## 1. Introduction

Now it is obvious that quantum mechanics enters in the 21st century into a principally new and important phase of its development which will cardinally change the currently used technical facilities in the areas of information and telecommunication technologies, exact measurements, medicine etc. Indisputably, all this on the whole will change the production potential of human civilization and influence its morality. Despite unquestionable success of quantum physics in the 20th century, including creation of lasers, nuclear energy use, etc. it seems that possibilities of the quantum nature are not yet studied and understood deeply, a fortiori, are used.

The central question which arises on the way of gaining a deeper insight into the quantum nature of various phenomena is the establishment of well-known accepted criteria of applicability of quantum mechanics. In particular, the major of them is the de-Broglie criterion, which characterizes any body-system by a wave the length of which is defined as

In order to answer this essentially important question for development of quantum physics, it is necessary to expand substantially the concepts upon which quantum mechanics is based. The necessity for generalization of quantum mechanics is also dictated by our aspiration to consider such hard-to-explain phenomena as spontaneous transitions between the quantum levels of a system, the Lamb Shift of energy levels, EPR paradox, etc. within the limits of a united scheme. In this connection it seems important to realize finally the concept according to which any quantum system is basically an open system, especially when we take into account the vacuum's quantum fluctuations [1- 3]. Specifically for a quantum noise coming from vacuum fluctuations we understand a stationary Wiener-type source with noise intensity proportional to the *vacuum power*

Thus, first of all we need such a generalization of quantum mechanics which includes nonperturbative vacuum as *fundamental environment* (FE) of a quantum system (QS). As our recent theoretical works have shown [4-9], this can be achieved by naturally including the traditional scheme of nonrelativistic quantum mechanics if we define quantum mechanics in the limits of a nonstationary complex stochastic differential equation for a wave function (conditionally named a stochastic Schrödinger equation). Indeed, within the limits of the developed approach it is possible to solve the above-mentioned traditional difficulties of nonrelativistic quantum mechanics and obtain a new complementary criterion which differs from de-Broglie's criterion. But the main achievement of the developed approach is that in the case when the de-Broglie wavelength vanishes and the system, accordingly, becomes classical within the old conception, nevertheless, it can have quantum properties by a new criterion.

Finally, these quantum properties or, more exactly, quantum-field properties can be strong enough and, correspondingly, important for their studying from the point of view of quantum foundations and also for practical applications.

The chapter is composed of two parts. The first part includes a general scheme of constructing the nonrelativistic quantum mechanics of a bound system with FE. In the second part of the chapter we consider the problem of a quantum harmonic oscillator with fundamental environment. Since this model is being solved exactly, its investigation gives us a lot of new and extremely important information on the properties of real quantum systems, which in turn gives a deeper insight into the nature of quantum foundations.

## 2. Formulation of the problem

We will consider the nonrelativistic quantum system with random environment as a closed united system QS *and* FE within the limits of a *stochastic differential equation* (SDE) of Langevin-Schrödinger (L-Sch) type:

In equation (2.1) the stochastic operator

where

We will suppose that when

We also assume that in the limit

where in the *(in)* asymptotic state
*.* In the *(out)* asymptotic state when the interaction potential tends to the limit:

Further we assume that the solution of problem (2.4) leads to the discrete spectrum of energy and wave functions which change adiabatically during the evolution (problem (2.3)). The latter implies that the wave functions form a full orthogonal basis:

where the symbol

Finally, it is important to note that an orthogonality condition similar to (2.5) can be written also for a stochastic wave function:

### 2.1. The equation of environment evolution

The solution of (2.1) can be represented,

Now substituting (2.6) into (2.1) with taking into account (2.3) and (2.5), we can find the following system of complex SDEs:

where the following designations are made:

Recall that in (2.7) dummy indices denote summations; in addition, it is obvious that the coefficients

For further investigations it is useful to represent the function

Now, substituting expression (2.8) into (2.7), we can find the following system of SDEs:

(10) |

where the following designations are made:

(11) |

Ordering a set of random processes

In the system of equations (2.10) the symbol

Assuming that random forces satisfy the conditions of white noise:

where

Now, using the system of equations (2.10) and correlation properties (2.11), it is easy to obtain the Fokker-Planck equation for the joint probability distribution of fields

where the operator

The joint probability in (2.12) is defined by the expression:

From this definition, in particular, it follows that equation (2.12) must satisfies to the initial condition:

where

Finally, since the function

where the function

### 2.2. Stochastic density matrix method

We consider the following bilinear form (see representation (2.6)):

where the symbol

After integrating (2.15) by the coordinates

where

.Now, using (2.16) we can construct an expression for a usual nonstationary density matrix [12]:

where

where

It is obvious that equation (2.18) is a nonlocal equation. Taking into account (2.12), one can bring equation (2.18) to the form:

where following designations are made;

Thus, equation (2.19) differs from the usual von Neumann equation for the density matrix. The new equation (2.19), unlike the von Neumann equation, considers also the exchange between the quantum system and fundamental environment, which in this case plays the role of a thermostat.

### 2.3. Entropy of the quantum subsystem

For a quantum ensemble, entropy was defined for the first time by von Neumann [11]. In the considered case where instead of a quantum ensemble one united system QS + FE, the entropy of the quantum subsystem is defined in a similar way:

In connection with this, there arises an important question about the behavior of the entropy of a multilevel quantum subsystem on a large scale of times. It is obvious that the relaxation process can be nontrivial (for example, absence of the stationary regime in the limit

A very interesting case is when the QS breaks up into several subsystems. In particular, when the QS breaks up into two fragments and when these fragments are spaced far from each other, we can write for a reduced density matrix of the subsystem the following expression:

Recall that the vectors

Now, substituting the reduced density matrix

where the following designations are made in expression (2.22):

Since at the beginning of evolution the two subsystems interact with each other, it is easy to show that

### 2.4. Conclusion

The developed approach allows one to construct a more realistic nonrelativistic quantum theory which includes fundamental environment as an integral part of the quantum system. As a result, the problems of spontaneous transitions (including decay of the ground state) between the energy levels of the QS, the Lamb shift of the energy levels, ERP paradox and many other difficulties of the standard quantum theory are solved naturally. Equation (2.12) - (2.13’) describes quantum peculiarities of FE which arises under the influence of the quantum system. Unlike the de-Broglie wavelength, they do not disappear with an increase in mass of the quantum subsystem. In other words, the macroscopic system is obviously described by the classical laws of motion; however, space-times structures can be formed in FE under its influence. Also, it is obvious that these quantum-field structures ought to be interpreted as a natural continuation and addition to the considered quantum (classical) subsystem. These quantum-field structures under definite conditions can be quite observable and measurable. Moreover, it is proved that after disintegration of the macrosystem into parts its fragments are found in the entangled state, which is specified by nonpotential interaction (2.22), and all this takes place due to fundamental environment. Especially, it concerns nonstationary systems, for example, biological systems in which elementary atom-molecular processes proceed continuously [13]. Note that such a conclusion becomes even more obvious if one takes into account the well-known work [14] where the idea of universal description for unified dynamics of micro- and macroscopic systems in the form of the Fokker-Planck equation was for the first time suggested.

Finally, it is important to add that in the limits of the developed approach the closed system QS + FE in equilibrium is described in the extended space

## 3. The quantum one-dimensional harmonic oscillator (QHO) with FE as a problem of evolution of an autonomous system on the stochastic space-time continuum

As has been pointed out in the first part of the chapter, there are many problems of great importance in the field of non-relativistic quantum mechanics, such as the description of the Lamb shift, spontaneous transitions in atoms, quantum Zeno effect [15] etc., which remain unsolved due to the fact that the concept of physical vacuum has not been considered within the framework of standard quantum mechanics. There are various approaches for investigation of the above-mentioned problems: the quantum state diffusion method [16], Lindblad density matrix method [17, 18], quantum Langevin equation [19], stochastic Schrödinger equation method (see [12]), etc. Recall that representation [17, 18] describes a priori the most general situation which may appear in a non-relativistic system. One of these approaches is based on the consideration of the wave function as a random process, for which a stochastic differential equation (SDE) is derived. However, the consideration of a reduced density matrix on a semi-group [20] is quite an ambiguous procedure and, moreover, its technical realization is possible, as a rule, only by using the perturbation method. For investigation of the inseparably linked closed system QSE, a new mathematical scheme has been proposed [5-8] which allows one to construct all important parameters of the quantum system and environment in a closed form. The main idea of the developed approach is the following. We suppose that the evolution time of the combined system consists of an infinite set of time intervals with different duration, where at the end of each interval a random force generated by the environment influences the quantum subsystem. At the same time the motion of the quantum subsystem within each time interval can be described by the Schrödinger equation. Correspondingly, the equation which describes the combined closed system QSE on a large scale of time can be represented by the *stochastic differential equation* of Langevin–Schrödinger (L–Sch) type.

In this section, within the framework of the 1D L–Sch equation an exact approach for the quantum harmonic oscillator (QHO) model with fundamental environment is constructed. In particular, the method of stochastic density matrix (SDM) is developed, which permits to construct all thermodynamic potentials of the quantum subsystem analytically, in the form of multiple integrals from the solution of a 2D second-order partial differential equation.

### 3.1. Description of the problem

We will consider that the 1D QHO+FE closed system is described within the framework of the L-Sch type SDE (see equation (2.1)), where the evolution operator has the following form:

In expression (3.1) the frequency

where

The constant of

where

where the function

for a harmonic oscillator on the stochastic space-time

The random solution

Taking into account (3.5) and the well-known solution of autonomous quantum harmonic oscillator (3.6) (see [28]) for stochastic complex processes which describe the 1D QHO+FE closed system, we can write the following expression:

The solution of (3.8) is defined in the extended space

Taking into account the orthogonal properties of (3.8), we can write the following normalization condition:

where the symbol

So, the initial L-Sch equation (2.1) - (3.1) which satisfies the asymptotic condition (3.4) is reduced to autonomous Schrödinger equation (3.6) in the stochastic space-time using the *etalon differential equation* (3.7). Note that equation (3.7) with taking into account conditions (3.2) and (3.3) describes the motion of FE.

### 3.2. The mean values of measurable parameters of 1D QHO

For investigation of irreversible processes in quantum systems the non-stationary density matrix representation based on the quantum Liouville equation is often used. However, the application of this representation has restrictions [11]. It is used for the cases when the system before switching on the interaction was in the state of thermodynamic equilibrium and after switching on its evolution is adiabatic. Below, in the frames of the considered model the new approach is used for the investigation of the statistical properties of an irreversible quantum system without any restriction on the quantities and rate of interaction change. Taking into account definition (2.15), we can develop SDM method in the framework of which it is possible to calculate various measurable physical parameters of a quantum subsystem.

Definition 1. The expression for a stochastic function:

will be referred to as stochastic density matrix. Recall that the *partial* SDM

Below we define the mean values of various operators. Note that at averaging over the extended space

Definition 2. The expected value of the operator

The mean value of the operator

Note that the operation

where

If we wish to derive an expression describing the irreversible behavior of the system, it is necessary to change the definition of entropy. Let us remind that the von Neumann non-stationary entropy (the measure of randomness of a statistical ensemble) is defined by the following form:

where

Let us note that the definition of the von Neumann entropy (3.15) is correct for the quantum information theory and agrees well with the Shannon entropy in the classical limit.

Definition 3. For the considered system of 1D QHO with FE the entropy is naturally defined by the form:

where the following designation

Finally, it is important to note that the sequence of integrations first in the functional space,

### 3.3. Derivation of an equation for conditional probability of fields. Measure of functional space
R { ξ }

Let us consider the stochastic equation (3.7). We will present the solution of the equation in the following form:

After substitution of (3.17) into (3.7) we can define the following nonlinear SDE:

The second equation in (3.18) expresses the condition of continuity of the function

the following system of SDEs can be finally obtained for the fields

The pair of fields

which is a non-factorable function. After differentiation of functional (3.21) with respect to time and using SDEs (3.18) and correlation properties of the random force (3.3), as well as making standard calculations and reasonings (see [29,30]), we obtain for a distribution of fields the following Fokker-Planck equation:

with the initial condition:

Thus, equation (3.22)-(3.23) describes the free evolution of FE.

Now, our purpose consists in constructing the measure of functional space, which is a necessary condition for further theoretical constructions. The solution of equation (3.22)-(3.23) for small time intervals can be presented in the form:

So, we can state that the evolution of fields

As follows from expression (3.26), the trajectory is continuous everywhere, and, correspondingly, the condition

(57) |

where

### 3.4. Entropy of the ground state of 1D QHO with fundamental environment

For simplicity we will suppose that

where the following designation

Now, we can calculate the reduced density matrix:

In expression (3.29) the function

which satisfies the following initial and boundary conditions:

Let us consider the expression for the entropy (3.17). Substituting (3.29) into (3.17) we can find:

After conducting integration in the space

where the following designations are made:

Similarly, as in the case with (3.29), using expressions (3.34) it is possible to calculate the functional trace in the expression

where the function

Recall that border conditions for (3.36) are similar to (3.31). Besides, if we assume that

which is solved by initial and border conditions of type (3.31).

Introducing the designation

Using (3.38) we can write the final form of the entropy of «ground state» in the limit of thermodynamics equilibrium:

It is simple to show that in the limit

Thus, at the reduction

### 3.5. Energy spectrum of a quantum subsystem

The energy spectrum is an important characteristic of a quantum system. In the considered case we will calculate the first two levels of the energy spectrum in the limit of thermodynamic equilibrium. Taking into account expressions (3.12) and (3.28) for the energy of the «ground state», the following expression can be written:

where the operator:

describes the Hamiltonian of 1D QHO without an environment.

Substituting (3.41) in (3.40) and after conducting simple calculations, we can find:

where the following designations are made:

In expression (3.43) the stationary solution

where

in addition:

In expression (3.45) the stationary solution

As obviously follows from expressions (3.42)-(3.46), the relaxation effects lead to infringement of the principle of equidistance between the energy levels of a quantum harmonic oscillator Fig.1. In other words, relaxation of the quantum subsystem in fundamental environment leads to a shift of energy levels like the well-known Lamb shift.

### 3.6. Spontaneous transitions between the energy levels of a quantum subsystem

The question of stability of the energy levels of a quantum subsystem is very important. It is obvious that the answer to this question may be received after investigation of the problem of spontaneous transitions between the energy levels. Taking into account (3.4) and (3.8), we can write an expression for the probability of spontaneous transition between two different quantum states:

where the wave function

It is obvious that in the considered formulation of the problem there might occur transitions between any energy levels, including transitions from the «ground state» to any excited states. Using expression (3.47), we can calculate the spontaneous decay of every quantum state. In particular, if

In (3.48)

(79) |

where

Let us note that in expressions (3.48) and (3.49) the functions

Comparing expressions (3.48) and (3.49) with taking into account the fact that equation (3.50) for a different number

### 3.7. Uncertainty relations, Weyl transformation and Wigner function for the ground state

According to the Heisenberg uncertainty relations, the product of the coordinate and corresponding momentum of the quantum system cannot have arbitrarily small dispersions. This principle has been verified experimentally many times. However, at the present time for development of quantum technologies it is very important to find possibilities for overcoming this fundamental restriction.

As is well-known, the dispersion of the operator

In the considered case the dispersion of the operator at the arbitrary time

Using expression (3.52), we can calculate the dispersions of operators, the coordinate,

The dispersions of operators at the moment of time

where average values of operators

It is obvious that expressions for operator dispersions (3.53)-(3.54) are different from Heisenberg uncertainty relations and this difference can become essential at certain values of the interaction parameter

Definition 4. We will refer to the expression:

as stochastic Wigner function and, correspondingly, to

Using the stochastic Wigner function, it is possible to calculate the mean values of the physical quantity, which corresponds to the operator

where the stochastic function

Now we can construct a Wigner function for the «ground state»:

(91) |

As one can see, function (3.61) describes distribution of the coordinate

Recall that for the Wigner function (3.61) in the general case the normalization condition of type (6.12) is not carried out.

### 3.8. Conclusion

Any quantum system resulting from the fact that all beings are immersed into a physical vacuum is an open system [1-3]. A crucially new approach to constructing the quantum mechanics of a closed non-relativistic system QS+FE has been developed recently by the authors of [5-8], based on the principle of *local equivalence of Schrodinger representation*. More precisely, it has been assumed that the evolution of a quantum system is such that it may be described by the Schrödinger equation on any small time interval, while the motion as a whole is described by a SDE for the wave function. However, in this case there arises a non-simple problem of how to find a measure of the functional space, which is necessary for calculating the average values of various parameters of the physical system.

We have explored the possibility of building the non-relativistic quantum mechanics of a closed system QS+FE within the framework of one-dimensional QHO which has a random frequency. Mathematically, the problem is formulated in terms of SDE for a complex-valued probability process (3.1) defined in the extended space

Finally, it is important to note that the developed approach is more realistic because it takes into account the shifts of energy levels, spontaneous transitions between the energy levels and many other things which are inherent to real quantum systems. The further development of the considered formalism in application to exactly solvable many-dimensional models can essentially extend our understanding of the quantum world and lead us to new nontrivial discoveries.

## 4. Appendix

### 4.1. Appendix 1

Theorem. *Let us consider a set of random processes*
*satisfying the set of SDE:*

*where*

*so that the Fokker-Planck equation for the conditional transition probability density:*

*is given by the equation:*

*are assumed to be Markovian processes and satisfy the condition*
* At the same time function* (4.1.2) *gives their exhaustive description*:

*where
*

*Under these assumptions we can obtain the following representation for an averaging procedure:*

*where
*

*which satisfies the following initial and boundary conditions:*

*where
*

Proof. The proof is performed formally under the assumption that all the manipulations are legal. We will expand into the Taylor series the quantity under the averaging in the left-hand side of (4.1.5):

where

(102) |

The designations

(103) |

Changing, where it is necessary, the order of integration, we can obtain the following representation for the

where the countable set of functions

where

i.e. the function

where

The representation (4.1.5) is thus obtained.

It remains to prove that the function

Taking into account the fact that

### 4.2. Appendix 2

Let us consider the bilinear form:

which can be represented,taking into account expressions (3.4) and (3.8), by the following form:

(111) |

After conducting functional integration of the expression

(112) |

where

The solution of equation (4.4) is useful to represent in the following form:

By substituting (4.2.5) into equation (4.2.4), it is possible to find the following two real-value equations for the real and complex parts of solution:

(115) |

The system of equations is symmetric in regard to the replacements:

Accordingly, for a complex solution

Now it is possible to pass to the calculation of the amplitude of transition between different quantum states. For simplicity we will compute the first two probabilities of transitions:

where

In a similar way it is possible to calculate the transition matrix element

As follows from expressions (4.2.9), (4.2.10) and (4.2.11), in the general case