Nonrelativistic Quantum Particles on the Minkowski Plane, These Solutions Receive a New Physical Interpretation

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1 Cartesian and polar coordinates on the Minkowski plane

The Minkowski plane is a two-dimensional space of zero curvature with a pseudo-Euclidean 2 2 2 metrics s = x1 x2 in Cartesian coordinates. Cartesian and polar coordinates in the regions I (x2 x2 > 0, x− > 0) and II (x2 x2 > 0, x < 0) are related by the formulas (see ﬁg. 1) 1 − 2 1 1 − 2 1 2 2 x1 = r ch ϕ tg r = x x > 0, ± 1 − 2 (1) x = r sh ϕ thϕ = x /x , 2 ± p2 1 where r [0, ), ϕ R. ∈ ∞ ∈ x1

M A

I φ N IV χ

0 B x2

III II

Figure 1: Polar coordinate r, ϕ , of point M and coordinate ρ, χ of point N on Minkowski plane. OM= r, ON= ρ. { } { }

2 2 2 2 In the regions III (x1 x2 < 0, x2 > 0) and IV (x1 x2 < 0, x2 < 0), instead of the − 2 2 − imaginary r, we introduce the real radius ρ = x2 x1, i.e. r = iρ, and the new angle χ, now counted from the axis x and connected with the angle− ϕ by the relation ϕ = χ i π , then 2 p − 2 x = ρ sh χ thρ = x2 x2 > 0, 1 ± 2 1 (2) x = ρ ch χ, thχ = x /x−, 2 ± p1 2 where ρ [0, ), χ R. ∈ ∞ ∈ 2 Schr¨odinger equation

The quantum-mechanical system on the Minkowski plane is described by the Schr¨odinger equation with the Hamiltonian H, the kinetic part of which is proportional to the Beltrami-Laplace

2 operator, and the potential U is described by a function of coordinates. The Schr¨odinger equation for the eigenvalues HΨ= EΨ in Cartesian coordinates has the form ~2 ∂2 ∂2 Ψ(x , x )+ U(x , x )Ψ(x , x )= EΨ(x , x ). (3) −2m ∂x2 − ∂x2 1 2 1 2 1 2 1 2 1 2 After moving to the polar coordinates (1), (2) the Hamiltonian is given by the expression

~2 ∂2 1 ∂ 1 ∂2 + + U(r, ϕ) in the regions I,II, −2m ∂r2 r ∂r − r2 ∂ϕ2 H =  2 2 2 (4)  ~ ∂ 1 ∂ 1 ∂  + U(ρ, χ) in the regions III,IV. 2m ∂ρ2 ρ ∂ρ − ρ2 ∂χ2 −  We consider the Schr¨odinger equation (3) in polar coordinates in the regions I,II

~2 ∂2 1 ∂ 1 ∂2 + Ψ(r, ϕ)+ U(r, ϕ)Ψ(r, ϕ)= EΨ(r, ϕ). (5) −2m ∂r2 r ∂r − r2 ∂ϕ2 If we introduce the operator Lˆ = i ∂ , then this equation can be written as follows − ∂ϕ ~2 ∂2 1 ∂ Lˆ2 2 + + 2 Ψ(r, ϕ)+ U(r)Ψ(r, ϕ)= EΨ(r, ϕ). (6) −2m ∂r r ∂r r !

The eigenvalue M of the angular momentum operator Lˆ corresponds to the solution 1 LˆΦ(ϕ)= MΦ(ϕ), Φ(ϕ)= eiMϕ, (7) √2π where M, generally speaking, can be either real or complex number. However, for purely imaginary values of M, the states of the system will be unnormalizable. For real values of M R, the angular part of the wave function is normalized to the delta function ∈ +∞ +∞ 1 ′ dϕΦ(ϕ)Φ(ϕ)∗ = dϕ ei(M−M )ϕ = δ(M M ′). (8) 2π − −∞Z −∞Z

We are looking for a solution to the equation (6) in the form u(r) Ψ(r, ϕ)= Φ(ϕ) (9) √r

with normalization of radial function u(r) u ′ (r) as follows ≡ n ,M +∞ ∗ ′ ′ 2π drun,M (r)un ,M (r)= δnn , (10) −∞Z i.e. for wave function 2 ′ Ψ(r, ϕ) rdrdϕ = δ(M M )δ ′ . (11) | | − nn Z 3 As a result, for the function u(r) we obtain the equation

2mE M 2 + 1 2m u′′(r)+ + 4 U(r) u(r)=0. (12) ~2 r2 − ~2 Thus, the problem of the behavior of a quantum-mechanical system on the Minkowski plane was reduced to the one-dimensional Schr¨odinger equation 2m u′′(r)+ (E U (r))=0 (13) ~2 − eff with the eﬀective potential

~2 M 2 + 1 U (r)= 4 + U(r). (14) eff −2m r2 We consider in detail three precisely solvable problems: a free particle, a harmonic oscillator, and a Coulomb particle, for which the potential does not depend on the angle ϕ

0, free particle U(r)= 1 mω2r2, oscillator . (15)  2 α , Coulomb particle  − r The harmonic oscillator potential and the Coulomb potential are shown in Fig. 2, and the corresponding eﬀective potentials (14) are shown in Fig. 3.

a) b)

Figure 2: Potentials (15) of a harmonic oscillator a) and Coulomb particle b) on the Minkowski plane.

2.1 Coulomb particle We will start by studying the Coulomb particle. After replacing (9) from the Schr¨odinger equation with the potential U(r)= α for the function u(r) from (12) we have the equation − r 2mE M 2 + 1 2mα u′′(r)+ + 4 + u(r)=0. (16) ~2 r2 ~2r 4 Ueff(r) Ueff(r) Ueff(r)

0 r 0 r

0 r

a) b) c)

Figure 3: Eﬀective potentials (14), (15) of a free particle a), harmonic oscillator b) and Coulomb particle c) on the Minkowski plane.

with boundary conditions u(0) = u( )=0. It appears in the relativistic equations for the Coulomb ﬁeld and is analyzed, for example,∞ in [20, 21]. For small r, the wave function u(r) behaves like u(r) √r sin(M ln r + γ), (17) ∼ since in this case the equation takes the form

M 2 + 1 u′′(r)+ 4 u(r)=0. (18) r2

1 1 − Its general solution is a superposition of particular solutions r 2 +iM and r 2 iM . The parameter γ in (17) is the phase of reﬂection from the singular cone r =0. If γ is a complex number, then in the usual one-dimensional problem [20] this means the presence of absorption at the point r = 0, and in our problem it can correspond to the passage of a particle through isotropic lines in regions III, IV in Fig. 1. A discrete spectrum is possible at negative energy values E. It is convenient enter a unit of length r0, parameter g and dimensionless variable z by formulas ~ mα r r0 = , g = , z = . (19) 2√ 2mE ~√ 2mE r0 − − Then the equation (16) can be rewritten in the form

M 2 + 1 g 1 u′′(z)+ 4 + u(z)=0. (20) z2 z − 4 For large z, a solution decreasing at inﬁnity behaves asymptotically as an exponential u(z) − z ∼ e 2 . Therefore, we can look for a solution to the equation (20) in the form

1 − z u(z)= z 2 +iM e 2 f(z). (21)

For the function f(z) we obtain the equation

1 zf ′′ + (2iM +1 z) f ′ g + + iM f =0. (22) − − 2

5 Its basic solutions will be degenerate hypergeometric functions [22] of the ﬁrst and second kind

a z a(a + 1) z2 f (z)= F (a, c, z)=1+ + + ..., 1 c 1! c(c + 1) 2!

f (z)=Φ(a, c, z)= z1−cF (a c +1, 2 c, z) , (23) 2 − − where c =1+2iM, a = iM + 1 g. 2 − We have three types of solutions to the equation (20)

− z 1 u (z)= C e 2 √z ziM F iM + g, 2iM +1, z , 1 1 2 −

− z − 1 u (z)= C e 2 √z z iM F iM g, 1 2iM, z , 2 2 2 − − − u(z)= C u (z)+ C u (z)= C u (z) e−2iγ u (z) . (24) 1 1 2 2 1 − 2 Consider the ﬁrst solution. For the function u1(z) to tend to zero for large z, it is necessary that the series (23) break oﬀ. This is possible with a = n, n =0, 1, 2 ..., then the eigenvalues − mα2 E(n, M)= , (25) − ~2 1 2 2 n + iM + 2 and the wave function

− z Ψ (z,ϕ)= C e 2 ziM F ( n, 2iM +1, z) eiMϕ. (26) 1 1 − For real M = 0, we have unstable decay states radiating to the singular center according to the interpretation6 of [23]. In our case, this corresponds to absorption by an isotropic cone or passage from regions I, II in regions III, IV in Fig. 1. For M = 0, we obtain singlet states with a discrete spectrum, the same as on the Euclidean plane with M = 0, which is completely natural, since in these cases the particle moves only in the radial direction. However, since the equation (5) with the Coulomb potential is invariant under the replacement r r, the particle passes from region I to region II in Fig. 1, in contrast to the Euclidean plane,→ where − it is absorbed by the center. It is easy to see that the first and second solutions (24) are connected by simply replacing M with M, therefore, the second solution is simply the complex conjugation of the first −∗ u2(z)= u1(z). Consider the third solution u(z), presented as a superposition of the first two (24). Now it is already impossible to cut of the series at u1(z) and u2(z) at the same time. Let us pay attention to the behavior of this solution for large and small values of the argument. When the z argument tends to zero, the wave function behaves like [20]

u √z ziM e−2iγ z−iM √r sin(M ln r + γ M ln r ). (27) ∼ − ∼ − 0 Phase β = γ M ln r depends on the energy β = β(E) and you can use the quantization − 0 condition [20] β(E ) β(E )= πn. (28) n − 0

6 The asymptotic behavior of the functions u (z) and u (z) for z has the form [22] 1 2 →∞ z − Γ(1+2iM) z − Γ (1 2iM) 2 g 2 g u1(z) e z , u2(z) e z − , (29) ∼ Γ 1 + iM g ∼ Γ 1 iM g 2 − 2 − − then the solution u(z) at inﬁnity behaves as follows

z − Γ(1+2iM) − Γ(1 2iM) u(r) e 2 z g e 2iγ − . (30) ∼ Γ 1 + iM g − Γ 1 iM g 2 − 2 − − ! For the function to go to zero, you need to require

Γ(1+2iM) − Γ (1 2iM) = e 2iγ − . (31) Γ 1 + iM g Γ 1 iM g 2 − 2 − − The quantization condition (28) takes the form γ(E ) M ln r (E )= γ(E ) M ln r (E )+ πn (32) n − 0 n 0 − 0 0 or f(En)= f(E0)+ πn, (33) where Γ 1 g(E)+ iM f(E)= M ln g(E)+arg 2 − . (34) − Γ(1+2iM) For large negative energy values (i.e., for small r) and M = 0, the function f(E) M ln g(E). Then for energy levels we get the same formula 6 ≈ − 2πn M En = E0e , E0 < 0, n =0, 1, 2,..., (35) as in [20]. Discrete energy levels of a Coulomb particle become more rare with E . Indeed, in this case, the effective potential is approximately equal to → −∞ ~2 M 2 + 1 U (r) 4 , (36) eff ≈ −2m r2 i.e. is determined only by orbital forces. In [24], such the solutions are called the ”event horizon fall mode”. Solution of the one-dimensional Schrödinger equation with potential U = β/r2 and the appearance of discrete energy levels was discussed in [20, 21, 25] and was confirmed− in [26] by numerical experiment. For negative energies tending to zero (large values of r and g(E)), the function f(E) πg(E) and energy levels are given by the formula ≈ − mα2 En = , n =0, 1, 2,... (37) 2 2 −2~ (n + g0) 1 For g0 = 2 we obtain a formula that describes the energy levels of the Coulomb particle on the Euclidean plane. Thus, when moving away from the isotropic cone, levels are condensed, just like in the ordinary Coulomb potential when moving away from the attracting center. Indeed, in both cases, the Coulomb potential is the leading one in this energy region. Since for large values of g(E) the quantum numbers n are also large (n g ), then for energy levels in this ≫ 0 area we can approximately write mα2 E = . (38) n −2~2n2 7 2.2 Free particle The energy levels for a free particle on the Minkowski plane can also be obtained directly from formulas (33), (34) for α = 0 (absence of the Coulomb potential). In this case

f(E)= M ln r (E)+ C(M), (39) − 0 where C(M) is a function of M. Substituting this expression in (33), we obtain the formula (35), coinciding with previously obtained energy levels for a free particle [8]. With increasing n, the distance between the levels increases and the discrete spectrum at E is becoming increasingly rare. For positive energies E > 0, the spectrum is continuous. → −∞

2.3 Harmonic oscillator The description of the harmonic oscillator on the Minkowski plane is obtained from the description of the Coulomb particle using the analogue of dion-oscillatory duality, which takes place α on the Euclidean plane [18, 19]. If in the equation (5) with the potential U(r) = r and the 2 − eigenvalue EC make a change of variables r0r = ̺ , ϕ =2φ, where r0 is the scale factor, having a dimension of length, we obtain the Schr¨odinger equation for the oscillator

~2 ∂2 1 ∂ 1 ∂2 1 + Ψ(̺, φ)+ mω2̺2Ψ(̺, φ)= E Ψ(̺, φ), (40) −2m ∂̺2 ̺ ∂̺ − ̺2 ∂φ2 2 osc where r E =4α, mω2r2 = 8E . Note that such a relativistic oscillator plays an important 0 osc 0 − C role in the theory of superstrings [11,12]. Thus, in order to go to the formulas for the oscillator in the corresponding formulas for solving the Coulomb problem, we need to make a substitution

r r = ̺2, ϕ =2φ, r E =4α, mω2r2 = 8E , M =2M . (41) 0 0 osc 0 − C osc C As a result, the energy levels (25) of the Coulomb particle for the ﬁrst solution (24) will go over into oscillator energy levels

Eosc = ~ω (2n +1+ iMosc) , (42) and the wave function (26) is converted into the wave function of the oscillator

− mω 2 mω Ψ (̺, φ)= C ̺iMosc e 2~ ̺ F n, 1+ iM ; ̺2 eiMoscφ. (43) osc osc − osc ~ This type of solutions for M = 0 on the Euclidean plane are interpreted as solutions for particles, falling on the center or outgoing6 from it. In the case of the Minkowski plane, such solutions can interpret not only as “disappearing” on the light cone or “emerging” from it, but also as passing into the region of III or IV . For M = 0, these solutions describe singlet, positive- norm, Lorentz-invariant physical states with discrete spectrum. For the second solution (24) everything is similar. The condition (31) of the tendency of the third solution (24) to zero with an unlimited increase in the argument can be rewrite as

Γ(1 + iM ) − Γ(1 iM ) mα E osc = e 2iγ − osc , g = = osc . (44) Γ 1+iMosc Eosc Γ 1−iMosc Eosc ~√ 2mE 2~ω 2 − 2~ω 2 − 2~ω − C 8 The quantization condition (33) with the function (34) retains its form. Given the relationship between the energies of the Coulomb particle and the oscillator

2αω√m Eosc = , (45) ±√ 2E − C which is easy to obtain from (41), we have at low positive energies (at low ̺) and M = 0 osc 6 oscillator spectrum

2πn Eosc = Eosce Mosc , Eosc > 0, n =0, 1, 2,..., (46) n 0 0 − − i.e. a formula similar to the case of a Coulomb particle (35). Discrete oscillator energy levels are condensed when approaching the zero value. In the limit of large positive energies (large values of ̺) from (37) we obtain

Eosc =2~ω(n + g0), n =0, 1, 2,.... (47)

For large quantum numbers (n g0), it takes a standard expression describing the discrete energy levels of a classical oscillator.≫

3 Conclusion

Eﬀective Coulomb potential [25] particles on the Euclidean plane (Fig. 4)

~2 1 M 2 α U (r)= 4 − , (48) eff −2m r2 − r differs from the Coulomb potential (14) on the Minkowski plane only by a sign at M 2 (Fig. 3). A similar sign-changing effect occurs in the relativistic Dirac and Klein-Gordon equations for Coulomb potential [24], where for particles with a charge Z less than a certain critical charge ZZcr — the case of the Minkowski plane.

Ueff(r)

0 r

Figure 4: Eﬀective potential of a Coulomb particle on the Minkowski plane.

For a harmonic oscillator on the Minkowski plane, unstable decay states (43), arising for real M = 0 are interpreted not only as ”disappearing” on an isotropic cone or “appearing” from it,6 but also as passing from regions I, II to regions III, IV in Fig. 1. For M = 0, we

9 obtain the same states with a discrete spectrum as on the Euclidean plane, which is completely natural, since in these cases the particle moves only in the radial direction. However, since the equation (5) with the Coulomb potential is invariant under the replacement r r, the → − particle passes from region I to region II in Fig. 1, in contrast to the Euclidean plane, where it is absorbed by the center. Thus, although solutions of the equations (12), (13) with potentials (15) have long been known, but they arose in other problems. In problems of nonrelativistic quantum particles on the Minkowski plane, these solutions receive a new physical interpretation.

References

[1] Remnev M.A., Klimov V.V. Metasurfaces: a new look at Maxwell’s equations and new ways of light control // Phys. Usp. 2018. Vol. 61. No. 2. P. 157–190.

[2] Davidovich M.V. Hyperbolic metamaterials: production, properties, applications, and prospects // Phys. Usp. 2019. Vol. 62. No. 12. P. 1173–1207.

[3] Smolyaninov I.I. Modelling of causality with metamaterials // J. Optics. 2013. Vol. 15. No. 2. 025101; arXiv:1210.5628.

[4] Smolyaninov I.I. Hyperbolic metamaterials; arXiv:1510.07137.

[5] Pimenov R.I. Uniﬁed axiomatics of spaces with maximal motion group // Lithuanian math. collection. 1966. Vol. 5. No. 3. P. 457–486 (in Russian).

[6] Gromov N.A. Contractions of classical and quantum groups. Moscow: FIZMATLIT, 2012. 320 p. (in Russian).

[7] Gromov N.A. Particles in the early Universe. World Scientiﬁc: Singapore, 2020. 160 p.

[8] Gromov N.A., Kuratov V.V. Quantum particle on Minkowski plane // Proc. Komi Sci. Centre, Ural Branch, RAS. 2018. No. 3(35). P. 5–7 (in Russian).

[9] Gromov N.A., Kuratov V.V., Kostyakov I.V. Quantum Coulomb particle on Minkowski plane // Proc. Komi Sci. Centre, Ural Branch, RAS. 2019. No. 1(37). P. 5–8 (in Russian).

[10] Gromov N.A., Kuratov V.V., Kostyakov I.V. Harmonic oscillator on Minkowski plane // Bull. Syktyvkar Univ. Ser. 1: Math., Mech., Inform., 2018. No. 2(27). P. 10–23 (in Russian).

[11] Green M.B., Schwarz J.H., Witten E. Superstring Theory. Vol. 1: introduction. Vol. 2: Loop Amplitudes, Anomalies and Phenomenology. Cambridge, UK: Univ. Pr., 1987. 596 p.

[12] Kaku M. Introduction to Superstrings. New York: Springer-Verlag. 1988.

[13] Bars I. Relativistic Harmonic Oscillator Revisited // Phys. Rev. D. 2009. Vol.79, No. 4, 045009; arXiv:0810.2075.

10 [14] Araya I.J., Bars I. Extended Rindler Spacetime and a New Multiverse Structure // Phys. Rev. D. 2018. Vol. 97. Iss. 8. 085009; arXiv:1712.01326 [gr-qc].

[15] Bars I. Wavefunction for the Universe close to its beginning with dynamically and uniquely determined initial conditions // Phys. Rev. D. 2018. Vol. 98. Iss. 10. 103510; arXiv:1807.08310 [gr-qc].

[16] Gorbatenko M.V., Neznamov V.P. Quantum mechanics of stationary states of particles in external singular spherically and axially symmetric gravitational and electromagnetic ﬁelds; arXiv:2003.11354 [physics.gen-ph].

[17] Gorbatenko M.V., Neznamov V.P. I. Proof of the absence of classical black holes // Problems of Atomic Science and Technology. Ser.: Theor. and Appl. Phys. 2019. No. 1. P. 3–19 (in Russian). II. Proof of the absence of classical black holes // Problems of Atomic Science and Technology. Ser.: Theor. and Appl. Phys. 2019. No. 2. P. 22–46 (in Russian). III. Proof of the absence of classical black holes // Problems of Atomic Science and Technology. Ser.: Theor. and Appl. Phys. 2019. No. 2. P. 47–56 (in Russian).

[18] Ter-Antonyan V. Dyon-oscillator duality. ArXiv:quant-ph/0003106.

[19] Grigoryan G.V., Grigoryan R.P., Tyutin I.V. Isomorphism between oscillator and Coulomb-like theories in one and two dimensions // Theor. Math. Phys. 2013. Vol. 176. P. 1115-1139.

[20] Perelomov A.M., Popov V.S. ”Fall to the center” in quantum mechanics // Theor. Math. Phys. 1970. Vol. 4. P. 664-677.

[21] Gitman D.M., Tyutin I.V., Voronov B.L. Self-Adjoint Extensions in Quantum Mechan- ics: General Theory and Applications to Schr¨odinger and Dirac Equations with Singular Potentials // Progress in Math. Phys. Vol. 62, Birkh¨auser, New York, 2012. 511 p.

[22] Bateman H., Erdelyi A. Higher Transcendental Functions. Vol. 1. Mc Graw-Hill: New York, 1953.

[23] Shabad A.E. Singular center as a nongravitational black hole // Theor. Math. Phys. 2014. Vol. 181. P. 1643-1651.

[24] Gorbatenko M.V., Neznamov V.P., Popov E.Yu. Some aspects of quantum mechanics of particle motion in static central-symmetric gravitational ﬁelds // Problems of Atomic Science and Technology. Ser.: Theor. and Appl. Phys. 2015. No. 2. P. 21–31 (in Russian); arXiv:1504.00306 [gr-qc].

[25] Landau L.D., Lifshitz E.M. Course of Theoretical Physics: Vol. 3, Quantum Mechanics: Non-Relativistic Theory. 1981. 689 p.

[26] Neznamov V.P., Safronov I.I. Particles falling on center. Landau-Lifshitz hypothesis and numerical calculations // Problems of Atomic Science and Technology. Ser.: Theor. and Appl. Phys. 2016. No. 4. P. 3–8 (in Russian).