Four-vector, Dirac spinor representation and Lorentz Transformations

Transcription

1 Available online at Advances in Applied Science Research, 2012, 3 (2): Four-vector, Dirac spinor representation and Lorentz Transforations S. B. Khasare 1, J. N. Rateke 2, Shashank S. Khasare 3 ISSN: CODEN (USA): AASRFC 1 Departent of Physics, Science College, Congress Nagar, Nagpur, India 2 Departent of Physics, Shri Maturadas Mohota College of Science, Nagpur, India 3 Departent of Coputer Science, College of Engineering, Pune, India _ ABSTRACT Author found a siple geoetrical representation of the general covariant for of Maxwell's equations and few related relations. We ention here briefly another equivalent way to forulate the geoetrization of the four fields, totally need Dirac spin atrices which are used as vectors, do not required tensors knowledge but leading to siplifications of the four vectors, three vector physical quantities. Key Words: Four-vector, Dirac spinor, Lorentz Transforations, SL (2,C). PACS: p (Special relativity);03.50.de (Classical electroagnetis, Maxwell equations); Gz (Spacetie topology, causal structure, spinor structure) _ INTRODUCTION Syetries are of fundaental iportances in the description of physical phenoena. The discovery of a non- Euclidean geoetry posed an extreely coplicated proble to physics, particularly explaining real space was Euclidean as have been believed earlier and if it was not, to what type of non-euclidean space it belonged. Thus, it is necessary to see the validity of the axios experientally or extension of Euclidean space to new space, so that the construction of spin-geoetries could be justified by the possibility of applying their conclusions to actually existing object and the observation. The fact that these conclusions are expressed in ter of geoetry is of no real consequence. As to the geoetry-structure of real space, coes within the doain of physics[1-3] and cannot be resolve by eans pure geoetry. Present forulation provides a better description of actual spatial relation than earlier workers. It is well established that the theory of relativity uses the forulas of non-euclidean geoetry but it never says that the Euclidean's geoetry ust be discarded. In the pseudo Euclidean space, tie coordinate have different footing than the space coordinates. Both geoetries are the tools for investigating spatial fors but the non-euclidean enables finer studies to be ade in the light of preconceived inforation. It is the confired fact that physical phenoena do not appear sae to the other observers in the relative otion, with respect to each other, although the physical law ust be sae for all observers (inertial frae). Thus the principle of relativity asserts that two observations/observers will describe a physical process by sae equation whenever they are stationary or in the unifor linear otion with respect to each other. A transforation of space-tie that aps any observers a reference in four-diensional spaces into an equivalent one cannot affect the description of the physical processes. Naturally such a transforation obviously fors a group. The principal of relativity does not deterine group distinctively because an additional postulate is required for such purpose. For unique identification there are three possibilities. 1. That two observers are equivalent only if they are at least rests with respect to each other, which consist of all 749

2 rotation and transforation in three- diensional space. 2. If in addition, equivalent observers are allowed to be in relative otion and tie reains absolute, we have a relativity group of Newtonian echanics i.e. Galilee group. 3. If instead of above the two possibilities light propagates with sae speed for every observer (inertial frae), which is now incopatible with absolute tie, we obtain the inhoogeneous Lorenz group also called Poincare group. This is the relativity group of Einstein's theory of special relativity. Relativity group adapts a classical echanics to the syetry properties inherent in electroagnetis the principle of relativity can be extended to the observers in gravitational field, but interpretation of general relativity in-group theoretic ter is no longer straightforward. Once the relativity group of the theory deterined, the principal of relativity ust be put in action. But every theory has its particular advantages and drawbacks. The ai of all continuu theories is to derive the atoic nature of the electricity fro the property that the differential equations expressing the physical law have only discrete nuber of solutions which are everywhere regular static and spherically syetric. In particular one such solution should exist for each of positive and negative kind of electricity. It is clear that the differential equations, which have this property, ust have coplicated structure. It sees that such coplexity of physical law itself speaks against the continuu theory. Thus it is required fro a physical point of view that the existence of an atoicity is itself so siple and basic, it should also be interpreted in siple and eleentary anner by theory and should not speak, appeals as a trick in analysis. The continuu theory forced to introduce a special forces, which keep the coulob repulsive forces in the interior of the electrical eleentary particle in equilibriu. If such forces are electrical in nature, then we have to assign an absolute eaning to the potential in the doain of four-vector and three-vector which leads to the different types of difficulties. 2. Four Vector Algebra: Indifference to location of the origin of coordinate syste is called hoogeneity of space and indifference to direction of axis is called isotropy of space. The requireent of hoogeneity of space is expressed by invariance of equations with respect to shift of origin. The equations are said to be invariant when they preserve their fors on transforation to another inertial reference frae. If the value of soe physical quantity reains the sae after transforation fro one frae of reference to an-other, then that quantity is said to be invariant. The requireent of isotropy of space is expressed by requiring covariance of our equations with respect to the rotation of the axis of reference frae. The equations that describe the physical law ust be covariant in for i.e. its for is independent of the choice of the inertial frae or both the side have the sae tensor character. Thus a scalar cannot be equated to a coponent of a vector nor can one ter in a su be a tensor of second rank, while another one, a tensor of first rank. If we express the physical law in four-vector/tensor for/equivalent for, then the resulting equations will be autoatically be covariant with respect to a given class of transforations. Fro the relativity, we have following basic quadratic expression as (P o / o c) 2 - (P/ o c) 2 = 1 (1) 3 Here we view above equation by defining any physical quantity say A, A ± = (A o, ± A) (2) and another physical B as, B = ( B o, B ) (3) Here we postulate that any physical vector quantity ust be represented through following apping given below A a A σ = A a σ a + A b σ b + A c σ c (4) And 750

3 B a B σ = B a σ a + B b σ b + B c σ c (5) and any physical scalar quantity through following apping given below A a A σ = A o σ o (6) Where we have following atrix algebra. σ a σ b = j σ c, j 2 = -1 (7) a,b,c are cyclic and σ a σ a = σ b σ b = σ c σ c = σ o σ o = σ o (8) So that one can handle siultaneously scalar, vector quantity, scalar product and vector product easily seen for following equation. A B = ( A B ) + j ( A B ) (9) Now we define product rule so that new physical quantity such that its existence would depend upon as A and B given by following equation A ± B ± = C ± (10) C ± = C scalar (C polar ± j C axial ) (11) C scalar = ( A o B o - A B ) (12) C polar = ( A o B - A B o ) (13) C axial = ( A B ) (14) This definition show that C contain scalar, polar-vector and axial-vector parts whereas A and B contain scalar and vector part. Let I o = γ And I = γ β So that we can have unitary quantity by following equation. I ± = ( I o ± I ) (15) I = ( I o I ) (16) so that it yield well known identity given below. I ± I = ( γ 2 - (γ β) 2 ) σ o = σ o (17) 751

4 Now using above postulate we have following expression for oentu as P ± = ( P o, ± P ) = ( P o ± P ) = P o σ o ± ( P a σ a + P b σ b + P c σ c ) (18) and if we use idea of quantu echanics the we have following apping ( P o, ± P ) j ћ ( o, ) (19) which can be used to obtain differential of physical quantity. 2.1 Invariant Physical Scalar Quantity and Four-vector representation: Now with the help of above set of postulate, definitions and equations only we can easily construct, different basic invariant physical quantity and its corresponding four-vector representation very easily. The coponents of fourvector velocity can be obtained fro an observed invariant velocity of light c as a transforation relation as U ± = I ± c = ( γ ± γ β ) c = (γ c, ± γ V) (20) Siilarly four-vector for oentu can be obtain as P ± = I ± o c = ( γ ± γ β ) o c = ( P o, ± P ) (21) It is known in the context of special relativity that a charge distribution that is static in one frae will appear to be a current distribution in another inertial frae. It iplies that the current and charge densities are not distinct entities and their relationship ay be presented with the help of rest charge density as given below. J ± = I ± ρ o c = ( γ ± γ β ) ρ o c = ( J o, ± J ) (22) Lastly we would construct four-vector potential fro scalar potential as given below. A ± = I ± φ = ( γ ± γ β ) φ = ( A o, ± A ) (23) 2.2 Invariant Physical Vector Quantity and Four-vector representation: If the oentu of a particle P = o V then its four-vector would be as P ± = I ± o V = ( γ ± γ β ) o V = ( ± β P, P ) (24) If the force on a particle is F then Minkowski force can be soothly obtained as K ± = I ± F = ( γ ± γ β ) F = ( ± γ β F, γ F ± j γ ( β F ) ) (25) Above equation shows that ( β F ) = Transforation of Four-vector Quantity: General transforation properties of any four-vector J (charge-current) are given below. I ± J = J * (26) 752

5 J * ± = J * scalar ( J * polar ± j J * axial) (27) J * scalar = ( I o Jo I J ) (28) J * polar = ( I o J I J o ) (29) J * axial = ( I J ) (30) 2.4 Electrodynaics: We can obtain definition of electric field, agnetic field and Lorentz condition siultaneously when we consider following expression for force which can be obtained with the help of four-differentiation of four-vector potential i.e. F ± = ( o ± ) ( A o ±A ) = ( G scalar, ± ( G polar ± j G axial ) ) ; (31) G scalar = ( o A o + A) = 0 (32) G polar = ( o A + A o ) = E (33) G axial = ( A ) = B (34) Hence an easy path to get Lorentz condition along with an regular definition of electric field and agnetic field. In the next step,using the four-differential operator we can easily obtain a set of hoogeneous/inhoogeneous Maxwell's equations siultaneously fro following single equation. ( o ± ) ( ± E + j B ) = ( 4 π c ( J o, ±J ) (35) ( 4 π c ) J o = E j B (36) ( 4 π c ) J = ( B o E ) j ( E + o B ) (37) So on separating real and iaginary part we have a stateent of Coulob's Law. ( 4 π c ) J o = E (38) and showing that absence of free agnetic poles. B = 0 (39) Siilarly we have a stateent for Apere's law. g ± * = g * scalar ( c g * polar j c g * axial ) (52) g * scalar = γ ( g o V g ) (53) 753

6 c g * polar = γ ( c g β g o ) (54) c g * axial = γ ( V g ) (55) But for electrostatic configuration the agnetic field in K is given by B = β E so that ( g o V g ) = ( E E B B ) (8 π ) (56) Let the interaction between electroagnetic field and four-vector potential eld a physical quantity be represented as a ± = A ± F ± = (A o ± A) ( ± E + j B) = ( A E ± (A o E + A B) (57) and iaginary part of four-vector is ( ± A B + ( A o B A E ) = 0 (58) which is zero. So that we can easily construct following very useful expression in a direction to construction of Lagrangian density for electroagnetic field-particle interaction as L ± = ( o ± ) ( a o ± a ) = ( L scalar, ± ( L polar ± j L axial ) ) (59) Now electroagnetic Lagrangian density for field- field and field-particle could be obtain fro single equation L scalar = ( o a o + a ) = 0 (60) along with following conditions. L polar = ( o a + a o ) = 0 (61) L axial = ( a ) = 0 (62) Invariant interaction between the charged particle and eld is J ± A = O (63) O ± = O scalar ± ( O polar ± j O axial ) (64) Following are the natural condition that the interaction between the charged particle and field to be invariant. O polar = ( J o A J A o ) = 0 (66) O axial = ( J A ) = 0 (67) 754

7 We can define generalize oentu using superposition principle as P ± = ( P o + e A o /c ) ± ( P + e A/c ) (68) Taking four-differentiation of above generalize oentu we have equivalent stateent of Newton's law for a particle- field interaction where ( P o c ) represent a potential energy and the ter o ( P c ) = F (69) is interpreted as a rate of change of oentu of a particle F ± = ( o ± ) ( P c ) = ( F scalar, ± ( F polar ± j F axial ) ) (70) with additional requireents in the for of an equations i.e. equation of continuity etc. F scalar = ( o ( P o c + e A o ) + ( P c + e A ) ) = 0 (71) and Newton's stateent under equilibriu condition as F polar = ( o ( P c + e A ) + ( P o c + e A o ) = 0 (72) Note that curl of a particle oentu and agnetic field are directly connected. F axial = ( ( P c + e A ) ) = 0 (73) The Lagrangian treatent of echanics is based on the principle of least action.in a nonrelativistic echanics the syste is describe by generalized coordinate and velocities i.e.(a displaceent, oentu representation). The Lagrangian L is a functional of coordinate and velocities and the action A is de ned as the tie integral of Lagragian L along a path of the syste i.e. in integral for we have a following definitions. P ± = ( ( P o + Q o ) ± ( P + Q ) ) (74) ds ± = ( dx o dx ) (75) P ± ds ± = Ps (76) Ps ± = Ps scalar ( Ps polar ± j Ps axia l ) (77) Ps scalar = ( ( P o + Q o ) dx o ( P + Q ) dx ) (78) Ps polar = ( ( P o + Q o ) dx ( P + Q ) dx o ) = 0 (79) Ps axial = ( P + Q ) dx = 0 (80) 755

8 L dt = P± ds (81) So that we have relativistic Lagrangian for a siple particle as given below. L = o c β ( Q o c Q v ) (82) ( Q o c Q v ) = φ(x) ( 1 β 2 ) (83) On siilar line the Lagrangian for a single particle in an electroagnetic field could be defined. CONCLUSION Author would like to ention that no references in particular are cited in the paper, and have only referred the well established theoretical developent reported in classical text book[1-3]. Hence found another equivalent way, an interesting siple representation technique with siple apping to obtain covariant for of physical quantity along with additional relations. REFERENCES [1] W. Pauli. Theory of Relativity, (Translated fro Geran by G.Field). [2] J.D.Jackson. Classical Electrodynaics, (Wiley Eastern Liited). [3]Siegfried Flugge. Practical Quantu Mechanics, (Springer International). 756