Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring

Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring

M. Schaden Department of Physics, Rutgers, The State University of New Jersey, 101 Warren Street, Newark, New Jersey - 07102, USA.
July 14, 2019
Abstract

Effects due to vacuum fluctuations in a semi-classical model of a massless scalar field interacting with a rotating ring are investigated by introducing a collective coordinate for the motion of the background potential. The model is solved for a repulsive periodic -distribution background of arbitrary strength. The Casimir energy of this system is calculated in the co-rotating and, by Legendre transformation, in the stationary laboratory frame. The zero-point contribution to the angular momentum in this model is bounded below by and the ground state of the entire system thus generally is non-rotating with a positive moment of inertia that decreases only slightly with increasing angular rotation frequency. There is no transfer between the zero-point and classical contributions to the total angular momentum and energy of this system at zero temperature.

pacs:
03.70.+k,11.10.Ef,11.30.Qc,42.50.Lc

Recently Chernodub observed Chernodub (2012a, b) that zero-point fluctuations of a scalar field contribute negatively to the moment of inertia and speculated that under favorable circumstances the ground state of some macroscopic devices could be self-rotating and thus realize a quantum time-crystal Shapere and Wilczek (2012); Wilczek (2012). Here the simplest of such models Chernodub (2012a, b) is examined in greater detail. I extend the original model to admit less singular interactions and conserve total angular momentum and energy by introducing a collective coordinate Bohr and Mottelson (1969) that describes the rotation dynamically.

Collective coordinates were used by Lord Rayleigh to describe classical vibrations of a droplet. Bohr and Mottelson Bohr and Mottelson (1969); Mottelson (1976) introduced them in the context of nuclear physics to represent collective aspects of many-body quantum mechanics. This approach also describes the dynamics of soliton moduli spaces in field theory Sutcliffe (1993). However, these methods do not appear to have been systematically exploited in dynamic Casimir effects. They are capable of resolving long-standing issues in this field and provide a general and rigorous framework for handling vacuum effects arising from dynamical classical systems.

Let us for example consider the relatively simple and instructive example model in which a scalar field on a circle of radius interacts with an everywhere positive background field (potential) whose position on the circle is referenced by the collective coordinate . The Lagrangian for this model is,

 (1)

where is the moment of inertia for the collective coordinate . Eq.(1) may be interpreted as describing quantum fluctuations to quadratic order of a scalar field on a circle in the background of a classical soliton solution located at . This only omits to specify the originally highly nonlinear model is the soliton of. All Casimir systems could (and perhaps should) be interpreted in this manner Graham et al. (2002). The presence of a “classical” contribution to the rotational energy in this case is due to the motion of the soliton on the circle (or that of a ring) and is not at all surprising. The original Chernodub model of a scalar on a circle subject to a rotating Dirichlet boundary condition Chernodub (2012a) is the limit of this extended model for very large and a “thin-wall” soliton proportional to a periodic -distribution at strong coupling . The main qualitative modification to Chernodub’s original model thus is the presence of a dynamical collective coordinate giving the location and dynamics of the wall.

The extended Lagrangian of eq.(1) does not explicitly depend on time and conserves total angular momentum. We are interested in the lowest (vacuum) energy of this model for a given total angular momentum . For a quantum time crystal, the energy is minimal at .

Written in terms of the relative angle , the Lagrangian of eq.(1) reads,

 L(ϕ,˙ϕ,˙θ)=I2˙θ2+∮S1Rdσ12((˙ϕ−˙θϕ′)2−R−2ϕ′2−V(σ)ϕ2) , (2)

and is a cyclical coordinate. The canonical momenta are,

 π(σ,t) =˙ϕ(σ,t)−˙θϕ′(σ,t) (3a) ℓ =I˙θ−∮S1Rdσπ(σ,t)ϕ′(σ,t) , (3b)

where defined by eq.(3b) is the conserved total angular momentum conjugate to . The Hamiltonian in the ()tationary frame in these coordinates is obtained from eq.(2) as,

 Hs =−L+ℓ˙θ+∮S1Rdσπ(σ,t)˙ϕ(σ,t) =12I(ℓ+∮S1Rdσπϕ′)2 (4) +∮S1Rdσ12(π2+R−2ϕ′2+V(σ)ϕ2) .

The quartic term of is linearized by a Legendre transformation to the energy in the co-rotating frame. With the angular rotation frequency given as,

 Ω=∂Hs∂ℓ=1I(ℓ+∮S1Rdσπ(σ,t)ϕ′(σ,t)) (5)

we have that (see footnote on p.74 of Landau and Lifshitz (1980)),

 Hc(Ω) =Hs(ℓ(Ω))−ℓ(Ω)Ω (6) =−I2Ω2+∮S1Rdσ12(π2+Ω(πϕ′+ϕ′π)+R−2ϕ′2+V(σ)ϕ2),

is quadratic in fields and momenta. Canonical quantization of this model proceeds by promoting fields and momenta to operators with equal time commutator,

 [ϕ(σ,t),π(σ′,t)]=iRδper(σ−σ′) , (7)

where denotes the periodic -distribution. With the ordering given in eq.(6), is hermitian and Hamilton’s equations,

 ˙ϕ =δHcδπ=π+Ωϕ′ (8a) ˙π =−δHcδϕ=Ωπ′+R−2ϕ′′−V(σ)ϕ , (8b)

coincide with Heisenberg’s equations of motion for the operators. Solving for in eq.(8a) and inserting in eq.(8b) gives the separable equation of motion in the co-rotating frame,

 ¨ϕ−2Ω˙ϕ′+(Ω2−R−2)ϕ′′+V(σ)ϕ=0 , (9)

whose general solution is,

 ϕ(σ,t)=∞∑ωm≥0 ame−iωmtum(σ)+a†meiωmt¯um(σ) . (10)

Upon quantization, the coefficients and are interpreted as annihilation and creation operators of quanta in mode . Inserting the general solution of eq.(10) in eq.(8a), the momentum is expressed in terms of operators and as,

 π(σ,t)=˙ϕ−Ωϕ′ =∞∑ωm≥0 ame−iωmt(−iωmum−Ωu′m) +a†meiωmt(iωm¯um−Ω¯u′m) . (11)

eq.(9) implies that the mode functions satisfy the homogeneous ODE,

 (Ω2−R−2)u′′m+2iωmΩu′m+(V(σ)−ω2m)um=0 . (12)

of Bloch waves Bloch (1929). The frequencies at which eq.(12) has non-trivial periodic solutions are discrete. For a finite periodic potential , the frequencies and corresponding solutions of eq.(12) are determined by requiring that,

 (um(σ)u′m(σ))=limε→0+(um(σ+2π−ϵ)u′m(σ+2π−ϵ)) (13)

The frequency spectrum is real and the complex conjugate mode function is a solution of eq.(12) to frequency . It thus suffices in eq.(10) to sum over non-negative frequencies only. Eq.(12) is consistent with the normalization conditions,

 R∮S1dσ[(ωm+ωn)¯umun−2iΩ¯umu′n] =δmn R∮S1dσ[(ωn−ωm)umun−2iΩumu′n] =0, (14)

and their complex conjugates. Mode functions to different frequencies are orthogonal in this sense and eq.(Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring) can be satisfied when some frequencies happen to be degenerate. Using eq.(Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring) in eq.(10) and eq.(Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring) the annihilation operators are related to field operators as,

 an =R∮S1dσ(ϕ(σ,0)ωn+iπ(σ,0))¯un , a†n =R∮S1dσ(ϕ(σ,0)ωn−iπ(σ,0))¯un . (15)

Eq.(Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring), eq.(7) and eq.(Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring) imply the usual commutation relations,

 [am,an]=[a†m,a†n]=0 ;  [am,a†n]=δmn , (16)

of creation and annihilation operators. Inserting eq.(10) and eq.(Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring) in eq.(6) and using eq.(Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring), the Hamiltonian of the co-rotating system is seen to be diagonal,

 Hc(Ω)=12∑ωn≥0ωn(Ω,R)(a†nan+ana†n) , (17)

and the construction of the Fock-space is analogous to the non-rotating case: at any given angular frequency , the lowest energy state of the co-rotating system is annihilated by all and has the zero-point energy given by the formal sum,

 EZP(R,Ω)="12∑ωn≥0ωn(R,Ω)  " . (18)

Since we are interested only in the dependence of this energy on external parameters like the radius and angular rotation frequency , we may instead compute the finite111A finite single-particle Casimir energy can be defined only if a certain coefficient of the asymptotic heat kernel expansion of the differential operator vanishes – for scalar fields on this is the case and the Casimir energy is finite for any positive potential as well as for a Dirichlet condition Kirsten (2001). difference,

 Ec(R,Ω)=EZP(R,Ω)−EZP(∞,0) , (19)

which one may refer to as the Casimir energy of the co-rotating system. Various methods have been developed to extract the parameter-dependent part of the infinite zero-point energy. Here this is straightforward only if the intermediate regularization of eq.(18) respects the symmetries of the co-rotating system. Many regularizations, ranging from the insertion of an exponential cutoff in the zero-point sum of eq.(18) to generalized zeta-function regularization, to point-splitting, meet this criterion. However, in the latter regularization method, the point-splitting should be invariant under the time-translation symmetry of the co-rotating frame. This is not the same as time-splitting in the lab-frame. The point-splitting regularization otherwise explicitly breaks rotational invariance, which would have to be explicitly restored for the total angular momentum to be conserved.

We further restrict our considerations to the example of a periodic -distribution background . The previous considerations for finite potentials are readily adapted to this singular case. Eq.(12) and the boundary conditions of eq.(13) for a periodic -distribution potential become,

 0=(1−β2)u′′m−2iαmβu′m+α2mum (20a) with  um(0)=um(2π) (20b) and u′m(0)−u′m(2π)=λR21−β2um(0) , (20c)

where and are the dimensionless frequency and rotation speed. Eq.(20c) ensures that the discontinuity in the derivative of compensates for the singular potential in eq.(12). The mode function satisfying eq.(20) to the dimensionless frequency is of the form,

 um(σ)∝(1−e−2πiαm1+β)eiσαm1−β−(1−e2πiαm1−β)e−iσαm1+β , (21)

with a solution to the secular equation,

 sinπα1−βsinπα1+β=λR24αsin2πα1−β2 . (22)

Note that for the mode function in eq.(21), satisfies the Dirichlet condition .

Using the generalized argument principle Kampen et al. (1968), eq.(22) gives the Casimir energy in the co-rotating frame of a scalar field interacting with a rotating ring by a periodic -distribution potential of strength for any222The generalized -function techniques of Kirsten (2001) allow one to numerically obtain this energy for any well-behaved potential . radius and angular rotation frequency as the finite integral,

 Ec(R,β=ΩR/c,λ) = (23) =ℏc2πR∫∞0dζln(1−4ζcosh(2πζβ1−β2)+(λR2−2ζ)e−2πζ1−β2(2ζ+λR2)e2πζ1−β2) .

One can perform the integral analytically in the limits of vanishing and very strong coupling: and . For vanishing interaction strength, the frequencies solving eq.(22) of left and right-moving modes are Doppler-shifted by factors . Their sum and thus the Casimir energy of the co-rotating frame do not depend on . In the limit of very strong interaction strength on the other hand, the spectrum of frequencies solving eq.(22) is and the dependence of the Casimir energy on is quadratic in the co-rotating frame,

 E∞c=−ℏc48R(1−β2)−I2Ω2=−ℏc48R+ℏRΩ248c−I2Ω2 . (24)

The inverse Legendre transform of in eq.(24) gives the dependence of the ground state energy on the total angular momentum,

 ℓ(λ∼∞)=−∂Ec∂Ω∣∣∣λ→∞=(I−ℏ24cR)Ω , (25)

as,

 Es =Ec(Ω)+ℓΩ (26) −−−−⟶λ→∞−ℏc48R+ℓ22(I−ℏ24cR)=−ℏc48R−ℏR48cΩ2+I2Ω2 .

Apart from the classical contribution proportional to , eq.(26) reproduces the zero-point energy of a scalar field with rotating Dirichlet boundary conditions obtained in Chernodub (2012a). Although the computation of Chernodub (2012a) in the stationary frame for general potentials does not conserve energy and angular momentum, our results do agree for a rotating Dirichlet boundary conditions. As Chernodub pointed out Chernodub (2012a), eq.(26) implies that zero-point fluctuations of a scalar field reduce the moment of inertia of the device. However, the classical contribution in general is not negligible and the semi-classical treatment of this system becomes questionable when the zero-point contribution to the total moment of inertia is larger in magnitude than the classical one. We argue below that this in fact never occurs.

The integral representation of eq.(23) for the Casimir energy with a -distribution potential in the co-rotating frame gives an equally explicit expression for the zero-point contribution to the total angular momentum ,

 ℓZP(β,λ) =−∂Ec∂Ω−IΩ=−ℏ∫∞0ξdξ× (27) ×ξ(1−β2)((1+β)2e−2πξβ−(1−β)2e2πξβ−4βe−2πξ)+2βλR2e−2πξλR2sinh(2πξ)+4ξ(1−β2)sinh(πξ(1+β))sinh(πξ(1−β)) .

Although the zero-point contribution to the moment of inertia is too lengthy to be reproduced here, one can deduce from eq.(27) that is always negative and a monotonically decreasing function of the angular rotation speed (see fig. 1). It thus is bounded below by its value when the ring is rotating at the speed of light. From eq.(27) one then finds,

 0 >IZP(R,β2,λ)>IZP(R,1,λ)= (28) =−ℏR24c(1−∫∞06λ(R/c)2ξ2dξe−πξ(λ(R/c)2+2ξ−2ξe−πξ)2)≥−ℏR24c ,

for any coupling .

Somewhat unexpected perhaps, the zero-point contribution to the total angular momentum in eq.(27) also is bounded in the accessible region by,

 (29)

for any coupling and radius .

Since the smallest observable change in the total angular momentum in this quantum system is , the upper bound of eq.(29) together with eq.(27) imply,

 IΔΩΔℓ=1−ΔℓZPΔℓ≥(1−1/24)>0 . (30)

We therefore have that either or that the moment of inertia of the collective coordinate is itself negative. Since the latter case would contradict the model assumptions, we arrive at the conclusion that the total effective moment of inertia of the simplest Chernodub-device is always positive and its ground state is non-rotating once the classical contribution to the total angular momentum of the device is included. This classical contribution is necessary to conserve the total angular momentum of this semi-classical model.

Noting that only the total moment of inertia of the entire device can be measured and not the contribution from quantum fluctuations alone, one can always renormalize and decompose the total moment of inertia of the device as,

 Itot(β2)=I+IZP(β2)=Itot(β20)+(IZP(β2)−IZP(β20)) , (31)

where is a reference rotation speed. If we choose , the second term in eq.(31) is positive for all due to the lower bound of eq.(28). Whether or not the total moment of inertia of the device is negative therefore depends exclusively on phenomenological input and can only be determined by a measurement. The renormalized form of eq.(31) pays tribute to the fact that the quantum fluctuations are not the whole story and also makes sense when but differences remain finite. Note that the negative contribution from quantum fluctuations in this model is irrelevant in eq.(31). The conclusion could be different only if quantum corrections to the total moment of inertia were unbounded below– in this case invariably turns negative for some value of and measuring determines only at which rotation speed this occurs.

This simple and transparent model thus demonstrates that a negative zero-point contribution to the moment of inertia does not imply that the ground state of the complete quantum system could be self-rotating – much as negative contributions to the mass from quantum fluctuations do not imply the existence of tachyons.

Note further that due to the relation in eq.(5), a self-rotating ground state would imply that at some finite value . Assuming that as well, would have to be multi-valued at . This is not the case for a quadratic Hamiltonian such as with a unique ground state.

The collective coordinate allows one to relate the Casimir energy in the stationary system to the one in the co-rotating frame by a Legendre transformation. Since of eq.(6) and the total angular momentum are commuting hermitian operators, one therefore can conclude that the Casimir energies of the co-rotating and stationary frame are both real and that total angular momentum is conserved. No vacuum friction slows the rotation of this device. There is no transfer of angular momentum between the zero-point and classical contributions to the total angular momentum of this device.

I would like to thank M. Chernodub for extensive discussions clarifying his model and for drawing my attention to the footnote inLandau and Lifshitz (1980). This work was partly supported by the National Science Foundation with Grant no. PHY0902054.

References

• Chernodub (2012a) M.N. Chernodub, “Permanently rotating devices: extracting rotation from quantum vacuum fluctuations?”  (2012a), arXiv:1203.6588 [quant-ph] .
• Chernodub (2012b) M.N. Chernodub, “Rotating Casimir systems: magnetic-field-enhanced perpetual motion, possible realization in doped nanotubes, and laws of thermodynamics,”  (2012b), arXiv:1207.3052 [quant-ph] .
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• Landau and Lifshitz (1980) L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Oxford, Pergamon, 1980).
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