GUP and Point Interaction

Home , Quantum group, Uncertainty principle

arXiv:2001.02850v2 [quant-ph] 27 Feb 2020 mndwe h oeta sone-dimensional is potential the when amined re fGPparameter. GUP of order a path-integral Feynman’s Schrödinger and the mechanics, ∗ [email protected] h o-eaiitcqatmmcaiswt generalized a with mechanics quantum non-relativistic The eeaie netit rnil n on Interaction Point and principle uncertainty Generalized 2 eateto hsc,Kuga nvriy hnwn631- Changwon University, Kyungnam Physics, of Department ynnmUiest,Cago 3-0,Korea 631-701, Changwon University, Kyungnam 1 eateto lcrncEngineering, Electronic of Department aKlPark DaeKil Abstract 1 , 2 ∗ δ 1 − n ye Jung Eylee and ucin ti hw htulk sa quantum usual unlike that shown is It function. poce r nqiaeta h first the at inequivalent are pproaches netit rnil GP sex- is (GUP) principle uncertainty 1 , 0,Korea 701, For a lone time it has been believed in the theories of quantum gravity[1–3] that there may exist a minimal observable distance at the Planck scale. The modern status of quantum gravity was reviewed in Ref. [4, 5]. The physical motivation for the existence of minimal distance (and/or momentum) is due to the conjecture that gravity may disturb the spacetime structure significantly at the Planck scale. It is also hoped that the introduction of minimal uncertainty may cure the nonrenormalizable character of quantum gravity. The existence of minimal observable distance and momentum modifies the Heisenberg uncertainty principle (HUP) to the generalized uncertainty principle (GUP). Thus, the commutation relation between position operator and its conjugate momentum operator should be modified[6, 7]. Although there are several expressions, we use in this paper the simplest form of the GUP proposed in Ref.[7]

2 Qî, Pˆj = i~ δij + αδijPˆ +2αPîPˆj (1) h i Qî, Qˆj = Pî, Pˆj =0 h i h i 2 where the GUP parameter α has a dimension (momentum)− . Of course, we return to the HUP if α = 0. Eq. (1) is solved up to the first order in α by defining the position and momentum operators as 2 Pî =p î 1+ αpˆ , Qî =q î, (2) wherep î andq î obey the usual commutation relations of HUP:

[ˆqi, pˆj]= i~δij, [ˆqi, qˆj]=[ˆpi, pˆj]=0. (3)

Then, Pˆi and Qˆi satisfy

2 Qî, Pˆj = i~ δij + αδijpˆ +2αpîpˆj . (4) h i The non-relativistic quantum mechanics with GUP-corrected Hamiltonian was examined by Schrödinger approach[7] and Feynman’s path-integral approach[8, 9]. In the usual quantum mechanics the transition amplitude K[qf , tf : q0, t0] from (t0, q0) to (tf , qf ), which is usually called Kernel[10], is calculated by a path-integral

(tf ,qf ) K[q , t : q , t ] q , t q , t = qe(i/~)S[q], (5) f f 0 0 ≡h f f | 0 0i D Z(t0,q0)

2 where S[q] is an actional functional and q is sum over all possible paths connecting (t , q ) D 0 0 and (tf , qf ) in spacetime. The Kernel also can be represented as

∞ (i/~)En(tf t0) K[qf , tf : q0, t0]= φn(qf )φn∗ (q0)e− − , (6) n=1 X

where φn(q) and En are eigenfunction and eigenvalue of Schrödinger equation. As far as we know, there is no rigorous mathematical proof that the path-integral in Eq. (5) always results in the right hand side of Eq. (6). Since, however, no counterexample is found in the usual quantum mechanics, it is asserted that Schrödinger and path-integral approaches are equivalent. In this short paper, however, we will show that Schrödinger and path-integral approaches are inequivalent in the non-relativistic GUP-corrected quantum mechanics when the potential is singular δ-function. In the quantum mechanics with HUP the Feynman propagator for the one-dimensional δ-function potential problem was exactly derived in Ref. [11]. If, however, one applies the computational technique used in Ref. [11] to the higher-dimensional δ-function potential problems, the propagators become infinity. This is due to the fact that the Hamiltonian with too singular potential loses its self-adjoint property. From the pure aspect of mathe- matics this difficulty can be overcome by incorporating the self-adjoint extension[12, 13] into the quantum mechanics. In this way the Schrödinger equation for the higher-dimensional δ-function potential problems were solved in Ref. [14]. Subsequently, the Feynman propagators and corresponding energy-dependent Green’s functions were explicitly derived[15]. From the aspect of physics this difficulty can be overcome by introducing the renormalization scheme to non-relativistic quantum mechanics. In this way same problems were reexamined in Ref. [14, 16]. Using renormalization and self-adjoint extension techniques the δ′ po- − tential problem was also solved[17, 18]. The equivalence of these two different methods was discussed in Ref. [19]. Now, we start with an one-dimensional free particle with GUP, whose Hamiltonian can be written as Pˆ2 pˆ2 α Hˆ = = + pˆ4 + (α2). (7) 0 2m 2m m O Thus, the time-independent Schrödinger equation can be written in a form

~2 d2 α~4 d4 + + (α2) φ(q)= φ(q), (8) −2m dq2 m dq4 O E

3 where is an energy eigenvalue. The solution of Eq. (8) is E 1 ~2k2 α~4 φ(q)= eikq = + k4 + (α2). (9) √2π E 2m m O If one extends Eq. (6) to the continuous variables, the corresponding Kernel (or Feynman propagator) can be derived as

∞ i ~ T K0[qf , tf : q0, t0]= dkφ(qf )φ∗(q0)e− E , (10) Z−∞ where T = t t . Within (α) the integral in Eq. (10) can be computed and the ﬁnal f − 0 O expression is m 3iα~m 6αm2(q q )2 K [q , t : q , t ]= 1+ f − 0 (11) 0 f f 0 0 2πi~T T − T 2 r im(q q )2 q q 2 exp f − 0 1 2αm2 f − 0 . × 2~T − T " ( )# This expression exactly coincides with the result of Ref. [8, 9], where the Kernel is derived by direct path-integral. Of course, when α = 0, Eq. (11) reduces to usual well-known free-particle propagator[10, 20]. By introducing the Euclidean time τ = iT , it is easy to derive the Brownian or Euclidean propagator in a form m 3α~m 6αm2(q q )2 G [q , q : τ]= 1 + f − 0 (12) 0 f 0 2π~τ − τ τ 2 r m(q q )2 q q 2 exp f − 0 1+2αm2 f − 0 . × − 2~τ τ " ( )#

Then, the energy-dependent Green’s function is deﬁned as a Laplace transform of G0[qf , q0 : τ]: ∞ ǫτ Gˆ [q , q : ǫ] G [q , q : τ]e− dτ, (13) 0 f 0 ≡ 0 f 0 Z0 where ǫ = E/~ and E is an energy parameter. Using an integral formula[21]

ν/2 ∞ ν 1 β γx β x − e− x − dx =2 K 2 βγ , (14) γ ν 0 Z p where Kν(z) is a modified Bessel function, one can derive the energy-dependent Green’s function up to first order in α. The final expression is

m 2mǫ Gˆ [q , q : ǫ]= (1+6α~mǫ) exp (1+2α~mǫ) q q . (15) 0 f 0 2~ǫ − ~ | f − 0| r " r #

4 Now, let us consider a problem of δ function potential, whose Hamiltonian is −

Hˆ1 = Hˆ0 + vδ(q). (16)

Then, the corresponding Schr¨odinger equation is

~2 d2 α~4 d4 + + vδ(q) φ(q)= Eφ(q). (17) −2m dq2 m dq4 Eq. (17) naturally imposes an boundary condition on φ(x) at the origin in a form

+ 2 + 2mv φ′(0 ) φ′(0−) 2α~ φ′′′(0 ) φ′′′(0−) = φ(0). (18) − − − ~2 If α = 0, Eq. (18) reduces to usual boundary condition introduced in the Kronig-Penney model. Also, one can show from Eq. (17) that if v < 0, there is a single bound state. Using d d x = ǫ(x), ǫ(x)=2δ(x), and δ′(x)f(x) = δ(x)f ′(x), where ǫ(x) is usual alternating dx | | dx − function, one can show that within (α) the normalized bound state φ (q) and bound state O B energy B are 2 3 4 a q mv αm v φ (q)= √ae− | | B = , (19) B − 2~2 − ~4 where a = mv/~2 2αm3v3/~4 = √ 2mB(1 2αmB)/~. It is straightforward to show − − − − that φB(q) satisﬁes the boundary condition (18).

Now, let us check whether the Kernel or energy-dependent Green’s function for Hˆ1 yields

the same results or not. Let the Brownian propagator for Hˆ1 be Gˆ[qf , q0 : τ]. Then,

Gˆ[qf , q0 : τ] satisﬁes an integral equation[10, 22]

τ v ∞ ∆G[qf , q0 : τ] = ~ ds dqG0[qf , q : τ s]δ(q)G[q, q0 : s] (20) − 0 − Z Z−∞ v τ = dsG [q , 0 : τ s]G[0, q : s], −~ 0 f − 0 Z0 where ∆G[q , q : τ]= G[q , q : τ] G [q , q : τ]. Taking a Laplace transform in Eq. (20), f 0 f 0 − 0 f 0 the energy-dependent Green’s function Gˆ[qf , q0 : ǫ] for Hˆ1 satisﬁes

v ∆Gˆ[q , q : ǫ]= Gˆ [q , 0 : ǫ]Gˆ[0, q : ǫ], (21) f 0 −~ 0 f 0 where ∆Gˆ[q , q : ǫ]= Gˆ[q , q : ǫ] Gˆ [q , q : ǫ]. Solving Eq. (21), one can derive f 0 f 0 − 0 f 0 v Gˆ [q , 0 : ǫ]Gˆ [0, q : ǫ] ∆Gˆ[q , q : ǫ]= 0 f 0 0 . (22) f 0 ~ v ˆ − 1+ ~ G0[0, 0 : ǫ]

5 Inserting Eq. (15) into Eq. (22), it is possible to derive ∆Gˆ[qf , q0 : ǫ] in a form m ~ 2 2~ǫ (1+6α mǫ) 2mǫ ∆Gˆ[qf , q0 : ǫ]= exp (1+2α~mǫ)( qf + q0 ) . (23) − ~ m α~mǫ − ~ | | | | v + 2~ǫ (1+6 ) " r # Since Eq. (23) is valid onlyp up to ﬁrst order of α, ∆Gˆ can be expressed as v m 2mǫ ∆Gˆ[q , q : ǫ]= exp ( q + q ) (24) f 0 −~ 2~ǫ − ~ | f | | 0| r " r #

1 2mǫ 6αvmǫ 1+12α~mǫ 2α~mǫ ( q + q ) + (α2) .  ~ f 0 2  × v 2~ǫ ( − | | | | ) − v 2~ǫ O ~ + m r ~ +  m   q q  Now, we derive the Brownian propagator for Hˆ1 by taking an inverse Laplace transform in Eq. (24). Using the following formulas of the inverse Laplace transform[23]

a√ǫ 2 1 e− 1 a ab+b2τ a − (τ)= e− 4τ be erfc + b√τ (25) L √ǫ + b √πτ − 2√τ 1/2 a√ǫ 1 ǫ− e− ab+b2τ a − (τ)= e erfc + b√τ L √ǫ + b 2√τ 1/2 a√ǫ 2 1 ǫ e− a 2bτ a 2 ab+b2τ a − (τ)= − e− 4τ + b e erfc + b√τ L √ǫ + b √ 3 2√τ 2 πτ a√ǫ 2 2 2 2 1 ǫe− 4b τ 2(ab + 1)τ + a a 3 ab+b2τ a − (τ)= − e− 4τ b e erfc + b√τ L √ǫ + b √ 5 − 2√τ 4 πτ a√ǫ 2 1 e− τ a 2 ab+b2τ a − (τ)= 2b e− 4τ + (2b τ + ab + 1)e erfc + b√τ L (√ǫ + b)2 − π 2√τ r where erfc(z) is an error function deﬁned as

2 ∞ t2 erfc(z)= e− dt, (26) √π Zz then, ∆G[q , q : τ] G[q , q : τ] G [q , q : τ] can be written in a form f 0 ≡ f 0 − 0 f 0

∆G[qf , q0 : τ] (27) mv mv vτ m q + q v = exp q + q + erfc | f | | 0| + √τ ~2 ~2 f 0 ~ ~ ~ −2 " | | | | 2 2 √τ h i r 12αm2v2 4αm3v3( q + q ) 3αm3v4 1+ + | f | | 0| + τ × ~2 ~4 ~5 2 2~ m( qf + q0 ) + α exp | |~ | | rπm − 2 τ 3 3 2 3m v 1/2 m v mv( qf + q0 ) 1/2 τ 9+ | | | | τ − × − ~4 − ~ ~2 h 3 3 2 mv( qf + q0 ) 3/2 m ( qf + q0 ) 5/2 +m ( q + q ) 7+ | | | | τ − | | | | τ − . f 0 ~2 ~ | | | | − # i 6 The Kernel for Hˆ can be explicitly derived from Eq. (27) by changing τ iT . 1 → If α =0, ∆G[qf , q0 : τ] reduces to mv mv vτ m q + q v ∆G[q , q : τ] = exp q + q + erfc | f | | 0| + √τ f 0 −2~2 ~2 | f | | 0| 2~ 2~ √τ ~ h i r mv ∞ mv z = dze− ~2 G [ q , q z ; τ] , (28) − ~2 F | f | −| 0|−| | Z0 where GF [qf , q0 : τ] is a Brownian propagator for free particle case in usual quantum mechanics, whose explicit form is

− 2 m m(qf q0) G [q , q : τ]= e− 2~τ . (29) F f 0 2π~τ r Eq. (28) exactly coincides with the result of Ref. [11]. In order to explore the equivalence of Schrödinger and path-integral approaches we first d 2 z2 examine the boundary condition at the origin. Using Eq. (27) and erfc(z) = e− , dz − √π one can show straightforwardly that within (α), G[q , q : τ] satisfies O f 0 3 3 ∂ + ∂ 2 ∂ + ∂ G[0 , q : τ] G[0−, q : τ] 4α~ G[0 , q : τ] G[0−, q : τ] ∂q 0 − ∂q 0 − ∂q3 0 − ∂q3 0 f f ( f f ) 2mv = G[0, q : τ]. (30) ~2 0

Since this is different from Eq. (18) at the first order of α, one can say that the Schrödinger and path-integral approaches are inequivalent at the same order of α. Of course, two boundary conditions are exactly the same when α = 0.

The diﬀerence also arises from the energy-dependent Green’s function Gˆ[qf , q0 : ǫ]. Since the pole and residue of the energy-dependent Green’s function have an information about the bound state energy and the corresponding bound state, Eq. (23) implies that if v < 0,

Hˆ1 generates a single bound state energy B′, whose explicit form is mv2 3αm3v4 B′ ~ǫ = . (31) ≡ − − 2~2 − ~4 This is also diﬀerent1 from second equation of Eq. (19) at the order of α. The corresponding normalized bound state is a′ q ΦB(q)= √a′e− | |, (32)

1 Since Eq. (23) is valid only within (α), one can change the denominator of ∆Gˆ[qf , q0 : ǫ], and hence O its pole. In this case, however, there are multiple bound states, which is also diﬀerent from the result of Schr¨odinger approach.

7 2 3 3 4 where a′ = mv/~ 4αm v /~ . This is also different from bound state derived from the − − Schrödinger equation at the first order of α. It is straightforward to show that ΦB(q) does not obey the boundary condition (18) but obeys (30). It is shown that the Schrödinger equation and Feynman path-integral do predict different results in the non-relativistic quantum mechanics with GUP when the potential is one-dimensional δ-function. This difference may be due to the singular nature of potential. However, one-dimensional δ function potential is well defined in the usual quantum − mechanics. Thus, the equivalence of Schrödinger and Feynman’s path-integral approaches should be checked in more detail when the GUP is involved in the non-relativistic quantum mechanics.

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