Path Integral Action of a Particle with the Generalized Uncertainty Principle and Correspondence with Noncommutativity

Path integral action of a particle with the generalized uncertainty principle and correspondence with noncommutativity

Sukanta Bhattacharyya∗, Sunandan Gangopadhyay Department of Physics, West Bengal State University ∗[email protected]

Introduction and Motivation • Therefore the Lagrangian of the particle moving in a potential V (q) is • The propagator for the harmonic oscillator therefore takes the form • The idea of an observer independent minimum length (Planck-length) given by this r " 2 implies a modification of the Heisenberg uncertainty principle (HUP) to hm 2 2 2 i mω 3βi~m 2 qf − q0 L = q˙ 1 − 2βm q˙ − V (q) . (7) hqf, tf|q0, t0i = 1 + − 6βm the generalized uncertainty principle (GUP). 2 2πi~ sin(ωT ) T T Free particle V (q) = 0 3 i • The possibility of the GUP has led to the investigation of various aspects 2 Sc − βm~ω T cot(ωT ) e~ . in theoretical physics like black hole thermodynamics, quantum gravity • For free particle, V (q)=0. The classical action reads 4 corrections in various quantum systems and so on. m 1 − 12βm2q˙2 q¨ = 0 =⇒ q¨ = 0 or 1 − 12βm2q˙2 = 0 . (8) • Ground state energy is given by • However the path integral formulation in the GUP framework would be • The upper bound on the velocity [2] is given by necessary to gain insight in our current understanding of quantum grav- 1 1 1 3 E0 = ~ω 1 + βm~ω . (14) ity phenomenology. q˙max = =⇒ p˙max = mq˙max = √ . (9) 2 2 p 2 2β Objective 2βm To derive the path integral representation and the classical action for • The classical action and the propagation kernel for the free particle in • The ground state energy depends on the mass of the particle. a particle in the presence of the GUP. the presence of the GUP take the forms • From an operatorial approach, the first order correction to the ground Methodology " 2# m qf − q0 state energy of the HO is S = (q − q )2 1 − 2βm2 . (10) • The standard form of GUP algebra [1] can be written as c 2T f 0 T 3 2 ˆ 2 2 [qî, pˆj] = i (δij + βδijpˆ + 2βpîpˆj). (1) hψ0|H1|ψ0i = βmw ~ (15) ~ r 2 2 4 m 3iβ~m 6βm (qf − q0) hqf, tf|q0, t0i = 1 + − • The position and momentum operators defined as ( first order in β) 2πi~T T T 2 2 Conclusion im 2 2qf −q0 2 (qf −q0) 1−2βm qˆ = qˆ , pˆ = pˆ (1 + βpˆ ) (2) 2~T T i 0i i 0i 0 × e . (11) • We have obtained the action of the particle in presence of GUP. We extend our analysis to second order in β. • The Hamiltonian of a non-relativistic particle (1-d) under V (qˆ) reads • The above form in NC plane reads [3] 2 pˆ β r " 2 2# • From the form of the action for free particle, we get an upper bound to Hˆ = 0 + pˆ4 + V (qˆ) + O(β2). (3) m imθ m θ ~xf − ~x0 2m m 0 (zf, tf|z0, t0) = 1 + − the velocity of the particle which depends on β and its mass m. 2πi~T 2~T 2~2 T • We obtain the propagation kernal for a particle in presence of GUP (apart • Calculate the propagator of a free particle and particle moving in a Har- m 2 from a constant factor) exp − (~xf − ~x0) . (12) monic Oscilltor potential. 2(i~T + mθ)  2 Z +∞ n n 2 • A surprising connection between the result of the free particle in the Y i X m qj+1 − qj • We also extend our analysis upto O(β ) in the GUP parameter β. hq , t |q , t i = dq exp τ GUP framework with the corresponding result in NC space. f f 0 0 j  2 τ Harmonic Oscillator V (q) = 1mω2q2 −∞ j=1 ~ j=0 2 ( ) #! • In this case the action of the particle reads • The ground state energy for HO depends on the mass of the particle and q − q 2 2 j+1 j T the GUP parameter. 1 − 2βm − V (qj) . (4) Z m 1 τ S = dt q˙2 1 − 2βm2q˙2 − mω2q2 (13) 0 2 2 • In the τ → 0 limit, the configuration space path integral representation • The classical action (Sc = Sc(β = 0) + Sc(β)) for the HO reads to be References Z 1 2 2 ˜ i S Sc(β = 0) = mω csc(ωT ) q0 + qf cos(ωT ) − 2q0qf hqf, tf|q0, t0i = N(T, β) Dq e~ (5) 2 [1] A. Kempf, G. Mangano, R. B. Mann, Phys. Rev. D 52 (1995) 1108.

1 3 3 4 4 4 • The action of the particle moving in the presence of a potential V (q) in Sc(β) = − βm ω csc (ωT ) (q + q ) sin(4ωT ) − 44q0qf [2] S. Das, S. Pramanik, Phys. Rev. D. 86 (2012) 085004. 32 0 f the conﬁguration space reads 2 2 2 2 q0 + qf sin(ωT ) − 12q0qf q0 + qf sin(3ωT ) [3] S. Gangopadhyay, F.G. Scholtz, Phys. Rev. Lett. 102 (2009) 241602. tf 2 2 2 2 Z hm i +24q q ωT cos(2ωT ) − 48q qfωT q + q cos(ωT ) S = dt q˙2 1 − 2βm2q˙2 − V (q) . (6) 0 f 0 0 f Published in: Phys.Rev.D 99 (2019) 10, 104010, Phys.Rev.D 101 +12ωT q4 + 4q2q2 + q4 + 4 2q4 + 15q2q2 + 2q4 sin(2ωT ) t0 2 0 0 f f 0 0 f f (2020) 6, 069903 (erratum)