Deformed special relativity based on α -deformed binary operations
aa r X i v : . [ phy s i c s . g e n - ph ] M a y Deformed special relativity based on α -deformed binary operations Won Sang Chung , † and Mahouton Norbert Hounkonnou , †† Department of Physics and Research Institute of Natural Science,College of Natural Science,Gyeongsang National University, Jinju 660-701, Korea International Chair in Mathematical Physics and Applications, ICMPA-UNESCO Chair,University of Abomey-Calavi, 072 BP 50, Cotonou, Rep. of Benin
E-mails: † [email protected]; †† [email protected], with copy [email protected] Abstract
In this paper, we define a new velocity having a dimension of (
Length ) α / ( T ime ) anda new acceleration having a dimension of (
Length ) α / ( T ime ) , based on the fractionaladdition rule. We then discuss the fractional mechanics in one dimension. We show theconservation of fractional energy, and formulate the Hamiltonian formalism for the frac-tional mechanics. As a matter of illustration, we exhibit some examples for the fractionalmechanics. May 25, 2020
Contents α − deformed Newton mechanics 53 α -deformed Galilean Relativity 6 α -deformed matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 α -deformed Galilei group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 α -translation symmetry 8 α -Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 α -Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Energy and α -momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Two body decay 12
Recently, a new analysis, called a pseud analysis, appeared in mathematics [1,2]. It is a gen-eralization of the classical analysis, where instead of the field of real numbers a semiring istaken on a real interval [ a, b ] ⊂ [ −∞ , + ∞ ] endowed with pseudo-addition ⊕ and with pseudo-multiplication ⊗ , with different applications in mathematics and physics, e.g. in modelingnonlinearity, uncertainty in optimization problems, nonlinear partial differential equations, non-linear difference equations, optimal control, fuzzy systems, decision making , game theory. Itsadvantage is that there covered with one theory, as universal mathematical theory, and so withunified methods, problems (usually nonlinear and uncertain) from various fields. It also givessolutions in the form which are not achieved by other approaches, e. g., Bellman differenceequation, Hamilton Jacobi equation with non-smooth Hamiltonians. For more details, see [1]and references therein. Definition 1.1
The pseudo binary operations are defined by the help of a monotonous bijectivemap f, called their generator, as: x ⊕ f y = f − ( f ( x ) + f ( y )) x ⊖ f y = f − ( f ( x ) − f ( y )) x ⊗ f y = f − ( f ( x ) f ( y )) x ⊘ f y = f − ( f ( x ) /f ( y )) . The simplest choice f ( x ) = x gives the ordinary binary operations, and x n ⊕ f y m = x n + y m , (1) x n ⊖ f y m = x n − y m ,x n ⊗ f y m = x n y m and x n ⊘ f y m = x n /y m . Furthermore, (cid:0) x ⊕ f y (cid:1) n = n X k =0 (cid:18) nk (cid:19) x k y n − k , x ⊖ f y (cid:1) n = n X k =0 (cid:18) nk (cid:19) ( − k x k y n − k , (cid:0) x ⊗ f y (cid:1) n = ( xy ) n , and (cid:0) x ⊘ f y (cid:1) n = ( x/y ) n . It can be easily checked that the operation ⊕ f and ⊗ f satisfy the commutativity andassociativity properties. Through the map f , we can perform many deformed binary operations[3,4]. The first use of this pseudo binary operations was made by Einstein [5] in the velocityaddition. The second use was made in constructing the q -additive entropy theory [6-8].For f ( x ) = x α , α > , we have: f − ( x ) = x α . Thus, x n ⊕ f y m = (cid:0) x n α + y m α (cid:1) α ,x n ⊖ f y m = (cid:0) x n α − y m α (cid:1) α ,x n ⊗ f y m = x n y m and x n ⊘ f y m = x n /y m . Moreover, (cid:0) x ⊕ f y (cid:1) n = (cid:0) x α + y α (cid:1) nα , (cid:0) x ⊖ f y (cid:1) n = (cid:0) x − y (cid:1) nα , (cid:0) x ⊗ f y (cid:1) n = ( x y ) n , and (cid:0) x ⊘ f y (cid:1) n = ( x/y ) n . f,f ( x ) = | x | α − x, α > , (2)so that the deformed multiplication and deformed division may be the same as the ordinaryones. Using this, these authors studied the anomalous diffusion process by using the α -deformedmechanics which possesses the α -translation in space x → x ⊕ δx . Inserting the equation (2) into(1), we have the α -deformed binary operations, i. e. α -addition, α -subtraction, α -multiplicationand α -division, as: a ⊕ α b = | a | a | α − + b | b | α − | /α − ( a | a | α − + b | b | α − ) (3) a ⊖ α b = | a | a | α − − b | b | α − | /α − ( a | a | α − − b | b | α − ) (4) a ⊗ α b = ab (5) a ⊘ α b = ab . (6)Besides, we get the following identities: a n ⊕ α b m = (cid:12)(cid:12)(cid:12) a n | a n | α − + b m | b m | α − (cid:12)(cid:12)(cid:12) α − (cid:0) a n | a n | α − + b m | b m | α − (cid:1) ,a n ⊖ α b m = (cid:12)(cid:12)(cid:12) a n | a n | α − − b m | b m | α − (cid:12)(cid:12)(cid:12) α − (cid:0) a n | a n | α − − b m | b m | α − (cid:1) ,a n ⊗ α b m = a n b m and a n ⊘ α b m = a n /a m . Furthermore, (cid:0) a ⊕ α b (cid:1) n = (cid:12)(cid:12)(cid:12) a | a | α − + b | b | α − (cid:12)(cid:12)(cid:12) nα − (cid:0) a | a | α − + b | b | α − (cid:1) n , (cid:0) a ⊕ α b (cid:1) n = (cid:12)(cid:12)(cid:12) a | a | α − − b | b | α − (cid:12)(cid:12)(cid:12) nα − (cid:0) a | a | α − − b | b | α − (cid:1) n , (cid:0) a ⊗ α b (cid:1) n = ( a b ) n , (cid:0) a ⊘ α b (cid:1) n = ( a/a ) n . Interestingly, the multiplication and division are invariant after α -deformation.The paper is organized as follows. In Section 2, we derive the Newton law of α − deformedNewton mechanics. Section 3 is devoted to the characterization of α -deformed Galilean rela-tivity. the α -deformed Galilei group is described, and energy conservation law is deduced. InSection 4, we study the special relativity with α -translation symmetry. Section 5 deals with ananalysis of two body decay. 4 α − deformed Newton mechanics In an ordinary Newtonian mechanics in one dimension, the Newton velocity is defined as v = dxdt , (7)where dx and dt denote the infinitesimal displacement and infinitesimal time interval, respec-tively. The infinitesimal displacement is invariant under spacial translation x → x + δx and theinfinitesimal time interval is invariant under temporal translation t → t + δt .If we impose new translation symmetry based on α -addition rule, we need to change thedefinition of velocity so that it may possess this new symmetry. Here we impose two translationsymmetries, i. e. , the α -translation in position, x → x ⊕ α δx, and α -translation in time, t → t ⊕ α δt .In [7], the authors defined the deformed velocity so that it is invariant under α -translationin position and ordinary translation in time. Then, the average velocity is given by v ave = f α ( x ′ ⊖ α x ) t ′ − t = ∆ α x ∆ t = | x ′ | α − x ′ − | x | α − xt ′ − t . (8)Taking t ′ → t , the velocity is given by v = d α xdt = α | x | α − dxdt . (9)If we impose the α -translation in both time and position, we have to change the definition ofthe velocity. In this case, the average α -velocity is furnished by the expression v α,ave = f α ( x ′ ⊖ α x ) f ( t ′ ⊖ α t ) = ∆ α x ∆ α t = | x ′ | α − x − | x | α − x | t ′ | α − t ′ − | t | α − t , (10)where we call ∆ α x and ∆ α t the α -displacement and α -time-interval, respectively. Taking t ′ → t ,the α -velocity is yielded by v α = d α xd α t = t − α | x | α − dxdt (11)Because v α is α -translation invariant, the α -acceleration is defined as a α = dv α d α t = 1 α t − α dv α dt (12)Since the α -velocity and α -acceleration have dimension [ Length ] α / [ T ime ] α and dimension[ Length ] α / [ T ime ] α , respectively, the Newton equation is obtained by the relation | F | α − F = m α a α (13)or, equivalently, F = m | a α | α − a α (14)In mechanics with α -translation symmetry, the α -velocity and α -acceleration have the fractionaldimensions which are different from the ordinary ones unless α = 1. But, for the force, weassumed that it has the same dimension as the one in the α = 1 − mechanics.5 α -deformed Galilean Relativity Based on the new definition of α -velocity and α -acceleration, we define the α -inertial framesof reference possessing the property that a body with zero net force acting upon these framesdoes not α -accelerate; that is, such a body is at rest or moving at a constant α -velocity. Herewe assume the physical laws must be the same in all α -inertial frames of reference. Now let us consider two inertial frames S ( t, x ) and S ′ ( t ′ , x ′ ) moving at a relative constant α -velocity u α with x -axes. The Newton equation is invariant under the transformations v ′ α = v α − u α , v ′ α = d α x ′ d α t (15)and x ′ = x ⊖ α | u α | α − u α t, t ′ = t. (16) α -deformed matrix Like the α -deformed binary operations for numbers based on the α -map, we can define the α -deformed binary operations for matrices based on the α -map. Now let us consider ( m × n )matrix A = a a · · · a n a a · · · a n ... ... . . . ... a m a m · · · a mn = ( a ij ) (17)We define the α -map of the matrix A as f ( A ) := f ( a ) f ( a ) · · · f ( a n ) f ( a ) f ( a ) · · · f ( a n )... ... . . . ... f ( a m ) f ( a m ) · · · f ( a mn ) = | a | α − a | a | α − a · · · | a n | α − a n | a | α − a | a | α − a · · · | a n | α − a n ... ... . . . ... | a m | α − a m | a m | α − a m · · · | a mn | α − a mn (18)and its inverse as f − ( A ) := f − ( a ) f − ( a ) · · · f − ( a n ) f − ( a ) f − ( a ) · · · f − ( a n )... ... . . . ... f − ( a m ) f − ( a m ) · · · f − ( a mn ) | a | /α − a | a | /α − a · · · | a n | /α − a n | a | /α − a | a | /α − a · · · | a n | /α − a n ... ... . . . ... | a m | /α − a m | a m | /α − a m · · · | a mn | /α − a mn (19)The scalar multiplication of the matrix is the same as the one in α = 1 − theory. We define the α -addition and α -subtraction of two matrices of same type A, B as A ⊕ α B := f − ( f ( A ) + f ( B )) (20) A ⊖ α B := f − ( f ( A ) − f ( B )) , (21)which imply ( A ⊕ α B ) ij = a ij ⊕ α b ij (22)( A ⊖ α B ) ij = a ij ⊖ α b ij (23)For n × p matrix A and p × m matrix B , the α -multiplication of A and B is defined as A ⊗ α B := f − ( f ( A ) f ( B )) (24)which implies ( A ⊗ α B ) ij = p M k =1 a ik b kj (25)where p M k =1 C k = C ⊕ α C ⊕ α · · · ⊕ α C p (26) α -deformed Galilei group Based on the α -operations for matrices, we can rewrite the eq.(16) as x ′ t ′ ! = T α ( u α ) ⊗ α xt ! = −| u α | α − u α ! ⊗ α xt ! (27)Here we know that the transformation matrix T α ( u α ) forms a Lie group with the α -multiplication.Indeed, the following properties are satisfied: • T α ( u α ) ⊗ α T α ( v α ) = T α ( u α + v α ). • The α -multiplication is associative. • The identity is T α (0). • The inverse is T α ( − u α ). 7 .3 Energy conservation Because dx is not invariant under the α -translation, we use α -translational invariant infinitesi-mal displacement as d α x = α | x | α − dx in defining the work, | W | α − W = − Z d α x | F | α − F, (28)having the same dimension asin the α = 1 − mechanics. We define the potential energy throughthe conservative force, | F | α − F = − d α Ud α x = −| x | − α | U | α − dUdx . (29)Thus, for the conservative force, we have | W → | α − W → = − ( | U | α − U − | U | α − U ) . (30)Inserting the Newton equation (13) into (28), we get | W → | α − W → = K − K (31)where the kinetic energy is given by K = 12 m α v α . (32)Considering the dimension, the conservation of energy is provided by | E | α − E = K + | U | α − U = 12 m α v α + | U | α − U = p α m α + | U | α − U, (33)where the linear momentum is expressed as p α = m α v α . (34)The energy has the same dimension as in the α = 1 − mechanics, while the linear momentumhas fractional dimension. α -translation symmetry The 3-position in non-relativistic mechanics is changed into 4-position (or event) in the rel-ativistic one. Let us consider the event P ( ct, x, y, z ), where c is the Newton speed of light,(i. e. speed with α = 1). Based on the definition of α -translation invariant infinitesimal dis-placement and α -translation invariant infinitesimal time interval, the α -translation invariantdistance ( α -distance) of infinitesimally close space-time events denoted by ds α is given by d α s = c α d α t − d α x − d α y − d α z . (35)The α -deformed proper-time τ α is d α τ = d α s c α . (36)8 .1 α -Lorentz transformations The α -Lorentz transformations making invariant the space-time interval(∆ α s ) = (cid:0) c α ( | t | α − t ) (cid:1) − (cid:0) ( | x | α − x ) (cid:1) (37)are given by | x | α − x = c α | t ′ | α − t ′ sh α ( ψ ) + | x ′ | α − x ′ ch α ( ψ ) c α | t | α − t = c α | t ′ | α − t ′ ch α ( ψ ) + | x ′ | α − x ′ sh α ( ψ ) , (38)where the little α -deformed hyperbolic functions are defined by sh α ( ψ ) := 12 ( e α ( ψ ) − e α ( − ψ )) = sinh (cid:0) | ψ | α − ψ (cid:1) (39) ch α ( ψ ) := 12 ( e α ( ψ ) + e α ( − ψ )) = cosh (cid:0) | ψ | α − ψ (cid:1) (40) th α ( ψ ) := sh α ( ψ ) ch α ( ψ ) = tanh (cid:0) | ψ | α − ψ (cid:1) (41) e α ( x ) := e | x | α − x . (42)The little α -deformed hyperbolic functions obey the relations ch α ( ψ ) − sh α ( ψ ) = 1 . (43)In terms of the α -deformed binary operations, we get x = ct ′ Sh α ( ψ ) ⊕ x ′ Ch α ( ψ ) ct = ct ′ Ch α ( ψ ) ⊕ x ′ Sh α ( ψ ) , (44)where the big α -deformed hyperbolic functions are Ch α ( ψ ) := | ch α ( ψ ) | α − ch α ( ψ ) (45) Sh α ( ψ ) := | sh α ( ψ ) | α − sh α ( ψ ) (46) T h α ( ψ ) := Sh α ( ψ ) Ch α ( ψ ) . (47)obeying | Ch α ( ψ ) | ⊖ | Sh α ( ψ ) | = 1 (48)Consider in the coordinate system ( ct, x ) the origin of the coordinate system ( ct ′ , x ′ ). Then, x ′ = 0 , and x = ct ′ Sh α ( ψ ) ct = ct ′ Ch α ( ψ ) . (49)9ividing the two equations gives xct = T h α ( ψ ) , (50)or, | x | α − xc α | t | α − t = th α ( ψ ) . (51)Since | x | α − x | t | α − t = v α is the relative uniform α -velocity (see [4]) of the two systems, we identify thephysical meaning of the imaginary ”rotation angle ψ ” as th α ( ψ ) = v α c α = β α . (52)Using the following identities ch α ( ψ ) = γ α , sh α ( ψ ) = γ α β α , , (53)where γ α = 1 p − β α , (54)we obtain the α -deformed Lorentz transformation of the form | x | α − x = γ α (cid:0) | x ′ | α − x ′ + v α | t ′ | α − t ′ (cid:1) | t | α − t = γ α (cid:16) | t ′ | α − t ′ + v α c α | x ′ | α − x ′ (cid:17) . (55)Expressing the eq.(4.1) in terms of the α -deformed binary operations, we get x = Γ α (cid:0) x ′ ⊕ v /αα t ′ (cid:1) t = Γ α t ′ ⊕ v /αα c x ′ ! (56)where Γ α = γ /αα = (1 − β α ) − α . (57)If we set u α = (cid:18) | x | α − | t | α − (cid:19) dxdt , u ′ α = (cid:18) | x ′ | α − | t ′ | α − (cid:19) dx ′ dt ′ (58)the addition of α -velocity becomes u α = u ′ α + v α v α u α c α (59)If we regard the α -speed of light as c α , the eq.(58) shows that the α -speed of light remainsinvariant, and, hence, the speed of light also remains invariant under the α -deformed Lorentztransformation. 10 .2 α -Lorentz group Now, let us introduce the four α -velocity. For that, we change the notation as: ct = x , x = x , y = x , z = x (60)Then, the four α -velocity is given by u aα = | x a | α − dx a ( ˜ dτ ) α (61)or, explicitly, u α = c α γ α (62) u iα = v iα γ α , i = 1 , , . (63)Therefore, we have η ab u aα u bα = c α . (64) α -momentum The four α -momentum is defined as p aα = m α u aα (65)explicitly giving p α = m α c α γ α (66) p iα = m α v iα γ α . (67)Thus, we have η ab p aα p bα = m α c α . (68)Here, we have p aα = ( E/c, ~p α ) because the energy in α -deformed mechanics has the same unitas in the undeformed case. Therefore, we set p aα = (cid:18)(cid:18) Ec (cid:19) α , ~p α (cid:19) (69)Thus, the eq.(68) gives E α = c α | ~p α | + m α c α (70)When | ~v α | ≪ c α , we have E α ≈ | ~p α | m α (71)which is the same as the non-relativistic case. 11 Two body decay
The simplest kind of particle reaction is the two-body decay of unstable particles. A wellknown example from nuclear physics is the alpha decay of heavy nuclei. In particle physics, oneobserves, for instance, decays of charged pions or kaons into muons and neutrinos, or decays ofneutral kaons into pairs of pions, etc.Consider the decay of a particle of mass M which is initially at rest. Then, its four α -momentum is P = ( M α , ~ c = 1. This reference frame is called the centre-of massframe (CMS). Denote the four α -momenta of the two daughter particles by p = ( E α , ~p α, ) , p =( E α , ~p α, ). From the momentum conservation, we get ~p α, + ~p α, = 0 (72)The energy conservation is M α = q | ~p α, | + m α + q | ~p α, | + m α (73)If we set p = | ~p α, | = | ~p α, | , (74)we have p = 12 M α p ( M α − ( m α − m α ) )( M α − ( m α + m α ) ) (75)Thus, we have M ≥ m ⊕ α m . (76) References [1] E Pap, Generalized real analysis and its applications, Internat. J. Approx. Reason. ,368–386 (2008) .[2] E. Pap, g − calculus, Novi Sad J. Math. , 145 (1993) .[3] N. Ralevi´c, Pseudo-analysis and applications on solution nonlinear equations , Ph.D. The-sis, PMF Novi Sad (1997).[4] W. Chung and H.Hassanabadi, The f -deformation I : f -deformed classical mechanics(2019).[5] W. Chung and H.Hassanabadi, The f -deformation II : f -deformed quantum mechanicsin one dimension (2019).[6] E. Borges, A possible deformed algebra and calculus inspired in nonextensive thermo-statistics, Physica. A. 340 , 95-101 (2004).127] W. Chung and H. Hassanabadi, Deformed classical mechanics with α − deformed trans-lation symmetry and anomalous diffusion, Mod. Phys. Lett. B33 , 1950368 (2019).[8] A. Einstein, Zur Elektrodynamik bewegter K¨orper, Annalen der Physik. , 891 (1905).[9] A. Nivanen, A. Le Mechaute, Q. Wang, Generalized algebra within a nonextensive statis-tics, Rep. Math. Phys. , 437 (2003) .[10] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys.52