Path integral action of a particle in $\kappa $-Minkowski spacetime

Path integral action of a particle in κ-Minkowski spacetime Ravikant Verma,1, ∗ Debabrata Ghorai,1, † and Sunandan Gangopadhyay1, ‡

arXiv:1802.00720v1 [physics.gen-ph] 1 Feb 2018

1 S. N. Bose National Centre for Basic Sciences, Block - JD, Sector - III, Salt Lake, Kolkata - 700098, India

In this letter, we derive the path integral action of a particle in κ-Minkowski spacetime. The equation of motion for an arbitrary potential due to the κ-deformation of the Minkowski spacetime is then obtained. The action contains a dissipative term which owes its origin to the κ-Minkowski deformation parameter a. We take the example of the harmonic oscillator and obtain the frequency of oscillations in the path integral approach as well as operator approach upto the first order in the deformation parameter a. For studying this, we start with the κ-deformed dispersion relation which is invariant under the undeformed κ-Poincar´ e algebra and take the non-relativistic limit of the κ-deformed dispersion relation to find the Hamiltonian. The propagator for the free particle in the κ-Minkowski spacetime is also computed explicitly. In the limit, a → 0, the commutative results are recovered.

The theory of general relativity describes the spacetime structure at the classical level and presents the notion of continuum spacetime. In contrast, quantum mechanics provides the behaviour of the physical system at the microscopic level, that is at very small length scales. These two profound theories of physics, which has changed the landscape of theoretical physics, have been very difficult to combine into a single theory. Attempts have been made in this direction to develop a unified theory and such theories are known as the quantum theories of gravity. Development of a consistent theory of quantum gravity is now one of the most active area of theoretical physics. Noncommutative geometry provides a possible way to capture the spacetime structure at the quantum level. The noncommutativity of spacetime was first introduced by Snyder in 1947 [1], with the aim of handling the UV divergences in quantum field theories. In this approach, coordinates of spacetime get quantized. This in turn leads to the modification of the symmetry of spacetime. One such deformed symmetry algebra that has attracted wide attention in the study of quantum gravity is the κPoincar´ e algebra and the associated spacetime is known as the κ-Minkowski spacetime [2]-[4]. This spacetime has emerged in the low energy limit of quantum gravity models [5] and is one of the examples of noncommutative spacetimes where the coordinates obey the following commutation relations 1 [ˆ xi , x ˆj ] = 0, [ˆ x0 , x ˆi ] = iaˆ xi , a = (1) κ where a is the deformation parameter and has the dimension of length. The coordinates xˆi and xˆ0 are mapped to functions of x0 , xi and ∂0 , ∂i (which are coordinates of the commutative spacetime and their derivatives)[6]. Such a realization is given by x ˆµ = xα Φµα(∂).

∗ † ‡

(2)

It is easy to see that coordinates obey [∂µ , x ˆν ] = Φµν (∂). This realization can map the function of noncommutative spacetime to the function of the commutative spacetime. This can be seen first by defining the constant function annihilated by ∂ as the vacuum |0i ≡ 1, one gets [7] F (ˆ xϕ )|0i = F (x).

(3)

In such class of realization, the form of the noncommutative coordinates x ˆi and xˆ0 define as[6] x ˆi = xi ϕ(A) xˆ0 = x0 ψ(A) + iaxi ∂i γ(A)

(4) (5)

where A = −ia∂0 . The motivation of this letter is to carry out a path integral formulation of quantum mechanics in κ-Minkowski spacetime which has so far been missing in the literature. Such a formulation has been carried out in noncommutative quantum mechanics [8] with a canonical structure of noncommutativity between coordinates, namely, [ˆ x, yˆ] = iθ. Our present investigation would therefore help us compare our results with those derived in [8] and would give an insight about the intricacies of quantum mechanics formulated in κ-Minkowski spacetime. The symmetry algebra of the κ-deformed spacetime known as the undeformed κ-Poincar´ e algebra reads [Mµν , Dλ ] = ηνλ Dµ − ηµλ Dν , [Dµ , Dν ] = 0

(6)

[Mµν , Mλρ ] = ηµρ Mνλ + ηνλ Mµρ − ηνρ Mµλ − ηµλ Mνρ . (7) The explicit form of the generators of the undeformed κ-Poincar´ e algebra are given as [7] Mij = xi ∂j − xj ∂i Mi0 = xi ∂0 ϕ

[email protected], [email protected] [email protected] [email protected],[email protected]

Di = ∂i

(8)

e2A − 1 1 γ 1 − x0 ∂i + iaxi ∇2 − iaxk ∂k ∂i 2A ϕ 2ϕ ϕ (9)

e−A sinh A e−A , D0 = ∂0 + ia∇2 2 . ϕ A 2ϕ

(10)

2 The twisted coproduct of the generators Mµν are given by △ϕ (Mij ) = Mij ⊗ I + I ⊗ Mij = △0 (Mij )

(11)

1 ⊗ Mij . (12) △ϕ (Mi0 ) = Mi0 ⊗ I + e ⊗ Mi0 + ia∂j ϕ(A) A

The undeformed κ-Poincar´ e algebra defined in eq.(s)(6,7) have the same form as the Poincar´ e algebra in commutative spacetime. But the generators of the algebra gets modified due to the κ-deformation of the spacetime. In this letter, we shall use this undeformed κ-Poincar´ e algebra. The Casimir of this undeformed κ-Poincar´ e algebra, Dµ Dµ reads Dµ Dµ = (1 +

a2 ). 4

(13)

The explicit form of the  operator is given by  = ∇2

e−A 2∂02 + (1 − cosh A). ϕ2 A2

(14)

In the κ-deformed Minkowski spacetime, the dispersion relation is obtained from the Casimir by µ

i

0

2

Pµ P = P0 P + Pi P = −m ,

Pµ = −iDµ

(15)

where the momenta P0 and Pi are the κ-deformed momenta. Explicitly, the κ-deformed dispersion relation is given as 4 a2

2

sinh

a2 + 4

"

 aE  2c

E

−

p2i

e−a c ϕ2 (a Ec ) E

 aE  e−a c 4 − p2i 2 E sinh2 2 a 2c ϕ (a c )

#2

= m2 c2 . (16)

In the limit a → 0, we get the dispersion relation in the commutative spacetime. With the above formalism in place, we now proceed to formulate path integral in the κ-deformed spacetime. To begin with, we take the non-relativistic limit of the κ-deformed dispersion relation given in eq.(16). After straight forward calculation for the choice of ϕ(A) = e−A , we obtain the following form of the Hamiltonian H=

p2 1 + acm p·p≡ 2m 2mef f

(17)

where mef f =

m . 1 + acm

(18)

This is the free particle Hamiltonian upto first order in the deformation parameter a and p is the momentum in

the commutative spacetime. Now from eq.(s)(4,5) and for the choice of ϕ(A) = e−A , we find x ˆ0 → x0 ,

x ˆi → xi e−A .

(19)

Using the eq.(s)(3,19), we obtain a x ˆ2i = x ˆi x ˆi → xi e−A xi e−A |0i = x2i + xi x˙ i . c

(20)

From the above relation, we get  1   21 p a ax˙ i 2 2 ˆi xˆi = xi + xi x˙ i = xi 1 + xˆi = x c cxi  api  ≈ xi + . (21) 2mc

Now the Hamiltonian (17) in the presence of an arbitrary potential V (ˆ xi ) can be written as    ap  1 + acm p2 + V x + . (22) H= 2m 2mc Using Taylor series expansion, we get   1 + acm ap ∂V (x) H= p2 + V (x) + . 2m 2mc ∂x

(23)

This is the Hamiltonian of the system upto first order in the deformation parameter a. Here note that V (x) is the potential of the system in the commutative spacetime. We can now write the Hamiltonian for the harmonic oscillator (V (x) = 12 mω 2 x2 ) upto first order in the deformation parameter a as   1 ap 1 + acm p2 + mω 2 x2 + ω 2 x H= 2m 2 2c  2 ap  1 p + mef f ω 2 x2 + x (1 + acm).(24) = 2mef f 2 2mc We shall now proceed to construct the path integral for this Hamiltonian written down in the commutative coordinates. For that we require the momentum eigenstate |pi satisfying p|pi = p|pi,

hp′ |pi = δ(p − p′ ).

(25)

The following completeness relations are also crucial for the construction of the path integral Z dp |pihp| = 1 (26) Z dx |x, tihx, t| = 1. (27) The kernel can be written as Z Z hxf tf |xi ti i = ... dx1 dx2 ...dxn hxf tf |xn tn i × hxn tn |xn−1 tn−1 i...hx1 t1 |xi ti i.

(28)

3 Now the propagator over the small segment in the path integral reads [9] hxj+1 tj+1 |xj tj i = hxj+1 |e−

iHτ ~

|xj i.

(29)

Using eq.(23) in the above equation, we get ( " Z i dp pj (xj+1 − xj ) exp hxj+1 tj+1 |xj tj i = 2π~ ~   1 + acm 2 a ∂V −τ pj − τ pj 2m 2mc ∂x !#) xj+1 + xj . (30) − τV 2 Substituting this in eq.(28), we obtain   n+1 2 m hxf tf |xi ti i = lim n→∞ ihτ (1 + acm) ( n Z Y n X iτ dxj exp × ~ j=1 j=0 !2 " xj+1 − xj a ∂V (x) m − 2(1 + acm) τ 2mc ∂x !#) xj+1 + xj −V . (31) 2 In the limit τ → 0, the above expression for the path integral representation of the propagator can be written as Z i hxf tf |xi ti i = N Dx e ~ S (32) where the action S is given by !2 # " Z a ∂V (x) m 1 x− ˙ −V (x) . (33) S = dt 2 1 + acm 2mc ∂x

The above equation gives the action of a particle in the κ-Minkowski spacetime. The form of the action differs from that derived in [8] where the action for the particle was non-local in form with the non-locality owing its origin to the noncommutative parameter θ appearing in the commutation relation [ˆ x, yˆ] = iθ.

The equation of motion following from this Lagrangian reads ! ! ! m 1 1 ax˙ ∂ ∂V (x) x ¨+ 1 + acm 2 1 + acm c ∂x ∂x +

which upto first order in the deformation parameter a simplifies to ! 1 ax˙ ∂V (x) m 2 L= x˙ − − V (x). (36) 2 1 + acm mc ∂x

(37)

It is interesting to note that a dissipative term arises in the equation of motion due to the κ-deformation of the spacetime. But this dissipative term vanishes in the commutative spacetime, that is in the limit a → 0. We now solve this equation of motion for two simple cases, namely, the free particle and the harmonic oscillator. For the free particle (V = 0), the equation of motion is x ¨ = 0 which is the same as the commutative case, that is the κ-deformation of the spacetime does not affect the equation of motion. But mass of the particle gets modified due to the κ-deformation of the spacetime. Now we calculate the propagator for the free particle in κ-Minkowski spacetime. For this, we start from eq.(31) and set V = 0 to get  n+1  2 m hxf tf |xi ti i = lim n→∞ ihτ (1 + acm) ( Z Y n im dxj exp 2~τ (1 + acm) j=1 ) n  2 X . (38) xj+1 − xj j=0

From straight forward calculation, we obtain the propagator for the free particle in κ-deformed spacetime as "

# 12

"

# imef f (xf − xi )2 exp . 2~ (tf − ti ) (39) The form of the propagator is the same as in commutative spacetime [9] but the mass of the particle does get affected due to the κ-deformation of the spacetime. For the harmonic oscillator, V = 12 mω 2 x2 . Using this potential in eq.(37), we find the equation of motion to be mef f hxf tf |xi ti i = ih(tf − ti )

x ¨+

(34)

From this action, we can write the Lagrangian of the particle to be !2 a ∂V (x) m 1 x˙ − L= − V (x) (35) 2 1 + acm 2mc ∂x

∂V (x) = 0. ∂x

2ω 2 x˙ + (1 + acm)ω 2 x = 0. 2c

(40)

The solution of the above equation reads x(t) = Be−γt sin(ω1 t + φ)

(41)

where aω 2 γ= , 4c

ω1 = ω

r

1 + acm −

and B is an arbitrary constant and  acm  ω ω1 = 1 + 2

a2 ω 2 16c2

(42)

(43)

4 is the modified frequency due to the κ-deformation of the spacetime upto first order in the deformation parameter a.

and noting that [ˆ a, a ˆ† ] = 1, we find the number operator † a ˆ a ˆ to be

We now proceed to obtain the frequency using the operator approach. The Hamiltonian (24) can be written as

(51)

H=

p2 1 + mef f ω 2 X ′2 . 2mef f 2

(44)

Here

1 p′2 + q ′2 − . a ˆ† a ˆ=  acm 2 2 1+ 2

Using this in eq.(47), we finally obtain  acm h † 1i H = ~ω 1 + a ˆ a ˆ+ 2 2 h 1i ′ † ˆ a ˆ+ ≡ ~ω a 2

(52)

where X ′2 = x2 +

ap x + acmx2 2mc

(45)

which in turn yields X′ = x +

ap acm + x 4mc 2

(46)

upto first order in the deformation parameter a. We can now rewrite eq.(44) as

H=

p ~mef f ω

where p′ = √

~ω ′2 (p + q ′2 ) 2

and q ′ = X ′

q

mef f ω . ~

(47)

From these,

we have the following commutation relations  acm  [X ′ , p] = i 1 + , 2

 acm  [q ′ , p′ ] = i 1 + . (48) 2

Introducing the annihilation and creation operators as 1 a ˆ= r  2 1+

 acm  ω′ = ω 1 + . 2

(53)

This matches exactly with the result that we have obtained from the path integral approach upto first order in the deformation parameter a. In this letter, we have carried out the path integral formulation of quantum mechanics in the κ-Minkowski spacetime. We have obtained the action of a particle for the generic potential in κ-Minkowski spacetime from the path integral representation of the propagator. From the action, we obtained the equation of the motion for the particle and observed that there is a dissipative term present in the equation of motion which owes its origin to the κ-deformed spacetime. This is one of the main finding in this letter. This term is absent in normal quantum mechanics. After that we studied the case of the free particle and the harmonic oscillator. In the free particle case, we computed the path integral propagator in the κ-deformed spacetime and noticed that only mass of the free particle gets affected due to the κ-deformation of the spacetime. We then obtained the frequency of the harmonic oscillator and found that it gets modified due to the κ-deformation of the spacetime. The result turns out to be the same as the κ-deformed frequency of the harmonic oscillator upto first order in the deformation parameter a computed using the operator approach.

′ ′  (q + ip )

(49)

Acknowledgments

′ ′  (q − ip )

(50)

DG would like to thank DST-INSPIRE for financial support. S.G. acknowledges the support by DST SERB under Start Up Research Grant (Young Scientist), File No.YSS/2014/000180. SG also acknowledges the support under the Visiting Associateship programme of IUCAA, Pune.

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acm 2

acm 2

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