Motion induced by asymmetric excitation of the quantum vacuum
Jeferson Danilo L. Silva, Alessandra N. Braga, Andreson L. C. Rego, Danilo T. Alves
aa r X i v : . [ qu a n t - ph ] S e p Motion induced by asymmetric excitation of the quantum vacuum
Jeferson Danilo L. Silva, ∗ Alessandra N. Braga, † Andreson L. C. Rego, ‡ and Danilo T. Alves § Campus Salinópolis, Universidade Federal do Pará, 68721-000, Salinópolis, Brazil Instituto de Estudos Costeiros, Universidade Federal do Pará, 68600-000, Bragança, Brazil Instituto de Aplicação Fernando Rodrigues da Silveira,Universidade do Estado do Rio de Janeiro, 20261-232, Rio de Janeiro, Brazil Faculdade de Física, Universidade Federal do Pará, 66075-110, Belém, Brazil (Dated: September 22, 2020)During the last fifty-one years, the effect of excitation of the quantum vacuum field induced by its couplingwith a moving object has been systematically studied. Here, we propose and investigate a somewhat invertedsetting: an object, initially at rest, whose motion becomes induced by an excitation of the quantum vacuumcaused by the object itself. In the present model, this excitation occurs asymmetrically on different sides of theobject by a variation in time of one of its characteristic parameters, which couple it with the quantum vacuumfield.
I. INTRODUCTION
In 1969, Gerald T. Moore published in his PhD thesis theprediction that a mirror in movement can excite the quan-tum vacuum, generating photons [1]. This effect is nowadaysknown as the dynamical Casimir effect (DCE) and was in-vestigated, during the 1970s, in other pioneering articles byDeWitt [2], Fulling and Davies [3, 4], Candelas and Deutsch[5], among others. Since then, many other authors have ded-icated to investigate the DCE (some excellent reviews on theDCE can be found in Ref. [6–9]).In his pioneering work, Moore remarked that “to practi-cal experimental situations, the creation of photons from thezero-point energy is altogether negligible” [1]. In an attemptto overcome this difficulty, several ingenious proposals havebeen made aiming to observe the particle creation from vac-uum by experiments based on the mechanical motion of amirror [10–15] , but this remains as a challenge [9]. How-ever, the particle creation from the vacuum occurs, in gen-eral, when a quantized field is submitted to time-dependentboundary conditions, with moving mirrors being a particu-lar case. Therefore, it is not necessary to move a mirror togenerate real particles from the vacuum. Within this moregeneral view, alternative ways to detect particle creation fromvacuum were inspired in the ideas of Yablonovitch [16] andLozovik et al. [17], which consist in exciting the vacuumfield by means of time-dependent boundary conditions im-posed to the field by a motionless mirror whose internal prop-erties rapidly vary in time. In this context, Wilson et al. [18]observed experimentally the particle creation from vacuum,using a time-dependent magnetic flux applied in a coplanarwaveguide (transmission line) with a superconducting quan-tum interference device (SQUID) at one of the extremities,changing the inductance of the SQUID, and thus yielding atime-dependent boundary condition [18]. Other experiments ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] have also been done [19–21], and other have been proposed[22–31].When one moves an object imposing changes in the vac-uum field, the latter offers resistance, extracting kinetic en-ergy from the object, which is converted into real particles.In this case, one can say that the net action of the vacuum isagainst the motion. Here, we propose and investigate a some-what inverted situation: an object, initially at rest, isolatedfrom everything and just interacting with the quantum vac-uum, whose motion becomes induced by an excitation of thevacuum field caused by the object itself. In other words, weare looking for a situation where the vacuum field acts in fa-vor of a motion in a preferred direction. With this in mind,we propose a model where an object imposes a change to thequantum field by the time variation of the properties of theobject. Resisting to this change, the vacuum field extracts en-ergy from the object, exciting the quantum field and convert-ing the energy into real particles. A fundamental point of themodel presented here is that, to get in motion in a preferreddirection, the object has to excite the quantum vacuum differ-ently on each side, which requires that an asymmetry must beintroduced in the object. Taking into account the same simpli-fied one-dimensional model considered in the pioneering ar-ticles on the DCE [1–5], we consider the coupling of a staticpoint object with a quantum real scalar field in (1+1)D via a δ − δ ′ potential, which simulates a partially reflecting objectwith asymmetric scattering properties on each side [32, 33].When the coupling parameters vary in time, this model simu-lates an object exciting asymmetrically the fluctuations of thequantum vacuum, which produces a non-null mean force act-ing on the object, so that it can get in motion. Then, instead ofagainst, the vacuum acts in favor of the motion in a preferreddirection. But, not completely in favor, since, once in motion,a dynamical Casimir force acts on the object, so that part ofits kinetic energy is extracted by the vacuum fluctuations andgoes to the field. II. THE INITIAL MODEL
We are interested in an object whose interaction with thefield is described by an asymmetric scattering matrix, intend-ing to excite asymmetrically the quantum vacuum fluctua-tions. The interaction between the object and the field de-scribed by a Dirac δ potential produces a (left-right) symmet-ric scattering matrix [34, 35]. Therefore, to generate an asym-metry, we also consider the presence of an odd δ ′ term in thedescription of the interaction, so that our starting point is thefollowing lagrangian for a real massless scalar field in (1+1)D: L = L − [ µ ( t ) δ ( x ) + λ δ ′ ( x )] φ ( t, x ) , (1)where L = [ ∂ t φ ( t, x )] − [ ∂ x φ ( t, x )] is the lagrangian forthe free field, and the real parameters λ and µ , together with δ and δ ′ functions, describe the coupling between the quan-tum field and a static object located at the point x = 0 .These coupling parameters are related to the properties of theobject. The parameter µ is a prescribed function of time: µ ( t ) = µ [1 + ǫf ( t )] , where µ ≥ is a constant, f ( t ) isan arbitrary function such that | f ( t ) | ≤ and ǫ ≪ . Inthis way, the parameter µ is a perturbation in time around thevalue µ . We consider that this time variation of µ occurs atthe expense of the internal energy of the object, with null heattransfer between the object and the environment. As we dis-cuss next, the point object described by this model is partiallyreflective, so that a modification in the parameter µ implies achange in the transmission and reflection coefficients. Here-after, we consider that c = ~ = 1 , tilde indicates the Fouriertransform, and the subscript + ( − ) indicates the right (left)side.The field equation for this model is ( ∂ t − ∂ x ) φ ( t, x ) +2 [ µ ( t ) δ ( x ) + λ δ ′ ( x )] φ ( t, x ) = 0 . It will be convenient tosplit the field as φ ( t, x ) = Θ( x ) φ + ( t, x ) + Θ( − x ) φ − ( t, x ) , where Θ( x ) is the Heaviside step function and the fields φ + and φ − are the sum of two freely counterpropagatingfields, namely φ +( − ) ( t, x ) = ϕ out ( in ) ( t − x ) + ψ in ( out ) ( t + x ) , where the labels “in” and “out” indicate, respectively,the incoming and outgoing fields with respect to the ob-ject (see Fig. 1.a). In terms of the Fourier transforms,we can write φ +( − ) ( t, x ) = R d ω π ˜ φ +( − ) ( ω, x ) e − iωt , where ˜ φ +( − ) ( ω, x ) = ˜ ϕ out ( in ) ( ω ) e iωx + ˜ ψ in ( out ) ( ω ) e − iωx . From thefield equation, we get the matching conditions ˜ φ ( ω, + ) =[(1+ λ ) / (1 − λ )] ˜ φ ( ω, − ) and ∂ x ˜ φ ( ω, + ) = [(1 − λ ) / (1+ λ )] ∂ x ˜ φ ( ω, − ) + [2 / (1 − λ )] R d ω ′ π ˜ µ ( ω − ω ′ ) ˜ φ ( ω ′ , − ) . Af-ter an algebraic manipulation in these equations, we obtain Φ out ( ω ) = S ( ω )Φ in ( ω ) + Z d ω ′ π δS ( ω, ω ′ )Φ in ( ω ′ )+ Z d ω ′ π Z d ω ′′ π δS ( ω, ω ′ , ω ′′ )Φ in ( ω ′′ ) , (2)with Φ out ( in ) ( ω ) = (cid:18) ˜ ϕ out ( in ) ( ω )˜ ψ out ( in ) ( ω ) (cid:19) , S ( ω ) = (cid:18) s + ( ω ) r + ( ω ) r − ( ω ) s − ( ω ) (cid:19) , where S ( ω ) is the scattering matrix, with s ± ( ω ) = ω (1 − λ ) / [ iµ + ω (1 + λ )] and r ± ( ω ) = − ( iµ ∓ ωλ ) / [ iµ + ω (1 + λ )] being the transmission and reflection coefficients,respectively. Notice that the change λ ↔ − λ leads to r + ( ω ) ↔ r − ( ω ) , i.e. the object shifts its properties from one side to the other. Moreover, S ( ω ) is analytic for Im ω > (as required by causality [36, 37]), unitary and real in the tem-poral domain. The terms δS ( ω, ω ′′ ) and δS ( ω, ω ′ , ω ′′ ) rep-resent the first-order and second-order corrections to S ( ω ) due to the time-dependence of µ via f ( t ) . They are givenby δS ( ω, ω ′ ) = ǫα ( ω, ω ′ ) S ( ω ′ ) and δS ( ω, ω ′ , ω ′′ ) = ǫ α ( ω, ω ′ ) α ( ω ′ , ω ′′ ) S ( ω ′′ ) , where α ( ω, ω ′ ) = − iµ ˜ f ( ω − ω ′ ) / [ iµ + ω (1 + λ )] and S ( ω ) = (cid:18) s + ( ω ) 1 + r + ( ω )1 + r − ( ω ) s − ( ω ) (cid:19) . (3)Particularly, µ → ∞ leads to the case of a perfectly reflect-ing object [ s ± ( ω ) → ] imposing to the field the Dirichletboundary condition in both sides, for which δS → and δS → , recovering the configuration of a perfectly reflect-ing object whose properties do not vary in time. On the otherhand, the limit λ → ( λ → − ) also leads to a perfectlyreflecting object, but imposing to the field the Dirichlet andRobin (Robin and Dirichlet) boundary conditions at the leftand right sides of the object, respectively. Figure 1. Illustration of the excitation of the quantum vacuum byan object fixed at x = 0 , whose asymmetry in its scattering matrix isrepresented by its two faces in gray and black. The dashed wavy linesrepresent the unperturbed “in” fields ϕ in ( t − x ) (left) and ψ in ( t + x ) (right). (a) Object for t < − τ ( τ > ), when µ ( t ) ≈ µ . The solid-gray and solid-dark wavy lines represent the unperturbed “out” fields ψ out ( t + x ) (left) and ϕ out ( t − x ) (right), respectively. (b) Object at aninstant t , with its parameter µ varying in time. The irregular parts ofthe solid wavy lines represent the perturbed parts of the “out” fields ψ out ( t + x ) and ϕ out ( t − x ) , respectively. (c) The object for t > τ ,when µ ( t ) ≈ µ , and we calculate the number of created particles(represented by the dark points). Note that we have more particlesproduced in the left side. III. SPECTRUM, ENERGY AND MOMENTUM
Let us consider the initial situation ( t < − τ ) ( τ > )when the characteristic parameters of the object are constant [ λ and µ ( t < − τ ) ≈ µ ] and the state of the field isthe quantum vacuum (see Fig. 1.a). At a certain instant − τ , the properties of the object start to vary [ µ → µ ( t ) ],changing the boundary conditions imposed to the field, ex-citing the fluctuations of the quantum vacuum in the interval − τ < t < τ (see Fig. 1.b). The final situation ( t > τ )is when the object recovers its constant characteristic param-eters [ λ and µ ( t > τ ) ≈ µ ] and real particles are cre-ated (see Fig. 1.c). The spectrum of created particles can becomputed by n ( ω ) = 2 ω Tr h in | Φ out ( − ω )Φ T out ( ω ) | in i [37].From Eq. (2), calculated at order up to O ( ǫ ) , we have that n ( ω ) = n + ( ω ) + n − ( ω ) , where n ± ( ω ) = ǫ π (1 ± λ ) (1 + λ ) Z ∞ d ω ′ η ( ω, ω ′ ) , (4)with η ( ω, ω ′ ) = Υ( ω )Υ( ω ′ ) | ˜ f ( ω + ω ′ ) | and Υ( ω ) = µ ω/ (cid:2) µ + ω (1 + λ ) (cid:3) . Therefore, we get n − ( ω ) =[(1 − λ ) / (1 + λ )] n + ( ω ) , which means that the spectrumfor one side of the object differs from the other one by afrequency-independent global factor. For λ > ( λ < ) n − ( ω ) is smaller (greater) than n + ( ω ) .The total number of created particles is given by N = R ∞ d ω n ( ω ) , and the number in each side of the objectis N ± = R ∞ d ω n ± ( ω ) , so that we can write N − =[(1 − λ ) / (1 + λ )] N + . Note that N is greater in the rightside of the object if λ > and smaller if λ < (this lattercase is illustrated in Fig. 1.c). Particularly, for the perfectlyreflecting cases where λ = 1 ( λ = − , we get N − = 0( N + = 0) , so that the particles are created only in one side ofthe object.The energy and momentum of the created particles in eachside are given, respectively, by E ± = R ∞ d ω ωn ± ( ω ) and P ± = ±E ± . The total energy E and momentum P are E = E + + E − and P = P + + P − . Specifically, P = 2 ǫ π λ (1 + λ ) Z ∞ d ω Z ∞ d ω ′ ωη ( ω, ω ′ ) , (5)which is negative for λ < and positive for λ > . Then, astatic object, initially fixed at x = 0 , with its properties vary-ing in time, can excite asymmetrically the fluctuations of thequantum vacuum, generating into the field a net momentum P 6 = 0 . For instance, for the perfectly reflecting case where λ = 1 ( λ = − , we obtain P − = 0 ( P + = 0) , so that mo-mentum is transferred to the field (by exciting it) just in oneof the sides of the object. This net momentum implies in a netforce acting on the object. IV. FORCE ON THE STATIC OBJECT
Let us now obtain the expression for the mean force act-ing on the object at x = 0 due to the field fluctuations(see Fig. 1.b). The components of the energy-momentumtensor for a scalar field in dimensions are given by T = T = [ ϕ ′ ( t − x )] + [ ψ ′ ( t + x )] ≡ E ( t, x ) , and T = T = [ ϕ ′ ( t − x )] − [ ψ ′ ( t + x )] ≡ P ( t, x ) , where E ( t, x ) and P ( t, x ) are the energy and momentum densi-ties respectively, and their mean values can be written as h E j ( t, x ) i = Tr[ ∂ t ∂ t ′ h Φ j ( t, x )Φ T j ( t ′ , x ) i ] t = t ′ and h P j ( t, x ) i = Tr [ diag (1 , − ∂ t ∂ t ′ h Φ j ( t, x )Φ T j ( t ′ , x ) i ] t = t ′ , where j = out , in. The force acting on the object due to the field fluc-tuations can be found as the difference between the radi-ation pressure ( T ), on the left and on the right sides ofthe object. Therefore, the mean force is given by F µ ( t ) = h P in ( t, − P out ( t, i and it can be written as F µ ( t ) ≈ F (1) µ ( t ) ǫ + F (2) µ ( t ) ǫ . (6)Taking its Fourier transform and considering Eq. (2), we ob-tain ˜ F µ ( ω ) ≈ ˜ F (1) µ ( ω ) ǫ + ˜ F (2) µ ( ω ) ǫ , where the mean valuewas taken considering a vacuum as the initial state of the field.The first-order term is ˜ F (1) µ ( ω ) = χ ( ω ) ˜ f ( ω ) where χ ( ω ) = 2 λ ω πρ ( λ + 1) (cid:26) ρ + iρ + 2 i [2 i arctan ρ − ln( ρ +1)] − iρ (cid:27) , and ρ = ( λ + 1) ω/µ . The second-order term is ˜ F (2) µ ( ω ) = R d ω ′ R d ω ′′ χ ( ω, ω ′ , ω ′′ ) α ( ω ′ , − ω ′′ ) α ( ω − ω ′′ , ω ′ ) where χ ( ω, ω ′ , ω ′′ ) = λ [ h ( − ω ′ ) Θ ( ω ′′ ) ω ′ (1 + λ ) − ω ′′ / × h ( ω ′′ )( ω ′′ − ω )sgn( ω ′′ ) /π , and h ( ω ) = 1 / [ iµ + ω (1 + λ )] . V. FREE TO MOVE
So far, the object has been assumed to be fixed at x = 0 ,as described by the langrangian (1). Now, let us consider thatfor t < − τ the object is kept at x = 0 (Fig. 2.a), but for t > − τ it is free to move. Even if µ ( t ) = µ for t > − τ , afluctuating force from the quantum vacuum field would act onthe object, so that it would start a Brownian motion [38–41].On the other hand, one can use one of the degrees of free-dom of the model, namely the initial mass of the object ( M ) ,to simplify the problem. Assuming M sufficiently large, themean-squared displacement in the position of the object, dur-ing the interval − τ < t < τ , can be neglected and, therefore,the object remains at x ≈ in the mentioned interval.Now, let us consider again µ ( t ) = µ [1+ ǫf ( t )] in the inter-val − τ < t < τ (Fig. 2.b), with M remaining large enoughto the Brownian motion be neglected, and the object free tomove for t > − τ (Fig. 2.c). For this case, the boundary con-dition imposed to the field on the static object at x = 0 mustnow be replaced by a boundary condition considered in the in-stantaneous position x = q ( t ) of the moving object, observedfrom the point of view of an inertial frame where the object isinstantaneously at rest (called tangential frame), and then bemapped into a boundary condition viewed by the laboratorysystem [33, 36, 42–44]. This means that the force F µ ( t ) [Eq.(6)] needs to be replaced by a modified F µq ( t, ˙ q ( t )) , whichnow can depend on the velocity of the object [for consistency,we consider that F µq ( t,
0) = F µ ( t ) ]. In addition to the force F µq ( t, ˙ q ( t )) , the motion of the object gives rise to an extra dis-turbance to the vacuum field, from which arises an additional(dynamical Casimir) force acting on the object. This forceis here represented by F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) , so that it actson non-uniformly accelerating objects (see, for instance, Ref.[3, 45]). For consistency, we assume that, for a static object, F q (0 , , ... ) = 0 . As an example, if one considers, for instance λ → − , the object imposes to the field on the left (right)side the Robin (Dirichlet) boundary condition. For this case, F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) ≈ / (6 π ) ∂ t q ( t ) − / (6 πµ ) ∂ t q ( t ) [36, 45, 46]. In summary, two forces act on the object free tomove, a force F µq ( t, ˙ q ( t )) (related to the time-varying proper-ties of the object) and a force F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) (relatedto the motion of the object). Figure 2. Illustration of an object, initially at rest (a) but free tomove [(b) and (c)], whose motion becomes induced by an excita-tion of the quantum vacuum caused by the object itself. The object,characterized by an asymmetric scattering matrix, is represented bya circle with two faces in gray and black. The dashed wavy linesrepresent the unperturbed “in” fields ϕ in ( t − x ) (left) and ψ in ( t + x ) (right). (a) The object for t < − τ , when µ ( t ) ≈ µ . The solid-gray and solid-dark wavy lines represent the unperturbed “out” fields ψ out ( t + x ) (left) and ϕ out ( t − x ) (right), respectively. (b) The objectat an instant t , when the parameter µ is varying in time. The irregu-lar parts of the solid-gray and of the solid-dark wavy lines representthe perturbed parts of the “out” fields ψ out ( t + x ) and ϕ out ( t − x ) ,respectively. Under this situation, the object is subjected to the force F (1) µq ( t, ˙ q ( t )) ǫ + F (2) µq ( t, ˙ q ( t )) ǫ + F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) and getsin motion. (c) The object for t > τ , when µ ( t ) ≈ µ , and we calcu-late the number of created particles (represented by the dark points),moving with a final constant mean velocity v f . Note that we havemore particles produced in the left side, coinciding in this illustra-tion with a larger flux of particle momentum. From the energy conservation, and assuming that the exci-tation of the quantum vacuum occurs at the expense of theenergy of the object [so that its initial mass M becomestime-dependent: M → M ( t ) ], we have M ( t ) ≈ M [1 −E field ( t ) /M ] / [1+ ˙ q ( t ) / , where E field ( t ) is the energy storedin the field, and the velocities of the object are considerednon-relativistic (remember that c = 1 ). We consider, now, theapproximation E field ( t ) /M ≪ , which means that the en-ergy E field ( t ) is negligible if compared to the initial energy ofthe object. We also consider the non-relativistic assumption ˙ q ( t ) ≪ . Then we have M ( t ) ≈ M .Considering all forces acting on the object, we have theequation of motion F µq ( t, ˙ q ( t )) + F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) ≈ M ¨ q ( t ) . We also consider that, up to second order in ǫ , the force F µq ( t, ˙ q ( t )) can be written as F µq ( t, ˙ q ( t )) ≈ F (1) µq ( t, ˙ q ( t )) ǫ + F (2) µq ( t, ˙ q ( t )) ǫ , which is an extension of Eq.(6), assuming F (1) µq ( t,
0) = F (1) µ ( t ) , and F (2) µq ( t,
0) = F (2) µ ( t ) . Then, we write the equation of motion as: F (1) µq ( t, ˙ q ( t )) ǫ + F (2) µq ( t, ˙ q ( t )) ǫ + F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) ≈ M ¨ q ( t ) . Now, onecan use two of the degrees of freedom of the model, namelythe value of ǫ and the initial mass M , to simplify the prob-lem. For instance, let us consider the changes ǫ → p ǫ ,and M → p M (with p > ) to build a new situa-tion for which the equation of motion is F (1) µq ( t, ˙ q ( t )) ǫ/ p + F (2) µq ( t, ˙ q ( t )) ǫ + F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) / p ≈ M ¨ q ( t ) . In-creasing the value of p , we can inhibit the effect of the firstand third terms in the last equation, so that we can set up asituation where these terms can be neglected in comparisonwith the second one, resulting in the approximate equationof motion F (2) µq ( t, ˙ q ( t )) ǫ ≈ M ¨ q ( t ) . In other words, for asuitable choice of the initial mass M , the force F (2) µq ( t, ˙ q ( t )) defines effectively the mean trajectory of the object. By keep-ing increasing the mass M , one can produce smaller accel-erations, so that the velocities of the object are of such mag-nitude that the boundary condition imposed by the object onthe field, considered by the tangential frame, after mappedinto the boundary condition viewed by the laboratory system,can be approximately given by F (2) µq ( t, ˙ q ( t )) ≈ F (2) µq ( t,
0) = F (2) µ ( t ) . With this approximation, we have the equationof motion given by F (2) µ ( t ) ǫ ≈ M ¨ q ( t ) . Integrating intime, R + ∞−∞ F (2) µ ( t )d t , using the mentioned formula shown for ˜ F (2) µ ( ω ) , the property η ( ω, ω ′ ) = η ( ω ′ , ω ) = η ( − ω, − ω ′ ) ,and also considering that ˙ q ( t < − τ ) = 0 , we get v f ≈ −P /M , (7)where v f = ˙ q ( t > τ ) is the mean final velocity. Note that −P is the opposite of the net momentum transferred fromthe object to the field [see Eq. (5)], so that the momentumtransferred to the object, caused by the action of the force F (2) µ ( t ) ǫ , is directly correlated with the particle creation pro-cess. This situation is illustrated in Figs. 2.c and 3.a.Let us give attention to another situation. The values of ǫ and p can be chosen in such way that the term related to F (1) µq ( t, ˙ q ( t )) becomes dominant in relation to F (2) µq ( t, ˙ q ( t )) ǫ .We consider [as done in a similar way for F (2) µq ( t, ˙ q ( t )) ǫ ] theapproximation F (1) µq ( t, ˙ q ( t )) ≈ F (1) µq ( t,
0) = F (1) µ ( t ) . It can beshown that R + ∞−∞ F (1) µ ( t )d t = 0 , which means that, althoughthis force makes the position of the object vary in time, thetotal net momentum transferred to the object is null. This sit-uation is illustrated in Fig. 3.b.Another situation can be obtained by manipulating the val-ues of ǫ and p in such way that F (1) µ ( t ) ǫ and F (2) µ ( t ) ǫ havesimilar magnitudes. The presence of F (1) µ ( t ) ǫ disturbs themean trajectory but does not change the final velocity v f ob-tained if only F (2) µ ( t ) ǫ was considered. This case is illus-trated in Fig. 3.c.Finally, we can set up the values of ǫ and p so that we have to consider all terms, F (1) µq ( t ) ǫ , F (2) µq ( t ) ǫ and F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) . One can see that R + ∞−∞ F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) ˙ q ( t )d t < , so that the net Figure 3. Mean trajectories q ( t ) (continuous lines) of an object ini-tially ( t < − τ ) at rest. (a) Illustration of q ( t ) when just the term F (2) µ ( t ) ǫ is considered. The mean final velocity v f is indicated bythe dashed line. (b) Illustration of q ( t ) when only the term F (1) µ ( t ) ǫ is taken into account. The null contribution of this force to the meanfinal velocity of the object is indicated by the vertical inclinationof the curve q ( t ) for t > τ . (c) Illustration of q ( t ) when the sum F (1) µ ( t ) ǫ + F (2) µ ( t ) ǫ is considered. The mean trajectory is drawnby introducing deformations on the curve shown in the case (a), butmaintaining the same mean final velocity v f . (d) Situation when allterms are considered: F (1) µ ( t ) ǫ + F (2) µ ( t ) ǫ + F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) .The net dissipation of the kinetic energy by F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) isindicated by the mean final velocity v ′ f , with | v ′ f | < | v f | . For com-parison purposes, the figure also shows, in gray lines, the situationdescribed in (a), when only F (2) µ ( t ) ǫ was considered. action of F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) is to dissipate energy of theobject. This leads to a mean final velocity v ′ f with a smallermagnitude if compared with v f ( | v f | > | v ′ f | ) obtained ifonly F (1) µ ( t ) ǫ + F (2) µ ( t ) ǫ was considered. This situation isillustrated in Fig. 3.d. VI. SUMMARY OF THE RESULTS AND FINALREMARKS
In the model proposed here, a static object, isolated fromeverything and just interacting with the quantum vacuum, gets in motion by exciting the vacuum. Then, the net action of thevacuum field is in favor of the motion, instead of against, asit occurs in the usual dynamical Casimir effect. This motionrequires a time variation of one of the parameters [ µ ( t )] of theobject, which couple it with the quantum vacuum field. Re-sisting to this change, the vacuum field extracts energy fromthe object, converting this energy into real particles. The mo-tion also requires an asymmetrical vacuum excitation on eachside, which can be achieved by an interaction field-object de-scribed by an asymmetric scattering matrix.The mean force acting on the object due to µ ( t ) can be di-vided in two parts, the forces F (1) µ ( t ) ǫ and F (2) µ ( t ) ǫ . Theseforces only exist (are non-null) owing to the asymmetry of theobject, which means that the asymmetry is fundamental to therise of these quantum forces.The part of the force related to F (1) µ ( t ) is a manifestation ofthe disturbed vacuum field in order ǫ , and the correspondentforce can remove the object from the rest, but gives no con-tribution to the net momentum. On the other hand, the termrelated to F (2) µ ( t ) is a manifestation of the disturbed vacuumfield in order ǫ , and it is a direct consequence of the momen-tum transferred to the object by the created particles.The mean forces, F (1) µ ( t ) ǫ and F (2) µ ( t ) ǫ , described in thepresent paper, are quantum forces emerged from the asym-metry and time-varying properties. In conjunction with thedynamical Casimir force F q ( ∂ t q ( t ) , ∂ t q ( t ) , ... ) , these threeforces define, in the approximation considered here, a meantrajectory for the object. Finally, the object, starting from therest, gets a non-null mean final velocity, so that, exciting thevacuum, it moves. VII. ACKNOWLEDGMENTS
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