}=1}^{n_{\scriptscriptstyle{\mathcal{V}}}}\mathscr{X}^{a}_{\bar{c}}\mathscr{X}^{b}_{\bar{c}}=G^{ab}$.

$\displaystyle
da^{\alpha}_{t}=\frac{1}{2}(\mu^{2}\kappa)\Bigl{[}\frac{1}{\sqrt{dH}}\frac{\partial}{\partial
Q^{\ast}{}^{A}}\Bigl{(}-\sqrt{dH}\,G^{EF}N^{A}_{E}N^{C}_{F}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\beta}_{C}}\Bigr{)}\bar{v}^{\alpha}_{\beta}$
$\displaystyle\;\;\;\;\;\;\;\;\;+\frac{1}{\sqrt{dH}}\frac{\partial}{\partial\tilde{f}^{b}}\Bigl{(}-\sqrt{dH}\,G^{EC}\Lambda^{\beta}_{E}\Lambda^{\mu}_{C}K^{b}_{\mu}\Bigr{)}\bar{v}^{\alpha}_{\beta}$
$\displaystyle\;\;\;\;\;\;\;\;\;+G^{BC}\Lambda^{\alpha^{\prime}}_{B}\Lambda^{\beta^{\prime}}_{C}\bar{v}^{\beta}_{\beta^{\prime}}\frac{\partial}{\partial
a^{\beta}}\bar{v}^{\alpha}_{\alpha^{\prime}}\Bigr{]}dt+\mu\sqrt{\kappa}\Lambda_{C}^{\beta}\bar{v}^{\alpha}_{\beta}\mathscr{X}^{C}_{\bar{M}}dw^{\bar{M}}_{t}.$
(23)

A short notation of the above stochastic differential equations is as follows:

$\displaystyle dQ^{\ast}_{t}{}^{B}$
$\displaystyle=(\mu^{2}\kappa)b^{B}_{t}dt+\mu\sqrt{\kappa}N^{B}_{C}\mathscr{X}^{C}_{\bar{M}}dw^{\bar{M}}_{t},$
$\displaystyle d\tilde{f}^{a}_{t}$
$\displaystyle=(\mu^{2}\kappa)b^{a}_{t}dt+\mu\sqrt{\kappa}(N^{a}_{C}\mathscr{X}^{C}_{\bar{M}}dw^{\bar{M}}_{t}+\mathscr{X}^{a}_{\bar{c}}dw^{\bar{c}}_{t}),$
$\displaystyle da^{\alpha}_{t}$
$\displaystyle=(\mu^{2}\kappa)a^{\alpha}_{t}dt+\mu\sqrt{\kappa}\Lambda_{C}^{\beta}\bar{v}^{\alpha}_{\beta}\mathscr{X}^{C}_{\bar{M}}dw^{\bar{M}}_{t}.$

The drift coefficients of the first two equations can be rewritten to obtain
the stochastic differential equations, which are used in stochastic theory to
describe diffusion on a manifold embedded in some ambient manifold (cf.
ref.[20]).

It can be shown that these drift coefficients are equal to the coefficients
standing at the first partial derivatives with respect to $Q^{\ast}{}^{A}$ and
$\tilde{f}^{b}$ in the Laplacian obtained by regrouping the terms of the
original Laplace-Beltrami operator in the following form:777Here and in what
follows, we adopt the notation in which subscripts (and superscripts),
indicated by capital letters with a tilde, denote multi-indices consisting of
two indices, for example $\tilde{B}=(B,b)$. It is also assumed that summation
occurs over these repeated indices.

$\displaystyle\frac{1}{2}{\tilde{G}}^{\tilde{C}\tilde{B}}\nabla_{\tilde{C}}\nabla_{\tilde{B}}=$
(24)
$\displaystyle\frac{1}{2}\Bigl{(}\tilde{\Pi}^{\tilde{A}}_{\tilde{C}}\,{\tilde{G}}^{\tilde{C}\tilde{B}}\nabla_{\tilde{A}}\nabla_{\tilde{B}}-K^{\tilde{A}}_{\alpha}d^{\alpha\beta}(\nabla_{\tilde{A}}K^{\tilde{B}}_{\beta})\nabla_{\tilde{B}}+K^{\tilde{A}}_{\alpha}d^{\alpha\beta}\nabla_{\tilde{A}}K^{\tilde{B}}_{\beta}\nabla_{\tilde{B}}\Bigr{)},$

where it is also necessary to replace the local coordinates $(Q^{A},f^{b})$
with the coordinates $(Q^{\ast}{}^{A},\tilde{f}^{b},a^{\alpha})$.

In the first term on the right side of this equation, the horizontal
projection operator

$\tilde{\Pi}=\left(\begin{array}[]{cc}\tilde{\Pi}^{D}_{C}&\tilde{\Pi}^{D}_{a}\\\
\tilde{\Pi}^{b}_{C}&\tilde{\Pi}^{b}_{a}\end{array}\right)$

such that $\tilde{\Pi}^{\tilde{D}}_{\tilde{C}}K^{\tilde{C}}_{\alpha}=0$, with
the components

$\displaystyle\tilde{\Pi}^{D}_{C}$
$\displaystyle=\delta^{D}_{C}-K^{D}_{\alpha}d^{\alpha\beta}K^{B}_{\beta}G_{BC},\;\;\tilde{\Pi}^{D}_{a}=-K^{D}_{\alpha}d^{\alpha\beta}K^{b}_{\beta}G_{ba},$
$\displaystyle\tilde{\Pi}^{b}_{C}$
$\displaystyle=-K^{b}_{\alpha}d^{\alpha\beta}K^{B}_{\beta}G_{BC},\;\;\;\;\;\;\;\;\,\tilde{\Pi}^{b}_{a}=\delta^{b}_{a}-K^{b}_{\alpha}d^{\alpha\beta}K^{q}_{\beta}G_{qa},$

has the following properties:

$\tilde{\Pi}^{\tilde{A}}_{\tilde{L}}N^{\tilde{L}}_{C}=\tilde{\Pi}^{\tilde{A}}_{C},\;\;\tilde{\Pi}^{E}_{B}N^{R}_{E}=N^{R}_{B},\;\;\tilde{\Pi}^{E}_{a}N^{R}_{E}=0.$

Note also that in our case

$\bigl{(}{\tilde{G}}^{\tilde{C}\tilde{B}}\nabla_{\tilde{C}}\nabla_{\tilde{B}}\bigr{)}(Q,f)=\bigl{(}G^{AB}\nabla_{A}\nabla_{B}\bigr{)}(Q)+G^{pq}\bigl{(}\nabla_{p}\nabla_{q}\bigr{)}(f).$

The symbol $\nabla_{A}$ denotes the covariant derivative defined using the
Christoffel symbols of the manifold $\mathcal{P}$. However, due to the special
choice of the initial metric on the manifold $\tilde{\mathcal{P}}$, in our
case we have $\nabla_{q}(f)=\frac{\partial}{\partial f^{q}}$.

As a result of the calculations, it follows that in each of the equations (21)
and (22), the drift coefficients are represented by the sum of two terms, $b$
and $b$, coming only from the first and the second terms on the right-hand
side of the equation (24). The third term does not contribute to the drift
coefficients of stochastic differential equations on the $\tilde{\Sigma}$.

###  4.1 The drift coefficient $b$ as a projection of the mean curvature
vector field of the orbit onto the submanifold $\tilde{\Sigma}$

In this subsection we consider the geometry of the drift term $b$. Our
calculation show that this term is given by the projection on the submanifold
$\tilde{\Sigma}$ of the mean curvature vector of the orbit.

In our case, the Killing vector fields which in the coordinates
$(Q^{A},f^{a})$ are given by the formula

$\tilde{K}_{\alpha}=K^{A}_{\alpha}(Q)\frac{\partial}{\partial
Q^{A}}+K^{a}_{\alpha}(f)\frac{\partial}{\partial f^{a}},$

are tangent to the group orbit.

The mean curvature vector field (mean curvature normal) of the orbit is
defined as follows:

$\displaystyle\vec{\mu}=\frac{1}{2}d^{\alpha\beta}\biggl{[}\Bigl{(}\tilde{\Pi}^{D}_{C}(\nabla_{K_{\alpha}}K_{\beta})^{C}+\tilde{\Pi}^{D}_{a}(\nabla_{K_{\alpha}}K_{\beta})^{a}\Bigr{)}\frac{\partial}{\partial
Q{}^{D}}$
$\displaystyle\;\;\;\;\;\;\;+\Bigl{(}\tilde{\Pi}^{b}_{C}(\nabla_{K_{\alpha}}K_{\beta})^{C}+\tilde{\Pi}^{b}_{a}(\nabla_{K_{\alpha}}K_{\beta})^{a}\Bigr{)}\frac{\partial}{\partial
f^{b}}\biggr{]}.$ (25)

By $(\nabla_{K_{\alpha}}K_{\beta})^{C}(Q)$, used in $\vec{\mu}$, we denote

$K^{A}_{\alpha}(Q)\frac{\partial}{\partial
Q^{A}}K^{C}_{\beta}(Q)+K^{A}_{\alpha}(Q)K^{B}_{\beta}(Q)\Gamma^{C}_{AB}(Q),$

where

$\Gamma^{C}_{AB}(Q)=\frac{1}{2}G^{CE}(Q)\Bigl{(}\frac{\partial}{\partial
Q^{A}}G_{EB}(Q)+\frac{\partial}{\partial
Q^{B}}G_{EA}(Q)-\frac{\partial}{\partial Q^{E}}G_{AB}(Q)\Bigr{)}.$

To obtain the projection of $\vec{\mu}$ onto the submanifold
${\tilde{\Sigma}}$, one must first find how the terms on the right-hand side
of the equation (25 ) are expressed in terms of the coordinates $(Q{}^{\ast
A},\tilde{f}^{a},a^{\alpha})$. It can be shown that
$\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{b}(f)$ is transformed as

$\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{b}(f)={\rho}^{\mu}_{\alpha}(a){\rho}^{\nu}_{\beta}(a)K^{q}_{\mu}(\tilde{f})(\bar{J}_{\nu})^{c}_{q}\bar{D}^{b}_{c}(a).$

In derivation of this representation it was used the formula

$D^{q}_{b}(a)(\bar{J}_{\nu})^{b}_{p}\bar{D}^{p}_{c}(a)={\rho}^{\beta}_{\nu}(a)(\bar{J}_{\beta})^{q}_{c}.$

For $\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{A}(Q)$ we have the
following expression:

$\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{A}(Q)=F^{A}_{D}K^{D}_{\mu}(Q^{\ast})c^{\nu}_{\alpha\beta}\rho^{\mu}_{\nu}+F^{A}_{D}\rho^{\sigma}_{\alpha}\rho^{\mu}_{\beta}(\nabla_{K_{\sigma}}K_{\mu}\bigr{)}^{D}(Q^{\ast}).$

As a result of the performed transformations, the mean curvature vector field
$\vec{\mu}(Q{}^{\ast},\tilde{f},a)$ is represented as

$\displaystyle\vec{\mu}=\frac{1}{2}d^{\alpha\beta}\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{B}(Q^{\ast})\Bigl{(}N^{R}_{B}\frac{\partial}{\partial
Q{}^{\ast
R}}+{\Pi}^{C}_{B}\Lambda^{\mu}_{C}\bar{v}^{\nu}_{\mu}(a)\frac{\partial}{\partial
a^{\nu}}+N^{b}_{B}\frac{\partial}{\partial\tilde{f}^{b}}\Bigr{)}$
$\displaystyle\;\;\;\;+\frac{1}{2}d^{\alpha\beta}\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{b}(\tilde{f})\Bigl{(}{\Pi}^{E}_{b}\Lambda^{\mu}_{E}\bar{v}^{\nu}_{\mu}(a)\frac{\partial}{\partial
a^{\nu}}+\frac{\partial}{\partial\tilde{f}^{b}}\Bigr{)}.$ (26)

(We recall that $N^{b}_{B}=-\Lambda^{\mu}_{B}K^{b}_{\mu}.$)

The projection of the vector field $\vec{\mu}$ $(Q{}^{\ast},\tilde{f},a)$ on
the submanifold $\tilde{\Sigma}\in\mathcal{P}\times\mathcal{V}$ is defined by
the following formula:

$\displaystyle{\tilde{G}}^{SL}{\tilde{G}}\Bigl{(}\vec{\mu},\frac{\partial}{\partial
Q{}^{\ast S}}\Bigr{)}\frac{\partial}{\partial Q{}^{\ast
L}}+{\tilde{G}}^{Sq}{\tilde{G}}\Bigl{(}\vec{\mu},\frac{\partial}{\partial
Q{}^{\ast S}}\Bigr{)}\frac{\partial}{\partial{\tilde{f}}^{q}}$
$\displaystyle+{\tilde{G}}^{aq}{\tilde{G}}\Bigl{(}\vec{\mu},\frac{\partial}{\partial{\tilde{f}}^{a}}\Bigr{)}\frac{\partial}{\partial{\tilde{f}}^{q}}+{\tilde{G}}^{aL}{\tilde{G}}\Bigl{(}\vec{\mu},\frac{\partial}{\partial{\tilde{f}}^{a}}\Bigr{)}\frac{\partial}{\partial
Q{}^{\ast L}}.$

Applying this formula, we get the following representation for
$\vec{b}(Q^{\ast},\tilde{f})$:

$\displaystyle\vec{b}=-\frac{1}{2}\,G^{CC^{\prime}}N^{L}_{C^{\prime}}N^{B^{\prime}}_{C}G_{BB^{\prime}}\,d^{\alpha\beta}\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{B}\frac{\partial}{\partial
Q{}^{\ast L}}$
$\displaystyle\;\;\;\;\;\;\;\;\;-\frac{1}{2}d^{\alpha\beta}\Bigl{(}N^{q}_{B}\,\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{B}+\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{q}\Bigr{)}\frac{\partial}{\partial{\tilde{f}}^{q}}.$
(27)

Since the first line of (27) can be also rewritten as

$-\frac{1}{2}\,d^{\alpha\beta}N^{L}_{B}(Q{}^{\ast})\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{B}\,\frac{\partial}{\partial
Q{}^{\ast L}},$

then, using the shorthand notation, $\vec{b}$ can be written in the following
form:

$\vec{b}=-\frac{1}{2}\,d^{\alpha\beta}N^{\tilde{L}}_{\tilde{B}}\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{\tilde{B}}\,\frac{\partial}{\partial\tilde{Q}{}^{{\tilde{L}}}},$

(that is, here $\tilde{L}=(L,a)$ and
$\tilde{Q}^{\tilde{L}}\equiv(Q{}^{{\ast}L},\tilde{f}^{a})$).

###  4.2 The drift coefficient $b^{A}$

In this subsection, we show that the drift coefficient $b^{A}$ can be obtained
using the corresponding stochastic differential equation given on the orbit
space $\tilde{\mathcal{M}}$ (which is locally isomorphic to $\tilde{\Sigma}$).
The differential generator of the semigroup associated with the stochastic
process described by this stochastic differential equation arises from the
first term on the right-hand side of the equation (24) after its restriction
to the manifold $\tilde{\mathcal{M}}$. The differential generator thus
obtained is the Laplace-Beltrami operator on $\tilde{\mathcal{M}}$.

Note that in our case, the local coordinates on the orbit space manifold
$\tilde{\mathcal{M}}$ are given by the invariant coordinates
$(x^{i},\tilde{f}^{a})$, where $x^{i},i=1,....,n_{\mathcal{M}},$ are
coordinates on $\mathcal{M}$. They are defined under the condition that the
local submanifold $\Sigma$ can also be represented in the parametric form:
$Q^{A}=Q^{\ast A}(x^{i})$. This implies that the coordinates of points on
$\Sigma$ must satisfy the equation $\chi^{\alpha}(Q^{\ast}{}^{A}(x^{i}))=0$.

The metric tensor of the Riemannian manifold $\tilde{\mathcal{M}}$ has the
following components in the coordinate basis $(\frac{\partial}{\partial
x^{i}},\frac{\partial}{\partial\tilde{f}^{a}})$:

$\left(\begin{array}[]{cc}\tilde{h}_{ij}&\tilde{G}^{\rm H}_{Bb}Q_{i}^{*B}\\\
\tilde{G}^{\rm H}_{Aa}Q_{j}^{*A}&\tilde{G}^{\rm H}_{ba}\\\
\end{array}\right),$ (28)

where $\tilde{h}_{ij}=Q^{\ast}{}^{A}_{i}\tilde{G}{}^{\rm
H}_{AB}(Q^{\ast}(x))Q^{\ast}{}^{B}_{j}$ with
$Q^{\ast}{}^{B}_{i}\equiv\frac{\partial}{\partial x^{i}}Q^{\ast}{}^{B}(x)$.
This means that the Riemannian manifold $\tilde{\mathcal{M}}$ can be
considered as a submanifold in the (Riemannian) manifold
$(\tilde{\mathcal{P}},\tilde{G}^{\rm H}_{\tilde{A}\tilde{B}})$ with degenerate
metric $\tilde{G}^{\rm H}_{\tilde{A}\tilde{B}}$.

The elements of the inverse matrix to matrix $(\ref{metric_h_ija})$ are given
by

$\displaystyle\tilde{h}^{ij}=h^{ij}:=h^{SD}T^{i}_{D}T^{j}_{S}=G^{EF}N^{S}_{E}N^{D}_{F}T^{i}_{D}T^{j}_{S},\;\;\tilde{h}^{jb}:=h^{bP}T^{j}_{P}=G^{EF}N^{b}_{F}N^{P}_{E}T^{j}_{P},$
$\displaystyle\tilde{h}^{ab}:=h^{ab}=G^{ab}+G^{EF}N^{a}_{E}N^{b}_{F},$

where the operator

$T^{i}_{D}=(P_{\bot})^{B}_{D}(Q^{\ast}(x))G^{\rm H}_{BL}(Q^{\ast}(x))Q^{\ast
L}_{m}(x)h^{mi}(x),$

with the properties $T^{i}_{D}Q^{\ast D}_{j}=\delta^{i}_{j}$ and
$T^{i}_{D}Q^{\ast B}_{i}=(P_{\bot})^{B}_{D}$, is defined by using the
projection operator $P_{\bot}$ on the tangent plane to the submanifold
$\Sigma$ of the manifold $\mathcal{P}$.

The diffusion on $\tilde{\mathcal{M}}$ is given by the stochastic process
locally represented by the processes $(x^{i}_{t},\tilde{f}^{a}_{t})$. But our
goal is to obtain a description of diffusion on $\tilde{\mathcal{M}}$ in terms
of the local stochastic differential equations for stochastic variables
associated with the ambient manifold, that is in our case for the stochastic
variable $Q_{t}^{\ast}{}^{A}$, which is associated with to the dependent
variable $Q^{\ast}{}^{A}$, and for $\tilde{f}^{a}_{t}$.

Since on $\;\tilde{\mathcal{M}}$ the variable $Q^{\ast}{}^{A}$ is now the
function of $x^{i}$, the stochastic differential equation for
$Q^{\ast}{}^{A}(x(t))$ can be obtained by means of the replacement of the
variables in the stochastic differential equations:

$dQ^{\ast}{}^{A}(x(t))=Q^{\ast}{}^{A}_{i}(x(t))dx^{i}_{t}+\frac{1}{2}Q^{\ast}{}^{A}_{ij}(x(t))<dx^{i}_{t},dx^{j}_{t}>.$
(29)

In turn, the drift coefficient $b^{i}(t)$ of the stochastic differential
equation for $x_{t}$

$dx^{i}_{t}=\mu^{2}\kappa\,b^{i}(t)dt+\mu\sqrt{\kappa}X^{i}_{\bar{n}}(t)dw^{\bar{n}}_{t}$

can be easily found using the Laplace-Beltrami operator of the manifold
$\tilde{\mathcal{M}}$:

$b^{i}=\frac{1}{2}\Bigl{(}\frac{1}{H^{1/2}}\frac{\partial}{\partial
x^{j}}\bigl{(}H^{1/2}h^{ij}\bigr{)}+\frac{1}{H^{1/2}}\frac{\partial}{\partial\tilde{f}^{a}}\bigl{(}H^{1/2}\tilde{h}^{ai}\bigr{)}\Bigr{)}.$

On the other hand,

$b^{i}=-\frac{1}{2}\bigl{(}h^{kn}{\Gamma}^{i}_{kn}+\tilde{h}^{kb}{\Gamma}^{i}_{kb}+\tilde{h}^{am}{\Gamma}^{i}_{am}+\tilde{h}^{ab}{\Gamma}^{i}_{ab}\bigr{)},$
(30)

where ${\Gamma}$ are Christoffel symbols of $\tilde{\mathcal{M}}$ with the
metric tensor (28).

The Christoffel symbol ${\Gamma}^{i}_{kn}$ is defined as

${\Gamma}^{i}_{kn}=\tilde{h}^{im}{\Gamma}_{knm}+\tilde{h}^{ia}{\Gamma}_{kna},$
(31)

where

$\displaystyle{\Gamma}_{knm}=\frac{1}{2}(\tilde{h}_{km,n}+\tilde{h}_{nm,k}-\tilde{h}_{kn,m}),$
$\displaystyle{\Gamma}_{kna}=\frac{1}{2}(\tilde{h}_{ka,n}+\tilde{h}_{na,k}-\tilde{h}_{kn,a}).$

Note that $\tilde{h}_{kn,a}$ in the previous formula is the partial derivative
of $\tilde{h}_{kn}$ with respect to $\tilde{f}^{a}$. Using explicit
expressions for the components of the metric given on the orbit space
manifold, one can obtain the following representations for the above
Christoffel symbols:

$\displaystyle{\Gamma}_{knm}={}^{\mathrm{H}}{\tilde{\Gamma}}_{BMT}Q^{\ast}{}^{B}_{k}Q^{\ast}{}^{M}_{n}Q^{\ast}{}^{T}_{m}+{\tilde{G}}^{H}_{AB}Q^{\ast}{}^{A}_{kn}Q^{\ast}{}^{B}_{m},$
$\displaystyle{\Gamma}_{kna}={}^{\mathrm{H}}{\tilde{\Gamma}}_{BMa}Q^{\ast}{}^{B}_{k}Q^{\ast}{}^{M}_{n}+{\tilde{G}}^{H}_{Aa}Q^{\ast}{}^{A}_{mn},$
(32)

where

${}^{\mathrm{H}}{\tilde{\Gamma}}_{BMD}\equiv\frac{1}{2}({\tilde{G}}^{\rm
H}_{BD,M}+{\tilde{G}}^{\rm H}_{MD,B}-{\tilde{G}}^{\rm H}_{BM,D}),$
${}^{\mathrm{H}}{\tilde{\Gamma}}_{BMa}\equiv\frac{1}{2}({\tilde{G}}^{\rm
H}_{Ba,M}+{\tilde{G}}^{\rm H}_{MD,B}-{\tilde{G}}^{\rm H}_{BM,a}).$

To determine the Christoffel symbol
${}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{BM}$ we use the following
equality:

${}^{\mathrm{H}}{\tilde{\Gamma}}_{BM\tilde{D}}={\tilde{G}}^{\mathrm{H}}_{\tilde{R}\tilde{D}}{}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{BM}.$
(33)

Note, however, that this defines
${}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{BM}$ only modulo such terms
$T^{\tilde{M}}_{BC}$ that satisfy
${\tilde{G}}^{\mathrm{H}}_{A\tilde{M}}T^{\tilde{M}}_{BC}=0$. In addition, note
that by our notation (33) also means that

${}^{\mathrm{H}}{\tilde{\Gamma}}_{BMa}={\tilde{G}}^{\mathrm{H}}_{\tilde{R}a}{}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{BM}.$
(34)

Substituting (33) and (34) into (32), and then the result in (31), we obtain,
after appropriate transformations, the following representation for
${\Gamma}^{i}_{kn}$:

${\Gamma}^{i}_{kn}=T^{i}_{S}N^{S}_{R}\bigl{(}{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{BM}Q^{\ast}{}^{B}_{k}Q^{\ast}{}^{M}_{n}+Q^{\ast}{}^{R}_{kn}\bigr{)}.$

Such a representation is obtained using the following identities:

$N^{\tilde{D}}_{F}{\tilde{G}}^{\rm H}_{\tilde{R}\tilde{D}}={\tilde{G}}^{\rm
H}_{\tilde{R}F},\;\;\;G^{EF}{\tilde{G}}^{\rm
H}_{\tilde{R}F}={\tilde{\Pi}}^{E}_{\tilde{R}},\;\;\;N^{A}_{E}{\tilde{\Pi}}^{E}_{\tilde{R}}=N^{A}_{R},\;\;\;N^{A}_{E}{\tilde{\Pi}}^{E}_{r}=0.$

In the same way one can get that

$\displaystyle{\Gamma}^{i}_{kb}=T^{i}_{S}N^{S}_{R}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{Ab}Q^{\ast}{}^{A}_{k},\;\;{\Gamma}^{i}_{am}=T^{i}_{S}N^{S}_{R}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{aB}Q^{\ast}{}^{B}_{m},\;\;{\Gamma}^{i}_{ab}=T^{i}_{S}N^{S}_{R}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{ab}.$

Using the obtained Christoffel symbols in (30), it can be shown that the
expression for the drift coefficient $b^{A}$ of the equation (29) is given by
the following formula:

$\displaystyle
b^{A}=-\frac{1}{2}N^{A}_{R}\Bigl{(}G^{EF}N^{B}_{E}N^{M}_{F}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{BM}+G^{EF}N^{b}_{F}N^{B}_{E}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{Bb}+G^{EF}N^{a}_{E}N^{B}_{F}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{aB}$
$\displaystyle\;\;\;\;\;\;\;\;+(G^{ab}+G^{EF}N^{a}_{E}N^{b}_{F})\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{ab}+h^{kn}Q^{\ast}{}^{R}_{kn}\Bigr{)}+\frac{1}{2}h^{ij}Q^{\ast}{}^{A}_{ij},$
(35)

where the terms depend on $Q^{\ast}(x)$ and $\tilde{f}$ and, therefore, in
(29) on $Q^{\ast}(x_{t})$ and $\tilde{f}_{t}$. Also note that, using a
shorthand notation, $b^{A}$ can be rewritten as

$b^{A}=-\frac{1}{2}N^{A}_{R}\Bigl{(}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{\tilde{B}\tilde{M}}+h^{kn}Q^{\ast}{}^{R}_{kn}\Bigr{)}+\frac{1}{2}h^{ij}Q^{\ast}{}^{A}_{ij}.$
(36)

###  4.3 The drift coefficient $b^{a}$

The drift coefficient $b^{a}$ of the stochastic differential equation for the
stochastic variable $\tilde{f}^{a}_{t}$ can also be found using the Laplace-
Beltrami operator of the manifold $\tilde{\mathcal{M}}$. This term is given by
the following expression:

$\displaystyle
b^{a}=\frac{1}{2}\Bigl{(}\frac{1}{H^{1/2}}\frac{\partial}{\partial
x^{i}}\bigl{(}H^{1/2}\tilde{h}^{ia}\bigr{)}+\frac{1}{H^{1/2}}\frac{\partial}{\partial\tilde{f}^{b}}\bigl{(}H^{1/2}\tilde{h}^{ab}\bigr{)}\Bigr{)}.$

This expression can be rewritten using Christoffel symbolls as follows:

$b^{a}=-\frac{1}{2}\bigl{(}h^{kn}{\Gamma}^{a}_{kn}+\tilde{h}^{kb}{\Gamma}^{a}_{kb}+\tilde{h}^{bm}{\Gamma}^{a}_{bm}+\tilde{h}^{bc}{\Gamma}^{a}_{bc}\bigr{)}.$

We omit here the steps with the necessary transformations to obtain the
representation of $b^{a}$, since they are similar to what we did when
obtaining the drift coefficient $b^{A}$. The relationships between Christoffel
symbols required for these transformations are given for reference in
Appendix.

As a result, we get the following representation for $b^{a}$:

$\displaystyle
b^{a}=-\frac{1}{2}N^{a}_{R}\bigl{(}G^{EF}N^{B}_{E}N^{M}_{F}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{BM}+G^{EF}N^{B}_{E}N^{b}_{F}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{bB}+G^{EF}N^{B}_{E}N^{b}_{F}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{Bb}$
$\displaystyle\;\;\;\;\;\;\;\;+G^{EF}N^{b}_{E}N^{c}_{F}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{bc}+G^{bc}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{R}_{bc}+h^{kn}Q^{\ast}{}^{R}_{kn}\bigr{)}$
$\displaystyle\;\;\;\;\;\;\;\;-\frac{1}{2}\bigl{(}G^{EF}N^{\tilde{B}}_{E}N^{\tilde{M}}_{F}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{a}_{\tilde{B}\tilde{M}}+G^{bc}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{a}_{bc}\bigr{)}.$
(37)

The same expression can be rewritten as

$b^{a}=-\frac{1}{2}N^{a}_{\tilde{R}}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{\tilde{B}\tilde{M}}-\frac{1}{2}N^{a}_{A}h^{kn}Q^{\ast}{}^{A}_{kn}.$

Note that when getting (37), the following identities were used:

$N^{T}_{F}{\tilde{G}}^{\mathrm{H}}_{\tilde{R}T}+N^{b}_{F}{\tilde{G}}^{\mathrm{H}}_{\tilde{R}b}={\tilde{G}}^{\mathrm{H}}_{\tilde{R}F},$
$G^{ab}{\tilde{G}}^{\mathrm{H}}_{Ab}=\tilde{\Pi}^{a}_{A},\;\;\;G^{EF}{\tilde{G}}^{\mathrm{H}}_{AF}=\tilde{\Pi}^{E}_{A},\;\;\;N^{a}_{E}\tilde{\Pi}^{E}_{A}+\tilde{\Pi}^{a}_{A}=N^{a}_{A}.$

From (35) and (37) we see that the expressions for $b^{A}$ and $b^{a}$ include
still “untransformed” terms with second partial derivatives of
$Q^{\ast}{}^{A}(x)$ with respect to $x^{k}$ and $x^{n}$. It turns out that the
solution to this problem can be found using the mean curvature normal of the
orbit space $\tilde{\mathcal{M}}$.

The mean curvature normal is defined as the trace of the second fundamental
form. In our case, the local basis (the frame) on the tangent plane to the
manifold of the orbit space (which is the submanifold in the ambient space
with the degenerated metric) is given by the coordinate vector fields
$(\frac{\partial}{\partial
x^{i}},\frac{\partial}{\partial\tilde{f}^{a}})\equiv(e_{i},e_{a})$. To
calculate the mean curvature normal we use the following representation for
these coordinate vector fields:
$(Q^{\ast}{}^{A}_{i}(x)\frac{\partial}{\partial
Q^{\ast}{}^{A}},\frac{\partial}{\partial\tilde{f}^{a}})=(Q^{\ast}{}^{A}_{i}e_{A},e_{a})$,
so the mean curvature normal $\vec{j}$ must be given as

$\vec{j}=j^{A}\frac{\partial}{\partial
Q^{\ast}{}^{A}}+j^{a}\frac{\partial}{\partial\tilde{f}^{a}}.$

The components of the mean curvature normal $\vec{j}$ are defined as follows:

$j^{\tilde{A}}=\frac{1}{2}\bigl{(}\delta^{\tilde{A}}_{\tilde{B}}-N^{\tilde{A}}_{\tilde{B}}\bigr{)}\bigl{(}\dots\bigr{)}^{\tilde{B}},$
(38)

where

$\bigl{(}\dots\bigr{)}^{\tilde{B}}=\Bigl{(}h^{ij}[\nabla_{e_{i}}e_{j}]+\tilde{h}^{ia}[\nabla_{e_{i}}e_{a}]+\tilde{h}^{ai}[\nabla_{e_{a}}e_{i}]+\tilde{h}^{ab}[\nabla_{e_{a}}e_{b}]\Bigr{)}^{\tilde{B}}.$

To calculate the mean curvature normal $\vec{j}$, we use the formula (38),
which states that $\vec{j}$ is defined as the projection of the vector field,
which in our case is represented by the above expression given in parentheses
(…), on the normal bundle to the orbit space[21]. This projection is performed
by an operator whose components are specified using
$(\delta^{\tilde{A}}_{\tilde{B}}-N^{\tilde{A}}_{\tilde{B}})$, where
$N^{\tilde{A}}_{\tilde{B}}=(N^{A}_{B},N^{a}_{B},0,N^{a}_{b}=\delta^{a}_{b})$.

Note that from (38) it follows that

$j^{A}=\frac{1}{2}\bigl{(}\delta^{A}_{B}-N^{A}_{B}\bigr{)}\bigl{(}\dots\bigr{)}^{B}\;\;\;{\rm
and}\;\;\;\;j^{a}=-\frac{1}{2}N^{a}_{B}\bigl{(}\dots\bigr{)}^{B}.$

In our local basis given by the tangent vector fields, we have

$\displaystyle\nabla_{e_{i}}e_{j}$ $\displaystyle=$
$\displaystyle\nabla_{Q^{\ast}{}^{A}_{i}(x)e_{A}}Q^{\ast}{}^{B}_{j}(x)e_{B}$
$\displaystyle=$ $\displaystyle
h^{ij}\bigl{(}Q^{\ast}{}^{A}_{i}Q^{\ast}{}^{B}_{j}\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{AB}+Q^{\ast}{}^{C}_{ij}\bigr{)}\frac{\partial}{\partial
Q^{\ast}{}^{C}}+h^{ij}Q^{\ast}{}^{A}_{i}Q^{\ast}{}^{B}_{j}\,{}^{\rm
H}{\tilde{\Gamma}}^{b}_{AB}\frac{\partial}{\partial\tilde{f}^{b}}$
$\displaystyle=$ $\displaystyle\bigl{(}G^{EF}N^{A}_{E}N^{B}_{F}\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{AB}+h^{ij}Q^{\ast}{}^{C}_{ij}\bigr{)}\frac{\partial}{\partial
Q^{\ast}{}^{C}}+G^{EF}N^{A}_{E}N^{B}_{F}\,{}^{\rm
H}{\tilde{\Gamma}}^{b}_{AB}\frac{\partial}{\partial\tilde{f}^{b}},$

where $G^{EF},N^{A}_{E}$, $N^{B}_{F}$ depend on $Q^{\ast}(x)$, ${}^{\rm
H}{\tilde{\Gamma}}$ depend on $Q^{\ast}(x)$ and $\tilde{f}$. The other
components of $\bigl{(}\dots\bigr{)}^{\tilde{B}}$ are found similarly.

As a result, we get that

$\displaystyle j^{a}$
$\displaystyle=-\frac{1}{2}N^{a}_{C}\Bigl{(}G^{EF}N^{A}_{E}N^{B}_{F}\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{AB}+G^{EF}N^{B}_{E}N^{b}_{F}\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{Bb}+G^{EF}N^{b}_{E}N^{B}_{F}\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{bB}$
$\displaystyle\;\;\;\;\;\;\;\;\;+(G^{db}+G^{EF}N^{d}_{E}N^{b}_{F})\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{db}+h^{ij}Q^{\ast}{}^{C}_{ij}\Bigr{)}$
$\displaystyle=-\frac{1}{2}N^{a}_{C}\bigl{(}h^{\tilde{E}\tilde{F}}\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{\tilde{E}\tilde{F}}+h^{ij}Q^{\ast}{}^{C}_{ij}\bigr{)}$
(39)

and

$\displaystyle j^{A}$
$\displaystyle=\frac{1}{2}\bigl{(}\delta^{A}_{C}-N^{A}_{C}\bigr{)}\Bigl{(}G^{EF}N^{A^{\prime}}_{E}N^{B^{\prime}}_{F}\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{A^{\prime}B^{\prime}}+G^{EF}N^{B}_{E}N^{b}_{F}\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{Bb}$
$\displaystyle\;\;\;\;\;\;\;\;\;+G^{EF}N^{b}_{E}N^{B}_{F}\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{bB}+(G^{db}+G^{EF}N^{d}_{E}N^{b}_{F})\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{db}+h^{ij}Q^{\ast}{}^{C}_{ij}\Bigr{)}$
$\displaystyle=\frac{1}{2}\bigl{(}\delta^{A}_{C}-N^{A}_{C}\bigr{)}\bigl{(}h^{\tilde{E}\tilde{F}}\,{}^{\rm
H}{\tilde{\Gamma}}^{C}_{\tilde{E}\tilde{F}}+h^{ij}Q^{\ast}{}^{C}_{ij}\bigr{)}.$
(40)

Comparing (35) and (40), we conclude that

$b^{A}=-\frac{1}{2}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{A}_{\tilde{B}\tilde{M}}+j^{A}.$
(41)

Similarly, from (37) and (39) it follows that

$b^{a}=-\frac{1}{2}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{a}_{\tilde{B}\tilde{M}}+j^{a}.$
(42)

Since the mean curvature normal can be also defined by the Weingarten map,
i.e. without using explicit coordinate expressions, $\vec{j}$ is the function
given on a submanifold $\tilde{\mathcal{M}}$, so $j^{A}\equiv
j^{A}(Q^{\ast}(x),\tilde{f})$ and $j^{a}\equiv j^{a}(Q^{\ast}(x),\tilde{f})$.

In our case, this can be shown by using in (40) the identity

$N^{A}_{B,D}\,Q^{\ast}{}^{B}_{i}Q^{\ast}{}^{D}_{j}+N^{A}_{B}Q^{\ast}{}^{B}_{ij}=Q^{\ast}{}^{A}_{ij},$

which is derived from the identity
$N^{A}_{B}Q^{\ast}{}^{B}_{i}=Q^{\ast}{}^{A}_{i}$, together with the relation
$h^{ij}Q^{\ast}{}^{A}_{i}Q^{\ast}{}^{B}_{j}=G^{EF}N^{A}_{E}N^{B}_{F}$. As a
result, we get the following representation for $j^{A}$:

$j^{A}=\frac{1}{2}h^{BM}N^{A}_{B,M}+\frac{1}{2}h^{\tilde{B}\tilde{M}}\Bigl{(}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{A}_{\tilde{B}\tilde{M}}-N^{A}_{C}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{C}_{\tilde{B}\tilde{M}}\Bigr{)}.$
(43)

Note that in $j^{A}$ not all $N^{A}_{\tilde{B},\tilde{M}}$ components are
nonzero, because in our case $N^{A}_{b}=0$ and then its derivatives are also
equal to zero.

It also follows from
$N^{a}_{R}Q^{\ast}{}^{R}_{ij}=N^{a}_{R}N^{R}_{B,D}\,Q^{\ast}{}^{B}_{i}Q^{\ast}{}^{D}_{j}$
that

$\displaystyle j^{a}$
$\displaystyle=-\frac{1}{2}N^{a}_{C}\Bigl{(}\,h^{BM}N^{C}_{B,M}+h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{C}_{\tilde{B}\tilde{M}}\Bigr{)}$
$\displaystyle=\;\;\;\frac{1}{2}h^{CM}N^{a}_{C,M}-\frac{1}{2}N^{a}_{C}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{C}_{\tilde{B}\tilde{M}}$
(44)

(the last line follows from differentiating $N^{a}_{C}N^{C}_{B}=0$ and taking
into account $h^{BM}N^{C}_{B}=h^{CM}$). Then we can conclude that the
resulting $b^{A}(Q^{\ast}(x),\tilde{f})$ and $b^{a}(Q^{\ast}(x),\tilde{f})$
should be used to determine the drift coefficients of the stochastic
differential equation for the stochastic process on $\tilde{\mathcal{M}}$,
which is locally represented by the processes
$(Q^{\ast}{}^{A}(x(t)),\tilde{f}^{a}(t))$.

###  4.4 Diffusion coefficients and stochastic differential equations

The diffusion coefficients of the equation (29) can be obtained if we first
assume that the equality
$Q^{\ast}{}^{A}_{i}X^{i}_{\bar{m}}dw^{\bar{m}}_{t}=\tilde{\mathscr{X}}^{A}_{\bar{M}}dw^{\bar{M}}_{t}$
holds for some matrix $\tilde{\mathscr{X}}^{A}_{\bar{M}}$. Taking the ‘square’
of the equality and using the main property of the Wiener process, according
to which $<dw^{\bar{m}}_{t},dw^{\bar{n}}_{t}>=\delta^{\bar{m}\bar{n}}dt$ and
also $<~{}dw^{\bar{M}}_{t},dw^{\bar{N}}_{t}>=\delta^{\bar{M}\bar{N}}dt$, we
find that the matrix $\tilde{\mathscr{X}}^{A}_{\bar{M}}$ satisfies the local
relation
$\sum_{\bar{M}}\tilde{\mathscr{X}}^{A}_{\bar{M}}\tilde{\mathscr{X}}^{B}_{\bar{M}}=G^{CD}N^{A}_{C}N^{B}_{D}$.
This can be verified by using a special representation for the projection
operator $N^{A}_{B}$:

$N^{A}_{B}(Q^{\ast}(x))=G^{\rm
H}_{BD}(Q^{\ast}(x))Q^{\ast}{}^{D}_{i}h^{ij}Q^{\ast}{}^{A}_{j}.$

In turn, the resulting relation allows us to define matrix
${\mathscr{X}}^{A}_{\bar{M}}$ so that
$\tilde{\mathscr{X}}^{A}_{\bar{M}}=N^{A}_{C}{\mathscr{X}}^{C}_{\bar{M}}$, and
hence
$\sum_{\bar{M}}{\mathscr{X}}^{A}_{\bar{M}}{\mathscr{X}}^{B}_{\bar{M}}=G^{AB}$.

Note that the relations that we have used to derive the diffusion coefficients
are possible only because these coefficients in stochastic differential
equations with Wiener processes are determined up to orthogonal
transformations.

To obtain a stochastic differential equation describing diffusion on a
submanifold $\tilde{\mathcal{M}}$ in terms of the variable given on the
external manifold (i.e., in our case, using dependent coordinates on charts of
the external manifold), it is necessary to redefine the coordinates
$Q^{\ast}(x(t))$ of local stochastic processes
$(Q^{\ast}{}^{A}(x(t)),\tilde{f}^{a}(t))$ for new coordinates
$Q^{\ast}(t)$888We denote new stochastic variable by the same letter.. In
addition, we require that the new process, represented by the local processes
$(Q^{\ast}{}^{A}(t),\tilde{f}^{a}(t))$, at the initial moment of time also be
given on the submanifold $\tilde{\mathcal{M}}$.

Thus, taking into account the new representations obtained for the drift and
diffusion coefficients, our local stochastic differential equations (21) and
(22) can be rewritten as follows:

$\displaystyle
dQ^{\ast}{}^{A}(t)=\mu^{2}\kappa\Bigl{(}-\frac{1}{2}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{A}_{\tilde{B}\tilde{M}}+j^{A}+j^{A}\Bigr{)}dt+\mu\sqrt{\kappa}N^{A}_{C}\mathscr{X}^{C}_{\bar{M}}dw^{\bar{M}}_{t},$
(45) $\displaystyle
d\tilde{f}^{a}(t)=\mu^{2}\kappa\Bigl{(}-\frac{1}{2}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{a}_{\tilde{B}\tilde{M}}+j^{a}+j^{a}\Bigr{)}dt+\mu\sqrt{\kappa}\bigl{(}N^{a}_{C}\mathscr{X}^{C}_{\bar{M}}dw^{\bar{M}}_{t}+\mathscr{X}^{a}_{\bar{b}}dw^{\bar{b}}_{t}\bigr{)}.$
(46)

The terms on the right-hand sides of the above equations now depend on
$Q^{\ast}(t)$ and $\tilde{f}(t)$. Also note, that in these equations we have
introduced a new notation for the drift coefficients $b$ defined in (27).
Hereinafter they will be denoted as $j$.

Solutions of the local equations (45), (46), and (23) are the coordinate
representatives of the local stochastic process
${\zeta}^{{\tilde{\varphi}}^{\tilde{\cal
P}}}(t)=(Q^{\ast}_{t}{}^{A},\tilde{f}^{a}_{t},a^{\alpha}_{t})$ given on a
chart of the principal fiber bundle. The set of solutions of such equations on
local charts determines the stochastic evolution family of mappings of the the
manifold $\tilde{\cal P}$, considered as the total space of the principal
fiber bundle $\pi^{\prime}$. As in [7, 8], these local evolution families of
mappings generate a global stochastic evolution family, which by definition of
the cited papers, is a global stochastic process $\zeta_{t}$ in the principal
fiber bundle $\rm P(\tilde{\mathcal{M}},\mathcal{G})$. The local stochastic
process ${\zeta}^{{\tilde{\varphi}}^{\tilde{\cal
P}}}(t)={\tilde{\varphi}}^{\tilde{\cal P}}({\zeta}_{t})$ is a local
representative of the global stochastic process $\zeta_{t}$ on the chart of
the principal fiber bundle with the coordinate homeomorphism
$\tilde{\varphi}^{\tilde{\cal
P}}=\tilde{\varphi}^{-1}\circ\varphi^{\tilde{\cal P}}$.

We also note that transition from the local stochastic processes
$(Q^{A}_{t},f^{b}_{t})$ to the local processes
$(Q^{\ast}_{t}{}^{A},\tilde{f}^{a}_{t},a^{\alpha}_{t})$, performed using the
mapping $\tilde{\varphi}^{-1}$, is the phase-space transformation of the
stochastic processes. It is known that such a transformation does not change
the probabilities and also the transition probabilities. This allows us to
rewrite the right-hand side of (8) as the expectation which is taken over the
distribution of the local process ${\zeta}^{{\tilde{\varphi}}^{\tilde{\cal
P}}}(t)$:

${\rm E}_{s,{\tilde{\varphi}}^{\tilde{\cal
P}}(p,v)}[\phi(\,({\varphi}^{\tilde{\cal
P}})^{-1}({\tilde{\varphi}}({\zeta}^{{\tilde{\varphi}}^{\tilde{\cal
P}}}(t))\,)]={\rm E}_{s,{\tilde{\varphi}}^{\tilde{\cal
P}}(p,v)}[\tilde{\phi}({\zeta}^{{\tilde{\varphi}}^{\tilde{\cal P}}}(t)\,)].$

When deriving such a representation, it was taken into account that
$\eta_{t}^{\varphi^{\tilde{\cal
P}}}=\tilde{\varphi}({\zeta}_{t}^{{\tilde{\varphi}}^{\tilde{\cal P}}})$ and
$({\varphi}^{\tilde{\cal
P}})^{-1}\circ{\tilde{\varphi}}=({\tilde{\varphi}}^{-1}\circ{\varphi}^{\tilde{\cal
P}})^{-1}\equiv({\tilde{\varphi}}^{\tilde{\cal P}})^{-1}$. Moreover, we have
replaced the function $\phi$ under the expectation sign with the function
$\tilde{\phi}=\phi\circ({\tilde{\varphi}}^{\tilde{\cal P}})^{-1}$.

The global evolution semigroup for the process $\zeta(t)$ is given by the
limit of the superposition of the local semigroups associated with local
stochastic processes ${\zeta}^{{\tilde{\varphi}}^{\tilde{\cal P}}}(t)$:

$\psi_{t_{b}}(p_{a},v_{a},t_{a})={\lim}_{q}\bigl{[}{\tilde{U}}_{{\zeta}^{{\tilde{\varphi}}^{\tilde{\cal
P}}}}(t_{a},t_{1})\cdot\ldots\cdot{\tilde{U}}_{{\zeta}^{{\tilde{\varphi}}^{\tilde{\cal
P}}}}(t_{n-1},t_{b}){\tilde{\phi}}_{0}\bigr{]}(Q^{\ast}_{a},\tilde{f}_{a},\theta_{a}),$
(47)

where the boundary values of ${\zeta}^{{\tilde{\varphi}}^{\tilde{\cal
P}}}(t_{a})\equiv(Q^{\ast}_{a},\tilde{f}_{a},\theta_{a})$ in the right-hand
side of the equation (47) should be expressed in terms of $(p_{a},v_{a})$ with
the help of inverse transformation $({\tilde{\varphi}}^{\tilde{\cal
P}})^{-1}$.

Note that the local semigroups
${\tilde{U}}_{{\zeta}^{{\tilde{\varphi}}^{\tilde{\cal P}}}}\tilde{\phi}$ are
now defined as

$\displaystyle{\tilde{U}}_{{\zeta}^{{\tilde{\varphi}}^{\tilde{\cal
P}}}}(s,t){\tilde{\phi}}(Q^{\ast}_{0},\tilde{f}_{0},\theta_{0})={\rm
E}_{s,(Q^{\ast}_{0},\tilde{f}_{0},\theta_{0})}[\tilde{\phi}(Q^{\ast}(t),\tilde{f}(t),a(t))],\;\;$
$\displaystyle\,\,s<t,\;\;Q^{\ast}(s)=Q^{\ast}_{0},\,\tilde{f}(s)=\tilde{f}_{0},\,a(s)=\theta_{0}.$
(48)

We may consider the global stochastic process $\zeta(t)$ as consisting of two
components: $\;\;\zeta(t)=\\{\xi_{\tilde{\Sigma}}(t),a_{\mathcal{G}}(t)\\}$.
The first process describes a special stochastic evolution (due to a certain
form of stochastic differential equations and their initial conditions) on the
submanifold ${\tilde{\Sigma}}$ (on the gauge surface), and the second one
describes the evolution on the orbits of the principal fiber bundle. Then,
taking into account the potential term, the global semigroup associated with
the stochastic process $\zeta(t)$ can be symbolically written in the following
form:

${\psi}_{t_{b}}(p_{a},v_{a},t_{a})={\rm
E}\Bigl{[}\tilde{\phi}_{0}({\xi}_{\tilde{\Sigma}}(t_{b}),a_{\mathcal{G}}(t_{b}))\exp\\{\frac{1}{\mu^{2}\kappa
m}\int_{t_{a}}^{t_{b}}\tilde{V}({\xi}_{\tilde{\Sigma}}(u))du\\}\Bigr{]},$

where ${\xi}_{\tilde{\Sigma}}(t_{a})$ is such a point on ${\tilde{\Sigma}}$
that has local coordinates $(Q^{\ast}_{a},\tilde{f}_{a})$ , and the point
$a_{\mathcal{G}}(t_{a})\in{\mathcal{G}}$ has the local coordinates
$\theta_{a}$. Also, ${{\tilde{\varphi}}^{\tilde{\cal
P}}}(p_{a},v_{a})=(Q^{*}_{a},\tilde{f}_{a},\theta_{a})$.

##  5 Factorization of the path integral measure

To perform the reduction procedure in the path integral, i.e. to transform the
original path integral into the path integral discribing the “quantum
evolution” on the orbit space of the principal bundle, we must first factorize
the path integral measure in the original path integral. For dynamical systems
with symmetry this can be done by the method developed in our works [5, 6, 2,
10, 11], and as well as in [9]. This method is based on using the optimal
nonlinear filtering stochastic differential equation from the stochastic
process theory [16, 17].

The equation deals with the evolution of the conditional mathematical
expectation of a function that depends on both an unknown signal process (the
process $a_{\mathcal{G}}(t)$ in our case) and the observation process (the
stochastic process $\xi_{\tilde{\Sigma}}(t)$) relative to the
sub-$\sigma$-algebra generated by the observation process.

When deriving such an equation in the stochastic theory, it is assumed that
the signal process $Z_{t}$ and the observation process $Y_{t}$ satisfy the
following stochastic differential equations:

$\displaystyle
dZ_{t}=\varphi(Y,Z,t)dt+X(Y,Z,t)\,dw_{1}(t)+X^{\prime}(Y,Z,t)\,dw_{2}(t),$
$\displaystyle dY_{t}=\varphi_{1}(Y,Z,t)dt+X_{1}(Y,t)\,dw_{2}(t),$

where $w_{1}(t)$ and $w_{2}(t)$ are independent Wiener processes.

These equations satisfy two main requirements. The first is that the diffusion
coefficient $X_{1}$ must not depend on the process $Z_{t}$. And the second
requirement is that in the equation for the observation process $Y_{t}$ there
should not be a term with $dw_{1}(t)$. Note that in our stochastic
differential equations, the second requirement will be satisfied due to the
presence of the corresponding projection operators in the diffusion
coefficients.

The nonlinear filtering stochastic differential equation for the conditional
mathematical expectation $\hat{f}=\rm
E[f(Y_{t},Z_{t},t)|{\mathcal{Y}}^{t}_{t_{0}}]$, (${\mathcal{Y}}^{t}_{t_{0}}$
is the sub-$\sigma$-algebra generated by the observation process $Y_{t}$), has
the following form:

$\displaystyle d\hat{f}(t)$ $\displaystyle=$ $\displaystyle{\rm
E}[f_{t}+f_{z}\varphi+\frac{1}{2}f_{zz}(XX^{T})|{\mathcal{Y}}^{t}_{t_{0}}]dt$
(49) $\displaystyle\\!\\!\\!+{\rm
E}\Bigl{[}f(\varphi_{1}-\hat{\varphi}_{1})+f_{z}(XX^{T}_{1})|{\mathcal{Y}}^{t}_{t_{0}}\Bigr{]}(X_{1}X^{T}_{1})^{-1}(dY_{t}-\hat{\varphi}_{1}dt),$

where $\hat{\varphi}_{1}\equiv{\rm
E}[\varphi_{1}(Y_{t},Z_{t},t)|{\mathcal{Y}}^{t}_{t_{0}}]$.

By the properties of the conditional mathematical expectation of Markov
processes, each local semigroup (48) of the global semigroup (47) can be
represented as follows:

${\tilde{U}}_{{\zeta}^{{\tilde{\varphi}}^{\tilde{\cal
P}}}}(s,t){\tilde{\phi}}(Q^{\ast}_{0},\tilde{f}_{0},\theta_{0})={\rm
E}\Bigl{[}{\rm E}\Bigl{[}\tilde{\phi}(Q^{\ast}(t),\tilde{f}(t),a(t))\mid({\cal
F}_{(Q^{\ast},\tilde{f})})_{s}^{t}\Bigr{]}\Bigr{]}.$ (50)

We are interested in the optimal nonlinear filtering equation for the
conditional expectation of $\tilde{\phi}$ with respect to the
sub-$\sigma$-algebra $({\cal F}_{(Q^{\ast},\tilde{f})})_{s}^{t}$:

$\hat{\widetilde{\phi}}(Q^{\ast}(t),\tilde{f}(t))\equiv{\rm
E}\Bigl{[}\tilde{\phi}(Q^{\ast}(t),\tilde{f}(t),a(t))\mid({\cal
F}_{(Q^{\ast},\tilde{f})})_{s}^{t}\Bigr{]}.$

Such an equation can be obtained from (49) provided that the process
$\xi_{\tilde{\Sigma}}$, which is locally determined by the processes
$Q^{\ast}{}^{A}(t)$ and $\tilde{f}^{a}(t)$, should be considered as an
observation process corresponding to the stochastic process $Y_{t}$ in the
above stochastic differential equations. Then, in our case, the local
stochastic differential equation for the observation process can be written in
the following form:

${dQ^{\ast}_{t}{}^{A}\choose d\tilde{f}^{a}_{t}}=(\mu^{2}\kappa){b^{A}\choose
b^{a}}dt+\mu\sqrt{\kappa}{N^{A}_{C}{\mathscr{X}}_{\bar{M}}^{C}\;\;0\choose{N^{a}_{C}{\mathscr{X}}_{\bar{M}}^{C}\;\;{\mathscr{X}}^{a}_{\bar{b}}}}{dw^{\bar{M}}_{t}\choose
dw^{\bar{b}}_{t}}.$ (51)

Note that the local stochastic process $(Q^{\ast}{}^{A}(t),\tilde{f}^{a}(t))$
is given on $\tilde{\Sigma}$ and is used to describe the stochastic evolution
on the base space $\tilde{\mathcal{M}}$ of the principal fiber bundle
$\rm{P}(\tilde{\mathcal{M}},\mathcal{G})$.

Also note that the stochastic process $a^{\alpha}(t)$ satisfying the equation

$da^{\alpha}_{t}=(\mu^{2}\kappa)b^{\alpha}dt+\mu\sqrt{\kappa}\Lambda^{\beta}_{C}{\bar{v}}^{\alpha}_{\beta}\mathscr{X}^{C}_{\bar{M}}dw^{\bar{M}}_{t}$

corresponds to the signal process $Z_{t}$. Therefore, in our case, instead of
the coefficient $X$ of the equation (49), we should use
$\Lambda^{\beta}_{C}{\bar{v}}^{\alpha}_{\beta}{\mathscr{X}}_{\bar{M}}^{C}$,
and $X_{1}$ must be given by the diffusion matrix of the equation (51):

$\displaystyle
X_{1}=\left(\begin{array}[]{cc}N^{A}_{C}{\mathscr{X}}_{\bar{M}}^{C}&0\\\
N^{a}_{C}{\mathscr{X}}_{\bar{M}}^{C}&{\mathscr{X}}^{a}_{\bar{b}}\end{array}\right).$

The remaining coefficients of the equation (49), which are necessary for the
derivation of the nonlinear filtering stochastic differential equation of our
problem, are easy to find, and they are given by the following expressions:

$X\cdot X^{\rm
T}=G^{CD}\Lambda^{\alpha^{\prime}}_{C}\Lambda^{\beta^{\prime}}_{D}{\bar{v}}^{\alpha}_{\alpha^{\prime}}{\bar{v}}^{\beta}_{\beta^{\prime}},$
$\displaystyle X\cdot X^{\rm
T}_{1}={\bar{v}}^{\alpha}_{\beta}\cdot\left(\begin{array}[]{cc}N^{B}_{C^{\prime}}\Lambda^{\beta}_{C}G^{CC^{\prime}}&N^{a}_{D}\Lambda^{\beta}_{C}G^{CD}\\\
0&0\\\ \end{array}\right),$ $\displaystyle X_{1}\cdot X^{\rm
T}_{1}=\left(\begin{array}[]{cc}G^{CE}N^{B}_{C}N^{D}_{E}&G^{CD}N^{B}_{C}N^{a}_{D}\\\
G^{EB}N^{D}_{E}N^{b}_{B}&G^{AB}N^{a}_{A}N^{b}_{B}+G^{ab}\\\
\end{array}\right),$

$\displaystyle(X_{1}\cdot X^{\rm
T}_{1})^{-1}=\left(\begin{array}[]{cc}(P_{\bot})^{A^{\prime}}_{A}{\tilde{G}}^{\rm
H}_{A^{\prime}B^{\prime}}(P_{\bot})^{B^{\prime}}_{B}&(P_{\bot})^{A^{\prime}}_{A}{\tilde{G}}^{\rm
H}_{A^{\prime}a}\\\ (P_{\bot})^{B^{\prime}}_{B}{\tilde{G}}^{\rm
H}_{bB^{\prime}}&{\tilde{G}}^{\rm H}_{ab}\\\ \end{array}\right).$

As a result, we get the following stochastic differential equation of the
optimal nonlinear filtering:

$\displaystyle
d\hat{\tilde{\phi}}(Q^{\ast}(t),\tilde{f}(t))=\mu^{2}\kappa\Bigr{\\{}-\frac{1}{2}\Bigl{[}d^{-1/2}H^{-1/2}\frac{\partial}{\partial
Q^{\ast}{}^{A}}(d^{1/2}H^{1/2}G^{EF}N^{A}_{E}N^{C}_{F}\,\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}^{\beta}_{C}})$
$\displaystyle+G^{EC}{\Lambda}^{\beta}_{E}{\Lambda}^{\mu}_{C}\,d^{-1/2}H^{-1/2}\frac{\partial}{\partial\tilde{f}^{b}}(d^{1/2}H^{1/2}K^{b}_{\mu})\Bigr{]}{\rm
E}\bigl{[}\bar{L}_{\beta}{\tilde{\phi}}(Q^{\ast}_{t},\tilde{f}_{t},a_{t})|(\mathcal{F}_{Q^{\ast},\tilde{f}})^{t}_{s}\bigr{]}$
$\displaystyle+\frac{1}{2}(G^{BC}{\Lambda}^{\alpha^{\prime}}_{B}{\Lambda}^{\beta^{\prime}}_{C}){\rm
E}\bigl{[}\bar{L}_{\alpha^{\prime}}\bar{L}_{\beta^{\prime}}{\tilde{\phi}}(Q^{\ast}_{t},\tilde{f}_{t},a_{t})|(\mathcal{F}_{Q^{\ast},\tilde{f}})^{t}_{s}\bigr{]}\Bigr{\\}}dt$
$\displaystyle+\mu\sqrt{\kappa}\,{\Lambda}^{\beta}_{C}{\tilde{\Pi}}^{C}_{E}{\mathscr{X}}^{E}_{\bar{M}}{\rm
E}\bigl{[}\bar{L}_{\beta}{\tilde{\phi}}(Q^{\ast}_{t},\tilde{f}_{t},a_{t})|(\mathcal{F}_{Q^{\ast},\tilde{f}})^{t}_{s}\bigr{]}dw^{\bar{M}}_{t}$
$\displaystyle+\mu\sqrt{\kappa}\,{\Lambda}^{\beta}_{C}{\tilde{\Pi}}^{C}_{a}{\mathscr{X}}^{a}_{\bar{b}}{\rm
E}\bigl{[}\bar{L}_{\beta}{\tilde{\phi}}(Q^{\ast}_{t},\tilde{f}_{t},a_{t})|(\mathcal{F}_{Q^{\ast},\tilde{f}})^{t}_{s}\bigr{]}dw^{\bar{b}}_{t}.$
(52)

To simplify this equation, we apply the Peter-Weyl theorem to the function
$\tilde{\phi}$, considered as a function given on a group $\mathcal{G}$.

According to this theorem, the function $\tilde{\phi}$ can be represented as

$\tilde{\phi}(Q^{\ast},\tilde{f},a)=\sum_{\lambda,p,q}c^{\lambda}_{pq}(Q^{\ast},\tilde{f})D^{\lambda}_{pq}(a),$

where $D^{\lambda}_{pq}(a)$999We have now introduced a different notation for
the matrix elements of an irreducible representation to distinguish them from
those used previously. are the matrix elements of an irreducible
representation $T^{\lambda}$ of a group $\cal G$:
$\sum_{q}D_{pq}^{\lambda}(a)D_{qn}^{\lambda}(b)=D_{pn}^{\lambda}(ab)$.

It follows from the properties of conditional mathematical expectations that

$\displaystyle{\rm
E}\bigl{[}\tilde{\phi}(Q^{\ast}(t),\tilde{f}(t),a(t))|(\mathcal{F}_{Q^{\ast},\tilde{f}})^{t}_{s}\bigr{]}$
$\displaystyle=$
$\displaystyle\sum_{\lambda,p,q}c^{\lambda}_{pq}(Q^{\ast}(t),\tilde{f}(t))\,{\rm
E}\bigl{[}D^{\lambda}_{pq}(a(t))|(\mathcal{F}_{Q^{\ast},\tilde{f}})^{t}_{s}\bigr{]}$
$\displaystyle\equiv$
$\displaystyle\sum_{\lambda,p,q}c^{\lambda}_{pq}(Q^{\ast}(t),\tilde{f}(t))\,\hat{D}^{\lambda}_{pq}(Q^{\ast}(t),\tilde{f}(t)),$

where

$c_{pq}^{\lambda}(Q^{\ast}(t),\tilde{f}(t))=d^{\lambda}\int_{\mathcal{G}}\tilde{\phi}(Q^{*}(t),\tilde{f}(t),\theta){\bar{D}}_{pq}^{\lambda}(\theta)d\mu(\theta),$

$d^{\lambda}$ is a dimension of an irreducible representation and
$d\mu(\theta)$ is a normalized
($\int_{\mathcal{G}}d\mu(\theta)=1$) invariant Haar measure on a group
$\mathcal{G}$.

Then, the equation (52) is reduced to the stochastic differential equation for
the conditional mathematical expectation
$\hat{D}_{pq}^{\lambda}(Q^{\ast}(t),\tilde{f}(t))$:

$\displaystyle
d\hat{D}_{pq}^{\lambda}=\mu^{2}\kappa\bigl{(}\Gamma_{1}^{\beta}\,(J_{\beta})_{pq^{\prime}}^{\lambda}\hat{D}_{q^{\prime}q}^{\lambda}dt+\Gamma_{2}^{\alpha\beta}\,(J_{\alpha})_{pq^{\prime}}^{\lambda}(J_{\beta})_{q^{\prime}q^{\prime\prime}}^{\lambda}\hat{D}_{q^{\prime\prime}q}^{\lambda}\bigr{)}dt$
$\displaystyle+\mu\sqrt{\kappa}(J_{\beta})_{pq^{\prime}}^{\lambda}\hat{D}_{q^{\prime}q}^{\lambda}\,\Lambda^{\beta}_{C}\bigl{(}{\tilde{\Pi}}^{C}_{E}{\mathscr{X}}^{E}_{\bar{M}}dw^{\bar{M}}_{t}+{\tilde{\Pi}}^{C}_{a}{\mathscr{X}}^{a}_{\bar{b}}dw^{\bar{b}}_{t}\bigr{)}.$
(53)

In this equation $(J_{\beta})_{pn}^{\lambda}$ are the infinitesimal generators
of the representation $D^{\lambda}(a)$. They are defined as
$(J_{\beta})_{pq}^{\lambda}\equiv({\partial D_{pq}^{\lambda}(a)}/{\partial
a^{\beta}})|_{a=e}$, and we also have

$\bar{L}_{\beta}D_{pq}^{\lambda}(a)=\sum\nolimits_{q^{\prime}}(J_{\beta})_{pq^{\prime}}^{\lambda}D_{q^{\prime}q}^{\lambda}(a).$

We do not write out explicit expressions for the coefficients
$\Gamma_{1}^{\beta}$ and $\Gamma_{2}^{\alpha\beta}$, since they can be easily
obtained using the equation (52).

Also note that the conditional expectations
$\hat{D}_{pq}^{\lambda}(Q^{\ast}(t),\tilde{f}(t))$ also depend on the initial
points $Q^{\ast}_{0}=Q^{\ast}(s)$, $\tilde{f}^{a}_{0}=\tilde{f}^{a}(s)$ and
$\theta^{\alpha}_{0}=a^{\alpha}(s)$. To avoid awkward notation, we have
omitted this dependence in $\hat{D}_{pq}^{\lambda}$.

The solution of the linear matrix stochastic differential equation (53) can be
written [18, 19] as

$\hat{D}_{pq}^{\lambda}(Q^{\ast}(t),\tilde{f}(t))=(\overleftarrow{\exp})_{pn}^{\lambda}(Q^{\ast}(t),\tilde{f}(t),t,s)\,{\rm
E}\bigl{[}D_{nq}^{\lambda}(a(s))\mid({\cal
F}_{Q^{\ast},\tilde{f}})_{s}^{t}\bigr{]},=$ (54)

where

$\displaystyle(\overleftarrow{\exp})_{pn}^{\lambda}(Q^{\ast}(t),\tilde{f}(t),t,s)=\overleftarrow{\exp}\int_{s}^{t}\Bigl{\\{}{\mu}^{2}\kappa\Bigl{[}\frac{1}{2}d^{\alpha\nu}(Q^{\ast}(u),\tilde{f}(u))(J_{\alpha})_{pr}^{\lambda}(J_{\nu})_{rn}^{\lambda}\Bigr{.}\Bigr{.}$
$\displaystyle-\Bigl{.}\Bigl{.}\frac{1}{2}\frac{1}{\sqrt{d\,H}}\frac{\partial}{\partial
Q^{\ast}{}^{A}}\left(\sqrt{d\,H}G^{EF}N^{A}_{E}N^{C}_{F}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}_{C}^{\nu}}(Q^{\ast}(u))\right)(J_{\nu})_{pn}^{\lambda}$
$\displaystyle-\frac{1}{2}(G^{EC}\Lambda^{\nu}_{E}\Lambda^{\mu}_{C})\frac{1}{\sqrt{d\,H}}\frac{\partial}{\partial\tilde{f}^{b}}\left(\sqrt{d\,H}K^{b}_{\mu}\right)(J_{\nu})_{pn}^{\lambda}\Bigr{]}du$
$\displaystyle+\mu\sqrt{\kappa}\Lambda^{\beta}_{C}(Q^{\ast}(u))(J_{\beta})_{pn}^{\lambda}\Bigl{[}{\tilde{\Pi}}^{C}_{E}{\mathscr{X}}^{E}_{\bar{M}}(Q^{\ast}(u))dw^{\bar{M}}(u)+{\tilde{\Pi}}^{C}_{a}{\mathscr{X}}^{a}_{\bar{b}}dw^{\bar{b}}(u)\Bigr{]}\Bigr{\\}}$
$\displaystyle(H,d,{\tilde{\Pi}}^{C}_{E},{\tilde{\Pi}}^{C}_{a}\;{\rm
depend}\,{\rm on}\;Q^{\ast}(u)\;{\rm and}\,\tilde{f}(u))$ (55)

is the multiplicative stochastic integral. This integral is defined as a limit
of the sequence of time–ordered multipliers that have been obtained as a
result of the piecewise breaking of the time interval $[s,t]$, $[s=t_{0}\leq
t_{1}\ldots\leq t_{n}=t]$. In (55), the time order of these multipliers is
indicated by the arrow directed to the multipliers given at greater times.

Using the solution of the matrix stochastic differential equation (53) defined
by (54) and (55), the local semigroup (50) can be represented as follows:

$\displaystyle{\tilde{U}}_{{\zeta}^{{\tilde{\varphi}}^{\tilde{\cal
P}}}}(s,t){\tilde{\phi}}(Q^{\ast}_{0},\tilde{f}_{0},\theta_{0})$
$\displaystyle\;\;\;\;=\sum_{\lambda,p,q,q^{\prime}}\\!\\!{\rm
E}\bigl{[}c_{pq}^{\lambda}(Q^{\ast}(t),\tilde{f}(t))(\overleftarrow{\exp})_{pq^{\prime}}^{\lambda}(Q^{\ast}(t),\tilde{f}(t),t,s)\bigr{]}D_{q^{\prime}q}^{\lambda}(\theta_{0}).$
(56)

Note that in this representation, the last multiplier on the right is due to

${\rm E}\bigl{[}D_{nq}^{\lambda}(a(s))\mid({\cal
F}_{Q^{\ast},\tilde{f}})_{s}^{t}\bigr{]}=D_{nq}^{\lambda}(a(s))=D_{nq}^{\lambda}(\theta_{0}).$

The global evolution semigroup (47) is now obtained in accordance with [7, 8]
as the limit (under the refinement of the subdivision of the time interval
$[t_{a},t_{b}]$) of the superposition of the local semigroups that are similar
to (54). We will write the global semigroup in the following symbolic form:

$\displaystyle{\psi}_{t_{b}}(p_{a},v_{a},t_{a})$ $\displaystyle=$
$\displaystyle\sum_{\lambda,p,q,q^{\prime}}{\rm
E}\bigl{[}c_{pq}^{\lambda}(\xi_{\tilde{\Sigma}}(t_{b}))(\overleftarrow{\exp})_{pq^{\prime}}^{\lambda}(\xi_{\tilde{\Sigma}}(t),t_{b},t_{a})\bigr{]}D_{q^{\prime}q}^{\lambda}(\theta_{a}),$
(57)
$\displaystyle\;\;\;\;\;\;\;\;\;(\xi_{\tilde{\Sigma}}(t_{a})=\pi^{\prime}_{(\tilde{\Sigma})}(p_{a},v_{a})),$

where the global process $\xi_{\tilde{\Sigma}}(t)=(\xi_{1}(t),\xi_{2}(t))$ is
defined on the submanifold $\tilde{\Sigma}$. The process
$\xi_{\tilde{\Sigma}}(t)$ is described locally by the stochastic equations
(51).

Thus, we have obtained that our initial path integral (3) is represented as
the sum of the matrix semigroups (the path integrals) given on the submanifold
$\tilde{\Sigma}$. The coordinate representation of the differential generator
(the Hamilton operator) of these matrix semigroups is

$\displaystyle\frac{1}{2}(\mu^{2}\kappa)\Bigl{\\{}\Bigr{[}h^{AB}\frac{\partial^{2}}{\partial
Q^{\ast}{}^{A}\partial Q^{\ast}{}^{B}}+2h^{Aa}\frac{\partial^{2}}{\partial
Q^{\ast}{}^{A}\partial\tilde{f}^{a}}+h^{ab}\frac{\partial^{2}}{\partial\tilde{f}^{a}\partial\tilde{f}^{b}}$
$\displaystyle-\bigl{(}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{A}_{\tilde{B}\tilde{M}}-2(j^{A}+j^{A})\bigr{)}\frac{\partial}{\partial
Q^{\ast}{}^{A}}-\bigl{(}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{a}_{\tilde{B}\tilde{M}}-2(j^{a}+j^{a})\bigr{)}\frac{\partial}{\partial\tilde{f}^{a}}\Bigr{]}(I)^{\lambda}_{pn}$
$\displaystyle+2N^{A}_{B}G^{EB}{\Lambda}^{\beta}_{E}(J_{\beta})^{\lambda}_{pn}\frac{\partial}{\partial
Q^{\ast}{}^{A}}-2d^{\beta\nu}K^{b}_{\nu}(J_{\beta})^{\lambda}_{pn}\frac{\partial}{\partial\tilde{f}^{b}}$
$\displaystyle-\Bigl{.}\Bigl{.}\Bigl{(}\frac{1}{\sqrt{d\,H}}\frac{\partial}{\partial
Q^{\ast}{}^{A}}\left(\\!\sqrt{d\,H}h^{AC}\\!\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}_{C}^{\nu}}\right)+G^{EC}{\Lambda}^{\nu}_{E}{\Lambda}^{\mu}_{C}\frac{1}{\sqrt{d\,H}}\frac{\partial}{\partial\tilde{f}^{b}}\left(\sqrt{d\,H}K^{b}_{\mu}\right)\Bigr{)}(J_{\nu})_{pn}^{\lambda}$
$\displaystyle+G^{CD}{\Lambda}^{\alpha}_{C}{\Lambda}^{\beta}_{D}(J_{\alpha})_{pq}^{\lambda}(J_{\beta})_{qn}^{\lambda}\Bigr{\\}},$
(58)

where $(I^{\lambda})_{pq}$ is a unity matrix.

The operator (58) acts in the space of the sections
$\Gamma({\tilde{\Sigma}},V^{*})$ of the associated covector bundle (we
consider the backward Kolmogorov equation). The scalar product in the space of
the sections of the associated co-vector bundle is given by

$(\psi_{n},\psi_{m})=\int_{\tilde{\Sigma}}\langle\psi_{n},\psi_{m}{\rangle}_{V^{\ast}_{\lambda}}\,d^{1/2}\,dv_{\tilde{\Sigma}},$
(59)

where $d=\det(d_{\alpha\beta})$ and $dv_{\tilde{\Sigma}}$ is the Riemannian
volume element on the submanifold ${\tilde{\Sigma}}$. In local coordinates, it
can be presented as

$dv_{\tilde{\Sigma}}(Q^{\ast},\tilde{f})=H^{1/2}(Q^{\ast},\tilde{f})\,dQ^{\ast}{}^{1}...dQ^{\ast}{}^{n_{\cal
P}}d\tilde{f}^{1}...\tilde{f}^{n_{\cal\mathcal{V}}}.$

To express the matrix semigroup under the sign of sum in (57) in terms of the
original semigroup defined on $\tilde{\mathcal{P}}$, it is necessary to
inverse this equality. In our previous papers [6, 2], such an inversion of the
analogous equality was made for the kernels of the corresponding local
semigroups. To do this, it was assumed that all the necessary restrictions for
the existence of semigroup kernels were fulfilled. This method is applicable
to the case we are now considering, if we take into account local isomorphism
of the principal bundle ${\rm P}(\tilde{\mathcal{M}},\mathcal{G})$ with the
trivial bundle $\tilde{\Sigma}\times\mathcal{G}\to\tilde{\Sigma}$. This
isomorphism, as shown in [4], leads to the existence of the relationship
between the charts of these principal fiber bundles. As a result, it becomes
possible to reverse the relation between local kernels (local Green’s
functions) which are defined on charts of the principal bundle ${\rm
P}(\tilde{\Sigma},\mathcal{G})$.

If the global semigroup on the left-hand side of (57) can be presented as

${\psi}_{t_{b}}(p_{a},v_{a},t_{a})=\int
G_{\tilde{\mathcal{P}}}(p_{b},v_{b},t_{b};p_{a},v_{a},t_{a})\phi_{0}(p_{b},v_{b})dv_{\tilde{\mathcal{P}}}(p_{b},v_{b}),$
(60)

then comparing the local expression of the right-hand side of (57) (i.e. what
is obtained on the charts of the principal fiber bundle ${\rm
P}(\tilde{\Sigma},\mathcal{G})$ ) with the local expression
$G_{\tilde{\mathcal{P}}}$ given on the corresponding charts from the atlas of
the manifold ${\tilde{\mathcal{P}}}$ (with account of a local isomorphism of
the bundles), one can find the following relation between the local Green’s
functions:

$\displaystyle\int_{\mathcal{G}}G_{\tilde{\mathcal{P}}}(\alpha_{b},F(Q^{\ast}_{b},{\theta}_{b}),\bar{D}({\theta}_{b})\tilde{f}_{b},t_{b};\beta_{a},F(Q^{\ast}_{a},{\theta}_{a}),\bar{D}({\theta}_{a})\tilde{f}_{a},t_{a})$
$\displaystyle\;\;\;\;\;\;\;\;\;\times
D^{\lambda}_{pq}({\theta}_{b})d{\mu}({\theta}_{b})=\sum_{q^{\prime}}G^{\lambda}_{q^{\prime}p}({\alpha}_{b},Q^{\ast}_{b},\tilde{f}_{b},t_{b};{\beta}_{a},Q^{\ast}_{a},\tilde{f}_{a},t_{a})D^{\lambda}_{q^{\prime}q}({\theta}_{a}),$

where $\alpha_{b}$ and $\beta_{a}$ are the labels of the charts. Note that to
obtain the above relation, $\phi_{0}(p_{b},v_{b})$ in (60) was also expanded
into a series using the Peter-Weyl theorem. This was done after introducing on
$\tilde{\mathcal{P}}$ the bundle coordinates.

The obtained relation between the local Green’s functions can be reversed:

$\displaystyle
G^{\lambda}_{mn}({\alpha}_{b},Q^{\ast}_{b},\tilde{f}_{b},t_{b};{\beta}_{a},Q^{\ast}_{a},\tilde{f}_{a},t_{a})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;=\int_{\mathcal{G}}G_{\tilde{\mathcal{P}}}(\alpha_{b},Q^{\ast}_{b},\tilde{f}_{b},{\theta},t_{b};\beta_{a},Q^{\ast}_{a},\tilde{f}_{a},e,t_{a})D^{\lambda}_{nm}({\theta})d{\mu}({\theta}),$
(61)

where $e$ is the unity element of the group $\mathcal{G}$. Also note that
there is the following invariance property:

$\displaystyle
G_{\tilde{\mathcal{P}}}(\alpha_{b},Q^{\ast}_{b},\tilde{f}_{b},{\theta}_{b},t_{b};\beta_{a},Q^{\ast}_{a},\tilde{f}_{a},{\theta}_{a},t_{a})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\equiv
G_{\tilde{\mathcal{P}}}(\alpha_{b},F(Q^{\ast}_{b},{\theta}_{b}),\bar{D}({\theta}_{b})\tilde{f}_{b},t_{b};\beta_{a},F(Q^{\ast}_{a},{\theta}_{a}),\bar{D}({\theta}_{a})\tilde{f}_{a},t_{a}).$

To extend the resulting equality (61) from local charts to the whole manifold,
we need to glue these local Green’s functions. In case of the trivial
principal fiber bundle this can be done using the transition coordinate
functions from the manifold atlas. As a result, we obtain the global integral
relation between the Green’s functions:

$\displaystyle
G^{\lambda}_{mn}(\pi^{\prime}_{(\tilde{\Sigma})}(p_{b},v_{b}),t_{b};\pi^{\prime}_{(\tilde{\Sigma})}(p_{a},v_{a}),t_{a})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;=\int_{\cal G}G_{\tilde{\cal
P}}(p_{b}\theta,v_{b}\theta,t_{b};p_{a},v_{a},t_{a})D_{nm}^{\lambda}(\theta)d\mu(\theta),$
(62)

The path integral for the Green’s function $G^{\lambda}_{mn}$ can be written
symbolically as

$\displaystyle
G^{\lambda}_{mn}(\pi^{\prime}_{(\tilde{\Sigma})}(p_{b},v_{b}),t_{b};\pi^{\prime}_{(\tilde{\Sigma})}(p_{a},v_{a}),t_{a})=$
$\displaystyle{\tilde{\rm
E}}_{{\xi_{\tilde{\Sigma}}(t_{a})=\pi^{\prime}_{(\tilde{\Sigma})}(p_{a},v_{a})}\atop{\xi_{\tilde{\Sigma}}(t_{b})=\pi^{\prime}_{(\tilde{\Sigma})}(p_{b},v_{b})}}\Bigl{[}(\overleftarrow{\exp})_{mn}^{\lambda}(\xi_{\tilde{\Sigma}}(t),t_{b},t_{a})\exp\\{\frac{1}{\mu^{2}\kappa
m}\int_{t_{a}}^{t_{b}}\tilde{V}(\xi_{1}(u),\xi_{2}(u))du\\}\Bigr{]}$
$\displaystyle=\int\limits_{{\xi_{\tilde{\Sigma}}(t_{a})=\pi^{\prime}_{(\tilde{\Sigma})}(p_{a},v_{a})}\atop{\xi_{\tilde{\Sigma}}(t_{b})=\pi^{\prime}_{(\tilde{\Sigma})}(p_{b},v_{b})}}d{\mu}^{\xi_{\tilde{\Sigma}}}\exp\\{\frac{1}{\mu^{2}\kappa
m}\int_{t_{a}}^{t_{b}}\tilde{V}(\xi_{\tilde{\Sigma}}(u))du\\}$
$\displaystyle\times\overleftarrow{\exp}\int_{t_{a}}^{t_{b}}\Bigl{\\{}{\mu}^{2}\kappa\Bigl{[}\frac{1}{2}d^{\alpha\nu}(\xi_{\tilde{\Sigma}}(u))(J_{\alpha})_{mr}^{\lambda}(J_{\nu})_{rn}^{\lambda}\Bigr{.}\Bigr{.}$
$\displaystyle-\Bigl{.}\Bigl{.}\frac{1}{2}\frac{1}{\sqrt{d\,H}}\frac{\partial}{\partial
Q^{\ast}{}^{A}}\left(\sqrt{d\,H}G^{EF}N^{A}_{E}N^{C}_{F}\underset{\scriptscriptstyle{(\gamma)}}{{\mathscr{A}}_{C}^{\nu}}\right)(J_{\nu})_{mn}^{\lambda}$
$\displaystyle-\frac{1}{2}(G^{EC}\Lambda^{\nu}_{E}\Lambda^{\mu}_{C})\frac{1}{\sqrt{d\,H}}\frac{\partial}{\partial\tilde{f}^{b}}\left(\sqrt{d\,H}K^{b}_{\mu}\right)(J_{\nu})_{mn}^{\lambda}\Bigr{]}du$
$\displaystyle-\mu\sqrt{\kappa}{\Lambda}^{\beta}_{C}(J_{\beta})_{mn}^{\lambda}\Bigl{[}{\tilde{\Pi}}^{C}_{E}{\mathscr{X}}^{E}_{\bar{M}}({\xi}_{1}(u))dw^{\bar{M}}(u)+{\tilde{\Pi}}^{C}_{a}{\mathscr{X}}^{a}_{\bar{b}}dw^{\bar{b}}(u)\Bigr{]}\Bigr{\\}}.$
(63)

The semigroup with this kernel acts in the space of the equivariant functions
given on $\tilde{\mathcal{P}}$:

$\tilde{\psi}_{n}(pg,vg)=D^{\lambda}_{mn}(g)\tilde{\psi}_{m}(p,v).$

In local coordinates, the isomorphism of these functions with the functions
$\psi_{n}\in\Gamma({{\tilde{\Sigma}}},V^{*})$ is represented as follows:

$\tilde{\psi}_{n}(F(Q^{\ast},e),\bar{D}^{b}_{c}(e)\tilde{f}^{c})=\psi_{n}(Q^{\ast},\tilde{f}).$

The semigroup with the Green’s function $G^{\lambda}_{mn}$ is used to describe
the “quantum evolution” of the reduced dynamical system on the manifold
$\tilde{\mathcal{M}}$ – the orbit space of the principal fiber bundle. This
path integral reduction corresponds to the case of the reduction of a
mechanical system with symmetry to a nonzero momentum level [12].

Note that in our case, in order to describe the evolution on the orbit space
$\tilde{\mathcal{M}}$ we use an additional surface $\tilde{\Sigma}$ (‘gauge
surface’) on which diffusion is given locally by the solution of the local
stochastic differential equation (51). In fact, this equation is represented
by two equations: (45) and (46). We see that in both of these equations, in
the drift coefficients, there are terms that we denoted as $j$ and $j$. The
presence of $j$-terms in the drift coefficients of our stochastic equation on
$\tilde{\Sigma}$ is essential for the correct description of the stochastic
process on the manifold $\tilde{{\Sigma}}$, considered as a submanifold in the
ambient manifold with the horizontal metric. On the other hand, the $j$-terms
in the drift coefficients of stochastic differential equations are not
necessary.

Transformation of a measure in a path integral by replacing the stochastic
process ${\xi}_{\tilde{\Sigma}}$, determined by solutions of local stochastic
differential equations (51), with the process
${\tilde{\xi}}_{\tilde{\Sigma}}$, described by the same local stochastic
equations, but without the $j$ terms,101010For the components of the local
stochastic process representing the global stochastic process
${\tilde{\xi}_{\tilde{\Sigma}}}(t)$ on charts, we will use the same notation
as before, namely $(Q^{\ast}_{t}{}^{A},\tilde{f}_{t}^{a})$. can be performed
using the Girsanov-Cameron-Martin transformation.

In our article, we consider such a transformation for the reduction onto a
zero-momentum level. In this case, i.e. when $\lambda=0$, instead of (62), we
get the integral relation between the scalar Green’s functions (semigroup
kernels) acting in the spaces of scalar functions given on
$\tilde{\mathcal{P}}$ and on $\tilde{\Sigma}$.

Note, that the standard Girsanov-Cameron-Martin formula for the path integral
measure transformation deals with the case when the matrix representing
diffusion coefficient of the stochastic differential equation is
nondegenerate. The elements of such a matrix in (51) are defined using
projection operators. Therefore, this matrix is degenerate. In spite of this,
an analogue of the Girsanov-Cameron-Martin formula can also be derived in our
case. Its derivation (using the Itô’s differentiation formula) follows from
the assumption that the solution of a parabolic differential equation with the
differential operator (58) (taken at $\lambda=0$) is unique (modulo those
ambiguities that we have in the problem under consideration).

The Radon-Nikodym derivative of the measure ${\mu}^{{\xi}_{\tilde{\Sigma}}}$
with respect to the measure ${\mu}^{\tilde{\xi}_{\tilde{\Sigma}}}$ is defined
as

$\frac{d{\mu}^{{\xi}_{\tilde{\Sigma}}}}{d{\mu}^{\tilde{\xi}_{\tilde{\Sigma}}}}({\tilde{\xi}_{\tilde{\Sigma}}}(t))=\exp\int^{t}_{t_{a}}\Bigl{[}\mu^{2}\kappa<A^{-1}\\!\cdot\vec{j},\,d\bar{w}_{s}>-\frac{1}{2}(\mu^{2}\kappa)^{2}||A^{-1}\\!\cdot\vec{j}||^{2}ds\Bigr{]},$
(64)

where $A^{-1}$ is the matrix which is (pseudo)inverse to the ‘diffusion’
matrix $X_{1}$ of the local stochastic differential equation (51), the Wiener
process $\bar{w}_{s}$ consists of two independent Wiener processes:
$\bar{w}_{s}=(w_{s}^{\bar{M}},w_{s}^{\bar{b}})$, and $\vec{j}$ is the
projection of the mean curvature vector field of the orbit onto the
submanifold $\tilde{\Sigma}$. It has the following components:

$\displaystyle
j^{A}=-\frac{1}{2}\,G^{CC^{\prime}}N^{A}_{C^{\prime}}N^{B^{\prime}}_{C}G_{BB^{\prime}}\,d^{\alpha\beta}\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{B},$
$\displaystyle
j^{a}=-\frac{1}{2}d^{\alpha\beta}\Bigl{(}N^{a}_{B}\,\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{B}+\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{a}\Bigr{)}.$
(65)

To get an explicit representation of $A^{-1}\\!\cdot\vec{j}$ in (64), we first
rewrite the terms on the right-hand side of the equations (65) using the
following identities:

$\displaystyle
d^{\alpha\beta}\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{B}=-\frac{1}{2}\Bigl{(}G^{BC}N^{A}_{C}{\sigma}_{A}+G^{BC}N^{a}_{C}\,{\sigma}_{a}\Bigr{)},$
$\displaystyle
d^{\alpha\beta}\bigl{(}\nabla_{K_{\alpha}}K_{\beta}\bigr{)}^{a}=-\frac{1}{2}G^{aq}{\sigma}_{q},$

where ${\sigma}_{A}=d^{\alpha\beta}\frac{\partial}{\partial Q^{\ast
A}}d_{\alpha\beta}\equiv\frac{\partial}{\partial Q^{\ast A}}\ln d$ and
${\sigma}_{a}=\frac{\partial}{\partial{\tilde{f}}^{a}}\ln d,$ $d=\det
d_{\alpha\beta}$.

As a result, we obtain

$\displaystyle\displaystyle\Bigl{(}\begin{array}[]{cc}j^{A}\\\
j^{a}\end{array}\Bigr{)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Bigl{(}\begin{array}[]{cc}G^{BD}N^{A}_{B}N^{C}_{D}&G^{BD}N^{A}_{B}N^{b}_{D}\\\
G^{BD}N^{a}_{B}N^{C}_{D}&G^{BD}N^{a}_{B}N^{b}_{D}+G^{ab}\\\
\end{array}\Bigr{)}\Bigl{(}\begin{array}[]{cc}{\sigma}_{C}\\\ {\sigma}_{b}\\\
\end{array}\Bigr{)}$ (72) $\displaystyle=$
$\displaystyle\frac{1}{4}\Bigl{(}X_{1}\cdot
X_{1}^{\top}\Bigr{)}\Bigl{(}\begin{array}[]{cc}{\sigma}_{C}\\\ {\sigma}_{b}\\\
\end{array}\Bigr{)}.$ (75)

In turn, this means that

$\displaystyle
A^{-1}\\!\cdot\vec{j}=X_{1}^{\top}\cdot\vec{j}=\frac{1}{4}\Bigl{(}\begin{array}[]{cc}N^{A}_{C}\mathscr{X}^{C}_{\bar{M}}&N^{b}_{D}\mathscr{X}^{D}_{\bar{M}}\\\
0&\mathscr{X}^{b}_{\bar{b}}\\\
\end{array}\Bigr{)}\Bigl{(}\begin{array}[]{cc}{\sigma}_{C}\\\ {\sigma}_{b}\\\
\end{array}\Bigr{)}.$

Thus, the terms under the integral in (64) have the following form:

$<A^{-1}\\!\cdot\vec{j},\,d\bar{w}_{s}>=\frac{1}{4}\bigl{[}(N^{A}_{C}\mathscr{X}^{C}_{\bar{M}}{\sigma}_{A}+N^{a}_{D}\mathscr{X}^{D}_{\bar{M}}{\sigma}_{a})dw^{\bar{M}}_{s}+\mathscr{X}^{a}_{\bar{b}}dw^{\bar{b}}_{s}\bigr{]}$

and

$\displaystyle||A^{-1}\\!\cdot\vec{j}||^{2}$
$\displaystyle\;=\frac{1}{16}\bigl{[}G^{CD}N^{A}_{C}N^{B}_{D}{\sigma}_{A}{\sigma}_{B}+2G^{CD}N^{a}_{C}N^{B}_{D}{\sigma}_{a}{\sigma}_{B}+(G^{CD}N^{a}_{C}N^{b}_{D}+G^{ab}){\sigma}_{a}{\sigma}_{b}\bigr{]}$
$\displaystyle\;=\frac{1}{16}\bigl{[}h^{AB}{\sigma}_{A}{\sigma}_{B}+2h^{aB}{\sigma}_{a}{\sigma}_{B}+h^{ab}{\sigma}_{a}{\sigma}_{b}\bigr{]}.$

For brevity, the quadratic form in square brackets of the above expression
will be further denoted as $<\partial\sigma,\partial\sigma>_{\tilde{\Sigma}}$.

The argument of the exponential on the right-hand side of the equation (64) is
represented by the sum of the stochastic and ordinary integrals. This
exponential function can be rewritten to include only ordinary integrals. This
can be done using the special identity (known as the Itô’s identity).

In our case, the identity is obtained from the solution of a local stochastic
differential equation having
$\exp\sigma(Q^{\ast}_{t}{}^{A},\tilde{f}_{t}^{a})$ as an unknown function. The
equation itself is derived by taking the stochastic Itô differential of this
function, provided that now the components $Q^{\ast}_{t}{}^{A}$ and
$\tilde{f}_{t}^{a}$ of the local stochastic process representing the global
stochastic process ${\tilde{\xi}_{\tilde{\Sigma}}}(t)$ must satisfy the
stochastic differential equations (45) and (46) without ${j}$ terms.

Performing calculations, one can find the following identity:

$\displaystyle\exp\int^{{t}}_{t_{a}}(\mu\sqrt{\kappa})\Bigl{[}(N^{A}_{C}\mathscr{X}^{C}_{\bar{M}}{\sigma}_{A}+N^{a}_{D}\mathscr{X}^{D}_{\bar{M}}{\sigma}_{a})dw^{\bar{M}}_{s}+\mathscr{X}^{a}_{\bar{b}}dw^{\bar{b}}_{s}\Bigr{]}$
$\displaystyle=\left(\frac{exp(\sigma(Q^{\ast}(t),\tilde{f}(t))}{exp(\sigma(Q^{\ast}(t_{a}),\tilde{f}(t_{a}))}\right)\exp\biggl{\\{}\int^{{t}}_{t_{a}}\frac{1}{2}(\mu^{2}{\kappa})\Bigl{[}h^{AB}\sigma_{AB}+2h^{Ab}\sigma_{Ab}+h^{ab}\sigma_{ab}$
$\displaystyle-\Bigl{(}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{A}_{\tilde{B}\tilde{M}}-2j^{A}\Bigr{)}\sigma_{A}-\Bigl{(}h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{a}_{\tilde{B}\tilde{M}}-2j^{a}\Bigr{)}\sigma_{a}\Bigr{]}ds\biggr{\\}}.$

From this identity for local representatives of the stochastic process
${\tilde{\xi}_{\tilde{\Sigma}}}(t)$ it follows that (64) can be rewritten as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{d{\mu}^{{\xi}_{\tilde{\Sigma}}}}{d{\mu}^{\tilde{\xi}_{\tilde{\Sigma}}}}({\tilde{\xi}_{\tilde{\Sigma}}}(t))$
$\displaystyle=$
$\displaystyle\left(\frac{exp(\sigma({\tilde{\xi}_{\tilde{\Sigma}}}(t)}{exp(\sigma({\tilde{\xi}_{\tilde{\Sigma}}}(t_{a})}\right)^{1/4}$
(76)
$\displaystyle\times\exp\Bigl{\\{}-\frac{1}{8}\mu^{2}\kappa\int^{t}_{t_{a}}\bigl{(}\tilde{\triangle}_{\tilde{\Sigma}}\sigma+\frac{1}{4}<\partial\sigma,\partial\sigma>_{\tilde{\Sigma}}\bigr{)}ds\Bigr{\\}},$

where by $\tilde{\triangle}_{\tilde{\Sigma}}\sigma$ we denote the expression
in the square brackets on the right-hand side of the previous formula. We see
that in addition to the standard differential expression used to define the
Laplace-Beltrami operator, it includes two more terms: $2j^{A}\sigma_{A}$ and
$2j^{a}\sigma_{a}$. It can be shown that the sum of these terms is equal to
zero.

Using the representations (43) and (44) for $j^{A}$ and $j^{a}$, we rewrite
this sum as follows:

$h^{BM}(N^{A}_{B,M}\sigma_{A}+N^{a}_{B,M}\sigma_{a})+(K^{A}_{\alpha}\sigma_{A}+K^{a}_{\alpha}\sigma_{a})\,\Lambda^{\alpha}_{C}\,h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{C}_{\tilde{B}\tilde{M}}.$

To prove that the expression in the second bracket of the above expression is
equal to zero, one must first apply the operator
$K^{A}_{\alpha}\partial_{A}+K^{a}_{\alpha}\partial_{a}$ to the metric
$d_{\mu\nu}=K^{C}_{\mu}G_{CD}K^{D}_{\nu}+K^{p}_{\mu}G_{pq}K^{q}_{\nu}$ and
then transform the resulting expression using Killing’s relations from
Appendix. This leads to

$K^{A}_{\alpha}\partial_{A}d_{\mu\nu}+K^{a}_{\alpha}\partial_{a}d_{\mu\nu}=d_{\nu\sigma}c^{\sigma}_{\alpha\mu}+d_{\mu\sigma}c^{\sigma}_{\alpha\nu}.$

By multiplying both sides of this equality by $d^{\mu\nu}$, we can conclude
that the right-hand side of the equality is equal to zero for semisimple Lie
groups.

The expression in the first bracket of the equality under study is also equal
to zero. This can be shown using the following representations:

$N^{A}_{B,M}=-K^{A}_{\alpha,M}{\Lambda}^{\alpha}_{B}-K^{A}_{\alpha}{\Lambda}^{\alpha}_{B,M}\;\;\mathrm{and}\;\;N^{a}_{B,M}=-K^{a}_{\alpha}{\Lambda}^{\alpha}_{B,M},$

and taking into account that $h^{BM}{\Lambda}^{\alpha}_{B}=0$ and
$K^{A}_{\alpha}\sigma_{A}+K^{a}_{\alpha}\sigma_{a}=0$.

The global integral relation between the Green’s functions for the reduction
onto the zero-momentum level is given by

$\displaystyle
d_{b}^{-1/4}d_{a}^{-1/4}G_{\tilde{\Sigma}}(\pi^{\prime}_{(\tilde{\Sigma})}(p_{b},v_{b}),t_{b};\pi^{\prime}_{(\tilde{\Sigma})}(p_{a},v_{a}),t_{a})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\displaystyle=\int_{\cal
G}G_{\tilde{\cal
P}}(p_{b}\theta,v_{b}\theta,t_{b};p_{a},v_{a},t_{a})d\mu(\theta),$

where $d_{b}$ and $d_{a}$ are the values of the $\det(d_{\alpha\beta})$ taken
at the points $\pi^{\prime}_{(\tilde{\Sigma})}(p_{b},v_{b})$ and
$\pi^{\prime}_{(\tilde{\Sigma})}(p_{a},v_{a})$. The Green’s function
$G_{\tilde{\Sigma}}$ is presented by the following path integral:

$\displaystyle
G_{\tilde{\Sigma}}(\pi^{\prime}_{(\tilde{\Sigma})}(p_{b},v_{b}),t_{b};\pi^{\prime}_{(\tilde{\Sigma})}(p_{a},v_{a}),t_{a})$
$\displaystyle=\int\limits_{{\tilde{\xi}_{\tilde{\Sigma}}(t_{a})=\pi^{\prime}_{(\tilde{\Sigma})}(p_{a},v_{a})}\atop{\tilde{\xi}_{\tilde{\Sigma}}(t_{b})=\pi^{\prime}_{(\tilde{\Sigma})}(p_{b},v_{b})}}d{\mu}^{\tilde{\xi}_{\tilde{\Sigma}}}\exp\Bigl{\\{}\frac{1}{\mu^{2}\kappa
m}\int_{t_{a}}^{t_{b}}\Bigl{[}\tilde{V}(\tilde{\xi}_{\tilde{\Sigma}}(u))+J(\sigma(\tilde{\xi}_{\tilde{\Sigma}}(u)))\Bigr{]}du\Bigr{\\}},$

where $J$ is the integrand in the reduction Jacobian (76):

$J=-\frac{1}{8}\mu^{2}\kappa\bigl{(}\triangle_{\tilde{\Sigma}}^{{\scriptscriptstyle\rm
H}}\sigma+\frac{1}{4}<\partial\sigma,\partial\sigma>_{\tilde{\Sigma}}\bigr{)}.$
(77)

In this expression,

$\triangle_{\tilde{\Sigma}}^{{\scriptscriptstyle\rm
H}}\sigma=h^{AB}\sigma_{AB}+2h^{Ab}\sigma_{Ab}+h^{ab}\sigma_{ab}-h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{A}_{\tilde{B}\tilde{M}}\sigma_{A}-h^{\tilde{B}\tilde{M}}\,{}^{\mathrm{H}}{\tilde{\Gamma}}^{a}_{\tilde{B}\tilde{M}}\sigma_{a}.$

The global semigroup determined by the Green’s function $G_{\tilde{\Sigma}}$
acts in the Hilbert space of the scalar functions on $\tilde{\Sigma}$ with the
following scalar product
$(\psi_{1},\psi_{2})=\int\psi_{1}\,\psi_{2}\,\,dv_{\tilde{\Sigma}}.$

The Green’s function $G_{\tilde{\Sigma}}$ satisfies the forward Kolmogorov
equation with the operator

$\hat{H}_{\kappa}=\frac{\hbar\kappa}{2m}\tilde{\triangle}_{\tilde{\Sigma}}-\frac{\hbar\kappa}{8m}\Bigl{[}\triangle_{\tilde{\Sigma}}^{{\scriptscriptstyle\rm
H}}\sigma+\frac{1}{4}<\partial\sigma,\partial\sigma>_{\tilde{\Sigma}}\Bigr{]}+\frac{1}{\hbar\kappa}\tilde{V}.$

Note that at $\kappa=i$ the forward Kolmogorov equation becomes the
Schrödinger equation with the Hamilton operator
$\hat{H}=-\frac{\hbar}{\kappa}{\hat{H}}_{\kappa}\bigl{|}_{\kappa=i}$.

Note also that the geometric properties of the operator
$\tilde{\triangle}_{\tilde{\Sigma}}$,

$\tilde{\triangle}_{\tilde{\Sigma}}=\triangle_{\tilde{\Sigma}}^{{\scriptscriptstyle\rm
H}}+2j^{A}\,\partial_{A}+2j^{a}\,\partial_{a},$

as well as the fact that this operator could be transformed into the Laplace-
Beltrami operator given on the orbit space $\tilde{\mathcal{M}}$ with the
Riemannian metric (28), if in charts of the reduced manifold it were possible
to find invariant local coordinates $(x^{i},\tilde{f}^{a})$, such that
$\chi^{\alpha}(Q^{\ast}(x^{i}))\equiv 0$, lead us to conlusion that the global
semigroup with the kernel $G_{\tilde{\Sigma}}$ can be used to describe the
diffusion of interacting scalar particles on the orbit space
$\tilde{\mathcal{M}}$ of the principal fiber bundle ${\rm
P}(\tilde{\mathcal{M}},\mathcal{G})$.

##  6 Conclusion

In this article we have considered the reduction procedure in the Wiener-type
path integral which represents the “quantum evolution” of a special finite-
dimensional mechanical system with symmetry, consisting of two interacting
scalar particles. The choice of such a system for study is due to its
geometric properties, similar to the properties of those gauge theories that
are used to describe the interaction of Yang-Mills fields with scalar fields.

Note that in order to apply the standard approach to the quantization of gauge
theories, the original gauge fields must first be rewritten in terms of the
constrained variables. In accordance with this, in the principal fiber bundle
associated with our problem, a special system of local coordinates (including
dependent coordinates) was introduced. The coordinates of this coordinate
system are defined using the local surface $\tilde{\Sigma}$ (a submanifold of
the total space of the principal fiber bundle $\rm
P(\tilde{\mathcal{M}},\mathcal{G})$). And the reduced motion on the orbit
space $\tilde{\mathcal{M}}$ is locally described in terms of these
coordinates.

The reduction procedure performed in the article leads to the integral
relation (62) between Green’s functions $G^{\lambda}_{mn}$ (the kernel of a
reduced semigroup acting in the space of sections
$\Gamma({{\tilde{\Sigma}}},V^{*})$ for the general case of reduction onto the
nonzero momentum level, $\lambda\neq 0$) and $G_{\tilde{\cal P}}$, which is
the kernel of the original semigroup. The representation of $G^{\lambda}_{mn}$
in terms of the path integral is given by the expression (63).

The result obtained essentially depends on what assumption about the
submanifold ${\tilde{\Sigma}}$ was made. Provided that the submanifold
${\tilde{\Sigma}}$ is a global section of the principal fiber bundle $\rm
P(\tilde{\mathcal{M}},\mathcal{G})$ (which is thus a trivial principal bundle
isomorphic to the trivial bundle $\rm P(\tilde{\Sigma},\mathcal{G}$), then in
this case it follows that the dependent coordinates $Q^{\ast A}$ (and the
coordinates $\tilde{f}^{a}$) can be thought of as global variables defined on
${\tilde{\Sigma}}$. This case is analogous to what is usually assumed in gauge
theories, when the evolution of the reduced system on the orbit space of the
gauge group is described in terms of variables with constraints.

However, in the most cases, the principal fiber bundles of gauge theories are
nontrivial. Therefore, using this method, we can actually describe the
‘quantum’ dynamics on the orbit space only locally. Note that the problem of
transition from a local description of the reduced dynamics given in terms of
dependent coordinates to a global description is not solved yet.

The main result of the article is the obtained expression (77) for the
integrand $J$ in the path integral reduction Jacobian for the case of
reduction onto a zero-momentum level. In particular, it is shown that $J$ is
generated by the projection of the mean curvature vector field of the orbit
onto the submanifold ${\tilde{\Sigma}}$.

In addition, we note that the resulting $J$ is a generalization to interacting
dynamical systems with symmetry of a similar expression obtained by J. Lott in
[22] while studying the quantum potential in pure Yang-Mills theory.

The Girsanov-Cameron-Martin transformation can also be performed in the path
integral (63). In this case, the exponential with the integral of $J$, defined
by the formula (77), will be the diagonal part of the reduction Jacobian.

## Appendix

## Relationship between the Christoffel symbols

$\displaystyle{\Gamma}^{a}_{kn}=\tilde{h}^{am}{\Gamma}_{knm}+\tilde{h}^{ab}{\Gamma}_{knb}$
$\displaystyle{\Gamma}^{a}_{kb}=\tilde{h}^{an}{\Gamma}_{kbn}+\tilde{h}^{ac}{\Gamma}_{kbc}$
$\displaystyle{\Gamma}^{a}_{bm}=\tilde{h}^{an}{\Gamma}_{bmn}+\tilde{h}^{ac}{\Gamma}_{bmc}$
$\displaystyle{\Gamma}^{a}_{bc}=\tilde{h}^{an}{\Gamma}_{bcn}+\tilde{h}^{ad}{\Gamma}_{bcd}$
$\displaystyle{\Gamma}_{knb}={}^{\mathrm{H}}{\tilde{\Gamma}}_{BMb}\,Q^{\ast
B}_{k}Q^{\ast M}_{n}+{\tilde{G}}^{H}_{Ab}Q^{\ast A}_{kn}$
$\displaystyle{\Gamma}_{kbn}={}^{\mathrm{H}}{\tilde{\Gamma}}_{AbB}\,Q^{\ast
A}_{k}Q^{\ast B}_{n}$
$\displaystyle{\Gamma}_{kbc}={}^{\mathrm{H}}{\tilde{\Gamma}}_{Abc}\,Q^{\ast
A}_{k}$
$\displaystyle{\Gamma}_{bmn}={}^{\mathrm{H}}{\tilde{\Gamma}}_{bAB}\,Q^{\ast
A}_{m}Q^{\ast B}_{n}$
$\displaystyle{\Gamma}_{bmc}={}^{\mathrm{H}}{\tilde{\Gamma}}_{bAc}\,Q^{\ast
A}_{m}$
$\displaystyle{}^{\mathrm{H}}{\tilde{\Gamma}}_{AbB}={\tilde{G}}^{H}_{\tilde{R}B}{}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{Ab}$
$\displaystyle{}^{\mathrm{H}}{\tilde{\Gamma}}_{Abc}={\tilde{G}}^{H}_{\tilde{R}c}{}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{Ab}$
$\displaystyle{}^{\mathrm{H}}{\tilde{\Gamma}}_{bAB}={\tilde{G}}^{H}_{\tilde{R}B}{}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{bA}$
$\displaystyle{}^{\mathrm{H}}{\tilde{\Gamma}}_{bAc}={\tilde{G}}^{H}_{\tilde{R}c}{}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{bA}$
$\displaystyle{\Gamma}^{a}_{kn}=G^{EF}N^{a}_{E}{\tilde{G}}^{H}_{\tilde{R}F}{}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{BM}Q^{\ast
B}_{k}Q^{\ast
M}_{n}+G^{ab}{\tilde{G}}^{H}_{\tilde{R}b}{}^{\mathrm{H}}{\tilde{\Gamma}}^{\tilde{R}}_{BM}Q^{\ast
B}_{k}Q^{\ast M}_{n}+N^{a}_{E}Q^{\ast A}_{kn}$
$\displaystyle{\Gamma}^{a}_{kb}=N^{a}_{\tilde{R}}\,{\tilde{\Gamma}}^{\tilde{R}}_{Ab}\,Q^{\ast
A}_{k}$
$\displaystyle{\Gamma}^{a}_{bm}=N^{a}_{\tilde{R}}\,{\tilde{\Gamma}}^{\tilde{R}}_{bA}\,Q^{\ast
A}_{m}$
$\displaystyle{\Gamma}^{a}_{bc}=N^{a}_{\tilde{R}}\,{\tilde{\Gamma}}^{\tilde{R}}_{bc}$

Killing relations

$\displaystyle\;\;\;\;\,K^{A}_{\alpha}G_{CD,A}=-G_{CR}K^{R}_{\alpha,D}-G_{RD}K^{R}_{\alpha,C},$
$\displaystyle-
G_{qb^{\prime}}K^{b^{\prime}}_{\alpha,a}-G_{ab^{\prime}}K^{b^{\prime}}_{\alpha,q}=0.$

Properties of the horizontal metric ${\tilde{G}}^{H}_{\tilde{A}\tilde{B}}$

$\displaystyle
N^{T}_{F}{\tilde{G}}^{H}_{\tilde{R}T}+N^{b}_{F}{\tilde{G}}^{H}_{\tilde{R}b}={\tilde{G}}^{H}_{\tilde{R}F},$
$\displaystyle
G^{ab}{\tilde{G}}^{H}_{Ab}=\tilde{\Pi}^{a}_{A},\;\;\;G^{EF}{\tilde{G}}^{H}_{AF}=\tilde{\Pi}^{E}_{A},\;\;\;N^{a}_{E}\tilde{\Pi}^{E}_{A}+\tilde{\Pi}^{a}_{A}=N^{a}_{A}.$

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