Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://xray.sai.msu.ru/~popova/papers/lit21.ps Äàòà èçìåíåíèÿ: Wed Jul 26 16:19:14 2006 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 22:43:43 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 8

NONSTANDARD VARIATIONAL PRINCIPLES
IN QUANTUM MECHANICS WITH SELF-INTERACTION
A. D. Popova
Sternberg Astronomical Institute, Moscow, Russia
Abstract
A nonlinear quantum mechanics with self-consistent interaction is constructed
by a way which provides some features of classical field theory, but does not preclude
from probability interpretation. The main feature of this theory is its invariance un-
der the rescaling transformations of wave functions. Some fictitious, unobtainable
exactly, Lagrangian function and the variational principles of its smoothed extremal-
ity, instead of those of extremality of an action, are the necessary elements of this
construction.
1 Introduction
The standard self-consistent way of including interaction in classical field theories, based
on the localization of symmetries, is well-known, it was introduced long ago [1] and gen-
eralized for gravitation in [2]. However this way is not directly suitable for theories
which one wants to interpret as quantum-mechanical. Quantum mechanics (first quan-
tization) remains up date linear theory, without included interaction, since one believes
that the superposition principle provides probability interpretation. Quantum theories
with interaction are just the second quantized field theories. Recall that the procedure
of second quantization differs from that of first quantization, the former operates with
the field space. However, the main drawback of quantum field theories is the appearance
of divergencies, often immovable. Constructions presented here give an alternative way
for inclusion of quantum (self-) interaction which perhaps could resolve the problems of
second quantized theories.
Thus, the inherent feature of quantum mechanics is its probability interpretation. This
means that the three postulates of the probability theory must be fulfilled i) there exists
a unit-valued functional making the sense of full probability, ii) the full probability is the
sum of individual probabilities for mutually exclusive events, and iii) the full probability is
the product of individual probabilities for independent events. These postulates are easily
incorporated in linear quantum mechanics. Indeed, the (linear) rescaling transformation
for the wave function #,
# # a# (1.1)
with a an arbitrary complex number, permits one to supply with unit value a conserved
norm-square functional. The second postulate is realized for the superposition of orthog-
onal states, and the third one is suitable for the standard many-particle problem.
The main problem of the inclusion of (self-)interaction into a theory which one wants
to interpret by a quantum-mechanical way is to preserve the invariance of the theory un-
der (1.1). In the paper [3], we outlined the program of such the inclusion for gravitational
self-interaction into one-particle quantum mechanics. (We use the term "self-interaction"
for one particle.) It was continued in [4] and accomplished in [5, 6] for the gravitational
1

interaction. The inclusion of electromagnetic interaction was considered in [7]. An ap-
proach to the many-particle problem of quantum mechanics was also described [7, 5, 6].
We already discussed interpretation features [3, 5] and argued for that all the postulates
of the probability theory can also survive in the framework of the nonlinear theory with
interaction.
Without claims to a mathematical rigor, we give here a general recipe of constructing
a theory invariant under (1.1) which gives just equations slightly different from those
in a classical field theory. The evolution equation for a wave function remains formally
the same. In equations for an interaction field, a source term is a "classical" source
divided by the conserved norm-square, that permits one to interpret the source as some
probability density. However, this is achieved by introducing exotic variational principles:
the principles of extremality of an action are replaced by some principles of smoothed
extremality. We make our generalizations step by step, paying attention to intermediate
stages and general facts. Below, we review them following the plan of the paper. Some
preliminary mathematics for this is adduced in [8, 9, 10].
In Section 2, we consider an evolution equation which is taken with an arbitrary order
of time derivatives, and the nature of the interaction field (or fields) is not specified. It
is important to note (especially if we deal with the inclusion of gravity [5, 6]) that wave
functions should be 3-dimensional scalar densities. We also assume that wave functions
are always square-integrable over any required 3-domain. We present the functionals
of a norm-square, a bilinear action and a bilinear current, which are associated with
the evolution equation. We assume that the norm-square functional is positive definite.
Certainly, the problem of nonpositive-definite norm-square functionals appears in theories
with higher time derivatives [11] but it has nothing to do with the inclusion of self-
interaction.
In Section 3, we represent a remarkable mathematical fact which, to our knowledge, is
not described earlier. We introduce a formal differentiation with respect to the operator
of time differentiation The fact is that, for any bilinear action with arbitrary operators
and of an arbitrary order n of time derivatives, without reference to any especially chosen
wave functions (and, moreover, to any solutions of the evolution equation), the derivative
of the bilinear action with respect to the time-differentiation operator coincides with the
norm-square functional of Section 2. This result is the most important in what follows.
It gets a realization in terms of stationary states in Section 4 and in terms of so-called
time-separated states in Section 5. It also lays the foundation of the constructions of
Sec.7 and Sec.8.
In Section 4, we recall the Rayleigh variational principle. It was introduced in [12].
This is just the principle for obtaining an eigenvalue problem. Its degenerate form ap-
pears in the standard Schrodinger quantum mechanics [13]. Further development of the
Rayleigh formalism one can find in the monograph [14]. We already gave the description
the Rayleigh formalism [3, 4] and here we also account it in brief.
Section 5 is devoted to the case when the evolution equation admits the separation
of time and space variables, although we do not formulate conditions of such the sepa-
ration. The separation of space variables for typical equations of mathematical physics
was considered in [15], some algorithm was constructed for obtaining a system of coor-
dinates where the separation can be done. We outline a problem which generalizes the
eigenvalue problem polynomial in a spectral parameter and introduce the concepts of
a time-separated state and a basic function which generalize those of an eigenfunction
and a spectral parameter, respectively. Generalization of the Rayleigh formalism is also
2

given. Some solvable examples are considered. On one hand, Section 5 represents only a
preparatory stage for inclusion of self-interaction because special form of time-separated
states precludes from it. On the other, it may be of interest by itself.
In Section 6, we recall the standard procedure of inclusion of interaction which is
suitable for classical field theories and write standard variational principles.
Section 7 is a central section of this paper. We present here some formal constructions,
just the variational principles of smoothed extremality leading to a theory with desired
features.
In Section 8, we give our formulation of the many-particle problem of quantum me-
chanics. It was not self-consistently formulated so far [16]. Traditionally it was restricted
by the non-relativistic motion of particles, and, moreover, self-interaction of particles
was excluded. We consider M quantum particles, no matter of various or the same
sorts, described by a wave function which is the product of M individual functions
# I (I = 1, 2, . . . , M) and extend our variational principles to the M-particle problem.
Formally, our equations resembles equations of a classical field theory which describes M
different matter fields with a distinction similar to above. However, our theory remain
invariant under the rescaling transformations
# I # a I # I (1.2)
for all I. The inclusion of interaction between particles exposes the problem of a relation
between the usual 3-dimensional space and the 3M-dimensional space and the impossi-
bility to include interaction in the only 3-space [17]. This problem is discussed.
2 Evolution Equation
Let the wave function #(t, x) satisfy the evolution equation of the order n
n
# k=0
“
A k # k # = 0 (2.1)
where # # i#/#t, “
A k are some formal differential operators which do not already contain
time differentiation but contain the differentiation with respect to the 3-space coordinates
x. Let “
A k also involve some sufficiently smooth real functions #(t, x) which will be treated
as physical fields,
“
A k = “
A k [#(t, x)].
Note that the presence of # supplies “
A k with a time dependence. We settle below that, in
any expression, # and “
A k act until an unclosed right round bracket or, otherwise, on the
whole right penetrating any closed brackets. The equation complex conjugate to Eq.(2.1)
is n
# k=0
“
A # k # #k # # = 0. (2.2)
As for the hermiticity properties of “
A k , the transposed ( “
A T
k ) and complex conjugate ( “
A # k )
operators ought to be connected by the relation:
n
#
k=0
“
A # k # #k . . . =
n
# k=0
# #k “
A T
k . . . (2.3)
3

which follows from the equality
“
A # k =
n
# l=k
C k
l (# #(l-k) “
A T
l )
where C k
l = l!/[k!(l - k)!] is the binomial coefficient. Clearly, the operators “
A k are
generally non-Hermitean: “
A # k #= “
A T
k ; the hermiticity is restored for time-independent
operators.
We associate with Eq.(2.1) the following functional which plays the role of a La-
grangian function bilinear in # # and #:
K(t) = K[# # , #] =
n
#
k=0
# d 3 x # # “
A k # k # #
n
#
k=0
< # # · “
A k # k # >, (2.4)
the complex conjugate functional associated with Eq.(2.2) being
K # (t) = K # [# # , #] =
n
# k=0
# d 3 x ( “
A # k # #k # # ) # #
n
# k=0
< “
A # k # #k # # · # > .
Using the relation (2.3), one can see that
K(t) -K # (t) = #N 2 (t) (2.5)
where the functional N 2 can be expressed in several equivalent forms, and here we give
the two of them:
N 2 (t) = N 2 [# # , #] =
n
# k=0
k-1
# l=0
< # #l “
A T
k # # · # k-l-1 # >
=
n
# k=1
k-1
# l=0
(-1) l C l+1
k # l < “
A T
k # # · # k-l-1 # > . (2.6)
Therefore, under the condition K = 0 (hence, K # = 0) which is the weaker condition
than the fulfillment of Eq.(2.1), the conservation law
#N 2 = 0 (2.7)
holds. The functional (2.6) makes the sense of a norm-square functional and gives the
standard Schrodinger norm square in the case of the first order Schrodinger equation.
Although it is not immediately seen from (2.6), N 2 = (N 2 ) # . Note that (2.7) is valid for
any # independently of its origin, thus (2.6) is also suitable for the case of self-interaction.
We can introduce the bilinear action functional which is associated with the La-
grangian (2.4):
J = # dt K(t) .
= # dt K # (t) (2.8)
where the dot equality denote neglecting of some initial and final terms, [cf. (2.5)], which
have no significance for functional differentiation.
We also define the bilinear current functional which corresponds to the field #:
Q bil (t, x) = Q bil [# # , #] =
#J
##(t, x)
. (2.9)
Note that Q bil is a real quantity due to the reality of #.
4

3 Differentiation with Respect to the Operator of Time
Differentiation
In the previous section, we introduced functionals which contain the operator of time
differentiation expressed as # or # # depending on what function it is applied. Here, we
give formal rules of differentiation with respect to # and # # for the functionals of the
type (2.4). One can construct analogous rules by a less formal way if to express the
operator of time differentiation via a time derivative of the #-function with a difference of
time arguments and to express the functionals of the type (2.8) via contractions of these
functions. This was done in [8]. However, the rules presented here are more simple and
clear.
Thus, define the derivatives with respect to # and # # which we denote by the letters
# in the symbol of differentiation (#/#):
#
##
< # # · ## >=
#
##
< # # # # · # >=< # # · # >, (3.1)
#
## #
< # # · ## >=
#
## #
< # # # # · # >=< # # · # > . (3.2)
It can be easily shown that these rules imply that the functional derivatives with respect
to # and # # of a total time derivative vanish. Indeed,
#
##
# < # # · # >=
#
##
(- < # # # # · # > + < # # · ## >) = 0.
The rules (3.1) and (3.2) should be completed by the rule of differentiation of arbitrary-
order time derivatives, that is by some generalization of the standard Leibnitz rule:
#
##
< # # · # k # >=< # # ·
#
##
## k-1 # > + . . . + < # #(p-1) # # ·
#
##
## (k-p) # >
+ . . . + < # #(k-1) # # ·
#
##
## >=< # # · # (k-1) # > + . . . + < # #(p-1) # # · # (k-p) # >
+ . . . + < # #(k-1) # # · # >=
k-1
# l=0
< # #l # # · # (k-l-1) # > (3.3)
where only the symbol # which is the nearest to the operator of differentiation is immedi-
ately differentiated. The rules analogous to (3.3) for #/## # and for another distribution
of # k between # and # can be restored without difficulty. The rule is clear: when # [# # ]
at the p-th place is differentiated, all the remaining (p-1) # [# # ] on the left are translated
to the left [right] function # [#] with acquiring [losing] asterisks. It follows that the result
of the above differentiation is independent of a concrete distribution of # between # and
#, the only summarized order of differentiation, k, is in essence.
Now let us differentiate the Lagrangian functional (2.4) with respect to # and # # , it
is easy to verify that
#
##
K[# # , #] =
#
## # K[# # , #] # K # = N 2 [# # , #]. (3.4)
Thus, the equality (3.4) contains an achievement of this section:
3.1 # : The functional derivative of the Lagrangian functional (2.4) with respect to the
time differentiation operator is equal to the norm-square functional (2.6).
In the next sections, we demonstrate that, for functions of some special types, the
operator of time differentiation is replaced by equivalent multiplication operators, and
that the proposition 3.1 # always has an equivalent counterpart.
5

4 Stationary Case and Rayleigh Variational Principles
Let the operators “
A k be independent of time and involve functions #(x), then they can
be assumed Hermitean: “
A # k = “
A T
k . Consider the stationary wave function
# s (t, x) = exp(-i#t)#(x), (4.1)
where the spectral parameter # is assumed to be a real number, and #(x) is an eigen-
function to be determined. After substituting (4.1) into (2.1), the latter reduces to the
eigenvalue problem with the nonlinear dependence on the spectral parameter # (see also
[11]):
n
# k=0
# k “
A k #(x) = 0. (4.2)
The Lagrangian functional (2.4) on (4.1) is associated with Eq.(4.2):
K(#) = K[# # s , # s ] = K[# # , #; #] =
n
# k=0
# k < # # · “
A k # > . (4.3)
We can calculate the derivative
#K(#)
## # K # (#) =
n
# k=1
k# k-1 < # # · “
A k # >, (4.4)
leading to the proposition
4.1 # . The functional (4.4) coincides with the norm-square functional (2.6) on the
function (4.1):
K # (#) = N 2 [# # s , # s ] = N 2 [# # , #; #].
The stationary bilinear #-current (2.9) on (4.1) can be rewritten as follows
Q bil [# # s , # s ] = Q bil [# # , #; #] =
# (3) K(#)
# (3) #(x)
=
n
# k=0
# k # (3)
# (3) #(x)
< # # · “
A k [#] # > . (4.5)
The index "(3)" in functional derivatives means that it is applied to functionals performed
by integration over the coordinates x [4, 9].
Let the functional R s be some real solution to the equation
K(R s ) = K[# # , #; R s ] =
n
# k=0
(R s ) k < # # · “
A k # >= 0 (4.6)
where the functional K(R s ) has the same form as (4.3); R s is called the Rayleigh func-
tional. Eq.(4.6) is an algebraic equation of the order n with real coefficients due to the
hermiticity of “
A k . Let also the condition
K # (R s ) #= 0 (4.7)
be fulfilled. Then:
4.2 # . The extremum condition of R s with respect to # # (x):
# (3) R s
# (3) # # (x)
= 0,
6

is equivalent to Eq.(4.2) in terms of R s :
n
#
k=0
(R s ) k “
A k #(x) = 0.
Indeed, the rule of differentiation of a complete function gives
# (3) R s
# (3) # # (x)
= -
# (3) K(R s )
# (3) # # (x) # K # (R s ) = - # n
# k=0
(R s ) k “
A k #(x) # # N 2 [# # , #; R s ]. (4.8)
The set of extremum values of some R s as a given solution to (4.6) must coincide with
some subset of eigenvalues, see [14, 4] for details, and the set of functions which supply
this R s with extrema must coincide with the corresponding subset of eigenfunctions. The
set of extrema of all the solutions R s covers the whole set of eigenvalues. The condition
(4.7) in terms of #: K # (#) #= 0, guarantees the absence of associated vectors in the space
of eigenfunctions [18, 11].
Define the Rayleigh #-current:
Q s = - # (3) R s
# (3) #(x)
. (4.9)
then:
4.3 # . The Rayleigh #-current (4.9) is equal to the bilinear #-current (4.5) divided by
the norm-square functional, both of them expressed in terms of R s ,
Q s = Q bil [# # , #; R s ] # N 2 [# # , #; R s ].
The proof is based on a formula similar to (4.8).
We note that, first, the extremum condition of real R s with respect to # gives a correct
equation for the function # # , and that, second, the difference between the currents (4.5)
and (4.9) is the only denominator with N 2 [# # , #; R s ]. The latter fact provides the rescaling
invariance of (4.9) under the transformations # # a# and gives a possibility to include
into a theory stationary self-interaction. For the case of gravity, it was done in [3].
5 Time-Separated States and Generalized
Rayleigh Variational Principles
Consider a wave function which possesses the specific property
## ts (t, x) = #(t) # ts (t, x), (5.1)
and which we call a time-separated (ts-)state; #, which is in general a complex function
of time, being called a basic function.
The integration of (5.1) shows that, for this function, a time variable is separated from
space variables, that justifies its name,
# ts (t, x) = exp[-i # dt #(t)] #(x)
where # is an arbitrary function of x. We assume henceforth that the operators in Eq.(2.1)
are such that they admit the separation of t and x, although there are no necessity to
know any special form of these operators.
7

In applying the time derivative of the order k to #, we can write
# k # ts (t, x) = ([#(t) + #] k 1) # ts (t, x) (5.2)
where the expression in the parentheses on the right-hand side of (5.2) is merely a multi-
plier which represents the k-fold application of the operator [# +#] k to unity. We further
denote
D k # = [# + #] k 1. (5.3)
For k = 1, 2, 3, . . . we have
D 0 # = [# + #] 0 1 = 1, D 1 # = [# + #] 1 1 = #, D 2 # = [# + #] 2 1 = # 2 + ##,
D 3 # = [# + #] 3 1 = # 3 + 3# ## + # 2 #, . . . .
The operators A k are commutative with #(t) and their time derivatives because they
involve the only differentiation with respect to x (l = 0, 1, 2, . . .):
[A k , # l #] = 0, (5.4)
The substitution of (5.3) into (2.1) and the use of (5.4) lead to the equation
n
#
k=0
D k # “
A k # ts = 0. (5.5)
The stationary function (4.1) is certainly a partial case of a ts-state. However, despite
that the problem of finding the ts-states and basic functions is some generalization of
the eigenvalue problem we can hardly expect any continuity between basic functions and
spectral parameters, because, unlike Eq.(4.2), Eq.(5.5) involve the derivatives of # of the
order n - 1. The exclusion is the degenerate case of the generalized Schrodinger theory
(n = 1) where the basic function can be treated as a time-dependent spectral parameter.
Now, we give an example of time separation in a second order theory (n = 2). Consider
the KleinGordon equation with the electromagnetic interaction
[# - e#] 2 #-
3
#
#=1
# i
#
#x #
+ eA #
# 2
#-m 2 # = 0.
In order to admit time-separation, let the scalar potential # be the only function of time,
# = #(t), and let the 3-vector potential A # (# = 1, 2, 3) be the only function of the
3-space coordinates x, A # = A # (x). Let # k (x) be an eigenfunction of the generalized
momentum square operator:
3
#
#=1
# i
#
#x #
+ eA #
# 2
# k (x) = k 2 # k (x)
with k 2 an eigenvalue (suitable boundary conditions are implied), then the quantity #,
which now acquires the index k, is a solution to the equation
#(# k - e#) + (# k - e#) 2
- (k 2 +m 2 ) = 0. (5.6)
Eq.(5.6) is easily integrable and has the two solutions with the opposite signs:
# k = e#(t) ± (k 2 +m 2 ) 1/2 1 + exp[±2(k 2 +m 2 ) 1/2 t + C]
1 - exp[±2(k 2 +m 2 ) 1/2 t + C]
(5.7)
8

where C is a constant. There also exist the isolated solutions
# k = e#(t) ± (k 2 +m 2 ) 1/2 . (5.8)
The solutions (5.8) can be obtained from (5.7) by tending C to minus infinity: C # -#.
If # is independent of time, # = Const, then # k coincides with the spectral parameter
# k , which formally have the same value as (5.8).
The Lagrangian functional (2.4) on a ts-state can be associated with Eq.(5.5):
K[#(t)] =
n
# k=0
D k # < # # ts · “
A k # ts > . (5.9)
We can calculate the following functional derivative
#
##(x)
# dt # K[#(t # )] # K # [#(t)]
=
n
# k=1
k-1
# l=0
(-1) l C l+1
k # l [D k-l-1 #(t) < # # ts · “
A k # ts >] (5.10)
which leads to the proposition analogous to 4.1 # .
5.1 # . The functional (5.10) coincides with the norm-square functional (2.6) on the
ts-state:
K # [#(t)] = N 2 [# # ts , # ts ].
The #-current (2.9) on the ts-state acquires the form
Q bil [# # ts , # ts ] =
#
##(t, x)
# dt # K[#(t # )] =
# dt #
n
# k=0
D k #(t # )
#
##(t, x)
< # # ts · “
A k [#] # ts > . (5.11)
Now, we construct the generalization of the Rayleigh functionals and principles. Let
R ts (t) be some nonzero solution to the differential equation:
K[R ts (t)] =
n
#
k=0
D k R ts < # # ts · “
A k # ts >= 0, (5.12)
and let the condition analogous to (4.7) be fulfilled:
K # [R ts (t)] #= 0.
Then:
5.2 # . The condition
# # dt # dt ## #K(t ## )
#R ts (t # ) ·
#R ts (t # )
## # ts (t, x)
# # K # [R ts (t)] = 0, (5.13)
is equivalent to Eq.(5.5) in terms of R s
n
# k=0
D k R ts
“
A k # ts = 0.
9

The proof is based of the functional identity
# dt # #K(t # )
## # ts (t, x)
+ # dt # dt ## #K(t ## )
#R ts (t # ) ·
#R ts (t # )
## # ts (t, x)
= 0 (5.14)
where the differentiation in the first term is meant as if R ts were independent of # ts .
We call the functional R ts the generalized Rayleigh functional and the condition (5.13)
the time-smoothed extremum condition of R ts with respect to # ts . It is worth noting that
the time-smoothed extrema of R ts with respect to # ts are not the set of discrete numerical
values as for the true Rayleigh functionals but some set of time-dependent functions.
The generalized Rayleigh current functionals should be defined with the use of time
smoothing
Q ts = - # # dt # dt ## #K(t ## )
#R ts (t # ) ·
#R ts (t # )
## # (t, x)
# # K # [R ts (t)]. (5.15)
5.3 # . The generalized Rayleigh #-current (5.15) is equal to the bilinear current func-
tional (2.9) divided by the norm-square functional (2.6):
Q ts = Q bil [# # ts , # ts ] # N 2 [# # ts , # ts ].
The generalized Rayleigh functional can play the role of some Lagrangian function. We
note, however, that the variational principle for obtaining an evolution equation is "one-
sided", i.e., the time-smoothed extremum of the functional R ts with respect to the function
# ts does not give a correct evolution equation for # # ts . This is because the time derivatives
of # ts replaced by factors involving R ts in the functional (5.12), cannot be converted into
the time derivatives of # # ts when integrating over t # in the first term of the identity (5.14).
The exclusion is the case of Hermitean operators “
A k (t) together with the real-valued
functional R ts (t).
Now we return to Eq.(5.12) which is the nonlinear differential equation of the order
n - 1 and of some special kind (for n = 2 it is a partial Riccati equation, see below).
The function R ts as the solution to (5.12) can hardly be found explicitly, however, the
principle of smoothed extremality and the definition of the generalized Rayleigh current
are formulated without reference to any explicit solutions of these equations. Eq.(5.12)
can be transformed into a linear equation of the order n using the transformation of
variables R ts (t) = # ln r(t), then
D k R ts (t) = [R ts (t) + #] k 1 =
# k r
r
.
After that Eq.(5.12) acquires the form
n
#
k=0
< # # ts · “
A k # ts > # k r = 0. (5.16)
The linear equation (5.16) must have n independent solutions r (i) , i = 1, 2, . . . , n, so that
its general solution is
r =
n
# i=1
c i r (i)
with n arbitrary constants c i . In terms of the variable R ts , the solution has the form
R ts = # ln
n
# i=1
c i exp # - # dt R (i)
ts # .
10

In the case n = 2, Eq.(5.12) acquires the form
K(R) =< # # ts · “
A 2 # ts > (R 2
ts + #R ts )+ < # # ts · “
A 1 # ts > R ts + < # # ts · “
A 0 # ts >= 0
which is the partial Riccati equation, and which has no, in general, explicit solutions.
However, there is a solvable example in the case “
A 0 = 0, namely, the Bernoulli equation
[19]:
a 2 (t)(R 2
ts + #R ts ) + a 1 (t)R ts = 0 (5.17)
where we have denoted
a 1 (t) =< # # ts · “
A 1 # ts >, a 2 (t) =< # # ts · “
A 2 # ts > .
Eq.(5.17) has the explicit solution
R ts = # ln # -i # dt exp{+i # dt[a 1 (t)] -1 a 2 (t)} # . (5.18)
Obviously, the integrals in (5.18) are indefinite and thus the solution contains implicitly
two integration constants. This example allows one to verify immediately the propositions
5.2 # and 5.3 # . The only point that should be stressed is that some accuracy should be
done in calculating functional derivatives, see [10].
6 Standard Variational Principles and the Inclusion of
Self-Interaction
For a classical gauge theory with self-interaction, we represent standard variational prin-
ciples of extremality in a form which will be convenient below for a comparison with our
modifications. Define the field # as a gauge field which enters extended derivatives in
order to make Eq.(2.1) invariant under a localized group of transformations suitable for
the theory which is considered. Let # # Inv(#) be a suitable invariant of the field #.
The procedure of such a kind was firstly introduced in [1]. For example, if we consider
the group of localized phase transformations, then # represents the electromagnetic four-
potential [1]. For obtaining gravitation, the localization of Poincare transformations in
the framework of the vierbein formalism requires the compensating connection field [2].
Consider the total action
J tot = -# -1 J int + J = # dt L tot (t) (6.1)
where # is an interaction constant,
J int = # d 3 x # #(t, x # )
and
L tot (t) = -# -1 # d 3 x # #(t, x # ) +K(t) (6.2)
is the total Lagrangian function. Then:
6.1 # . The extremum condition of (6.1) with respect to # # (t, x) [#(t, x)]:
#J tot
## # (t, x)
= # dt # #L tot (t # )
## # (t, x)
= 0
11

# #J tot
##(t, x)
= # dt # #L tot (t # )
##(t, x)
# = 0,
is equivalent to Eq.(2.1) [Eq.(2.2)].
6.2 # . The extremum condition of (6.1) with respect to #(t, x):
#J tot
##(t, x)
= # dt # #L tot (t # )
##(t, x)
= 0,
is equivalent to the following interaction equation
#
##(t, x)
# dt # d 3 x # #(t # , x # ) = # Q bil (t, x) (6.3)
where the Q bil (t, x) is the bilinear current (2.9).
It should be stressed that, first, although our consideration is based on the privileged
role of time, the procedure of the inclusion of interaction in this Section is indeed rela-
tivistically invariant. In second, we couple replace K by K # in (6.2), but this fact does
not affect the propositions 6.1 # and 6.2 # . In this connection we note that Q bil is a real
quantity due to the reality of #.
7 Another Version of Variational Principles with Self-
interaction
Let us construct the functional
K[t; R] =
n
#
k=0
# d 3 x # # ( “
A k D k R #) (7.1)
which resembles the functional (5.9) but differs from it by some unspecified as yet quantity
R which is a function of t and x: R = R(t, x). The multiplier D k R is defined similar to
(5.3)
D k R = [R + #] k 1.
We shall need also the time-dependent functional of the type (5.10):
# dt # d 3 x
#K(t # )
#R(t, x)
=
n
#
k=1
k-1
# l=0
(-1) l C l+1
k # l # d 3 x ( “
A T
k # # ) D k-l-1 R # = K # . (7.2)
Further, we require the condition
## = R # (7.3)
which entails the two following preliminary propositions.
7.1 # . Under the condition (7.3), a) the functional (7.1) coincides with (2.4), b) the
functional (7.2) coincides with (2.6):
K # (t)| (7.3) = N 2 (t).
The validity of 7.1 # a can be verified using the relation similar to (5.2)
# k # = D k R # (7.4)
12

which is the consequence of (7.3); 7.1 # b is evident after the comparison of (7.2) and (2.6)
with the use of (7.4).
We should distinguish now local and nonlocal functionals. We call a functional F [f ]
local if its functional derivative with respect to the function f contains no primitive
functions of the #-functions, otherwise this functional is called nonlocal. For example, K
is the local functional of R.
Henceforth, let R be some solution to the integro-differential equation
K[t; R] = 0, (7.5)
where the left-hand side of (7.5) is defined by (7.1). Then R is, in general, a nonlocal
functional of # # , # and #, hence it is some nonlocal function of t and x and may carry
explicit dependences on t and x. However, the knowledge of the explicit form of R is
unnecessary for further constructions.
7.2 # . If Eq.(7.5) is fulfilled simultaneously with the condition (7.3), then the conser-
vation law (2.7) takes place for (7.2): #K # = 0.
This is true due to 7.1 # b and the relation (2.5).
Now we start a new inclusion of self-interaction. Let us for a moment leave aside the
condition (7.3). Instead of (6.2), consider another (nonlocal) Lagrangian function with K
replaced by -•hR:
L tot (t, x) = -# -1 # d 3 x # #(t, x # ) - •
h R(t, x). (7.6)
(The Planck constant • h is added to R to get the dimensionality of energy since the
dimensionality of R is always inverse time independently of that of #. Note here that
the dimensionality of K depends on that of #; by intention, the latter is chosen by such
a way to absorb any dimensional factor by K.)
Instead of the variational principles given by 6.1 # and 6.2 # , we construct the principles
where the functional derivatives are replaced by some smoothed quantities. Consider the
condition similar to that in Sec.5: Let the functional (7.2) be nonzero for any R:
K # (t) #= 0. (7.8)
For any functional F [t, x; f ] which can carry an explicit dependence on t and x, define the
smoothed functional derivative with respect to f always keeping in mind (7.8) as follows:
# #F
#f(t, x)
# sm
= # # dt ## dt # d 3 x # #K(t ## )
#R(t # , x # ) ·
#F (t # , x # )
#f(t, x)
# # K # . (7.9)
After the integration over the argument t ## the expression in the numerator of (7.9) can
be represented in the form
# dt # d 3 x #
n-1
# l=0
# # (3)
K(t # )
# (3) # #l R(t # , x # )
# #l # #F (t # , x # )
#f(t, x)
where # # # i#/#t # and the index (3) as before indicates the "3-dimensional" functional
derivative taken for fixed t # . This form is the most general in the sense that it is valid both
for local and nonlocal functionals. The integrand of (7.9) is not generally the product of
K(t # ) and the usual functional derivative of F with respect to f ; this is the case for local
functionals only. Equating (7.9) to zero can be called the smoothed extremum condition
of F with respect to f .
13

The basic propositions are the following.
7.3 # . The smoothed extremum condition of (7.6) with respect to # # (t, x):
# #L tot
## # (t, x)
# sm
= 0, (7.10)
with the simultaneous fulfillment of the condition (7.3) is equivalent to Eq.(2.1).
To prove 7.3 # note that the first term in (7.6) is independent of # # thus the smoothed
extremum is taken from the second term. Substituting R which is a solution to (7.5)
back to (7.5) and integrating the obtained identity over time, the usual functional differ-
entiation of the result with respect to # # (t, x) gives
# dt # #K(t # )
## # (t, x)
+ # dt # dt ## d 3 x # #K(t ## )
#R(t # , x # ) · #R(t # , x # )
## # (t, x)
= 0. (7.11)
Dividing (7.11) by the nonzero functional (7.2), we can see that
# #R
## # (t, x)
# sm
= - # n
# k=0
“
A k D k R# # # K #
whence 7.3 # becomes evident after using (7.3).
7.4 # . The smoothed extremum condition of (7.6) with respect to #(t, x):
# #L tot
##(t, x)
# sm
= 0,
with the simultaneous fulfillment of the condition (7.3) is equivalent to the following in-
teraction equation
#
##(t, x)
# dt # d 3 x # #(t # , x # ) = • h#
Q bil (t, x)
N 2
. (7.12)
It is worth noting that the left-hand side of (7.12) is just that of (6.3), however, unlike
(6.3) the bilinear current (2.9) on the right-hand side of (7.12) is divided by the conserved
norm square.
Turn to the left-hand side of (7.12). When taking the smoothed functional derivative
of the first term with respect to #, due to the locality of # as the function of #, these
derivatives involve the only functions #(t-t # ), #(x-x # ) and their partial derivatives. That
is why, in an expression of the type (7.10), the functional (7.2) which is independent of
t # due to the proposition 7.2 # can be extracted as the multiplier, put out the sign of an
integral over t # and contracted with the same denominator, i.e.,
# #
##(t, x)
# dt # d 3 x # #(t # , x # ) # sm
=
#
##(t, x)
# dt # d 3 x # #(t # , x # ).
As for the right-hand side of (7.12), consider the identity similar to (7.11)
# dt # #K(t # )
##(t, x)
+ # dt # dt ## d 3 x # #K(t ## )
#R(t # , x) ·
#R(t # , x # )
##(t, x)
= 0.
The first term of (7.13), after using the relation (7.4), the proposition 7.1 # a and the fact
that the operation of taking #/## and that of substituting (7.4) commute, is the bilinear
current (11). Therefore, recalling (7.8) , 3 # b and 7.2 # ,
# #R
##(t, x)
# sm
= -
Q bil
N 2
14

and 7.4 # becomes evident.
Propositions 7.3 # and 7.4 # show that we need not the notion of action but rather that
of Lagrangian function. This is because the "smoothing functions" are disposed under
the sign of time integration which usually makes action from a Lagrangian function.
Now, we have formulated the variational principles of smoothed extremality. We have
achieved the desired result, namely, the invariance under the transformations (1.1). The
only distinction from classical field theory is the appearance of the conserved norm-square
functional which is, on one hand, a functional and, on the other hand, merely a number as
a denominator on the right-hand side of interaction equations. However, this is done at
the expense of losing the three important properties of the standard procedure, namely,
(i) the principles of extremality, (ii) the non-privileged position of the time coordinate
and (iii) the unambiguous way of constructing the Lagrangian function. The latter means
that since R is a complex functional, we have the two possibilities: to choose R or R #
as the matter Lagrangian function. In the case of R # , Eq.(2.2) would be obtained as a
smoothed extremum of R # with respect to #, whereas Eq.(7.12) is obtained as before: it is
insensitive to the above choice. However, Eq.(2.2) [Eq.(2.1)] can never be obtained as the
smoothed extremum of R [R # ] with respect to # [# # ]. Thus our variational principles are
"one-sided" by the same reason as in the Section 5, but this has no significance because
one complex evolution equation is always enough for complete description of a physical
situation.
Emphasize that the condition (7.3) is necessary to compensate introduced superflu-
ous quantity R, this condition generalizes an analogous condition for stationary states:
## s (t, x) = # # s (t, x), and the condition (5.1) for ts-states. It is sufficient to only use
(7.3) in the smoothed extrema.
8 Many-Particle Problem and Inclusion of Interaction
Consider M particles each of which is described by the wave function # I (t I , x I ) where
the index I = 1, 2, . . . , M shows that the function and its arguments belong to the I th
particle. Construct the wave function in the (3 + 1)M-space
#(t 1 , t 2 , . . . , t M , x 1 , x 2 , . . . , xM ) =
M
#
I=1
# I (t I , x I ) (8.1)
which is reduced to the usual wave function •
# of the M-particle system defined in the
3M + 1-space for coinciding time arguments
•
#(t, x 1 , x 2 , . . . , xM ) = #(t, t, . . . , t, x 1 , x 2 , . . . , xM ). (8.2)
Let each # I satisfy the equation of the type (2.1)
n I
#
k=0
“
A Ik (t I , x I ) # k
I # I (t I , x I ) = 0 (8.3)
where # I = i#/#t I , the equation's order n I and the operators “
A Ik are attached to the
I th particle, however, we suppose as before that these operators contain some field #, the
same for all the particles, but expressed as a function of the I th arguments:
“
A Ik = “
A Ik [#(t I , x I )].
15

As the consequence of (8.3), the function (8.1) satisfies the equation
# M
#
I=1
n I
#
k=0
“
A Ik (t I , x I )# k
I
# #(t 1 , t 2 , . . . , t M , x 1 , x 2 , . . . , xM ) = 0. (8.4)
Define henceforth
d 3M x = d 3 x 1 · d 3 x 2 · . . . · d 3
xM ,
d M t = dt 1 · dt 2 · . . . · dt M .
For the Lagrangian function consider the following one suggested by Eq.(8.4) which
has the dimensionality of (energy) M :
K(t 1 , t 2 , . . . , t M ) = # d 3M x # # (t 1 , t 2 , . . . , t M , x 1 , x 2 , . . . , xM )â
â # M
#
I=1
n I
#
k=0
“
A Ik (t I , x I ) # k
I
# #(t 1 , t 2 , . . . , t M , x 1 , x 2 , . . . , xM ), (8.5)
and which can certainly be decomposed to the product of M Lagrangian functions:
K(t 1 , t 2 , . . . , t M ) = K 1 (t 1 ) K 2 (t 2 ) . . . KM (t M ) (8.6)
where
K I (t I ) =
n I
#
k=0
# d 3 x # # I (t I , x) “
A Ik (t I , x) # k
I # I (t I , x) .
Define also the functional which can be interpreted as the norm square functional for
the function (8.1)
N 2
(t 1 , t 2 , . . . , t M ) = N 2
1 (t 1 ) N 2
2 (t 2 ) . . . N 2
M (t M ) (8.7)
where
N 2
I (t I ) =
n I
#
k=1
k-1
# l=0
# d 3 x # # #l
I
“
A T
Ik (t I , x) # # I (t I , x) # # k-l-1
I # I (t I , x) (8.8)
is the norm square for the I th particle [it is possible but rather cumbersome to express
(8.7) like (8.5) via the function (8.1) without the above decomposition]. If we require
K I = 0 for all I, then the conservation law for (8.8) of the type (2.7) holds. In this case,
the functional (8.7) is conserved in each time argument:
# I N 2 (t 1 , t 2 , . . . , t M ) = 0.
The conserved functional (8.7) is a real number as the product of real numbers. We can
also define the norm square functional for the function (8.2)
•
N 2 (t) = N 2 (t, t, . . . , t) (8.9)
which is certainly conserved in t:
# •
N = 0. (8.10)
Evidently, the conserved functionals (8.7) and (8.9) are equal numerically.
The definition (8.9) generalizes that in the usual M-particle Schrodinger theory. Such
the definition reflects the third postulate of the probability theory: the probability of
independent events is equal to the product of individual probabilities. This definition is
16

also relevant for the system with interaction because the conservation law (8.10) is valid
for any # as before.
Using the Lagrangian function (8.5), it is possible to construct an M-particle bilinear
action which is the product of M bilinear actions, however, any attempt to include an
interaction term into such an action meets with a failure. This is because of the difficulties
to bring in correlation the dimensionalities of matter and interaction terms in a total action
and to get a sum of M sources in the interaction equation.
Similarly to previous constructions, consider the functional
K[t 1 , t 2 , . . . , t M ; R 1 , R 2 , . . . , RM ] = K 1 [t 1 ; R 1 ]K 2 [t 2 ; R 2 ] . . . KM [t M ; RM ] (8.11)
where
K 1 [t I ; R I ] =
n I
#
k=0
# d 3 x # # I (t I , x) # “
A Ik (t I , x) # k
I # I (t I , x) # ,
and as before
D IK R I = [R I + # k
I ]1.
We also need the functional
# d M t d 3M x
# M
K(t 1 , t 2 , . . . , t M )
# M
I=1 #R I (t, x I )
=
M
#
I=1
# dt I d 3 x I
#K I (t I )
#R I (t, x I ) # K #
M . (8.12)
which depends on t. Note that all the R I in (8.12) have the same time argument t which
takes no part in the integration.
Let, as before, the following conditions be satisfied
# I # I (t I , x) = R I (t I , x) # I (t I , x) (8.13)
for all I, then the following propositions extend 7.1 # a, 7.1 # b and 7.2 # to the M-particle
case.
8.1 # . Under the conditions (8.13) a) the functional (8.11) coincides with (8.6), b) the
functional (8.12) coincides with (8.7):
K # M (t)| (8.13) = •
N 2 (t).
Let each R I be the solution to the equation
K I [t I ; R I ] = 0, (8.14)
as the consequence
K[t 1 , t 2 , . . . , t M ; R 1 , R 2 , . . . , RM ] = 0.
Similarly to 7.2 # ,
8.2 # . If Eqs.(8.14) are fulfilled simultaneously with the conditions (8.13), then the
conservation law for (8.12) of the type (8.10) takes place: #K # M = 0.
It worth noting that the conditions (8.13) yield the following condition for the function
(8.2):
# •
#(t, x 1 , x 2 , . . . , xM ) = R(t, x 1 , x 2 , . . . , xM ) •
#(t, x 1 , x 2 , . . . , xM ) (8.15)
where
R(t, x 1 , x 2 , . . . , xM ) = R(t, x 1 ) +R(t, x 2 ) + . . . +R(t, xM ). (8.16)
17

Now, independently of the conditions (8.13) but taking (8.16) as the definition of R,
consider the nonlocal Lagrangian function
L tot (t, x 1 , x 2 , . . . , xM ) = -# -1 # d 3 x # #(t, x # ) - •
h R(t, x 1 , x 2 , . . . , xM ). (8.17)
Let for all the R I satisfying (8.14) the functional (8.12) be nonzero
K # M [t; R 1 , R 2 , . . . , RM ] #= 0. (8.18)
For any functional F [t, x 1 , x 2 , . . . , xM ; f ] which can carry an explicit dependences on t and
x I , we can define the M-smoothed (i.e., smoothed over M particles) functional derivative
with respect to the function f , keeping in mind (8.18):
# #F
#f(t, x)
# M-sm
= # # dt # dt M d 3M x
# M
K I (..., t I , ...)
# M
I=1 #R I (t # , x I ) ·
#F (t # ..., x I , ...)
#f(t, x)
# # K # M . (8.19)
Equating (8.19) to zero can be called the M-smoothed extremum condition of F with
respect to f . Then, we have the following generalizations of the propositions 7.3 # and
7.4 # .
8.3 # . The M-smoothed extremum condition of (8.17) with respect to any # # K (t, x)[(K =
1, 2, . . . , M )]:
# #L tot
## # K (t, x)
# M-sm
= 0,
with the simultaneous fulfillment of the conditions (8.13) for all I = 1, 2, . . . , M is equiv-
alent to the evolution equation:
nK
#
k=0
“
AKk (t, x) # k
#K (t, x) = 0.
Note that here we take the arguments t and x instead of t I and x I , cf. (8.3). Since
# # K (t, x) enters only RK , we can rewrite the definition of the required M-smoothed deriva-
tive in the form:
# # dt # dt 1 d 3 x 1
#K 1 (t 1 )
#R 1 (t # , x 1 )
. . . dt K d 3 xK
#KK (t K )
#RK (t # , xK ) · #RK (t # , xK )
## # K (t, x)
. . . â
â dt M d 3
xM #KM (t M )
#RM (t # , xM )
# # M
#
I=1
# dt I d 3 x I
#K I (t I )
#R I (t, x I )
.
That all the factors #K I /#R I integrated over t I and x I for I #= K are independent of
t # due to 8.2 # leads to the fact that they can be put out the sign of the integral over t #
contracting with the same factors in the denominator. As the result, the M-smoothed
extremum here is reduced to the "one-particle" smoothed extremum in 7.3 # :
# #L tot
## # K (t, x)
# M-sm
= # #RK
## # K (t, x)
# sm
.
Thus, 8.3 # is valid as the consequence of 7.3 # .
8.4 # . The M-smoothed extremum condition of (8.17) with respect to #(t, x):
# #L tot
##(t, x)
# M-sm
= 0,
18

with the simultaneous fulfillment of the conditions (8.13) for all I = 1, 2, . . . , M is equiv-
alent to the interaction equation:
#
##(t, x)
# dt # d 3 x # #(t # , x # ) = •
h#
M
#
I=1
Q bil
I (t, x)
N 2
I
. (8.20)
The proof of the validity of the left-hand side of (8.20) repeats mutatis mutandis that
in 7.4 # , in particular, one should refer to 8.2 # instead of 7.2 # . On the right-hand side of
(8.20), by the same reason as in 8.3 # , we have the sum of one-particle extrema,
# #R
##(t, x)
# M-sm
=
M
# I=1
# #R I
##(t, x)
# sm
,
leading to the validity of 8.4 # .
Thus, we have formulated the variational principles of M-smoothed extremality which
reduce to those formulated for a single particle when M = 1. We stress that, just for the
M-particle system, our way seems more natural and advantageous than any generalization
of a "bilinear" way and leads to equations invariant under the rescaling transformation
(1.2) applied to any wave function # I . Eq.(8.20) is very similar to a classical interaction
equation for M matter fields, however the source contribution of each field on the right-
hand side is divided by its own conserved norm square.
Concerning the mentioned relation between different spaces, we can see that the in-
teraction indeed occurs in the 3 + 1-space: this is guaranteed by the same arguments t
and x for all the #K and #. Nevertheless, the consistent formulation of our principles
requires to deal with the 3M+1-space at the level of the Lagrangian function and with the
(3 + 1)M-space at the level of smoothing. We can imagine this by such a way that each
I th particle moves in its own I th (3+1)-space, but these motions are not now independent
and are determined by the interactions with other particles and the self-interaction in the
common (3 + 1)-space. We have already shown how to come from our approach to the
standard formulation of a many-particle problem with gravitational coupling by means
of some inconsistent assumptions [6].
In conclusion, it is worth noting the succession between the conditions (7.3) and (8.15)
both involving a usual wave function suitable for a given system, they always mean that
(wave function) # exp[-i # dt R(t)],
exiting associations with a quasiclassical wave function where, however, the exact La-
grangian function -•hR is taken instead of a classical one.
Acknowledgment
The author is very grateful to Dr. B.G. Sidharth for the hospitality in the B.M. Birla
Science Centre, Hyderabad, during FebruaryMarch 1995. The author also appreciates
the financial travel support of ISF, Washington, grant no. 4692/2, for visiting the Con-
ference "Differential Equations  Theory, Methods and Applications", February 2023,
1995, Hyderabad.
19

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