Quantum implications of non-extensive statistics

TL;DR

This paper develops a unified framework linking quantum mechanics and non-extensive statistical mechanics, deriving propagators for various entropy types, and exploring their implications for quantum wave functions and non-linear quantum phenomena.

Contribution

It introduces an integrated Quantropy functional approach to derive quantum propagators for non-additive entropies, extending the q-Schrödinger equation to interacting systems.

Findings

Derived propagators for Tsallis and $S_{\pm}$ entropies.

Obtained analytical solutions for probability-energy relations.

Explicitly calculated corrections for free particles and harmonic oscillators.

Abstract

Exploring the analogy between quantum mechanics and statistical mechanics we formulate an integrated version of the Quantropy functional [1]. With this prescription we compute the propagator associated to Boltzmann-Gibbs statistics in the semiclassical approximation as $K=F(T) \exp\left(i S_{cl}/\hbar\right)$. We determine also propagators associated to different non-additive statistics; those are the entropies depending only on the probability $S_{\pm}$ [2] and Tsallis entropy $S_q$ [3]. For $S_{\pm}$ we obtain a power series solution for the probability vs. the energy, which can be analytically continued to the complex plane, and employed to obtain the propagators. Our work is motivated by [4] where a modified q-Schr\"odinger equation is obtained; that provides the wave function for the free particle as a q-exponential. The modified q-propagator obtained with our method, leads to the same q-wave function for that case. The procedure presented in this work allows to calculate q-wave functions in problems with interactions; determining non-linear quantum implications of non-additive statistics. In a similar manner the corresponding generalized wave functions associated to $S_{\pm}$ can also be constructed. The corrections to the original propagator are explicitly determined in the case of a free particle and the harmonic oscillator for which the semi-classical approximation is exact.

Equations

$$S_{+}=\sum _l(1-p_1^{p_l}),S_{-}=\sum _l(p_l^{-p_l}-1),$$

$${\Phi }_{BP}$$

$$-\lambda {\int }_{X}a(x)S(x)dx.$$

$${\Phi }_{+}=\sum _l\left(1-p_l^{p_l}\right)-\gamma \sum _l{p}_l-\beta \sum _l{E}_l{p}_l^{p_l+1},$$

$$\beta E=\frac{(-\gamma p^{-p}-1-\ln p)}{(1+p+p\ln p)}.$$

$$e^{-\beta E}$$

$$1/6p^4(-3\ln p^2-8\ln p^3-9\ln p^4-6\ln p^5-\ln p^6)$$

$$1/24p^5(12\ln p^2+44\ln p^3+70\ln p^4+68\ln p^5$$

$$+42\ln p^6+12\ln p^7+\ln p^8)+....$$

$$p=e^{-\beta E}(1+\sum _{n=1}{c}_n{e}^{-n\beta E}).$$

$$c_1$$

$$c_2$$

$$c_3$$

$$c_4$$

$$+30x^4-300x^5+125x^6).$$

$$p_{+}$$

$$+\frac{1}{6}{e}^{-4x}{x}^2(3+4x-6x^2-24x^3+16x^4)$$

$$+\frac{1}{24}{e}^{-5x}{x}^2(-12-16x+60x^2+116x^3$$

$$\left(+30x^4-300x^5+125x^6\right))+....$$

$${\Phi }_{-}=\sum _l\left(p_l^{-p_l}-1\right)-\gamma \sum _l{p}_l-\beta \sum _l{E}_l{p}_l^{1-p_l}.$$

$$p_{-}$$

$$\frac{1}{6}{e}^{-3x}(-3x^2-4x^3+6x^4+24x^5-16x^6))+....$$

$${\Phi }_{BP,+}$$

$$-\lambda \int A(x)a(x{)}^{a(x)+1}dx.$$

$$\frac{A}{i\hslash }=\frac{(-\gamma a^{-a}-1-\ln a)}{(1+a+a\ln a)}=F\left(a\left(\frac{A}{i\hslash }\right)\right).$$

$$a_{+}\left(\frac{A}{i\hslash }\right)$$

$$-\frac{1}{2}{A}^2/{\hslash }^2(-1+2iA/\hslash -3A^2/{\hslash }^2)e^{i2A/\hslash }$$

$$-\frac{1}{6}{A}^2/{\hslash }^2(3-4iA/\hslash +6A^2/{\hslash }^2$$

$$-24i{A}^3/{\hslash }^3+16A^4/{\hslash }^4)e^{3iA/\hslash }+...).$$

$$a_{+}(A)$$

$$+\frac{1}{2}(\lambda A{)}^2(-1-2\lambda A+3(\lambda A{)}^2)e^{-2\lambda A}$$

$$+\frac{1}{6}(\lambda A{)}^2(3+4\lambda A-6(\lambda A{)}^2$$

$$-24(\lambda A{)}^3+16(\lambda A{)}^4)e^{-3\lambda A}+...).$$

$${\left(\frac{\lambda }{n}\right)}^m\frac{{\partial }^m}{\partial {\lambda }^m}{e}^{-n\lambda A}=(-1{)}^m(\lambda A{)}^m{e}^{-n\lambda A},$$

$$a_{+}$$

$$+\frac{1}{2}\left(-\frac{{\lambda }^2}{3^2}\frac{{\partial }^2}{\partial {\lambda }^2}+2\frac{{\lambda }^3}{3^3}\frac{{\partial }^3}{\partial {\lambda }^3}+3\frac{{\lambda }^4}{3^4}\frac{{\partial }^4}{\partial {\lambda }^4}\right)e^{-3\lambda A}$$

$$+\frac{1}{6}(3\frac{{\lambda }^2}{4^2}\frac{{\partial }^2}{\partial {\lambda }^2}-4\frac{{\lambda }^3}{4^3}\frac{{\partial }^3}{\partial {\lambda }^3}-6\frac{{\lambda }^4}{4^4}\frac{{\partial }^4}{\partial {\lambda }^4}+24\frac{{\lambda }^5}{4^5}\frac{{\partial }^5}{\partial {\lambda }^5}+$$

$$16\frac{{\lambda }^6}{4^6}\frac{{\partial }^6}{\partial {\lambda }^6})e^{-4\lambda A}+....$$

$$a_{-}$$

$$-\frac{1}{2}(A/\hslash {)}^2(-1+2iA/\hslash -3(A/\hslash {)}^2)e^{2iA/\hslash }$$

$$+\frac{1}{6}(3(A/\hslash {)}^2-4i(A/\hslash {)}^3+6(A/\hslash {)}^4-24i(A/\hslash {)}^5$$

$$+16(A/\hslash {)}^6)e^{3i(A/\hslash )}+...).$$

$${\Phi }_0$$

Full text

PACS numbers

03.65.-w; 05.70.Ln; 05.90.+m.

keywords

quantropy, non linear quantum systems, propagator, non-extensive entropies.

I Introduction

Non-extensive entropies depending only on the probabilities have been obtained in Obregon:2010zz . They belong to a family of non-extensive statistical mechanics, relevant for non-equilibrium systems. Renowned examples are: Rényi renyi , Tsallis ($S_{q}$) tsallis ; TsallisQM1 and Sharma-Mital sharma , all of them can be obtained in the frame of Superstatistics super .

For the entropies depending only on the probability there are two entropy functionals :

[TABLE]

which can be considered as building blocks for non-extensive entropies without parameters. For example one can consider $S_{1}=(S_{+}+S_{-})/2$. These entropies are noticeably distinct to Boltzmann-Gibbs(BG) entropy for systems with few degrees of freedom; however when the number of degrees of freedom goes to the thermodynamical limit, they match perfectly BG statistics jesus19 . They belong to the class of Superstatistics super with an intensive parameter $\chi^{2}$ distribution Obregon:2010zz .

There is a universality of the Superstatistics family super . As it has been shown: for several distributions of the temperature the Boltzmann factor essentially coincides up to the first expansion terms. This has as a consequence that also the entropies associated to these Boltzmann factors have all of them basically the same first corrections to the usual entropy. Furthermore, the three entropies listed here that depend only on the probability are expanded only on the parameter $y=p\ln p$, this is always smaller than 1 giving correction terms to the entropy which at any order are smaller than the previous ones. So, that any function of $y$ proposed as another generalized entropy, depending only on this parameter, when expanded in $y$ will basically coincide with one of the three ones studied here. Clearly demanding that the first term in the expansion is $-y$ giving BG entropy. Thus the entropies $S_{+}$ , $S_{-}$, and their linear combination can be considered as building blocks to compute any possible modified entropies depending only on the probability.

Motivated by the concept of Quantropy developed by Baez and Pollard baez and by non linear quantum systems with modified wave functions based on Tsallis statistics TsallisQM1 ; TsallisQM2 ; QMT , we develop a version of Quantropy in terms of the propagator of a quantum mechanical theory. The work Chavanis19 , appeared recently, it studies non linear quantum equations with the harmonic oscillator potential. Our generalized propagators could possibly be connected to the appropriate quantum equations. Baez and Pollard’s Quantropy is a functional of the amplitude on the path integral $a$, with the same functional form as the entropy in terms of the probability $Q=-\int_{X}a(x)\ln a(x)dx$. Giving the functional:

[TABLE]

From the search of extrema of this functional, restricted to values of $a$ normalized and an average of the action, Baez and Pollard obtain the relation $a=\exp(iS/\hbar-1-\alpha)$ with $\lambda=\frac{1}{i\hbar}$. Then $a\sim\exp(iS/\hbar)$ with the normalization fixed by the Lagrange multiplier $\alpha$.

In their approach the energy is mapped to the action $S$ and the temperature to $i\hbar$. We consider the same identification, but instead we identify $E$ with the classical action $S_{cl}$. Thus we consider as the analogue of the entropy a functional in terms of the propagator, instead of the amplitude $a$. This is an extrapolation of the Quantropy baez to an integration over all classical paths. The standard expression for the propagator is given semi-classically by $K\sim\exp(iS_{cl}/\hbar)$. We use this fact to define a kind of integrated Quantropy functional now in terms of the propagator for BG statistics, which we extend to the modified statistics $S_{+}$, $S_{-}$ and $S_{q}$.

This paper is organized as follows: In Section II we obtain a series expansion for the probabilities versus $\beta E$ for the generalized entropies depending only on the probabilities $S_{+}$ and $S_{-}$. In Section III we extend this expansions to the complex plane. In Section IV we present a version of Quantropy for BG statistics, $S_{+}$ and $S_{-}$ and $S_{q}$. In Section V we study in particular the case of the free particle propagator, obtained from the extrema of the Quantropy in the cases of $S_{+}$ and $S_{-}$ and $S_{q}$ for $q<1$ and $q>1$. We show that the $K_{q}$ propagator results exactly in the q-exponential that defines the q-wave function for the free particle TsallisQM1 . In a similar manner we argue that the corresponding generalized propagators $K_{+}$ and $K_{-}$ provide us with a procedure to construct $\Psi_{+}$ and $\Psi_{-}$ for the free particle. Our method however gives the possibility to construct $K_{q}$, $K_{+}$ and $K_{-}$ also for problems with interactions and by this mean to identify the corresponding wave functions. We also provide a way to perform the normalization inspired in Feynman and Hibbs work FeynmanHibbs . Section VI is devoted to the analysis of the harmonic oscillator, we exemplify with the case corresponding to $S_{+}$. In Section VII we summarize our results and present the conclusions. An Appendix A describes a numerical study of the propagators.

II Probability distributions for systems with maximal $S_{+}$ and $S_{-}$

We start by developing a recurrent solution for the probability distribution of the generalized entropy $S_{+}$ Obregon:2010zz . On the contrary to BG statistics, for a system subject to $S_{+}$ extremization, probability normalization and energy conservation there is not a simple inverse function of the probabilities $p_{l}$ v.s. the energy of the state $\beta E_{l}$. We overcome this difficulty by finding a series solution to the extremum equation. There are other possible series solutions, but we discuss here one that has a good convergence. At the end of the section we give also the probability expansion for the entropy $S_{-}$, which is obtained by an equivalent Ansatz.

The functional to maximize in terms of $S_{+}$ entropy and for a certain distribution of probabilities $p_{l}$ is given by PhysRevE.88.062146 ; obregon15

[TABLE]

$\beta$ and $\gamma$ are Lagrange multipliers, and $E_{l}$ is the energy of the state $l$. The average values of energy and the normalization value have been omitted for simplicity. This gives a relation between the energy and the probabilities, which we now denote $p$ without the index:

[TABLE]

Setting the Lagrange multiplier $\gamma$ to $-1$, we first expand the previous equation around $p=0$. That accounts to consider the expansion around $y=p\ln p=0$.

[TABLE]

We make the following Ansatz to solve the equation (4)

[TABLE]

Plugging (5) in (4), to have the l.h.s. equal to the r.h.s. the coefficients multiplying the powers of $e^{-n\beta E}$ with $n>1$ have to vanish. This gives a recurrent expression for the coefficients $c_{n}$, which for the first four coefficients is solved as

[TABLE]

We have denoted $\beta E$ as $x$. The coefficients (6) give the following approximate solution for the probabilities versus $\beta E$

[TABLE]

In Figure 1 we compare the exact value of $p$ vs. $\beta E$ with the power series solution (7) till 3rd order, and with the Boltzmann distribution $e^{-\beta E}$.

Entropy $S_{-}$

Consider the other generalized entropy dependent only on the probabilities is $S_{-}$. For this entropy, defining the functional $\Phi_{-}$ as

[TABLE]

Finding the extrema of (8) and proposing the same Ansatz (5) we obtain a set of equations that can be solved to give the recursive probability solution:

[TABLE]

III Modified amplitude expansions

In this Section we use the series solutions for the probabilities in terms of the energy obtained in the previous Section, to perform an analytic continuation into the complex plane. Consider $a$ as the amplitude of a path, new complex variable substituting the probability $p_{l}$, and $A$ as the action replacing $\beta E_{l}$. This will allow to study modified quantropy functionals, for the definition of Baez and Pollard (1), as well as for our definition (18). The usual Quantropy solution will give an exponential $a\sim e^{\frac{iA}{\hbar}}$. In our approach this would be $K\sim e^{\frac{iS_{cl}}{\hbar}}$. We want to analyze the new statistics $S_{+}$ and $S_{-}$. We will find a functional dependence of $a$ vs. $A$ ($K$ vs. $S_{cl}$) to deviate from the exponential.

The main idea is to complexify the power expansion solution (7) since the amplitude is a complex number, such that we have a solution to the extrema of the modified Quantropy

[TABLE]

Finding the extrema of (10) w.r.t to $a$ one gets

[TABLE]

The range of validity of the propagators computations depends on the convergence of the imaginary series solution to this equation. The series is obtained by doing the replacement $\beta E$ by $\frac{A}{i\hbar}$ and $p$ by $a$ in (3). Also the Lagrange multipliers have to me mapped as $\beta$ to $\lambda$ and $\alpha$ to $\gamma$. The minus sign gives the right sign after a rotation on the argument of the exponential (similar to a Wick rotation). As the series solution continuation of (7) we obtain the following expression

[TABLE]

Since $A$ has units of action the argument of the exponentials and the terms on the expansion are adimensional. Substituting this expression on the constraint equation (11) we obtain the real and imaginary parts of $F(a\left(\frac{A}{i\hbar}\right))$. Beginning in Figure 3, the relevant difference of the propagators $K_{+}$ obtained with (12) with respect to the standard one can be observed in the region of $S_{cl}\approx\hbar$.

Using the parameter $\lambda=\frac{1}{i\hbar}$ in (12) the expression becomes

[TABLE]

Is not difficult to observe that any term of this expansion can be written as derivatives with respect to the parameter $\lambda$. If we derive with respect to $\lambda$ the usual amplitude we obtain: $\frac{\partial}{\partial\lambda}e^{-\lambda A}=-Ae^{-\lambda A}$. Higher derivates can be written as

[TABLE]

where $m$ and $n$ are positive integers. Thus we rewrite (III) as

[TABLE]

One can compute the corrections to any order. In the case of $a_{0}=e^{iS/\hbar}$ those corrections can be interpreted as higher order interactions of the action at different frequencies of the usual amplitude 111For example for a massive particle those will be contributions from multiples of the particle mass. For the harmonic oscillator also there will be contributions with a tower of masses and frequencies..

Now one can apply the same method to determine the distribution arising from the Quantropy with statistics $S_{-}$. We also have to perform the extension to the complex plane. The distribution for the modified Quantropy coming from $S_{-}$ is given by:

[TABLE]

IV Quantropy in terms of the propagator

In this section we present as an alternative proposal a kind of integrated version of the Quantropy baez . First we do it for the BG entropy, then for $S_{+},S_{-}$ and $S_{q}$. The change in distribution probabilities which arise from modified entropies in statistics is now reflected in the quantum arena as modifications to the propagators. The propagators between points in space-time $(\mathbf{x}_{a},t_{a})$ and $(\mathbf{x}_{b},t_{b})$ in quantum mechanics determine the probability amplitude of particles to travel from certain position to another position in a given time. As modified entropies in statistical physics lead to modified probability distributions, distinct probabilities of propagation over all paths from $(\mathbf{x}_{a},t_{a})$ and $(\mathbf{x}_{b},t_{b})$ will arise from a modified Quantropy.

In the work baez the Quantropy functional associated with BG statistics was formulated, and its maximization leads to the weight on the path integral $a\sim\exp(-\lambda S)$ with $\lambda=\frac{1}{i\hbar}$. We propose another functional, which in a sense constitutes an integrated version of Quantropy. Its maximization leads to the propagator $K(x)\sim\exp(-\lambda S_{cl}(x))$. The Wentzel-Kramers-Brillouin(WKB) method WKB allows to compute the wave function in a semiclassical approximation. In a sense this is linked to our approach, on which we maximize a functional which determines the propagator with a semiclassical approximation in terms of the classical action $S_{cl}$. This is exact for the free particle, the harmonic oscillator as well as for other cases SCA ; SCA2 . The procedure is applied to generalized entropy functionals, giving a modified propagator. For the Tsallis statistics we obtain $K_{q}(x)\sim\exp_{q}(-\lambda S_{cl}(x))$. This structure is the same that the wave function for the free particle that the Tsallis statistic possesses $\Psi_{q}(x)=\exp_{q}(i(kx-wt))$, which has been proposed as solution to the non linear quantum equations of QMT . According to Feynman arguments, one can start with the free particle propagator and determine the corresponding wave function FeynmanHibbs . Thus our procedure allows to find a propagator which can be identified with the wave function of interest. We should note that the propagator resulting from our procedure will not only describe the free particle but to a good approximation any other problem with its corresponding classical action. Our method should give the wave function solution for the problem of interest.

With the same method we write functionals for $S_{+}$, $S_{-}$ and obtain probability distributions, we can write the corresponding Quantropies and obtain the propagators $K_{+}$ and $K_{-}$, and correspondingly extrapolate them to the wave functions $\Psi_{+}$ and $\Psi_{-}$. This would give us the quantum behavior for the corresponding action. The given propagators, can be related to non linear quantum systems studied in the literature Chavanis19 .

To define our functionals we use the semiclassical limit to compute the propagator, this is $K(x)=F(a,b)e^{\frac{iS_{cl}(x)}{\hbar}}$, denoting the classical action as $S_{cl}(x)$, and being $F(a,b)$ a constant depending on the time difference $t_{b}-t_{a}$. For the free particle and the harmonic oscillator as well as oder physical problems SCA ; SCA2 this is an exact result.

For the BG statistics we define the Quantropy functional

[TABLE]

The extrema condition $\frac{\delta\Phi_{0}}{\delta K(x)}=0$ gives as solution the propagator dependence $K(x)=e^{-1-\alpha-\lambda S_{cl}(x)}$, $\lambda=\frac{1}{i\hbar}$, where the normalization constant $\alpha$ determines $F(a,b)$.

The integrated Quantropy functional for the new $S_{+}$ statistic is given by

[TABLE]

The extrema condition $\frac{\delta\Phi_{+}}{\delta K(x)}=0$ gives the equation:

[TABLE]

Using our knowledge to resolve this type of equation from the statistical physics case, presented in Section II, this gives for the modified propagator the series solution:

[TABLE]

This is obtained by taking the normalization $\alpha=-1$. A different normalization would change the coefficients in the expansion (20).

The maximization constraint for the new $S_{-}$ statistic is given by

[TABLE]

The extrema condition $\frac{\delta\Phi_{-}}{\delta K(x)}=0$ gives the equation:

[TABLE]

Using our knowledge of this type of equation from the statistical physics case, we obtain for the modified propagator the series solution:

[TABLE]

In the case of Tsallis statistics the functional is given by:

[TABLE]

and the solution is

[TABLE]

We have still to discuss the normalization of the different Kernels. This q-propagator is related to the q-wave function for the free particle non linear quantum mechanics of TsallisQM1 . We will specify to this case which has been studied by other means, in the literature TsallisQM1 ; QMT .

V Free particle propagators

In this section we write a modified propagator up to third order for the free particle in the case of the statistics $S_{+}$, $S_{-}$ and $S_{q}$ for $q=1-\delta$ and $q=1+\delta$ with $\delta>0$. The values of $q$ less or equal than one are considered in order to compare the different propagators. We determine when the corrections to the usual propagator play an important role, which turns to be in the quantum regime characterized by $S_{cl}\approx\hbar$. First we describe the procedure, then the normalization and in the last subsection we summarize our results.

V.1 Superposition of Kernels

Now we proceed to describe a generalized Kernel. The generalized complex probability distribution given by the expansion (20), can be regarded as a superposition of Kernels. Furthermore, the superposition will carry to the wave functions. In order to normalize the superposition we consider that the total Kernel expansion integration is the same of the usual (1 for the free particle), as is explicit in the Quantropy functional (18). We show that this coincides with the result for the normalization obtained from propagating the wave function FeynmanHibbs .

For the free particle the unnormalized Kernel is:

[TABLE]

$n$ is the number of divisions of the time interval and $\epsilon$ is an infinitesimal time parameter that satisfies $t=\epsilon n$. This expression arises from computing the path integral to get:

[TABLE]

with $A=\frac{m}{2\epsilon\hbar}$. The normalization constant is given by $N=(\frac{2\pi i\hbar\epsilon}{m})^{-\frac{n}{2}}$, hence the normalized propagator is

[TABLE]

We define the unnormalized Kernel for the free particle as

[TABLE]

and the first two corrections in $K_{+}$ are given by

[TABLE]

Thus the generalized Kernel associated with $S_{+}$ entropy is given by

[TABLE]

The normalization constant is determined by the requirement $\int_{-\infty}^{\infty}K_{+}(x,t)dx=1$, and up to the first corrections is given by $N_{+}=\frac{1}{1+3/(16\sqrt{2})}=0.883...$. The reason for this normalization is also understood by an argument presented in the following, motivated by Feynmann and Hibbs procedure FeynmanHibbs and discussed next.

Let us also discuss the normalization used in the modified propagators, as shown in the standard case FeynmanHibbs . We start considering the original unnormalized Kernel for the free particle, computed from the path integral:

[TABLE]

To determine the normalization constant in the Feynman and Hibbs method we can apply formulae (2-34) and (4-3) on their book FeynmanHibbs , to write the new infinitesimal Kernel between position $x_{i}$ and $x_{i+1}$, with $\Delta x_{i}=x_{i+1}-x_{i}$, in a time $\epsilon$ as follows

[TABLE]

The method consists in writing the wave function at a position $x$ at a time $t+\epsilon$ in terms of the wave function at position $y=x+\eta$ at a time $t$, explicitly

[TABLE]

In the quantum standard theory the normalization constant can be determined by expanding the l.h.s, of (V.1) $\psi(x,t+\epsilon)=\psi(x,t)+\epsilon\partial_{t}\psi$, the r.h.s. $\psi(x+\eta)=\psi(x,t)+\eta\partial_{x}\psi+\frac{\eta^{2}}{2}\partial_{x}^{2}\psi$ and $\exp(-i\epsilon V/\hbar)=1-\frac{i\epsilon V}{\hbar}+...$, then we compare the leading $\epsilon^{0}$ term. This implies that

[TABLE]

In a similarly fashion one would get for the first correction to $K_{+}$ written in (29)

[TABLE]

It is worth to mention that the more important contribution to (V.1) is given for small $\eta$’s, as well as in our generalized case. It is necessary to check this argument, in order to verify let us consider the following integrals:

[TABLE]

The first correction gives the relation:

[TABLE]

The previous normalization factor is a general feature to apply to any potential $V(x,t)$, in particular is valid for both cases discussed here: the free particle and the harmonic oscillator. A similar expression holds for $K_{-}$ normalization, this will be calculated in next Section.

V.2 Analysis of the propagators

Here we summarize the propagators obtained with the normalization methods described in previous subsections. The results for $K_{+}$ can be extrapolated to $K_{-}$ and $K_{q}$ because the method applied to obtain all of the propagators is basically the same.

Recall the standard propagator of the free particle from the space-time point $(0,0)$ to $(x,t)$ is given by

[TABLE]

The constant w.r.t. to x: $N_{0}=\sqrt{\frac{m}{2\pi i\hbar t}}$. For the case of the $S_{+}$ and $S_{-}$ statistics the first two contributions to the modified propagators (20) and (23) read:

[TABLE]

with $N_{+}\sim N_{0}$. For the Tsallis statics the associated propagator (25) is given by the expression:

[TABLE]

We calculate the normalization constants for $K_{\pm}$ up to the first correction and exactly $K_{q}$ to get:

[TABLE]

In the quantum regime $S_{cl}\approx\hbar$ the differences between the propagators $K_{+},K_{-},K_{q}$ and $K_{0}$ are shown in Figures 2, 3 and 4. Figure 2 compares $K_{+}$ propagator with the usual one. Also Fig. 3 compares $K_{-}$ propagator with the standard one. The last Figure 4 does a comparison between the propagators for the $S_{q}$ statistics $K_{q}$ for $q<1$ and $q>1$ with the usual one. The region of interest is the quantum regime with $S_{cl}\approx\hbar$. Furthermore in the classical regime the oscillations of the standard propagator grow averaging to zero FeynmanHibbs . Thus we are interested in comparing the corrections arising from different statistics in the quantum region of interest.

VI The Harmonic Oscillator

In this Section we apply the formulation of a modified Quantropy of Section IV for the case of the harmonic oscillator. We compute the modified propagator constructed by a superposition as previously. The extension of quantum systems employing the modified q-statistics has only been made for the case of the free particle QMT with different arguments. Our proposal allows to search the manifestation of non-extensive statistics in quantum systems (non linear) for generic potentials. We illustrate the procedure calculating only $K_{+}$, the $K_{-}$ and $K_{q}$ cases could be similarly calculated.

For the harmonic oscillator with Lagrangian $\mathcal{L}=\frac{m}{2}\dot{x}^{2}-\frac{m\omega^{2}}{2}x^{2}$ the path integral Kernel reads

[TABLE]

Next, following a similar procedure as for the free particle we compute the generalized Kernel and normalize it. The unnormalized Kernel is given by:

[TABLE]

now we compute the next terms of the Kernel up to third order, those are:

[TABLE]

Thus the total normalized propagator up to third order reads

[TABLE]

where $N_{+}=\frac{1}{\sqrt{\cos(\omega T)}}\frac{1}{(1+\frac{3}{16\sqrt{2}}+\frac{1}{96\sqrt{3}})}$. The relative normalization of the modified to the usual propagator is a result obtained in Section V, see formulas (V.1-32). This result is universal i.e. independent of the action. In Figure 6 we compare the propagator for the harmonic oscillator for the standard Quantropy and for the one based on $S_{+}$ and $S_{-}$ statistics. We are interested in the quantum regime given by $S_{cl}\approx\hbar$. There are noticeably effects in that regime. Outside the quantum region oscillations grow as in the usual case FeynmanHibbs . This behavior occurs in the classical region in which the modified Kernels will also not contribute.

VII Final Remarks

In this work we studied a newly proposed concept in the quantum theory inspired in the entropy of statistical mechanics. It fills a previous gap on the analogy between both disciplines. This is the main object developed by Baez and Pollard named Quantropy. It can be regarded as an analytical continuation of the Entropy in Statistical Mechanics (S.M.) to Quantum Mechanics (Q.M.); where probabilities of states change into complex amplitudes of paths. The identification performed is of energy in S.M. to action in Q.M., and from temperature to Planck constant, it reads: $E\rightarrow S$ and $T\rightarrow i\hbar$. We have considered an analogous definition at a macroscopic level with energy mapped to classical action $E\rightarrow S_{cl}$, i.e. considering as the main entity instead of the amplitude $a(x)$ of the path the propagator between space-time points $K(x)$. We constructed from the propagator a kind of integrated version of the Quantropy $Q_{0}=-\int_{X}K(x)\ln K(x)$. In the standard BG case the maximization of the functional leads to $K_{0}\sim\exp(iS_{cl})$. This functional can be generalized for modified entropies as $S_{+}$, $S_{-}$ and $S_{q}$ and the extrema of it will lead to modified propagators.

The result for the Tsallis statistics, leads to the propagator $K_{q}\sim\exp_{q}(iS_{cl})$, this makes contact with the Tsallis result of modified wave function for the free particle, because such propagator will lead to $\psi_{q}\sim\exp_{q}(i(kx-\omega t))$. The connection is made by arguments of Feynmann and Hibbs FeynmanHibbs , in the discussion of the propagator for the free particle $K_{0}\sim\exp(iS_{cl})$. They show that $K_{0}$ corresponds to the free particle wave function $\psi_{0}\sim\exp(i(kx-\omega t))$. Thus analogously $K_{q}$ will lead to $\Psi_{q}$. As a further work we need to explore the relations in the case of the modified propagators $K_{+}$ and $K_{-}$. They will give rise to wave functions which dependence are $\Psi_{\pm}=\exp_{\pm}(i(kx-\omega t))$, also for the free particle. In this case we have a recurrent series solution but we do not have exact expressions for these generalized exponentials. As discussed our proposal provides also generalized propagators $K_{+},K_{-}$ and $K_{q}$ for problems with interactions; we illustrated this by considering the $K_{+}$ associated with the harmonic oscillator.

There are hints from previous studies that the modified entropies considered here can be interpreted as linked with modified effective potentials. Therefore these modifications to the free particle could be related to a usual quantum mechanics with an effective potential OGT . However, this effects could also lead to non linear quantum equations explored in the literature with modified wave functions TsallisQM1 ; TsallisQM2 ; QMT ; Chavanis19 . Furthermore what we found here based on the concept of Quantropy, could be linked to results for quantum systems in terms of usual entropy vs. the density-matrix CaboObregon18 . A system governed by a modified statistics ($S_{+}$, $S_{-}$ or $S_{q}$) will lead to modified density-matrix distributions.

Moreover the modified “propagators” $K_{+},K_{-}$ and $K_{q}$ actually are strictly no longer standard propagators, because they lack the usual propagation property. This means that is not equivalent to propagate the particle from $(0,0)$ to $(t_{2},x_{2})$, than to first propagate it from $(0,0)$ to $(t_{1},x_{1})$ and then from $(t_{1},x_{1})$ to $(t_{2},x_{2})$. This is seen because for example $\int K_{+}(0,0;x_{1},t_{1})K_{+}(x_{1},t_{1};x_{2},t_{2})dx_{1}\neq K_{+}(0,0;x_{2},t_{2})$. This relates to the fact that in a quantum open systems where this generalized entropies are motivated, the nature of the processes are Non-Markovian. Those systems in consideration are modeled with Master Equations(Stochastic) RevModPhys.89.015001 .

We would like to further explore processes where the modified statistics in Quantropy play a central role. This could be done via the implied modified wave functions, which could also be interpreted as coming from standard quantum mechanics with an effective interaction or from non-linear quantum equations. The modified wave functions will correspond to the modified propagators here obtained. One should then explore in detail their quantum mechanic evolution. For some physical systems, in particular the harmonic oscillator, the $K_{+}$, $K_{-}$ and $K_{q}$ will illustrate its modified quantum behavior. That is the matter of future work.

VIII Acknowledgments

We thank Alejandro Cabo, Vishnu Jejjala, Oscar Loaiza-Brito, Miguel Sabido, Marco Ortega, Nelsón Flores-Gallegos and Pablo López-Vázquez for useful discussions and comments. NCB thanks PRODEP NPTC UGTO-515 Project, CIIC 181/2019 UGTO Project and CONACYT Project A1-S-37752. RSS thanks to CONACYT and PRODEP NPTC UDG-PTC-1368 Project for supporting this work. OO thanks CONACYT Project 257919 and CIIC 188/2019 UGTO Project.

Appendix A Numerical Approach

In this appendix we shall develop a numerical approach to study the propagator $K_{+}$ vs. the action $S_{cl}/\hbar$. To construct a relation between the action and the Kernel it is proposed an interpolating function which is forced to satisfy (19). This is, given a real value of the action ($S_{cl}/\hbar$), the interpolating function, finds a complex value of the Kernel that satisfies (19). The numerical function is expected to improve the convergence for larger values of the action with respect to the exponential expansion proposed in (12). In Figure 7 we show the plot obtained thorough the numerical approach. The X axis represents the numerical value of the action, and the Y axis represents the real (yellow line) and imaginary (blue line) parts of the function $F(K(\frac{S_{cl}}{i\hbar}))=-iS_{cl}/\hbar$ in (11).

For the case of the free particle the results are shown in Figure 8, we show the imaginary and real part of the Kernel.

The numerical function denoted is represented as $exp_{+}$ which is the behavior found from the quantropy introduced in Section IV. Thus, the free particle Kernel takes the form

[TABLE]

Using this definition we are able to compute the Kernel in a numerical manner. In Figure 8 we present the modified versus the usual Kernel. Thus as is observed the modified Kernel has a similar behavior that the standard Kernel for $S_{cl}<<\hbar$, but it deviates in already in the quantum regime. However, since the free particle wave function is not normalizable we shall not proceed further to check the probability distribution.

Now, let us proceed to check the numerical approach of the Kernel for the case of the harmonic oscillator. In order to match with the standard results known for the harmonic oscillator we shall employ the expansion

[TABLE]

where as usual

[TABLE]

and $H_{n}$ are the Hermite polynomials defined by the generating function

[TABLE]

For the case of the numerical function, the Kernel is written in a perturbative form as

[TABLE]

where $p_{i}$ are the Taylor polynomials of the functions

[TABLE]

As in the free particle case the modified entropy shall provide a new function $\exp_{+}$. In Figure 9 we show the comparison of the usual propagator and the modified propagator employing the numerical function.

For the harmonic oscillator, the modified Kernel deviates from the standard Kernel in the quantum region.