Physics Letters B 681 (2009) 96–99

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Physics Letters B

www.elsevier.com/locate/physletb

Noncommutative quantum mechanics of a test particleunder linearized gravitational waves

Anirban Saha a,b, Sunandan Gangopadhyay a,c,∗a Department of Physics and Astrophysics, West Bengal State University, Barasat, North 24 Paraganas, West Bengal, Indiab Visiting Associate in Inter University Centre for Astronomy and Astrophysics, Pune, Indiac Visiting Associate in S.N. Bose National Centre for Basic Sciences, Kolkata, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 August 2009Received in revised form 18 September2009Accepted 24 September 2009Available online 29 September 2009Editor: T. Yanagida

PACS:11.10.Nx03.65.Ta11.10.Ef04.30.Nk42.50.Dv

We consider the quantum dynamics of a test particle in noncommutative space under the influence oflinearized gravitational waves in the long wave-length and low-velocity limit. A prescription for quan-tizing the classical Hamiltonian for the interaction of gravitational wave with matter in noncommutativespace is proposed. The Hamiltonian (and hence the system) is then exactly solved by using standardalgebraic methods. The solutions show prominent signatures of the noncommutative nature of space.Computation of the expectation value of the particle’s position reveals the inherent quantum nature ofspacetime noncommutativity.

© 2009 Elsevier B.V. All rights reserved.

Gravitational waves (GWs) are tiny disturbances in spacetime.Their effect on matter is expected to be measured by the relativeoptical phase shift between the light paths in two perpendicularkm-length arm cavities caused by the tiny displacement of two testmass mirrors (hung at the end of each cavity) induced by passingGWs. The typical amplitude of GWs h ∼ δL

L emitted by binary sys-tems in the VIRGO cluster of galaxies, at a distance ∼20 Mpc, at100 Hz which the ground based Laser-interferometric GW detec-tors like LIGO [1], VIRGO [2], GEO [3] TAMA [4], etc., are designedto probe is ∼10−21. Since the cavity arm length L ∼ 1 km, δL willbe ∼10−18 m. It is, therefore, at the quantum mechanical level thatexperimental evidence for the GWs is likely to appear [5].

Interestingly, in recent developments of noncommutative (NC)quantum mechanics and NC quantum field theory, where the coor-dinates xμ satisfy the NC algebra[xμ, xν

] = iθμν (1)

the upperbounds on various NC parameters appearing in the lit-erature [6–12] are quite close to this length scale. A wide rangeof theories have been constructed in a NC framework including

* Corresponding author at: S.N. Bose National Centre for Basic Sciences, Kolkata,India.

E-mail addresses: anirban@iucaa.ernet.in (A. Saha),sunandan.gangopadhyay@gmail.com, sunandan@bose.res.in (S. Gangopadhyay).

0370-2693/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2009.09.063

various gauge theories [13], gravity [14] and even encompassingcertain possible phenomenological consequences [15–19]. The up-perbound on the value of the coordinate commutator θ i j foundin [6] is �(10 TeV)−2 which corresponds to 4 × 10−40 m2 forh = c = 1. Whereas such upperbounds on timespace NC parameterθ0i is �9.51 × 10−18 m2. However, recent studies in NC quantummechanics revealed that the NC parameter associated with differ-ent particles are not same [21] and this bound could be as high asθ � (4 GeV)−2–(30 MeV)−2 [10]. These upperbounds correspondto the length scale ∼10−20–10−17 m.

With the prospect of the direct detection of GWs of such tinyamplitude as ∼10−18 in the near future, a sharp possibility ofdetecting the NC structure of spacetime would be in the GW de-tection experiments. Therefore, analyzing the interplay of classicalGWs with a test particle in a NC quantum mechanical frameworkbecomes important in its own right as it may predict some de-tectable signature of noncommutativity.

In light of these observations, we would like to construct thequantum mechanics of a test particle in NC spacetime, interact-ing with a linearized GW in the long wave-length and low velocitylimit. Since it has been demonstrated in various formulations ofNC general relativity [14,20] that any NC correction in the grav-ity sector is second order in the NC parameter, therefore, in afirst order theory in NC space, the linearized GW remains unal-tered by NC effects and any NC correction appearing in the system

A. Saha, S. Gangopadhyay / Physics Letters B 681 (2009) 96–99 97

will be through the particle sector only. This is true not only withthe canonical (i.e. constant) form of noncommutativity but also forthe Lie-algebraic NC spacetime [20]. We shall incorporate the NCeffect in the problem by writing the NC Hamiltonian and then re-expressing it in terms of the commutative coordinates and theirmomenta by the well-known Bopp-shift transformations [10,12].Then we shall quantize the system following [22]. As a first steptowards the formulation of a NC quantum mechanics of GW inter-acting with a test particle, we shall concentrate on the simplest ofthe GWs, namely the linearly polarized ones in this analysis.

In a linearized theory of gravity, once we choose the transverse-traceless (TT) gauge, all the gauge redundancies of the theory getremoved and the GW is characterised by the only non-zero com-ponents [23] h11 = −h22, and h12 = h21, called the + and × po-larizations, respectively. The only non-trivial components of thecurvature tensor in TT-gauge are1

R j0,k0 = −∂Γ j

0k

∂t= −1

2

∂2h jk

∂t2(2)

and the geodesic deviation equation in the proper detector framebecomes [23]

md2x j

dt2= −mR j

0,k0xk (3)

which governs the response of a scalar spin-zero test particle tothe passage of a GW. Here x j is the proper distance of the test par-ticle from the origin and m is its mass. For convenience, we haveassumed that GW detectors can be reasonably isolated so that anyexternal forces other than the GW interaction are negligible, i.e.the particles are considered to be free otherwise. Also, the GW istreated as an external classical field. Eq. (3) can be used to describethe evolution of proper distance in TT-gauge frame as long as thespacial velocities involved are non-relativistic. Also, |x j| has to bemuch smaller than the typical scale over which the gravitationalfield changes substantially, i.e. the reduced wavelength λ

2π of GW.This situation is referred to as the small-velocity and long wavelengthlimit. Thus, with Eq. (3) we can analyze the interaction of GW witha detector2 which has a characteristic linear size L � λ

2π .The Lagrangian for the system, whose time evolution is de-

scribed by Eq. (3), can be written, up to a total derivative term[22] as

L = 1

2mx2 − mΓ j

0kx jxk. (4)

The canonical momentum corresponding to x j is p j = mx j −mΓ j

0kxk and the Hamiltonian becomes

H = 1

2m

(p j + mΓ j

0kxk)2. (5)

If we take GWs propagating along the z-axis, then due to the trans-verse nature of GWs, Γ j

0k has non-zero components only in thex–y plane and the particle undergoes free motion along the z-di-rection. Hence we can essentially focus on the planar motion ofthe particle.

1 As usual, Latin indices run from 1–3.2 Note that this condition is satisfied by resonant bar detectors and earth bound

interferometers but not by the proposed space-borne interferometers such as LISA[24] or by the Doppler tracking of spacecraft.

To “quantize” this system in the NC plane, we now replace x j

and p j in the above Hamiltonian by operators x j and p j satisfyingthe NC Heisenberg algebra3

[xi, p j] = ihδi j, [xi, x j] = iθεi j, [pi, p j] = 0. (6)

It is well known that this can be mapped to the standard (θ = 0)

Heisenberg algebra spanned by Xi and P j using [10,12]

xi = Xi − 1

2hθεi j P j, pi = Pi . (7)

Using the traceless property of the GW and rewriting the NC ver-sion of Eq. (5) in terms of the operators Xi and P j , we obtain

H = P j2

2m+ Γ j

0k X j Pk − θ

2hε jm Pm PkΓ

j0k. (8)

Observe that a direct coupling between the GW and spatial non-commutativity appears here. Since we are dealing with linearizedgravity, a term quadratic in Γ has also been neglected in Eq. (8).

It is to be noted that the above Hamiltonian can be obtainedfrom the following Lagrangian (which can be computed from theHamiltonian (8) by an inverse Legendre transformation)

L = m

2

[(X2

i − 2Γ j0i Xi X j

) + θm

hε jkΓ

j0i Xi Xk

]. (9)

This is reassuring since we do have both the Lagrangian and theHamiltonian of the problem in hand as in the commutative (θ = 0)case. Further, the above form of the Lagrangian can be obtainedfrom the noncommutative version of the Lagrangian (4) by makingthe following Bopp shift

xi = Xi − 1

2hθεi j P j, ˆxi = Xi (10)

and then putting Pi = mXi since both the quadratic terms in θ

and Γ can be neglected. This can be regarded as the Lagrangianversion of the Bopp shift (7) applied at the Hamiltonian level.

Defining raising and lowering operators

X j =(

h

2m

)1/2(a j + a†

j

), (11)

P j = −i

(hm

2

)1/2(a j − a†

j

)(12)

where is determined from the initial uncertainty in either theposition or the momentum of the particle, we write the Hamilto-nian (8) as

H = h

4

(2a†

ja j + 1 − a2j − a†

j

2) − ih

4h jk

(a jak − a†

ja†k

)+ mθ

8ε jmh jk

(amak − ama†

k + C.C.)

(13)

where C.C. means complex conjugate. Working in the Heisenbergrepresentation, the time evolution of a j(t) is given by

da j

dt= −i

2

(a j − a†

j

) + 1

2h jka†

k

+ imθ

8h(εl j hlk + εlkhlj)

(ak − a†

k

)(14)

3 We would like to mention that it is possible to shift the noncommutativity fromthe coordinates to the momenta leading to a dual description as shown in the liter-ature [25].

98 A. Saha, S. Gangopadhyay / Physics Letters B 681 (2009) 96–99

and that of a†j(t) is the C.C. of Eq. (14). Next, noting that the raising

and lowering operators must satisfy the commutation relations[a j(t),ak(t)

] = 0,[a j(t),a†

k(t)] = δ jk (15)

we write them in terms of a j(0) and a†j(0), the free operators at

t = 0, by the time-dependent Bogoliubov transformations

a j(t) = u jkak(0) + v jka†k(0),

a†j(t) = a†

k(0)ukj + ak(0)vkj (16)

where the bar denotes the C.C. and u jk and v jk are the generalizedBogoliubov coefficients. They are 2 × 2 complex matrices which,due to Eq. (15), must satisfy uv T = uT v , uu† − v v† = I , writtenin matrix form where T denotes transpose, † denotes conjugatetranspose and I is the identity matrix. Since a j(t = 0) = a j(0), u jkand v jk have the boundary conditions u jk(0) = I and v jk(0) = 0.Then, from Eq. (14) and its C.C. we get the equations of motions interms of ξ = u + v† and ζ = u − v†:

dξ jk

dt= −iζ jk + h jl

2ξlk + Θ jlζlk, (17)

dζ jk

dt= −1

2h jlζlk (18)

where Θ jl is the new term reflecting the interplay of noncommu-tativity with GW

Θ jl = imθ

4h(h jmεml − ε jmhml). (19)

Eqs. (17), (18) are difficult to solve analytically for general h jk .However, our goal, in the present Letter, is to investigate to whatextent spatial noncommutativity affects the interaction of GWs withspin-zero test particle in the simplest of settings. Therefore we shallsolve Eqs. (17), (18) for the special case of linearly polarized GWs.

In the two-dimensional plane, the GW, which is a 2 × 2 ma-trix h jk , is most conveniently written in terms of the Pauli spinmatrices as

h jk(t) = 2 f (t)(ε×σ 1

jk + ε+σ 3jk

) = 2 f (t)εAσ Ajk. (20)

Note that the index A runs from 1–3, however, no contributionfrom σ 2 is included. 2 f (t) is the amplitude of the GW whereasε×(t) and ε+(t) represent the two possible polarization states ofthe GW and satisfy the condition ε2× + ε2+ = 1 for all t . In case oflinearly polarized GWs however, the polarization states εA are in-dependent of time and f (t) is arbitrary. To set a suitable boundarycondition we shall assume that the GW hits the particle at t = 0so that

f (t) = 0, for t � 0. (21)

We now move on to solve Eq. (18) by noting that any 2 × 2 com-plex matrix can be written as a linear combination of the Paulispin matrices and identity matrix. Hence we make the ansatz:

ζ jk(t) = A(t)I jk + B(t)εAσ Ajk. (22)

Substituting for h jk and ζ jk from Eqs. (20) and (22) respectivelyin Eq. (18) and comparing the coefficients of I and εAσ A , we getfirst order differential equations for A and B which can be readilyintegrated to obtain

ζ jk = cosh[

f (t)]I jk − sinh

[f (t)

]εAσ A

jk (23)

once we have used the boundary conditions Eq. (21).Now that we have the solution for ζ , we can take a closer

look at the NC correction term in Eq. (19), before proceeding to

solve Eq. (17). Computation shows that the coupling between non-commutativity and GW not only deforms the effects of the twopolarization states ε×σ 1

jk and ε+σ 3jk present in the GW but also

generates a new term proportional to the Pauli spin matrix σ 2jk:

Θ jlζlk = mθ

hf (t)

{i A

(ε+σ 1 − ε×σ 3)

jk + Bσ 2jk

}. (24)

Hence, as a trial solution we take a more general form for ξ jk thataccommodates all the Pauli matrices along with the identity matrix

ξ jk(t) = C I jk + D1σ1jk + D2σ

2jk + D3σ

3jk. (25)

Substituting for h jk , ζ jk and ξ jk from Eqs. (20), (22) and (25) re-spectively in Eq. (17) and comparing the coefficients of I and σ A ,we get the following set of first order differential equations

C = −i A + f (D1ε× + D3ε+),

D1 = −iBε× + f

(Cε× − iD2ε+ + imθ

hAε+

),

D2 = i f (D1ε+ − D3ε×) + mθ

hf B,

D3 = −iBε+ + f

(Cε+ + iD2ε× − imθ

hAε×

). (26)

With the recombination

G = D1ε× + D3ε+, Q = D1ε+ − D3ε× (27)

we recast the set of Eqs. (26) in the following form

C = −i A + f G, G = −i B + f C, (28)

D2 = i f Q + mθ

hf B, Q = − f D2 + mθ

hi f A (29)

Integrating Eqs. (28) and (29) and imposing the boundary condi-tion Eq. (21) we get

C = cosh f (t) − i

2F+(t); G = − i

2F−(t);

D2 = −mθ

hf (t) sinh

[f (t)

];Q = − imθ

hf (t) cosh

[f (t)

];F±(t) =

[e f (t)

t∫0

dt′ e−2 f (t′) ± e− f (t)

t∫0

dt′ e2 f (t′)].

Inverting the set of equations in (27), we can express D1 and D3in terms of G and Q . Combining the expressions for C , D1, D2 andD3 we can write the solution for ξ using Eq. (25) which along withζ in Eq. (23), give ul and vl , where l stands for linear polarization:

ul(t) = I

[cosh f − i

4F+

]− i

4

(ε×σ 1 + ε+σ 3)F−

+(

mθ

2h

)f{

i(ε+σ 1 − ε×σ 3) cosh f + σ 2 sinh f

},

(30)

vl(t) = Ii

4F+ + (

ε×σ 1 + ε+σ 3)[sinh f + i

4F−

]

−(

mθ

2h

)f{

i(ε+σ 1 − ε×σ 3) cosh f + σ 2 sinh f

}.

(31)

A. Saha, S. Gangopadhyay / Physics Letters B 681 (2009) 96–99 99

The system has now been essentially solved for the linearly polar-ized GWs once one specifies the initial expectation values of theparticle position and momentum and also the initial uncertaintyin either the position or the momentum of the particle to deter-mine . For example, one can assume the initial wave-function4

of the particle to be a Gaussian wave-packet

ψ(x, y) = 1√π l0

e− 1

2l20[(x−x0)2+(y−y0)2]

ei�p.�r

h (32)

located at (x0, y0) and moving with momentum �p0 = (px0 , p y0 )

when the GW just hits the system at time t = 0. This will give

uncertainties in momentum and position as√

hm2 and

√h

2m ,respectively. Either can be used to compute . From the initialposition and momentum expectation values, i.e. 〈�r0〉 = (x0, y0) and〈�p0〉 = (px0 , p y0 ), we get the raising and lowering operator at time

t = 0. We then use Eqs. (16), (30) and (31) to find a j(t) and a†j(t)

at a general time. This in turn gives us the expectation value of theposition of the particle at any arbitrary time t

⟨x(t)

⟩ = cosh f (t)x0 + (ε+x0 + ε× y0) sinh f (t)

+ (F+ + ε+ F−)px0

2m

+ ε× F−p y0

2m+ θ px0

hε× f (t) cosh f (t)

+ θ p y0

h

[f (t) sinh f (t) − ε+ f (t) cosh f (t)

], (33)⟨

y(t)⟩ = cosh f (t)y0 + (ε×x0 − ε+ y0) sinh f (t)

+ (F+ − ε+ F−)p y0

2m+ ε× F−

px0

2m

− θ p y0

hε× f (t) cosh f (t)

− θ px0

h

[f (t) sinh f (t) + ε+ f (t) cosh f (t)

]. (34)

Interestingly, we find that the NC nature of spacetime affects onlythose particles which are in motion at time t = 0. Further, thepresence of 1

h factor in the NC correction even after the com-putation of the expectation value indicates that the NC effect isinherently quantum mechanical in nature. To make the connectionwith the well-known classical result, we first set f (t) = f0 ( f0 be-ing a small constant amplitude of GW) and y0 = p y0 = 0. FromEqs. (33) and (34), we can immediately see

⟨x(t)

⟩ = (x0 + px0t

m

)+ ε+ f0

(x0 − px0t

m

)+ θ px0

hε× f0, (35)

⟨y(t)

⟩ = ε× f0

(x0 − px0t

m

)− θ px0

h(ε+ + f0) f0. (36)

Now putting θ = 0 gives us the well-known classical result [26]in the low-velocity, long-wavelength limit. From the above expres-sions for the expectation values, we find that the NC effect for aparticle interacting with GW appears in the form of a product of θ

and f0 and hence it would be difficult to detect this effect as it is

4 x is the eigenvalue of the position operator X and �p is the eigenvalue of the

momentum operator �P and l0 =√

hm sets the length scale.

very small compared to the term showing the effect of the passingGW. This is due to the fact that noncommutativity has been in-troduced by Bopp-shifting the coordinates which couple with theΓ -term in the Hamiltonian and hence the NC correction in theHamiltonian (8) contains a Γ -term. However, the present exercisegives us a substantial idea of what to expect if the same computa-tional scheme is applied for a harmonic oscillator system. The NCeffect is expected to be more prominent in this case since alongwith the coupling term of GW with noncommutativity, there willalso be NC corrections coming from the harmonic oscillator poten-tial which may give rise to a NC effect of comparable magnitudewith GW interaction. Interaction of GW with a Hydrogen atom inthe NC setting will be more involved [10] and is yet another in-teresting case to study where a detectable NC effect should occur.Work in this direction is in progress and will be taken up in thesubsequent papers.

Acknowledgements

The authors would like to acknowledge the hospitality of IITKanpur where a considerable part of this work was completed. Theauthors would also like to thank the referee for useful comments.

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