The so-called μ-bracket of a number or operator X, X μ ≡ X 1 + μ X ; X μ → X , i f μ → 0 , (1) and the related μ-deformed oscillator have been introduced 3 decades ago in (Jannussis, 1993). More recently, there appeared some papers (Gavrilik et al., 2010; Gavrilik and Mishchenko, 2012; Gavrilik et al., 2013) in which the μ-deformation based approach was initiated and developed.

For our purposes we define the μ-deformed trigonometric function (see (Gavrilik et al., 2013; Gavrilik et al., 2019) and references therein) as sin μ x = ∑ n = 0 ∞ − 1 n x 2 n + 1 2 n + 1 μ ! , cos μ x = ∑ n = 0 ∞ − 1 n x 2 n 2 n μ ! , (2) where [n] _μ ! = [1] _μ [2] _μ … [n] _μ . Clearly, at μ → 0 one recovers customary sine and cosine.

For our purposes, we introduce the μ-deformed analogs of spherical Bessel functions, namely, j 0 μ x = sin μ x x , y 0 μ x = cos μ x x . (3) At μ = 0 these reduce to the familiar Bessel functions.

The physical motivation for introducing these functions is two-fold: the first one in Eq. (3) describes the density profile of the dark matter halo and also leads to the rotation curves within the μ-deformed extension (Gavrilik et al., 2019) of the Bose-condensate model, while both functions, taken jointly, are of importance for constructing the representation space of the position and momentum operators, see Sections 3.1–3.2 below.

Since the applied deformation concerns mainly the basic trigonometric functions, let us study sin _μ x and cos _μ x in detail. Contracting the corresponding series to the Gaussian hypergeometric function, we can then represent them in the analytic form sin μ x = I x , μ sin ⁡ φ x , cos μ x = I x , μ cos ⁡ φ x (4) where I x , μ ≡ 1 + μ 2 x 2 − 1 + μ 2 μ , φ x ≡ 1 + μ μ arctan μ x . (5)

Therefore, in the case of μ-deformation, the main trigonometric identity is written as follows: sin μ 2 x + cos μ 2 x = I 2 x , μ , I x , μ ≤ 1 . (6)

The behavior of the μ-deformed trigonometric and spherical functions is shown in Figure 1.

In principle, it is possible to express the deformed trigonometric functions in terms of the μ-deformed exponential function. The μ-analogs of exponential and logarithmic functions are e μ x = 1 − μ x − 1 + μ μ , ln μ x = 1 μ 1 − x − μ 1 + μ , (7) which give us the known functions at μ → 0⁻ due to the asymptotic formulas: e x = lim n → ∞ 1 + x n n , ln x = lim n → ∞ n x 1 / n − 1 . (8)

Note that the μ-deformed functions exhibit a non-trivial property at μ > 0: e μ x n = e μ 1 − 1 − μ x n μ , ln μ x n = 1 − 1 − μ ⁡ ln μ x n μ . (9)

Focusing on the problems with spherical symmetry, we need to define an inner product ⟨f|g⟩ in terms of which the real functions u 1 ( x ) = j 0 ( μ ) ( x ) and u 2 ( x ) = y 0 ( μ ) ( x ) become orthonormal on finite interval x ∈ [0; R(μ)]: 〈 f | g 〉 = ∫ 0 R μ f * x g x w μ x d x , 〈 u i | u j 〉 = δ i , j , (10) where the asterisk means complex conjugation; the Latin indexes i, j run from 1 to 2.

For the orthogonality of sin φ and cos φ on the interval φ ∈ [0; π], we constitute ad hoc w μ x d x = 2 π x 2 I − 2 x , μ d φ x , π = φ R μ , (11) and obtain w μ x = 2 1 + μ π x 2 1 + μ 2 x 2 1 μ , R μ = 1 μ tan μ π 1 + μ , (12) where R(μ) coincides with the first zero of sin _μ x.

Expanding these as. w μ x = 2 π x 2 1 + 1 + x 2 μ + x 2 + x 4 2 μ 2 + O μ 3 , (13) R μ = π − π μ + π + π 3 3 μ 2 + O μ 3 , (14) We see that the known quantities are restored at μ = 0.

We would like to define the μ-deformed derivative D x ( μ ) with respect to the positive variable x, and its inverse. Let the functions f(x) and ϕ(x) admit expansion in the Taylor series and satisfy the relation D x μ f x = ϕ x . (15)

The actions of D x ( μ ) and antiderivative D x ( μ ) − 1 are respectively given as. ϕ x = d d x f x − x − 1 μ ∫ 0 x f ′ s s 1 μ d s , (16) f x = μ x ϕ x + ∫ 0 x ϕ s d s + f 0 , (17) Were the prime means ordinary differentiation.

We see that ϕ(x) = f′(x) at μ → 0 due to vanishing (s/x)^1/μ for s < x. By definition, the derivative D x ( μ ) lowers the exponent of the monomial x ⁿ by one, namely, D x ( μ ) x n = [ n ] μ x n − 1 . However, the operator D x ( μ ) violates the Leibniz rule: D x ( μ ) ( f ( x ) g ( x ) ) ≠ g ( x ) D x ( μ ) f ( x ) + f ( x ) D x ( μ ) g ( x ) .

The μ-deformed derivatives of some functions are collected in Table 1. To derive expressions 4–7, we have used the known auxiliary integrals: ∫ sin p − 1 ⁡ x sin p + 1 x cos p + 1 x d x = 1 p sin p ⁡ x sin p x cos p x . (18)

On the base of relations 4–5 (not 6–7) in Table 1, we define the Hermitian momentum operator P ̂ as P ̂ = − i x D x μ x , (19)

so that ⟨ u i | P ̂ | u i ⟩ = 0 , and ⟨ u 1 | P ̂ | u 2 ⟩ = ⟨ u 2 | P ̂ * | u 1 ⟩ = i (see Eq. 10), using the imaginary unit i. This operator will play an important role in the study of the deformed Heisenberg algebra further on.

To demonstrate the action of P ̂ on some functions, note that P ̂ x n = − i [ n + 1 ] μ x n − 1 for n ≥ 0, and then P ̂ ψ p x = p ψ p x , ψ p x = e μ i p x x . (20)

In addition, we consider the radial part Δ r ( μ ) of μ-deformed Beltrami–Laplace operator and its inverse (up to the additive constant C ∼ f (0)). Δ r μ f r ≡ 1 r 2 D r μ r 2 D r μ f r , (21) Δ r μ − 1 f r = μ 2 r 2 f r + 1 + μ ∫ 0 r f s s d s − 1 − μ r ∫ 0 r f s s 2 d s + C . (22)

It is easy to verify for positive n that Δ r μ r n = n μ ⋅ n + 1 μ r n − 2 ; Δ r μ − 1 r n = r n + 2 n + 2 μ ⋅ n + 3 μ , C = 0 . (23)

We also verify that Δ r μ − 1 j 0 μ r + j 0 μ r = 0 , C = − j 0 μ 0 , (24) by the use of the integrals ∫ cos p − 1 ⁡ x sin p + 1 x cos p + 1 x d x = 1 p cos p ⁡ x − cos p x sin p x . (25)

Here we are going to present the equations of some models using deformed differential calculus. The main model for us, from which the deformed Heisenberg algebra will follow, is described by the deformation of the Lane–Emden equation (LEE) for finite density function ρ(r) in the two possible formulations. Δ r μ ρ r + k 2 ρ r = 0 , (26) D r μ D r μ + g μ r 2 r D r μ + h μ r k 2 ρ r = 0 , (27)

Where g μ r = 1 1 − 2 μ 1 − 1 − μ 1 + μ μ 2 k 2 r 2 , h μ r = 1 + 2 μ 1 − 2 μ − 2 μ 2 1 − μ 2 k 2 r 2 1 + μ 1 − 2 μ . (28)

Note that the version Eq. 26 of μ-deformed LEE was already dealt with earlier in (Gavrilik et al., 2019), whereas the version in Eq. 27 is completely new, unpublished one. As seen, g _μ (r) → 1 and h _μ (r) → 1 at μ → 0.

It is important that, due to the special form of g _μ (r) and h _μ (r), these two μ-deformed versions of LEE have the same physical solution j 0 ( μ ) ( k r ) (along with y 0 ( μ ) ( k r ) ) at μ < 0.5, which means that the two versions are equivalent. To display this equivalence we have to explicitly transform Eq. 26 into Eq. 27. Setting kr ≡ x for simplicity, we assume the permutation rule as D x ( μ ) x = σ ( x ) x D x ( μ ) + λ ( x ) , apply it twice to the operator D r ( μ ) r 2 D r ( μ ) of the μ-Laplace operator in Eq. 21, and find the functions σ(x) and λ(x). Then, the equivalence of Eq. 26 and Eq. 27 is seen, with the non-trivial commutation relation: σ x x D x μ − D x μ x = − λ x , (29) σ x = 1 h μ = 1 − 2 μ 1 + μ 1 + μ 3 + 2 μ 3 x 2 1 / 2 , λ x = 2 g μ 1 + σ h μ = 1 + μ − 1 − μ μ 2 x 2 μ 2 + μ 1 + μ 2 x 2 1 − σ . (30)

As result, we have come to the new ( μ -deformed) generalization of Heisenberg algebra.

The functions λ(x) and σ(x) are real for 0 < μ < 0.5, tend to 1 at μ → 0, and are shown in Figure 2. We have 0 ≤ σ(x) ≤ 1, while the maximum of λ(x) is determined by λ(0) and is equal to λ max μ = 1 + μ 2 + μ μ 1 − 1 − 2 μ 1 + μ 1 + 3 μ . (31)

Although the function λ(x) has a tail in negative values for x > x max μ = 1 μ 1 + μ 1 − μ , (32) as shown in Figure 2, consideration of the problem over finite interval of x ∈ [0; R(μ)] with R(μ) ≤ x _max(μ) for μ ∈ 0 ; 0.5 guarantees a positive value of λ(x). Therefore, R(μ) varies between R _min ≃ 2.886 and R max = 2 3 ≃ 3.464 , and it is the minimum positive number that satisfies the condition sin _μ R(μ) = 0 (see Eq. 12).

It seems to be of interest to consider, elsewhere, the quantum-mechanical problem of the propagation of a particle, viewed as a spherical wave Ψ(r) in a space curved due to μ-deformation. Without specifying the boundary condition, it can be formulated as follows: P ̂ 2 Ψ r = k 2 Ψ r , (33)

where the momentum operator Eq. 19 for x = r is used. Let us remark again that the operator D r ( μ ) in P ̂ is a pseudohermitian one, see e.g., (Mostafazadeh, 2002; Bagchi and Fring, 2009; Gavrilik and Kachurik, 2016b; Gavrilik and Kachurik, 2019), but the “sandwiching” η − 1 D r ( μ ) η with η = r transforms it into Hermitian form as in Eq. 33.

In view of the definition Eq. 19 of the momentum operator, we formulate our μ-deformed Heisenberg algebra: σ x x P ̂ − P ̂ x = i λ x . (34)

In what follows, we will focus on the study of the uncertainty principle (relation) which follows from the algebra Eq. 34.

Denoting the standard deviations as Δ x = 〈 x 2 〉 − 〈 x 〉 2 , Δ P = 〈 P ̂ 2 〉 − 〈 P ̂ 〉 2 , (35) we proceed to the analysis of the generalized uncertainty relation (GUP) Δ x Δ P ≥ 1 2 | ⟨ x , P ̂ ⟩ | , (36) where the commutator is taken from Eq. 34.

To gain insight into the general properties of Eq. 36 for the μ-deformed Heisenberg algebra Eq. 34, let us combine Eq. 34 with its Hermitian conjugate to obtain 1 + σ x x , P ̂ = 2 i λ x . (37)

Applying Eq. 36 to this commutation relation, we get Δ 1 + σ x Δ P ≥ ⟨ λ x ⟩ . (38)

Taking into account that 1 ≥ σ(x) > 0 for μ < 0.5 in the left hand side, we come to the GUP Δ x Δ P ≥ 1 2 ⟨ λ x ⟩ . (39)

To evaluate the averages, we specify the states similarly to quantum ones. So, let us consider a normalized mixed state |ξ⟩ for ξ ∈ 0 ; 2 π in a Hilbert space basis Eq. 3 endowed with the inner product from Eq. 10: | ξ 〉 = cos ⁡ ξ | u 1 〉 + sin ⁡ ξ | u 2 〉 . (40) Here u 1 ( x ) = j 0 ( μ ) ( x ) and u 2 ( x ) = y 0 ( μ ) ( x ) as before.

In fact, the mixed state |ξ⟩ represents a general solution to the μ-deformed LEE, given by Eq. 26 and Eq. 27. Since the μ-deformed LEE is formulated for the local density of matter and, therefore, basically differs from the complex-valued Schrödinger equation, it is natural to describe its solution from Eq. 40 in terms of real-valued functions. Although the state |0⟩ for ξ = 0, such that ⟨ x | 0 ⟩ = j 0 ( μ ) ( x ) , serves to describe the finite DM distribution in (Gavrilik et al., 2019), the case ξ ≠ 0 admits the contribution of the cuspidal distribution y 0 ( μ ) ( x ) at x → 0.

Thus, we define the mean: 〈 … 〉 = 〈 ξ | … | ξ 〉 (41) for fixed ξ ∈ [0; 2π] and 0 < μ < 0.5.

In contrast to quantum mechanics, Eq. 41 suggests to evaluate a mean of some operator (…) in the basis generated by the μ-deformed LEE. There is no mathematical incorrectness in choosing basis functions coinciding with physical distributions. Only in turning to an interpretation, does one face the averaging (of powers) of the distribution function itself (this also happens in multifractal analysis).

The necessary matrix elements are given by. 〈 ξ | f x | ξ 〉 = A f μ − B f μ cos 2 ξ + C f μ sin 2 ξ , (42) 〈 ξ | f x | ξ + π / 2 〉 = B f μ sin 2 ξ + C f μ cos 2 ξ , (43) A f B f C f = 1 π ∫ 0 π 1 cos 2 φ sin 2 φ f X φ d φ , X φ = 1 μ tan μ φ 1 + μ . (44)

It is immediately seen that 〈 P ̂ 〉 = 〈 ξ | P ̂ | ξ 〉 = − i 〈 ξ | ξ + π / 2 〉 = 0 , 〈 P ̂ 2 〉 = 〈 ξ | P ̂ 2 | ξ 〉 = − 〈 ξ | ξ + π 〉 = 1 , (45) Therefore, the standard deviation ΔP = 1 is fixed for the set of states {|ξ⟩}.

On the other hand, let us introduce the functions Δ x ξ , μ ≡ ⟨ ξ | x 2 | ξ ⟩ − ⟨ ξ | x | ξ ⟩ 2 , Λ ξ , μ ≡ ⟨ ξ | λ x | ξ ⟩ , (46) which represent the averages Δx and ⟨λ(x)⟩, respectively.

Thus, in the basis of the μ-deformed spherical waves, one has ΔP = 1, and it is required that Δ x ξ , μ ≥ 1 2 Λ ξ , μ . (47)

This relation can be analyzed with the help of Figure 3A.

Let us introduce the auxiliary momentum variance, accounting for Eq. 47: δ P ξ , μ = Λ ξ , μ 2 Δ x ξ , μ ≤ 1 . (48) We see that δP (ξ, μ) ≤ ΔP = 1 and δP (ξ, μ) Δx (ξ, μ) = Λ(ξ, μ)/2 by definition. The behavior of δP (ξ, μ) is shown in Figure 3B.

The admissible domain of variety of the running values of Δx and ΔP is shown in Figure 4. We see that the black and pink curves are in antiphase regime, as it should be. For comparison, the violet curve describes the change in the deviation Δx according to the hyperbolic law in accordance with the standard Heisenberg algebra.

Let us remind the connection between the operators in terms of the dimensionless variable x = kr and the operators of the physical radial coordinate r and the momentum P ̂ r : r = x k , P ̂ r = ℏ k P ̂ , (60) where k is the parameter of Eqs 26 and 27 and has the dimension of inverse length; the operator P ̂ is given by Eq. 19.

The most successful results of paper (Gavrilik et al., 2019) for describing the dark matter halo of dwarf galaxies based on the μ-deformed Lane–Emden equation were obtained in the following range of parameters: μ = 0.151 … 0.18 , k = 0.17 … 2.64 k p c − 1 . (61)

Using the turning points A ((Δx)_min; (δP)_max) and B ((Δx)_max; (δP)_min) for fixed μ as in Figure 3B, we relate extreme physical values Δr and ΔP _r with dimensionless ones Δx and δP as. Δ r = Δ x k 1 k p c − 1 − 1 k p c , (62) Δ P r = δ P k 1 k p c − 1 × 6.394 × 1 0 − 27 e V c . (63)

The calculation results are collected in Table 2. Therein, we present the obtained data for five dwarf galaxies (from the eight ones given in Table 1 of (Gavrilik et al., 2019)), because just for these galaxies the μ-deformation based description of the rotation curves is most successful with respect to earlier approaches, as it provides the best agreement with observational data (certainly better then if one uses the profile from the usual Bose-condensate model of DM being the solution of non-deformed Lane–Emden equation as in (Harko, 2011), or uses the famous Navarro–Frenk–White profile (Navarro et al., 1997)).

Of course, the remaining three galaxies can also be considered, but the choice of five ones is both sufficient, trustful, and best suited for our treatment and conclusions.

Note that both the Lane–Emden equation and its μ-deformed extensions determine the distribution function ρ(r) of the matter, not the wave function of single particle. Therefore, the mean Eq. 41 is a quadratic form in the distribution, related here with |0⟩ which differs by a multiplicative constant defining ρ(0) (Gavrilik et al., 2019). Generally speaking, the state |0⟩ may not determine the turning points of a banana-like curve in the (Δx, δP) plane in Figure 3B, along with the extreme values of the deviations Δr and ΔP _r . Nevertheless, the mathematically correct mean Eq. 41 can be used to obtain new additional information about dark matter, even by means of considering the moments ⟨ρ ⁿ ⟩ of the distribution similarly to multifractal analysis. Besides, the extreme deviations set the limits for the fluctuations of physical quantities at ξ = 0.

Without a deep study of the structure of averages here, let us analyze the physical consequences of the data in Table 2. We see that in the non-relativistic theory the momentum deviation ΔP _r = m Δυ _r , where m is the particle mass, Δυ _r is the deviation of particle radial velocity υ _r . Since the original work (Gavrilik et al., 2019) was using bosons with m ∼ 10^–22 eV/c ², we obtain from Table 2 that Δυ _r ∼ 10⁻⁵ c in units of the speed of light c. Moreover, deviation of the kinetic energy Δ E K = m ( Δ υ r ) 2 / 2 can be used to determine the effective temperature of dark matter, namely, T eff = ( Δ P r ) 2 / ( 2 m ) ∼ 1 0 − 32 e V . This value is much smaller than the critical temperature of the Bose–Einstein condensation, as it should be in such a paradigm.

Due to the GUP given by Eq. 39, we relate the temperature T _eff of the spherical layer in the vicinity of ⟨r⟩ to its width 2Δr: Δ r 2 T eff ≥ ℏ 2 8 m ⟨ λ ⟩ 2 . (64) This formula holds for a macroscopic system of finite volume when Δr does not exceed the radius of the system, and it shows that a smaller domain may have a higher temperature, and vice verse.

It is worth to note that the mean ⟨λ⟩ in Eq. 64 takes values in the limited interval ⟨λ⟩ = Λ(ξ, μ) ∈ [Λ_min, Λ_max], where the positive Λ_min and Λ_max depend on μ (see Eq. 46; Figure 3A). For μ = 0.18, we have Λ_min ≃ 0.818 and Λ_max ≃ 0.875.