Is the Free Vacuum Energy Infinite?

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1 Is the Free Vauum Energy Infite? H. Razmi () an S. M. Shirazi () Department of Physis, the University of Qom, Qom, I. R. Iran. () & () Abstrat Consierg the funamental utoff applie by the unertaty relations limit on virtual partiles frequeny the quantum vauum, it is shown that the vauum energy ensity is proportional to the verse of the forth power of the imensional istane of the spae uner onsieration an thus the orrespong vauum energy automatially regularize to zero value for an fitely large free spae. This an be use regularizg a number of unwante fities happen the Casimir effet, the osmologial onstant problem an so on without usg alreay nown mathematial (not so reasonable) tehniques an tris. Keywors: Quantum Vauum; the Unertaty relations; Virtual Partiles PACS: ;.50.Pq

2 Introution In the stanar quantum fiel theory, not only the vauum (zero pot) energy has an absolute fite value, but also all the real exite states have suh an irregular value; this is beause these energies orrespons to the zero-pot energy of an fite number of harmoni osillators ( W ). We usually get ri of this, irregularity via simple tehnique of normal orerg by onsierg the energy ifferene relative to the vauum state [-6]; but of ourse, there are some important situations where one eals iretly with the absolute vauum energy as the osmologial onstant problem [7], or the regularization of the vauum energy the Casimir effet [8]. After a half entury of nowg an "livg" with the vauum energy, there are only some mathematial tehniques an approahes regularizg its fite value without payg enough oneptual attention to the "ontent" of the quantum vauum "struture". In this paper, onsierg the funamental assumption that the vauum energy origates from the motion of virtual partiles the quantum vauum, it is shown that the free vauum energy an be regularize base on the unertaty relations limit on these partiles frequeny. Inee, the free vauum (or any fitely large vauum spae) energy is automatially regularize to zero value without usg any presupposition (e.g. normal orerg) usually use gettg ri of the fity of the quantum vauum energy (the vauum atastrophe). The quantum vauum, virtual partiles an the unertaty relations The quantum vauum is not really empty. It is fille with virtual partiles whih are a ontuous state of flutuation. Virtual partile-antipartile pairs are reate from vauum an annihilate ba to it. These virtual partiles exist for a time itate by Heisenberg unertaty relation. Base on the unertaty relations, for any virtual partile, there is a limit on the timesale of beg reate from the vauum flutuations an then annihilate ba to vauum (its lifetime ); thus, there shoul be a limit on the frequeny of the virtual partiles whose total energies is onsiere as the vauum energy. In quantum (fiel) theory, it is well-nown that the reason for namg the quantum vauum partiles as virtual partiles is that although they are existene an an have observable effets (e.g. the Casimir effet, spontaneous emission, Lamb shift), they annot be iretly etete (i.e. they are unobservable). For these unobservable (virtual) partiles, the energy an lifetime values are onstrae ue to the unertaty relation an an tae, at most, the mimum values of unertaties for real partiles. This an be written as the followg relation: virtual ( E ) (), max. where is a onstant whih annot have a value muh more than to guarantee that we are ealg with virtual partiles than real ones. As we now, the unertaties energy an lifetime of real (etetable) partiles satisfy the relation: E ().

3 The vauum energy ensity of fitely large free spaes Although attribution of physial parameters an quantities to the virtual partiles as the same as what we now for the real partiles isn t a ompletely nown an prove fat, the ma reason of irregularity/fity of the vauum energy QFT is beause of attribution of the frequeny to the virtual partiles an summg on the fite moes for them. Also, attributg istane to virtual partiles is a nown fat; the Casimir effet, we say about the onfement of virtual partiles a fite istane between the two plates an the Casimir fore epens on this istane. The only nown pot about the attenane of the virtual photons a fite istane QFT is that these termeiate partiles (as the Feynman iagrams) have nonzero masses with fite range of ation ; this maes them have a veloity of v where we shall onsier it our alulation. For a free spae of imensional length D, usg the relations (), the frequeny of virtual partiles shoul satisfy: virtual (). D D E,, an Consierg this limit on an the followg well-nown relation for the vauum energy ensity orrespong to an fitely large spae: it is foun: E V V (), (5). 0 E D V For an fitely large free spae, the vauum energy is zero; it is automatially regularize as the followg: E free ( )( Volume) ( )( D ) 0 ( D ) (6). Disussion This result that the vauum energy of the free fitely large spaes is zero may be terprete as that the fite vauum is a potentially resoure ontag fitely free virtual partiles of negligible frequeny where an tae higher values of frequeny (energy) uner the fluene of the restritions mae on them by the presene of external bounaries that onstra their fite freeom; this terpretation seems to be more reasonable than that the vauum energy for the free fitely large (or even fite) spae has an fite (irregular) value.

4 We shoul mention that the ifferent meang (terpretation) of energy-time unertaty relation from the well-nown momentum-position unertaty priple oesn't affet what we have alulate here. Inee, the relation (), means as a "lifetime" with quantity than as an unertaty time. It is also mentione that the result of this paper isn t onflit to the response of quantum vauum to a fite boune restrition (the Casimir effet). Inee, although the ma soun of this paper is that the Casimir energy for free spaes or fitely large outer spaes the stanar geometries well-nown the Casmir effet beomes zero spite of alreay aepte fite (irregular) values, it is possible to f out the expete Casimir fore for the well-nown problem of two parallel onutg plates base on the regularization troue here (see Appenix). Appenix For two plates of istane from eah other, there is a freeom of x ~ for ner virtual partiles an x ~ D ( D ) for the partiles the two (left an right) outer spaes. As is well-nown, the Casimir energy orrespong to the famous geometry of two parallel onutg plates is: E Casimir E E ( E E E ) E (A-). boune free left sie right free Consierg the resultg relation (6), all three terms thus: E left, E right, an E free vanish an ECasimir E sie (A-). For the ase of a salar fiel [9], usg the well-nown energy-momentum tensor fiel T g (A-), an the followg relation between vauum to vauum expetation value of the fiel operators at two spae-time pots an the time epenent Green funtion (the propagator) 0 T{ ( x) ( x)}0 ig( x, x) (A-), an the relation ˆ 0 T 0 i lim ( ' g ) G( x, x') xx (A-5), by means of i( tt) i.( xx) G( x, x) e e s( z )s ( z ) ( ) s (A-6),, x y we arrive at this result that:

5 T i ( ) s s z s ( z ) osz os( z ) (A-7). With the appliation of omplex frequeny rotation ( i), T ( ) sh sh z sh ( z ) oshz osh( z ) (A-8). After appropriate hange of variables an simple tegral alulation, the sie energy per unit area is foun as: E area area 0 whih Tˆ I( ) x 0 Tˆ z 6( ) 0 ( )oth 5 (A-9), x.75 e I( ) x x 0 x, x x 0 e (A-0). As we now, even the preise measurements (e.g. [0]), there is no iret experiment onfirmg the exat numerial oeffiient the alreay nown result ( ) for the salar fiel Casimir pressure beause there are experimental 0 iffiulties mag two plates parallel at the sales an preisions neee the moern experiments an unavoiable errors ue to worg with goo real materials than perfetly ieal onutors. By the way, one an reover the ieal result by puttg I( ) whih an be ahieve by hoosg (A-9); this is an aeptable value for base on what explae followg the relation (). Referenes [] W. Greer, Quantum Mehanis: Speial Chapters, p. (Sprger, 998). [] W. Greer, Quantum Mehanis: Speial Chapters, p. (Sprger, 998). [] C. Itzyson, J. B. Zuber, Quantum Fiel Theory. p. 0 (Dover Publiations, 6). [] F. Manl, G. Shaw, Quantum fiel theory, p. (Wiley, ).

6 [5] V. B. Berestetsii, L. P. Pitaevsii, E. M. Lifshitz, Quantum Eletroynamis, p. 0 (Butterworth-Heemann, 98). [6] M. Guiry, Gauge Fiel Theories, p. 67 (Wiley-Intersiene, 999). [7] S. Weberg, Rev. Mo. Phys. 6, (989). [8] H. B. G. Casimir, Pro. K. Ne. Aa. Wet. 5, 79 (98). [9] K. A. Milton, The Casimir Effet: Physial Manifestations of Zero-pot Energy, p. -5 (Worl Sientifi, Sgapore, ). [0] S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (997); U. Mohieen an A. Roy, Phys. Rev. Lett. 8, 59 (998).

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