The Quaternion Structure of Space-Time and Arrow of Time

Home , Local reference frame

Journal of Modern Physics, 2012, 3, 570-580 http://dx.doi.org/10.4236/jmp.2012.37078 Published Online July 2012 (http://www.SciRP.org/journal/jmp)

Yingqiu Gu Fudan University, Shanghai, China Email: [email protected]

Received April 6, 2012; revised May 8, 2012; accepted June 1, 2012

ABSTRACT In fundamental theories of physics, the dynamical equations all have time inversion invariance. Except for the evolution of some simple system which has realistic inverse processes, but for a slightly more complicated system, the evolution processes are irreversible. This is the problem of arrow of time, which is always warmly debated. In different point of view, we find there may have some conceptual misunderstanding in the controversy: 1) The realization of an inverse process does not mean the time of the system goes backward; 2) The principles of relativity and covariance are the constraints to physical laws, but not constraints to specific solutions. The equations must be covariant, but the solutions are not definitely symmetric; 3) Time is a global property of the universe, which is a measurement of the evolution process of the universe. The internal time of a matter system reflecting its internal evolution speed also takes this cosmic time as a unified background and standard of measurement; 4) The universe has a unified cosmic time T and a cosmic space related to this cosmic time. They are objective and absolute; 5) The eigensolution of a spinor is a critical state losing time concept, which responses the interaction of environment with some uncertainty, then the evolution process of the world is not uniquely determined; 6) The non-uniqueness of the evolution process means that the inverse process is absent. So for a world including spinors, the evolution is essentially irreversible. In this paper, according to the widely accepted principles and direct calculations of transformation, we reveal the misunderstandings in the usual controversy, and then give more natural and reasonable explanations for structure of space-time and arrow of time.

Keywords: Arrow of Time; Invariance; The Principle of Relativity; Quaternion Structure of Space-Time; Simultaneity

1. Introduction backward. The expansion or contraction is just a dynamical effect of the space-time. In fundamental theories of physics such as Newtonian 2) The principles of relativity and covariance are the mechanics, electrodynamics, quantum mechanics, the constraints to physical laws, but not constraints to con- dynamical equations all have time inversion invariance. crete solutions. The specific matter system is only a But except for the evolution of some simple system special solution of physical laws, so the equations must which has realistic inverse processes, such as swing of a be covariant, but the solutions are not definitely sym- pendulum, dissolution and crystallization of crystal, metric. emission and absorption of photons, for a slightly more 3) Time is a global property of the universe, which is a complicated system, the evolution processes are essen- measurement of the evolution process of the universe. tially irreversible. This is the problem of arrow of time, The universe has a unique and unified cosmic time, and it which is always warmly debated [1-9]. always goes unidirectionally forward. Related to this In different point of view, we find there may have cosmic time we have the specific state of universe at some conceptual misunderstanding in the controversy: some time, which forms the cosmic space and the exis- 1) The realization of an inversion process does not tences. There is an internal time to reflect the evolution mean that time of the system goes backward. We make process of a system, just like the history and lifetime of a two clocks with mirror image. One rotates clockwise, man, which is the proper time  of the system. How- and the other rotates anticlockwise. But this does not ever, this proper time also takes the cosmic time as a mean time of the clockwise one goes forward, and time unified background and a measuring standard. of the other goes backward. Similarly, the expansion of 4) The universe has a unified cosmic time T and a the universe does not mean the cosmic time goes forward, cosmic space related to this cosmic time. The cosmic and the contraction does not mean the cosmic time goes space is the total space described by the coordinate

Copyright © 2012 SciRes. JMP Y. Q. GU 571 system relatively at rest to the cosmic background be accepted as a fundamental postulate. The selection of radiation. The cosmic space and time are objective and coordinate system and the local reference frame or tetrad absolute. Since the total universe is also a realistic at each event is subjective and arbitrary. However, the solution of all physical laws. So this solution is objective, event and the 4 dimensional distance or interval between and it does not definitely have the symmetry of the two events are objective. For the same physical system, equations. different researchers can choose quite different coor- 5) The eigensolution of spinor is a critical state losing dinate system and local tetrad, but their coordinates and time concept, which responses the interaction of environ- physical parameters should be in one-to-one corres- ment with some uncertainty, then the evolution process pondence, and the equations should take the same of the world is not uniquely determined and has some structure. This is the essential meaning of the principles randomness. The evolution of the world is not uniquely of relativity and covariance. However, the usual under- determined. standing and explanations in textbooks have some 6) The non-uniqueness of the evolution process means problems. that the inverse process is absent. So for a world in- Spinor equation is the most basic equation to describe cluding spinors, the evolution is essentially irreversible. matter. Detailed analysis shows that, to this kind of These opinions have been discussed in other papers equation, the relativity principle is valid only if the [10-12]. In what follows, we give some detailed discus- space-time has 3 + 1 dimensions and has a quadratic sion for the structure of the space-time and relative interval ds2 . In this case, the space-time has elegant symmetries of the transformation through specific calcu- quaternion structure, and the physical fields and their lations and explanations. dynamic equations can be expressed in quaternion form. Their transformation laws can be realized only for such 2. The Quaternion Structure of Space-Time quaternion differential operator. This feature can be found in the following calculations. Further more, for Relativity is a theory of space-time of the universe, this first order differential field equation system, the which is based on the principle of constant vacuum light solutions evolve in one direction in time. speed and the principles of relativity and generalized In this paper, we use notations ab,0,1,2,3 covariance. Though this treatment is reasonable is phy-  stands for the indexes in flat space-time and sics, there is an underlying risk in epistemology. Because ,0,1,2,3  stands for the indexes in the curved the structure of space-time is a problem involving much   space-time, but ( jkl, , {1,2,3}) for the indexes in broader contents than the vacuum light speed [13], and space. Let x =,,,txyz be a smooth enough coor- we can't accurately measure light speed in true vacuum. dinate system labeled in the space-time and Using the property of a specific matter system to describe X a xTXYZ =,,,  is an orthogonal reference sys- general concepts, it is easy to fall into the trap of   tem in the tangent space-time at given event x . This language. The logical procedure should be to establish tetrad can be represented by the basis of Clifford algebra the geometry of space-time at first, and then to use the C which is isomorphic to the quaternion basis. The geometry and electrodynamics explaining the invariance 1,3 usual Dirac matrix is a realization. Denote of the vacuum light speed. In this way, a lot of plausible a  paradoxes will disappear, and one will not be puzzled by d=Xxa dX ,d=d, x (2.1) superluminal neutrino phenomena and problems of the  a time’s arrow [11,12]. The theory of space-time is a in which coordinate increments dx and dX are vec- 0 geometry, so like the Euclidian geometry, it will be more tors satisfying linear transformation law. d=dx t and 0  effective and convenient to describe the structure of d=dX T are time-like vectors.   is a set of qua- space-time by the method of geometry, and then explain ternion basis in curvilinear coordinate system. They are the corresponding relations between realistic world and not definitely orthogonal, but satisfy the following rela- geometric concepts. In this way, the problems become tion simple and clear in logic. In a sense, the principle of ab  ba=2  ab ,    =2g , constant light speed is a constraint for the metric of the (2.2) space-time, and the principles of relativity and covari- where ab =diag 1,1,1,1  is Minkowski metric, and ance explain the rules of transformation between coor- g is the metric in curvilinear coordinate system, dinate systems. A point or event in the space-time can be labeled by 3 =,hlhaab  =, lh =bl k =, b  aa a kaa independent spatial coordinates xk =,,xyz and one  aa d=d,xhX aa = l , (2.3) temporal coordinate x0 = t , that is to say the space-time xX     ab a b is 3 + 1 dimensional. This is a basic fact, which can only ghhgll=,=. ab    ab

The calculation and explanation for the tetrad coeffi- then the base vector  a of 4 order can be represented  a cients hl a ,  refer to [14,15]. by the following matrices with symmetry, The interval between two infinitely adjacent events 0  a 00I σ  dx2 or dX2 is an objective quantity independent of a     a =,  . (2.8) coordinate system, we have  0 I 00σ  1 This representation is equivalent to the usual Dirac d=d=s22x    ddxx=g dd xx     matrix. But in contrast with Dirac matrix, (2.8) can dis- 2 (2.4) 1 play the symmetry of the fields. The coefficient matrix of =dX2 =  dX ab dXXX= dab d . 2 ab ba ab (2.5) can be constructed by the block matrix of (2.8). Dynamical Equation (2.5) is a first order symmetric (2.1) and (2.4) are the fundamental postulates on the hyperbolic evolution equation system. Its general solu- space-time, and all results of the special relativity can be tion can be expressed by the following integral, derived from them [12]. kjkt The properties of a fundamental physical system can tx,=d  k FKj , d,  (2.9) tx0  ,   be described by fields and interactive coefficients. For example, a particle can be described by spinor  , where t0 is any given time smaller than the present time  t , tt< ,  , xk is the dependent domain of electromagnetic field by vector potential A and field 0 tt0 ,    k intensity E, B , and gravitational field by metric g .  tx,  , which is included in the light cone with vertex For a relatively isolated system, we can express all fields tx, k  , and to describe the system by a column vector T FF=,,=,.kjjkKK   =,12,, n , then the dynamical equation of the     system certainly take the following form [16], The representation (2.9) is solved by characteristic  =,f      , (2.5) cone method. The geometrical meaning is shown in Figure 1. The representation (2.9) manifestly reveals in which  i s a quaternion differential operator, f    some philosophical features such as the arrow of time, consists of some tensorial products of  , such th at the causal and historical connection. This is the reason why total equation satisfies covariance. The following calcu- to choose the formula as a fundamental physical postu- lations show that, the principles of relativity and covari- late to replace (2.5) [16]. ance completely depend on the perfect and marvellous In Figure 1, the cosmic space is absolute space of the quaternion structure of (2.5). The covariance is invalid in universe, which is described by the coordinate system other dimensional space-time. relatively rest to the cosmic background radiation. Cos-   The quaternion basis  in operator =   can mic time is the absolute time. The absolute space is a be represented by matrices. The minimum order is 4, and simultaneous hyperplane relative to the cosmic time. The the usual Dirac matrices are a specific realization. It can cosmic time has unitarity and uniqueness. P stands for be proved that the order of the matrices is an integral a particle moving in the cosmic space, but A,,,BOQ multiple of 4, so the dimension n of field T stand for points relatively rest to the cosmic space. The    =,12,, n is also the integral multiple of 4. coordinates x =,,,txyz is not definitely orthogonal, In fact, all physical phenomena we have observed can be but we can establish orthogonal local tetrad described by spinor  with 4 components and spin dX a =d,d TXYZ,d,d at each event x , and (3) gives 1  the relation between dX a and dx . tt,,Q stands s = , vector A, A with 8 components and spin s =1, 0 2 for dependent domain of point Q from time t0 to t ,  and tensor Gg=, G with 32 components and spin which means the state an d phys ical p arameters at point s =2 together with their combinations. These fields all Q is determined by the historica l data in this d omai n . exist in couples form. Between different representations, tt,,0 A B is the dependent domain of region A  B the operator  and  should make a double linear from t0 to t . Due to the local correlation of interaction, transformation. This tr an sformation reflects the different t0 can be any given time smaller than present time t , selection of coordinates and measuring methods of fields. such as t1 or t2 . Denote the Pauli matrices by From the space-time diagram, we can get the following conclusions: a 10 01 0 i 1 01) The unidirectionality of time: The general solution   ,,,, (2.6) 01 10 i 0 0 1 (2.9) reflects the causal connections between parameters in field Equation (2.5). It evidently shows the unidirec-  a 01,,σσ=,,.23 (2.7) tionality and sequentiality of time. The future is always  

Figure 1. Space-time diagram: The cosmic space is the absolute space of the universe, which is describe by the coordinate system relatively at rest to the cosmic background radiation. Cosmic time is the absolute time. The absolute space is a simultaneous hyperplane relative to the cosmic time. The cosmic time has unitarity and uniqueness. P stands for a mass point moving in the cosmic space, but A, B, O, Q stand for points relatively at rest to the cosmic space. The coordinates xtx =,,,y z is not definitely orthogonal, but we can establish orthogonal local tetrad dX a =TXYZ d ,d ,d ,d  at each event x μ , and (3) gives the a μ relation between dX and dx . tt,,0 Q stands for dependent domain of point Q from time t0 to t , and

tt,,0 A B for depe ndent domain of region AB from t0 to t . determined by the past. The fields of Q at time t are rather than the entropy decides the irreversible process only determined by the history in the ba ckward charac- [11]. teristic cone. Future and past can not be mixed. The 2) The absoluteness of space-time: By Figure 1 we particle P moves in left direction at t0 and t2 , and in learn, the evolution of the universe has globality and right direction at t1 and t . The motion can be reversed, synchronism. There exists a global cosmic time scale  , but the cosmic ti me  goes always unidirectionally whose meaning is close to the concept of Newtonian forward. The inversion of process just means the reflec- absolute time. When  evolves to some specific motion of motion, rather than inversion of time. The arrow ment t0 , it corresponds to the fact that universe evolves of time has nothing to do with the expansion or con- to a specific state. This specific state is a simultaneous traction of the universe, which is just a dynamical effect hyperplane which defines the cosmic space. The meaning of the space-time. The unidirectionality of time is an of the cosmic space is close to the concept of Newtonian objective property of the universe. absolute space. The space-time itself and the matter Since the eigen state of a spinor is a critical state inside it all evolve from one hyperplane to next hyper- without unique evolutionary direction, the non-unique- plane. The cosmic time can neither go back, nor be ness of the process makes the inverse of evolution be- transcended. come impossible. A thermodynamics system consists of 3) The subjectivity of choice coordinate and tetrad a tremendous number of microscopic particles all system: Does the absoluteness of the space-time con- described by spinors. And the evolution of each particle tradicts relativity? The answer is NO. Different from the has some uncertainty or randomness, so the thermody- usually understanding that the relativity denied the namic process is essentially irreversible. Entropy can be objectivity of time and space, the significance of rela- only clearly defined for a system satisfying large number tivity theory is that it emphasizes the correlation between statistical assumption. But for a system with internal space and time, and the universality of the principle of structure, the entropy can not be defined. It is the ran- relativity and covariance. To understand this, we must domness of a statistical system which defines the entropy, firstly understand “The realistic world is just a solution

Copyright © 2012 SciRes. JMP 574 Y. Q. GU of physical laws, and this solution is objective existence”. worlds. In this sense, the proposition “all inertial frames The laws of physics can also have many other reasonable of reference are equivalent” is invalid. and realizable solutions, but they are not objective 5) The internal time of matter system: To under- existence. What the covariance emphasizes is that, the stand time, we should distinguish the different meanings coordinate system is just an identification system which of cosmic time  and the internal time or proper time can be quite arbitrary chosen. Between different coor-  of a specific matter system.  is a measurement for dinate systems there is 1-1 corresponding transformation. the evolution process of the whole universe, and it is Only in the coordinate system relatively rest to the objective and acts as a global standard of time. But  is cosmic background radiation we have objective simul- a measurement of the evolution speed of a specific matter taneous hyperplane and realistic global simultaneity. In system. Relativity provides the method to treat the the local frame and curvilinear coordinate system moving relationship between  and  , that is, relative to the cosmic background radiation, the simul- T taneous hyperplane is only of theoretical sense, rather  =1d.v 2  (2.10) T0 than objective existence. 4) The uniqueness of simultaneity: This problem has been explained in [11,12], which can be illustrated by 3. Transformation Law and the Physical Figure 2. Meaning Assume that we have realistic simultaneous hyper- Dynamical Equation (2.5) simultaneously has general plane in the static reference frame STXYZ,,, , and at covariance under curvilinear coordinate transformation moment T =0, we have simultaneous hyperplane (i.e. and Lorentz invariance under transformation of local the space of the world) AB , which evolves into CD  frame  a . For vectors and tensors, since they can be  at moment TT= 1 and EF, . By Lorentz transfor- directly projected to the natural basis of the coordinate mation, this real space corresponds to the i nclined plane system  , the representation is simple. It is enough that AB,, CD EF, respectively in Otxyz,,, . The si- the equation takes tensorial form. However, for the multaneous hyperplane in Otxyz,,, is the dash lines. spinors the problem becomes complicated and abstruse, The dash lines can not be realistic simultaneous because the spinor is related to concrete local tetrad. In hyperplane, because the part with x <0 has become what follows, we examine the physical meaning of the history in view of STXYZ,,,, but not yet been transformation and the quaternion structure of spinor and determined in view of Otxy ,,,z and could be space-time via specific calculation. designed and chosen. On the other hand, the part x >0 in STXYZ,,, does not take place ye t, but it has 3.1. Dirac Equation in Spherical Coordinate become history in Otxyz,,, . The situation is absurd,  System because the evolution of the universe is not uniquely determined. The simultaneous hyperplane in In [15], we derived the covariant form of the spinor ST,,XYZ, and Otxyz ,,, describe two different equation in spherical coordinate system,

Figure 2. The real space is an evolution simultaneous hyperplane: In the static reference frame S, the simultaneous hyperplane is A'B', C'D', E'F' etc. But in moving reference frame O, it corresponds to inclined hyperplane AB, CD, EF. The simultaneous hyperplane in reference frame O is the dash lines. The simultaneous hyperplanes in two reference frame can not be all realistic for a world.

23We usually solve the Dirac equation with central 0111  i tr    cot    potential in this form (3.5). However, this form is not rr2sin  r covariant, because the tetrad is transformed following the =.m coordinate system, but the spinor still remains the ori- (3.1) ginal without transformation. The covariant form should But some readers told that it is different from the be equation directly transformed from the Cartesian Coor-  11 dinate system. They felt that it is not understandable and 01 2 3   im tr       =, even doubted whether there is error. Now we derive the rrsin   covariant equation by direct calculation, and show the 11 a  01 2 3    meaning of tetrad basis  . We represent the bispinor im tr       =,  rrsin  in the Weyl form  and  . This is the so called  chiral form. Then the Dirac equation in Cartesian coor- (3.6) dinate system becomes, in which  and  is the spinor connection. When making tetrad transformation, the spinor also imt σ  =,  (3.2)  needs a linear transformation, imt σ  =,  =,=. *  (3.7) in which =,,x yz. Transform the Cartesian coordinates x,,yz into spherical coordinates r,,  , by Then the equation can keep covariance. For pure space straightforward calculation we have coordinate transformation, we have 11 a *1 σ =,       (3.3) =,wwak 0  , wi and = , where the rr rr sin  superscript  stands for the transposed conjugate where  ,, reads r   matrix. Substituting (3.7) into (3.5), and then compared it cos sin ei with (3.6), we get i , sine  cos 1  (3.4) wiiia = sin , sin , cos , cos , (3.8)  sincos eii0 ie 2 ,.  ii  cosei sin e0  in which Then the dynamical Equation (3.2) becomes 1 π    ,  11 22  (3.9) imtrr     =, rrsin  1 π  (3.5)    . 11 22   imtrr     =.  rrsin Then we get

iiπ π  i exp  , exp   1 22 22  ==* , (3.10) 2 iiππ  exp  ,i exp   22 22  iiππ  i exp  , exp  1 22 22  1 =. (3.11) 2 iiππ   exp  ,i exp   22 22  Substituting (3.7)-(3.11) into original Equation (3.5), we can get the following form by some arrangement.  11 1 1 012   3  im tr     cot    = , rr 2sin  r   (3.12) 11 1 1  012   3  im tr    cot    = .  rr 2sin  r 

(3.12) is just the same equation calculated in [15]. The   112  LieAcwAA=,          above calculation and equation show that, 22 1) The tetrad coefficient matrix is given by (3.18)   11 in which A is the electromagnetic potential, hdiag a =1,1,,, rrsin   (3.13)  =,=.   (3.19) ld a =1iagrr,1,,sin.  According to the transformation of  , it is easy to    The following vectors are orthogonal, prove that,  is a contravariant vector, and  a true scalar. The corresponding dynamical equation is given 11 by, atr=,,, , X rrsin  (3.14)   a  ieAcw=,   (3.20) d=d,d,d,sind.Xtrrr   2) By spinor transformation (3.7), the direction of the  A =,eq (3.21) 0 1  tetrad basis tu rns into:  along dt ,  along dr , in which q    is the current of spinor. The Hamil- 2 3  along d , and  along d direction, because tonian form of (3.20) is given by, in the tangent s pace  a iHHceApcw =,ˆˆ = α ˆ  . (3.22) dXTXYZtr  d ,d ,d ,d = d ,d ,rr d, sin d  , t  0  and (3.12) becomes where pieˆ =   A is the momentum operator. For (3.22) we have normalizing condition,  01 23  im TXYZ     =,  (3.15)  2 d=1.3 x (3.23) i 01   23  =.mR3  TXYZ Now we check the invariance of (3.22) under space 3) The transformation law between displacement vec- reflection transformation. Making double linear transfor- a a tor d=x  a dx and d=dX  a X , and between the dif- mation ferential operators =  a and =  a is given x xa XaX tt=, xx =  ,t , x =t,,x (3.24) by [12],    where an indeterminate gauge factor ei is ignored. Easy d=xX 00 d , =00  , (3.16) xX to check αα=, 2 =,=I  and ppˆ =  ˆ , a where  = wa , and the operat or  X does not make derivatives on  . qq==,,    012qq,,q3 (3.25)  3 3.2. Transformation Law under the Reflection of  ==. Tetrad That is qq00= and qqkkk=,=1,2,3 . Substi- Under curvilinear coordinate transform ation x  x , tuting (3.25) into (3.21) we get the transformation of a  the transformation of tetra d  = h a is realized via A ,  the index  of coefficient matrix h , but the spinor 00 a AA=,A =A. (3.26) does not transform in form. However, the definition of covariant deri vative of spinor actually implies a trans- Of course, (3.25) and (3.26) are just the trans form ation formation like (3.7). Under proper Lorentz transforma - of a contravariant vector under space reflection. tion of tetrad, the transformation of spinor is similar to Substituting the above relations into (3.22), we get (3.7), and ca n be easily derived. In w hat follows, we  iceApcw =,α ˆ    check the invariance and transformation laws of the t 0   (3.27)  ˆ spinor equation with interactions under the tetrad reflec- =.ceA0 α p  c  w tion transformation. Denote 44 Hermitian matrices by The Equation (3.27) takes the same form of (3.22). This m eans the law for this spinor is independent of  II00σ  0I 0  reference frame t, x or t, x , and the transfor- =,,=,=,   00I σ II 00  mation for field satisfies invertible linear relation (3.24). (3.17) This is true meaning of principle of relativity. Now we check the invariance under time inversion we examine the following Lagrangian transformation. Making double linear transformation,

even no charge is an objective fact, which can not be tt=, xx =, t , x=2 t ,.x (3.28) transformed. In this point of view, the so called CPT Easy to prove invariance theorem is simply not a physical law, but a  result due to misusing concepts. For a specific spinor =,  =  , =   ,22 =  equation, the change of charge means the object being and described also changed, and the equation is describing 123 12 3 another system. Of course, as a mathematical theorem to 22,, =  ,,  , (3.29) show the relation between solutions and coefficients is   ==.22 reasonable. 3) The function of quaternion basis is similar to the q   =       22 unit of imaginary number i . They are concrete numbers,  0123 ==22   ,,,     so they can not be transformed under coordinate or tetrad (3.30) transformation. In quantum field theory we have trans- =, 0123,,  formations such as TiTˆˆ11=, i Tˆαα Tˆ = . These relations should be a results of misunderstanding the mean- =,,,qqqq0123 .  ing of principle of relativity and misusing concepts. The From (3.30) we learn, the transformation law of con- reason why we have such transformations seems to be we travariant vector q under time inversion is different need them. Nobody can really understand such illegal from th at under curvilinear coordinate transformation relations [17].  a 00 qq=,= qq. It takes the same form under space The tetrad coefficients h a or l is different from a a reflection transformation.  . They are fields similar to metric g , so  = h a Substituting (3.30) into (3.21) we get the transfor- is transformed under curvilinear coordinate transforma- mation law of potential AA00=,=AA . For operator tion. However in this case the transformation of spinors pˆ we have is implicit, and the possible transformation is included in the definition of spinor covariant derivative [15]. pieAˆ == ieAp='. ˆ (3.31) 4) Chiral Equation (3.2) is equivalent to the usual Substituting the above relations into (3.22), we get Dirac equation, and the transformation just means the method of measuring field is different. Just like changing  i t coordinates does not influence the property of space-time,  ˆthe transformation of field also does not influence the =,ceApcw20α    2 (3.32)  property of this concrete part icle. T h e representation  =,,'ceA12  3 pˆ   c  w  . 0 and  ,  only needs an invertible linea r transfor - mation, Take the complex conjugate of (3.32), we get II   1  ic=eApˆ cw. (3.33)  =. (3.34) t 0    2 II  (3.33) also takes the same form of (3.22), so the spinor Such transformation can not increase any new and vector equations all keep the invariance under the phenomenon. For the same equation, it is impossible that space-time inversion transformation. the equation describes electron if the spinor is denoted by From the above calculations we learn,  , and describes neutrino when  is transformed into 1) For the first order Equation (2.5), the principle of  ,  . Because physical phenomenon is objective, but relativity and covariance is a strong constraints, which the description is subjective. Electrons and neutrino have can be realized only in 3 + 1 dimensional space-time due different int ernal structure, they should be described by to the quaternion structure. In this point of view, the different equations with different interacting coefficients. theories with other dimensions are almost impossible in physics. The fact that the unverse chooses the 3 + 1 4. Some Problems Relative to the Structure dimensional space-time with elegant quaternion structure of Space-Time has profound philosophical implications, which is the 4.1. Hyperdrive Neutrino sapiential design of the Creator. 2) The philosophical reason of the principle of rela- It was reported that the hyperdrive neutrino phenomenon tivity and covariance is the arbitrariness and subjective- was observed in supernova explosion and OPERA ness to choose the coordinates and measurement method experiment [18]. This leads to the doubt of the effec- of fields. Charge is a property of a specific matter system. tiveness of the theory of relativity. However, the problem One particle carries a positive or a negative charge or may be not in the foundation of relativity, but in the

Copyright © 2012 SciRes. JMP 578 Y. Q. GU expression of the theory. From the above calculation we dynamical equation of fields. In [20-22] we found the find that, the 3 + 1 dimensional space-time has special mass-speed relation for a spinor with interaction potential advantages, which should be described by quaternion is not the simple Einstein mass-energy relation Emc= 2 , geometry. The traditional theory of relativity starts from which has fine structure, the principle of constant vacuum speed of light. This M Mv2 W 1 isn’t illegal in principle. But for the following reasons it Eln=,0 1 F (4.1) is easy to cause difficulty in understanding: 1) The elec- 1111vvvv2222 tromagnetic interaction is of “viscosity”. The light where M ,,MW are positive constants with mass always slows down when it goes through medium due to 01F dimension, and 0

ee 0 . The encounter of particle and anti-particle, simultaneously display coordinates and tetrad, and this like the encounter of electron and proton, will form a form is more convenient for understanding and calcu- tight neutral core. Such neutral core only has extremely lations. In fact, other physical fields also can be ex- weak interaction with other particles, which forms a kind pressed in quaternion form. of pseudo dark matter dispersed in the universe. In For vector field with spin s =1, we have the first appropriate conditions and interacted by high-energy order field equation in quaternion form, photons, the neutral core will be broken again. So for   AA=, electromagnetic interaction we can never get the absolute  (4.2)   2 vacuum. How can we get the accurate vacuum speed of   A =,aA eq light? where q is a source term,  0 = I and

4.2. The Fine Structure of Mass-Energy Relation 01 0 0 001 0  Only in Newtonian mechanics, mass is a basic concept, 10 0 0 000i 12=, =, and the value is determined by empirical data. However, 00 0 i 100 0 mass is not a basic concept in other fundamental theory  00ii 0 0 0 0 of physics, which must be derived from other equations (4.3) or relations, and the value depends on the context theory. 0001  This problem leads to a lot of misunderstanding and 00i 0  3 =, controversy. In the author's point of view, only fields are 000i the basic concept for elementary particles, and other  1000 properties and relations should be derived from the 

T source term. AAA=,01,,,AA23   Obviously, (3.2), (4.2) and (4.10) all take the form of   0123T (2.5). Alm ost all known physical laws and relations can AH=,H,,,HH  (4.4) be derived from some combination of these equation s, so T qqq=,,01qq23,, (2.5) is the most fundamental and universal law of  physics, and it actually forms a kind of unified field    =,0123,,.   theory. In (4.2) and (4.10), the components of fields are By current conservation or normalizing condition we projected to the ortho gonal tetrad. How to transform the have equation into curved space-time and establish the relation 0 qH=0, =A =0, (4.5) between the components is still an open problem. Of course, the transformation is just a choice of description. Expanding (4.2), we get the total Maxwell equation We can make deep understanding into the structure of system in 3 dimensional vector form, space-time and fields, and find new internal symmetries.  2  qA==0,   AaA=eq, However, like the chiral transformation, the transfor-  0 mation can not produce new physical phenomenon, be- EAB=,A  0 =  A,  (4.6) cause the physical phenomenon is objective, but the EE=, aA20  eq0 =,  B  0 transformation is subjective.  2 B =0, BEAq = 0 ae  , in which 5. Conclusions HEBiAAA,,, A 123,qq123,,.qq (4.7) From the above specific discussion and calculation, we can get the following conclusions: For long distance interaction such as electromagnetic 1) The space-time has a quaternion structure, which field, the distance factor a =0 in (4.6). From (4.2), we should be naturally described by the basic relations (2.1)- find magnetic char ge should be the imaginary part of (2.4). electric charge. Unfortunately, this part leads to the vio- 2) The evolution of a given physical system is gov- lation of normalizing condition, that is to say, the charge erned by the universal dynamical Equation (2.5), and in conservation is broken. This is absurd, so the monopole principle, all properties of the system can be logically can not exist in the nature. derived from the dynamical equation together with the The strong interaction also should be described by concrete initial and boundary conditions. The solution of (4.2). In this case, we should have a >0. the first order dynamics (2.5) determines the arrow of  The transformation law of the comp lex field A is time, and the non-uniqueness of evolution of spinor 1 interesting. Make boosting transformation along X , we fields breaks the reversibility of the process. have Lorentz transformation X  = X , 3) The invariance and covariance of the dynamics are cosh sinh different concepts from the symmetry of the solutions,  =,1,1.diag  (4.8) although there are some connections between them. The sinh cosh  physical laws must be invariant and covariant, but the Then the transformation of A becomes A =  A , solutions or the configurations of concrete matter system where are not definitely symmetric.

 coshi sinh  =1,1,diag . (4.9) REFERENCES isinhcosh  [1] S. F. Savitt, “Time’s Arrows Today,” Cambridge Univer- When making space rotation transformation, A has sity Press, Cambridge, 1995. the same transformation law of coordin ate system, that is doi:10.1017/CBO9780511622861  = . [2] J. L. Lebowitz, “Boltzmann’s Entropy and Time’s Ar- Similar to (4.2), for tensor field wit h spin s =2, we row,” Physics Today, Vol. 46, No. 9, 1993, pp. 32-38. have dynamics [3] P. M. Harman, “The Natural Philosophy of James Clerk    GG=, Maxwell,” Cambridge University Press, Cambridge, 1998.  (4.10)   [4] D. Ruelle, “Chance and Chaos,” Princeton University  GG=,T  Press, New Jersey, 1991.  in which GG=  is contravariant tensor in matrix [5] D. Albert, “Time and Chance,” Harvard University Press, form, G is the complex field intensity, and T is Cambridge, 2000.

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