Comment on "Path integral action of a particle with the generalized uncertainty principle and correspondence with noncommutativity"
aa r X i v : . [ qu a n t - ph ] F e b Comment on “Path integral action of a particle with thegeneralized uncertainty principle and correspondence withnoncommutativity”
DaeKil Park , ∗ and Eylee Jung , Department of Electronic Engineering,Kyungnam University, Changwon 631-701, Korea Department of Physics, Kyungnam University, Changwon 631-701, Korea
Abstract
Recently in [Phys. Rev. D 99 (2019) 104010] the non-relativistic Feynman propagator forharmonic oscillator system is presented when the generalized uncertainty principle is employed.In this short comment it is shown that the expression is incorrect. We also derive the correctexpression of it. ∗ [email protected] Q i , P j ] = i ~ ( δ ij + βδ ij P + 2 βP i P j ] , (1)where β is a GUP parameter, which has a dimension (momentum) − . The modified Heisen-berg algebra can be readily represented up to first order of β as Q i = q i , P i = p i (1 + β p ),where { p i , q i } satisfies the usual Heisenberg algebra [ q i , p j ] = i ~ δ ij .The authors of Ref.[1] have considered the harmonic oscillator system, whose Hamiltonianis ˆ H = 12 m P + 12 mω X = p m + βm p + 12 mω x + O ( β ) . (2)Without any explicit explanation they presented the following Feynman propagator (see Eq.(41) of Ref. [1]) of this system in a form: h q f , t f | q , t i (3)= r mω πi ~ sin ωT " iβ ~ mT − βm (cid:18) q f − q T (cid:19) − βm ~ ω T cot ωT + O ( β ) e i ~ S cl , where T = t f − t and S cl is a classical action. The only comment the authors presented isthat the ω → K F [ q f , t f : q , t ] = r m πi ~ T (cid:18) iβ ~ mT − βm ( q − q f ) T + O ( β ) (cid:19) (4) × exp " im ~ T ( q − q f ) ( − βm (cid:18) q − q f T (cid:19) ) , which is Feynman propagator for free particle. Since Eq. (3) is one of the main results ofRef. [1] and it can be used in other GUP-related issues, it is worthwhile to check the validityof Eq. (3) more carefully. Unfortunately, it is incorrect although it approaches to a correct ω → h q f , t f | q , t i = r mω πi ~ sin ωT (cid:2) βf ( q , q f : T ) + O ( β ) (cid:3) e i ~ ( S + βS ) , (5)2here S = mω ωT (cid:2) ( q + q f ) cos ωT − q q f (cid:3) (6) S = − m ω
32 sin ωT " { ωT + 8 sin 2 ωT + sin 4 ωT } ( q + q f ) − { ωT cos ωT + 11 sin ωT + 3 sin 3 ωT } q q f ( q + q f )+12 { ωT + 2 ωT cos 2 ωT + 5 sin 2 ωT } q q f f ( q , q f : T ) = 3 i ~ mω ωT (2 ωT + 5 sin ωT cos ωT + ωT cos 2 ωT ) − m ω ωT " ωT (cid:8) ωT ( q + q f ) − ωT ) q q f (cid:9) +10 sin ωT ( q + q f − q q f cos ωT ) − ωT ( q + q f ) . Of course, S + βS is a classical action. It is straightforward to show that the ω → K F [ q f , t f : q , t ].In order to show Eq. (5) explicitly we note that the Feynman propagator h q f , t f | q , t i can be derived from Schr¨odinger equation as h q f , t f | q , t i = X n ψ n ( q f ) ψ ∗ n ( q ) e − ( i/ ~ ) E n ( t f − t ) , (7)where ψ n ( q ) and E n are n th -order eigenfunction and eigenvalue of Schr¨odinger equation. TheSchr¨odinger equation for the harmonic oscillator system is given by (cid:20) − ~ m ∂ ∂x + β ~ m ∂ ∂x + 12 mω x + O ( β ) (cid:21) ψ n ( x ) = E n ψ n ( x ) . (8)If we treat the GUP term β ~ m ∂ ∂x as small perturbation, one can derive ψ n ( x ) and E n in aform: ψ n ( x ) (9)= φ n ( x ) + ( βm ~ ω ) (cid:20) (2 n + 3) p ( n + 1)( n + 2)4 φ n +2 ( x ) − (2 n − p n ( n − φ n − ( x )+ p n ( n − n − n − φ n − ( x ) − p ( n + 1)( n + 2)( n + 3)( n + 4)16 φ n +4 ( x ) (cid:21) + O ( β ) E n = (cid:18) n + 12 (cid:19) ~ ω (cid:20) n + 2 n + 1)2(2 n + 1) ( βm ~ ω ) (cid:21) + O ( β ) , n = 0 , , , · · · and φ n ( x ) = 1 √ n n ! (cid:16) mωπ ~ (cid:17) / H n (cid:18)r mω ~ x (cid:19) exp h − mω ~ x i . (10)In Eq. (10) H n ( z ) is a n th -order Hermite polynomial. We assume φ m ( z ) = 0 for m < h q f , t f | q , t i as h q f , t f | q , t i = J + ( βm ~ ω )( K + K ) + O ( β ) (11)where J = ∞ X n =0 φ n ( q f ) φ n ( q ) exp (cid:20) − i ~ (cid:18) n + 12 (cid:19) ~ ωT (cid:26) n + 2 n + 1)2(2 n + 1) ( βm ~ ω ) (cid:27)(cid:21) (12) K = " ∞ X n =0 (2 n + 3) p ( n + 1)( n + 2)4 [ φ n ( q f ) φ n +2 ( q ) + φ n ( q ) φ n +2 ( q f )] − ∞ X n =2 (2 n − p n ( n − φ n ( q f ) φ n − ( q ) + φ n ( q ) φ n − ( q f )] exp (cid:20) − i ~ (cid:18) n + 12 (cid:19) ~ ωT (cid:21) K = " ∞ X n =4 p n ( n − n − n − φ n ( q f ) φ n − ( q ) + φ n ( q ) φ n − ( q f )] − ∞ X n =0 p ( n + 1)( n + 2)( n + 3)( n + 4)16 [ φ n ( q f ) φ n +4 ( q ) + φ n ( q ) φ n +4 ( q f )] . × exp (cid:20) − i ~ (cid:18) n + 12 (cid:19) ~ ωT (cid:21) . Using the extended Mehler’s formula[3] ∞ X k =0 t k k ! H k + m ( x ) H k + n ( y ) = (1 − t ) − ( m + n +1) / exp (cid:20) txy − t ( x + y )1 − t (cid:21) (13) × min( m,n ) X k =0 k k ! mk nk t k H m − k (cid:18) x − ty √ − t (cid:19) H n − k (cid:18) y − tx √ − t (cid:19) , one can show J = r mωπ ~ e − i ωT (cid:20)(cid:18) − i βm ~ ω T ) (cid:19) + 3 i βm ~ ω T ) ∂ ∂µ + 32 ( βm ~ ω T ) ∂∂µ (cid:21) F ( µ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ = ωT K = − r mω πi ~ e − iωT sin ωT (cid:18) i ∂∂µ + 3 (cid:19) G ( µ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ = ωT , (14)where F ( µ ) = e i µ √ i sin µ exp (cid:20) imω ~ sin µ (cid:8) ( q + q f ) cos µ − q q f (cid:9)(cid:21) (15) G ( µ ) = √ ie iµ sin µ (cid:20) imω ~ sin µ (cid:8) ( q + q f ) cos µ − q q f (cid:9) + 1 (cid:21) F ( µ ) . J = r mω πi ~ sin ωT e i ~ S ˜ J K = r mω πi ~ sin ωT e i ~ S ˜ K , (16)where˜ J = 1 − iβmω T ~ sin ωT " − i ~ mω ( q + q f ) sin 2 ωT + m ω ( q + q f − q q f cos ωT ) +4 i ~ mω sin ωT (2 + cos 2 ωT ) q q f − ~ sin ωT (2 + cos 2 ωT ) (17)˜ K = − i ~ sin ωT " − m ω q q f ( q + q f )(3 + cos 2 ωT ) + 3 ~ (cos 3 ωT − cos ωT )+4 mω cos ωT (cid:8) mω ( q + 6 q q f + q f ) + 12 i ~ q q f sin ωT (cid:9) − i ~ mω ( q + q f )(5 sin ωT + sin 3 ωT ) . Using Eq. (13) again and H ( z ) = 16 z − z + 12, one can show again K = p mω πi ~ sin ωT e i ~ S ˜ K , where˜ K = − i cos ωT ~ sin ωT " m ω q q f − ~ (1 − cos 2 ωT ) + 2 mω (cid:26) mω cos 2 ωT ( q + q f )(18) − mωq q f ( q + q f ) cos ωT − i ~ sin ωT (cid:8) ( q + q f ) cos ωT − q q f (cid:9) (cid:27) . Inserting J , K , and K into Eq. (11), it is possible to show that the Feynman propagatorbecomes Eq. (5). [1] S. Gangopadhyay and S. Bhattacharyya, Path integral action of a particle with the general-ized uncertainty principle and correspondence with noncommutativity , Phys. Rev.
D 99 (2019)104010 [arXiv:1901.03411 (quant-ph)].[2] A. Kempf, G. Mangano, and R. B. Mann,
Hilbert Space Representation of the Minimal LengthUncertainty Relation , Phys. Rev.
D 52 (1995) 1108 [hep-th/9412167].[3] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev,
Integrals and Series vol. 2 , (Gordon andBreach Science Publishers, 1983, New York)., (Gordon andBreach Science Publishers, 1983, New York).