Application of Bi-Quaternions in Physics

Transcription

1 Appliation of Bi-Quaternions in Physis André Waser * First issued: 9.7. Last update: This paper introdues a new bi-quaternion notation and applies this notation to eletrodynamis. A set of extended MAXWELL equations and other fundamental equations of eletrodynamis are derived. By applying the LORENTZ ondition, these equations redue to the lassial form. Additionally the bi-quaternion notation allows a ompat formulation of SRT. Furthermore an appliation of bi-quaternions in other disiplines of physis as mehanis (dynamis) is shown. Introdution One of the most emotional disputes in the late nineteenth-entury was about the mathematial notation to use with eletrodynamis equations []. Today s vetor notation was not fully developed at that time and many physiists one of them was James Clerk MAXWELL promoted the quaternion notation. The quaternion was invented in 1843 by Sir William Rowan HAMILTON [6]. Peter Guthrie Tait [11] was the most outstanding promoter of quaternions. On the other side Oliver HEAVISIDE [7] and Josiah Willard GIBBS [13] both deided independently that they ould better apply a part of the quaternion number than use the entire number, why they proeeded further with that, what today is alled the vetor notation. Generally the vetor notation used in pre-einstein [4] eletrodynamis uses three-dimensional vetors. The quaternion on the other hand is a fourdimensional number. To make the quaternion ompatible to the three-dimensional eletrodynamis of MAXWELL, HAMILTON and TAIT, the salar part of the quaternion was indiated with the prefix S and the vetor part with the prefix V. By doing this, the quaternion has been vetorized. This notation was also used by Maxwell in his Treatise [9], where he published some equations with vetorized quaternion notation. But with applying this prefixes the whole benefit of quaternions is not used. Maxwell didn t use any quaternion alulus. He only onverted some summarized end results into vetorized quaternion notation. The quaternion notation was simply not as powerful and suitable as the vetor notation. Some time ago we have shown, that the introdution of bi-quaternions a omplex expansion of quaternions makes a ompat formulation of eletrodynamis in spirit of TAIT s and HAMILTON s work possible. This artile is a ontinuation of TAIT s and HAMILTON s ideas and shows the broad appliability of bi-quaternions in physis. * André Waser, Birhli 35, CH-884 Einsiedeln opyright (7) by André Waser; Page 1

2 HAMILTON s Quaternions A general quaternion has both a salar (real) and a vetor (imaginary) part. In the example below a is the salar part and ib + j + kd is the vetor part. Q = a + bi + j + dk (1.1) Where a, b,, and d are real numbers and i, j, k are so-alled Hamilton ian unit vetors with the magnitude of 1. They satisfy the equations i = j = k = ijk = 1 (1.) ij = k jk = i ki = j ij = ji jk = kj ki = ik A nie explanation about the rotation apabilities of the HAMILTON ian units in a threedimensional ARGAND diagram was published by GOUGH [5]. A quaternion is a hyperomplex number. The quaternion radii (or magnitude) in four-dimensional spae is defined similar as for ordinary omplex numbers as: Q x + x + x + x (1.3) 1 3 as also shown by WALKER [1]. By introduing a onjugate quaternion number Q* = a - bi - j - dk (1.4) the quaternion magnitude is also * Q QQ = x + x + x + x. (1.5) 1 3 Suh a four-dimensional quaternion is very suitable to represent an event in fourdimensional spae: X = x + x i+ x j+ x k. (1.6) 1 3 From now on we use the following onvention for indies and unit vetors: Indies k = 1,, 3 Indies j =, 1,, 3 =, k The unit vetors of three-dimensional spae are e 1, e, e 3 = e k The HAMILTON ian units i, j, k are written in itali letters and the imaginary unit i= 1 in normal letter. Then we an summarize the HAMILTON ian units and the unit vetors to i ie, je, ke ( ) 1 3 and we find the following notation for a quaternion X j X = x + i x j k (1.7) (1.8) Page opyright (7) by André Waser;

3 Calulus wit Quaternions We introdue the following two quaternions X = x + i x ( ) und Y = ( y + i y) A onjugate quaternion X * is then: * X = ( x i x ) The salar multipliation X Y is: X Y = Y X = x y + x y ( ) ( ) i ( ) ( ) i ( ). (1.9) (1.1) The multipliation XY is: XY = x y x y + x y + xy + x y YX = x y x y + x y + xy x y (1.11) The magnitude of the quaternion X is: X = X X = x + x x Then we have also * * XX = X X = x + x x = X = X X (1.1) (1.13) The inverse quaternion X -1 is * * 1 X X x i ix = = = * (1.14) X XX X x + x x And the division Y/X is: * Y YX ( y + i iy)( x i ix) = = * (1.15) X XX x y + x x opyright (7) by André Waser; Page 3

4 Introdution of Bi-Quaternions An expansion of quaternions to bi-quaternions is made by replaing the real numbers a, b, and d with omplex numbers. A (omplex) quaternion or a bi-quaternion is then: = x + ix + i ix + x k (1.16) ( ) ( k ) This new introdued quaternion is different from the otionions (known form LIE algebra) sine the HAMILTON ian units i, j, k are still valid and no new imaginary units are introdued. Suh a bi-quaternion now represents a superposition of two four-dimensional numbers: = + j = x + i ix + ix + i x k (1.17) ( ) ( ) j k The later shown examination with this eight-fold number shows, that suh a biquaternion is very useful for a ompat desription of eletrodynamis and other disiplines in physis. But first we have to deide whether we use j or j as our representation of a physial four-vetor. Easy alulations show, that both lead to the same result. Therefore we are free to hoose between j and j. To still have a real term, we deide for j and apply this part of the bi-quaternion to physial four-vetors. As we will se later, all basi equations of lassial and relativisti eletrodynamis an be written just by applying bi-quaternion multipliations without using any tensor multipliation. A multipliation of two bi-quaternions produes 16 or 64 terms, whih in turn build again a bi-quaternion aording to (1.17). The mapping of these terms to physial fourvetor j gives in parallel j =. As we show later, this nullifying of the other four-vetors has it s origin in physial laws of onservation. The bi-quaternion (1.17) an be lassified as: Inomplete bi-quaternion j k j = x + i ixk = x + i ix (1.18) (1.19) Complete bi-quaternion = j ( ) ( k j + = x + ix + i ixk + i x ) Page 4 opyright (7) by André Waser;

5 Calulus with inomplete bi-quaternions We have the following two inomplete bi-quaternions = x + i ix = y + i iy. ( k) und ( k) A onjugate inomplete bi-quaternion * is then: * = x i ix (1.) ( k) The salar multipliation of two inomplete bi-quaternions is: = = xy x y (1.1) and * is: ( ) k k (1.) ( xy x y) * * = = + k k The multipliation of two inomplete bi-quaternions is: = ( xy + xk yk) + i ixy ( k + xy k ) xk yk (1.3) = ( xy + xk yk) + i ixy ( k + xy k ) + xk yk The magnitude if an inomplete bi-quaternion is: = = x x x (1.4) Then we have also k k k k * * = = x x x = = (1.5) Calulus with omplete bi-quaternions We have the following two omplete bi-quaternions k = x + ix + i ( ixk + x ) und = y ( k + iy + i iyk + y ). where the supersripts and subsripts represent the indies k=1,,3. A onjugate omplete bi-quaternion * is then: * ( x ) ( k ix i ixk x ) (1.6) The salar multipliation is: = = + and * is: k k ( xy xk yk xy x y) k k ( k k) + i x y + x y + x y + x y = = xy + x y + xy + x y (1.7) * * k k k k (1.8) The multipliation is: k k k k = xy + xk yk xy x y + ixy xy k + xy xyk k k k k i + xy xyk + yx yxk + ixy k + xy + yx k + yx k k k k + ( x y xk yk) + i( xk y + x yk) ( ) ( ) ( ) ( ) (1.9) The magnitude of a omplete bi-quaternion is: = = xx x x + xx x k x k + ixx + x x k (1.3) ( k k ) ( k ) opyright (7) by André Waser; Page 5

6 The multipliation * is: * ( x k k ) ( k ) ( k x x x xk xk x x i = + + xx + x xk + i xk x ) (1.31) The multipliation * is: * = i k k k k ( xy xy xk yk x y) ixy ( xyk xy xy k ) k k k k ( xy xyk yx yxk) ixy ( yx k xy k yx ) k k k k ( xk yk + x y ) + i( xk y + x yk) (1.3) Derivations The bi-quaternion NABLA operator is: 1 1 j = + i i+ j+ k = + i i j x1 x x3 The bi-quaternion d ALEMBERT operator is: * 1 1 Δj j = j ( j ) = = Δ x x x 1 3 (1.33) (1.34) Total time derivative The operator for the total time derivative of a bi-quaternion is (see Appendix A): d i v i = + v + + ( ) dt v (1.35) The speial multipliation symbol indiates, that on applying this operator, the salar multipliation must be used for the salar part and the ross produt must be used for the vetor part. The operator in equation (1.35) is analogue to the known operator of two-dimensional derivations: d = + v dt With the following multipliation we get the same result as with (1.35): v ( ) = + v i i ( ) t + + t v (1.36) Applying this to the event vetor X we find d 1 = = ( ) (1.37) dt 4 Page 6 opyright (7) by André Waser;

7 Bi-Quaternions in four-dimensional spae An event in four-dimensional spae an be expressed diretly with a bi-quaternion aording to t + i ix. (1.38) Aording (1.1), the magnitude (distane of an event to fulrum) is: = t x (1.39) In speial theory of relativity SRT this magnitude is invariant for eah inertial system. The same is valid for the differential d. A division of d by gives another invariant: 1 1 v d = dt ( dx1 + dx + dx3) = dt 1 = dτ (1.4) This is the time dilatation known of speial relativity. For the differential of an event vetor we also have d = dt+ i idx (1.41) and we find the well-known relativisti four-veloity as: d i iv = = +, (1.4) dτ v v 1 1 where v is the veloity between two relatively moving inertial systems. The magnitude of the relativisti four-veloity is known to be always equal to speed of light. = (1.43) From (1.4) we find the well-known fators from speial relativity: 1 v γ 1 v γ= γ = γ =, 1, v γ 1, (1.44)a, b, Now we build the total time differential of an event vetor and we get another, different non-relativisti four-veloity (oordination veloity) d = = + i iv (1.45) dt whih is onneted to the relativisti four-veloity aording to = γ (1.46) The addition of two veloities 1 and shall further omply with equation (1.43): + = + + i v + v (1.47) ( ) In next setion we drive the law of veloity addition. opyright (7) by André Waser; Page 7

8 The Lorentz transformation in bi-quaternion form The relativisti four-veloity is espeially helpful for the formulation of oordination transformation between relative moving systems S and S (see Appendix B). Shall = + i iu be the relative veloity between S and S, then we have: γ * = (1.48) And inversely we have γ = (1.49) And solved: v x t =γ t, x =γ( x vt) (1.5)a, b v x t =γ t +, x =γ ( x + vt ) (1.51)a, b The four-veloities and * are the operators to transform a bi-quaternion (four-vetor) from one inertial system into another relative moving inertial system. The general form of the transformation operator of values in inertial system S and S with the relative veloity = + i iu is: = = γ * (1.5) γ (1.53) Beside the normal way via the differentials (1.5) or (1.51), the theorem of veloity addition an be derived with above transformation equations. We have the systems S and S moving against eah other with relative veloity. A body moves in S with the veloity = + i iv. Then we have γ uv = = γ + +γi i( u+ v ) (1.54) The salar part in four veloity must always be equal to speed of light. This we ahieve in (1.54) with a division by 1 1 u+ v = = = + i i (1.55) uv uv γ and thereof u+ v v = uv 1+ (1.56) U+ V = (1.57) The normalisation with has it s origin in the designation of veloity outside of S. It s impossible to measure primed veloity in un-primed system S diretly. Page 8 opyright (7) by André Waser;

9 Bi-quaternion eletrodynamis for linear & isotropi medium (vauum) The bi-quaternion eletri and magneti field Analogue to the veloity we define the bi-quaternion potentials with the omponents ϕ [V] and A [Vs / m]: ϕ + i ia (.1) or with ϕ A = v (.) also ϕ ϕ = ( + i iv) (.3) Then we have for the derivation of the bi-quaternion potentials (.1): 1 ϕ i A = + A + i ϕ+ ( A) (.4) We use the substitutions 1 ϕ F q A s = A, E = = ϕ, B = A (.5)a, b, q t and find the equation for the eletri fore field [V / m]: = = s+ i ie+ B = i ie + B with s= (.6) ( ) or ( ) and the equation for the magneti indution [Vs / m ]: 1 1 = i = is+ i ib E or = i ib E with s= (.7) and find thereof i = i, = (.8) Now we look loser to the real salar term s. It is 1 ϕ 1 dϕ = s = + ( ϕ v) = (.9) dt s is known as the LORENTZ ondition (s = ). Thus the LORENTZ ondition is a demand for a soure-free bi-quaternion potential, beause we have 1 ϕ ϕ s = = + A = + ( ϕ v) = (.1) It shows, that the LORENTZ ondition is a demand for onservation of salar potentials ϕ; i.e. the LORENTZ ondition is a onservation law. From (.9) we find further and independently of s also the general LORENTZ ondition: 1dϕ + = (.11) dt opyright (7) by André Waser; Page 9

10 The bi-quaternion potential density Analogue to the potentials we define the bi-quaternion harge- and urrent density with the omponents ρ [ As / m 3 ] and J [A / m ]: ρ+ i ij (.1) or with also J =ρ v= γρ v (.13) ρ = ρ = ρ + i iρv (.14) Then we have for the derivation of the urrent density (.1) ρ 1 J = + J+ i i ρ+ ( J) Introduing the substitutions = ρ σ J, μ J A = μ ρ, we get the equations for a volume urrent density [A / m 3 ]: 1 = i = iσ+ i ij A μ and for a potential density [Vs / m 4 ]: = μ =μσ + i +μ and find thereof = iμ, ( ia J ) J = J (.15) (.16)a, b (.17) (.18) i = (.19) μ Looking more losely to the real salar term σ we find ρ 1dρ = σ= ( ρ v) = (.) dt σ is known from harge onservation (σ = ). Thus the harge onservation law is the demand for a soure-free bi-quaternion potential density, beause we have ρ ρ σ = μ = + J = + ( ρ v) = (.1) This onservation law is analogue to onservation of eletri salar potential ϕ. With (.1) we get for the bi-quaternion potential density and volume urrent density: 1 = i ( A+ ij), = i ij A (.) μ Page 1 opyright (7) by André Waser;

11 MAXWELL s equations for free harge und free urrent densities The following equation diretly leads to Maxwell s equations in bi-quaternion form: 1 * * * = i= =Δ =μ (.3) or expanded: 1 s 1 E 1 B μ = E i B+ i i B + s + E (.4) respetively s E B μρ 1 μ = + i = + i μj 1 E B B + s (.5) + E Similar equations have been published by HONIG [8]. Using (.5) we an print Maxwell s equations in differential form straight forward: AMPERE s law B= B + E = Extended COULOMB law s ρ E = ε Extended FARADAY law 1 E B + s =μj (.6) (.7) (.8) (.9) With onservation of eletri salar potential (.1), the last two equations redue to COULOMB s law ε E =ρ (.3) 1 E FARADAY s law B =μj (.31) From (.3) we find also the relation between the eletri field and urrent density 1 * * = = ε (.3) μ opyright (7) by André Waser; Page 11

12 The wave equations of potentials and fore fields To derive Maxwell s equations we applied the d Alembert operator to the eletri potentials. This is expliitly: 1 1 ϕ 1 A Δ = ϕ + i i A =μ (.33) This leads to the wave equations for eletri potentials 1 ϕ ρ ϕ = (.34) t ε 1 A A =μj (.35) Applying the derivative to (.3), we find * i= Δi = μ (.36) and thereof 1 s i 1 E 1 B s i E B + = ρ 1 J μ + J + i μ i ρ+ ( J) and finally the wave equations of fore fields 1 s ρ s = μ + J = (.37) 1 E J E = μ ρ+ (.38) 1 B B=μ J (.39) Page 1 opyright (7) by André Waser;

13 Bi-quaternion LORENTZ fore density and LORENTZ fore We define the bi-quaternion power- and fore density with the omponents P [W / m 3 ] and F [N / m 3 ], and get 1 P + i if (.4) Then we hoose the bi-quaternion identities 1 d = i = = = ρ = ρ (.41) dt and use the substitutions (.1) and (.13) to get the fore density on a harge density ρ, whih moves with veloity in an external potential field as 1 P 1 ρv E+ ρs ρv B = + i = i + (.4) F ρb ρ v E ρ v B+ρ E+ρvs We have finally ρv B= v E B = ρ v E+ s = P ρ v B+ E+ vs = F extended power density ( ) extended LORENTZ fore density ( ) Notable is also J B= and thereof (.43) (.44) (.45) (.46) = P (.47)a, b With onservation of eletri salar potentials (.1), above equations (.45) and (.46) redue to Power density (s=) ρv E = P (.48) ρ v B+ E = F LORENTZ fore density (s=) ( ) (.49) For a point harge q in potential field we find analogue to (.4) and (.41) i d P+ i F= q = q (.5) dt q v E+ s = P (.51) extended Power density ( ) q v B+ E+ v = F extended LORENTZ fore ( s) (.5) an wit the onservation of eletri salar potential (.1) Power density (s=) qv E = P q v B+ E = F LORENTZ fore (s=) ( ) (.53) (.54) opyright (7) by André Waser; Page 13

14 Bi-quaternion Poynting theorem (energy flow density) First we define the bi-quaternion energy flow density with the omponents of energy density w [J / m 3 ] and energy flow density S [Js / m 4 ] as 1 w + S i (.55) Now we onsider a urrent density J and a potential field (MAXWELL equations) aused by this urrent density. We find the equilibrium of both field s fore densities by multipliations of bi-quaternion MAXWELL equation (.3) on both sides with. Then, by using (.41) we get: 1 1 ( Δ) = ( Δ) + = (.56) μ μ The alulation (Appendix C) shows for the salar part: 1 E B s 1 ε +μ + μ + ( E B s E ) + E J +ρ s = (.57) μ Now we insert the substitutions for the energy density w and the energy flow density S w = εe E+ B B+ s, S= E B s E (.58)a, b μ μ μ into (.57) and use (.45). Then we get the well known Poynting theorem w + S+ P = (.59) Or with (.59) and (.55) follows the expanded Poynting theorem (with s ) + = (.6) Inidentally we may also write for the energy density w (1.8) 1 * w = ε (.61) Inserting (.3) into (.41), we find also 1 * = = ε( ) (.6) The imaginary vetor term is (Appendix C) E ( E) + ( B) B+ ( E) E ( E B+ se) + sb F= (.63) There the first three terms orrespond to the divergene of Maxwell stress tensor : = 1 1 E ( E) ( B) B ( E) E + + (.64) followed by the partial time derivative of expanded Poynting vetor. Page 14 opyright (7) by André Waser;

15 Bi-quaternion Lorentz transformation of eletromagneti fields In (.14) we have used the relation =ρ = γρ (.65) In relativisti ase we have expliitly for the transformation of harge and urrent densities ρ = γρ und ρ v= J = γ J = γρv (.66) The Lorentz transformation of bi-quaternion potentials is: ϕ γ * ϕ u A ϕ u A + i ia = = = γ + i ia i u and with omparison of oeffiients we find: ϕ = γ ϕ ( A u) ϕ A =γ A u (.67) (.68) (.69) The Lorentz transformation of bi-quaternion urrent density is: γ * γ * u J u J ρ+ ' i ij' = = = ( ρ+ i ij) =γ ρ + i ij ρ i u and again with omparison of oeffiients we find: J u ρ=γ ρ (.7) J =γ J ρu (.71) ( ) For the Lorentz transformation of fore fields E and B we have: γ * = u u s + i ( ie + B ) = γ s E + iu B + i ie + iu B + B E i su and thereof E u s =γ s (.7) E =γ E+ u B s u (.73) ( ) u B =γ B E (.74) opyright (7) by André Waser; Page 15

16 Dynamis & Kinematis The bi-quaternion gravity field In this setion we desribe the dynamis of moving bodies analogue to previous formulas used for eletrodynamis. As a starting point we need two fields, whereof the deeper meaning shall not be disussed now: U (X): Veloity field (analogue to A ) [m / s] φ (X): Gravity potential (analogue to ϕ) [m / s ] These fields form the kinemati bi-quaternion potential field (veloity field) φ = + i iu Then we have for the derivation of 1 φ i U = + U + i φ+ ( U) (3.1) (3.) Defining the following substitutions: 1 φ s m = U F U, m G = = φ m, T = U (3.3)a, b, with F m: Fore on a point mass m (test mass) [N] = [kg m / s ] G : Gravity field, aeleration field [m / s ] T : Rotations field, dynami indution [1 / s] whereas the gravity field onsists of two (established) parts: U G G = φ and GT = (3.4)a, b t G G: Gravity field of a gravity potential [m / s ] G T: Aeleration field, indued gravity field [m / s ] Then we find for the bi-quaternion gravity field [m / s ] on a test mass m = = s + i ig+ T m ( ) (3.5) and for the dynami indution [s -1 ] we have: 1 = i = ism + i it G (3.6) Thereof we find i = i, = (3.7) The salar term of (3.) is if nullified again a onservation law: 1 φ φ s m = = + U = + ( φ U) = (3.8) Page 16 opyright (7) by André Waser;

17 Bi-quaternion momentum density Analogue to harge- and urrent density of eletrodynamis we define the bi-quaternion mass- and momentum density m with the omponents m [kg / m 3 ] and p [kg / sm ]: m m+ i ip (3.9) m m or with also p = mv =γm v m m m= m+ i imv (3.1) (3.11) Then we have for the derivation of the momentum density [kg / sm 3 ]: m 1 pm m = + pm + i i m+ ( pm) (3.1) Within salar part we identify the ontinuity equation of mass density and the onservation law of mass respetively. m + pm = Instead of mass density m we ould also use the energy density w m /. Equation (3.9) hanges then to: w m m = + i ipm (3.13) Using point masses instead of mass densities we also have W ip m m = + i The relativisti momentum m of a point mass m with bi-quaternion veloity is m (3.14) = m γ (3.15) This ontemplation is analogue to (3.59) and orresponds with the total self-energy or self-momentum of a mass (or harge, respetively). From (3.15) we find immediately v Wm =γm m + mv + m + (3.16) 8 opyright (7) by André Waser; Page 17

18 Field equations of dynamis Equipped with the bi-quaternion momentum density m and the potential field we ould start again analogue to eletrodynamis (.3): 1 * * * g = i= =Δ = m (3.17) where g is the gravity onstant with [N m / kg ]. From (3.17) we have g 1 sm 1 G 1 T m = G i T+ i i T + s m + G (3.18) respetively sm G T g g m 1 t m i = i + = + p 1 G T T + s + G m (3.19) Thereof we extrat the Maxwell equations of dynamis: T = T AMPERE s law + G = sm Expanded COULOMB law G = gm 1 G g Expanded FARADAY law T + s m = p (3.) (3.1) (3.) (3.3) With the onservation of gravity potential (3.8), the last two equations redue to COULOMB s law 1 G m g = 1 G g T = p FARADAY s law (3.4) (3.5) From (3.17) we also find the relation between the gravity field and the momentum density m * m = (3.6) g Page 18 opyright (7) by André Waser;

19 The wave equations of the potentials To derive above equations we impliitly applied the d Alembert operator to the potentials. Expliitly this is: 1 1 φ 1 U g Δ = φ i i U + = m (3.7) Thereof follows the wave equations of the potentials 1 φ gm φ= 1 U g U = p (3.8) (3.9) With the derivation of (3.17) we find * g i= Δi = m (3.3) and thereof 1 s m i 1 G 1 T s m i G T + = g m g 1 p + p + i i m+ ( p) And finally we have the wave equations for the fore fields 1 s m g m s m = p + = 1 G p 1 G = g m+ 1 T g T = p (3.31) (3.3) (3.33) Analogue to eletrodynamis without harge- and urrent densities, these equations desribe a transverse gravity wave in the absene of mass densities, having the propagation veloity equal to the speed of light. 1 G G = (3.34) 1 T T = (3.35) opyright (7) by André Waser; Page 19

20 The bi-quaternion fore density and inertia (reation fore) With (.4) we use again the bi-quaternion power- and fore density with the omponents P [W / m 3 ] and F [N / m 3 ] 1 P + i if and hoose the bi-quaternion equation 1 d = m= im= m = m = m (3.36) dt Using the substitutions (3.8) and (3.1), we find the fore density on a mass density m, whih is moving with veloity in an external potential field aording to 1 P 1 mv G+ m s m mv T = + i = + i F m T mv G (3.37) mv T + mg + mv s m and finally mv T = v G T = m v G+ = P Expanded power density ( sm ) Expanded reation fore density ( sm ) m v T+ G+ v = F Notable is also p T = (3.38) (3.39) (3.4) (3.41) and thus m = P (3.4)a, b With onservation of gravity potential (3.8), equations (3.4) and (3.41) redue to Power density (s m =) mv G = P m v T+ G = F Reation fore density (s m =) ( ) (3.43) (3.44) Analogue to (.4) and (3.36) we find for a point mass m in a potential field i d P+ i F= m = m dt m v G+ = P Expanded power ( sm ) m v T+ G+ v = F Expanded reation fore ( sm ) and with the onservation of gravity potential (3.8) this redues to Power (s m =) mv G = P m v T+ G = F Reaktions-Kraft (s m =) ( ) (3.45) (3.46) (3.47) (3.48) (3.49) Page opyright (7) by André Waser;

21 Bi-quaternion aeleration and inertia (ation fore) The ation fore to aelerate a point mass m is opposite to inertia fore (reation fore) (3.45). The external veloity field is replaed by the oordinate veloity and we have the equation d m = m= m = m (3.5) dt The alulation of yields: v v v + v ( v e ) = + i = + i a v v v v (3.51) + v( v) v ( v ) and thereof v v = ( ) v v = v Flow rate e [s -1 ]: v v v+ = e Aeleration [m / s ]: v + v( v) v ( v) = a or v v + v ( v) = a or v + v v ( v ) v = a (3.5) (3.53) (3.54) (3.55)a (3.55)b (3.55) Equation (3.55)b is known from fluid mehanis. Additionally we have newly found an equation for the flow rate e with the unit s -1. To find the fore equations, we multiply (3.49) with a point mass m or a mass density m. For example we find for a point mass m Ation fore: F= ma (3.56) Power: v v P = m e= m v+ (3.57) opyright (7) by André Waser; Page 1

22 Bi-quaternion self-energy density and mass density Calulating the salar produt between bi-quaternion urrent density and of assoiated bi-quaternion potential field, we find the self-energy density as: ϕ = = ρ (3.58) With the bi-quaternion momentum density of a mass density m: = m (3.59) we find = = (3.6) Thereof and with (.34) follows for a mass density and for stati ase ( ϕ/=) only: ϕ ε m = ρ= ϕδϕ (3.61) The integration over the whole volume V is ε ε ε m = m dv = ϕδϕ dv = ϕ dv = E E dv ( ) (3.6) For a spherial potential field this beomes r q 1 q 1 m = dr = (3.63) 4πε r 4πε r m m With the elementary harge q= [As] and the eletron mass m e = [kg] we find for example the following value for the eletron radius r e : q 1 15 re = = m 4πε me [ ] (3.64) From (3.6) and with ϕ/= we also find the relation =ε ( ) (3.65) m dv E E dv Page opyright (7) by André Waser;

23 Summary All important basi equations of eletrodynamis an be ast in a very ompat form by using bi-quaternions. Beside the known text-book equations we find some possible expansions to Maxwell s equations and other fundamental equations of eletrodynamis. This expanded equations redue to lassial form if the LORENTZ ondition s= is applied. An interpretation of the salar field s beyond the interpretation as pure LORENTZ ondition shall be disussed at another plae. Beside eletrodynamis, bi-quaternions are also very useful for appliation in other disiplines of physis as for example in mehanis (dynamis) or as only indiated in the appendix in quantum mehanis. Referenes [1] VAN VLAENDEREN Koen and André WASER, Generalisation of lassial eletrodynamis to admit a salar field and longitudinal waves, Hadroni Journal 4 (1) [] BORK Alfred M., Vetors Versus Quaternions The Letters of Nature, Amerian Journal of Physis 34 (1966) -11 [3] CONWAY A. W., Quaternion Treatment of the Relativisti Wave Equation, Proeedings of the Royal Soiety London, Series A 16 (15 September 1937) [4] EINSTEIN Albert, Zur Elektrodynamik bewegter Körper, Annalen der Physik und Chemie 17 (3. Juni 195) [5] GOUGH W., Quaternions and spherial harmonis, European Journal of Physis 5 (1984) [6] HAMILTON William Rowan, On a new Speies of Imaginary Quantities onneted with a theory of Quaternions, Proeedings of the Royal Irish Aademy (13 November 1843) [7] HEAVISIDE Oliver, On the Fores, Stresses and Fluxes of Energy in the Eletromagneti Field, Philosophial Transations of the Royal Soiety 183A (189) 43 [8] HONIG William M., Quaternioni Eletromagneti Wave Equation and a Dual Charge-Filled Spae, Lettere al Nuovo Cimento, Ser. 19 No.4 (8 Maggio 1977) [9] MAXWELL James Clerk, A Treatise on Eletriity & Magnetism, (1893) Dover Publiations, New York ISBN (Vol. 1) & (Vol. ) [1] SILBERSTEIN L., Quaternioni Form of Relativity, Philosophial Magazine, Ser. 6 3 (191) [11] TAIT Peter Guthrie, An elementary Treatise on Quaternions, Oxford University Press 1 st Edition (1875) [1] WALKER Marshall J., Quaternions as 4-Vetors, Amerian Journal of Physis 4 (1956) [13] WILSON E. B., Vetor Analysis of Josiah Willard Gibbs The History of a Great Mind, Charles Sribner s Sons New York (191) opyright (7) by André Waser; Page 3

24 Appendix A We have a bi-quaternion field = (), where is a funtion of spae and time, and we have = ((t, x 1, x, x 3 )). The total time derivation of is then: ( ) d = d = dt+ dx= dt d t + x x t (A.1) x with = and = X x Now we expand the operator in first term of (A.1) and get dt = + i i dt (A.) (A.3) The operator in seond term of (A.1) has the following expansion: { dx} dx = x x = e + e + e 1 3 [ t+ i( x ie + x je + x ke ) ] x x x 1 3 = i dx + dx + dx 1 3 x x x 1 3 { i ( 1 3) } 1 = idx + idx + idx + i i + + x x x i = i dx + ( i idx)( i i ) t i = dx + i dx dx 1 3 and finally i dx = dx + i dx dx (A.4) x The addition of (A.3) and (A.4) and the afterwards division by dt gives the total time differential in bi-quaternion form: d v = + v + i i + ( v ) (A.5) dt The speial multipliation symbol indiates, that on applying this operator the salar multipliation must be used for the salar part and the ross produt must be used for the vetor part. Page 4 opyright (7) by André Waser;

25 Appendix B We start with the transformation: γ * v = =γ 1 i i t+ i ix and derive ( i ) ( ) v x v x t + ix = γ t + i ix ivt With omparison of oeffiients we find: v x t =γ t x =γ x vt ( ) (B.1) (B.) (B.3) (B.4) The first equation (B.3) is well-known and orresponds to Lorentz transformation of time in vetor form. The following derivation shows, that this is also valid for equation (B.4): v γ[ x vt] =γv x γv t = ( γ 1vx ) + vx γvt vv x = v x γ vt+ ( γ 1) v v In many text-books the last term in square brakets is desribed as Lorentz transformation of position vetor x. By omparison of both terms in square brakets we find the identity γ x vt = x γ vt+ γ 1 (B.5) v v ( ) ( ) vv x This proves, that (B.4) is indeed the Lorentz transformation of x. If v is ollinear to x, both equations redue to the well-known one-dimensional transformation equations vx t =γ t (B.6) x =γ( x vt) (B.7) opyright (7) by André Waser; Page 5

26 Appendix C We start with 1 ( Δ) = μ ( Δ) μ = (C.1) Therein we have the terms: 1 i = = s+ i E + B (C.) 1 s 1 E 1 B Δ = E i B+ i i B + s + E (C.3) 1 1 μ = μρ ( v E s) iv B i iv B ie ivs ( B v E) (C.4) Equations (C.4) is already the seond term in (C.1). Therefore we need to alulate the first term only: 1 s E i B+ s t + ( Δ) = i 1 E 1 B i E + B i i B + s + E (C.5) Imaginary salar term We find for the imaginary salar term of (C.5): 1 1 B 1 E s B B ( B) + E ( E) + E + B B s = B ( B) = = ` = E `= B+ s Together with the imaginary salar term of (C.4) we get: -μρv B = -μj B in (C.1): μj B= (C.6) Real salar term Calulating the real salar term of (C.5) we find 1 s 1 E B s( E) s + E ( B) E + E s+ B + B ( E) And expanded: s 1 E B 1 s( E) s + E ( B) E + E s+ B + B ( E) + E ( B) E ( B) Using FARADAY s law 1 E B= +μj s we get (C.7) (C.8) Page 6 opyright (7) by André Waser;

27 s 1 E B s( E) s E ( B) E E s B B ( E) (C.9) E + E + μe J E s E ( B ) or 1 1 E B s E + B + B ( E) E ( B) + μe J E + s s( E) s ( E B) Then, with ρ s E = + ε follows 1 1 E B ρ E + B + ( E B) + μe J E s+ s (C.1) ε Inserting the real salar part of (C.4): 1 i { ( v E )} s μρ + = μj E+ s ρ ε and (C.9) in (C.1), we get finally Poynting s theorem in different notations 1 E B E + B + ( E B) + E ( μj s) = (C.11) 1 1 E B + + ( E B) + E ( μj s) = 1 εe B B E + E J s = μ μ μ ε +μ + ( E H) + E J s = μ 1 E H 1 Inserting Maxwell s equation s ρ s ρ = s E = ( se) ( s) E s s ε ε in (C.1), we further find E B E + E J+ = 1 s 1 E B ρ ( s ) s ε (C.1) (C.13) (C.14) (C.15) opyright (7) by André Waser; Page 7

28 Imaginary vetor term We find for the imaginary vetor term of (C.5): s E 1 E B( B) + s( B) + s s+ ( B) B B 1 B 1 1 s 1 + s B E E E E+ E E ( ) ( ) We take the bi-quaternion MAXWELL equations 1 E B= und B= +μj s to redue (C.16) to 1 E 1 B 1 1 s 1 sμ J+ B B B+ B E E E E+ E E ( ) s ( ) ( ) or E B 1 s E ( E) + ( B) B ( E) E B+ E E+ s B+ sμj To hange the sign of third term, we add the last term `1 T (C.16) E 1 B 1 s E ( E) + ( B) B+ ( E) E B E E+ s B+ sμj ( E) E use AMPERE s law and get `1 T B E = t E 1 B 1 s E ( E) + ( B) B+ ( E) E B+ E E+ s B+ sμj Therof we have E B 1 s E ( E) + ( B) B+ ( E) E B+ E E+ s B+ sμj 1 T ( E B) (C.17) With two vetor identities and with the FARADAY equation, multiplied by s s E E ( se) s 1 E =, ( s) B= ( sb) s( B), s = B+μJ equations (C.17) beomes E ( E) + ( B) B+ ( E) E ( E B+ s E) + s B (C.18) Inserting (C.18) and the imaginary vetor term of (C.4) i μρ ( E+ v B) + vs = μ i ρ E+ J B+ Js into (C.1), we finally get 1 1 E ( E) + ( B) B+ ( E) E (C.19) 1 ( E B+ s E) + sb=μρ E+μ J B+μsJ Page 8 opyright (7) by André Waser;

29 Appendix D: Quantum mehanis Relativisti wave equation In 1937 CONWAY [3] has shown a possible notation for the relativisti wave equation, if the Hamilton ian units are used as pre- and post fators in his formulas. In this setion we derive the relativisti wave equation with the bi-quaternion notation used already in previous hapters. We start with the momentum law: = m (4.1) m and with the definition of total energy of a mass E (4.) we an derive EINSTEIN s formula by using (1.4) with k = 1..3: E = m + p (4.3) 4 k k Until this point we have used a salar value representing the energy, whih for example takes a onstant value for a resting body. But quantum physis (experiments) have shown, that this is not quite orret but that energy is merely an osillating phenomenon and therefore must satisfy a wave equation. Therefore equation (4.3) an be seen as a stati average equation of a olletion of many energy osillators. But for a single partile the osillatory behavior of its energy an learly be observed. Equation (4.3) is already in a quadrati form as this is the ase also for the differential operators of a wave equation. To find the wave equation behind the energy equation, we use the established substitutions for the differentials as known in quantum mehanis: E i und p k i x k, (4.4)a, b where is PLANCK s onstant divided by π. Now this differentials must be applied to a new, unknown funtion. This is nothing else than the (dimensionless) wave funtion Ψ. This wave funtion is again a bi-quaternion: Ψ= ψ + i iψ (4.5) k By inserting (4.5) into (4.3) we get diretly DIRAC s relativisti wave equation: 1 m Ψ Δ + = Ψ Ψ (4.6) Partile without external potential fields DIRAC has solved the energy equation (4.3) E =± m + p (4.7) k k by taking the square root with the introdution of 4x4 matrixes. On the other side equation (4.) offers an other possible derivation without taking a square root, if diretly taking the quaternion momentum. But the sign of the quaternion momentum should not hange. This an be ahieved with a modifiation of equation (4.): ± then it is E = = ± (4.8) opyright (7) by André Waser; Page 9

30 The two possible sign for the energy state in (4.7) and (4.8) has motivated DIRAC to postulate the existene of anti-partiles espeially the positron. By inserting of the substitutions (4.4) into (4.8) another DIRAC equation a be obtained: Ψ Ψ Ψ 1 Ψ i+ j+ k+ mψ =, (4.9) x1 x x3 On a first glane this equation differs form the original DIRAC equation beause no matrixes are used. But the HAMILTON ian units an also be written as matrixes (see Appendix E). The multipliation of (4.9) with this HAMILTON ian matrixes gives the following equation system: ψ ψ1 ψ ψ 3 mψ + + = i x1 x x3 ψ 1 ψ ψ3 ψ mψ 1 + i + = x1 x x3, (4.1) ψ ψ3 ψ ψ 1 mψ + + i = x1 x x3 ψ 3 ψ ψ1 ψ mψ3 i = x1 x x3 This system ontains four equations for a partile without external fields as originally proposed by DIRAC. Now, in an analogue way the equation for a partile within external fields an be desribed. Partile within external potential field The momentum on a harged partile hanges with an external potential field. If the momentum of an external field on a harged partile q is defined as then it follows for its energy and The total energy is then q = q (4.11) q E = q (4.1) q q = q (4.13) ( q q ) = ± (4.14) Again the extended DIRAC equation follows with the substitution of energy and momentum to: 1 ϕ qa1 i+ qa j+ qa3 k+ Ψ m + q Ψ = (4.15) x1 x x3 Again quaternions an be used instead of matrixes. Page 3 opyright (7) by André Waser;

31 Appendix E: Matries in quaternion form Aording Arthur CAYLEY, omplex numbers an be expressed with matries: a 1 a+ ib= with i = (4.16) b 1 Example: i = ii = = = 1 (4.17) The HAMILTON ian units of a quaternion build together with the numbers 1 and 1 a non ABEL ian group of eighth order. Its first four positive elements are: i 1 i 1 i = j = k = 1 = i 1 i 1 Replaing the imaginary unit i with the orresponding matrix (4.16) gives: i = 1 j = (4.18) 1 1 k = = 1 1 (4.19) The square of this matrixes always return the value 1 as requested by the definition of the HAMILTON ian unit vetors (1.). opyright (7) by André Waser; Page 31