Absolute Zero in Casimir Vacuum

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Original Research Papers Fundamental Journals Open Access Journals International Journal of Fundamental Physical Sciences (IJFPS) ISSN: 2231-8186

IJFPS, Vol 10 , No 1, pp 01- 04 March, 2020 Xingwu Xu DOI: 10.14331/ijfps.2020.330133

Absolute zero in Casimir vacuum

Xingwu Xu

Research Institute of Hefei Gotion High-tech Power Energy Co, Ltd 599 Daihe Road, Hefei, Anhui 230012, P. R. China

E-mail addresses: [email protected]

Received Dec 2019 Received in revised: Feb 2020 Published: March 2020

ABSTRACT

Casimir vacuum has many peculiar properties, e.g. the speed of light can exceed c. This paper investigated the behavior of gas molecules in Casimir vacuum. Analysis and calculation indicate that we can get a new absolute zero in Casimir vacuum. This is due to the Hamiltonian, which equals the kinetic energy of the ensemble of gas molecules that is reduced in Casimir vacuum; therefore, the gas constant R is changed. This conclusion has great significance for the research of the vacuum properties.

Keywords: Casimir vacuum, Absolute zero

INTRODUCTION 푛 (0) = 1 (2) Casimir vacuum has many peculiar properties. For example, ‖ Scharnhorst found that the speed 푐⊥ of light normal to the 11π2 α2 Casimir parallel mirrors could exceed 푐 = 1 (Barton, 1990; n (0) = 1– (3) ⊥ (90)2 (mL)4 Scharnhorst, 1990). In fact, the vacuum structure has been modified in Casimir vacuum. The propagation of light depends Where L is the distance of two plates. Therefore, 푛 (0) < 1, on the structure of the vacuum (Scharnhorst, 1990). The main ⊥ and the propagation velocity (c/n⊥) of light perpendicular to factor is that the refractive index (n) in Casimir vacuum is the mirrors will be faster than c. This conclusion does not changed, contradict the special theory of relativity (Scharnhorst, 1998).

44훼2 This research stimulated a broader interest in the study of the 푛(0) = 1 − 푘휌 푘 = (1) 135푚4 properties of vacuum. The question raised in this paper is: Will Casimir vacuum affect the value of absolute zero? Where 휌 is the vacuum energy density, 훼 is the fine structure constant, m is the electron mass. The results of Scharnhost’s calculation are,

https://doi.org/10.14331/ijfps.2020.330133 ISSN: 2231-8186/ ©2020 The Authors. Published by Fundamental Journals This is an open access article under the CC BY-NC https://creativecommons.org/licenses/by-nc/4.0/ 1

IJFPS, Vol 10 , No 1, pp 01-04, March, 2020 Xingwu Xu

THE INTERACTION BETWEEN MOLECULES OR Where HA is the atomic Hamiltonian, HF is the free field ATOMS AND THE VACUUM FIELD Hamiltonian, and HAF is the atom-field interaction Hamiltonian. Now, we consider an ensemble of gas molecules This issue is related to the reaction between molecules or in Casimir vacuum. The total kinetic energy of gas molecules atoms of gas and the vacuum field. The molecules or atoms is, moving in the ordinary vacuum will change their momentum 퐸푘𝑖푛−푠푦푠 = ∑ 퐻 (9) due to the reaction between molecules or atoms and the vacuum. Bercegol and Lehoucq (2015) state that there is a In fact, the Casimir vacuum is a space where some k modes braking torque in a rotating pair of atoms when reacting with (longer wavelength) are excluded, as shown in Fig. 1 (Chown, quantum vacuum, which decreases the kinetic energy and the 1990). angular momentum of the atoms. Recently, Sonnleitner found that an excited two-level atom moving through a vacuum sees a tiny friction force (Sonnleitner, Trautmann, & Barnett, 2017). The friction force arises from the interaction between atoms and the vacuum field. The interaction can be described by the so-called “Rontgen term (HAF)” (Wilkens, 1994).

1 퐻 =– 퐝̂ ∙ 푬̂⊥(푹̂) − {푷̂ ∙ [푩̂(푹̂) × 풅̂] + [푩̂(푹̂) × 풅̂] ∙ 푷̂} 퐴퐹 2M

(4)

Here 풅̂ is the atomic dipole operator, 푷̂ and 푹̂ are the atomic center-of-mass canonical momentum and position operator, 푬̂⊥ is the electric field, and 푩̂ is the magnetic field:

̂⊥ ̂ i퐤∙퐑̂ † −i퐤∙퐑̂ 푬 (푹) = i ∑k,λ εk퐞퐤,훌 (âk,λe – âk,λe ) (5)

̂ ̂ i i퐤∙퐑̂ † −i퐤∙퐑̂ Fig 1. Casimir effect: The vacuum is full of virtual photons, but 푩(푹) = ∑k,λ εk(풏푘 × 퐞k,λ)(âk,λe – âk,λe ) c photons with wavelength (λ) more than twice the separation of (6) the plates are excluded from the space between them.

Where εk = √ℏωk/(2ω0 V) is the electric field strength per photon in quantization volume 푉, 풏푘 = 풌/|풌| is a unit vector The Rontgen term is actually the friction term with negative in the direction of propagation, 풆푘,휆 is a polarization vector value. For the sum operation, if the value of term is positive, perpendicular to the mode propagating in a direction 푲 = the greater the number of terms, the greater the sum. † Conversely, if the value of term is negative, the larger the 퐤c/ωk (Sonnleitner et al., 2017), 푎̂푘,휆 and 푎̂푘,휆 denote the ′′ photon annihilation and creator operators of the 푘푡ℎ mode with number of terms, the smaller the sum. Now let us use 푘 for ′ the polarization λ (λ = 1, 2). In fact, the “Rontgen term” is a the number of modes (terms) in Casimir vacuum, and 푘 for negative energy dissipation term caused by the displacement the number of modes in the ordinary vacuum. From equations ′′ ′ current. The interaction mentioned above results in the (4)～(6), we know that for 푘 < 푘 (Fig. 1), we have decrease of the momentum of atoms. Because the kinetic ′′ ′ momentum 푝 = 푀푣 (푀: mass, 푣: velocity), the change of 퐻퐴퐹 > 퐻퐴퐹 (10) momentum will be (Sonnleitner et al., 2017). Equation (10) indicates that in Casimir vacuum, the mean 푝̇ = 푀푣̇ + 푀̇ 푣 (7) coupling energy between the ensemble of gas molecules and the field is lower than in ordinary vacuum. Therefore, we have The conclusion is that the velocities of atoms do not change; ′′ ′ the atom’s decay in momentum comes from the mass defect 퐸푘𝑖푛−푒푛푠 < 퐸푘𝑖푛−푒푛푠 (11) caused by the photon’s emission. Where 퐸푘𝑖푛−푒푛푠 denotes the kinetic energy of the ensemble of THE KINETIC ENERGY OF GAS MOLECULES IN gas molecules. From another viewpoint, one can divide the CASIMIR VACUUM kinetic energy of the gas ensemble into two parts-the self- energy of the gas molecules and the energy of the vacuum However, if we put the gas molecules in the Casimir vacuum, field. A coupling exists between the gas molecules and the the kinetic energy will change. For a two-level atom, the sum vacuum field; the total energy of gas molecules is the coupling of the center-of-mass kinetic energy is (Wilkens, 1994). effect of two parts of energies. Thus,

퐏̂2 H = + 퐻 + 퐻 + 퐻 (8) 퐸푘𝑖푛−푠푦푠 = 퐸푠푒푙푓 + 퐸푣푎푐 (12) 2M 퐴 퐹 퐴퐹

IJFPS, Vol 10 , No 1, pp 01-04, March, 2020 Xingwu Xu

From (Bercegol & Lehoucq, 2015) we know that, In Casimir vacuum, the total average kinetic energy of gas molecules becomes π2 퐸푣푎푐 =– (13) 720L3 퐸̅퐶푎푠𝑖 = 퐸̅ + 퐸 = 1.5198105 × 10−2 J (23) 푘𝑖푛 푘𝑖푛 푣푎푐

Where L is the distance between two plates, ℎ = 푐 = 1. The average kinetic energy for a single molecule in Casimir Therefore, the kinetic energy of gas molecules in Casimir vacuum is vacuum is lower than that in an ordinary vacuum

̅퐶푎푠푖 2 ′′ 퐸푘푖푛 −21 ′′ π ′ 퐸̅ = = 5.655928 × 10 퐽 (24) 퐸 (= 퐸 – ) < 퐸 (= 퐸 ) (14) 푘𝑖푛 푛푁 푘𝑖푛−푒푛푠 푠푒푙푓 720L3 푘𝑖푛−푒푛푠 푠푒푙푓 퐴

Therefore, the new 푘퐵 is A NEW ABSOLUTE ZERO IN CASIMIR VACUUM ̅′′ 퐶푎푠𝑖 2 퐸푘푖푛 −23 푘퐵 = = 1.38042 × 10 (25) According to (Fischer et al., 2007), temperature is proportional 3 푇0 to the mean translational kinetic energy of atoms or molecules; it is just the well-known equation The new gas constant is

3 퐶푎푠𝑖 퐸̅ = 푘 T (15) 푅퐶푎푠𝑖 = 푘퐵 푁퐴 = 8.31289 (26) 푘𝑖푛 2 퐵

The absolute zero in Casimir vacuum is Where 퐸̅ can be considered same as 퐸 . Now, the 푘𝑖푛 푘𝑖푛−푒푛푠 paradox arises: in Casimir vacuum at absolute zero 푉0푃0 temperature, 퐸′′ < 퐸′ , which term will change- 푇퐶푎푠𝑖 = − = −273.20 K (27) 푘𝑖푛−푒푛푠 푘𝑖푛−푒푛푠 푅퐶푎푠푖 kB or T? We have two options: 1) T decreases or 2) kB decreases. ∆T = 푇퐶푎푠𝑖 − 푇퐾 = −273.20 − (−273.15) = −0.05 K 1) If we put the Casimir vacuum as well as the ensemble of gas molecules into absolute zero, the temperature in Casimir (28) vacuum will be less than absolute zero, The conclusion shows that the change of vacuum structure will 푇퐶푎푠𝑖 < 0 K (16) result in a new absolute zero temperature. It is possible that in another planet, where the vacuum structure is different from 2) If the kB decreases, according to the relation: earth, there will be a new absolute zero, which will obey the 푅 푘퐵 = (17) third law of thermodynamics there. 푁퐴

PROOF AND MEASUREMENT Where 푅 is the gas constant, 푁퐴 is the Avogadro constant, then R will decrease. Back to the gas law, As a “thought experiment”, the new absolute zero can be proved and measured by designing a design of Casimir 푃 푉 0 0 = 푅 (18) vacuum pump (Fig. 2). As illustrated in Fig.2, the “modern 푇0 Magdeburg hemispheres” consist of several hollow spheres. Where 0 here denotes the standard condition, 푃0 = 1.01325 × There are many Casimir slits in the shells. In order to shield 5 −3 3 10 푃푎 is the pressure, 푉0 = 22.414 × 10 푚 is the volume the photons or other field particles with longer wavelengths, of gas, 푇0 = 273.15퐾 is absolute temperature, and 푅 = these slits can be designed in staggered patterns, which are in 8.314. If 푅 changes, then 푇0 will change. In standard twists and turns. Moreover, these shells have to have enough conditions, the average kinetic energy for a single molecule is, strength to bear the huge pressure. Moreover, the shell material itself also has a weak shielding function. The next step is to 3 퐸̅′ = 푘 푇 = 5.656864 × 10−21 J (19) “pump” out these photons as well as field particles with longer 푘𝑖푛 2 퐵 0 wavelengths in the hollow space. Two Casimir plates are

Suppose the distance between two conductive plates is 0.1μ, installed in this hollow space. Both the plates have one of their the size of the plate is 1푚 × 1푚; then, the mole number of the ends separated by a small distance (Casimir slit), and the other gas in Casimir vacuum is end can open or close. When opening, there are many photons as well as other field particles in the containing space. Then, 푉 the two plates move together, leaving a Casimir slit between 푛 = 퐶푎푠푖 = 4.4615 × 10−6 (20) 푉0 them. Meanwhile, the doors in the shells close synchronously. By repeating this operation, the larger space with Casimir The total average kinetic energy of gas molecules in Casimir vacuum can be obtained. When opening, there are many vacuum is photons as well as other field particles in the containing space. Then, the two plates move together, leaving a Casimir slit ̅ ̅′ −2 퐸푘𝑖푛 = 퐸푘𝑖푛푛푁퐴 = 1.519838 × 10 J (21) between them. Meanwhile, the doors in the shells close synchronously. By repeating this operation, the larger space The negative vacuum energy is with Casimir vacuum can be obtained.

hcπ2 퐸 = – = −2.75 × 10−7 J (22) 푣푎푐 720L3 3

IJFPS, Vol 10 , No 1, pp 01-04, March, 2020 Xingwu Xu

T 푉 = 푉 (1 + ) (α 273.15) (29) 푇 0 α

Therefore, the absolute zero in Casimir vacuum is lower than the ordinary one.

CONCLUSION Casimir vacuum has different vacuum structure. The energy density in Casimir vacuum is lower than ordinary one. Therefore, the change of vacuum structure will result in change of the kinetic energy of gas molecules. In this paper, a the kinetic energy of gas molecules in Casimir vacuum proved to be reduced in two aspects:

1) The number of wave modes in Casimir vacuum is less than that in ordinary vacuum. Therefore, the Rontgen term (a negative energy dissipation term) in Casimir vacuum is smaller than that in ordinary vacuum, which results in reducing the kinetic energy of gas molecules.

2) The kinetic energy of gas molecules can be divided into two parts, 퐸푘𝑖푛−푠푦푠 = 퐸푠푒푙푓 + 퐸푣푎푐. Because of the negative value of 퐸푣푎푐, the kinetic energy of gas molecules 퐸푘𝑖푛−푠푦푠 is less than that of ordinary vacuum. According equation 퐸̅푘𝑖푛 = (3/2)푘퐵푇, the lower value of 퐸푘𝑖푛 must result in the lower value of 푇 or 푘퐵, this is just the new absolute zero temperature. Casimir vacuum-the kind of “modern Magdeburg b hemispheres” can be used for utilization of the “zero-point energy.” This is just as the air vacuum pump can utilize the air energy. If the new absolute zero is found, the “zero-point

Fig 2. Casimir vacuum pump: The “modern Magdeburg energy” could be used. hemispheres” consisting of the hollow spheres with many However, what we are worried about is if it violates the law of Casimir slits in the shells, which can shield the photons with conservation of energy. The answer is no. Because the “zero- longer wavelength. Two Casimir plates are installed in the inner point energy” is just like the lake on the top of mountain; the hollow space, which can squeeze the inner photons with longer water in the lake has high potential energy, but if there is no wavelength out by moving these plates. There are some doors in altitude difference, this energy cannot be used. We need to dig the shells. As soon as the longer waves are squeezed out, the a very deep hole to use this water’s potential energy. As we doors will close. know, we still need energy to dig a deep hole. Meanwhile, the vacuum energy is a hopeful candidate for “Dark Energy.” The In the space with Casimir vacuum, one can do the experiment relation between the “new absolute zero’’ and the “Dark for getting the new absolute zero temperature. One can put n Energy” is an interesting topic for further investigation. gas molecules into the Casimir vacuum and test the parameters of Charles’s law. The Charles’s law in Casimir vacuum is

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