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FIELD EQUATIONS Kdward Raport nr 1275/PH ON TUK •RivLATION BLTVivEN CALI Iii AN, POINCARE AND EUCLIDliAI.' FIELD EQUATIONS Kdward KAl'USCIi: Institute of Nuclear Physics, Department of Thooroćieal Physics 152, 31-3^2 KraKów, Poland .Jsriuary 198 5 Abstracts A nev kind of relation between Galilean, Poinearc and Kuelidean field equations is demonstrated. 1. Introduction The aim of the present paper is to show that tliore exists a new kind of relation between the Galilean*and rolativistic field equations which is different from the relation discussed in Ref.£i], Tho material presented definitely contradicts the standard understanding of the non-relativistic physics as the limiting case of the relativistic one. In our procedure the Galilean covariant field equations are the primary objects and form an universal basis for all other field equations. In par¬ ticular, from the Galilean field equations wo obtain both the Poincare and Euclidean field equations and this is true not only for free fields but also for the coupled fields in the abelian gauge theory. Needless to say, tho considei'ed problem is very important for n deeper understanding of the meaning of the relativity principles in physics. After presenting the main results wo try to interpret them in a more general way usin{j the modern lan¬ guage of fibre bundles. In our formulation, various relativity principles are implemented by different maps from a universal bundle space to tho physical space-tines and these maps in turn are determined by particular choices of classes of solutions of tho Galilean field eqations. 2, The l:iolati-i'is t.ic" properties of the Galiler-n field equa tion In Hef . [3.1 we hnvo introduced the most fjenonU Galilean invariant field equation in the form vhere y. U . C/ aro the five coordinates of the Galilean space-time and oi } ft ^ V are arbitrary dimensional constants which should bo expressed in terms of primary param¬ eters of the particle described by the field 'VJ/ . In Ref. [2] we adnittcd only the rest energy Sc anc* the I'laiick constant ji as such primary constants and this led to the choice it turns out, however, that raore interesting- results nay be de rived if we include into the set of primary constants also the r.iciss rfl of the particle. Vith such set of primary constants instead of (2.2) we mrxy then make the choice fi'-O which improves the behaviouc X>£ (,2.0 with rospact to time inversion and also leads to the usual Schrodinger equation -when we pass from tho one-parameter exteusion of the Galilei group to the usual Galilei group. Indeed, taking in (2.1) wo get for tho wave function tp (*>£,) the SchrSdinger equa¬ tion with rest energy jc and mass m • The field equation (2.1) with the choice (2.3) besides tho solutions of the type (2,^0 possesses many other solutions. In particular, let us consider- the solutions of the type where U.(X,T) are twice differentiable functiona. Substituting (2.5) into (2,1) and taking the relativistic relation = me1 (2.6) vo got tho Klein-Gcrcion wave equation where O = Z*vŹ*-~k' (2.8) Sijnilarly, considering solutions of (2.1) of the type nd taking (2.6) we get the liuclidean field equation whore /^i. is the four-dimensional Laplace operator i We therefore see that equation (2.0 leads to all free field equations used in physics for scalar particles and we should somehow explain the physical weaning of this situation. Defore doing this let us show that the same situation is vnlid for other Galilean field equations. As the first example, lot us take the Dirac equation which in the extended Galilean space-time has the form (see Appendix) where Y V and \/c. are the; usual Dlrac matrices. Talcing in this equation 'W in the form (2.'0 we f;et the non-relativistic Dirac equation for the spinor which, however, is different from the non-relativistic Dirac equation derived in Ref,\3\• Ths reason for such a discrepancy is that in Ref. [3] only the singular case Si. - 0 j Yft ^ 0 i<af considered. Me shall not consider this singular case here because it has no reJ.a tivis tic counterpart. It is easily seen that taking in (.2.12) 'U/ in the form of (2.5) ve sot for the spinor txC*)^) the usual relativ¬ is tic Dirac equation where while taking \U in the form of (2,8) we &ct for the spi nor Z C*;^) the Euclidean lijrao equation This shows that the relativistic wave equations follow from the Galilean equations not only for the case of scalar fields but also for the case of spinor fields. It is the choice of tho solutions of the basic Galilean field theory which determines the character of the wave equation t It is easy to see that the saftic is t>lso true for inter¬ acting fields, provided the particular choice of the solutions of the Galilean field equations does not introduce constraints for the fields. An example of the theory where such constraints arise is the Galilean gnuge field theory and we shall discuss this case in detail. The field equations for the Galilean £au#e theory arc of the form £2]: and where C is an arbitrary constant of the dimension of the squ¬ are of velocity. In the relativistic case we shall identify this constant vith the square of the velocity of light. For the solutions of the fielf equation (2.18) of the type (2.5) we have the identity a. (2.19) - C and in the order to preserve this identity also for tho co-variant derivatives we need tho following relation between the gauge fields Vith this relation the left-hand aide of (2.17b) vanishes, while the right-hand side in general does not. We must there¬ fore include the constraint relation (3.20) in the Lagxangian for the gauge fields by means of a Lagrange multiplier, '.;e shall do this by introducing into the total Lagrangian of the gauge theory a gauge fixing term of the form where X(y,^$) is a Galilean scalar auxiliary field and ( rt. n* no) is an arbitrary constant fivo-vector in the ex- tonded Galilean space-timo. The expression (2.2i) is then a Galilean scalar and tho theory with this term in the Lagran- gian remains to be Galilean covariant. Tho presence of (2,21) in the Lagrangian adds to the left-hand sides of (2.1?a), (2.17b) and (2.17c) the terms -} n. , } fl± and ^ n.© , re¬ spectively, and the variation with respect to /k gives the gauge fixing condition (2.22) Obviously, for the Galilean gauge theory we may always choose O,= KU= YIQ-0 . Now, choosing fl<* 0 j fi^-~C f HQ - d we get the constraint (2.20) and from (2.17b) wo get Substituting this into (2.17c) we get tho equation • D W - itf i.hich is tho con-ect relativistic wave equation for the field W^t). The remaining equations (2.17a) and (2.18) also automatically tijko the rolativistic form. X similar discussion can be made for the Euolidean type of solutions. Here the fields do not depend on "t and therefore we need in the gauge covariant derivative the relation V,Te shall take into account this relation by chosing in (2.22) (2.17b") •and(2.17c) we then get (2.26) and the Euclidean gauge theory is obtained. f 3. l"he interpretation of the results, We shall now interpret our results in the modern language of fibre bundles. As it is well-known [h] , in order to define a fibre bundle we must indicate three objoctsj the total space of the bundle P, the base space M and the projection fT ' T5-^ M » For our purpose we shall define the total space P as the five dimensional vector space R-. with coordinates rev us ( H4. ••» Hif') » P i° iy denoted by x( t and Q . We assume that each phyisical elementary event p is repre¬ sented by a point in P and the field phenomena are described by some fields \KV(HI ... L(<"^ which satisfy the equations of a given Galilean field theory. The space-time coordinates of 10 the event p are represented by a point in the four-dimensional i c A i 1 \ veotor space M with coordinates (X ,A , # ,X } which are linear combinations of the coordinates u« . The pro jeotion fT is implo- mented as the linear relation 1^^ --0,1,1,1). (3.0 Since the physical coordinates X^" depend on the chosen referen¬ ce system the projeotion TT and therefore the coefficients oC^ and ot' change with the reference system. Ve are therefore deal¬ ing not with a single fibre bundle ("P,1/^ M) but with a oolleo- tion of fibre bundles ("p 1t(o)}t^) with fixed total and base spaces and variable projections, and the notation lt{o)indicates the dependenoe of the projections on the observers or referenoe systems. If the observer 0' with coordinates x'^ is related to the observer 0 with the coordinates x^* by a given relati¬ vity principle desoribed by a symmetry group of the base space, we must have X'^rL^x'+a/', (3.2) where L v represent the homogenous part of the symmetry group and Q. desoribe the translations. Combining (3.1) with (3.2) we see that the coefficients oC ^ and oL *" of the projection Tł{o') ar® related to the coefficients oćr ^ and otr of the projection TT(O) by the foroulaet (3.3) * and these formulae show that in order to specify the pro joctj <-,: TrC^) it -'-^ sufficient to .specify Lho coefficient:; dL an<"' ol^ in ono arbitrarily eho'-cn I'ofnrorifc systom.
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