Classical Trajectories in Rindler Space and Restricted Structure of Phase Space
aa r X i v : . [ g r- q c ] A ug Classical Trajectories in Rindler Space and Restricted Structure of Phase Space withNon-Hermitian
P T -Symmetric Hamiltonian
Soma Mitra a, and Somenath Chakrabarty a, a Department of Physics, Visva-Bharati,Santiniketan 731235, India [email protected] [email protected] The nature of single particle classical phase space trajectories in Rindler space with non-hermitian
P T -symmetric Hamiltonian have been studied both in the relativistic as well as in the non-relativisticscenarios. It has been shown that in the relativistic scenario, both positional coordinates and thecorresponding canonical momenta are real in nature and diverges with time. Whereas the phasespace trajectories are a set of hyperbolas in Rindler space. On the other hand in the non-relativisticapproximation the spatial coordinates are complex in nature, whereas the corresponding canonicalmomenta of the particle are purely imaginary. In this case the phase space trajectories are quitesimple in nature. But the spatial coordinates are restricted in the negative region only.
PACS numbers: 03.65.Ge,03.65.Pm,03.30.+p,04.20.-q
I. INTRODUCTION
Exactly like the Lorentz transformations of space time coordinates in the inertial frame [1, 2], the Rindler coordinatetransformations are for the uniformly accelerated frame of references [3–9]. From the references [3–9], it can veryeasily be shown that the Rindler coordinate transformations are given by: ct = (cid:18) c α + x ′ (cid:19) sinh (cid:18) αt ′ c (cid:19) and x = (cid:18) c α + x ′ (cid:19) cosh (cid:18) αt ′ c (cid:19) (1)Hence it is a matter of simple algebra to prove that the inverse transformations are given by: ct ′ = c α ln (cid:18) x + ctx − ct (cid:19) and x ′ = ( x − ( ct ) ) / − c α (2)Here α indicates the uniform acceleration of the frame. Hence it can very easily be shown from eqns.(1) and (2) thatthe square of the four-line element changes from ds = d ( ct ) − dx − dy − dz to ds = (cid:18) αx ′ c (cid:19) d ( ct ′ ) − dx ′ − dy ′ − dz ′ (3)where the former line element is in the Minkowski space. Hence the metric in the Rindler space can be written as g µν = diag (cid:18)(cid:16) αxc (cid:17) , − , − , − (cid:19) (4)whereas in the Minkowski space-time we have the usual form g µν = diag(+1 , − , − , −
1) (5)It is therefore quite obvious that the Rindler space is also flat. The only difference from the Minkowski space is thatthe frame of the observer is moving with uniform acceleration. It has been noticed from the literature survey, thatthe principle of equivalence plays an important role in obtaining the Rindler coordinates in the uniformly acceleratedframe of reference. According to this principle an accelerated frame in absence of gravity is equivalent to a frameat rest in presence of a gravity. Therefore in the present scenario, α may be treated to be the strength of constantgravitational field for a frame at rest.Now from the relativistic dynamics of special theory of relativity [1], the action integral is given by S = − α c Z ba ds ≡ Z ba Ldt (6)where α = − m c [1] and m is the rest mass of the particle and c is the speed of light in vacuum. The Lagrangianof the particle may be written as L = − m c (cid:20)(cid:16) αxc (cid:17) − v c (cid:21) / (7)where ~v is the three velocity vector. Hence the three momentum of the particle is given by ~p = ∂L∂~v , or (8) ~p = m ~v h(cid:0) αxc (cid:1) − v c i / (9)Then from the definition, the Hamiltonian of the particle may be written as H = ~p.~v − L or (10) H = m c (cid:16) αxc (cid:17) (cid:18) p m c (cid:19) / (11)Hence it can very easily be shown that in the non-relativistic approximation, the Hamiltonian is given by H = (cid:16) αxc (cid:17) (cid:18) p m + m c (cid:19) (11 a )In the classical level, the quantities H , x and p are treated as dynamical variables. Further, it can very easily be verifiedthat in the quantum mechanical scenario where these quantities are considered to be operators, the Hamiltonian H is not hermitian. However the energy eigen spectrum for the Schr¨odinger equation has been observed to be real[10]. This is found to be solely because of the fact that H is P T -invariant. Now it is well know that
P xP − = − x , P pP − = − p , whereas, T pT − = − p and P αP − = − α but T αT − = α , therefore it is a matter of simple algebra toshow that P T H ( P T ) − = H P T = H . As has been shown by several authors [11] that if H is P T -invariant, then theenergy eigen values will be real. Here P and T are respectively the parity and the time reversal operators. Further ifthe Hamiltonian is P T symmetric, then H and P T should have common eigen states. In [10] we have noticed thatthe solution of the Schr¨odinger equation is obtained in terms of the variable u = 1 + αx/c , which is P T -symmetric.Hence any function, e.g., Whittaker function M k,µ ( u ) or Associated Laguerre function L nm ( u ), the solution of theSchr¨odinger equation are P T -symmetric. These polynomials are also the eigen functions of the operator
P T . Ofcourse with the replacement of hermiticity of the Hamiltonian with the
P T -symmetry, we have not discarded theimportant quantum mechanical key features of the system described by this Hamiltonian and also kept the canonicalquantization rule invariant, i.e.,
T iT − = − i . This point was also discussed in an elaborate manner in reference [11]and in some of the references cited there.In this article we have investigated the time evolution for both the space and the momentum coordinates of theparticle moving in Rindler space. We have considered both the relativistic and the non-relativistic form of the RindlerHamiltonian (eqns.(11) and (11a) respectively). Hence we shall also obtain the classical phase space trajectories forthe particle in the Rindler space. We have noticed that in the relativistic scenario, both the spatial and the momentumcoordinates are real in nature and diverge as t −→ ∞ . For both the variables the time dependencies are extremelysimple. Hence we have obtained classical trajectories p ( x ) by eliminating the time dependent part.However, in the non-relativistic approximation, the spatial coordinates are quite complex in nature, whereas themomentum coordinates are purely imaginary. Since the mathematical form of the phase space trajectories are quitecomplicated, we have obtained p ( x ) numerically in the non-relativistic scenario.In the first part of this article, we have considered the relativistic picture and obtained the phase space trajectories,whereas in the second part, the classical phase space structure is obtained for non-relativistic case. To the best of ourknowledge such studies have not been done before II. RELATIVISTIC PICTURE
The classical Hamilton’s equation of motion for the particle is given by [12]˙ x = [ H, x ] p.x and ˙ p = [ H, p ] p,x (12)where [ H, f ] p,x is the Poisson bracket and is defined by [12][ f, g ] p,x = ∂f∂p ∂g∂x − ∂f∂x ∂g∂p (13)In this case f = x or p . In eqn.(12) the dots indicate the derivative with respect to time. Now using the relativisticversion of Rindler Hamiltonian from eqn.(11), the explicit form of the equations of motion are given by˙ x = (cid:16) αxc (cid:17) pc ( p c + m c ) / and ˙ p = − αc ( p c + m c ) / (14)The parametric form of expressions for x and p represent the time evolution of spatial coordinate and the correspondingcanonical momentum. The analytical expressions for time evolution for both the quantities can be obtained afterintegrating these coupled equations and are given by x = c α [ C cosh( ωt − φ ) −
1] and p = − m c sinh( ωt − φ ) (15)where C and φ are the integration constants, which are real in nature and ω = α/c is the frequency defined for somekind of quanta in [10]. Hence eliminating the time coordinate, we can write (cid:16) αxc (cid:17) C − p m c = 1 (16)This is the mathematical form of the set of classical trajectories of the particle in the phase space. Or in other wards,these set of hyperbolas are the classical trajectories of the particle in the Rindler space. This is consistent with thehyperbolic motion of the particle in a uniformly accelerated frame. These set of hyperbolic equations can also bewritten as p = m c (cid:18) αxc (cid:19) (cid:16) xω c (cid:17) (17)It is quite obvious from the parametric form of the variation of x and p with time that both the quantities areunbound. This is also reflected from the nature of phase space trajectories as shown in fig.(1) for the scaled x and p . The scaling factors are α/c for x and ( m c ) − for p . For the sake of illustration, we have chosen the arbitraryconstant C = 1. In this figure we have also taken both the scaling factors identically equal to unity. Then obviouslyeqn.(16) reduces to ( x + 1) − p = 1We shall get the other set of trajectories by choosing different values for the scaling factors. It is obvious that in thiscase the centre of the hyperbola is at ( − , α , the centre −→ (0 , ,
0) and ( − , α large enough, both the vertices coincide at the centre (0 , α , these two curves touch each other at (0 , III. NON-RELATIVISTIC PICTURE
We next consider the non-relativistic form of Rindler Hamiltonian given by eqn.(11a). Now following eqn.(12), theequations of motion for the particle in Rindler space in the non-relativistic approximation are given by˙ x = (cid:16) αxc (cid:17) pm and ˙ p = − αc (cid:18) p m + m c (cid:19) (18)On integrating the second one we have p = i / m c cot (cid:18) / ωt + φ (cid:19) = ip I (19)The particle momentum is therefore purely imaginary in nature with its real part p R = 0. Here φ is a real constantphase. Next evaluating the first integral analytically, we have x = cω (cid:20) − (cid:26) ln (cid:18) sin (cid:18) / ωt − φ (cid:19)(cid:19)(cid:27)(cid:21) + i cω (cid:20) sin (cid:26) ln (cid:18) sin (cid:18) / ωt − φ (cid:19)(cid:19)(cid:27)(cid:21) = x R + ix I (20)The spatial part is therefore complex in nature, where the real part x R = cω (cid:20) − (cid:26) ln (cid:18) sin (cid:18) / ωt − φ (cid:19)(cid:19)(cid:27)(cid:21) (21)and the corresponding imaginary part is given by x I = cω (cid:20) sin (cid:26) ln (cid:18) sin (cid:18) / ωt − φ (cid:19)(cid:19)(cid:27)(cid:21) (22)Here again eliminating the time part, we have the mathematical form of phase space trajectories for the imaginaryparts only p I = 2 / m c (cid:2) − exp (cid:8) sin − (cid:0) ωc x I (cid:1)(cid:9)(cid:3) / exp (cid:8) sin − (cid:0) ωc x I (cid:1)(cid:9) (23)Which gives the phase space trajectories of the particle in the Rindler space in non-relativistic scenario. It should benoted here that since the real part of the particle momentum is zero, we have considered the imaginary parts only.Since p I is real, therefore | ωx I /c |≤
1, i.e., can not have all possible values.In fig.(2) we have plotted the scaled x R , i.e. ( ωx R /c ) with scaled time ( ωt/ / ) for φ = 0. Since the constant phase φ is completely arbitrary, for the sake of illustration we have chosen it to be zero. In this diagram the scaling factorsare also taken to be unity. Now if we consider variation of the scaling factors, the qualitative nature of the graphs willnot change but there will be quantitative changes. In fig.(3) we have plotted the scaled x I , i.e., ( ωx I /c ) with scaledtime ( ωt/ / ) for φ = 0. In this case also same type of changes as has been mentioned for x R will be observed. Infig.(4) we have plotted the scaled p I , which is actually ( p I / / m c ) with scaled time ( ωt/ / ) for φ = 0. In thiscase also the scaling factors are exactly equal to one. Further the same kind of variation as mentioned above will beobserved for p I with the change of scaling parameters. Finally in fig.(5) the phase space trajectory for scaled x I andscaled p I is shown Since the physically accepted domain for scaled x I is from − x I and scaled p I with scaled time. IV. CONCLUSION
Finally in conclusion we would like to mention that to the best of our knowledge this is the first time the phasespace trajectories are obtained in Rindler space using non-hermitian