Adv. Studes Theor. Phys., Vol. 8, 014, no. 1, 7-34 HIKARI Ltd, www.-hkar.co http://dx.do.org/10.1988/astp.014.311136 Quantu Partcle Moton n Physcal Space A. Yu. Saarn Dept. of Physcs, Saara State Techncal Unversty Molodogvardeyskaya 44, 443100 Saara, Russa Copyrght 013 A. Yu. Saarn. Ths s an open access artcle dstrbuted under the Creatve Coons Attrbuton Lcense, whch perts unrestrcted use, dstrbuton, and reproducton n any edu, provded the orgnal work s properly cted. Abstract Usng Feynan s representaton of the quantu evoluton and consderng a quantu partcle as a atter feld, t s shown that ndvdual partcles of the feld have unque paths of the oton. Ths allows descrbng oton of the quantu partcle contnuous edu by Lagrange s ethod. It s shown that the for of a real quantu partcle path s deterned by classcal nu acton prncple. Keywords: quantu evoluton, physcal space, atter feld, contnuous edu, echancal oton 1 Introducton If a wavefuncton expresses propertes of the atter feld, then a reducton process s a real nonlocal transforaton of the contnuous edu. In the paper [1] the wavefuncton reducton has been descrbed as such real process. Suppose a quantu partcle s the atter feld n physcal space. Then, the set of physcal space coordnates that are applcable doan of the wavefuncton has cardnalty of contnuu. Ths eans that echancal oton of the quantu partcle contnuous edu cannot be descrbed by observables (any observables collecton s countable []).
8 A. Yu. Saarn Evoluton of the wavefuncton n physcal space s consdered n Feynan s nterpretaton of quantu echancs [3, 4]. Note that there s no necessty to use the observables for evoluton descrpton n ths nterpretaton per se and t can be consdered as Euleran ethod of contnuu oton descrpton. It reans to check that Lagrangan ethod exsts. Ths s the an goal of the paper. Consder one-densonal oton of the quantu contnuu. Denote by ( x ) t1 1 and t ( x ) wavefunctons of the ntal and fnal states. Let x 1, x are the ntal and fnal space coordnates and t 1, t are respectve nstants of te. The dynac low n Feynan theory has the for of an ntegral wave equaton: ( x ) K ( x, x ) ( x ) dx, (1) t t, t1 1 t1 1 1 Kernel of the ntegral evoluton operator K ( x, x ) consdered as a functon of space t, t1 1 varables has paraetrcal te dependence (as the wavefunctons []). It s transton apltude that has the for of a path ntegral: K ( x, x ) [ dx( )]exp S [ x( )]. () t, t1 1 1 The acton functonal S [ ( )] 1 x s the te ntegral of Lagrangan taken along the vrtual path x(). The ntegraton s to be taken over all vrtual paths x() ; t has the for of a contnual ntegral [5]. The transton apltudes for dfferent values of the space varable are ndependent on each other. Ths allows consderng these transtons as the result of echancal oton of the ndvdual partcles (the ndvdual partcles are the contnuu eleent such that there s bunque correspondence between these eleents and coordnates of the space occuped by the contnuous edu). Suppose that the ndvdual partcles conserve dentty n the transton process. Then the physcal quanttes have to exst such that the echancal oton s descrbed by these quanttes at any transton te. These quanttes have to descrbe the oton state of the ndvdual partcle at any nstant of transton te. Generalzng (), we get for the oton characterstcs (the transton quanttes) followng expresson [4]: f ( ) [ dx( )] f ( )exp S [ x( )], (3) 1 x( ) 1 where f () x( ) s a classcal kneatc quantty for the vrtual path x(). If to attrbute these quanttes to the real paths of the ndvdual partcles, t reans to check that the unque path exsts.
Quantu partcle oton n physcal space 9 The real path The followng theore s useful for ths goal. Theore. If the physcal quantty f f ( x) s functon of the ndvdual partcle coordnate x for each vrtual path, then the unque path x () exsts such that the transton quantty f s deterned by ths path. Let a classcal physcal quantty be the functon of the ndvdual partcle coordnate: f f ( x). Suppose the coordnate values are attrbutes of the path x( ), then values of the quantty f s attrbute of the sae vrtual path too: f f ( x ). Usng expresson (3), we have f [ d] f ( x )exp S[ ]. Suppose x s set of the varables havng contnuu cardnal nubers. Then the quantty f can be consdered as a functon of these varables. If to expand t nto power seres, then ters of the expanson are contnual ntegral. For lnear ter we have: x [ d ] f ( x)exp S[ ] x[ d ]. For the ter correspondng to the quadratc for we obtan 1 x [ d ] x [ d ] f ( x )exp S[ ] [ d ] x x. Snce the expanson ters, correspondng to dfferent varables, are taken nto account n the path ntegral doubly, then the correspondng ters have to be dvded by (the ters correspondng to the sae varable have to be dvded by n lne wth the usual for the Taylor expanson reason). Other ters of the expanson have the sae structure. Therefore f [ d ] f ( x )exp S[ ] x [ d ] f ( x )exp S[ ] [ d ] x
30 A. Yu. Saarn 1 x [ d ] x [ d ] f ( x )exp S[ ] [ d ]... x x Substtutng the dervatve x by x n last expresson, we get x f [ d ] f ( x )exp S[ ] [ d ] f ( x )exp S[ ] x [ d ] x x 1 x x xx x x (4) [ d ] f ( x )exp S[ ] x [ d ] x [ d ]... Every set of the kneatcs quanttes corresponds to the unque vrtual path for the oton te. In order to express ths atheatcally, we can ntroduce the followng object: 0, when [ ], when. Takng nto account the physcal sense of ths functonal and the foral slarty wth Drac s -functon we have [ ] f [ ][ d ] [ ][ d ] 1 f [ ]. It now follows that Usng (3) for the transton quantty Then f [ d] f ( x )exp S[ ] x x. [ d ] [ ] f ( x )exp S[ ] x x we get x [ d] x exp S[ ].
Quantu partcle oton n physcal space 31 x x [ d ] x [ d ] x exp S[ ] [ d ] x x x [ d ] [ ] x exp S[ ] [ d ] x x[ d ] [ ]exp S[ ] [ d ] [ d ] xexp S[ ] x. Cobnng these results and expanson (4), we obtan f 1 f f f x x.... x x (5) Snce the path ntegral s over all vrtual path then there s the vrtual path x () such that In ths case x ( ) ( ) x. x( ) x( )exp S[ x( )] [ d ] x ( )exp S[ ] x( ) exp, where the ntegral [ d ] x ( )exp S[ ] s over all vrtual paths, exceptng the path ; s phase of the transton coordnate. Snce ths expresson ust hold dentcally for any te and any full set of vrtual paths, t follows that Sx [ ( )], and [ d ] x ( )exp S[ ] 0. Therefore x( ) x ( )exp S[ x ( )]. (6) Substtutng x ( )exp S[ x ( )] for x() n (5), we get f 1 f x x f f x x....
3 A. Yu. Saarn Snce then f [ d ] f ( x)exp S[ ] [ d ] [ x( )] f ( x)exp S[ ] x x x, f ( x)exp S[ x( )] x Therefore f ( x ) 1 f ( x ) f f ( x) 0 x x... exp [ ( )] S x x x 0 0 f f ( x )exp S[ x ( )] (7) Thus, transton quanttes for any ndvdual partcle are deterned by the unque vrtual path x (). Ths corresponds to Lagrange s descrpton of the contnuu oton and the path x () can be consdered as a real path. In order to realze ths descrpton t s necessary to fnd the dynac law that deternes for of the real path. 3 Dynac low Let the ndvdual partcle oves fro the pont wth the coordnate x 1 of physcal space at the nstant of te t 1 to the pont wth the coordnate x of physcal space at the nstant of tet. Consder all vrtual paths for ths oton. Let be varaton of the vrtual paths path such that 0. t1 t Varatons of the dfferent vrtual path are the sae. Snce the vrtual paths set are exhaustve, we see that
Quantu partcle oton n physcal space 33 [ ]exp d S[ ] [ d( )]exp S[ ]. Expandng nto seres the coplex exponent n the second ntegral, we obtan exp S[ ] exp S[ ] S[ ]exp S[ ].... exp S[ ] 1 S[ ]... Takng nto account the ters accurate wthn the frst order of vanshng, we obtan [ d ]exp S[ ] [ d ]exp S[ ] [ d ] S[ ]exp S[ ]; [ d ] S[ ]exp S[ ] 0. Therefore S[ ] 0. Usng (7), we get S[ ] L( x, x, ) d L( x, x, ) d exp S[ ] S[ ]exp S[ ] 0. Fnally we have Sx [ ( )] 0. Thus the ndvdual partcles oton obeys classcal nu acton prncple.
34 A. Yu. Saarn 4 Concluson These results can be suarzed as follows. If the quantu partcle s consdered as a atter contnuu then descrpton of the oton can be represented n Lagrange s for as echancal oton of every ndvdual partcle along the unque real path. Ths path s deterned by classcal nu acton prncple. Ths descrpton does not use observables and t s ore general then conventonal quantu echancs. Snce there are no restrctons to ths descrpton, then, probably, t could be used for vrtual processes hdden by uncertanty prncple. References [1] A. Yu. Saarn, Space localzaton of the quantu partcle, Vestn. Saar. Gos. Tekhn. Unv. Ser. Fz.- Mat. Nauk, 30 (013), 387 397. [] von Johann V. Neuann Matheatsche grundlagen der quantenechank, Verlag von Julus Sprnger, Berln, 193. [3] R. P. Feynan, Space-te approach to non-relatvstc quantu echancs, Rev. Mod. Phys., 0 (1948), 1-36. [4] R. P. Feynan and A. R. Hbbs, Quantu Mechancs and Path Integrals, McGraw-Hll, New York, 1965. [5] J. Znn Justn, Path Integrals n Quantu Mechancs, Oxford Unversty Press, Oxford, 004. Receved: Noveber 7, 013