Hindawi Publishing Corporation Advances in High Energy Physics Volume 2015, Article ID 278502, 3 pages http://dx.doi.org/10.1155/2015/278502

Research Article Is the Free Vacuum Energy Infinite?

H. Razmi and S. M. Shirazi

Department of Physics, The University of Qom, Qom 3716146611, Iran

Correspondence should be addressed to H. Razmi; [email protected]

Received 12 February 2015; Revised 14 April 2015; Accepted 16 April 2015

Academic Editor: Chao-Qiang Geng

Copyright © 2015 H. Razmi and S. M. Shirazi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Considering the fundamental cutoff applied by the uncertainty relations’ limit on virtual particles’ frequency in the quantum vacuum, it is shown that the vacuum energy density is proportional to the inverse of the fourth power of the dimensional distance of the space under consideration and thus the corresponding vacuum energy automatically regularized to zero value for an infinitely large free space. This can be used in regularizing a number of unwanted infinities that happen in the Casimir effect, the cosmological constant problem, and so on without using already known mathematical (not so reasonable) techniques and tricks.

1. Introduction 2. The Quantum Vacuum, Virtual Particles, and the Uncertainty Relations In the standard quantum field theory, not only does the vacuum (zero-point) energy have an absolute infinite value, The quantum vacuum is not really empty. It is filled with but also all the real excited states have such an irregular value; virtual particles which are in a continuous state of fluctuation. this is because these energies correspond to the zero-point Virtual particle-antiparticle pairs are created from vacuum 𝑊= energy of an infinite number of harmonic oscillators ( and annihilated back to it. These virtual particles exist for (1/2) ∑ ℏ𝜔 →∞ 𝑘,𝜎 𝑘 ). We usually get rid of this irregularity a time dictated by Heisenberg uncertainty relation. Based via simple technique of normal ordering by considering the on the uncertainty relations, for any virtual particle, there energy difference relative to the vacuum state [1–6]; but, of is a limit on the timescale of “being” created from the course, there are some important situations where one deals vacuum fluctuations and then annihilated back to vacuum directly with the absolute vacuum energy as in the cosmo- (its “lifetime”); thus, there should be a limit on the frequency logical constant problem [7] or in the regularization of the of the virtual particles whose total energies are considered vacuum energy in the Casimir effect8 [ ]. After a half century as the vacuum energy. In quantum (field) theory, it is well of knowing and “living” with the vacuum energy, there are known that the reason for naming the quantum vacuum only some mathematical techniques and approaches in reg- particles as virtual particles is that although they are in ularizing its infinite value without paying enough conceptual “existence” and can have observable effects (e.g., the Casimir attention to the “content” of the quantum vacuum “structure.” effect, spontaneous emission, and Lamb shift), they cannot In this paper, considering the fundamental assumption that be directly detected (i.e., they are unobservable). For these thevacuumenergyoriginatesfromthe“motion”ofvirtual unobservable (virtual) particles, the energy and lifetime particles in the quantum vacuum, it is shown that the free values are constrained due to the uncertainty relation and can vacuum energy can be regularized based on the uncertainty take, at most, the minimum values of uncertainties for real relations’ limit on these particles’ frequency. Indeed, the free particles. This can be written as the following relation: vacuum (or any infinitely large vacuum space) energy is ℏ automatically regularized to zero value without using any (𝐸𝜏)|virtual ≈𝛼 , max. (1) presupposition (e.g., normal ordering) usually used in getting 2 ridoftheinfinityofthequantumvacuumenergy(thevacuum where 𝛼 isaconstantwhichcannothaveavaluemuchmore catastrophe). than 1 to guarantee that we are dealing with virtual particles 2 Advances in High Energy Physics instead of real ones. As we know, the uncertainties in energy constrain their infinite freedom; this interpretation seems to and lifetime of real (detectable) particles satisfy the relation be more reasonable than that the vacuum energy for the free infinitely large (or even finite) space has an infinite (irregular) ℏ Δ𝐸Δ𝜏 ≥ . (2) value. 2 We should mention that the different meaning (interpre- tation) of energy-time uncertainty relation from the well- 3. The Vacuum Energy Density of known momentum-position uncertainty principle does not Infinitely Large Free Spaces affect what we have calculated here. Indeed, in relation (2), Δ𝜏 means a “lifetime” width quantity rather than an uncertainty Although attribution of physical parameters and quantities in time. It is also mentioned that the result of this paper is not to the virtual particles, the same as what we know for the in conflict with the response of quantum vacuum to a finite real particles, is not a completely known and proved fact, bounded restriction (the Casimir effect). Indeed, although the main reason of irregularity/infinity of the vacuum energy the main “sound” of this paper is that the Casimir energy in QFT is because of attribution of the frequency 𝜔 to the forfreespacesorinfinitelylargeouterspacesinthestandard virtual particles and summing on the infinite modes for them. geometries well known in the Casmir effect becomes zero Also, attributing “distance” to virtual particles is a known in spite of already accepted infinite (irregular) values, it is fact; in the Casimir effect, we talk about the confinement of possible to find out the expected Casimir force for the well- virtual particles in a finite distance between the two plates and known problem of two parallel conducting plates based on the Casimir force depends on this distance. The only known the regularization introduced here (see the appendix). point about the “attendance” of the virtual photons in a finite distance in QFT is that these intermediate particles (as in the Feynman diagrams) have nonzero masses with finite range of Appendix V <𝑐 “action”; this makes them have a velocity of ,wherewe For two plates of distance 𝑑 from each other, there is a will consider it in our calculation. freedom of 𝑥∼𝑑for inner virtual particles and 𝑥∼𝐷 Forafreespaceofdimensionallength𝐷,usingthe 𝐷→∞ 𝐸=ℏ𝜔𝜏≤𝐷/𝑐 ( ) for the particles in the two (left and right) outer relations , ,and(1),thefrequencyofvirtual spaces. As is well known, the Casimir energy corresponding particles should satisfy to the famous geometry of two parallel conducting plates is 𝛼𝑐 𝜔|virtual ≤ . 2𝐷 (3) 𝐸 =𝐸 −𝐸 Casimir bounded free (A.1) Considering this limit on 𝜔 and the following well-known =(𝐸 +𝐸 +𝐸 )−𝐸 . relation for the vacuum energy density corresponding to an left inside right free infinitely large space 𝐸 Considering the resulting relation (6),allthreeterms left, 𝐸 1 3 𝐸 𝐸 = ∑ ℏ𝜔 󳨀→ 2𝜋ℏ𝑐 ∫ 𝑘 𝑑𝑘, right,and free vanish and thus 𝑉 2𝑉 𝑘 (4) 𝑘 𝐸 =𝐸 . (A.2) it is found that Casimir inside 𝐸 𝛼/2𝐷 𝜋ℏ𝑐𝛼4 = 2𝜋ℏ𝑐 ∫ 𝑘3𝑑𝑘 = . For the case of a scalar field9 [ ], using the well-known energy- 4 (5) 𝑉 0 8𝐷 momentum tensor field

For an infinitely large free space, the vacuum energy is zero; 1 𝑇𝛼𝛽 =𝜕𝛼𝜑𝜕𝛽𝜑− 𝑔𝛼𝛽𝜕 𝜑𝜕𝜇𝜑 it is automatically regularized as in the following: 2 𝜇 (A.3) 𝜋ℏ𝑐𝛼4 𝜋ℏ𝑐𝛼4 𝐸 =( ) ( ) ≈( )(𝐷3) free 8𝐷4 Volume 8𝐷4 and the following relation between vacuum to vacuum expec- (6) tation value of the field operators at two space-time points 𝜋ℏ𝑐𝛼4 and the time dependent Green function (the propagator) = 󳨀→ 0 (𝐷󳨀→∞) . 8𝐷 󵄨 󵄨 󵄨 󸀠 󵄨 󸀠 ⟨0 󵄨𝑇{𝜑(𝑥) 𝜑(𝑥 )}󵄨 0⟩ = −𝑖ℏ𝑐𝐺 (𝑥, 𝑥 ) (A.4) 4. Discussion This result that the vacuum energy of the free infinitely large and the relation spaces is zero may be interpreted as that the infinite vacuum is 󵄨 󵄨 󵄨̂𝛼𝛽󵄨 a potential resource containing infinitely free virtual particles ⟨0 󵄨𝑇 󵄨 0⟩ of negligible frequency which can take higher values of (A.5) 𝛼 󸀠󸀠𝛽 1 𝛼𝛽 󸀠𝜇 󸀠 frequency (energy) under the influence of the restrictions = −𝑖ℏ𝑐 lim (𝜕 𝜕 − 𝑔 𝜕𝜇𝜕 )𝐺(𝑥,𝑥 ), made on them by the presence of external boundaries that 𝑥→𝑥󸀠 2 Advances in High Energy Physics 3 by means of Conflict of Interests

󸀠 −1 𝑑𝜔 2 −𝑖𝜔(𝑡−𝑡󸀠) 𝑖𝑘 ⋅(𝑥−𝑥󸀠) 𝐺(𝑥,𝑥 ) = ∫ 𝑑 𝑘𝑒 𝑒 ⊥ The authors declare that there is no conflict of interests 3 in (2𝜋) 𝑐 regarding the publication of this paper. 1 ⋅ (𝜆𝑧 ) 𝜆(𝑧 −𝑑), 𝜆 𝜆𝑑 sin < sin > References sin (A.6) 𝜔2 𝜆2 = −𝑘2, [1] W. Greiner, Quantum Mechanics: Special Chapters,Springer, 𝑐2 1998. ⇀󳨀 ⇀󳨀 ⇀󳨀 ⇀󳨀 [2] W. Greiner, Quantum Mechanics: Special Chapters,Springer, 𝑘 ⊥ = 𝑘 𝑥 + 𝑘 𝑦 ≡ 𝑘, Berlin, Germany, 1998. [3] C. Itzykson and J. B. Zuber, Quantum Field Theory,Dover we arrive at this result that Publications, 2006. 𝑖ℏ𝑐 𝑑𝜔 1 ⟨𝑇00⟩ = ∫ 𝑑2𝑘 [4] F. Mandl and G. Shaw, Quantum Field Theory,Wiley,2010. 3 in 2 (2𝜋) 𝑐 𝜆 sin 𝜆𝑑 [5] V. B. Berestetskii, L. P. Pitaevskii, and E. M. Lifshitz, Quantum Electrodynamics, Butterworth-Heinemann, 1982. 𝜔2 ×{( +𝑘2) 𝜆𝑧 𝜆 (𝑧−𝑑) +𝜆2 𝜆𝑧 [6] M. Guidry, Gauge Field Theories, Wiley-Interscience, 1999. 𝑐2 sin sin cos (A.7) [7] S. Weinberg, “The cosmological constant problem,” Reviews of Modern Physics,vol.61,no.1,pp.1–23,1989. ⋅ cos 𝜆 (𝑧−𝑑)}. [8] H. B. G. Casimir, “On the attraction between two perfectly conducting plates,” Proceedings of the Koninklijke Nederlandse With the application of complex frequency rotation (𝜔→ Akademie van Wetenschappen,vol.51,p.793,1948. 𝑖𝜔), [9] K.A.Milton,The Casimir Effect: Physical Manifestations of Zero- Point Energy, World Scientific, Singapore, 2001. ℏ𝑐 𝑑𝜔 1 ⟨𝑇00⟩ =− ∫ 𝑑2𝑘 [10] S. K. Lamoreaux, “Demonstration of the casimir force in the 0.6 3 in 2 (2𝜋) 𝑐 𝜆 sinh 𝜆𝑑 to 6 𝜇mrange,”Physical Review Letters,vol.78,article5,1997. 𝜔2 [11] U. Mohideen and A. Roy, “Precision measurement of the ×{(− +𝑘2) 𝜆𝑧 𝜆 (𝑧−𝑑) +𝜆2 Casimir force from 0.1 to 0.9 𝜇m,” Physical Review Letters,vol. 2 sinh sinh (A.8) 𝑐 81, p. 4549, 1998.

⋅ cosh 𝜆𝑧 cosh 𝜆 (𝑧−𝑑)}.

After appropriate change of variables and simple integral calculation, the inside energy per unit area is found as 𝐸 1 𝑑 in = ∫ ⟨𝑇̂00⟩ 𝑑3𝑥=∫ ⟨𝑇̂00⟩ 𝑑𝑧 area area in 0 in ℏ𝑐 𝛼/2𝑑 =− ∫ 𝜆2 ((𝜆𝑑) 𝜆𝑑) +5 𝑑𝜆 2 coth (A.9) 6 (2𝜋) 0 𝜋2ℏ𝑐 =− 𝐼 (𝛼) , 1440𝑑3 in which 3.75 𝛼 𝑒𝑥 +1 𝐼 (𝛼) = ∫ 𝑥2 (𝑥 + 10) 𝑑𝑥, 𝑥 = 2𝜆𝑑. 4 𝑥 (A.10) 𝜋 0 𝑒 −1 Asweknow,evenintheprecisemeasurements(e.g.,[10, 11]), there is no direct experiment confirming the exact numerical 2 3 coefficient in the already known result (−𝜋 ℏ𝑐/1440𝑑 )for the scalar field Casimir pressure because there are exper- imental difficulties in making two plates parallel at the scales and precisions needed in the modern experiments and unavoidable errors due to working with good real materials rather than perfectly ideal conductors. By the way, one can recover the ideal result by putting 𝐼(𝛼) =1 which can be achieved by choosing 𝛼 ≈ 1.874786 in (A.9);thisisan acceptable value for 𝛼 based on what was explained following relation (1). Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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