Alam, Md Shah

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Indian J. Phys. 77A (1), 47-49 (2003)U P  Aan intcrnationai journalComparative study of mixed product and quaternion productMd. Sihali AlaniIDepartment of Physics, Shahjalal jUnivcrsiiy of Science and Technology, Syihet»|6angiudcshE-inail «lamC»'sust.eduR e c e i v e d  12 S e / t f em h ei  2 ^ 2 .  a c c e p t e d  2 9  O c t o b e r  2 0 0 2A b strac t Mixed iiuinl^i is the sum of a scalar and a veclor The qqaicimon cun also be written as the sum ot a scalar and a vector but the productol mixed numbers and the product of quaternions are different. Here, we titudied the mixed product which is derived from the pioduet ot mixed numlK'rs and the quaternion product which is derived from the product of quaternions It was observed that mixed product is more consistent with Physics than that ot quaternion product.K eyw ords . Mixed number, mixed product, quaternion product PACS No. 02 SH) +p1. IntroductionLI. Mixed product:M ix e d  number [ 11 a  is the sum of a scalar x and a vector A  ; t.e. a  = .V"bA.The product of two mixed numbers is defined as [ 11a/3 = (.V + A) (,v + i?) = xy A.B-¥ xB + \A->tiAxB. ( I)Taking .v = >- = 0, we get from eq. (1)A® B = A.B + tA x fi. (2)This product is called mixed product 12] and the symbol ® IS chosen for it.L2 Quaternion product:The quaternion can also be written as the sum of scalar and a vector[31/>. A = ^i-bA.The multiplication of any two quaternions A = + A and is given by [3-5JCorresponding AuthorAB  = {a + A){h + B) = c i h - A , B ^  aB -f M  + A x Z?. (3)Taking a - h -  0, we get from cq. (3)^  = - A .B ^ A x B .  (4)This product we call quaternion product.2. Consistency of mixed product and quaternion product with physics2. / Consistency w ith Pauli matrix alRchra :It can be shown that |61ia .A )(a .B )  = A,£? +/<T.( A x B ) , (5)where A and B arc two vectors and a  is the Pauli matrix.From cqs. (2) and (5), we can say that the mixed product is directly consistent with Pauli matrix algebra. From eqs. (4) and (5), wc can also say that the quaternion product is not directly consistent with Pauli matrix algebra.2.2 Consistency with Dirac equation :Dirac equation ( E - a .P  - PnOy/ = 0 can be operated by the Dirac operator (t -  a.V  -  p n ), then wc geti t - a .V  - p n ){ (E -a ,F  - p m )\i/}^0 . (6)©2003IACS48 Md Shah AlamFor mass-less particles i.e. for m = = 0, we get [7] (r-a.V)(E-a.P)v' = {/£’+V.P-f/a.(VxP)}v/i+{-(ra .F  + Ea.V  )}v^ 2 = 0 )where Y  is the wave function and y/  ^ and Y i  the components of V .Pulling f = 0 and E = 0 in eq* (7), we gel(a .V )(a.P)\!f = \y .P + ia .(V 'K P )}\|/^  = 0 . (8)From eqs. (2) and (8), it is clear that Mixed product is consistent with Dirac equation.From eqs. (4) and (8), it is also clear that quaternion productIt can be shown that V0(V(8>E) = V^E. (13)From eqs. (12) and (13), we can writeV^E = HQE^{d^E)Kdt^)  which is exactly same as shown in cq. (10).Similarly using mixed product, it can also be shown thatTherefore, mixed product can be used successfully in dealing with differential operators. Using the definition of quaternion product (eq. 4), it can be shown that quaternion product can not be used in dealing with differential operators.4. Elementary properties of these productsis not consistent with Dirac equation. 4. ] Elementary' properties o f mixed product:3. Applications of these products in dealing with differential operators(i) Mixed product of two perpendicular vectors is equal to the imaginary of the vector product of the vectorsIn region ofspacc where there is no charge or current. Maxwell's equation can be written as(ii) Mixed product of two parallel vectors is simply the scalar product of the vectors.(i) V.E = 0, (ii) V x E ^ - ( d B ) K d t ) .(iii)(iv)It sati.sfics the distribution law of multiplication. It is associative.(iii) V.B = 0, (iv) V x B  -  ! (dt).  (9)From these equations, it can be written as [8]V -B = tlo C o (d ^B )n d tb .  (10)Using eqs. (2) and (9), we can write V ® £  = V.E + i V x £= 0 + - /(5 f i) /(5 f) .or V ® E - - H d B ) / ( d t ) ,  (11)V ® ( V ® £ )  = V ® { - i (^ B ) / (5 / ) }= - i i d f d t )  {V® £}- - i ( a / a / ) { V .£  + /V x £ }- - i { d / d t ) { 0+i f i o£Q{ dE) / { 9 t ) } ,V ® (V ® £ ) = /io£„(<9^£)/(^f2^ ^j2)oror4,2 Elementary^ properties o f quaternion product:(i) Quaternion product of two perpendicular vectors is simply the vector product of the vectors.(ii) Quaternion product of two parallel vectors is negative of the scalar product of the vectors.(iii) It satisfies the distribution law of multiplication.(iv) It is non-associative.5. ConclusionA comparison of Quaternion and mixed products is given hi Table 1. From the table we may conclude that mixed producl i^  more consistent with different laws of Physics than that Quaiernion product. From the table we may conclude that mixetl product is more consistent with different laws of Physics than the Quaternion product.AcknowledgmentsI am grateful to Mushfiq Ahmad, Department of Physics. University of Rajshahi, Rajshahi, Bangladesh and Prof. Habibul Ahsan, Department of Physics, Shahjalal University of Science and Technology, Sylhet, Bangladesh and Prof, Anwarul Haque Sharif, Department of Mathematics, Jahangir Nagar University Dhaka, Bangladesh for their help and advice.Comparative study o f mixed product and (luaternion product 49T able  1. Compun. (^>n of cjualcrnion product and mixed pioduclQuaternion product1. Mathematical expression AB -  A B - A x B2. C onsistency  w ith Pauli matrix algebra3. C onsistency with Dirac equation4. In dealing with differential operators5. Elementary propertiesMixed productIt is not directly consistent with Pauli matrix algebraIt is not conijSstent with Dirac equation .It can not b^ used in dealing with differeitiai operators(i) Ouatemion product of tw o Ip e rp e n d ic u la r  vec lo ri is sim ply the vectorfproduct o f the v ec to ri(ii) Quaternion product of two parallel vectors is negative of the scalar product of the vectors.(iii) It sa tis f ie s  the d is tr ib u tio n  law  of multiplicalitMi.(iv) It IS lion-associative.A ^ B -  A .B + /A x f lIt is directly consistent with Pauli matrix algebraIt is consistent with Dirac equationIt can be used successlully in dealing with differential operators(i) M ixed product ol two perpendicular vectors is equal to the imaginary of the vector product ot the vecliirs(ii) Mixed |)rodiict of two p a ra lle l v ec to rs  is sim ply  the sc a la r  product of the vectors.(lii) It sa tis f ie s  the d is tr ib u tio n  law ol multiplication.(iv) It is associativeKi'krcnccs111 Md vShah Alain Siiuly nf Mixed Numhei Pkh Pakislan Ann! Sa 37 119(2000)|2 | Md Shah Alain Mixed Funhut of Verlois J Theoielus Vol 3 ( 2001 )I M A Kyrala Theoietical Physics (Philadelphia & London • W B Saunders Company) (Toppan Company Limited Tokyo. Japan) (1907)|4 | hup //malhwoild wolliam eom/Quatnmon himl[51 htlp://uww cs appslalc edu/-'S)g/class/^ 110/in.UhlcsUilj^2000/ quaternions 1 htinl|6] L I SchiO Qiumtimi Meihanus (New York M tdiau Hill)(7] Md Shah Alam Sfiohhir Plwsi^  (Doparlment ol Physics, lunversily of Ra)shaln, ka)shahi, Bangladesh) (1994)|8 | David J Ciriflitlis Innodui ikoi Elei tuHhiuimu \ scsoiid cd (New Delhi Prentice Hall ol India) (1994)

Comparative Study of Mixed Product and Quaternion Product

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Comparative Study of Mixed Product and Quaternion Product

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