The mathematical representation of physical objects and relativistic Quantum Mechanics.

Transcription

1 The mathematcal representaton of physcal obects and relatvstc Quantum Mechancs. Enrque Ordaz Romay 1 Facultad de Cencas Físcas, Unversdad Complutense de Madrd Abstract The mathematcal representaton of the physcal obects determnes whch mathematcal branch wll be appled durng the physcal analyss n the systems studed. The dfference among non-quantum physcs, lke classc or relatvstc physcs, and quantum physcs, especally n quantum feld theor s nothng else than the dfference between the mathematcs that s used on both branches of the physcs. A common physcal and mathematcal orgn for the analyss of the dfferent systems brngs both forms (quantum and classc) of understandng the nature mechansms closer to each other. 1 eorgazro@cofs.es

2 Introducton Insde any physcal system there are groups of elements that are usually consdered as ndvdual unts that nteract wth each other, but n whose nteror other forces ntervene that are dfferent from those that are usually consdered at the level of the system as a whole. Accordngl for example, when we study the nteractons among solds we usually don't keep n mnd the ntermolecular forces or when we study the gravtatonal nteractons among planets we do not take nto account the geologc planetary forces. Usuall ths smplfcaton s carred out usng transformatons (smplfcatons that lead us to symmetres and changes of coordnates based on such symmetres) that allow us to obtan later on, smple and convenent expressons and solutons for our systems under study. To carry out these transformatons (smplfcatons plus changes of coordnates) we should make a model of the physcal obects we work wth, by means of an approprate mathematcal representaton. We call these transformatons that allow us to make a model of an obect, the mathematcal representaton of the physcal obect. Ths wa our physcal system wll be formed by a set of equatons that represent the real physcal obects. In quantum physcs, the physcal obects are the partcles of the system and the mathematcal representaton of these physcal obects s the wave functon. However, a narrow relatonshp exsts between the wave functon and the acton functon of the system, snce both of them are maxmal representatons of the system. Ths observaton allows us to obtan, from the classc analyss (relatvstc) perspectve, the complete development of the quantum theory of felds.

3 Physcal systems. A physcal system s defned by a delmted regon of 4-space. All the groups of ponts X = (x 0, x 1, x 2, x 3 ) compose ths regon that we consder by defnton to conform the system. A physcal magntude s a degree 3 or 4 scalar, vector, matrx or tensor feld that s defned nsde the system 2. When the feld s a scalar, ths magntude m(x) s called mass; when the feld s a vector, ths magntude s called 4-potental vector e A(X); Let P(X) be the lnear momentum due to a group of magntudes. That s to sa for scalar and vector magntudes [1], the form of the lnear momentum s: P(X) = m(x) v + ea(x) On the other hand, a physcal system s defned by a functon that we call acton S (tradtonally scalar, although of tensor orgn [2], at the moment we wll not look at ts degree) that descrbes the system at ts maxmal way. That s to sa f S represents a system defned n the regon and P, as lnear momentum, represents the magntudes, then [3] (the negatve sgn only s tradtonal): Ω S = P (1) Ths equaton comes to say that, the complete defnton of a physcal system, to whch we call acton, s defned by the space-tme evoluton of the physcal magntudes that the system contans represented by the lnear momentum. If we reverse the expresson (1) we obtan: P = (2) x 2 The maxmum degree that a feld could have n a space of four dmensons s four (Refer to [1] physcs/ The acton as a dfferental n-form and the analytc deducton of the nuclear potentals ).

4 Ths expresson concdes wth the four-mpulse n classc theory of felds [4], when the acton s expressed as a functon of the coordnates. Ths second equaton says that, the physcal magntudes of a system are the varatons or non-homogenetes that there are n the system n respect to ther four-coordnates. Mathematcal representaton of physcal obects. Usuall the expressons of physcal systems use non-cartesan coordnate systems. These coordnates are called generalzed coordnates [5]. The reason for the coordnates change s to obtan smpler expressons for the system equatons. We get ths observng the symmetres. 0 1 A coordnate change mples the electon of some equatons Ψ = Ψ ( x, x,...) U that appled over the sets U ( Ω ) conform a mappng on Ù. Substtutng n (2) U we obtan: P k =. k x Usng the tensor form of the acton [1] of range one, we fnd: P k = (3) k x From the expresson (3) we deduce that, at frst, there are two ways to understand the physcal form of the Ø coordnates: Knowng the relatonshp Knowng k. x k, ust as we have made t at the begnnng.

5 The frst case s the tradtonal one. In ths case the geometrc symmetres nduce a change of coordnates that smplfes the equatons of the system. The cylndrcal, sphercal and hyperbolc coordnate transformatons belong to ths type. In the second case we can smplfy the equatons of the system f we look for a transformaton that s smple from the acton pont of vew. Let our system be formed by physcal obects n such a way that, P takes postve values nsde the physcal obects, whle ts value s zero outsde of such obects. In ths case, the functon P(x) for each pont of Ù, turns nto a dstrbuton functon for the magntudes that the system contans at that pont. In lne wth the expresson (3) the lnear momentum P has the form P(x) = P(Ø(x)). Snce P characterzes the physcal magntude, the functon Ø(x) represents a dstrbuton functon that characterzes the obect or obects of the system s magntude. That s to say: n the transformaton that smplfes the expresson k, the functon Ø(x) s the mathematcal representaton of the system s physcal obects. Quantum theory of felds. In quantum theory the obects (partcles) come to be represented by a functon Ø that s called wave functon. The square of ths functon s n fact a dstrbuton functon. The wave functon of a system s a maxmal representaton of the system, same as the acton functon s. That s to say that both, the acton S and the wave functon Ø represent the same physcal concept, wth the dfference that the acton has unts of power, whle the wave functon lacks of unts. An mportant property n quantum feld theory s that the wave functon has a tensoral character. Ths fact, far from beng a restrcton, actually t s a generalzaton of the concept of wave functon n classc quantum mechancs, besdes beng the natural form of deducng the spn concept.

6 Ths wa the degree of the wave functon tensor speaks to us of the spn of the system [6]. A system of zero spn wll be defned by a scalar functon, that s to sa t s a tensor wave functon of zero degree (wthout ndexes). A spn system ½ wll come defned by a vector functon, that s to sa a degree-one tensor wave functon (wth a sngle ndex). A spn one system wll be defned by a matrx functon, that s to sa a degree-two tensor wave functon (wth two ndexes). For the sake of smplcty n the expressons we wll use vector notaton, keepng nto account that ths restrcton n the notaton does not suppose a restrcton n the valdty of our expressons whose generalzaton to a system of generc spn s trval. makng The smplest form of relatng the acton S and the wave functon Ø would be k = cte, meanng a lneal relatonshp S ~ Ø. The proportonalty constant wll depend on the relatonshp exstng between the system and the wave functon. Such dfferences are the propertes that we observe n quantum partcles that characterze them aganst the tradtonal physcal systems. These propertes are: 1. The wave functon represents a very small obect compared to a tradtonal physcal system; therefore we wll use a small proportonalty constant havng unts of energ whch s represented by h. 2. The partcles are characterzed by an oscllatng behavour compared to the systems that are tradtonally consdered statc. Therefore, such a constant should contan the magnary factor = 1. Substtutng these two consderatons n the relatonshp between S and Ø, we obtan [1]: S = h Ψ (4) The proportonalty constant h has been named the Planck constant [7], t has unts of power and t s dependent of the unts system that we use. In nternatonal unts system h = J s The expresson (4) tells us that the wave functon represents a quantum system as a whole, wth the partcularty that the partcles, compared to our tradtonal

7 macroscopc observatons, have a lower magntude, equvalent to the Planck constant and a vbratory behavour. We also see that the expresson (4) forces us to consder the acton as a tensoral obect whose degree s equvalent to that of the wave functon of the system. Ths consderaton, same as n the case of the wave functon, t s a generalzaton of the partcular case of the scalar acton seen n tradtonal classc mechancs. Ths s even more ustfed than n the equaton (3) of the general form of felds theor whereas n quantum theory ths tensor expresson has a concrete physcal realty: same as the tensoral wave functon, the tensoral acton represents the spn of the system. The method for passng from a tensor representaton to a scalar one s performng the correspondng contracton [2]: S = = S S S. Applyng the property of tensors whereby the product of two tensors s constant and ndependent of the change n reference system and keepng n mnd that a varaton n the acton when ths s a functon of the coordnates s really a coordnates transform, we deduce that ä S = ä = 0 ; mathematcal expresson of the prncple of least acton. S Mathematcal representaton of physcal obects n quantum theory of felds. The form by whch the contaned obects n a system are represented n quantum feld theory s by means of the wave functon. Ths wa the exstent relatonshp among the mathematcal representaton of the quantum physcal obect (that s frequently called partcles assembly) and the system s: 1. The assembles of partcles represent very small obects n comparson wth the usual unts of the macroscopc systems. 2. The assembles of partcles are characterzed by a vbratory behavour, compared to the classc systems n whch, n the fundamental state of energy the physcal obects reman at rest.

8 These propertes, as we already saw, lead to the equaton (4). If we combne ths expresson wth equaton (3) generc for feld theor we deduce: P x k... k... k... ( Ψ ) = h (5) These equatons represent the canoncal quantzaton. From the tradtonal quantum pont of vew, the expresson (5) represents the dualty between the representaton of the partcle as a corpuscle n the frst term of the expresson and the partcle as a wave n the second term. The partcularzaton for lnear vector momentums, that s to sa wth a sngle ndex, leads to the tradtonally wellknown canoncal quantzaton [8]. The step from the expresson (5) to the transformatons of canoncal quantzaton of non-relatvstc quantum mechancs requres two addtonal consderatons: On one hand, the lnear momentum n the non relatvstc verson s deduced from the physcalgeometrc macroscopc condtons and not startng from the wave functon. On the other hand the wave functon does not represent to the complete system but only the partcle that s beng studed, consderng the rest of the system as a macroscopc obect. The result of these two consderatons becomes the transformaton of the expresson (5) nto the well-known transformatons of canoncal quantzaton. If we express them n components and makng c P 0 = - H beng c the lght speed and H the Hamltonan of the system, we obtan: ø Px ( x, ø ( x, = h, x ø Pz ( x, ø( x, = h, z ø Py ( x, ø( x, = h y ø H ( x, ø( x, = h t (6) When we ntroduce here as wave functon for non relatvst mechancs the functon ø nstead of Ø we want to emphasze the dfference among the non relatvst wave functon that represents a sngle partcle nsde the system and the relatvstc one whch represents the assembly of partcles that consttute the complete system.

9 Hstorcal reflecton. The development of the quantum and relatvty theores took place n a contemporary way durng the frst years of the XX century. Ths wa the expresson (2), for any type of metrc could not be deduced n a general way before 1915 (formulaton of the General Relatvty), whle the expressons (6) became known from 1923 (formulaton of the equaton of Schrödnger). Ths small tme dfference caused that both theores were developed n an ndependent wa ust as f dd not exst any relatonshp among them. However, f the expresson (2) had been deduced long before, the reasonng that leads us from ths equaton to the expresson (6) would have become natural, n lack of knowng the value of the Planck constant (ust as t happened wth the law of Avogadro n whch the value of the constant was dscovered much later). Regardng the expresson (4), the relaton between S and Ø, where the acton s a magntude related wth the space-tme, whle the wave functon represents the matter, can only be deduced n a natural way consderng the dualty between both concepts. If the space s represented n a real wa the matter should be represented n a non-real wa that s to say magnary. Ths form of reasonng s completely smlar to the one used by Fermat whle he was establshng the geometrc optcs n whch the conc (?) functon ψ representng the propertes of the lght beam s related wth the matter by means of a relaton that nvolves the magnary constant: ψ f = ae [4]. Concluson. From a mathematcal pont of vew the analyss of a physcal system supposes to choose an expresson for each physcal magntude and to relate such expressons to obtan the equatons that descrbe the system to us.

10 The expressons that are used for the physcal magntudes wll determne the mathematcs branch that wll be appled and therefore the nterpretve apparatus that wll allow to translate the mathematcal results nto physcal results. Consequentl the unfcaton of the dfferent branches of physcs, especally the classc and quantum physcs, should be focused on the search for common and general mathematcal representatons whch smplfcatons, n functon of the propertes of the systems, are the source of the present dversty of physcal theores. References [1] E. Orda The acton as a dfferental n-form and the analytc deducton of the nuclear potentals. Physcs/ (2003) [2] E. Orda Tensoral orgn of the acton functon, nexus between Quantum physcs and General Relatvty. Physcs/ (2003) [3] E. Orda Causalty and tme-space evoluton of physcal systems. Physcs/ (2003) [4] L. D. Landau and E. M. Lfsht The classcal theory of felds. (1975) [5] L. D. Landau and E. M. Lfsht Mechancs. (1976) [6] F. J. Ynduran, Relatvstc quantum mechancs. (1996) [7] M. Alonso and E. J. Fnn, Quantum and statstcal physcs. (1968) [8] B. Felsager, Geometr partcles and felds. (1981)